2) Local Admissible Representation at p=5.sagews

cf = ModularForms(Gamma0(50),2).cuspidal_subspace()
f = cf.newforms()[0]

MS = ModularSymbols(GammaH(50,[crt(1+5, 1, 25, 2)]))
sigmaf = (MS.T(7) - f.coefficient(7)).kernel().sign_submodule(1).basis()[0]
sigmaf_c = MS.coordinate_vector(sigmaf)


mat = MS._action_on_modular_symbols([5,1,0,5])
MS_c = MS.free_module()

translates = mat.maxspin(sigmaf_c)

typespace = MS_c.submodule(translates)

print "\n"

tt = mat.restrict(typespace)


ch = matrix([[1,1,1,1],[2,-3,2,2], [-2,3,-2,3], [-1,-1,4,-1]])

ch * tt* ~ch

[ 0 0 0 -1] [ 1 0 0 -1] [ 0 1 0 -1] [ 0 0 1 -1]

Slift = [25, 24, 26, 25]
bSbi = [5*25, 24, 25 * 26, 5 * 25]
ss = MS._action_on_modular_symbols(bSbi).restrict(typespace).transpose()
5* ch * ss * ~ch

[-4 3 0 -3] [-3 6 0 -1] [-2 4 -5 1] [-6 2 0 3]

Rlift = [25, 51, 24, 49]
bRbi = [5 * 25, 51, 25 * 24, 5 * 49]

rr = MS._action_on_modular_symbols(bRbi).restrict(typespace).transpose()
5 * ch * rr * ~ch

[ 3 0 -3 4] [ 6 0 -1 -2] [ 4 -5 1 2] [ 2 0 3 1]

ss * tt

[ 1 -2 2 -1] [ 0 -1 2 -1] [ 0 0 1 0] [ 0 1 0 0]

T = matrix([[1,1],[0,1]])
S =  matrix([[0,1],[-1,0]])
tor2 = matrix([[2,0],[0,1]])

tor2i = matrix([[3,0],[0,1]])

tor2 * S * tor2i

[ 0 2] [-3 0]

c = matrix([[10,7],[7,5]])
-T*S*~T^3*S*~T^2*S*~T^2*S

[10 7] [ 7 5]

ss_conj = tt* ss * ~tt^3 * ss * ~tt^2 * ss * ~tt^2 * ss
ss_conj

print "\n"

ss

print "\n"

tt_conj = tt^2
tt_conj

print "\n"

tt

[-1 2 -1 0] [ 0 1 0 0] [ 0 0 1 0] [ 0 -1 2 -1] [ 0 1 -2 1] [ 0 0 -1 0] [ 0 -1 0 0] [ 1 -2 1 0] [-1 0 1 -1] [ 0 -1 2 -1] [ 1 -2 2 -1] [ 1 -1 1 -1] [ 0 0 0 -1] [ 0 0 -1 0] [ 0 1 -2 1] [ 1 0 -1 1]

MatSp = ss.parent()
rows = []
for m in MatSp.basis():
    to_add_1 = (ss_conj * m - m * ss).list()
    to_add_2 = (tt_conj * m - m * tt).list()
    to_add = to_add_1 + to_add_2
    rows.append(to_add)
themat = matrix(rows)
poss_tor = themat.left_kernel()
tor2 = MatSp(poss_tor.gens()[0].list())
ch*tor2*~ch

[ 0 1 0 -1] [ 0 0 0 -1] [ 0 0 1 -1] [ 1 0 0 -1]

matrix([[0,2],[1,0]])

[0 2] [1 0]

from sage.modular.local_comp.liftings import *
lift_matrix_to_sl2z([4,8,1,1],5)

[4, 3, 1, 1]

~tor2 * ss

[-1 0 0 0] [-1 0 1 -1] [-1 1 0 -1] [ 0 0 0 -1]

G = GL(2,5)
sG = G([0,-1,1,0])
sT = G([1,1,0,1])
sgam2 = G([2,0,0,1])

a = G([3,-2,-1,3])
a.word_problem([sG,sT,sgam2])

([2 0] [0 1]) * ([1 1] [0 1]) * ([0 4] [1 0])^-1 * ([1 1] [0 1])^2

f25gen = tor2 * tt * ~ss * tt^2
for i in range(25): 
    print (f25gen^i).trace()

4 1 1 -2 1 1 4 1 1 -2 1 1 4 1 1 -2 1 1 4 1 1 -2 1 1 4

(~tt)^-2 * ~ss * ~tt * ~tor2

[ 1 -2 1 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1]

KK.<cc> = GF(25, 'cc')
KK
sqrt2 = 4*cc - 2

Finite Field in cc of size 5^2

cc.multiplicative_order()

24

3 - sqrt2

cc

cc^3

4*cc + 3