# q-expansion of newform 63.2.f.a, downloaded from the LMFDB on 02 May 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 = 63
weight = 2
poly_data = [1, 0, 0, -1, 0, 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], 1], [[0, 0, 0, 1, 0, 0], 1], [[0, 1, 0, 0, 0, 1], 1], [[0, 1, 1, 0, -1, 0], 1], [[0, 0, 0, 0, 1, -1], 1], [[0, 1, 0, 0, -1, -1], 1]]

hecke_ring_character_values = [[29, [0, -1, 0, 0, 0, 0]], [10, [1, 0, 0, 0, 0, 0]]]
aps_data = [[-1, 1, 0, 0, 0, 1], [0, 0, 0, -1, 0, 1], [0, -1, 1, -1, -1, 1], [-1, 1, 0, 0, 0, 0], [-2, 2, 1, -1, 0, 1], [0, 1, 4, -2, -2, 2], [2, 0, 0, -1, 0, 0], [-1, 0, 0, -1, -2, 0], [0, -4, -2, -1, -1, 1], [-3, 3, -4, 4, 0, -5], [0, 1, -3, -3, -3, 3], [-1, 0, 0, 6, 3, 0], [0, 0, -1, -1, -1, 1], [1, -1, -1, 1, 0, 1], [-1, 1, 3, -3, 0, -2], [2, 0, 0, -1, -3, 0], [0, 1, 0, 5, 5, -5], [-2, 2, 0, 0, 0, -3], [0, 4, -3, 0, 0, 0], [3, 0, 0, -3, 3, 0], [-7, 0, 0, 5, 4, 0], [7, -7, 3, -3, 0, 0], [6, -6, -1, 1, 0, 4], [4, 0, 0, 4, -3, 0], [1, -1, 8, -8, 0, 7], [7, -7, 1, -1, 0, -2], [0, -2, 0, 3, 3, -3], [1, 0, 0, 4, 0, 0], [5, 0, 0, -3, 0, 0], [0, -12, -4, 2, 2, -2], [-4, 0, 0, -8, -13, 0], [0, 5, 10, -7, -7, 7], [-5, 5, 7, -7, 0, 7], [0, -5, -5, 10, 10, -10], [0, 2, 7, -7, -7, 7], [1, -1, -12, 12, 0, -9], [0, 1, 5, -10, -10, 10], [2, 0, 0, -3, 0, 0], [0, 3, -4, 14, 14, -14], [-13, 13, -3, 3, 0, 7], [-7, 0, 0, -1, 0, 0], [-7, 0, 0, -6, -3, 0], [-1, 1, 9, -9, 0, 1], [0, -5, 0, 3, 3, -3], [15, 0, 0, -3, 3, 0], [2, 0, 0, -3, -3, 0], [0, 1, -2, -2, -2, 2], [-5, 5, -6, 6, 0, -6], [7, -7, -2, 2, 0, 1], [0, -5, 0, 12, 12, -12], [7, 0, 0, 4, 9, 0], [0, -3, 2, 8, 8, -8], [10, -10, 0, 0, 0, 3], [-6, 0, 0, 15, 6, 0], [0, -16, -2, -4, -4, 4], [1, -1, -17, 17, 0, -14], [-7, 0, 0, -7, -9, 0], [8, 0, 0, 0, -3, 0], [-2, 2, 3, -3, 0, -6], [-18, 18, -7, 7, 0, -8], [0, -5, -3, 15, 15, -15], [0, -3, 2, -7, -7, 7], [2, 0, 0, 1, 8, 0], [0, 16, -3, -1, -1, 1], [1, -1, 10, -10, 0, 17], [8, -8, -3, 3, 0, -11], [-14, 14, 1, -1, 0, -4], [0, -5, -12, -3, -3, 3], [0, -5, -15, 8, 8, -8], [-8, 8, 8, -8, 0, 10], [-8, 8, -2, 2, 0, -5], [-2, 0, 0, -11, -6, 0], [-5, 5, -5, 5, 0, 5], [0, -5, 5, -1, -1, 1], [-1, 0, 0, -5, 5, 0], [0, 11, -11, -1, -1, 1], [7, -7, 13, -13, 0, 4], [-19, 0, 0, -6, -3, 0], [0, -4, 7, 11, 11, -11], [0, 4, 18, -12, -12, 12], [0, 12, 8, -13, -13, 13], [13, -13, -15, 15, 0, -6], [5, 0, 0, -19, -15, 0], [17, 0, 0, 4, -10, 0], [-11, 11, -8, 8, 0, 11], [2, -2, -6, 6, 0, 4], [0, 0, 0, 3, 6, 0], [19, -19, -15, 15, 0, -12], [-6, 6, -7, 7, 0, -2], [0, 4, -6, 3, 3, -3], [-23, 0, 0, -8, 3, 0], [25, -25, 7, -7, 0, 4], [-7, 0, 0, 17, 1, 0], [0, 9, -7, -13, -13, 13], [0, -2, 18, -6, -6, 6], [-12, 0, 0, 18, 9, 0], [0, 8, -2, 11, 11, -11], [-13, 0, 0, -4, -6, 0], [11, 0, 0, 3, 15, 0], [11, 0, 0, 8, 10, 0], [-8, 8, 20, -20, 0, 1], [-1, 0, 0, 17, 21, 0], [0, 10, 21, -13, -13, 13], [17, -17, 6, -6, 0, 10], [0, 13, 14, -13, -13, 13], [-16, 0, 0, -9, -9, 0], [-18, 18, -13, 13, 0, 7], [19, 0, 0, 13, 6, 0], [0, -14, -12, 14, 14, -14], [22, -22, -1, 1, 0, -5], [0, -17, 8, -22, -22, 22], [-1, 0, 0, -18, 6, 0], [0, -24, 11, 5, 5, -5], [22, -22, 0, 0, 0, -15], [2, 0, 0, 16, -1, 0], [-14, 14, 10, -10, 0, 13], [0, 19, -2, 13, 13, -13], [-7, 0, 0, -1, 18, 0], [0, 8, -23, 2, 2, -2], [3, -3, 5, -5, 0, 28], [0, -5, -22, 5, 5, -5], [-14, 14, -3, 3, 0, -9], [-1, 1, 24, -24, 0, 1], [32, 0, 0, 2, 0, 0], [4, -4, 15, -15, 0, 9], [6, 0, 0, -6, -12, 0], [-2, 2, 12, -12, 0, 6], [-11, 0, 0, -5, -21, 0], [13, -13, -19, 19, 0, -17], [0, -11, -12, -6, -6, 6], [-31, 0, 0, 3, 3, 0], [0, 6, 23, -10, -10, 10], [0, -11, 11, -16, -16, 16], [14, 0, 0, -5, 8, 0], [0, -10, 16, 2, 2, -2], [0, -20, 12, -9, -9, 9], [-1, 0, 0, -4, 12, 0], [0, 19, -9, 15, 15, -15], [0, -24, -19, 2, 2, -2], [8, 0, 0, 5, 12, 0], [-4, 0, 0, -15, 9, 0], [5, -5, -3, 3, 0, 22], [0, -32, 12, 0, 0, 0], [3, 0, 0, 21, 3, 0], [5, 0, 0, 11, 10, 0], [-9, 9, -10, 10, 0, -8], [-11, 11, -15, 15, 0, 0], [4, -4, -14, 14, 0, -17], [0, 31, 0, 12, 12, -12], [-2, 0, 0, 1, 12, 0], [0, -2, 9, 0, 0, 0], [33, 0, 0, 6, 21, 0], [5, 0, 0, -18, 3, 0], [0, -4, -26, 17, 17, -17], [-17, 17, -6, 6, 0, 18], [-9, 9, -22, 22, 0, -11], [8, 0, 0, 12, 3, 0], [-25, 25, 12, -12, 0, 19], [2, 0, 0, 1, -16, 0], [0, 31, -15, -1, -1, 1], [8, -8, -12, 12, 0, -17], [18, 0, 0, -12, -24, 0], [0, 22, -11, 28, 28, -28], [-26, 0, 0, 1, -18, 0], [0, -17, 24, -16, -16, 16], [-32, 32, 4, -4, 0, 13], [35, 0, 0, 15, 0, 0], [-2, 2, 24, -24, 0, 12]]