
# q-expansion of newform 300.3.c.f, downloaded from the LMFDB on 23 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 = 300
weight = 3
poly_data = [256, -192, 112, -48, 12, -12, 7, -3, 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, 0, 0], 1], [[-160, 368, 264, 108, -30, 5, -13, -3], 224], [[608, 80, -96, 4, -12, -19, -1, 3], 224], [[-320, 400, -4, -8, -11, 10, -12, 1], 112], [[408, -160, 24, -22, 10, -11, 9, 1], 56], [[-736, 528, -152, 60, 2, 23, -15, 3], 224], [[928, -656, 244, -72, 41, -36, 18, -5], 112], [[384, -592, 324, -52, 37, -26, 20, -11], 112]]

hecke_ring_character_values = [[151, [-1, 0, 0, 0, 0, 0, 0, 0]], [101, [1, 0, 0, 0, 0, 0, 0, 0]], [277, [1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 1, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, -1, 0, 1, 0, 0], [-1, 0, 0, 0, 1, 0, 1, -3], [0, 0, 1, 1, 0, 3, 0, 0], [0, -4, 1, 1, 0, -1, 0, 0], [-2, 0, 0, 0, 2, 0, -6, 2], [0, 0, 6, 2, 0, -2, 0, 0], [23, 0, 0, 0, 3, 0, 3, 3], [0, 0, 0, 0, 0, 0, -8, 8], [0, 0, 3, 3, 0, 9, 0, 0], [-32, 0, 0, 0, 6, 0, 6, 6], [0, 0, 4, -8, 0, 8, 0, 0], [0, 0, -12, -8, 0, 8, 0, 0], [0, 0, -3, -3, 0, -9, 0, 0], [-9, 0, 0, 0, 9, 0, -23, 5], [38, 0, 0, 0, 0, 0, 0, 0], [0, 0, 18, 14, 0, -14, 0, 0], [-10, 0, 0, 0, 10, 0, -6, -14], [0, 8, -2, -2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, -8, 8], [0, 0, -24, -4, 0, 4, 0, 0], [-70, 0, 0, 0, 4, 0, 4, 4], [0, 8, 20, 20, 0, 68, 0, 0], [37, 0, 0, 0, 1, 0, 1, 1], [0, 0, -5, 31, 0, -31, 0, 0], [0, 0, 20, 16, 0, -16, 0, 0], [76, 0, 0, 0, -10, 0, -10, -10], [0, -4, 3, 3, 0, 5, 0, 0], [0, 0, -53, 23, 0, -23, 0, 0], [5, 0, 0, 0, -5, 0, 11, -1], [0, 20, -21, -21, 0, -43, 0, 0], [0, 0, 0, 0, 0, 0, 8, -8], [11, 0, 0, 0, 3, 0, 3, 3], [8, 0, 0, 0, -8, 0, -16, 32], [0, -16, 21, 21, 0, 47, 0, 0], [0, 0, 66, 22, 0, -22, 0, 0], [0, 0, -42, -38, 0, 38, 0, 0], [0, 8, -13, -13, 0, -31, 0, 0], [23, 0, 0, 0, -23, 0, 57, -11], [-52, 0, 0, 0, 22, 0, 22, 22], [14, 0, 0, 0, -14, 0, 34, -6], [0, -16, -10, -10, 0, -46, 0, 0], [0, -40, 13, 13, 0, -1, 0, 0], [24, 0, 0, 0, -24, 0, 48, 0], [-16, 0, 0, 0, 16, 0, -8, -24], [0, 0, -129, -29, 0, 29, 0, 0], [0, 0, 64, -68, 0, 68, 0, 0], [-128, 0, 0, 0, -18, 0, -18, -18], [0, 4, 5, 5, 0, 19, 0, 0], [2, 0, 0, 0, -2, 0, -50, 54], [194, 0, 0, 0, -36, 0, -36, -36], [5, 0, 0, 0, -5, 0, -21, 31], [0, -36, -17, -17, 0, -87, 0, 0], [0, 0, 62, -22, 0, 22, 0, 0], [161, 0, 0, 0, -11, 0, -11, -11], [4, 0, 0, 0, -4, 0, 84, -76], [0, 64, -45, -45, 0, -71, 0, 0], [-134, 0, 0, 0, -12, 0, -12, -12], [0, 0, 114, -26, 0, 26, 0, 0], [0, -8, -35, -35, 0, -113, 0, 0], [0, 0, 64, -52, 0, 52, 0, 0], [6, 0, 0, 0, -6, 0, -70, 82], [0, -16, 6, 6, 0, 2, 0, 0], [0, 56, 3, 3, 0, 65, 0, 0], [18, 0, 0, 0, -18, 0, -10, 46], [0, 40, 10, 10, 0, 70, 0, 0], [0, 0, 136, 52, 0, -52, 0, 0], [-110, 0, 0, 0, -12, 0, -12, -12], [0, 20, 17, 17, 0, 71, 0, 0], [-24, 0, 0, 0, 24, 0, -72, 24], [0, 0, -157, 47, 0, -47, 0, 0], [0, 64, 9, 9, 0, 91, 0, 0], [4, 0, 0, 0, -4, 0, -108, 116], [0, 0, -256, -52, 0, 52, 0, 0], [-1, 0, 0, 0, 63, 0, 63, 63], [0, -48, -21, -21, 0, -111, 0, 0], [88, 0, 0, 0, 34, 0, 34, 34], [64, 0, 0, 0, 58, 0, 58, 58], [-19, 0, 0, 0, 19, 0, -29, -9], [176, 0, 0, 0, 14, 0, 14, 14], [40, 0, 0, 0, -40, 0, 72, 8], [0, 32, -74, -74, 0, -190, 0, 0], [8, 0, 0, 0, -8, 0, -128, 144], [0, 0, 132, -88, 0, 88, 0, 0], [-40, 0, 0, 0, 18, 0, 18, 18], [0, -24, -56, -56, 0, -192, 0, 0], [331, 0, 0, 0, 3, 0, 3, 3], [0, 0, -95, -59, 0, 59, 0, 0], [0, 0, 360, 4, 0, -4, 0, 0], [-14, 0, 0, 0, 14, 0, 14, -42], [0, 0, -315, -79, 0, 79, 0, 0], [-35, 0, 0, 0, 35, 0, 35, -105], [-26, 0, 0, 0, 26, 0, 98, -150], [0, 0, 40, 116, 0, -116, 0, 0], [-49, 0, 0, 0, -21, 0, -21, -21], [-728, 0, 0, 0, 2, 0, 2, 2], [0, 0, 146, -26, 0, 26, 0, 0], [-388, 0, 0, 0, -74, 0, -74, -74], [0, 0, 132, -64, 0, 64, 0, 0], [0, -136, 61, 61, 0, 47, 0, 0], [0, 0, 168, -148, 0, 148, 0, 0], [20, 0, 0, 0, -10, 0, -10, -10], [82, 0, 0, 0, -82, 0, 214, -50], [0, 16, -64, -64, 0, -176, 0, 0], [0, 0, 156, 40, 0, -40, 0, 0], [0, -4, 115, 115, 0, 341, 0, 0], [88, 0, 0, 0, -88, 0, 232, -56], [476, 0, 0, 0, 26, 0, 26, 26], [0, 0, -39, -187, 0, 187, 0, 0], [0, -80, 35, 35, 0, 25, 0, 0], [0, -76, 77, 77, 0, 155, 0, 0], [60, 0, 0, 0, -60, 0, -20, 140], [48, 0, 0, 0, -48, 0, 24, 72], [-704, 0, 0, 0, 54, 0, 54, 54], [0, 0, -306, -14, 0, 14, 0, 0], [0, 0, -76, -8, 0, 8, 0, 0], [0, 128, -81, -81, 0, -115, 0, 0], [41, 0, 0, 0, -41, 0, -73, 155], [26, 0, 0, 0, 44, 0, 44, 44], [0, 112, -38, -38, 0, -2, 0, 0], [0, 48, 23, 23, 0, 117, 0, 0], [0, 0, 468, 104, 0, -104, 0, 0], [54, 0, 0, 0, -54, 0, 146, -38], [103, 0, 0, 0, 131, 0, 131, 131], [172, 0, 0, 0, -78, 0, -78, -78], [44, 0, 0, 0, -44, 0, 132, -44], [0, 0, -267, -31, 0, 31, 0, 0], [0, 64, -105, -105, 0, -251, 0, 0], [-14, 0, 0, 0, 14, 0, -42, 14], [0, 0, 96, -4, 0, 4, 0, 0], [-44, 0, 0, 0, 44, 0, -92, 4], [0, 112, -37, -37, 0, 1, 0, 0], [-428, 0, 0, 0, -90, 0, -90, -90], [-176, 0, 0, 0, 54, 0, 54, 54], [0, -40, 163, 163, 0, 449, 0, 0], [0, 0, -394, 146, 0, -146, 0, 0], [0, 88, -141, -141, 0, -335, 0, 0], [-4, 0, 0, 0, 42, 0, 42, 42], [-104, 0, 0, 0, 104, 0, -240, 32], [739, 0, 0, 0, -57, 0, -57, -57], [0, 0, 393, -131, 0, 131, 0, 0], [0, 0, -52, 184, 0, -184, 0, 0], [-440, 0, 0, 0, 74, 0, 74, 74], [-124, 0, 0, 0, 124, 0, -196, -52], [0, 192, -49, -49, 0, 45, 0, 0], [0, -84, -7, -7, 0, -105, 0, 0], [-100, 0, 0, 0, 100, 0, -4, -196], [0, 0, -386, 170, 0, -170, 0, 0], [0, 80, 141, 141, 0, 503, 0, 0], [-14, 0, 0, 0, -148, 0, -148, -148], [0, 0, -466, -62, 0, 62, 0, 0], [0, 0, 478, -14, 0, 14, 0, 0], [0, 0, 168, -316, 0, 316, 0, 0], [-16, 0, 0, 0, 16, 0, -272, 240], [-52, 0, 0, 0, 52, 0, -228, 124], [-1312, 0, 0, 0, -22, 0, -22, -22], [0, 64, -148, -148, 0, -380, 0, 0], [1183, 0, 0, 0, 63, 0, 63, 63], [0, 0, 84, -32, 0, 32, 0, 0], [0, 132, 3, 3, 0, 141, 0, 0], [0, 0, 591, 155, 0, -155, 0, 0], [-3, 0, 0, 0, 3, 0, -141, 135], [0, -36, -107, -107, 0, -357, 0, 0], [0, 0, 468, -208, 0, 208, 0, 0], [100, 0, 0, 0, -100, 0, 132, 68], [0, -304, 113, 113, 0, 35, 0, 0]]
