
# q-expansion of newform 1260.2.s.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 = 1260
weight = 2
poly_data = [49, 0, 7, 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], 1], [[0, 1, 0, 0], 1], [[0, 0, 1, 0], 7], [[0, 0, 0, 1], 7]]

hecke_ring_character_values = [[631, [1, 0, 0, 0]], [281, [1, 0, 0, 0]], [757, [1, 0, 0, 0]], [1081, [0, 0, 1, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, -1], [1, 1, 1, 0], [0, 0, 0, -1], [1, 1, 1, 0], [0, -2, 3, -2], [0, 1, 1, 1], [-5, 0, 0, -1], [-1, -2, -1, 0], [0, -1, -2, -1], [3, 0, 0, -3], [2, 0, 0, -3], [0, 0, -6, 0], [-2, -2, -2, 0], [3, -3, 3, 0], [0, 0, 8, 0], [2, 1, 2, 0], [-11, 0, 0, -1], [0, 5, 0, 0], [0, -2, -3, -2], [3, 0, 0, 3], [0, -5, 1, -5], [8, 0, 0, 0], [3, 3, 3, 0], [0, 1, 12, 1], [0, -3, 9, -3], [13, 0, 13, 0], [-6, 0, 0, 0], [-10, 0, 0, 3], [0, 2, -16, 2], [-7, 5, -7, 0], [5, 0, 0, 0], [0, -4, -4, -4], [-6, -4, -6, 0], [-4, -2, -4, 0], [0, 0, -16, 0], [-17, 0, 0, -1], [0, 6, 6, 6], [-10, -4, -10, 0], [1, 0, 0, -2], [0, 8, 2, 8], [16, 3, 16, 0], [-3, 0, 0, 3], [2, -8, 2, 0], [-2, 0, 0, 4], [2, 0, 0, 6], [17, -1, 17, 0], [0, 0, -7, 0], [0, 2, -16, 2], [-6, 0, 0, 0], [18, 2, 18, 0], [3, 0, 0, 3], [0, 7, 13, 7], [-2, 10, -2, 0], [12, -6, 12, 0], [0, 4, 18, 4], [18, -1, 18, 0], [-24, 0, 0, 0], [12, -1, 12, 0], [0, 0, 0, 0], [20, 0, 0, 3], [11, -1, 11, 0], [0, 5, -8, 5], [0, 7, -5, 7], [0, 8, 1, 8], [-20, 0, 0, -5], [-8, -8, -8, 0], [-10, 0, 0, 0], [-7, 11, -7, 0], [0, -9, -3, -9], [6, 5, 6, 0], [0, 3, 8, 3], [-21, 0, 0, 2], [0, -2, 22, -2], [-19, -7, -19, 0], [0, -1, -14, -1], [0, -2, -14, -2], [15, 8, 15, 0], [-16, 0, 0, -2], [-9, 0, 0, -4], [24, -6, 24, 0], [6, 0, 0, -1], [0, -2, 0, -2], [0, 6, 6, 6], [2, 0, 0, 4], [0, 1, -12, 1], [17, 0, 0, 1], [-8, 0, 0, 7], [0, -12, 6, -12], [4, 4, 4, 0], [0, -1, 0, 0], [-18, 0, 0, 6], [0, -10, 1, -10], [4, 0, 0, -10], [0, -6, 12, -6], [-18, -6, -18, 0], [0, -5, 18, -5], [0, -12, 5, -12], [-38, 0, 0, -2], [-18, 0, -18, 0], [-26, 4, -26, 0], [0, 5, 17, 5], [-39, 2, -39, 0], [24, -7, 24, 0], [-9, 0, 0, -3], [0, 3, 33, 3], [18, 6, 18, 0], [-27, 0, 0, -4], [0, -5, -30, -5], [20, -8, 20, 0], [-4, 0, 0, -2], [19, 0, 19, 0], [-4, 0, 0, -12], [17, 11, 17, 0], [2, 0, 0, -9], [15, -3, 15, 0], [0, -1, -31, -1], [24, 0, 0, 0], [25, -6, 25, 0], [-8, 0, 0, 1], [0, 1, 25, 1], [-3, -3, -3, 0], [0, -4, 13, -4], [-27, 0, 0, -3], [0, 4, -12, 4], [0, 2, 20, 2], [-10, 0, 0, -9], [0, -5, -6, -5], [-29, 0, -29, 0], [21, 0, 0, -3], [0, -8, 3, -8], [-30, 0, 0, 8], [0, -11, -17, -11], [-9, 0, 0, 2], [-7, -13, -7, 0], [10, 6, 10, 0], [3, 0, 0, -3], [-2, -8, -2, 0], [12, 0, 0, 2], [0, -17, 7, -17], [8, 10, 8, 0], [4, 0, 0, -10], [-25, -2, -25, 0], [-19, 0, 0, 1], [30, 0, 0, 5], [-11, 1, -11, 0], [0, 10, 0, 10], [0, -10, 8, -10], [0, 0, 14, 0], [20, 0, 0, -8], [2, 0, 0, -9], [0, -1, 17, -1], [18, -1, 18, 0], [-13, 0, 0, 7], [0, 4, 21, 4], [0, 7, -17, 7], [-6, 0, 0, 17], [-3, 3, -3, 0], [0, 1, 43, 1], [26, 0, 0, 10], [6, 0, 0, -13], [0, -8, -32, -8], [-33, -3, -33, 0], [-15, -15, -15, 0], [-39, 2, -39, 0], [-30, 5, -30, 0], [-9, 0, 0, 14], [0, -15, 21, -15], [-10, 14, -10, 0], [-6, 0, 0, 14], [0, 13, -5, 13], [12, 11, 12, 0], [-11, 12, -11, 0], [-12, 0, 0, 12], [3, 0, 0, 14], [0, -6, 0, 0], [-20, 0, 0, -8], [0, -12, -13, -12], [0, -1, 22, -1], [9, 0, 0, -3], [12, 0, 0, -1], [0, 10, -14, 10], [18, -12, 18, 0], [42, 6, 42, 0], [16, -18, 16, 0], [-10, -5, -10, 0], [0, 12, -25, 12], [15, 3, 15, 0], [0, 3, 14, 3], [29, 0, 0, 1], [0, 8, 16, 8], [0, 0, -60, 0], [-13, 11, -13, 0], [-39, -3, -39, 0], [1, -24, 1, 0], [0, -12, 32, -12], [40, 0, 0, 2], [0, -9, 15, -9], [0, 18, 0, 0], [-21, 0, 0, 2], [0, 6, -16, 6], [0, -16, 0, 0], [3, 0, 0, -15], [16, 10, 16, 0], [0, 4, 36, 4], [0, 0, 6, 0], [-26, 0, 0, 2], [4, -18, 4, 0], [0, 3, -28, 3], [-51, 0, 0, -3], [54, 0, 0, 2], [0, 4, -44, 4], [7, 1, 7, 0], [0, 4, -27, 4], [0, 8, 0, 0], [24, 0, 24, 0], [0, -10, 2, -10], [6, 0, 0, 12], [0, -6, 29, -6], [-5, 0, 0, -2], [0, 15, 27, 15], [0, 16, -24, 16], [-16, 0, 0, -14], [-8, 0, 0, 4], [0, -12, 0, -12], [-12, -6, -12, 0], [0, -11, 18, -11], [0, -1, 35, -1], [-18, -13, -18, 0], [12, 8, 12, 0], [27, 0, 0, 8], [35, -13, 35, 0], [-14, 0, 0, 1], [7, 1, 7, 0], [0, 14, -5, 14], [0, -9, -39, -9], [-20, 0, 0, -4], [-1, 0, 0, 13], [-27, -9, -27, 0], [0, 10, 3, 10], [-34, 7, -34, 0], [0, -18, 2, -18], [-54, 0, 0, -6], [0, -18, -18, -18], [-44, 0, 0, 1], [5, -19, 5, 0], [-44, 6, -44, 0], [20, 0, 0, -20], [24, 0, 0, -1], [0, 8, -52, 8], [22, 4, 22, 0], [-53, 0, 0, -8], [14, -10, 14, 0], [0, -5, -23, -5], [3, 26, 3, 0], [-54, -7, -54, 0], [-25, 0, 0, 19], [0, 12, 14, 12], [30, -13, 30, 0], [-6, 0, 0, -18], [-53, 0, -53, 0], [50, 0, 0, 3], [-24, -18, -24, 0], [0, 4, -21, 4], [13, 0, 0, 11], [6, 0, 0, -12], [56, 0, 0, -6], [33, 3, 33, 0], [0, -20, -15, -20], [-24, -10, -24, 0], [28, 9, 28, 0], [0, 6, 35, 6], [12, 0, 0, -22], [0, -2, 30, -2], [0, -5, -11, -5], [25, 12, 25, 0], [0, -2, 27, -2], [0, -20, -8, -20], [-10, 14, -10, 0], [49, 0, 49, 0], [-6, 0, 0, 12], [37, 0, 0, -8], [0, -6, 26, -6], [0, 1, 37, 1], [-2, -9, -2, 0], [-59, 0, 0, -7], [-17, 12, -17, 0], [11, 0, 0, 1], [-10, -10, -10, 0], [26, 2, 26, 0], [0, -9, -39, -9], [-29, 0, 0, -1], [16, 0, 0, 7], [-6, 24, -6, 0], [0, -16, -35, -16], [1, 0, 0, -1], [0, 7, 43, 7], [14, 0, 0, -9], [0, -20, -30, -20], [0, 22, -8, 22], [-4, -14, -4, 0]]
