
# q-expansion of newform 625.2.d.m, downloaded from the LMFDB on 24 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 = 625
weight = 2
poly_data = [1, 0, 1205, 0, 3809, 0, 4820, 0, 3166, 0, 1165, 0, 239, 0, 25, 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, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-1474, 0, -12623, 0, -4230, 0, 15463, 0, 13949, 0, 4515, 0, 619, 0, 29, 0], 801], [[1601, 0, -25961, 0, -63150, 0, -52142, 0, -20887, 0, -4422, 0, -470, 0, -19, 0], 801], [[673, 70883, 12623, 237289, 4230, 297480, -15463, 183139, -13949, 59873, -4515, 10389, -619, 886, -29, 29], 1602], [[431, 10906, -1224, 29676, -4869, 39643, -4282, 32176, -1388, 15210, -142, 3907, 6, 488, 1, 22], 534], [[431, -10906, -1224, -29676, -4869, -39643, -4282, -32176, -1388, -15210, -142, -3907, 6, -488, 1, -22], 534], [[-1556, 159574, -77159, 474718, -150381, 534120, -110467, 297029, -38668, 88283, -6717, 13968, -541, 1082, -16, 32], 1602], [[1556, 159574, 77159, 474718, 150381, 534120, 110467, 297029, 38668, 88283, 6717, 13968, 541, 1082, 16, 32], 1602], [[3891, 98578, -17926, 240239, -90759, 227865, -120384, 100622, -70055, 18841, -19770, 240, -2583, -316, -119, -23], 1602], [[164, 17682, -1224, 91029, -4869, 161716, -4282, 136696, -1388, 60887, -142, 14429, 6, 1678, 1, 72], 534], [[-950, 89311, -138110, 221134, -371667, 217803, -377263, 116507, -185932, 38195, -46914, 7668, -5683, 833, -250, 35], 1602], [[-21, 11052, -149932, 11761, -375897, -19152, -361800, -41577, -171983, -27682, -42399, -8280, -5064, -1113, -221, -52], 1602], [[-2335, 0, 95085, 0, 241140, 0, 230851, 0, 108723, 0, 26487, 0, 3124, 0, 135, 0], 801], [[950, 89311, 138110, 221134, 371667, 217803, 377263, 116507, 185932, 38195, 46914, 7668, 5683, 833, 250, 35], 1602], [[1634, -43770, -121014, -100789, -352830, -99777, -379880, -54951, -195717, -17948, -51003, -3441, -6320, -351, -282, -14], 1602], [[2391, -55321, 245786, -285710, 665577, -489378, 689538, -394625, 346813, -168712, 88980, -38772, 10914, -4415, 484, -187], 1602]]

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