
# q-expansion of newform 4680.2.l.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 = 4680
weight = 2
poly_data = [64, 0, 176, 0, 97, 0, 18, 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, 0, 0, 0, 0], 1], [[-80, 184, 2, 15, 20, -10, 2, -1], 64], [[0, -168, 0, -179, 0, -46, 0, -3], 64], [[40, 176, -1, 97, -10, 18, -1, 1], 32], [[0, -104, 0, -89, 0, -18, 0, -1], 16], [[272, 120, 358, 15, 92, -10, 6, -1], 64], [[8, 0, -37, 0, -14, 0, -1, 0], 8], [[-272, 120, -358, 15, -92, -10, -6, -1], 64]]

hecke_ring_character_values = [[3511, [1, 0, 0, 0, 0, 0, 0, 0]], [2341, [1, 0, 0, 0, 0, 0, 0, 0]], [2081, [1, 0, 0, 0, 0, 0, 0, 0]], [937, [-1, 0, 0, 0, 0, 0, 0, 0]], [1081, [1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [2, 1, -1, -1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 1, -1, 1, -1, -1, 0, -1], [4, -1, 1, 1, 0, 1, -1, -1], [0, 1, 1, 1, 1, -1, 0, -1], [2, 0, 0, 0, 0, 0, -1, 0], [-4, 0, 0, 0, 0, 0, 2, 0], [0, 0, -2, 0, 0, 2, 0, 2], [0, 0, 0, 0, 0, -1, 0, 1], [0, 1, -1, 1, 2, 1, 0, 1], [0, 1, 3, 1, 0, -2, 0, -2], [0, -2, 0, -2, 0, 0, 0, 0], [-2, -1, 1, 1, 0, 0, 0, 0], [4, 1, -1, -1, 0, 1, 2, -1], [0, 2, -6, 2, 0, 0, 0, 0], [2, 2, -2, -2, 0, -1, 0, 1], [0, 0, 6, 0, 1, -2, 0, -2], [6, 1, -1, -1, 0, 3, 0, -3], [0, 3, -3, 3, 2, 2, 0, 2], [-4, 0, 0, 0, 0, -3, 2, 3], [0, 0, -2, 0, -1, -2, 0, -2], [2, 0, 0, 0, 0, 0, -5, 0], [0, -2, 0, -2, 0, 4, 0, 4], [0, 2, 6, 2, 2, 0, 0, 0], [-2, -1, 1, 1, 0, -1, -1, 1], [0, 1, -1, 1, -1, -3, 0, -3], [0, -3, -3, -3, -2, 3, 0, 3], [0, 2, -2, -2, 0, -2, 3, 2], [0, -1, 7, -1, 0, -2, 0, -2], [-4, 2, -2, -2, 0, -2, 2, 2], [-4, -1, 1, 1, 0, 0, 2, 0], [4, -2, 2, 2, 0, 2, -4, -2], [0, 5, -3, 5, 0, -1, 0, -1], [0, 0, 4, 0, 4, 0, 0, 0], [0, 3, 9, 3, -2, 2, 0, 2], [0, 0, -2, 0, 6, 0, 0, 0], [0, 4, -4, -4, 0, 2, -1, -2], [10, 2, -2, -2, 0, -2, -2, 2], [12, 2, -2, -2, 0, 2, 2, -2], [0, -4, 6, -4, 3, 4, 0, 4], [0, 3, 3, 3, 4, 2, 0, 2], [6, 1, -1, -1, 0, 3, 0, -3], [10, 1, -1, -1, 0, -1, -2, 1], [0, 2, -6, 2, 3, 0, 0, 0], [0, 1, 15, 1, 0, 2, 0, 2], [-10, 5, -5, -5, 0, -3, 3, 3], [0, 3, -3, 3, -3, -1, 0, -1], [-18, 0, 0, 0, 0, -1, 0, 1], [-2, 0, 0, 0, 0, -2, -2, 2], [0, 0, 0, 0, 0, 4, 1, -4], [0, 3, 13, 3, 1, -1, 0, -1], [0, -1, -1, -1, 3, 5, 0, 5], [10, -2, 2, 2, 0, -2, -3, 2], [4, -4, 4, 4, 0, 2, -4, -2], [0, 1, 5, 1, 4, 3, 0, 3], [8, 4, -4, -4, 0, 1, -2, -1], [0, 0, 4, 0, -4, -4, 0, -4], [0, -1, -13, -1, -4, 0, 0, 0], [0, -4, 0, -4, -2, -6, 0, -6], [-4, 0, 0, 0, 0, 2, -2, -2], [0, -1, 9, -1, -4, -3, 0, -3], [0, 3, 3, 3, 0, 2, 0, 2], [8, 5, -5, -5, 0, 1, 5, -1], [0, -1, 1, -1, -8, -3, 0, -3], [0, 2, -2, 2, 6, -2, 0, -2], [-18, -3, 3, 3, 0, -1, -5, 1], [0, 3, 3, 3, 0, 4, 0, 4], [10, 4, -4, -4, 0, 5, 0, -5], [0, 8, -6, 8, 4, 2, 0, 2], [0, 2, 12, 2, 2, -6, 0, -6], [4, -1, 1, 1, 0, 1, 5, -1], [0, 3, -19, 3, 2, 4, 0, 4], [6, -4, 4, 4, 0, -2, -3, 2], [0, 2, -8, 2, -6, 0, 0, 0], [-12, 2, -2, -2, 0, -1, -4, 1], [6, -4, 4, 4, 0, -2, 0, 2], [-4, -4, 4, 4, 0, 0, 1, 0], [-2, -5, 5, 5, 0, -5, -1, 5], [10, -6, 6, 6, 0, -1, 0, 1], [0, 1, -1, 1, -4, -5, 0, -5], [-8, -4, 4, 4, 0, 0, 4, 0], [0, -4, -4, -4, 2, -6, 0, -6], [4, -2, 2, 2, 0, 1, 2, -1], [0, -6, -12, -6, -1, -4, 0, -4], [-8, -3, 3, 3, 0, 4, 0, -4], [0, 0, 0, 0, -5, 4, 0, 4], [0, -2, 18, -2, -2, 0, 0, 0], [-2, 0, 0, 0, 0, -1, -4, 1], [0, 0, 4, 0, 3, 6, 0, 6], [-8, 2, -2, -2, 0, -2, 11, 2], [0, 1, -1, -1, 0, -3, -7, 3], [0, -3, 5, -3, -5, -1, 0, -1], [16, 5, -5, -5, 0, 6, 4, -6], [22, 0, 0, 0, 0, 2, -4, -2], [0, 1, 23, 1, -2, 1, 0, 1], [10, -1, 1, 1, 0, -5, 3, 5], [0, -1, 5, -1, 2, 7, 0, 7], [0, 3, -1, 3, 8, -2, 0, -2], [0, -8, 0, -8, -2, 4, 0, 4], [18, 8, -8, -8, 0, 2, 8, -2], [10, 5, -5, -5, 0, -1, 4, 1], [0, -6, -16, -6, 1, -2, 0, -2], [0, -1, 1, -1, -4, -6, 0, -6], [0, 1, 5, 1, -2, 2, 0, 2], [-16, -10, 10, 10, 0, -4, -4, 4], [20, 3, -3, -3, 0, -1, 6, 1], [0, 8, -2, 8, 4, -2, 0, -2], [0, -10, -4, -10, 0, 4, 0, 4], [0, 5, 5, 5, 4, 2, 0, 2], [16, 3, -3, -3, 0, 3, -3, -3], [0, 0, 0, 0, 0, 4, -4, -4], [6, 2, -2, -2, 0, 6, -6, -6], [0, 2, -6, 2, 8, 4, 0, 4], [0, 3, 11, 3, -7, 3, 0, 3], [0, -6, 0, -6, 0, -2, 0, -2], [-20, -2, 2, 2, 0, -8, 5, 8], [34, 1, -1, -1, 0, 3, -1, -3], [0, -6, 2, -6, -4, 6, 0, 6], [0, -2, -16, -2, 0, 4, 0, 4], [0, -11, 3, -11, -2, 6, 0, 6], [4, -1, 1, 1, 0, 7, -7, -7], [-10, -4, 4, 4, 0, -2, 3, 2], [-18, 1, -1, -1, 0, 1, -9, -1], [-8, -6, 6, 6, 0, 4, 4, -4], [0, -7, 25, -7, -4, -1, 0, -1], [0, -4, -6, -4, -2, 4, 0, 4], [12, -9, 9, 9, 0, 3, 1, -3], [0, 5, -5, 5, 2, 4, 0, 4], [16, -4, 4, 4, 0, -4, -4, 4], [0, -4, -22, -4, 0, -4, 0, -4], [16, -4, 4, 4, 0, -3, -4, 3], [-10, 0, 0, 0, 0, 0, 8, 0], [0, 1, -19, 1, -4, -8, 0, -8], [0, 8, 4, 8, 0, 6, 0, 6], [0, -6, -24, -6, 2, -4, 0, -4], [10, -4, 4, 4, 0, -10, 2, 10], [0, -7, 7, 7, 0, 3, 1, -3], [4, 1, -1, -1, 0, 6, 2, -6], [0, 2, 16, 2, -8, 0, 0, 0], [0, -7, -1, -7, 6, -2, 0, -2], [4, -1, 1, 1, 0, 11, -2, -11], [-2, -2, 2, 2, 0, 9, 0, -9], [0, -2, -8, -2, -6, 0, 0, 0], [0, 1, -13, 1, -3, -11, 0, -11], [2, 1, -1, -1, 0, 3, -14, -3], [0, -1, -11, -1, 6, 2, 0, 2], [0, 0, -2, 0, -2, -6, 0, -6], [-6, -4, 4, 4, 0, -10, -2, 10], [0, -6, -10, -6, -4, -6, 0, -6], [0, -1, 7, -1, 3, -13, 0, -13], [0, -2, -6, -2, 8, 2, 0, 2], [12, -10, 10, 10, 0, -2, 8, 2], [6, -3, 3, 3, 0, -5, -2, 5], [8, -8, 8, 8, 0, 7, -2, -7], [0, 1, 7, 1, 10, 3, 0, 3], [12, 5, -5, -5, 0, 0, 14, 0], [0, 3, 5, 3, -2, 2, 0, 2], [0, 1, 27, 1, 9, 1, 0, 1], [0, -4, -4, -4, -5, 6, 0, 6], [32, -2, 2, 2, 0, 0, 1, 0], [0, -3, -7, -3, -6, 4, 0, 4], [0, -3, -5, -3, -2, -12, 0, -12], [24, 8, -8, -8, 0, 0, 2, 0], [0, -1, -13, -1, -4, 9, 0, 9], [14, -6, 6, 6, 0, 0, -6, 0], [0, -6, -4, -6, -2, -6, 0, -6], [-10, 1, -1, -1, 0, -2, -10, 2], [18, 9, -9, -9, 0, 7, 9, -7], [12, -4, 4, 4, 0, 10, -4, -10], [0, -8, -14, -8, -1, -6, 0, -6], [0, 4, -4, -4, 0, 4, 2, -4], [-18, -2, 2, 2, 0, -6, -6, 6], [8, -9, 9, 9, 0, -9, 1, 9], [4, 1, -1, -1, 0, -8, -6, 8], [0, -8, -18, -8, 2, -6, 0, -6], [-32, -3, 3, 3, 0, -7, 4, 7], [0, 2, 26, 2, 7, -2, 0, -2], [-16, 0, 0, 0, 0, -10, -1, 10], [0, 12, -10, 12, -4, -12, 0, -12], [0, 3, -25, 3, 2, -4, 0, -4], [0, -1, 1, -1, -2, 8, 0, 8], [6, -14, 14, 14, 0, -4, -1, 4], [0, -3, 5, -3, 6, 3, 0, 3], [0, 0, 12, 0, -4, 4, 0, 4], [-22, 2, -2, -2, 0, -8, 10, 8], [2, -8, 8, 8, 0, 5, 4, -5], [0, -10, 14, -10, 0, 6, 0, 6], [0, -3, -5, -3, -8, -14, 0, -14], [20, -8, 8, 8, 0, 0, 4, 0], [-12, 5, -5, -5, 0, 8, -2, -8], [0, 2, -6, 2, 12, 10, 0, 10], [0, 1, -1, 1, -1, -15, 0, -15], [10, -2, 2, 2, 0, 6, 0, -6], [0, 8, -2, 8, 6, 12, 0, 12], [0, 3, -41, 3, -6, -4, 0, -4], [0, -5, -17, -5, -9, 3, 0, 3], [8, -9, 9, 9, 0, 8, -8, -8], [-16, 4, -4, -4, 0, -8, 4, 8], [0, 4, 10, 4, 6, -8, 0, -8], [-6, 6, -6, -6, 0, -6, -4, 6], [14, 3, -3, -3, 0, 4, 12, -4], [0, -8, -18, -8, 2, -10, 0, -10], [12, -4, 4, 4, 0, 0, -10, 0], [0, 6, -6, 6, -6, 0, 0, 0], [4, -2, 2, 2, 0, 9, 0, -9], [6, 5, -5, -5, 0, -1, -10, 1], [0, -5, -3, -5, -4, 1, 0, 1], [-18, -4, 4, 4, 0, 10, 5, -10], [0, 5, -3, 5, 12, -1, 0, -1], [0, -9, 17, -9, -14, -2, 0, -2], [-10, 12, -12, -12, 0, -1, 0, 1], [-2, 2, -2, -2, 0, -6, 10, 6], [0, 11, 27, 11, -4, 5, 0, 5], [50, 0, 0, 0, 0, 0, 4, 0], [0, 3, -3, 3, 6, 8, 0, 8], [0, 5, 9, 5, 8, -2, 0, -2], [18, -4, 4, 4, 0, -8, -2, 8], [8, -16, 16, 16, 0, -2, -2, 2], [-12, -10, 10, 10, 0, 7, 0, -7], [0, -2, -26, -2, 5, 8, 0, 8], [0, 0, 20, 0, 0, -2, 0, -2], [-10, 4, -4, -4, 0, -8, 14, 8], [0, -5, 25, -5, -11, -7, 0, -7], [-40, 6, -6, -6, 0, 0, 4, 0], [0, -3, -19, -3, 0, 3, 0, 3], [-10, -7, 7, 7, 0, -14, 4, 14], [0, -1, -9, -1, 4, -11, 0, -11], [20, 0, 0, 0, 0, -4, 8, 4], [-16, -14, 14, 14, 0, 0, -4, 0], [46, 4, -4, -4, 0, 4, 2, -4], [0, -9, -19, -9, -6, -9, 0, -9], [0, 1, 19, 1, -6, 2, 0, 2], [10, 6, -6, -6, 0, 16, 0, -16], [0, 1, 13, 1, 16, 0, 0, 0], [-8, -4, 4, 4, 0, 10, -5, -10], [-8, 8, -8, -8, 0, 0, -2, 0], [0, -1, 9, -1, 6, 2, 0, 2], [6, -3, 3, 3, 0, -13, 2, 13], [0, -5, 11, -5, 4, -3, 0, -3], [2, -11, 11, 11, 0, -9, 1, 9], [0, -5, -25, -5, -2, 6, 0, 6], [44, -4, 4, 4, 0, -2, 6, 2], [0, -10, 22, -10, -11, -6, 0, -6], [-34, -5, 5, 5, 0, -8, -2, 8], [-28, 3, -3, -3, 0, 1, -13, -1], [0, 7, -9, 7, -5, -3, 0, -3], [0, 6, -4, 6, 2, -2, 0, -2], [8, 6, -6, -6, 0, -1, -6, 1], [0, 3, 13, 3, 0, -8, 0, -8], [-12, 1, -1, -1, 0, -3, -6, 3], [0, 6, -24, 6, 4, 14, 0, 14], [-2, 3, -3, -3, 0, 12, -6, -12], [-28, -11, 11, 11, 0, 1, -10, -1], [0, 2, 10, 2, -8, -2, 0, -2], [0, 2, -24, 2, 0, 0, 0, 0], [0, 4, -42, 4, 3, 0, 0, 0], [0, 5, -19, 5, 8, -1, 0, -1], [0, 0, 20, 0, -12, 0, 0, 0], [22, 5, -5, -5, 0, 5, -1, -5], [0, -13, 7, -13, -10, 1, 0, 1], [0, 3, -17, 3, 14, 2, 0, 2], [-14, -3, 3, 3, 0, -13, 12, 13], [-8, 13, -13, -13, 0, 8, -4, -8], [24, -14, 14, 14, 0, -1, 2, 1], [0, 6, 30, 6, -4, 0, 0, 0], [0, 0, 34, 0, -12, 0, 0, 0], [40, 7, -7, -7, 0, 3, 4, -3], [0, -2, -10, -2, 10, -4, 0, -4], [0, -2, 12, -2, 15, -4, 0, -4], [0, 0, 0, 0, 0, 0, -4, 0], [0, -3, 15, -3, -10, 3, 0, 3], [0, -2, -30, -2, -3, 6, 0, 6], [0, -1, -23, -1, -8, 2, 0, 2], [-34, 3, -3, -3, 0, 1, -11, -1], [18, 10, -10, -10, 0, 10, 12, -10], [-4, 10, -10, -10, 0, -6, 5, 6], [0, -5, 19, -5, -1, 3, 0, 3], [36, 2, -2, -2, 0, 8, 6, -8], [0, 5, -7, 5, -7, -7, 0, -7], [-26, -1, 1, 1, 0, -3, -3, 3], [0, 0, 28, 0, 10, -2, 0, -2], [-12, -10, 10, 10, 0, 2, -14, -2], [0, 9, -37, 9, 8, 3, 0, 3], [0, -5, -9, -5, -16, -10, 0, -10], [8, -6, 6, 6, 0, 2, -6, -2], [22, 0, 0, 0, 0, 14, 0, -14], [-54, 2, -2, -2, 0, 4, 1, -4], [0, 4, -12, 4, -18, -4, 0, -4], [0, 3, -17, 3, 6, 8, 0, 8], [-6, -1, 1, 1, 0, 6, 10, -6], [0, 13, -3, 13, 14, 3, 0, 3], [-6, -2, 2, 2, 0, -8, -3, 8], [16, 8, -8, -8, 0, 16, -2, -16], [0, -4, 38, -4, -4, 2, 0, 2], [32, -2, 2, 2, 0, 0, -11, 0], [0, 4, 32, 4, 6, -6, 0, -6], [0, 2, 14, 2, 6, 6, 0, 6], [0, 9, 1, 9, 4, -12, 0, -12], [-8, 0, 0, 0, 0, 4, -6, -4], [0, 0, 36, 0, 16, 0, 0, 0], [-4, -2, 2, 2, 0, -10, 0, 10], [0, 10, -8, 10, 1, 12, 0, 12], [0, 2, -26, 2, 2, -2, 0, -2], [-18, 0, 0, 0, 0, 4, -2, -4], [-6, -4, 4, 4, 0, 1, -4, -1], [0, -2, 0, -2, 8, -2, 0, -2], [0, -5, -9, -5, 9, -11, 0, -11], [16, -5, 5, 5, 0, 6, -4, -6], [18, 0, 0, 0, 0, 6, 6, -6], [0, 8, 12, 8, 4, 8, 0, 8], [0, 15, -19, 15, 8, 6, 0, 6], [24, 11, -11, -11, 0, 3, 2, -3], [38, -1, 1, 1, 0, 0, 8, 0], [10, 2, -2, -2, 0, 3, 12, -3], [0, 8, 30, 8, 1, 2, 0, 2], [-54, 6, -6, -6, 0, -10, 10, 10], [-20, -16, 16, 16, 0, -4, -10, 4], [0, 4, 30, 4, -1, -4, 0, -4], [22, -4, 4, 4, 0, -10, 7, 10], [0, 2, -26, 2, -5, 8, 0, 8], [0, 3, 31, 3, 6, -12, 0, -12], [4, -11, 11, 11, 0, 9, -14, -9], [-24, 1, -1, -1, 0, -5, -3, 5], [0, 2, -14, 2, -18, 0, 0, 0], [0, 5, 33, 5, 9, -3, 0, -3], [0, 8, -34, 8, 2, -4, 0, -4], [58, -7, 7, 7, 0, 13, -13, -13], [0, 10, -36, 10, 10, 2, 0, 2], [2, 7, -7, -7, 0, -11, -2, 11], [0, 15, -7, 15, 8, 6, 0, 6], [-40, 5, -5, -5, 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