# q-expansion of newform 625.2.d.g, downloaded from the LMFDB on 21 June 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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, -1, 1, -1, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[2, [0, 0, 0, -1]]]
aps_data = [[0, 1, -1, 0], [1, -1, 0, -2], [0, 0, 0, 0], [3, 0, 0, 0], [3, -3, 3, -3], [3, -3, 3, 0], [0, 2, -3, 2], [0, 1, -3, 1], [2, 0, 0, -2], [-6, 6, 0, 3], [0, -5, 2, -5], [-6, 0, -6, 0], [0, 3, 0, 0], [9, 0, 0, 0], [-7, 7, 0, 3], [1, -1, 0, 3], [-3, -6, -3, 0], [-2, -3, 3, 2], [0, -3, 12, -3], [0, 0, 0, 3], [-3, 0, 0, 3], [4, -4, 0, -7], [0, 4, 2, 4], [-6, -6, 6, 6], [3, -3, 0, 3], [-3, 0, 0, 0], [6, -6, 0, 3], [8, 0, 5, -5], [7, -1, 7, 0], [8, 7, 8, 0], [0, 6, -6, 0], [0, 15, -3, 15], [-1, 5, -1, 0], [9, -11, 11, -9], [-15, 0, 0, 0], [-13, 0, 5, -5], [-12, 0, -15, 15], [3, -18, 3, 0], [0, 2, -8, 2], [-13, 5, -5, 13], [-6, 6, 0, 3], [0, -10, 12, -10], [0, -12, 0, 0], [9, 0, 15, -15], [8, -8, 0, -17], [0, 0, -10, 10], [-12, 2, -2, 12], [-3, 0, 0, 3], [-5, 6, -6, 5], [-1, 1, 0, 3], [0, -16, 17, -16], [-21, 24, -24, 21], [-10, 13, -10, 0], [-18, 0, -15, 15], [3, 0, 20, -20], [-7, 12, -7, 0], [0, -9, -3, -9], [-5, 5, 0, -12], [15, -9, 9, -15], [0, 0, 12, 0], [0, -6, 12, -6], [-11, 0, -20, 20], [18, 0, 15, -15], [3, 12, -12, -3], [-12, 12, -12, 0], [0, -8, 22, -8], [0, 5, -18, 5], [24, -15, 24, 0], [8, -8, 0, 3], [15, 0, 20, -20], [16, -16, 0, -22], [-3, 9, -3, 0], [0, -3, -3, -3], [-3, 0, 0, 3], [9, -9, 0, -7], [0, 4, -33, 4], [-21, 9, -9, 21], [-12, 12, 0, 3], [12, 0, 0, 0], [2, 24, 2, 0], [0, 6, -3, 6], [5, -5, 0, -17], [0, 15, -3, 15], [0, -6, -18, -6], [-1, 19, -19, 1], [9, 0, 10, -10], [-15, 0, 15, -15], [18, 0, 0, 0], [-12, 27, -27, 12], [18, -3, 18, 0], [0, 2, -38, 2], [9, -9, 0, 18], [-6, 0, -6, 0], [15, -27, 15, 0], [10, 0, -20, 20], [11, -11, 0, 8], [12, -6, 12, 0], [-15, 15, 0, 18], [-18, 30, -30, 18], [-10, 28, -10, 0], [3, -3, 0, -27], [8, 0, -20, 20], [8, 12, 8, 0], [0, -24, 12, -24], [0, 0, 0, 13], [15, -24, 24, -15], [4, 0, 4, 0], [9, 0, -20, 20], [0, 0, 15, -15], [-33, 0, -20, 20], [18, 0, 30, -30], [-12, -3, -12, 0], [0, 2, 27, 2], [0, -4, -23, -4], [0, 0, 2, 0], [15, 18, 15, 0], [-6, 0, -15, 15], [-2, 2, 0, -17], [-14, 14, 0, 18], [-3, 24, -3, 0], [18, 2, -2, -18], [12, 15, -15, -12], [30, -14, 14, -30], [0, -26, 27, -26], [-10, -32, -10, 0], [27, 0, 15, -15], [-13, 9, -13, 0], [0, -9, 42, -9], [0, -24, 24, 0], [0, 9, -18, 9], [-21, 19, -19, 21], [34, 0, 5, -5], [17, 0, 20, -20], [-27, 0, 0, 0], [18, -18, 18, -18], [0, 16, -3, 16], [-18, 20, -20, 18], [-6, -15, -6, 0], [23, -23, 0, -2], [-18, -6, -18, 0], [28, -18, 18, -28], [-15, 15, 0, 3], [-3, 15, -15, 3], [-30, 31, -31, 30], [19, -19, 0, -37], [24, 9, -9, -24], [-24, 24, 0, 33], [18, 0, -10, 10], [-18, 14, -18, 0], [-2, -8, -2, 0], [15, 6, -6, -15], [0, -30, 27, -30], [0, 24, -3, 24], [4, -40, 4, 0], [18, 0, -15, 15], [-12, -18, 18, 12], [0, 26, -8, 26], [-6, 6, 0, -27], [9, 15, 9, 0], [-15, 18, -15, 0], [-22, 22, 0, 13], [31, -31, 0, -37], [0, -3, -18, -3], [30, -30, 0, 3], [25, -24, 24, -25], [0, 19, 17, 19], [10, 33, 10, 0], [3, -3, 0, 48]]