  
  [1X10 [33X[0;0YCongruence Subgroups, Cuspidal Cohomology and Hecke Operators[133X[101X
  
  [33X[0;0YIn  this  chapter  we  explain  how HAP can be used to make computions about
  modular forms associated to congruence subgroups [22XΓ[122X of [22XSL_2( Z)[122X.[133X
  
  
  [1X10.1 [33X[0;0YEichler-Shimura isomorphism[133X[101X
  
  [33X[0;0YWe begin by recalling the Eichler-Shimura isomorphism [Eic57][Shi59][133X
  
  
  [24X[33X[0;6YS_k(\Gamma)  \oplus  \overline{S_k(\Gamma)}  \oplus  E_k(\Gamma)  \cong_{\sf
  Hecke} H^1(\Gamma,M_{k-2})[133X
  
  [124X
  
  [33X[0;0Ywhich  relates  the  cohomology  of  groups  to  the theory of modular forms
  associated  to a finite index subgroup [22XΓ[122X of [22XSL_2( Z)[122X. In subsequent sections
  we  explain  how to compute with the right-hand side of the isomorphism. But
  first, for completeness, let us define the terms on the left-hand side.[133X
  
  [33X[0;0YLet  [22XN[122X  be  a  positive  integer.  A  subgroup [22XΓ[122X of [22XSL_2( Z)[122X is said to be a
  [13Xcongruence  subgroup[113X  of  level [22XN[122X if it contains the kernel of the canonical
  homomorphism  [22Xπ_N: SL_2( Z) → SL_2( Z/N Z)[122X. So any congruence subgroup is of
  finite index in [22XSL_2( Z)[122X, but the converse is not true.[133X
  
  [33X[0;0YOne  congruence  subgroup of particular interest is the group [22XΓ(N)=ker(π_N)[122X,
  known  as  the  [13Xprincipal congruence subgroup[113X of level [22XN[122X. Another congruence
  subgroup  of  particular interest is the group [22XΓ_0(N)[122X of those matrices that
  project to upper triangular matrices in [22XSL_2( Z/N Z)[122X.[133X
  
  [33X[0;0YA  [13Xmodular  form[113X of weight [22Xk[122X for a congruence subgroup [22XΓ[122X is a complex valued
  function  on  the  upper-half  plane,  [22Xf:  frakh}={z∈  C  :  Re(z)>0}  →  C[122X,
  satisfying:[133X
  
  [30X    [33X[0;6Y[22Xdisplaystyle f(fracaz+bcz+d) = (cz+d)^k f(z)[122X for [22X(beginarraylla&b c &d
        endarray) ∈ Γ[122X,[133X
  
  [30X    [33X[0;6Y[22Xf[122X  is `holomorphic' on the [13Xextended upper-half plane[113X [22Xfrakh^∗ = frakh ∪
        Q  ∪  {∞}[122X  obtained from the upper-half plane by `adjoining a point at
        each cusp'.[133X
  
  [33X[0;0YThe  collection  of  all  weight  [22Xk[122X  modular forms for [22XΓ[122X form a vector space
  [22XM_k(Γ)[122X over [22XC[122X.[133X
  
  [33X[0;0YA  modular form [22Xf[122X is said to be a [13Xcusp form[113X if [22Xf(∞)=0[122X. The collection of all
  weight  [22Xk[122X  cusp  forms  for  [22XΓ[122X  form  a  vector  subspace [22XS_k(Γ)[122X. There is a
  decomposition[133X
  
  
  [24X[33X[0;6YM_k(\Gamma) \cong S_k(\Gamma) \oplus E_k(\Gamma)[133X
  
  [124X
  
  [33X[0;0Yinvolving  a  summand  [22XE_k(Γ)[122X known as the [13XEisenstein space[113X. See [Ste07] for
  further introductory details on modular forms.[133X
  
  [33X[0;0YThe  Eichler-Shimura  isomorphism  is  more  than  an  isomorphism of vector
  spaces.  It  is an isomorphism of Hecke modules: both sides admit notions of
  [13XHecke  operators[113X,  and the isomorphism preserves these operators. The bar on
  the  left-hand  side  of  the  isomorphism  denotes complex conjugation. See
  [Wie78] for a full account of the isomorphism.[133X
  
  [33X[0;0YOn  the  right-hand  side  of  the  isomorphism, the [22XZΓ[122X-module [22XM_k-2⊂ C[x,y][122X
  denotes  the  space  of  homogeneous degree [22Xk-2[122X polynomials with action of [22XΓ[122X
  given by[133X
  
  
  [24X[33X[0;6Y\left(\begin{array}{ll}a&b\\   c   &d   \end{array}\right)\cdot   p(x,y)   =
  p(ax+cy,bx+dy)\ .[133X
  
  [124X
  
  [33X[0;0YIn particular [22XM_0= C[122X is the trivial module.[133X
  
  [33X[0;0YIn  the  following sections we explain how to use the right-hand side of the
  Eichler-Shimura  isomorphism  to  compute eigenvalues of the Hecke operators
  restricted to the subspace [22XS_k(Γ)[122X of cusp forms.[133X
  
  
  [1X10.2 [33X[0;0YGenerators for [22XSL_2( Z)[122X[101X[1X and the cubic tree[133X[101X
  
  [33X[0;0YThe  matrices [22XS=(beginarrayrr0&-1 1 &0 endarray)[122X and [22XT=(beginarrayrr1&1 0 &1
  endarray)[122X  generate  [22XSL_2( Z)[122X and it is not difficult to devise an algorithm
  for  expressing  an arbitrary integer matrix [22XA[122X of determinant [22X1[122X as a word in
  [22XS[122X, [22XT[122X and their inverses. The following illustrates such an algorithm.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:=[[4,9],[7,16]];;[127X[104X
    [4X[25Xgap>[125X [27Xword:=AsWordInSL2Z(A);[127X[104X
    [4X[28X[ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, 1 ], [ -1, 0 ] ], [ [ 1, -1 ], [ 0, 1 ] ], [128X[104X
    [4X[28X  [ [ 0, 1 ], [ -1, 0 ] ], [ [ 1, 1 ], [ 0, 1 ] ], [ [ 0, 1 ], [ -1, 0 ] ], [128X[104X
    [4X[28X  [ [ 1, -1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 0, 1 ] ], [128X[104X
    [4X[28X  [ [ 0, 1 ], [ -1, 0 ] ], [ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XProduct(word);[127X[104X
    [4X[28X[ [ 4, 9 ], [ 7, 16 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is  convenient  to  introduce  the  matrix [22XU=ST = (beginarrayrr0&-1 1 &1
  endarray)[122X.  The  matrices  [22XS[122X and [22XU[122X also generate [22XSL_2( Z)[122X. In fact we have a
  free presentation [22XSL_2( Z)= ⟨ S,T | S^4=U^6=1 ⟩[122X.[133X
  
  [33X[0;0YThe  [13Xcubic  tree[113X  [22Xcal T[122X is a tree ([13Xi.e.[113X a [22X1[122X-dimensional contractible regular
  CW-complex)  with  countably  infinitely many edges in which each vertex has
  degree  [22X3[122X.  We can realize the cubic tree [22Xcal T[122X by taking the left cosets of
  [22Xcal  U=⟨ U⟩[122X in [22XSL_2( Z)[122X as vertices, and joining cosets [22Xxcal U[122X and [22Xycal U[122X by
  an  edge  if,  and  only  if, [22Xx^-1y ∈ cal U Scal U[122X. Thus the vertex [22Xcal U[122X is
  joined  to  [22XScal  U[122X, [22XUScal U[122X and [22XU^2Scal U[122X. The vertices of this tree are in
  one-to-one correspondence with all reduced words in [22XS[122X, [22XU[122X and [22XU^2[122X that, apart
  from the identity, end in [22XS[122X.[133X
  
  [33X[0;0YFrom  our  realization  of the cubic tree [22Xcal T[122X we see that [22XSL_2( Z)[122X acts on
  [22Xcal  T[122X  in  such  a  way that each vertex is stabilized by a cyclic subgroup
  conjugate  to  [22Xcal  U=⟨  U⟩[122X and each edge is stabilized by a cyclic subgroup
  conjugate to [22Xcal S =⟨ S ⟩[122X.[133X
  
  [33X[0;0YIn  order  to  store this action of [22XSL_2( Z)[122X on the cubic tree [22Xcal T[122X we just
  need to record the following finite amount of information.[133X
  
  
  [1X10.3  [33X[0;0YOne-dimensional  fundamental  domains  and  generators  for congruence[101X
  [1Xsubgroups[133X[101X
  
  [33X[0;0YRecall  that  a [13Xcell structure[113X on a space [22XX[122X is a partition of [22XX[122X into subsets
  [22Xe_i[122X  such  that  each  [22Xe_i[122X is homeomorphic to an open Euclidean ball of some
  dimension.  We  say  that [22Xe_i[122X is an [22Xn[122X-[13Xcell[113X if it is homeomorphic to the open
  [22Xn[122X-dimensional  ball.  We  say  that  the cell structure is [13Xreduced[113X if it has
  precisely  one  [22X0[122X-cell.  A  CW-complex  is  a  cell complex satisfying extra
  conditions.[133X
  
  [33X[0;0YSuppose  that  we  wish  to  compute  a  set  of generators for a congruence
  subgroup  [22XΓ[122X.  The required generators correspond to the [22X1[122X-cells of a reduced
  classifying CW-complex (or free resolution) [22XB(Γ)[122X. Such a classifying complex
  can  be  constructed,  using perturbation techniques, from [22Xcal T[122X and reduced
  classifying  CW-complexes  [22XB(stab(e^0))[122X,  [22XB(Stab(e^1))[122X  for  the  stabilizer
  groups  of  a vertex and edge of [22Xcal T[122X. In this construction, the [22X1[122X-cells of
  [22XB(Γ)[122X are in one-one correspondence with generators for the first homology of
  the quotient graph [22XΓ ∖ cal T[122X together with the [22X1[122X-cells of [22XB(stab(e^0))[122X. If [22XΓ[122X
  acts freely on the vertices of [22Xcal T[122X then the [22X1[122X-cells of [22XB(Γ)[122X are in one-one
  correspondence with just the generators for the first homology of [22XΓ ∖ cal T[122X.
  To determine the quotient [22XΓ ∖ cal T[122X we need to determine a cellular subspace
  [22XD_Γ  ⊂ cal T[122X whose images under the action of [22XΓ[122X cover [22Xcal T[122X and are pairwise
  either  disjoint  or identical. The subspace [22XD_Γ[122X will not be a CW-complex as
  it  won't  be  closed,  but  it  can  be  chosen  to be connected, and hence
  contractible.  We call [22XD_Γ[122X a [13Xfundamental region[113X for [22XΓ[122X. We denote by [22Xmathring
  D_Γ[122X  the  largest CW-subcomplex of [22XD_Γ[122X. The vertices of [22Xmathring D_Γ[122X are the
  same  as  the  vertices  of [22XD_Γ[122X. Thus [22Xmathring D_Γ[122X is a subtree of the cubic
  tree  whose  vertices  correspond to coset representatives of [22XΓ[122X in [22XSL_2( Z)[122X.
  For  each  vertex [22Xv[122X in the tree [22Xmathring D_Γ[122X define [22Xη(v)=3 - degree(v)[122X. Then
  the  number  of homology generators for [22XΓ ∖ cal T[122X will be [22X(1/2)∑_v∈ mathring
  D_Γ  η(v)[122X. The role of tree diagrams in the study of congruence subgroups of
  [22XSL_2( Z)[122X is explained in detail in [Kul91].[133X
  
  [33X[0;0YSuppose  that  we  wish  to  calculate a set of generators for the principal
  congruence  subgroup  [22XΓ(N)[122X  of  level [22XN[122X. Note that [22XΓ(N)[122X intersects trivially
  with  [22Xcal  U[122X, and hence [22XΓ(N)[122X acts freely on the vertices of the cubical tree
  [22Xcal  T[122X.  The  following  commands  determine generators for [22XΓ(6)[122X and display
  [22Xmathring D_Γ(6)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=HAP_PrincipalCongruenceSubgroup(6);;[127X[104X
    [4X[25Xgap>[125X [27Xgens:=GeneratorsOfGroup(G);[127X[104X
    [4X[28X[ [ [ -65, 18 ], [ 18, -5 ] ], [ [ -41, 18 ], [ 66, -29 ] ], [128X[104X
    [4X[28X  [ [ -29, 12 ], [ 12, -5 ] ], [ [ -17, -12 ], [ -24, -17 ] ], [128X[104X
    [4X[28X  [ [ -17, -6 ], [ -48, -17 ] ], [ [ -5, 6 ], [ -6, 7 ] ], [128X[104X
    [4X[28X  [ [ -5, 18 ], [ -12, 43 ] ], [ [ 1, -6 ], [ 0, 1 ] ], [128X[104X
    [4X[28X  [ [ 1, 0 ], [ -6, 1 ] ], [ [ 7, -18 ], [ -12, 31 ] ], [128X[104X
    [4X[28X  [ [ 7, 12 ], [ 18, 31 ] ], [ [ 7, 18 ], [ 12, 31 ] ], [128X[104X
    [4X[28X  [ [ 13, -18 ], [ -18, 25 ] ], [ [ 19, -30 ], [ -12, 19 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XHAP_SL2TreeDisplay(G);[127X[104X
    [4X[28X[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe congruence subgroup [22XΓ_0(N)[122X does not act freely on the vertices of [22Xcal T[122X.
  However,  we  can replace [22Xcal T[122X by a double cover [22Xcal T'[122X which admits a free
  action  of  [22XΓ_0(N)[122X  on its vertices. The following commands display [22Xmathring
  D_Γ(39)[122X for a fundamental region in [22Xcal T'[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=HAP_CongruenceSubgroupGamma0(39);;[127X[104X
    [4X[25Xgap>[125X [27XHAP_SL2TreeDisplay(G);[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo compute [22XD_Γ[122X one only needs to be able to test whether a given matrix lies
  in  [22XΓ[122X  or  not.  However,  for  speed,  the  above  calculations of [22XD_Γ[122X take
  advantage  in GAP's facility for iterating over elements of [22XSL_2( Z/N Z)[122X. An
  algorithm  that  does not use this facility is also implemented but seems to
  be a bit slower in general.[133X
  
  [33X[0;0YGiven  an  inclusion [22XΓ'⊂ Γ[122X of congruence subgroups, it is straightforward to
  use  the  trees  [22Xmathring D_Γ'[122X and [22Xmathring D_Γ[122X to compute a system of coset
  representative for [22XΓ'∖ Γ[122X.[133X
  
  
  [1X10.4 [33X[0;0YCohomology of congruence subgroups[133X[101X
  
  [33X[0;0YTo  compute  the  cohomology  [22XH^n(Γ,A)[122X  of  a  congruence  subgroup  [22XΓ[122X  with
  coefficients  in  a  [22XZΓ[122X-module  [22XA[122X  we  need to construct [22Xn+1[122X terms of a free
  [22XZG[122X-resolution  of  [22XZ[122X.  We can do this by first using perturbation techniques
  (as  described in [BE14]) to combine the cubic tree with resolutions for the
  cyclic  groups of order [22X4[122X and [22X6[122X in order to produce a free [22XZG[122X-resolution [22XR_∗[122X
  for  [22XG=SL_2( Z)[122X. This resolution is also a free [22XZΓ[122X-resolution with each term
  of rank[133X
  
  
  [24X[33X[0;6Y{\rm  rank}_{\mathbb Z\Gamma} R_k = |G:\Gamma|\times {\rm rank}_{\mathbb ZG}
  R_k\ .[133X
  
  [124X
  
  [33X[0;0YFor  congruence  subgroups  of lowish index in [22XG[122X this resolution suffices to
  make computations.[133X
  
  [33X[0;0YThe following commands compute[133X
  
  
  [24X[33X[0;6YH^1(\Gamma_0(39),\mathbb Z) = \mathbb Z^9\ .[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSL2Z_alt(2);[127X[104X
    [4X[28XResolution of length 2 in characteristic 0 for SL(2,Integers) .[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xgamma:=HAP_CongruenceSubgroupGamma0(39);;[127X[104X
    [4X[25Xgap>[125X [27XS:=ResolutionFiniteSubgroup(R,gamma);[127X[104X
    [4X[28XResolution of length 2 in characteristic 0 for [128X[104X
    [4X[28XCongruenceSubgroupGamma0( 39)  .[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XCohomology(HomToIntegers(S),1);[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  commands  show  that [22Xrank_ ZΓ_0(39) R_1 = 112[122X but that it is
  possible  to  apply  `Tietze  like'  simplifications to [22XR_∗[122X to obtain a free
  [22XZΓ_0(39)[122X-resolution  [22XT_∗[122X  with [22Xrank_ ZΓ_0(39) T_1 = 11[122X. It is more efficient
  to  work with [22XT_∗[122X when making cohomology computations with coefficients in a
  module [22XA[122X of large rank.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS!.dimension(1);[127X[104X
    [4X[28X112[128X[104X
    [4X[25Xgap>[125X [27XT:=TietzeReducedResolution(S);[127X[104X
    [4X[28XResolution of length 2 in characteristic 0 for CongruenceSubgroupGamma0([128X[104X
    [4X[28X39)  . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XT!.dimension(1);[127X[104X
    [4X[28X11[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  computations  establish  that the space [22XM_2(Γ_0(39))[122X of weight [22X2[122X
  modular forms is of dimension [22X9[122X.[133X
  
  
  [1X10.5 [33X[0;0YCuspidal cohomology[133X[101X
  
  [33X[0;0YTo define and compute cuspidal cohomology we consider the action of [22XSL_2( Z)[122X
  on the upper-half plane [22Xfrak h[122X given by[133X
  
  
  [24X[33X[0;6Y\left(\begin{array}{ll}a&b\\ c &d \end{array}\right) z = \frac{az +b}{cz+d}\
  .[133X
  
  [124X
  
  [33X[0;0YA standard `fundamental domain' for this action is the region[133X
  
  
  [24X[33X[0;6Y\begin{array}{ll}  D=&\{z\in  {\frak  h}\  :\  |z|  >  1,  |{\rm  Re}(z)|  <
  \frac{1}{2}\}   \\   &  \cup\  \{z\in  {\frak  h}  \  :\  |z|  \ge  1,  {\rm
  Re}(z)=-\frac{1}{2}\}\\  &  \cup\  \{z \in {\frak h}\ :\ |z|=1, -\frac{1}{2}
  \le {\rm Re}(z) \le 0\} \end{array}[133X
  
  [124X
  
  [33X[0;0Yillustrated below.[133X
  
  [33X[0;0YThe   action   factors   through   an   action  of  [22XPSL_2(  Z)  =SL_2(  Z)/⟨
  (beginarrayrr-1&0  0  &-1  endarray)⟩[122X.  The  images of [22XD[122X under the action of
  [22XPSL_2(  Z)[122X  cover  the  upper-half  plane, and any two images have at most a
  single  point in common. The possible common points are the bottom left-hand
  corner  point which is stabilized by [22X⟨ U⟩[122X, and the bottom middle point which
  is stabilized by [22X⟨ S⟩[122X.[133X
  
  [33X[0;0YA  congruence  subgroup [22XΓ[122X has a `fundamental domain' [22XD_Γ[122X equal to a union of
  finitely  many  copies  of  [22XD[122X,  one  copy for each coset in [22XΓ∖ SL_2( Z)[122X. The
  quotient  space  [22XX=Γ∖  frak  h[122X  is  not  compact, and can be compactified in
  several ways. We are interested in the Borel-Serre compactification. This is
  a space [22XX^BS[122X for which there is an inclusion [22XX↪ X^BS[122X and this inclusion is a
  homotopy  equivalence.  One  defines the [13Xboundary[113X [22X∂ X^BS = X^BS - X[122X and uses
  the  inclusion  [22X∂  X^BS  ↪ X^BS ≃ X[122X to define the cuspidal cohomology group,
  over the ground ring [22XC[122X, as[133X
  
  
  [24X[33X[0;6YH_{cusp}^n(\Gamma,M_{k-2}) = \ker (\ H^n(X,M_{k-2}) \rightarrow H^n(\partial
  X^{BS},M_{k-2}) \ ).[133X
  
  [124X
  
  [33X[0;0YStricly   speaking,   this   is   the   definition  of  [13Xinterior  cohomology[113X
  [22XH_!^n(Γ,M_k-2)[122X   and   not  cuspidal  cohomology.  However,  for  congruence
  subgroups   of   [22XSL_2(   Z)[122X   there   is   an   equality   [22XH_!^n(Γ,M_k-2)  =
  H_cusp^n(Γ,M_k-2)[122X.[133X
  
  [33X[0;0YWorking over [22XC[122X has the advantage of avoiding the technical issue that [22XΓ[122X does
  not  necessarily  act  freely  on  [22Xfrak h[122X since there are points with finite
  cyclic  stabilizer groups in [22XSL_2( Z)[122X. But it has the disadvantage of losing
  information  about  torsion  in  cohomology.  So  HAP confronts the issue by
  working  with  a  contractible CW-complex [22Xtilde X^BS[122X on which [22XΓ[122X acts freely,
  and  [22XΓ[122X-equivariant  inclusion  [22X∂  tilde X^BS ↪ tilde X^BS[122X. The definition of
  cuspidal  cohomology  that we use, which coincides with the above definition
  when working over [22XC[122X, is[133X
  
  
  [24X[33X[0;6YH_{cusp}^n(\Gamma,A)     =     \ker    (\    H^n({\rm    Hom}_{\,    \mathbb
  Z\Gamma}(C_\ast(\tilde  X^{BS}),  A)\,  )  \rightarrow  H^n(\  {\rm Hom}_{\,
  \mathbb Z\Gamma}(C_\ast(\tilde \partial X^{BS}), A)\, \ ).[133X
  
  [124X
  
  [33X[0;0YThe  following  data is recorded and, using perturbation theory, is combined
  with free resolutions for [22XC_4[122X and [22XC_6[122X to constuct [22Xtilde X^BS[122X.[133X
  
  [33X[0;0YThe following commands calculate[133X
  
  
  [24X[33X[0;6YH^1_{cusp}(\Gamma_0(39),\mathbb Z) = \mathbb Z^6\ .[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgamma:=HAP_CongruenceSubgroupGamma0(39);;[127X[104X
    [4X[25Xgap>[125X [27Xc:=CuspidalIntegralCohomology(gamma,1);[127X[104X
    [4X[28X[ g1, g2, g3, g4, g5, g6, g7, g8, g9 ] -> [ g1^-1*g3, g1^-1*g3, g1^-1*g3, [128X[104X
    [4X[28X  g1^-1*g3, g1^-1*g2, g1^-1*g3, g1^-1*g4, g1^-1*g4, g1^-1*g4 ][128X[104X
    [4X[25Xgap>[125X [27XAbelianInvariants(Kernel(c));[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFrom the Eichler-Shimura isomorphism and the already calculated dimension of
  [22XM_2(Γ_0(39))≅  C^9[122X,  we  deduce from this cuspidal cohomology that the space
  [22XS_2(Γ_0(39))[122X  of  cuspidal  weight  2  forms  is  of  dimension  [22X3[122X,  and the
  Eisenstein space [22XE_2(Γ_0(39))≅ C^3[122X is of dimension [22X3[122X.[133X
  
  
  [1X10.6 [33X[0;0YHecke operators[133X[101X
  
  [33X[0;0YA  congruence  subgroup  [22XΓ le SL_n( Z)[122X and element [22Xg∈ SL_n( Q)[122X determine the
  subgroup [22XΓ' = Γ ∩ gΓ g^-1[122X and homomorphisms[133X
  
  
  [24X[33X[0;6Y\Gamma\  \hookleftarrow\  \Gamma'\  \  \stackrel{\gamma \mapsto g^{-1}\gamma
  g}{\longrightarrow}\ \ g^{-1}\Gamma' g\ \hookrightarrow \Gamma\ .[133X
  
  [124X
  
  [33X[0;0YThese homomorphisms give rise to homomorphisms of cohomology groups[133X
  
  
  [24X[33X[0;6YH^k(\Gamma,\mathbb  Z)\  \ \stackrel{tr}{\leftarrow} \ \ H^k(\Gamma',\mathbb
  Z)  \ \ \stackrel{\alpha}{\leftarrow} \ \ H^k(g^{-1}\Gamma' g,\mathbb Z) \ \
  \stackrel{\beta}{\leftarrow} H^k(\Gamma, \mathbb Z)[133X
  
  [124X
  
  [33X[0;0Ywith  [22Xα[122X, [22Xβ[122X functorial maps, and [22Xtr[122X the transfer map. We define the composite
  [22XT_g=tr  ∘ α ∘ β: H^k(Γ, Z) → H^k(Γ, Z)[122X to be the [13X Hecke operator [113X determined
  by [22Xg[122X. Further details on this description of Hecke operators can be found in
  [Ste07, Appendix by P. Gunnells].[133X
  
  [33X[0;0YThe       following       commands       computs      [22XT_g[122X      for      [22Xk=1[122X,
  [22Xg=(beginarraycc2&00&frac12endarray)[122X and [22XΓ=Γ_0(39)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgamma:=HAP_CongruenceSubgroupGamma0(39);;[127X[104X
    [4X[25Xgap>[125X [27Xp:=2;;N:=1;;h:=HeckeOperator(gamma,p,N);;[127X[104X
    [4X[25Xgap>[125X [27XAbelianInvariants(Source(h));[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XA:=HomomorphismAsMatrix(h);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(A);[127X[104X
    [4X[28X[ [  -4,   2,   4,   4,  -3,   0,   3,   3,  -3 ],[128X[104X
    [4X[28X  [  -4,  -2,   6,   4,   1,   0,   1,   1,  -1 ],[128X[104X
    [4X[28X  [  -3,   1,   3,   4,   0,   0,   1,   1,  -1 ],[128X[104X
    [4X[28X  [  -3,   1,   4,   2,  -4,   2,   4,   4,  -4 ],[128X[104X
    [4X[28X  [  -5,  -1,   7,   4,   2,   0,  -1,  -1,   1 ],[128X[104X
    [4X[28X  [  -7,  -3,   6,   6,   0,   2,   2,   2,  -2 ],[128X[104X
    [4X[28X  [  -1,   7,   2,  -4,  -5,   2,   5,  -1,   1 ],[128X[104X
    [4X[28X  [  -2,  -2,   4,   4,   0,   0,  -4,   2,   4 ],[128X[104X
    [4X[28X  [   0,   4,   1,  -4,  -5,   2,   2,   2,   4 ] ][128X[104X
    [4X[25Xgap>[125X [27XEigenvalues(Rationals,A);[127X[104X
    [4X[28X[ 6, -2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
