
# q-expansion of newform 735.2.q.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 = 735
weight = 2
poly_data = [4, -8, 8, -24, 20, 20, -8, 16, 4, -8, 2, -2, 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, 0, 0, 0, 0], 1], [[-164, 164, -170, 482, 338, -276, 112, -62, 3, 5, -5, 3], 460], [[-68, 50, -50, 150, -154, -96, 21, -13, 33, -8, 2, -2], 230], [[8, -8, -298, 18, -58, 230, -130, -80, 61, -21, 21, -8], 460], [[-28, -20, 20, 216, 80, 20, 263, -50, -50, -6, -10, 10], 230], [[-142, -154, 154, 1148, 524, 246, 560, -109, -201, 9, -31, 31], 230], [[48, -84, 84, 967, 290, 130, 318, -72, -118, 7, -19, 19], 115], [[296, 296, 1062, -2246, -1158, -78, -1384, 52, 505, -61, 105, -74], 460], [[-12, 32, -38, 86, 74, -12, 46, 4, -23, 5, -5, 3], 20], [[-1028, 1028, 688, 2908, 2232, 966, 1042, -162, -674, 88, -88, 85], 460], [[516, 316, -316, -1490, -1034, -546, -572, 100, 330, -34, 43, -43], 230], [[-900, 756, -2940, 6540, 3948, -630, 3630, 374, -1680, 330, -378, 225], 460]]

hecke_ring_character_values = [[491, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [442, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [346, [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 2, -2, 1, 0, 0, 0, -1, 0], [0, 0, 0, 2, 0, 0, -1, 0, 0, -1, 0, 1], [0, -2, 2, 0, 0, 0, 0, 0, -2, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, -2, 0, -1, -1, 1], [-2, 0, 0, 0, 0, 0, -2, 0, 0, 0, -2, 0], [2, 2, 0, 0, 0, 0, 1, 0, -2, -1, 0, -1], [0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, -1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, -2, -2, 2], [0, 0, 0, 2, 0, 6, 1, 6, 0, 1, 0, -1], [-4, -4, 0, 0, 0, 0, 4, 0, 4, -4, 0, -4], [0, -4, 4, 0, 0, 0, 0, 0, -2, -2, -2, -2], [0, 0, 0, 0, 0, -4, -2, -4, 0, -2, 0, 2], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 6, 1, 6, 0, 1, 0, -1], [0, 4, -4, 0, 0, 0, 0, 0, 4, 2, 2, 2], [0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 1, 1, 1], [0, 0, 0, 0, -2, 10, 3, 0, 0, 0, -3, 0], [-8, 4, 0, 0, 0, 0, -3, 0, 8, 3, 0, 3], [0, 0, 0, -4, -4, 0, 0, 0, 0, 2, 2, -2], [0, 0, 0, -4, -4, 0, 0, 10, 0, 1, 1, -1], [-2, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, -6, -2, -3, 0, 0, 0, 3, 0], [0, 0, 0, 0, 4, -4, 2, 0, 0, 0, -2, 0], [0, 4, -4, 0, 0, 0, 0, 0, 8, 4, 4, 4], [0, 0, 0, -2, 0, -14, -3, -14, 0, -3, 0, 3], [6, 0, 2, 0, 0, 0, -3, 0, 0, 0, -3, 0], [0, 8, -8, 0, 0, 0, 0, 0, -6, 0, 0, 0], [-8, -4, 0, 0, 0, 0, 2, 0, 8, -2, 0, -2], [0, 0, 0, -2, 0, -6, 1, -6, 0, 1, 0, -1], [0, 0, 0, -4, -4, 0, 0, -12, 0, 2, 2, -2], [0, 0, 0, 0, 4, -4, 6, 0, 0, 0, -6, 0], [0, 0, 0, -2, -2, 0, 0, 4, 0, -3, -3, 3], [10, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0], [2, 0, 4, 0, 0, 0, -4, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 4, 4], [0, 0, 0, -8, 0, -4, 6, -4, 0, 6, 0, -6], [0, 0, 0, 0, 2, 2, -1, 0, 0, 0, 1, 0], [2, -6, 0, 0, 0, 0, -1, 0, -2, 1, 0, 1], [8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, -8, -2, -8, 0, -2, 0, 2], [0, -4, 4, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, -6, -6, 0, 0, -10, 0, 3, 3, -3], [6, 0, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0], [-6, 0, 0, 0, 0, 0, 2, 0, 6, -2, 0, -2], [-20, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 2, 0, 0, -4, 0, -1, -1, 1], [0, 0, 0, 0, 0, -10, -1, -10, 0, -1, 0, 1], [4, -4, 0, 0, 0, 0, -3, 0, -4, 3, 0, 3], [0, 2, -2, 0, 0, 0, 0, 0, -2, -5, -5, -5], [0, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, -4], [10, 0, 8, 0, 0, 0, 2, 0, 0, 0, 2, 0], [0, 0, 0, -4, 0, 4, -2, 4, 0, -2, 0, 2], [0, 0, 0, 0, -2, 4, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, -12, -4, -6, 0, 0, 0, 6, 0], [0, -12, 0, 0, 0, 0, 2, 0, 0, -2, 0, -2], [0, 0, 0, 10, 10, 0, 0, -10, 0, -1, -1, 1], [0, 0, 0, -6, -6, 0, 0, 6, 0, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -16, -4, -4, -4], [0, 0, 0, 0, 0, 8, -4, 0, 0, 0, 4, 0], [0, 0, 0, 4, 0, 6, -5, 6, 0, -5, 0, 5], [-10, 0, -4, 0, 0, 0, -2, 0, 0, 0, -2, 0], [0, 0, 0, 2, 0, 0, 7, 0, 0, 7, 0, -7], [0, -8, 8, 0, 0, 0, 0, 0, -18, -4, -4, -4], [0, 0, 0, 8, 0, -16, -4, -16, 0, -4, 0, 4], [0, 0, 0, 0, 0, 0, 0, -16, 0, 0, 0, 0], [-4, 0, 4, 0, 0, 0, -2, 0, 0, 0, -2, 0], [0, 0, 0, 4, 4, 0, 0, -16, 0, 0, 0, 0], [-10, 8, 0, 0, 0, 0, -6, 0, 10, 6, 0, 6], [0, 0, 0, -2, -2, 0, 0, 2, 0, -7, -7, 7], [0, 8, -8, 0, 0, 0, 0, 0, 18, 6, 6, 6], [18, 8, 0, 0, 0, 0, 2, 0, -18, -2, 0, -2], [16, 0, -8, 0, 0, 0, 2, 0, 0, 0, 2, 0], [-26, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -8, 0, 0, 0, 0, 4, 0, -6, -4, 0, -4], [0, 0, 0, 0, -2, -2, 3, 0, 0, 0, -3, 0], [0, -2, 2, 0, 0, 0, 0, 0, 18, -1, -1, -1], [0, 0, 0, 12, 12, 0, 0, -2, 0, -5, -5, 5], [6, 0, -16, 0, 0, 0, 8, 0, 0, 0, 8, 0], [0, 0, 0, 4, 4, 0, 0, -12, 0, -2, -2, 2], [4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 4, -12, 2, 0, 0, 0, -2, 0], [0, 0, 0, 4, 4, 0, 0, 8, 0, 2, 2, -2], [0, -4, 0, 0, 0, 0, -2, 0, 0, 2, 0, 2], [0, 0, 0, -12, 0, 8, 4, 8, 0, 4, 0, -4], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 4, 0, 0, 0, 0, 0, 12, -6, -6, -6], [0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0], [0, -12, 12, 0, 0, 0, 0, 0, 4, -5, -5, -5], [-28, 4, 0, 0, 0, 0, -3, 0, 28, 3, 0, 3], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0], [0, 4, -4, 0, 0, 0, 0, 0, -2, 8, 8, 8], [0, 0, 0, 0, 0, -20, -2, 0, 0, 0, 2, 0], [0, 0, 0, -10, 0, -6, 11, -6, 0, 11, 0, -11], [0, 0, 0, -8, 0, 16, -2, 16, 0, -2, 0, 2], [0, 8, -8, 0, 0, 0, 0, 0, -10, 6, 6, 6], [-4, 0, 0, 0, 0, 0, 8, 0, 4, -8, 0, -8], [0, 0, 0, 10, 0, 14, 7, 14, 0, 7, 0, -7], [0, 0, 0, 0, 16, -16, -2, 0, 0, 0, 2, 0], [0, 0, 0, -6, -6, 0, 0, 0, 0, 11, 11, -11], [-10, -8, 0, 0, 0, 0, -4, 0, 10, 4, 0, 4], [2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0], [0, 0, 0, -8, -8, 0, 0, 0, 0, 8, 8, -8], [0, 0, 0, -4, 0, -4, -2, -4, 0, -2, 0, 2], [0, 0, 0, 0, -6, 26, -3, 0, 0, 0, 3, 0], [-22, -10, 0, 0, 0, 0, 3, 0, 22, -3, 0, -3], [4, 0, 4, 0, 0, 0, -6, 0, 0, 0, -6, 0], [6, -16, 0, 0, 0, 0, 10, 0, -6, -10, 0, -10], [0, 0, 0, 0, -12, 20, -10, 0, 0, 0, 10, 0], [0, 0, 0, 4, 0, 4, -6, 4, 0, -6, 0, 6], [0, 0, 0, 6, 6, 0, 0, -34, 0, -5, -5, 5], [2, 0, -16, 0, 0, 0, 4, 0, 0, 0, 4, 0], [6, 20, 0, 0, 0, 0, -6, 0, -6, 6, 0, 6], [0, 0, 0, 0, -12, 0, -8, 0, 0, 0, 8, 0], [0, 0, 0, -14, -14, 0, 0, 12, 0, 7, 7, -7], [0, 0, 0, 4, 0, -14, -3, -14, 0, -3, 0, 3], [0, 2, -2, 0, 0, 0, 0, 0, 2, -1, -1, -1], [10, 0, -16, 0, 0, 0, 10, 0, 0, 0, 10, 0], [0, 8, -8, 0, 0, 0, 0, 0, 2, 8, 8, 8], [0, 4, -4, 0, 0, 0, 0, 0, 4, 2, 2, 2], [0, 0, 0, 0, -12, -8, -10, 0, 0, 0, 10, 0], [0, 0, 0, -2, -2, 0, 0, -10, 0, -7, -7, 7], [12, -8, 0, 0, 0, 0, 8, 0, -12, -8, 0, -8], [0, 0, 0, 0, -8, -10, -5, 0, 0, 0, 5, 0], [0, -16, 16, 0, 0, 0, 0, 0, -12, -12, -12, -12], [0, 0, 0, 0, 16, 24, 4, 0, 0, 0, -4, 0], [0, 8, -8, 0, 0, 0, 0, 0, 20, 9, 9, 9], [6, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 14, 0, -4, -11, -4, 0, -11, 0, 11], [0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, -2, 12, -5, 0, 0, 0, 5, 0], [6, 8, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0], [2, 0, 6, 0, 0, 0, -13, 0, 0, 0, -13, 0], [0, -8, 8, 0, 0, 0, 0, 0, 26, -8, -8, -8], [0, 0, 0, 4, 0, 4, -10, 4, 0, -10, 0, 10], [0, 0, 0, 0, 4, -22, 9, 0, 0, 0, -9, 0], [-22, 4, 0, 0, 0, 0, 0, 0, 22, 0, 0, 0], [4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0], [0, 0, 0, 0, -2, 6, -9, 0, 0, 0, 9, 0], [0, 0, 0, -18, 0, 20, 9, 20, 0, 9, 0, -9], [0, 2, -2, 0, 0, 0, 0, 0, 2, 1, 1, 1], [0, 0, 0, -16, -16, 0, 0, -2, 0, 11, 11, -11], [0, 0, 0, 20, 20, 0, 0, 12, 0, -10, -10, 10], [-16, 0, 8, 0, 0, 0, -5, 0, 0, 0, -5, 0], [0, 0, 0, 0, -8, 16, -4, 0, 0, 0, 4, 0], [0, 0, 0, 24, 24, 0, 0, -4, 0, -10, -10, 10], [0, 0, 0, 8, 0, 16, 12, 16, 0, 12, 0, -12], [-10, 0, 16, 0, 0, 0, -4, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 12, -8, -8, -8], [0, 16, -16, 0, 0, 0, 0, 0, -12, -5, -5, -5], [0, 0, 0, 0, 6, -2, 7, 0, 0, 0, -7, 0], [16, 8, 0, 0, 0, 0, -7, 0, -16, 7, 0, 7], [0, 0, 0, -4, -4, 0, 0, -26, 0, -5, -5, 5], [0, 0, 0, 0, 10, -34, -3, 0, 0, 0, 3, 0], [0, 0, 0, 0, 24, 4, 10, 0, 0, 0, -10, 0], [0, -16, 16, 0, 0, 0, 0, 0, 4, -10, -10, -10], [0, 0, 0, -14, 0, 6, 11, 6, 0, 11, 0, -11], [0, 0, 0, 0, 0, 24, 0, 24, 0, 0, 0, 0], [12, 8, 0, 0, 0, 0, -12, 0, -12, 12, 0, 12], [0, 0, 0, 10, 0, 14, 7, 14, 0, 7, 0, -7]]
