
# q-expansion of newform 1728.2.s.f, downloaded from the LMFDB on 16 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 = 1728
weight = 2
poly_data = [16, -24, 20, -12, 6, -6, 5, -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, 0, 0], 1], [[-8, 4, 4, 2, 0, 1, 1, -1], 8], [[-4, 4, -4, 2, -1, 1, -1, 0], 4], [[16, -12, 12, -2, 4, -3, 1, -1], 8], [[-32, 28, -24, 10, -6, 7, -5, 3], 8], [[20, -16, 10, -4, 4, -5, 3, -1], 4], [[-40, 20, -24, 10, -8, 9, -3, 3], 8], [[-48, 36, -28, 10, -6, 11, -5, 3], 8]]

hecke_ring_character_values = [[703, [-1, 0, 0, 0, 0, 0, 0, 0]], [325, [1, 0, 0, 0, 0, 0, 0, 0]], [1217, [0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 0], [0, 0, 0, -1, 0, 0, -1, -1], [-1, 0, 1, 0, -2, 2, 0, 0], [1, 0, -1, 0, 2, -1, 0, 0], [0, 2, 0, -2, 0, 0, -1, -1], [0, 0, 0, 1, 0, 0, 0, -1], [1, 0, 0, 0, 0, -1, 0, 0], [0, 2, 0, -1, 0, 0, 2, -3], [2, 0, 2, 0, 0, -2, 0, 0], [-4, 0, -2, 0, -1, 0, 0, 0], [0, 2, 0, 3, 0, 0, 1, -3], [0, 1, 0, -4, 0, 0, -3, -4], [4, 0, -2, 0, 8, -2, 0, 0], [0, -1, 0, 4, 0, 0, 1, -3], [1, 0, -2, 0, 2, -1, 0, 0], [0, 3, 0, 2, 0, 0, 3, -1], [0, 2, 0, 2, 0, 0, 4, -4], [1, 0, 1, 0, 0, -1, 0, 0], [0, 3, 0, 4, 0, 0, 1, -4], [0, 3, 0, -4, 0, 0, -1, -4], [4, 0, 2, 0, 8, 2, 0, 0], [3, 0, -8, 0, 7, -4, 0, 0], [3, 0, 0, 0, -8, 5, 0, 0], [0, 2, 0, 5, 0, 0, 2, 3], [0, -2, 0, -2, 0, 0, 5, -5], [-4, 0, -4, 0, 0, 4, 0, 0], [-5, 0, 7, 0, -6, 0, 0, 0], [0, 6, 0, -6, 0, 0, -6, -6], [0, 4, 0, -3, 0, 0, -4, -1], [5, 0, 0, 0, -9, 4, 0, 0], [0, 3, 0, -4, 0, 0, 3, -7], [-13, 0, -1, 0, -6, 0, 0, 0], [0, -3, 0, -2, 0, 0, 1, 2], [5, 0, -5, 0, 10, -10, 0, 0], [0, -2, 0, 2, 0, 0, 4, 4], [0, 10, 0, -7, 0, 0, -10, -3], [7, 0, 0, 0, 0, -7, 0, 0], [0, 4, 0, 4, 0, 0, -4, 4], [4, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 2, 0, 0, -3, 2], [-2, 0, 2, 0, -1, 4, 0, 0], [8, 0, 4, 0, 16, 4, 0, 0], [0, 0, 0, 0, 0, 0, 6, 6], [0, -4, 0, 3, 0, 0, -4, 7], [0, -1, 0, 2, 0, 0, 3, -2], [0, 4, 0, 3, 0, 0, 7, 3], [-1, 0, 1, 0, -8, 2, 0, 0], [13, 0, -7, 0, 26, -7, 0, 0], [0, -8, 0, 5, 0, 0, 8, 3], [-7, 0, 8, 0, -11, 4, 0, 0], [0, -6, 0, -6, 0, 0, -3, 3], [-2, 0, -8, 0, 3, 0, 0, 0], [0, 9, 0, -2, 0, 0, 7, -2], [8, 0, -14, 0, 16, -14, 0, 0], [0, -10, 0, 10, 0, 0, 2, 2], [3, 0, 2, 0, 2, 1, 0, 0], [11, 0, 0, 0, -6, -5, 0, 0], [0, -4, 0, -3, 0, 0, -4, 1], [-25, 0, -1, 0, -12, 0, 0, 0], [0, -6, 0, 6, 0, 0, 9, 9], [0, -6, 0, -5, 0, 0, 6, 11], [5, 0, 8, 0, 1, 4, 0, 0], [-11, 0, 0, 0, -6, 17, 0, 0], [0, -11, 0, 4, 0, 0, 15, -4], [2, 0, -2, 0, -7, -4, 0, 0], [0, -1, 0, -2, 0, 0, 1, 3], [-9, 0, 2, 0, -10, 1, 0, 0], [-3, 0, 0, 0, -1, 4, 0, 0], [0, 8, 0, 8, 0, 0, -2, 2], [0, -5, 0, 0, 0, 0, 5, 0], [-5, 0, 5, 0, -2, 10, 0, 0], [0, 0, 0, 0, 0, 0, -3, -3], [0, 2, 0, -3, 0, 0, -2, 1], [15, 0, 0, 0, -8, -7, 0, 0], [14, 0, 2, 0, 0, -2, 0, 0], [10, 0, -8, 0, 9, 0, 0, 0], [-2, 0, 2, 0, 5, 4, 0, 0], [0, 4, 0, -9, 0, 0, -4, 5], [13, 0, -14, 0, 20, -7, 0, 0], [0, -8, 0, -8, 0, 0, -10, 10], [-11, 0, 1, 0, 0, -1, 0, 0], [0, -11, 0, -6, 0, 0, 5, 6], [0, -4, 0, 1, 0, 0, -3, 1], [-1, 0, -7, 0, -2, -7, 0, 0], [-13, 0, -16, 0, -5, -8, 0, 0], [-3, 0, 0, 0, 10, -7, 0, 0], [0, -4, 0, -7, 0, 0, -4, -3], [0, 6, 0, 6, 0, 0, -3, 3], [0, 5, 0, -6, 0, 0, -1, -6], [0, 10, 0, -10, 0, 0, 4, 4], [0, -11, 0, 6, 0, 0, 11, 5], [0, -1, 0, 0, 0, 0, -1, 1], [0, -6, 0, -6, 0, 0, -6, 6], [19, 0, -1, 0, 10, 0, 0, 0], [-1, 0, 11, 0, -2, 11, 0, 0], [0, -4, 0, 4, 0, 0, 2, 2], [-10, 0, 8, 0, 0, -8, 0, 0], [0, 4, 0, -5, 0, 0, -9, 5], [-10, 0, 4, 0, -20, 4, 0, 0], [0, 7, 0, -6, 0, 0, -7, -1], [11, 0, 0, 0, -3, -8, 0, 0], [0, -1, 0, -6, 0, 0, -1, -5], [1, 0, 13, 0, 0, -13, 0, 0], [0, 2, 0, -5, 0, 0, -3, -5], [-8, 0, 2, 0, -16, 2, 0, 0], [0, -6, 0, 1, 0, 0, 6, 5], [11, 0, -16, 0, 19, -8, 0, 0], [0, 6, 0, 13, 0, 0, 6, 7], [10, 0, -6, 0, 0, 6, 0, 0], [-2, 0, 4, 0, -3, 0, 0, 0], [0, 10, 0, 1, 0, 0, -9, -1], [0, -10, 0, 10, 0, 0, -4, -4], [-13, 0, 0, 0, 15, -2, 0, 0], [0, 9, 0, -4, 0, 0, 9, -13], [0, 16, 0, 16, 0, 0, 14, -14], [7, 0, -1, 0, 4, 0, 0, 0], [0, 5, 0, -8, 0, 0, -3, -8], [-1, 0, 1, 0, -20, 2, 0, 0], [-13, 0, -10, 0, -8, -5, 0, 0], [3, 0, 0, 0, -14, 11, 0, 0], [0, 2, 0, 2, 0, 0, 7, -7], [0, -5, 0, 4, 0, 0, 9, -4], [-16, 0, -2, 0, -32, -2, 0, 0], [7, 0, 10, 0, 2, 5, 0, 0], [0, 0, 0, 0, 0, 0, 6, -6], [0, -3, 0, 10, 0, 0, 13, -10], [1, 0, -1, 0, 4, -2, 0, 0], [0, 0, 0, 0, 0, 0, 3, 3], [0, 10, 0, 5, 0, 0, -10, -15], [0, 0, 0, -17, 0, 0, 0, -17], [-14, 0, 0, 0, 0, 0, 0, 0], [1, 0, -11, 0, 6, 0, 0, 0], [-11, 0, 11, 0, -40, 22, 0, 0], [20, 0, -14, 0, 40, -14, 0, 0], [0, 12, 0, 5, 0, 0, 12, -7], [41, 0, -7, 0, 24, 0, 0, 0], [7, 0, -19, 0, 14, -19, 0, 0], [0, 6, 0, -6, 0, 0, -9, -9], [-11, 0, 0, 0, 12, -1, 0, 0], [0, 0, 0, -11, 0, 0, 0, -11], [0, 8, 0, 8, 0, 0, 16, -16], [34, 0, 6, 0, 0, -6, 0, 0], [0, 17, 0, -24, 0, 0, -7, -24], [21, 0, 14, 0, 14, 7, 0, 0], [-13, 0, 0, 0, -6, 19, 0, 0], [0, 5, 0, 12, 0, 0, 5, 7], [0, 4, 0, 4, 0, 0, 14, -14], [-1, 0, 1, 0, -26, 2, 0, 0], [-16, 0, -10, 0, -32, -10, 0, 0], [0, 14, 0, -14, 0, 0, -13, -13], [0, -6, 0, 7, 0, 0, 6, -1], [0, 10, 0, -5, 0, 0, -15, 5], [0, -13, 0, 12, 0, 0, -1, 12], [0, -2, 0, 2, 0, 0, -14, -14], [-13, 0, 0, 0, 18, -5, 0, 0], [22, 0, 10, 0, 0, -10, 0, 0], [29, 0, -7, 0, 18, 0, 0, 0], [0, 0, 0, -13, 0, 0, -13, -13], [13, 0, -13, 0, 26, -13, 0, 0], [0, 20, 0, 17, 0, 0, 20, -3], [0, 2, 0, 2, 0, 0, -14, 14], [44, 0, 10, 0, 17, 0, 0, 0], [0, -11, 0, 2, 0, 0, -9, 2], [0, -6, 0, 6, 0, 0, 6, 6], [-41, 0, 10, 0, -46, 5, 0, 0], [-14, 0, -14, 0, 0, 14, 0, 0], [7, 0, -13, 0, 10, 0, 0, 0], [0, 20, 0, -23, 0, 0, -3, -23], [-7, 0, 7, 0, 16, 14, 0, 0], [0, -16, 0, 21, 0, 0, 16, -5], [11, 0, -4, 0, 13, -2, 0, 0], [0, 0, 0, 7, 0, 0, 0, 7], [11, 0, 7, 0, 2, 0, 0, 0], [0, -20, 0, 1, 0, 0, 21, -1], [2, 0, 4, 0, 4, 4, 0, 0], [0, 26, 0, -26, 0, 0, -16, -16], [19, 0, -2, 0, 20, -1, 0, 0], [0, 17, 0, -4, 0, 0, -21, 4], [0, -16, 0, 1, 0, 0, -15, 1], [-1, 0, 1, 0, -32, 2, 0, 0], [-13, 0, 29, 0, -26, 29, 0, 0], [0, 6, 0, -11, 0, 0, -6, 5], [-23, 0, 0, 0, 6, 17, 0, 0], [-16, 0, -10, 0, 0, 10, 0, 0], [0, -1, 0, 0, 0, 0, 1, 0], [-4, 0, 4, 0, -1, 8, 0, 0], [0, 8, 0, 8, 0, 0, -14, 14], [31, 0, -5, 0, 0, 5, 0, 0], [0, -6, 0, 17, 0, 0, 11, 17], [0, 2, 0, -2, 0, 0, -19, -19], [-17, 0, 0, 0, 30, -13, 0, 0], [0, 6, 0, 6, 0, 0, -9, 9], [-4, 0, -38, 0, 17, 0, 0, 0], [4, 0, -4, 0, -19, -8, 0, 0], [-9, 0, -22, 0, 2, -11, 0, 0], [-13, 0, 0, 0, 21, -8, 0, 0], [0, 12, 0, 12, 0, 0, 12, -12], [23, 0, 23, 0, 0, 0, 0, 0], [0, -21, 0, -2, 0, 0, 19, 2], [13, 0, -13, 0, 22, -26, 0, 0], [5, 0, -28, 0, 19, -14, 0, 0], [0, 6, 0, 6, 0, 0, 9, -9], [-20, 0, -2, 0, -40, -2, 0, 0], [0, 6, 0, -6, 0, 0, 12, 12], [0, -17, 0, 24, 0, 0, 17, -7], [3, 0, 0, 0, -13, 10, 0, 0], [0, -19, 0, -6, 0, 0, -19, 13], [-37, 0, 3, 0, 0, -3, 0, 0], [47, 0, 11, 0, 18, 0, 0, 0], [0, -11, 0, 12, 0, 0, 23, -12], [0, 17, 0, 4, 0, 0, 21, 4], [0, 24, 0, -17, 0, 0, -24, -7], [-51, 0, 4, 0, -53, 2, 0, 0], [0, -10, 0, 23, 0, 0, -10, 33], [-19, 0, 0, 0, 21, -2, 0, 0], [0, -20, 0, -20, 0, 0, -4, 4], [11, 0, 23, 0, -6, 0, 0, 0], [-1, 0, 1, 0, 22, 2, 0, 0], [0, -16, 0, -13, 0, 0, -16, 3], [16, 0, -14, 0, 15, 0, 0, 0], [0, -14, 0, 14, 0, 0, 4, 4], [0, -7, 0, 10, 0, 0, 7, -3], [-41, 0, -2, 0, -40, -1, 0, 0], [9, 0, 0, 0, -10, 1, 0, 0], [0, 16, 0, 16, 0, 0, 2, -2], [0, 11, 0, -4, 0, 0, -15, 4], [0, -11, 0, 12, 0, 0, 1, 12], [11, 0, -11, 0, 52, -22, 0, 0], [0, -22, 0, 22, 0, 0, 20, 20], [0, -16, 0, -7, 0, 0, -16, 9], [10, 0, 28, 0, -9, 0, 0, 0], [0, -11, 0, -20, 0, 0, -9, 20], [0, 19, 0, 2, 0, 0, 21, 2], [-5, 0, 5, 0, -4, 10, 0, 0], [10, 0, -20, 0, 20, -20, 0, 0], [0, 5, 0, -14, 0, 0, -5, 9], [0, -16, 0, -16, 0, 0, 4, -4], [0, -8, 0, 9, 0, 0, 1, 9], [0, -14, 0, 14, 0, 0, 16, 16], [0, 4, 0, 15, 0, 0, 4, 11], [-8, 0, 12, 0, 0, -12, 0, 0], [14, 0, 4, 0, 5, 0, 0, 0], [0, -23, 0, 32, 0, 0, 9, 32], [0, 14, 0, -14, 0, 0, 2, 2], [0, -3, 0, -16, 0, 0, 3, 19], [0, 4, 0, 7, 0, 0, 4, 3], [0, 0, 0, 0, 0, 0, -24, 24], [-19, 0, 19, 0, -32, 38, 0, 0], [13, 0, -37, 0, 26, -37, 0, 0], [0, 24, 0, -11, 0, 0, -24, -13], [29, 0, -22, 0, 40, -11, 0, 0], [-5, 0, 0, 0, -18, 23, 0, 0], [0, -4, 0, -4, 0, 0, -11, 11], [-28, 0, -4, 0, 0, 4, 0, 0], [0, -15, 0, -28, 0, 0, -13, 28], [14, 0, -14, 0, 28, -14, 0, 0], [17, 0, -3, 0, 0, 3, 0, 0], [0, -7, 0, -22, 0, 0, -15, 22], [0, 8, 0, -23, 0, 0, -15, -23], [-7, 0, 7, 0, -2, 14, 0, 0], [46, 0, -6, 0, 0, 6, 0, 0], [17, 0, -11, 0, 14, 0, 0, 0], [0, -22, 0, -15, 0, 0, 7, 15], [8, 0, -2, 0, 16, -2, 0, 0], [-13, 0, 0, 0, 3, 10, 0, 0], [0, -3, 0, 8, 0, 0, -3, 11], [-67, 0, 5, 0, -36, 0, 0, 0], [-1, 0, 1, 0, -50, 2, 0, 0], [0, 10, 0, -10, 0, 0, 4, 4], [-51, 0, 16, 0, -59, 8, 0, 0], [0, 8, 0, -1, 0, 0, 8, -9], [-14, 0, 14, 0, 11, 28, 0, 0], [0, -14, 0, 14, 0, 0, -2, -2], [0, 4, 0, 9, 0, 0, -4, -13], [13, 0, 10, 0, 8, 5, 0, 0], [1, 0, -11, 0, 0, 11, 0, 0], [0, -18, 0, 17, 0, 0, -1, 17], [0, 10, 0, -19, 0, 0, -10, 9], [0, -2, 0, -3, 0, 0, -2, -1], [0, -3, 0, -2, 0, 0, -5, -2], [-41, 0, -14, 0, -34, -7, 0, 0], [0, -7, 0, 30, 0, 0, -7, 37], [0, -26, 0, -26, 0, 0, -10, 10], [-26, 0, -2, 0, 0, 2, 0, 0], [31, 0, 23, 0, 4, 0, 0, 0], [0, -1, 0, -10, 0, 0, -9, 10], [-23, 0, 29, 0, -46, 29, 0, 0], [-15, 0, 0, 0, -8, 23, 0, 0], [0, 8, 0, 8, 0, 0, 7, -7], [-50, 0, -8, 0, -21, 0, 0, 0], [0, -25, 0, 10, 0, 0, 25, 15], [19, 0, -26, 0, 32, -13, 0, 0], [5, 0, -31, 0, 18, 0, 0, 0], [0, 1, 0, -6, 0, 0, -7, 6], [-11, 0, 0, 0, 0, 11, 0, 0], [0, 16, 0, 16, 0, 0, 20, -20], [0, 23, 0, 12, 0, 0, -11, -12], [16, 0, -16, 0, 35, -32, 0, 0], [16, 0, 10, 0, 32, 10, 0, 0], [0, 6, 0, -6, 0, 0, 0, 0]]
