
# q-expansion of newform 960.2.v.f, downloaded from the LMFDB on 25 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 = 960
weight = 2
poly_data = [1, 0, 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], 1], [[0, 0, 1, 0], 1], [[0, 1, 0, 1], 1], [[0, 1, 0, -1], 1]]

hecke_ring_character_values = [[511, [1, 0, 0, 0]], [901, [1, 0, 0, 0]], [641, [-1, 0, 0, 0]], [577, [0, 1, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, 1, 0, -1], [1, 2, 0, 0], [-1, -1, 1, 1], [0, -2, -2, 0], [-1, 1, -2, 2], [1, -1, -2, 2], [0, -2, -2, 0], [3, 3, -1, -1], [2, 0, 0, 4], [0, 0, 0, -4], [3, 3, -2, -2], [0, 0, 4, 0], [-1, 1, 1, -1], [5, -5, -1, 1], [3, 3, 0, 0], [4, 0, 0, 0], [-6, 0, 0, -4], [-3, -3, 5, 5], [0, -2, 6, 0], [1, -1, -4, 4], [0, -2, -2, 0], [1, 1, 3, 3], [10, 0, 0, -4], [1, 1, 0, 0], [0, -4, 4, 0], [-7, 7, 1, -1], [11, -11, -1, 1], [0, -4, 0, 0], [1, 1, -6, -6], [7, 7, 1, 1], [0, 6, -2, 0], [-7, 7, -2, 2], [0, 6, -2, 0], [2, 0, 0, -8], [-8, 0, 0, 4], [3, 3, 6, 6], [-1, 1, 1, -1], [5, -5, 3, -3], [-5, -5, -4, -4], [12, 0, 0, 0], [-6, 0, 0, -8], [0, -10, -10, 0], [9, -9, 4, -4], [-9, 9, -4, 4], [0, -10, 6, 0], [-4, 0, 0, 12], [-7, 7, 1, -1], [-5, 5, 7, -7], [0, -20, -4, 0], [9, 9, -2, -2], [16, 0, 0, 0], [-6, 0, 0, -4], [0, -18, 6, 0], [1, -1, 6, -6], [3, 3, 7, 7], [-6, 0, 0, -16], [16, 0, 0, 4], [11, 11, 2, 2], [0, 0, -4, 0], [-1, 1, -3, 3], [11, 11, -4, -4], [-3, -3, -7, -7], [0, 14, 6, 0], [9, -9, -8, 8], [-1, 1, 4, -4], [4, 0, 0, -4], [1, 1, 0, 0], [-5, 5, -5, 5], [0, 4, -4, 0], [-7, -7, 6, 6], [-24, 0, 0, 8], [-9, -9, -3, -3], [-9, 9, 6, -6], [0, -10, 14, 0], [11, 11, -1, -1], [18, 0, 0, 8], [-5, -5, 2, 2], [0, -16, 0, 0], [0, 16, -4, 0], [-4, 0, 0, -24], [-14, 0, 0, 12], [0, 6, -10, 0], [-15, 15, 0, 0], [0, 6, 14, 0], [-7, -7, -5, -5], [10, 0, 0, 8], [1, 1, 0, 0], [0, -12, 12, 0], [-23, 23, 1, -1], [-5, 5, -13, 13], [0, 0, 0, -16], [-1, -1, -11, -11], [0, 30, 6, 0], [0, -2, -26, 0], [-13, -13, -5, -5], [2, 0, 0, 4], [0, 24, 0, 0], [-17, 17, -7, 7], [18, 0, 0, -8], [-3, -3, 9, 9], [7, -7, -16, 16], [17, 17, 11, 11], [34, 0, 0, -8], [-12, 0, 0, -12], [9, 9, -4, -4], [-5, 5, 11, -11], [-7, -7, 6, 6], [-8, 0, 0, 24], [2, 0, 0, 12], [-9, -9, 17, 17], [-1, 1, -6, 6], [-15, 15, 2, -2], [0, 6, -10, 0], [-8, 0, 0, -28], [0, -8, -20, 0], [-1, 1, 21, -21], [5, -5, 15, -15], [-21, -21, 4, 4], [12, 0, 0, 16], [2, 0, 0, -4], [-15, 15, 8, -8], [7, -7, 20, -20], [-7, -7, 11, 11], [28, 0, 0, -4], [0, -4, 0, 0], [0, -4, 20, 0], [-16, 0, 0, 0], [-17, -17, -3, -3], [-9, 9, 10, -10], [0, -50, -2, 0], [19, 19, -13, -13], [16, 0, 0, 20], [11, 11, 18, 18], [0, 24, 8, 0], [0, -16, 24, 0], [-5, -5, 4, 4], [29, 29, 1, 1], [-1, 1, 4, -4], [-38, 0, 0, -4], [-28, 0, 0, 4], [0, 4, -24, 0], [-7, 7, 9, -9], [-5, 5, 23, -23], [0, -4, 8, 0], [-24, 0, 0, -8], [-1, 1, 18, -18], [25, -25, 6, -6], [0, -10, 30, 0], [-5, -5, -1, -1], [-13, -13, -18, -18], [0, 8, 12, 0], [-33, 33, 1, -1], [5, -5, -9, 9], [-27, -27, 1, 1], [0, 38, -2, 0], [0, 6, -2, 0], [-22, 0, 0, -24], [-31, -31, 0, 0], [0, 4, -28, 0], [11, -11, -9, 9], [-7, -7, 6, 6], [-17, -17, 1, 1], [0, -34, 6, 0], [9, -9, 10, -10], [3, 3, -21, -21], [-32, 0, 0, 12], [11, 11, -14, -14]]
