# 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
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
weight k = 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.
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
(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
x: cancel
> ? c
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
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
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
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
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
[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\$

```