
# q-expansion of newform 462.2.i.f, downloaded from the LMFDB on 23 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 = 462
weight = 2
poly_data = [4, -18, 27, -19, 12, -3, 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, 0, 0], 1], [[3, -1, 1, 0, 0, 0], 1], [[-18, 43, -28, 22, -5, 2], 2], [[22, -56, 37, -31, 7, -3], 1], [[50, -121, 82, -64, 15, -6], 2], [[24, -62, 44, -33, 8, -3], 1]]

hecke_ring_character_values = [[155, [1, 0, 0, 0, 0, 0]], [199, [-1, 0, -1, 0, 0, 0]], [211, [1, 0, 0, 0, 0, 0]]]
aps_data = [[-1, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, -1, 0, 0, 1, 0], [0, -1, 0, -1, 1, -1], [0, 0, 1, 0, 0, 0], [0, 2, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [1, 1, 0, -2, 0, -1], [-3, 0, -4, -2, 1, -1], [4, 0, 0, -1, 0, 1], [-2, -2, 0, -2, 4, -4], [-3, 1, -4, -2, 0, -1], [5, 0, 0, -1, 0, 1], [2, -2, 0, 1, 0, -1], [5, 0, 4, -2, 1, -1], [0, 0, 0, 0, -2, 0], [1, 1, 4, 1, 2, 2], [2, 3, 4, 4, -5, 2], [-1, -1, 1, -1, 2, -2], [2, 0, 0, -1, 0, 1], [2, 2, 2, 2, -2, 4], [-4, -3, -4, 0, 3, 0], [-3, -4, 0, 1, 0, -1], [12, -2, 12, 0, 2, 0], [1, 0, 0, 2, 0, -2], [-1, -1, -6, -1, -4, -2], [4, 2, 2, -4, 0, -2], [-8, -5, -9, -2, 6, -1], [2, 2, 4, 2, -5, 4], [-6, 0, 0, 2, 0, -2], [2, -3, 0, 1, 0, -1], [6, 6, 8, 4, -8, 2], [0, 0, 0, 0, -2, 0], [12, 0, 0, -3, 0, 3], [-9, -1, -8, 2, 0, 1], [-1, -1, 0, -1, 5, -2], [-3, -3, -6, -3, 0, -6], [4, 5, 5, 2, -6, 1], [6, 6, 0, -2, 0, 2], [14, 2, 16, 4, -4, 2], [-1, -1, -10, -1, 4, -2], [12, -2, 0, -2, 0, 2], [-8, -4, -8, 0, 4, 0], [2, 2, 0, 2, 2, 4], [4, -2, 0, 3, 0, -3], [2, 2, -10, 2, 0, 4], [8, 2, 0, -4, 0, 4], [-2, 2, 0, 2, 0, -2], [1, 1, 5, 1, -2, 2], [2, 0, 0, -4, 2, -2], [-7, 2, -9, -4, 0, -2], [0, 2, 0, -4, 0, 4], [2, 2, 2, 2, 4, 4], [-8, -6, 0, 1, 0, -1], [16, -2, 16, 0, 2, 0], [-4, -4, -14, -4, -2, -8], [-2, -2, -12, -2, 7, -4], [-8, -4, -8, 0, 4, 0], [0, 0, 8, 0, -6, 0], [-11, 0, 0, 2, 0, -2], [-4, -4, 0, -4, 10, -8], [-10, 0, 0, 3, 0, -3], [-2, -4, 0, 2, 0, -2], [-1, -1, -4, -1, -3, -2], [-25, -1, -26, -2, 2, -1], [4, 5, 8, 8, -9, 4], [0, 5, 5, 10, -10, 5], [0, 10, 0, -2, 0, 2], [-1, -1, -15, -1, 6, -2], [12, 5, 0, 0, 0, 0], [-2, -2, -18, -2, 6, -4], [10, 8, 10, 0, -8, 0], [0, 0, 20, 0, -6, 0], [-2, 5, -4, -4, -3, -2], [-1, 4, 0, 1, 0, -1], [-5, 1, 0, 10, -6, 5], [0, 0, -4, 0, -1, 0], [-13, -1, -10, 6, -2, 3], [2, -4, 0, -4, 6, -2], [6, 6, -12, 6, -10, 12], [0, -6, 0, -1, 0, 1], [-20, -2, 0, 3, 0, -3], [4, 4, 8, 4, 6, 8], [11, -4, 0, -2, 0, 2], [1, -4, 4, 6, 1, 3], [19, -5, 18, -2, 6, -1], [4, -2, 0, -4, 0, 4], [10, -4, 14, 8, 0, 4], [-28, -4, 0, 5, 0, -5], [-22, -2, 0, 2, 0, -2], [-21, 1, -24, -6, 2, -3], [0, 0, -2, 0, -6, 0], [-2, -2, -30, -2, -2, -4], [-3, -12, 0, 5, 0, -5], [-2, -10, 0, 4, 8, 2], [4, 6, 0, -2, 0, 2], [4, 2, 0, -8, 2, -4], [0, 0, -2, 0, 8, 0], [-20, 6, -16, 8, -10, 4], [-16, -1, -12, 8, -3, 4], [2, -2, 0, 3, 0, -3], [-3, -3, -26, -3, 4, -6], [0, 0, 0, 0, 0, 0], [-15, -7, -18, -6, 10, -3], [1, 1, 10, 1, -2, 2], [-3, -3, 9, -3, 10, -6], [-30, -2, 0, -2, 0, 2], [11, -5, 6, -10, 10, -5], [-4, -4, -8, -4, 5, -8], [24, 0, 0, -2, 0, 2], [-12, -1, -8, 8, -3, 4], [-4, -4, -4, -4, 9, -8], [4, 2, 0, -6, 0, 6], [3, 3, 19, 3, 6, 6], [2, 2, 0, 6, 0, -6], [-2, -2, -12, -2, 0, -4], [-16, 0, 0, 6, 0, -6], [4, 4, 16, 4, 1, 8], [0, -3, -4, -8, 7, -4], [35, 4, 0, 1, 0, -1], [1, 1, -10, 1, 4, 2], [-2, -4, 0, 6, 0, -6], [25, -3, 28, 6, 0, 3], [-7, -7, -12, -7, 14, -14], [0, 7, 3, 6, -10, 3], [2, 2, 0, 3, 0, -3], [15, 1, 16, 2, -2, 1], [9, 4, 8, -2, -3, -1], [2, -8, 0, 4, 0, -4], [-6, 1, -8, -4, 1, -2], [6, 6, -10, 6, -12, 12], [-32, 2, 0, -2, 0, 2], [10, 8, 8, -4, -6, -2], [4, 4, 0, 3, 0, -3], [-20, -1, -21, -2, 2, -1], [28, 6, 0, -10, 0, 10], [-2, -2, -28, -2, 11, -4], [9, 9, 16, 9, -8, 18], [0, -7, 0, -2, 0, 2], [1, 1, -15, 1, -2, 2], [-42, 0, 0, 2, 0, -2], [-10, 6, -8, 4, -8, 2], [-6, -6, -12, -6, 6, -12], [21, 0, 0, -5, 0, 5], [1, 1, 48, 1, 0, 2], [0, 9, 0, -8, 0, 8], [12, 3, 0, 0, 0, 0], [4, 4, 9, 4, -4, 8], [2, 3, 1, -2, -2, -1], [28, 1, 32, 8, -5, 4], [0, 7, 4, 8, -11, 4], [-2, 0, 0, 2, 0, -2], [17, 0, 0, -9, 0, 9], [-12, 0, -6, 12, -6, 6], [-3, -3, -9, -3, -6, -6], [14, -15, 0, 1, 0, -1], [19, -6, 24, 10, 1, 5], [-10, 8, -4, 12, -14, 6], [-14, 0, 0, 6, 0, -6], [3, 3, -30, 3, -6, 6], [11, -9, 6, -10, 14, -5], [15, -12, 0, 5, 0, -5], [30, -5, 0, 1, 0, -1], [12, 2, 18, 12, -8, 6], [8, 8, -4, 8, -12, 16], [-3, -3, -36, -3, -1, -6], [0, 0, 8, 0, -10, 0], [0, 0, 40, 0, -2, 0]]
