
# q-expansion of newform 450.2.e.m, downloaded from the LMFDB on 18 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 450
weight = 2
poly_data = [4, 0, -2, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0], 1], [[0, 1, 0, 0], 1], [[0, 0, 1, 0], 2], [[0, 0, 0, 1], 2]]

hecke_ring_character_values = [[101, [0, 0, -1, 0]], [127, [1, 0, 0, 0]]]
aps_data = [[1, 0, -1, 0], [-1, 0, 0, 1], [0, 0, 0, 0], [-2, 1, 2, -2], [0, 2, 0, -4], [0, 1, -2, 1], [0, 4, 0, -2], [5, 2, 0, -1], [0, 1, 0, 1], [0, 1, 0, -2], [0, -1, -2, -1], [-4, -6, 0, 3], [0, 0, -9, 0], [-5, 1, 5, -2], [6, 2, -6, -4], [-6, 2, 0, -1], [0, 1, 3, 1], [-8, 0, 8, 0], [0, -3, 7, -3], [6, 6, 0, -3], [-1, 0, 0, 0], [-2, -6, 2, 12], [3, 1, -3, -2], [-9, 0, 0, 0], [1, -4, -1, 8], [-6, -1, 6, 2], [0, -6, -2, -6], [-3, 10, 0, -5], [8, -2, 0, 1], [0, -2, 9, -2], [-4, 6, 0, -3], [0, 4, -6, 4], [-3, 0, 3, 0], [0, 0, 4, 0], [0, 0, 6, 0], [-20, 0, 20, 0], [0, 4, 10, 4], [5, 2, 0, -1], [0, -8, 0, -8], [0, -4, 0, 8], [-3, 10, 0, -5], [8, 8, 0, -4], [6, -2, -6, 4], [0, 0, -20, 0], [12, -10, 0, 5], [-4, -8, 0, 4], [0, 1, 1, 1], [-8, -4, 8, 8], [-21, 3, 21, -6], [0, -7, 4, -7], [-9, -12, 0, 6], [0, -1, -12, -1], [13, -4, -13, 8], [-3, 2, 0, -1], [0, 4, 3, 4], [-18, -1, 18, 2], [12, 2, 0, -1], [8, 6, 0, -3], [4, -1, -4, 2], [-24, 2, 24, -4], [0, 5, 1, 5], [0, -9, 6, -9], [8, 12, 0, -6], [0, -2, 6, -2], [1, 2, -1, -4], [0, -7, 0, 14], [1, -3, -1, 6], [0, 2, 16, 2], [0, 0, -12, 0], [-14, 0, 14, 0], [9, 0, -9, 0], [-24, 8, 0, -4], [-2, 6, 2, -12], [0, 8, 4, 8], [-4, -4, 0, 2], [0, 1, -24, 1], [-6, -8, 6, 16], [2, -16, 0, 8], [0, -6, -24, -6], [0, 2, -5, 2], [0, 7, 9, 7], [4, 9, -4, -18], [18, 6, 0, -3], [-10, 16, 0, -8], [4, -3, -4, 6], [-18, 4, 18, -8], [-9, -16, 0, 8], [1, -6, -1, 12], [-12, 3, 12, -6], [0, 3, -2, 3], [-3, -6, 0, 3], [12, -5, -12, 10], [-22, -2, 0, 1], [0, 5, 15, 5], [0, 3, 1, 3], [24, -2, 0, 1], [0, 0, 0, 0], [0, -24, 0, 12], [-13, -6, 0, 3], [-28, -8, 0, 4], [-23, 3, 23, -6], [12, 24, 0, -12], [0, 11, -3, 11], [-24, -4, 24, 8], [0, -11, 1, -11], [11, 24, 0, -12], [12, 6, -12, -12], [-3, -4, 0, 2], [0, -3, 30, -3], [-8, 10, 8, -20], [0, -3, 4, -3], [-10, 18, 0, -9], [0, -12, -15, -12], [-5, 15, 5, -30], [-16, 8, 0, -4], [-15, 12, 15, -24], [0, -3, 25, -3], [12, 10, 0, -5], [0, -5, -6, -5], [-27, 7, 27, -14], [0, -12, 22, -12], [-8, -6, 8, 12], [-6, 13, 6, -26], [15, 22, 0, -11], [-17, -9, 17, 18], [24, 8, 0, -4], [-2, -1, 2, 2], [30, -18, 0, 9], [16, 0, -16, 0], [0, 0, -26, 0], [-19, 18, 0, -9], [0, -3, -42, -3], [0, -9, 28, -9], [-16, -18, 0, 9], [0, -2, -9, -2], [0, 16, -11, 16], [-30, 16, 0, -8], [0, -6, -20, -6], [0, 5, 18, 5], [-21, 12, 0, -6], [-31, 2, 0, -1], [12, -16, -12, 32], [0, -2, 22, -2], [-9, -22, 0, 11], [-10, -30, 0, 15], [-30, -3, 30, 6], [46, 0, -46, 0], [45, 2, -45, -4], [0, 3, 13, 3], [-12, -26, 0, 13], [0, 4, 22, 4], [24, 12, 0, -6], [38, 8, 0, -4], [0, 10, -30, 10], [-29, -3, 29, 6], [0, 3, 0, -6], [-4, 6, 0, -3], [12, 16, -12, -32], [17, -8, 0, 4], [0, -17, -12, -17], [-9, -7, 9, 14], [12, -8, 0, 4], [0, 0, -32, 0], [-9, -26, 0, 13], [0, 0, 9, 0], [18, -2, -18, 4], [-16, 0, 0, 0], [-14, 9, 14, -18]]
