
# q-expansion of newform 960.3.l.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 = 3
poly_data = [9, 0, -4, 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], [[-2, 0, 1, 0], 1], [[0, -2, 0, 1], 1]]

hecke_ring_character_values = [[511, [1, 0, 0, 0]], [901, [1, 0, 0, 0]], [641, [-1, 0, 0, 0]], [577, [1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [1, -1, -1, 0], [0, 0, -1, 0], [-2, -5, 0, 1], [0, 2, 0, 2], [10, 0, 0, 0], [0, -4, 6, -4], [8, -10, 0, 2], [0, 1, 6, 1], [0, 0, -12, 0], [-8, 0, 0, 0], [22, 20, 0, -4], [0, 8, -6, 8], [14, 15, 0, -3], [0, 5, 30, 5], [0, 4, -6, 4], [0, -12, -12, -12], [16, -20, 0, 4], [-82, 15, 0, -3], [0, -10, -12, -10], [50, -20, 0, 4], [28, -10, 0, 2], [0, 3, -6, 3], [0, 8, -12, 8], [74, 20, 0, -4], [0, 24, 24, 24], [-74, 35, 0, -7], [0, -7, 6, -7], [-20, 60, 0, -12], [0, 24, 42, 24], [106, -15, 0, 3], [0, -14, 72, -14], [0, 32, 18, 32], [56, 70, 0, -14], [0, 24, -78, 24], [-56, -20, 0, 4], [10, 60, 0, -12], [74, 45, 0, -9], [0, -31, 54, -31], [0, 32, 66, 32], [0, 28, 84, 28], [-38, -120, 0, 24], [0, 6, 60, 6], [-130, 20, 0, -4], [0, -12, 54, -12], [-92, -30, 0, 6], [-172, -60, 0, 12], [-14, 45, 0, -9], [0, -29, 42, -29], [-98, 120, 0, -24], [0, -28, -30, -28], [0, 84, -24, 84], [104, -180, 0, 36], [0, -18, 72, -18], [0, -72, 6, -72], [0, 37, 54, 37], [0, -96, 54, -96], [400, -40, 0, 8], [-302, -60, 0, 12], [0, 48, 102, 48], [170, -15, 0, 3], [0, -80, 78, -80], [74, 45, 0, -9], [0, -78, 84, -78], [-274, -120, 0, 24], [0, 44, 66, 44], [-28, 220, 0, -44], [98, -120, 0, 24], [0, 47, 66, 47], [-206, -160, 0, 32], [0, 8, 270, 8], [0, -52, 24, -52], [22, 255, 0, -51], [454, -20, 0, 4], [-88, 90, 0, -18], [0, -101, 42, -101], [0, 8, -258, 8], [214, 0, 0, 0], [0, 32, 264, 32], [-460, 60, 0, -12], [0, -80, -108, -80], [-224, 260, 0, -52], [0, 126, 108, 126], [146, -180, 0, 36], [-428, -90, 0, 18], [0, 9, -114, 9], [0, 40, -186, 40], [-130, 360, 0, -72], [0, 80, -96, 80], [-110, -295, 0, 59], [0, 61, 102, 61], [0, -148, 48, -148], [178, 45, 0, -9], [0, 22, -360, 22], [80, -30, 0, 6], [0, -57, 90, -57], [0, -32, -156, -32], [0, -64, -120, -64], [350, -325, 0, 65], [262, 40, 0, -8], [-262, 25, 0, -5], [0, 80, 150, 80], [0, -65, -198, -65], [0, 40, 6, 40], [-220, -40, 0, 8], [-226, -400, 0, 80], [0, 67, 42, 67], [0, 72, -18, 72], [0, 128, -72, 128], [680, 180, 0, -36], [826, 5, 0, -1], [-302, -280, 0, 56], [0, 180, -138, 180], [128, 110, 0, -22], [304, 420, 0, -84], [0, 32, 90, 32], [-514, -45, 0, 9], [0, -49, 114, -49], [0, -116, -30, -116], [0, -100, -108, -100], [520, 20, 0, -4], [602, 220, 0, -44], [0, 52, -378, 52], [0, -31, 294, -31], [932, 0, 0, 0], [0, -88, 438, -88], [-110, -200, 0, 40], [0, 76, -96, 76], [634, -255, 0, 51], [334, 180, 0, -36], [-880, -30, 0, 6], [0, -53, -102, -53], [-920, 140, 0, -28], [370, -260, 0, 52], [0, 8, -240, 8], [-514, 120, 0, -24], [0, 188, 78, 188], [914, 305, 0, -61], [0, 76, -102, 76], [0, -64, -372, -64], [-124, 60, 0, -12], [0, -32, -90, -32], [190, 375, 0, -75], [0, -179, 102, -179], [-68, 140, 0, -28], [0, 96, 240, 96], [58, 600, 0, -120], [0, -232, -126, -232], [-376, -50, 0, 10], [0, 83, -30, 83], [-662, 220, 0, -44], [0, 72, -198, 72], [386, 665, 0, -133], [0, 59, -342, 59], [926, -305, 0, 61], [0, -154, 156, -154], [268, -270, 0, 54], [0, -8, 474, -8], [-370, 20, 0, -4], [0, 272, 216, 272], [0, 47, -630, 47], [0, 224, 114, 224], [1438, 215, 0, -43], [0, -262, 24, -262], [0, 28, 378, 28], [0, 175, -126, 175], [280, -800, 0, 160], [-446, 240, 0, -48]]
