
# q-expansion of newform 770.2.n.f, downloaded from the LMFDB on 20 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 = 770
weight = 2
poly_data = [121, 0, 66, 0, 16, 0, 1, 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], [[0, 1, 0, 0, 0, 0, 0, 0], 1], [[-363, 0, 629, 0, -37, 0, 7, 0], 1991], [[-539, 0, -525, 0, 148, 0, -28, 0], 1991], [[0, -539, 0, -525, 0, 148, 0, -28], 1991], [[2761, 0, 750, 0, 73, 0, 40, 0], 1991], [[0, -1672, 0, -646, 0, 38, 0, -61], 1991], [[0, 3300, 0, 1275, 0, -75, 0, 68], 1991]]

hecke_ring_character_values = [[617, [1, 0, 0, 0, 0, 0, 0, 0]], [661, [1, 0, 0, 0, 0, 0, 0, 0]], [211, [-1, 0, -1, -1, 0, 1, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, -1, 0, 0, -1], [0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 1, 0, -1, 0, 0], [0, 1, 0, 0, 2, 0, -1, 0], [0, 2, -1, -1, 1, -3, -1, -1], [0, 0, 3, 0, 0, 0, -1, -1], [1, 0, 0, 0, -2, -1, 0, -2], [0, -2, 0, 0, 0, 0, 0, 2], [0, 0, 0, -5, -1, 5, 1, 0], [0, 4, 3, 3, 3, 3, -1, -1], [-2, 0, -2, -7, -1, 7, 1, 0], [0, 1, 0, 6, -1, 0, -1, -1], [3, 3, 1, 0, -2, -1, 0, -3], [-4, -2, 0, -4, 2, 4, 2, 2], [0, 1, 4, 4, -1, -7, -2, -2], [1, -2, 1, 6, -2, -6, 2, 0], [3, 0, 5, 3, 0, 0, -1, -2], [1, -1, 2, 0, 1, -2, 0, 1], [3, 0, -2, 3, 0, 0, -1, -1], [-11, -5, -11, -4, -3, 4, 3, 0], [0, -1, 6, 6, -1, -5, 0, 0], [-6, 0, -1, -6, 0, 0, 3, 0], [-4, -3, -6, 0, -1, 6, 0, 3], [0, -5, 1, 1, -3, 2, 2, 2], [0, -1, 2, 2, -1, 1, 0, 0], [7, -3, 7, -1, -3, 1, 3, 0], [11, -1, 0, 6, 2, -11, 1, 2], [-5, -1, 0, 0, 4, 0, 0, 1], [-3, -5, 0, -1, 4, 3, 5, 4], [4, 0, 5, 4, 0, 0, 6, -3], [7, -2, 9, 0, 2, -9, 0, 2], [2, 0, 7, 2, 0, 0, -5, 0], [0, 4, 0, 8, 2, -8, -2, 0], [4, 0, 6, 4, 0, 0, -2, 2], [7, -2, 0, 0, -1, -7, 2, -1], [-3, -4, 0, 0, -1, 3, 4, -1], [0, -4, 10, 10, -1, -3, 3, 3], [0, 5, 3, 3, 8, -4, 3, 3], [-2, 1, 0, -1, -4, 2, -1, -4], [-6, -6, 0, -10, 7, 6, 6, 7], [2, 0, 3, 2, 0, 0, -4, -7], [-13, 9, -13, -10, 4, 10, -4, 0], [-4, 0, -2, -4, 0, 0, -4, 6], [12, 2, 4, 0, 0, -4, 0, -2], [11, 3, 2, 0, 2, -2, 0, -3], [0, 6, -12, -12, 1, 6, -5, -5], [0, 3, 0, 1, -6, 0, -3, -6], [4, -1, 4, -1, -1, 1, 1, 0], [0, -9, -3, -3, -10, -2, -1, -1], [0, -3, 0, 0, 2, -1, 5, 5], [7, -5, 0, 9, 7, -7, 5, 7], [-6, 6, -8, 0, 1, 8, 0, -6], [0, -8, 6, 6, -4, -4, 4, 4], [7, 7, 7, 8, 1, -8, -1, 0], [22, 1, 3, 0, -1, -3, 0, -1], [2, 0, -13, 2, 0, 0, -2, 3], [-16, 12, -16, -9, 9, 9, -9, 0], [0, 3, 3, 3, 8, -8, 5, 5], [-7, 0, -12, -7, 0, 0, 2, 6], [10, -4, 0, 12, -2, -10, 4, -2], [-13, -3, -13, -11, -3, 11, 3, 0], [-1, -2, -10, 0, -3, 10, 0, 2], [-6, 6, 0, -14, -8, 6, -6, -8], [-6, 0, -3, -6, 0, 0, -9, 3], [0, -4, -7, -7, -5, -1, -1, -1], [-8, -1, -4, 0, 9, 4, 0, 1], [6, -11, 6, 3, -5, -3, 5, 0], [8, 0, 7, 8, 0, 0, 3, 1], [-6, -6, 0, 2, -2, 6, 6, -2], [-1, -5, 6, 0, 3, -6, 0, 5], [-14, -4, -14, -12, 0, 12, 0, 0], [2, 10, 2, 3, 1, -3, -1, 0], [-5, 2, 11, 0, -3, -11, 0, -2], [-2, 0, -2, -2, 0, 0, 9, -7], [0, -8, 9, 9, 1, -11, 9, 9], [3, 3, 3, 0, -2, 0, 2, 0], [5, -3, 20, 0, -2, -20, 0, 3], [3, 0, 12, 3, 0, 0, 4, 6], [0, 0, -20, -20, 2, 10, 2, 2], [14, -10, 11, 0, 2, -11, 0, 10], [0, -6, 0, -12, 8, 0, 6, 8], [0, -2, 10, 10, -6, -2, -4, -4], [1, 4, 1, 3, 5, -3, -5, 0], [3, 5, -6, 0, -8, 6, 0, -5], [-5, 3, 0, 2, -6, 5, -3, -6], [0, 8, 13, 13, -2, -10, -10, -10], [6, 0, -3, 6, 0, 0, -3, -10], [11, 5, 2, 0, 0, -2, 0, -5], [-3, 13, -8, 0, -2, 8, 0, -13], [10, 0, 0, 10, 0, 0, -8, -6], [1, 0, 0, 1, 0, 0, -1, -3], [-7, -12, 0, -10, 5, 7, 12, 5], [4, 6, 4, 16, -1, -16, 1, 0], [4, -12, 4, -8, -3, 8, 3, 0], [-10, 2, 0, -2, 8, 10, -2, 8], [1, 3, 0, -7, 9, -1, -3, 9], [-10, 7, -10, -6, 10, 6, -10, 0], [-11, 0, -16, -11, 0, 0, -2, -1], [0, -4, 5, 5, -9, -7, -5, -5], [14, -9, 0, 19, 11, -14, 9, 11], [-21, 17, -21, -5, 7, 5, -7, 0], [0, -3, -20, -20, -2, -9, 1, 1], [3, -4, 0, 10, 2, -3, 4, 2], [-6, -12, 0, 0, 10, 0, 0, 12], [-4, 0, -3, -4, 0, 0, -7, 1], [-8, 1, -8, -17, 1, 17, -1, 0], [37, 5, 8, 0, 5, -8, 0, -5], [-16, 0, -21, -16, 0, 0, 0, -1], [-2, -14, -2, 2, -9, -2, 9, 0], [0, -6, -3, -3, -3, -17, 3, 3], [-18, -3, 0, 1, -2, 18, 3, -2], [18, -1, -5, 0, 8, 5, 0, 1], [-23, -2, 0, -26, 6, 23, 2, 6], [24, 16, 24, 16, 6, -16, -6, 0], [1, 2, 0, 13, -10, -1, -2, -10], [10, 0, 19, 10, 0, 0, -5, -10], [0, -7, -12, -12, -1, 19, 6, 6], [3, -9, 3, -20, 4, 20, -4, 0], [22, -8, 12, 0, -5, -12, 0, 8], [13, -8, 15, 0, 1, -15, 0, 8], [0, 15, -17, -17, 13, 3, -2, -2], [-23, 0, -6, -23, 0, 0, -9, -7], [24, -2, 2, 0, -6, -2, 0, 2], [0, -14, -3, -3, 2, -2, 16, 16], [7, -4, 0, -16, 5, -7, 4, 5], [-2, 0, -17, -2, 0, 0, -6, -9], [7, -3, 7, 18, 6, -18, -6, 0], [0, 6, 26, 0, -10, -26, 0, -6], [26, -2, 26, 12, 2, -12, -2, 0], [0, -3, 2, 2, -10, -16, -7, -7], [0, 0, 24, 0, 0, 0, 0, -6], [-1, 5, 0, 17, -3, 1, -5, -3], [0, -10, 1, 1, -13, -3, -3, -3], [0, -13, 6, 6, -6, -20, 7, 7], [-12, -2, -2, 0, -12, 2, 0, 2], [20, -1, 0, 15, 16, -20, 1, 16], [3, 0, -18, 3, 0, 0, 6, 7], [-19, 0, -7, -19, 0, 0, 3, 4], [-13, 0, -15, -13, 0, 0, 10, -8], [5, 4, 0, 7, -18, -5, -4, -18], [-27, -7, -27, -27, -11, 27, 11, 0], [0, 10, 19, 19, 9, -1, -1, -1], [0, 8, 12, 12, -3, -13, -11, -11], [8, -4, 8, -3, -3, 3, 3, 0], [-11, -1, 0, -3, 13, 11, 1, 13], [-23, 0, -7, -23, 0, 0, 9, 8], [-17, 13, -13, 0, -4, 13, 0, -13], [-9, -4, 7, 0, -10, -7, 0, 4], [13, 0, 26, 13, 0, 0, -5, -7], [-3, -11, 0, -7, -9, 3, 11, -9], [-4, -17, 8, 0, 9, -8, 0, 17], [-5, -1, 0, -4, 8, 5, 1, 8], [7, 7, 7, 14, 6, -14, -6, 0], [-12, 0, -13, -12, 0, 0, 7, 9], [0, -9, 1, 1, -2, -4, 7, 7], [4, 0, 15, 4, 0, 0, -10, 7], [-2, 0, -16, -2, 0, 0, 5, -10], [0, -12, -18, -18, 7, 25, 19, 19], [15, 0, -12, 15, 0, 0, -13, -5], [5, 21, -4, 0, -9, 4, 0, -21], [5, 17, 5, -7, 20, 7, -20, 0], [-23, -7, -20, 0, -4, 20, 0, 7], [6, 0, 0, 14, 0, -6, 0, 0], [0, 14, -2, -2, -2, 4, -16, -16], [7, -11, 7, 22, -2, -22, 2, 0], [-19, 8, 5, 0, 1, -5, 0, -8], [-7, 5, -7, 12, 4, -12, -4, 0]]
