
# q-expansion of newform 980.2.e.f, downloaded from the LMFDB on 18 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 = 980
weight = 2
poly_data = [25, -5, -4, -1, 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], [[-25, 16, 4, 1], 20], [[15, 4, -4, -1], 10], [[5, 4, 0, -1], 4]]

hecke_ring_character_values = [[491, [1, 0, 0, 0]], [197, [-1, 0, 0, 0]], [101, [1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1], [0, 0, 0, 0], [-2, -1, 0, -1], [0, 2, 2, -2], [0, 1, 2, -1], [0, -1, 0, -1], [0, -2, 1, 2], [-1, -1, 0, -1], [0, 1, 0, 1], [0, 1, -4, -1], [-7, 1, 0, 1], [0, 1, 3, -1], [0, -1, 2, 1], [0, -3, -2, 3], [0, 1, 0, 1], [-7, -2, 0, -2], [0, -4, -5, 4], [2, -2, 0, -2], [0, 1, -2, -1], [-4, -1, 0, -1], [0, -3, -1, 3], [7, 0, 0, 0], [0, 0, -4, 0], [-7, -2, 0, -2], [0, 2, -3, -2], [0, -4, -5, 4], [5, 2, 0, 2], [0, -2, -6, 2], [0, 4, -2, -4], [-14, 1, 0, 1], [0, 7, 0, -7], [0, -4, 0, -4], [1, 2, 0, 2], [-16, 1, 0, 1], [0, 3, 4, -3], [0, 3, 2, -3], [0, 3, 5, -3], [0, 9, 4, -9], [4, 1, 0, 1], [-21, -1, 0, -1], [10, -3, 0, -3], [0, 7, 0, -7], [0, 4, 12, -4], [-14, -1, 0, -1], [6, 6, 0, 6], [0, -4, -2, 4], [0, 3, -6, -3], [0, -1, 0, -1], [0, -1, -10, 1], [6, -2, 0, -2], [0, 3, 0, 3], [-14, -2, 0, -2], [0, -3, -12, 3], [0, -6, -11, 6], [21, -2, 0, -2], [0, 3, 0, 3], [0, 1, 10, -1], [6, 0, 0, 0], [0, 7, 6, -7], [0, 0, -4, 0], [0, -7, 3, 7], [0, -3, 0, -3], [0, -11, 0, 11], [0, -5, 6, 5], [8, 3, 0, 3], [0, -2, -6, 2], [0, 0, 7, 0], [7, -1, 0, -1], [0, -3, -10, 3], [-20, -5, 0, -5], [0, -2, -7, 2], [0, 1, -4, -1], [2, 6, 0, 6], [0, -6, -7, 6], [-16, -5, 0, -5], [0, 1, 8, -1], [3, 0, 0, 0], [7, 4, 0, 4], [0, 4, 0, 4], [-1, -1, 0, -1], [-2, -5, 0, -5], [0, 2, 14, -2], [-14, -3, 0, -3], [0, 0, 7, 0], [13, -3, 0, -3], [0, 5, 8, -5], [-14, 0, 0, 0], [0, -1, -3, 1], [0, -8, -5, 8], [0, -3, 0, -3], [0, 7, 14, -7], [-22, -2, 0, -2], [-18, 5, 0, 5], [0, 11, 1, -11], [21, -2, 0, -2], [0, 3, 0, 3], [0, -1, -6, 1], [-11, -2, 0, -2], [0, 13, 11, -13], [0, -1, 4, 1], [0, -8, -1, 8], [8, -5, 0, -5], [4, 1, 0, 1], [0, -1, -4, 1], [0, -8, -2, 8], [0, 3, -14, -3], [-2, -1, 0, -1], [14, 0, 0, 0], [0, -10, -11, 10], [0, -15, -10, 15], [0, 8, -4, -8], [-14, 7, 0, 7], [20, 4, 0, 4], [11, -4, 0, -4], [0, -16, -10, 16], [0, -6, 5, 6], [0, 15, 10, -15], [34, 2, 0, 2], [-7, 2, 0, 2], [0, -8, 4, 8], [0, -5, 18, 5], [0, 8, 3, -8], [14, 5, 0, 5], [-1, 3, 0, 3], [17, -2, 0, -2], [14, 5, 0, 5], [0, -5, -7, 5], [0, 1, -20, -1], [-2, 7, 0, 7], [0, 3, 9, -3], [-18, 1, 0, 1], [0, -6, 10, 6], [0, 7, 0, 7], [14, 0, 0, 0], [0, -3, 18, 3], [0, 0, -1, 0], [0, 4, -20, -4], [-9, 8, 0, 8], [14, -2, 0, -2], [-4, -5, 0, -5], [0, 18, 5, -18], [0, -7, -21, 7], [28, 7, 0, 7], [-28, -4, 0, -4], [0, 10, 10, -10], [0, -1, 20, 1], [-14, 5, 0, 5], [0, 2, 13, -2], [0, -5, -22, 5], [-7, -11, 0, -11], [0, 8, -18, -8], [0, 18, 9, -18], [0, 20, 11, -20], [-12, -4, 0, -4], [24, 7, 0, 7], [35, 4, 0, 4], [0, -8, 0, 8], [0, -1, 0, -1], [0, 12, 15, -12], [0, 12, 8, -12], [0, -1, -3, 1], [-14, -7, 0, -7], [0, 3, -12, -3], [0, 2, -23, -2], [4, 9, 0, 9], [0, 1, 2, -1]]
