  
  [1X2 [33X[0;0YDescription[133X[101X
  
  [33X[0;0YThe   group   [23X\mathrm{SL}_2(\mathbb{Z})[123X   is  generated  by  [23X\mathfrak{s}[123X  =
  [10X[[0,1],[-1,0]][110X and [23X\mathfrak{t}[123X = [10X[[1,1],[0,1]][110X (which satisfy the relations
  [23X\mathfrak{s}^4   =  (\mathfrak{st})^3  =  \mathrm{id}[123X).  Thus,  any  complex
  representation  [23X\rho[123X  of  [23X\mathrm{SL}_2(\mathbb{Z})[123X on [23X\mathbb{C}^n[123X (where [23Xn
  \in \mathbb{Z}^+[123X is called the [13Xdegree[113X or [13Xdimension[113X of [23X\rho[123X) is determined by
  the [23Xn \times n[123X matrices [23XS = \rho(\mathfrak{s})[123X and [23XT = \rho(\mathfrak{t})[123X.[133X
  
  [33X[0;0YThis      package      constructs     irreducible     representations     of
  [23X\mathrm{SL}_2(\mathbb{Z})[123X           which           factor           through
  [23X\mathrm{SL}_2(\mathbb{Z}/\ell\mathbb{Z})[123X for some [23X\ell \in \mathbb{Z}^+[123X; the
  smallest  such  [23X\ell[123X is called the [13Xlevel[113X of the representation, and is equal
  to  the  order  of  [23XT[123X.  One  may  equivalently  say  that  the kernel of the
  representation  is  a  congruence  subgroup. Such representations are called
  [13Xcongruent[113X   representations.  A  congruent  representation  [23X\rho[123X  is  called
  [13Xsymmetric[113X  if  [23XS = \rho(\mathfrak{s})[123X is a symmetric, unitary matrix and [23XT =
  \rho(\mathfrak{t})[123X  is  a diagonal matrix; it was proved by the authors that
  every congruent representation is equivalent to a symmetric one (see [14X2.1-4[114X).
  Any  representation  of  [23X\mathrm{SL}_2(\mathbb{Z})[123X  arising  from  a modular
  tensor category is symmetric [DLN15].[133X
  
  [33X[0;0YWe  therefore  present  representations  in  the  form of a record [10Xrec(S, T,
  degree, level, name)[110X, where the name follows the conventions of [NW76].[133X
  
  [33X[0;0YNote  that  our  definition  of  [23X\mathfrak{s}[123X follows that of [Nob76]; other
  authors  prefer the inverse, i.e. [23X\mathfrak{s}[123X = [10X[[0,-1],[1,0]][110X (under which
  convention    the    relations    are    [23X\mathfrak{s}^4    =    \mathrm{id}[123X,
  [23X(\mathfrak{s}\mathfrak{t})^3  =  \mathfrak{s}^2[123X).  When  working  with  that
  convention, one must invert the [23XS[123X matrices output by this package.[133X
  
  [33X[0;0YThroughout,  we  denote  by  [23X\mathbf{e}[123X  the map [23Xk \mapsto e^{2 \pi i k}[123X (an
  isomorphism from [23X\mathbb{Q}/\mathbb{Z}[123X to the group of finite roots of unity
  in  [23X\mathbb{C}[123X). For a group [23XG[123X, we denote by [23X\widehat{G}[123X the character group
  [23X\operatorname{Hom}(G, \mathbb{C}^\times)[123X.[133X
  
  
  [1X2.1 [33X[0;0YConstruction[133X[101X
  
  [33X[0;0YAny  representation [23X\rho[123X of [23X\mathrm{SL}_2(\mathbb{Z})[123X can be decomposed into
  a  direct  sum of irreducible representations (irreps). Further, if [23X\rho[123X has
  finite  level,  each irrep can be factorized into a tensor product of irreps
  whose  levels  are  powers  of  distinct primes (using the Chinese remainder
  theorem).  Therefore, to characterize all finite-dimensional representations
  of [23X\mathrm{SL}_2(\mathbb{Z})[123X of finite level, it suffices to consider irreps
  of  [23X\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})[123X  for primes [23Xp[123X and positive
  integers [23X\lambda[123X.[133X
  
  
  [1X2.1-1 [33X[0;0YWeil representations[133X[101X
  
  [33X[0;0YSuch  representations  may  be  constructed  using  Weil  representations as
  described  in  [Nob76,  Section  1].  We give a brief summary of the process
  here.  First, if [23XM[123X is any additive abelian group, a [13Xquadratic form[113X on [23XM[123X is a
  map [23XQ : M \to \mathbb{Q}/\mathbb{Z}[123X such that[133X
  
  [30X    [33X[0;6Y[23XQ(-x) = Q(x)[123X for all [23Xx \in M[123X, and[133X
  
  [30X    [33X[0;6Y[23XB(x,y)  =  Q(x+y)  -  Q(x)  - Q(y)[123X defines a [23X\mathbb{Z}[123X-bilinear map [23XM
        \times M \to \mathbb{Q}/\mathbb{Z}[123X.[133X
  
  [33X[0;0YNow  let  [23Xp[123X  be  a  prime  number  and  [23X\lambda  \in  \mathbb{Z}^+[123X. Choose a
  [23X\mathbb{Z}/p^\lambda\mathbb{Z}[123X-module  [23XM[123X  and  a  quadratic form [23XQ[123X on [23XM[123X such
  that  the  pair [23X(M,Q)[123X is of one of the three types described in Section [14X2.2[114X.
  Each  such  [23XM[123X  is  a  ring,  and has at most 2 cyclic factors as an additive
  group.  Those with 2 cyclic factors may be identified with a quotient of the
  quadratic  integers,  giving  a  norm  on [23XM[123X. Then the [13Xquadratic module[113X [23X(M,Q)[123X
  gives          rise          to          a         representation         of
  [23X\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})[123X   on  the  vector  space  [23XV  =
  \mathbb{C}^M[123X  of  complex-valued  functions  on  [23XM[123X.  This  representation is
  denoted  [23XW(M,Q)[123X. Note that the [13Xcentral charge[113X of [23X(M,Q)[123X is given by [23XS_Q(-1) =
  \frac{1}{\sqrt{|M|}} \sum_{x \in M} \mathbf{e}(Q(x))[123X.[133X
  
  
  [1X2.1-2 [33X[0;0YCharacter subspaces and primitive characters[133X[101X
  
  [33X[0;0YA  family  of subrepresentations [23XW(M,Q,\chi)[123X of [23XW(M,Q)[123X may be constructed as
  follows. Denote[133X
  
  
  [24X[33X[0;6Y\operatorname{Aut}(M,Q)  =  \{  \varepsilon  \in  \operatorname{Aut}(M) \mid
  Q(\varepsilon x) = Q(x) \text{ for all } x \in M\}~.[133X
  
  [124X
  
  [33X[0;0YWe   then   associate   to  [23X(M,Q)[123X  an  abelian  subgroup  [23X\mathfrak{A}  \leq
  \operatorname{Aut}(M,Q)[123X; the structure of this group depends on [23X(M,Q)[123X and is
  described  in  Section  [14X2.2[114X.  Note  that [23X\mathfrak{A}[123X has at most two cyclic
  factors,  whose  generators we denote by [23X\alpha[123X and [23X\beta[123X. Now, let [23X\chi \in
  \widehat{\mathfrak{A}}[123X  be  a  1-dimensional  representation  ([13Xcharacter[113X) of
  [23X\mathfrak{A}[123X, and define[133X
  
  
  [24X[33X[0;6YV_\chi = \{f \in V \mid f(\varepsilon x) = \chi(\varepsilon) f(x) \text{ for
  all } x \in M \text{ and } \varepsilon \in \mathfrak{A}\}~,[133X
  
  [124X
  
  [33X[0;0Ywhich  is a [23X\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})[123X-invariant subspace
  of  [23XV[123X.  We  then  denote  by  [23XW(M,Q,\chi)[123X the subrepresentation of [23XW(M,Q)[123X on
  [23XV_\chi[123X. Note that [23XW(M,Q,\chi) \cong W(M,Q,\overline{\chi})[123X.[133X
  
  [33X[0;0YFor  the  abelian  groups [23X\mathfrak{A} \leq \operatorname{Aut}(M,Q)[123X, we will
  frequently  refer  to  a  character [23X\chi \in \widehat{\mathfrak{A}}[123X as being
  [13Xprimitive[113X.  With  the exception of a single family of modules of type [23XR[123X (the
  [13Xextremal[113X  case,  for  which  see  Section [14X2.2-4[114X), primitivity amounts to the
  following:   there  exists  some  [23X\varepsilon  \in  \mathfrak{A}[123X  such  that
  [23X\chi(\varepsilon)  \neq  1[123X  and [23X\varepsilon[123X fixes the submodule [23XpM \subset M[123X
  pointwise.  There  exists  a  subgroup [23X\mathfrak{A}_0 \leq \mathfrak{A}[123X such
  that  a non-trivial [23X\chi \in \widehat{\mathfrak{A}}[123X is primitive if and only
  if  [23X\chi[123X is injective on [23X\mathfrak{A}_0[123X (or, equivalently, if [23X\mathfrak{A}_0
  \cap \operatorname{ker} \chi[123X is trivial).[133X
  
  [33X[0;0YExplicit descriptions of the group [23X\mathfrak{A}_0[123X for each type are given in
  Section [14X2.2[114X and may be used to determine the primitive characters.[133X
  
  
  [1X2.1-3 [33X[0;0YIrrep Types[133X[101X
  
  [33X[0;0YAll irreps of prime-power level and finite degree may then be constructed in
  one of three ways ([NW76, Hauptsatz 2]):[133X
  
  [30X    [33X[0;6YThe  overwhelming  majority  are  of  the  form  [23XW(M,Q,\chi)[123X  for [23X\chi[123X
        primitive and [23X\chi^2 \neq 1[123X; we call these [13Xstandard[113X. This includes the
        primitive characters from the extremal case.[133X
  
  [30X    [33X[0;6YA  finite  number,  and  a  single  infinite  family  arising from the
        extremal  case  (Section  [14X2.2-4[114X),  are  instead  constructed  by using
        non-primitive characters or primitive characters [23X\chi[123X with [23X\chi^2 = 1[123X.
        We call these [13Xnon-standard[113X.[133X
  
  [30X    [33X[0;6YFinally,  18  [13Xexceptional[113X irreps are constructed as tensor products of
        two  irreps  from  the  other  two  cases. A full list of these may be
        constructed by [2XSL2IrrepsExceptional[102X ([14X4.3-1[114X).[133X
  
  
  [1X2.1-4 [33X[0;0YS and T matrices[133X[101X
  
  [33X[0;0YThe   images  [23XW(M,Q)(\mathfrak{s})(f)[123X  and  [23XW(M,Q)(\mathfrak{t})(f)[123X  may  be
  calculated for any [23Xf \in V[123X (see [Nob76, Satz 2]). Thus, to construct [23XS[123X and [23XT[123X
  matrices  for  the  irreducible subrepresentations of [23XW(M,Q)[123X, it suffices to
  find  bases  for the [23XW(M,Q)[123X-invariant subspaces of [23XV[123X. Choices for such bases
  are  given  by  [NW76];  however,  these  often  result  in  non-symmetric [23XS[123X
  matrices.  It  has  been proven by the authors of this package that, for all
  standard and non-standard irreps, there exists a basis for the corresponding
  subspace  of  [23XV[123X  such  that  [23XS[123X  is  symmetric  and unitary and [23XT[123X is diagonal
  ([NWW21],  in  preparation). In particular, [23XS[123X is always either a real matrix
  or  [23Xi[123X  times  a  real  matrix. It follows that these properties hold for the
  exceptional  irreps  as  well. This package therefore produces matrices with
  these properties.[133X
  
  [33X[0;0YAll     the     finite-dimensional     irreducible     representations    of
  [23X\mathrm{SL}_2(\mathbb{Z})[123X  of  finite level can now be constructed by taking
  tensor   products   of   these   prime-power   irreps.  Note  that,  if  two
  representations  are  determined by pairs [10X[S1,T1][110X and [10X[S2,T2][110X, then the pair
  for   their   tensor   product   may  be  calculated  via  the  GAP  command
  [10XKroneckerProduct[110X,                          namely                         as
  [10X[KroneckerProduct(S1,S2),KroneckerProduct(T1,T2)][110X.[133X
  
  
  [1X2.2 [33X[0;0YWeil representation types[133X[101X
  
  
  [1X2.2-1 [33X[0;0YType D[133X[101X
  
  [33X[0;0YLet  [23Xp[123X  be  prime. If [23Xp=2[123X or [23Xp=3[123X, let [23X\lambda \geq 2[123X; otherwise, let [23X\lambda
  \geq 1[123X. Then the Weil representation arising from the quadratic module with[133X
  
  
  [24X[33X[0;6YM  =  \mathbb{Z}/p^\lambda\mathbb{Z}  \oplus  \mathbb{Z}/p^\lambda\mathbb{Z}
  \qquad \text{and} \qquad Q(x,y) = \frac{xy}{p^\lambda}[133X
  
  [124X
  
  [33X[0;0Yis  said  to  be  of  type [23XD[123X and denoted [23XD(p,\lambda)[123X. Information on type [23XD[123X
  quadratic   modules   may   be   obtained   via   [2XSL2ModuleD[102X   ([14X3.1-1[114X),  and
  subrepresentations  of  [23XD(p,\lambda)[123X with level [23Xp^\lambda[123X may be constructed
  via [2XSL2IrrepD[102X ([14X3.1-2[114X).[133X
  
  [33X[0;0YThe group[133X
  
  
  [24X[33X[0;6Y\mathfrak{A} \cong (\mathbb{Z}/p^\lambda\mathbb{Z})^\times[133X
  
  [124X
  
  [33X[0;0Yacts  on  [23XM[123X by [23Xa(x,y) = (a^{-1}x, ay)[123X and is thus identified with a subgroup
  of  [23X\operatorname{Aut}(M,Q)[123X; see [NW76, Section 2.1]. The group [23X\mathfrak{A}[123X
  has  order [23Xp^{\lambda-1}(p-1)[123X and [23X\mathfrak{A} = \langle\alpha\rangle \times
  \langle\beta\rangle[123X.  The  relevant information for type [23XD[123X quadratic modules
  is as follows:[133X
  
        [23Xp[123X    [23X\lambda[123X                        [23X\alpha[123X                            [23X\beta[123X           [23X\mathfrak{A}_0[123X       
        ────   ─────────   ──────────────────────────────────────────────────   ───────────────   ────────────────────────  
        [23X>2[123X      [23X1[123X                             [23X1[123X                           [23X|\beta| = p-1[123X     [23X\langle 1 \rangle[123X      
        [23X>2[123X     [23X>1[123X      [23X|\alpha| = p^{\lambda-1}[123X (e.g. [23X\alpha = 1 + p[123X)   [23X|\beta| = p-1[123X   [23X\langle \alpha \rangle[123X   
        [23X2[123X       [23X2[123X                             [23X1[123X                                [23X-1[123X           [23X\langle 1 \rangle[123X      
        [23X2[123X      [23X>2[123X        [23X|\alpha| = 2^{\lambda-2}[123X (e.g. [23X\alpha = 5[123X)          [23X-1[123X         [23X\langle \alpha \rangle[123X   
  
  [33X[0;0YWhen  [23X\mathfrak{A}_0[123X  is  trivial,  every  non-trivial  character  [23X\chi  \in
  \widehat{\mathfrak{A}}[123X is primitive.[133X
  
  
  [1X2.2-2 [33X[0;0YType N[133X[101X
  
  [33X[0;0YLet  [23Xp[123X be prime and [23X\lambda \geq 1[123X. If [23Xp \neq 2[123X, let [23Xu[123X be a positive integer
  so  that  [23Xu  \equiv 3[123X mod 4 with [23X-u[123X a quadratic non-residue mod [23Xp[123X; if [23Xp = 2[123X,
  let [23Xu=3[123X. Then the Weil representation arising from the quadratic module with[133X
  
  
  [24X[33X[0;6YM  =  \mathbb{Z}/p^\lambda\mathbb{Z}  \oplus  \mathbb{Z}/p^\lambda\mathbb{Z}
  \qquad \text{and} \qquad Q(x,y) = \frac{x^2 +xy+\frac{1+u}{4}y^2}{p^\lambda}[133X
  
  [124X
  
  [33X[0;0Yis  said  to  be  of  type [23XN[123X and denoted [23XN(p,\lambda)[123X. Information on type [23XN[123X
  quadratic   modules   may   be   obtained   via   [2XSL2ModuleN[102X   ([14X3.2-1[114X),  and
  subrepresentations  of  [23XN(p,\lambda)[123X with level [23Xp^\lambda[123X may be constructed
  via [2XSL2IrrepN[102X ([14X3.2-2[114X).[133X
  
  [33X[0;0YThe additive group [23XM[123X is a ring with multiplication given by[133X
  
  
  [24X[33X[0;6Y(x_1, y_1) \cdot (x_2, y_2) = (x_1x_2 - \frac{1+u}{4}y_1y_2, x_1y_2 + x_2y_1
  + y_1y_2)[133X
  
  [124X
  
  [33X[0;0Yand  identity element [23X(1,0)[123X. We define a norm [23X\operatorname{Nm}(x,y) = x^2 +
  xy + \frac{1+u}{4}y^2[123X on [23XM[123X; then the multiplicative subgroup[133X
  
  
  [24X[33X[0;6Y\mathfrak{A}       =       \{\varepsilon       \in       M^\times       \mid
  \operatorname{Nm}(\varepsilon) = 1 \}[133X
  
  [124X
  
  [33X[0;0Yof [23XM^\times[123X acts on [23XM[123X by multiplication and is identified with a subgroup of
  [23X\operatorname{Aut}(M,Q)[123X; see [NW76, Section 2.2].[133X
  
  [33X[0;0YThe  group  [23X\mathfrak{A}[123X  has  order  [23Xp^{\lambda-1}(p+1)[123X  and [23X\mathfrak{A} =
  \langle   \alpha   \rangle   \times  \langle  \beta  \rangle[123X.  The  relevant
  information for type [23XN[123X quadratic modules is as follows:[133X
  
        [23Xp[123X    [23X\lambda[123X            [23X\alpha[123X                [23X\beta[123X           [23X\mathfrak{A}_0[123X       
        ────   ─────────   ──────────────────────────   ───────────────   ────────────────────────  
        [23X>2[123X      [23X1[123X               [23X(1,0)[123X             [23X|\beta| = p+1[123X   [23X\langle (1,0) \rangle[123X    
        [23X>2[123X     [23X>1[123X      [23X|\alpha| = p^{\lambda-1}[123X   [23X|\beta| = p+1[123X   [23X\langle \alpha \rangle[123X   
        [23X2[123X       [23X1[123X               [23X(1,0)[123X              [23X|\beta| = 3[123X    [23X\langle (1,0) \rangle[123X    
        [23X2[123X       [23X2[123X               [23X(1,0)[123X              [23X|\beta| = 6[123X    [23X\langle (-1,0) \rangle[123X   
        [23X2[123X      [23X>2[123X      [23X|\alpha| = p^{\lambda-2}[123X    [23X|\beta| = 6[123X    [23X\langle \alpha \rangle[123X   
  
  [33X[0;0YWhen  [23X\mathfrak{A}_0[123X  is  trivial,  every  non-trivial  character  [23X\chi  \in
  \widehat{\mathfrak{A}}[123X is primitive.[133X
  
  
  [1X2.2-3 [33X[0;0YType R, generic cases[133X[101X
  
  [33X[0;0YThe  structure  of  the  quadratic module [23X(M,Q)[123X of type [23XR[123X depends upon three
  additional parameters: [23X\sigma[123X, [23Xr[123X, and [23Xt[123X. Details are as follows:[133X
  
  [30X    [33X[0;6YIf [23Xp[123X is odd, let [23X\lambda \geq 2[123X, [23X\sigma \in \{1, \dots, \lambda\}[123X, and
        [23Xr,t \in \{1,u\}[123X with [23Xu[123X a quadratic non-residue mod [23Xp[123X. Then define[133X
  
  
  [24X      [33X[0;6YM           =           \mathbb{Z}/p^\lambda\mathbb{Z}          \oplus
        \mathbb{Z}/p^{\lambda-\sigma}\mathbb{Z}   \qquad   \text{and}   \qquad
        Q(x,y) = \frac{r(x^2 + p^\sigma t y^2)}{p^\lambda}~.[133X
  
  [124X
  
        [33X[0;6YWhen [23X\sigma = \lambda[123X, the second factor of [23XM[123X is trivial, and [23X(M,Q)[123X is
        said to be in the [13Xunary[113X family; otherwise, it is called [13Xgeneric[113X.[133X
  
  [30X    [33X[0;6YIf [23Xp=2[123X, let [23X\lambda \geq 2[123X, [23X\sigma \in \{0, \dots, \lambda-2\}[123X and [23Xr,t
        \in \{1,3,5,7\}[123X. Then define[133X
  
  
  [24X      [33X[0;6YM          =         \mathbb{Z}/2^{\lambda-1}\mathbb{Z}         \oplus
        \mathbb{Z}/2^{\lambda-\sigma-1}\mathbb{Z}   \qquad  \text{and}  \qquad
        Q(x,y) = \frac{r(x^2 + 2^\sigma t y^2)}{2^\lambda}~.[133X
  
  [124X
  
        [33X[0;6YWhen  [23X\sigma  =  \lambda  - 2[123X, the second factor of [23XM[123X is isomorphic to
        [23X\mathbb{Z}/2\mathbb{Z}[123X,  and  [23X(M,Q)[123X  is  said  to  be  in the [13Xextremal[113X
        family; otherwise, it is called [13Xgeneric[113X.[133X
  
  [33X[0;0YIn  all  cases,  the  resulting  representation  is said to be of type [23XR[123X and
  denoted   [23XR(p,\lambda,\sigma,r,t)[123X.  The  additive  group  [23XM[123X  admits  a  ring
  structure with multiplication[133X
  
  
  [24X[33X[0;6Y(x_1, y_1) \cdot (x_2, y_2) = (x_1x_2 - p^\sigma ty_1y_2, x_1y_2 + x_2y_1)[133X
  
  [124X
  
  [33X[0;0Yand  identity element [23X(1,0)[123X. We define a norm [23X\operatorname{Nm}(x,y) = x^2 +
  xy + p^\sigma t y^2[123X on [23XM[123X.[133X
  
  [33X[0;0YIn  this section, we detail generic type [23XR[123X quadratic modules. Information on
  the unary and extremal cases is covered in Section [14X2.2-4[114X.[133X
  
  [33X[0;0YLet  [23X(M,Q)[123X be a generic type [23XR[123X quadratic module. Information on [23X(M,Q)[123X can be
  obtained    via    [2XSL2ModuleR[102X    ([14X3.3-1[114X),    and    subrepresentations    of
  [23XR(p,\lambda,\sigma,r,t)[123X   with   level  [23Xp^\lambda[123X  may  be  constructed  via
  [2XSL2IrrepR[102X ([14X3.3-2[114X).[133X
  
  [33X[0;0YThe multiplicative subgroup[133X
  
  
  [24X[33X[0;6Y\mathfrak{A}       =       \{\varepsilon       \in       M^\times       \mid
  \operatorname{Nm}(\varepsilon) = 1 \}[133X
  
  [124X
  
  [33X[0;0Yof [23XM^\times[123X acts on [23XM[123X by multiplication and is identified with a subgroup of
  [23X\operatorname{Aut}(M,Q)[123X;  see  [NW76,  Section  2.3  -  2.4].  The  relevant
  information is as follows:[133X
  
  [30X    [33X[0;6YIf    [23Xp[123X   is   odd,   [23X\mathfrak{A}   =   \langle\alpha\rangle   \times
        \langle\beta\rangle[123X  with order [23X2p^{\lambda-\sigma}[123X. In this case, for
        fixed  [23Xp[123X,  [23X\lambda[123X,  [23X\sigma[123X,  each pair [23X(r,t)[123X gives rise to a distinct
        quadratic  module  [Nob76,  Satz  4].  The  following  table  covers a
        complete  list  of  representatives  of  equivalence  classes  of such
        modules.[133X
  
          [23Xp[123X      [23X\lambda[123X          [23X\sigma[123X                 [23X(r,t)[123X                       [23X\alpha[123X                   [23X\beta[123X          [23X\mathfrak{A}_0[123X       
        ────────   ─────────   ──────────────────────   ──────────────────────   ─────────────────────────────────   ─────────────   ────────────────────────  
          [23X3[123X         [23X2[123X               [23X1[123X               [23Xr,t \in \{1,2\}[123X               [23X|\alpha| = 3[123X               [23X(-1,0)[123X      [23X\langle \alpha \rangle[123X   
          [23X3[123X      [23X\geq 3[123X             [23X1[123X             [23Xt=1[123X, [23Xr \in \{1,2\}[123X   [23X|\alpha| = 3^{\lambda-\sigma-1}[123X   [23X|\beta| = 6[123X   [23X\langle \alpha \rangle[123X   
          [23X3[123X      [23X\geq 3[123X             [23X1[123X             [23Xt=2[123X, [23Xr \in \{1,2\}[123X    [23X|\alpha| = 3^{\lambda-\sigma}[123X      [23X(-1,0)[123X      [23X\langle \alpha \rangle[123X   
          [23X3[123X      [23X\geq 3[123X     [23X2,\dots,\lambda-1[123X       [23Xr,t \in \{1,2\}[123X       [23X|\alpha| = 3^{\lambda-\sigma}[123X      [23X(-1,0)[123X      [23X\langle \alpha \rangle[123X   
        [23X\geq 5[123X   [23X\geq 2[123X    [23X1, \dots,\lambda - 1[123X     [23Xr,t \in \{1,u\}[123X       [23X|\alpha| = p^{\lambda-\sigma}[123X      [23X(-1,0)[123X      [23X\langle \alpha \rangle[123X   
  
  [30X    [33X[0;6YIf  [23Xp=2[123X,  then  the generic case occurs when [23X\lambda \geq 3[123X and [23X\sigma
        \in  \{0,\dots,\lambda-3\}[123X. Again, [23X\mathfrak{A} = \langle\alpha\rangle
        \times  \langle\beta\rangle[123X;  the order is [23X2^{\lambda-\sigma-k}[123X with [23Xk
        \in  \{0,1,2\}[123X  (as  specified  below).  In  this  case,  for fixed [23Xp[123X,
        [23X\lambda[123X,  [23X\sigma[123X,  two  pairs [23X(r_1,t_1)[123X and [23X(r_2,t_2)[123X may give rise to
        equivalent  quadratic  modules  [Nob76,  Satz  4]. The following table
        covers  a  complete  list of representatives of equivalence classes of
        such modules.[133X
  
        [23X\lambda[123X   [23X\sigma[123X      [23Xr[123X         [23Xt[123X                                 [23X\alpha = (x,y)[123X                                    [23X\beta[123X              [23X\mathfrak{A}_0[123X         
        ─────────   ────────   ─────────   ─────────   ──────────────────────────────────────────────────────────────────────   ───────────────────   ───────────────────────────  
           [23X3[123X        [23X0[123X        [23X1,3[123X       [23X1,5[123X                                    [23X(1,0)[123X                                   [23X(\frac{t-1}{2},1)[123X    [23X\langle (-1,0) \rangle[123X     
           [23X3[123X        [23X0[123X         [23X1[123X        [23X3,7[123X                                    [23X(1,0)[123X                                        [23X(-1,0)[123X          [23X\langle (-1,0) \rangle[123X     
           [23X4[123X        [23X0[123X        [23X1,3[123X        [23X5[123X         [23Xx=2, y \equiv 3 \operatorname{mod} 4, |\alpha| = 2^{\lambda-2}[123X           [23X(-1,0)[123X         [23X\langle -\alpha^2 \rangle[123X   
        [23X\geq 4[123X      [23X0[123X        [23X1,3[123X        [23X1[123X        [23Xx \equiv 1 \operatorname{mod} 4, y = 4, |\alpha| = 2^{\lambda-3}[123X           [23X(0,1)[123X          [23X\langle \alpha \rangle[123X     
        [23X\geq 4[123X      [23X0[123X         [23X1[123X        [23X3,7[123X       [23Xx \equiv 1 \operatorname{mod} 4, y = 4, |\alpha| = 2^{\lambda-3}[123X          [23X(-1,0)[123X          [23X\langle \alpha \rangle[123X     
        [23X\geq 5[123X      [23X0[123X        [23X1,3[123X        [23X5[123X         [23Xx=2, y \equiv 3 \operatorname{mod} 4, |\alpha| = 2^{\lambda-2}[123X           [23X(-1,0)[123X          [23X\langle \alpha \rangle[123X     
        [23X\geq 3[123X     [23X1,2[123X     [23X1,3,5,7[123X   [23X1,3,5,7[123X   [23Xx\equiv 1 \operatorname{mod} 4, y=2, |\alpha| = 2^{\lambda-\sigma-2}[123X        [23X(-1,0)[123X          [23X\langle \alpha \rangle[123X     
        [23X\geq 3[123X    [23X\geq 3[123X   [23X1,3,5,7[123X   [23X1,3,5,7[123X   [23Xx\equiv 1 \operatorname{mod} 4, y=1, |\alpha| = 2^{\lambda-\sigma-1}[123X        [23X(-1,0)[123X          [23X\langle \alpha \rangle[123X     
  
  
  [1X2.2-4 [33X[0;0YType R, unary and extremal cases[133X[101X
  
  [33X[0;0YThis section covers the unary and extremal cases of type [23XR[123X.[133X
  
  [33X[0;0YFirst,  in  the  unary  family, we have [23Xp[123X odd and [23X\sigma = \lambda[123X. Then the
  second  factor  of  [23XM[123X is trivial (and hence [23Xt[123X is irrelevant). We then denote
  [23XR_{p^\lambda}(r)  =  R(p,\lambda,\lambda,r,t)[123X.  In  this  case,  we  do  not
  decompose  [23XW(M,Q)[123X  using characters: instead, if [23X\lambda \leq 2[123X, then [23XW(M,Q)[123X
  contains  two  distinct  irreducible  subrepresentations of level [23Xp^\lambda[123X,
  denoted   [23XR_{p^\lambda}(r)_{\pm}[123X;  otherwise,  it  contains  a  single  such
  subrepresentation,  denoted  [23XR_{p^\lambda}(r)_1[123X. The unary family is handled
  by  [2XSL2IrrepRUnary[102X  ([14X3.3-3[114X)  (which  is  called  by  [2XSL2IrrepR[102X  ([14X3.3-2[114X) when
  appropriate).[133X
  
  [33X[0;0YSecond,  in  the  extremal family, we have [23Xp=2[123X, [23X\lambda \geq 2[123X, and [23X\sigma =
  \lambda   -   2[123X.   Then   the   second   factor   of   [23XM[123X  is  isomorphic  to
  [23X\mathbb{Z}/2\mathbb{Z}[123X,  and  collapses in [23X2M[123X. Here, [23X\operatorname{Aut}(M,Q)[123X
  is  itself  abelian,  so we let [23X\mathfrak{A} = \operatorname{Aut}(M,Q)[123X. This
  group has order 1, 2, or 4, with the following structure:[133X
  
  [30X    [33X[0;6YFor  [23X\lambda  =  2[123X  and [23Xt=1[123X, [23X\mathfrak{A} = \langle \tau \rangle[123X where
        [23X\tau  :  (x,y)  \mapsto  (y,x)[123X,  and  [23X\mathfrak{A}_0  = \mathfrak{A} =
        \langle\tau\rangle[123X.[133X
  
  [30X    [33X[0;6YFor  [23X\lambda  =  2[123X  and  [23Xt  = 3[123X, [23X\mathfrak{A}[123X is trivial; there are no
        primitive characters.[133X
  
  [30X    [33X[0;6YFor  [23X\lambda  =  3[123X  or  [23X4[123X,  [23X\mathfrak{A}  =  \{\pm  1\}[123X acting on [23XM[123X by
        multiplication; there are no primitive characters.[133X
  
  [30X    [33X[0;6YFinally,  for [23X\lambda \geq 5[123X, [23X\mathfrak{A} = \operatorname{Aut}(M,Q) =
        \langle  \alpha \rangle \times \langle -1 \rangle[123X with [23X\alpha[123X of order
        2,  and  [23X\mathfrak{A}_0  =  \langle\alpha\rangle[123X.  Note that, for this
        special  case,  the  usual  test for primitivity (described in Section
        [14X2.1[114X)  fails,  as  there  are  no  elements  of  [23X\mathfrak{A}[123X fixing [23X2M[123X
        pointwise.[133X
  
  [33X[0;0YThe  extremal family is handled by [2XSL2ModuleR[102X ([14X3.3-1[114X) and [2XSL2IrrepR[102X ([14X3.3-2[114X),
  just like the generic case.[133X
  
