# Index

Symbols | A | B | C | D | E | F | G | H | K | L | M | N | O | P | R | S | T | U | V | W

## Symbols

 $\cF$ $\Gamma$-invariant on the left $\Mat_2(\Z)_n$ $\Q$-rank $\sM_k(G)$ $\sM_k(N,\eps)$ $\sS_k(\Gamma)$ $k$-sharblies $q$-expansion \Delta cusp forms \Gamma(N) \Gamma_0(N) \Gamma_1(N) \h^* \h \SL_2(\Z), [1] \sM_k(G R) \Z/N\Z Sage f^{[\gamma]_k} G_k(z) j-function M_k M_{36} Sage

## A

 abelian variety attached to $f$ action of \SL_2(\Z) cusps action of Hecke operators algorithm to enumerate Eisenstein series and Bernoulli numbers Eisenstein series and boundary map cusps and boundary space Manin symbols and cuspidal subspace Manin symbols and cusps Dirichlet character and Manin symbols boundary modular symbols cuspidal modular symbols antiholomorphic are eigenforms Eisenstein series arithmetic group associate proper $\Q$-parabolic subgroups of $G$ associated period map newform automorphic form automorphy factor

## B

 basis for M_{24} Sage basis for S_2(\Gamma_0(N)) Sage basis of Eisenstein series Bernoulli numbers Sage Bernoulli numbers modulo p Sage Bianchi groups Birch and Swinnerton-Dyer conjecture Borel conjecture boundary map, [1], [2] Sage computing boundary modular symbols and Manin symbols bounded domains

## C

 cellular decomposition character of the modular form Cholesky decomposition codimension complex upper half plane computation of period mapping compute Eisenstein series compute using Sage Eisenstein series computing boundary map conductor congruence subgroup, [1] congruence subgroup problem congruent number problem conjecture Maeda Shimura-Taniyama Connected continued fraction convergents Sage criterion for vanishing cusps critical integers cross polytope cusp form cusp forms \Delta for \Gamma higher level dimension, [1] cusp forms of higher level dimension, [1] cuspidal cuspidal automorphic form cuspidal cohomology cuspidal modular symbols, [1] and Manin symbols cuspidal submodule Sage cusps action of \SL_2(\Z) and boundary map criterion for vanishing cusps for a congruence subgroup \Gamma

## D

 Defined over $\Q$ degeneracy map, [1] diamond-bracket action diamond-bracket operators, [1] dimension cusp forms of higher level, [1] dimension S_k(\Gamma_0(N)) Sage dimension S_k(\Gamma_1(N)) Sage dimension formulas Sage dimension with character Sage, [1] Diophantine equations Dirichlet character, [1] and cusps Dirichlet character tutorial Sage Dirichlet group Sage divisor

## E

 echelon form Sage eigenforms Eilenberg--Mac Lane Eisenstein arithmetic Sage Eisenstein cohomology Eisenstein series, [1] Fourier expansion Sage algorithm to enumerate and Bernoulli numbers are eigenforms basis of compute compute using Sage Eisenstein subspace evaluation of character Sage extended modular symbols, [1] extended upper half plane

## F

 fan Farey tessellation Fermat's last theorem finite presentation modular symbols for \Gamma cusp forms formal power series Fourier expansion Eisenstein series

## G

 generalized Bernoulli numbers Sage generalized modular symbol generators over \Z Hecke algebra Grothendieck motive group cohomology

## H

 Hecke algebra, [1], [2] generators over \Z Hecke correspondence Hecke operator, [1], [2], [3], [4] Hecke operators \sM_2(\Gamma_0(39)) Sage Hecke operators \sM_2(\Gamma_0(6)) Sage Hecke polynomials height Heilbronn matrices, [1], [2], [3], [4], [5] Sage Hermite normal form, [1] Hermitian symmetric spaces higher level dimension cusp forms, [1] holomorphic holomorphic at $\infty$ holomorphic at the cusp $\alpha$ Humbert forms hypersimplices

## K

 Krylov methods Krylov subspace, [1]

## L

 Laplace--Beltrami--Casimir operator lattices left action of $\GL_2(\Q)$ left action of $\SL_2(\Z)$ left action of $G$ left translations level $1$, [1] level of $\Gamma$ linear fractional transformations

## M

 Maass forms Maeda conjecture Maeda's conjecture Manin symbol Manin symbols Sage and boundary space and cuspidal subspace meromorphic meromorphic at $\infty$ Miller basis Sage modular complex modular elliptic curves modular form, [1] modular function modular group modular symbols Sage finite presentation new and old subspace of modular symbols algorithm modular symbols for \Gamma_0(N) modular symbols of level 11 Sage modular symbols over a ring $R$ modular symbols printing Sage

## N

 new and old subspace of modular symbols new modular symbols, [1] new subspace, [1] newform, [1], [2], [3] associated period map nonnormalized weight $k$ Eisenstein series normalized Eisenstein series number field sieve

## O

 old modular symbols, [1] old subspace opposite

## P

 partitions perfect perfection period mapping computation of Petersson inner product, [1] pivot column plus one quotient primitive, [1] primitive character associated to principal congruence subgroup

## R

 Ramanujan function Ramanujan graphs rational Jordan form rational period mapping rational reconstruction Sage real-analytic reduced reducing point regular relative to the cusps restriction of scalars right action of $\SL_2(\Z)$, [1] right action of \GL_2(\Q) right translation

## S

 Sage Bernoulli numbers Bernoulli numbers modulo p Dirichlet character tutorial Dirichlet group Eisenstein arithmetic Eisenstein series Hecke operators \sM_2(\Gamma_0(39)) Hecke operators \sM_2(\Gamma_0(6)) Heilbronn matrices Manin symbols Miller basis M_{36} \Z/N\Z basis for M_{24} basis for S_2(\Gamma_0(N)) boundary map continued fraction convergents cuspidal submodule dimension S_k(\Gamma_0(N)) dimension S_k(\Gamma_1(N)) dimension formulas dimension with character, [1] echelon form evaluation of character generalized Bernoulli numbers modular symbols modular symbols of level 11 modular symbols printing rational reconstruction satisfies condition $C_n$ self-adjoint homogeneous cone Semisimple Serre's conjecture set of cusps Set of real points sharbly complex Shimura-Taniyama conjecture Shimura-Taniyama conjecture sigma function slowly increasing split form of $\SL_{n}$ split symplectic group standard fundamental domain star involution strong deformation retract symplectic sharbly complex

## T

 tilings topological cell transportable

## U

 unimodular

## V

 valence formula virtual cohomological dimension Voronoi polyhedron Voronoi reduction algorithm Voronoi decomposition

## W

 weakly modular function, [1] Weierstrass $\wp$-function weight, [1], [2], [3], [4] weight $k$ modular symbols for $G$ weight $k$ right action well-rounded retract width of the cusp