CoCalc Public Fileswww / tables / screenshot.html
Author: William A. Stein
Hecke "Screen Shots

# Hecke "Screen Shots"

[[email protected] web]$hecke HECKE: Modular Forms Calculator (old version) W. Stein Send bug reports and suggestions to [email protected] Type ? for help. HECKE> ? Modes: m: Modular symbols calculator M: 'm' mode but with more features f: Formula calculator g: Method of graphs calculator t: Table making routines a: About q: Quit HECKE> a This program was written by William Stein ([email protected]). Kevin Buzzard and Hendrik Lenstra contributed to the design. HECKE> m Define a space M_k(N,chi;K) of modular symbols. level N = 100 (100 = 2^2*5^2) character chi = 0 Loading 0_Chi weight k = 2 Loading Manin symbols list (size=180, name = 0_MSymbols_2): 0% 7% 14% 21% 28% 36% 43% 50% 57% 65% 72% 79% 86% 93% ***************************************************** Initializing M_2(Gamma_0(100=2^2*5^2))^+ ***************************************************** Making binomial table. Done. Now computing M_k = (Manin Symbols) / (S, T, and I Relations). Step 1: mod out by S and I relations. Done. [loading 28x18 matrix: ]Precomputing Heilbronn matrices. Loading from database. Done. Initialization of Manin symbols data complete. Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=M_2, dim=18 --------------------------------------------------------------- M_2(100) ? t Tn: Enter values of n, then q when done. 2 [loading 18x18 matrix: ]f2=[charpoly 18] (x + 2)*(x + 1)^2*(x -2)^2*(x -1)^4*(x )^9; x Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=M_2, dim=18 --------------------------------------------------------------- M_2(100) ? v Select one: c: CUSP forms m: MODULAR forms n: NEW cusp forms e: Eisenstein part s: save V l: load V x: cancel > ? c [loading 18x7 matrix: ] Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_2, dim=7 --------------------------------------------------------------- M_2(100) ? t Tn: Enter values of n, then q when done. 2 Computing Tn_On for n = 2 ..[loading 18x18 matrix: ][simple echelon] 0.02s f2=[charpoly 7] (x -1)*(x + 1)*(x )^5; x Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_2, dim=7 --------------------------------------------------------------- M_2(100) ? m matrix display on Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_2, dim=7 --------------------------------------------------------------- M_2(100) ? t Tn: Enter values of n, then q when done. 2 Computing Tn_On for n = 2 ..[loading 18x18 matrix: ][simple echelon] 0.02s T2=[0,0,0,0,0,0,0; 0,-1,0,0,0,1,1; 1,1,0,0,0,0,0; 0,-1,0,0,0,1,1; 0,1,0,0,0,-1,-1; 0,0,0,0,0,0,0; 1,0,0,0,0,1,1]; f2=[charpoly 7] (x -1)*(x + 1)*(x )^5; x Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_2, dim=7 --------------------------------------------------------------- M_2(100) ? v Select one: c: CUSP forms m: MODULAR forms n: NEW cusp forms e: Eisenstein part s: save V l: load V x: cancel > ? n Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_k^{new}, dim=1 --------------------------------------------------------------- Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_k^{new}, dim=1 --------------------------------------------------------------- M_2(100) ? t Tn: Enter values of n, then q when done. 2 Computing Tn_On for n = 2 ..[loading 18x18 matrix: ][simple echelon] 0s T2=[0]; f2=[charpoly 1] x ; x Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_k^{new}, dim=1 --------------------------------------------------------------- M_2(100) ? ? Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_k^{new}, dim=1 --------------------------------------------------------------- a: ap list -- Hecke eigenvalues b: basis of rational q-expansions c: toggle charpoly display [on] d: disc(T|V) - discriminant of Hecke algebra e: compute subspace with given Tp eigenvalues f: fast computation of Tp|subspace g: compute subspace with f(Tp)=0, f some polynomial h: display the basis for M_k in terms of Manin and modular symbols. i: compute the winding element -{0,oo} on the free basis j: compute the complex conjugation involution. l: computes if the winding element vanishes on the dual of V. m: toggle matrix display [on] n: basis (up to conjugates) of q-expansions of eigenforms o: L(A_f,1)/Omega p: reduce modulo p. q: Quit r: toggle trace display [off] s: display current space t: Hecke operator Tn on V u: convert a modular symbol to a sum of Manin symbols v: change v to the full space of modular forms, newforms, or cusp forms. w: Atkin-Lehner involution Wn on V x: Exit z: integral basis for for H_1(X_0(N),Z). Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_k^{new}, dim=1 --------------------------------------------------------------- M_2(100) ? b number of terms? 10 ** Loading dual eigenbasis from datafile. Loading MatVecList, size=1 [1] Finding e. Computing the values 1/. [[NTL XGCD] 0% (0s)] Computing ap: [2 3 5 7 ][Recursively computing an... f0: [ 1 2 3 4 5 6 7 8 9 10] ] F-rational basis of q-expansions (NOT eigenforms) for V f[1] = q+2*q^3-2*q^7+q^9+O(q^10); Top Level --------------------------------------------------------------- Current space: M_2(Gamma_0(100=2^2*5^2); Q)^+, dim=18 Hecke action on: V=S_k^{new}, dim=1 --------------------------------------------------------------- M_2(100) ? x HECKE> x Bye bash$