def algebraic_part_of_standard_l(f, l, space_of_cuspforms, verbose=False):
r'''f: (vector valued) cuspidal eigenform of degree 2 of weight det^k Sym(j).
l: positive even integer such that 2 le l < k - 2.
space_of_cuspforms: space of cusp form that f belongs to.
Return the algebriac part of the standard L of f at l
cf. Katsurada, Takemori Congruence primes of the Kim-Ramakrishnan-Shahidi lift. Theorem 4.1.
'''
k = f.wt
j = f.sym_wt
t0 = f._none_zero_tpl()
D = tpl_to_half_int_mat(t0)
if not (l % 2 == 0 and 2 <= l < k - 2):
raise ValueError
u3_val, u4_val, f_t0_pol_val = _u3_u4_nonzero(f, t0)
pull_back_vec = _pullback_vector(
l + ZZ(2), D, u3_val, u4_val, space_of_cuspforms, verbose=verbose)
T2 = space_of_cuspforms.hecke_matrix(2)
d = space_of_cuspforms.dimension()
vecs = [(T2 ** i) * pull_back_vec for i in range(d)]
ei = [sum(f[t0] * a for f, a in zip(space_of_cuspforms.basis(), v))
for v in vecs]
if j > 0:
ei = [a._to_pol() for a in ei]
chply = T2.charpoly()
nume = first_elt_of_kern_of_vandermonde(chply, f.hecke_eigenvalue(2), ei)
denom = f[t0] * f_t0_pol_val
if j > 0:
denom = denom._to_pol()
return f.base_ring(nume / denom)
def _assert(d):
alphas = []
while len(set(alphas)) < d and all(a != 0 for a in alphas):
alphas.append(random_prime(100000))
alphas = list(set(alphas))
A = matrix([[alpha ** i for alpha in alphas] for i in range(d)])
v = [random_prime(100000) for _ in range(d)]
x = PolynomialRing(QQ, names="x").gens()[0]
chpy = mul(x - alpha for alpha in alphas)
self.assertEqual(first_elt_of_kern_of_vandermonde(chpy, alphas[0], v),
(A ** (-1) * vector(v))[0])
