
# q-expansion of newform 1680.2.f.f, downloaded from the LMFDB on 25 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 = 1680
weight = 2
poly_data = [1, 0, 3, 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], [[1, 1, 1, 0], 1], [[0, 4, 0, 2], 1], [[-1, 1, -1, 0], 1]]

hecke_ring_character_values = [[1471, [1, 0, 0, 0]], [421, [1, 0, 0, 0]], [1121, [-1, 0, 0, 0]], [337, [1, 0, 0, 0]], [241, [-1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [-1, -1, 0, 0], [-1, 0, 0, 0], [1, -1, -1, 0], [0, 1, -1, 1], [0, 1, 0, 1], [2, 2, 0, -2], [0, -1, -3, -1], [0, 3, 2, 3], [0, 0, -2, 0], [0, 1, 1, 1], [2, 2, 0, -2], [-6, 2, 0, -2], [6, 1, 0, -1], [2, 1, 0, -1], [0, -1, -4, -1], [-4, -4, 0, 4], [0, -4, 0, -4], [-6, -3, 0, 3], [0, -5, -3, -5], [0, 1, -4, 1], [-8, -2, 0, 2], [6, 1, 0, -1], [-10, -2, 0, 2], [0, -5, -4, -5], [-2, -4, 0, 4], [0, -1, 6, -1], [0, 3, -2, 3], [2, -4, 0, 4], [0, -1, -8, -1], [10, 9, 0, -9], [-8, -6, 0, 6], [0, 1, 0, 1], [0, -1, -7, -1], [0, -6, -4, -6], [-4, -4, 0, 4], [0, -5, 4, -5], [-10, -7, 0, 7], [10, -1, 0, 1], [2, 0, 0, 0], [0, 1, -7, 1], [0, 2, 2, 2], [0, -1, -3, -1], [-6, 0, 0, 0], [0, 1, 12, 1], [0, 1, -5, 1], [8, 0, 0, 0], [0, 3, 4, 3], [-22, -1, 0, 1], [0, 8, 2, 8], [0, -5, 4, -5], [0, -5, 3, -5], [0, 10, 2, 10], [8, 10, 0, -10], [-14, -2, 0, 2], [0, -5, -8, -5], [22, -2, 0, 2], [0, 9, 5, 9], [-6, 6, 0, -6], [0, -6, -2, -6], [0, 7, 0, 7], [-14, 0, 0, 0], [0, -5, -6, -5], [20, 10, 0, -10], [0, -3, 4, -3], [0, 5, 0, 5], [16, 4, 0, -4], [2, 0, 0, 0], [0, 7, 12, 7], [0, 4, 2, 4], [18, 6, 0, -6], [0, -13, -9, -13], [0, 7, 8, 7], [10, -2, 0, 2], [-4, 2, 0, -2], [14, 1, 0, -1], [0, 2, 0, 2], [0, 3, 0, 3], [0, -4, -12, -4], [0, 0, 6, 0], [28, 0, 0, 0], [30, 2, 0, -2], [0, -1, 1, -1], [0, -7, -12, -7], [0, 5, -1, 5], [0, -13, -4, -13], [0, -8, -10, -8], [-14, 4, 0, -4], [-22, -2, 0, 2], [22, -1, 0, 1], [18, 15, 0, -15], [-8, -4, 0, 4], [2, 1, 0, -1], [0, 9, 15, 9], [4, 10, 0, -10], [-10, -7, 0, 7], [-18, -4, 0, 4], [2, -8, 0, 8], [0, 3, -6, 3], [-34, -8, 0, 8], [-14, -3, 0, 3], [0, 5, 4, 5], [-2, -13, 0, 13], [0, -8, -6, -8], [8, -12, 0, 12], [0, -13, -8, -13], [10, -3, 0, 3], [-22, 2, 0, -2], [0, 7, 11, 7], [0, 14, 14, 14], [0, -13, -12, -13], [10, 10, 0, -10], [0, -15, -4, -15], [0, 3, 1, 3], [-4, -4, 0, 4], [0, 6, -10, 6], [0, 3, -14, 3], [2, -9, 0, 9], [0, 11, 4, 11], [0, 1, -19, 1], [0, 12, 4, 12], [10, 4, 0, -4], [-6, 0, 0, 0], [0, 3, -12, 3], [0, -5, -1, -5], [0, -4, 4, -4], [-18, -4, 0, 4], [16, -8, 0, 8], [0, 3, 0, 3], [0, -7, -20, -7], [-28, -10, 0, 10], [0, 7, 8, 7], [44, 0, 0, 0], [18, 14, 0, -14], [14, 14, 0, -14], [0, -6, -12, -6], [18, 0, 0, 0], [0, 3, -10, 3], [-14, -12, 0, 12], [0, 10, 0, 10], [0, 3, -1, 3], [0, 12, 16, 12], [-2, 11, 0, -11], [0, 3, 18, 3], [0, -10, -4, -10], [8, 4, 0, -4], [0, 9, 4, 9], [26, 2, 0, -2], [0, 7, -11, 7], [0, 7, 22, 7], [-30, -2, 0, 2], [-2, -12, 0, 12], [22, 11, 0, -11], [10, 5, 0, -5], [26, 1, 0, -1], [0, -13, -15, -13], [32, 2, 0, -2], [-10, -8, 0, 8], [0, -1, 8, -1], [-22, -18, 0, 18], [0, -1, -12, -1], [0, 3, 0, 3], [-38, -5, 0, 5], [8, -6, 0, 6], [0, 21, 12, 21], [22, 15, 0, -15], [-20, -16, 0, 16], [0, -13, 0, -13], [-2, -6, 0, 6], [42, 8, 0, -8], [0, -7, 9, -7], [0, -14, -6, -14], [0, -5, -15, -5], [10, 0, 0, 0], [0, 5, -1, 5], [18, 4, 0, -4], [-32, 8, 0, -8], [0, 2, 26, 2], [0, -13, 4, -13], [0, 2, -8, 2], [-6, -13, 0, 13], [8, 14, 0, -14], [-30, -6, 0, 6], [-22, 6, 0, -6], [0, 19, 12, 19], [22, 4, 0, -4], [-14, -2, 0, 2], [0, -1, -2, -1], [14, 16, 0, -16], [44, -2, 0, 2], [0, 5, 24, 5], [0, 7, -6, 7], [24, -8, 0, 8], [34, 8, 0, -8], [0, -5, -4, -5], [18, -14, 0, 14], [-18, -8, 0, 8], [26, -2, 0, 2], [2, 14, 0, -14], [38, 13, 0, -13], [0, -12, 14, -12], [0, 1, -23, 1], [0, 11, 16, 11], [0, -2, -24, -2], [-20, -4, 0, 4], [26, -12, 0, 12], [0, 5, 11, 5], [0, 3, 2, 3], [0, 6, 24, 6], [0, -5, 19, -5], [34, 16, 0, -16], [54, 2, 0, -2], [46, 13, 0, -13], [-38, 7, 0, -7], [-8, -12, 0, 12], [0, 2, 34, 2], [-38, -3, 0, 3], [22, 14, 0, -14], [0, 3, 0, 3], [0, 7, 12, 7], [-18, 10, 0, -10], [0, -19, -5, -19], [0, -2, 16, -2], [-50, 5, 0, -5], [26, 9, 0, -9], [-6, -4, 0, 4], [10, 22, 0, -22], [0, -5, -1, -5], [0, 11, 6, 11], [0, 9, 11, 9], [34, 18, 0, -18], [0, 19, 13, 19], [-12, -12, 0, 12], [0, 6, 14, 6], [0, 23, 14, 23], [42, -3, 0, 3], [0, -8, -14, -8], [0, -9, 16, -9], [0, -7, 17, -7], [-44, -10, 0, 10], [0, -13, -10, -13], [0, -1, 19, -1], [0, -5, 14, -5], [10, -6, 0, 6], [26, -6, 0, 6], [-56, -8, 0, 8], [0, 7, -4, 7], [8, 2, 0, -2], [52, 10, 0, -10], [0, -1, 18, -1], [-22, -22, 0, 22], [-42, -2, 0, 2], [0, -9, -2, -9], [0, -32, -14, -32], [2, 12, 0, -12], [0, 13, 29, 13], [18, 26, 0, -26], [0, 11, 20, 11], [34, 0, 0, 0], [0, 11, -8, 11], [14, -1, 0, 1], [0, -13, 12, -13], [0, 8, -14, 8], [0, -7, 4, -7], [-14, -2, 0, 2], [0, 7, 29, 7], [0, 20, 10, 20], [-30, -8, 0, 8], [-2, -1, 0, 1], [0, 11, 12, 11], [0, 10, 6, 10], [18, 11, 0, -11], [0, -7, 24, -7], [-24, 6, 0, -6], [0, 11, 28, 11], [0, -13, -12, -13], [0, 3, -14, 3], [38, 20, 0, -20], [-14, 14, 0, -14], [-16, 10, 0, -10], [30, 3, 0, -3], [28, 8, 0, -8], [-14, 13, 0, -13], [0, 10, 14, 10], [0, -5, -10, -5], [0, 3, -7, 3], [2, -12, 0, 12], [0, 1, 12, 1], [0, -15, -5, -15], [54, 24, 0, -24], [0, -16, -8, -16], [58, 11, 0, -11], [0, 3, 12, 3], [16, -14, 0, 14], [-6, 14, 0, -14], [2, -16, 0, 16], [0, -23, 5, -23], [-38, 8, 0, -8], [-36, -28, 0, 28], [0, 3, -14, 3], [0, -15, 0, -15], [0, -3, -16, -3], [0, 10, 0, -10]]
