
# q-expansion of newform 507.2.f.f, downloaded from the LMFDB on 10 March 2026.

# 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 = 507
weight = 2
poly_data = [13, -38, 70, -74, 63, -34, 16, -4, 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], [[2, -1, 1, 0, 0, 0, 0, 0], 1], [[-55, 28, -132, -5, -46, -25, -1, -5], 37], [[-29, 41, 71, -117, 115, -67, 21, -6], 37], [[-126, 344, -295, 261, -100, 47, -7, 2], 37], [[278, -601, 793, -634, 361, -136, 36, -5], 37], [[61, -395, 546, -684, 419, -238, 63, -18], 37], [[283, -617, 805, -758, 456, -238, 63, -18], 37]]

hecke_ring_character_values = [[170, [-1, 0, 0, 0, 0, 0, 0, 0]], [340, [0, 0, 0, -1, 0, 0, 0, 0]]]
aps_data = [[0, 1, 1, 0, 1, 0, 0, 0], [1, 0, -1, 0, -1, 0, 0, 0], [0, 1, 1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0], [-1, -2, -1, 0, 0, 1, 1, -1], [0, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 2, -1, 4, 2, -1, -1], [3, 0, 0, -3, 0, -4, 0, 0], [2, 0, -5, -3, 0, 0, 0, 0], [-1, -1, -1, 1, -1, 0, 2, 0], [3, 0, -3, -6, 0, -3, 0, 0], [-4, -4, 0, 0, 4, 4, 0, 0], [2, 0, -1, -1, -2, -1, -1, -1], [2, 2, 0, 0, -2, -2, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, -4, 0, -6, 0, 0], [0, -2, -2, 0, -2, 0, 0, 0], [-2, 0, -3, -5, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, -2, 0], [5, 2, 0, -3, -2, -2, 0, -6], [7, 0, 0, -7, 0, -6, 0, 0], [-4, -8, -4, 0, 0, 4, -5, 5], [4, 0, -4, -4, 0, -4, 0, 0], [-10, 0, 2, 8, 4, 2, 8, 8], [-5, 0, 0, 5, 0, -6, 0, 0], [-7, 0, 2, 5, 4, 2, 5, 5], [-7, 0, 7, 10, 0, 7, 0, 0], [-1, 0, -1, 2, -2, -1, 2, 2], [3, 1, 0, -2, -1, -1, 0, -4], [8, 0, -6, 0, 0, 6, 0, 0], [1, 3, 3, -1, 3, 0, -2, 0], [2, 0, 4, 6, 0, 0, 0, 0], [-10, 0, -3, 0, 0, 3, 0, 0], [0, 0, 8, 8, 0, 0, 0, 0], [-6, -2, 0, 4, 2, 2, 0, 8], [0, 0, 0, 0, 0, 0, -6, 6], [7, 14, 7, 0, 0, -7, -4, 4], [0, 0, 0, 3, 0, 0, 0, 0], [-7, 0, 5, 2, 10, 5, 2, 2], [1, 0, 3, 4, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, -2, 0], [4, 0, -4, 2, 0, -4, 0, 0], [-7, 0, -3, 0, 0, 3, 0, 0], [-9, 0, 0, 9, 0, 20, 0, 0], [2, 8, 8, -2, 8, 0, -4, 0], [-9, 0, 14, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 6], [3, -2, -2, -3, -2, 0, -6, 0], [7, 0, -9, -2, 0, 0, 0, 0], [5, 10, 5, 0, 0, -5, 4, -4], [-6, -12, -6, 0, 0, 6, 3, -3], [-1, 0, 5, -4, 10, 5, -4, -4], [2, 0, -4, 2, -8, -4, 2, 2], [-4, 0, 4, 0, 0, 0, 0, 0], [9, 0, -9, 3, 0, -9, 0, 0], [11, 5, 0, -6, -5, -5, 0, -12], [9, 0, -9, -18, 0, -9, 0, 0], [-10, -7, 0, 3, 7, 7, 0, 6], [4, 0, -12, -8, 0, 0, 0, 0], [2, 4, 2, 0, 0, -2, -2, 2], [2, 0, 0, 0, 0, 0, 0, 0], [-5, 3, 3, 5, 3, 0, 10, 0], [16, 0, 0, -16, 0, 6, 0, 0], [-2, 0, 2, -13, 0, 2, 0, 0], [14, 0, -4, -10, -8, -4, -10, -10], [-15, 0, 14, -1, 0, 0, 0, 0], [-6, 7, 7, 6, 7, 0, 12, 0], [-9, -2, 0, 7, 2, 2, 0, 14], [20, 0, -6, 0, 0, 6, 0, 0], [-4, 0, -9, 0, 0, 9, 0, 0], [5, 0, 0, -5, 0, 4, 0, 0], [-8, 0, 0, 8, 0, 0, 16, 0], [1, 2, 1, 0, 0, -1, -7, 7], [-9, 0, 2, -7, 0, 0, 0, 0], [-12, -11, 0, 1, 11, 11, 0, 2], [2, 0, 0, -2, 0, -17, 0, 0], [5, 0, -1, -4, -2, -1, -4, -4], [-1, 0, 0, 1, 0, -5, 0, 0], [7, 6, 6, -7, 6, 0, -14, 0], [-11, 0, 11, -1, 0, 11, 0, 0], [1, 0, -1, -4, 0, -1, 0, 0], [-10, 0, 8, 2, 16, 8, 2, 2], [-5, 0, 0, 5, 0, 0, 0, 10], [15, 0, 0, -15, 0, -13, 0, 0], [-7, 5, 5, 7, 5, 0, 14, 0], [15, 0, -22, -7, 0, 0, 0, 0], [-9, -18, -9, 0, 0, 9, 0, 0], [-5, 0, 0, 5, 0, 0, 0, 10], [-15, 0, 0, 15, 0, 8, 0, 0], [3, 6, 3, 0, 0, -3, 12, -12], [3, 0, 0, -3, 0, -4, 0, 0], [11, 0, -7, -4, -14, -7, -4, -4], [-1, -7, -7, 1, -7, 0, 2, 0], [-13, 0, 8, 5, 16, 8, 5, 5], [-13, 0, 15, 0, 0, -15, 0, 0], [-5, 0, 21, 16, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0], [17, 17, 0, 0, -17, -17, 0, 0], [-2, -4, -2, 0, 0, 2, 2, -2], [-4, -8, -4, 0, 0, 4, -2, 2], [-11, 0, 11, 32, 0, 11, 0, 0], [-2, 0, 0, 2, 0, -15, 0, 0], [-6, 4, 4, 6, 4, 0, 12, 0], [6, 7, 0, 1, -7, -7, 0, 2], [5, 0, -7, 2, -14, -7, 2, 2], [8, 0, -9, 0, 0, 9, 0, 0], [5, 0, 3, 0, 0, -3, 0, 0], [-16, 0, 0, 16, 0, -11, 0, 0], [1, -15, -15, -1, -15, 0, -2, 0], [-28, 0, 10, -18, 0, 0, 0, 0], [12, 0, -10, 2, 0, 0, 0, 0], [9, 18, 9, 0, 0, -9, -9, 9], [14, 0, 0, -14, 0, -14, 0, 0], [-5, -10, -5, 0, 0, 5, 2, -2], [14, 0, -10, -4, -20, -10, -4, -4], [8, 0, -10, 2, -20, -10, 2, 2], [-8, 0, -3, -11, 0, 0, 0, 0], [16, 0, -16, -1, 0, -16, 0, 0], [14, 0, -10, -4, -20, -10, -4, -4], [-5, -12, 0, -7, 12, 12, 0, -14], [27, 0, 0, -27, 0, -10, 0, 0], [6, 12, 6, 0, 0, -6, 0, 0], [-5, 0, 9, 4, 0, 0, 0, 0], [2, 4, 2, 0, 0, -2, 4, -4], [-13, 0, 13, 16, 0, 13, 0, 0], [-30, 0, 0, 30, 0, 17, 0, 0], [14, 0, 16, 30, 0, 0, 0, 0], [8, -16, -16, -8, -16, 0, -16, 0], [-3, 0, 3, 42, 0, 3, 0, 0], [-4, 0, -12, 0, 0, 12, 0, 0], [-1, -4, 0, -3, 4, 4, 0, -6], [-5, 0, 0, 5, 0, -18, 0, 0], [9, 4, 4, -9, 4, 0, -18, 0], [11, 0, -2, 9, 0, 0, 0, 0], [-8, -16, -8, 0, 0, 8, 8, -8], [-13, 0, 2, 11, 4, 2, 11, 11], [19, 0, 0, -19, 0, 0, 0, 0], [9, 4, 4, -9, 4, 0, -18, 0], [2, 0, -2, -14, 0, -2, 0, 0], [0, -8, 0, -8, 8, 8, 0, -16], [-7, 0, 7, -29, 0, 7, 0, 0], [3, 4, 0, 1, -4, -4, 0, 2], [-22, 0, 1, -21, 0, 0, 0, 0], [-8, -16, -8, 0, 0, 8, -7, 7], [8, 0, -18, 0, 0, 18, 0, 0], [-7, 2, 2, 7, 2, 0, 14, 0], [2, 0, 0, -2, 0, -17, 0, 0], [4, 8, 4, 0, 0, -4, 11, -11], [21, 0, -21, -24, 0, -21, 0, 0], [26, 0, -10, -16, -20, -10, -16, -16], [10, 0, -10, -10, 0, -10, 0, 0], [-4, 0, 2, 2, 4, 2, 2, 2], [29, 0, 9, 0, 0, -9, 0, 0], [2, -19, -19, -2, -19, 0, -4, 0], [-37, 0, 0, 0, 0, 0, 0, 0], [-9, 16, 16, 9, 16, 0, 18, 0], [19, 12, 0, -7, -12, -12, 0, -14], [-10, -20, -10, 0, 0, 10, -8, 8], [-6, 0, 0, 6, 0, -16, 0, 0], [-13, 0, 17, -4, 34, 17, -4, -4], [-1, -13, -13, 1, -13, 0, 2, 0], [5, -4, 0, -9, 4, 4, 0, -18], [-16, 0, 24, 0, 0, -24, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0]]
