
# q-expansion of newform 810.2.c.f, downloaded from the LMFDB on 09 September 2025.

# 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 = 810
weight = 2
poly_data = [9, 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, 1, 0, 0], 1], [[0, 0, 1, 0], 3], [[0, 0, 0, 1], 3]]

hecke_ring_character_values = [[731, [1, 0, 0, 0]], [487, [-1, 0, 0, 0]]]
aps_data = [[0, 0, -1, 0], [0, 0, 0, 0], [1, 1, -1, 0], [0, -1, 2, -1], [-1, -1, 0, 1], [0, -1, 0, -1], [0, -2, -1, -2], [3, 1, 0, -1], [0, -2, -2, -2], [6, 0, 0, 0], [4, -1, 0, 1], [0, 0, 8, 0], [-1, 0, 0, 0], [0, 1, -5, 1], [0, -1, -2, -1], [0, 1, -6, 1], [-1, -5, 0, 5], [2, 1, 0, -1], [0, -1, 7, -1], [0, -1, 0, 1], [0, -4, -5, -4], [0, -3, 0, 3], [0, 0, 4, 0], [-8, 2, 0, -2], [0, 0, 13, 0], [-8, 0, 0, 0], [0, 5, 2, 5], [0, 3, 9, 3], [8, 0, 0, 0], [0, 2, 0, 2], [0, 2, 2, 2], [0, 2, 0, -2], [0, 0, 3, 0], [-1, -5, 0, 5], [-4, 5, 0, -5], [-2, 2, 0, -2], [0, 0, 16, 0], [0, -2, -4, -2], [0, 5, -12, 5], [0, 4, 2, 4], [-4, 2, 0, -2], [8, -1, 0, 1], [6, 5, 0, -5], [0, -6, 1, -6], [0, 0, -8, 0], [-18, 1, 0, -1], [-6, 4, 0, -4], [0, 2, -14, 2], [0, -1, -1, -1], [16, 1, 0, -1], [0, 6, -1, 6], [-14, 6, 0, -6], [1, 0, 0, 0], [9, -1, 0, 1], [0, -2, 15, -2], [0, 1, 10, 1], [6, -9, 0, 9], [4, -8, 0, 8], [0, -8, -10, -8], [12, 0, 0, 0], [0, 0, -4, 0], [0, 0, -18, 0], [0, 11, 3, 11], [18, -2, 0, 2], [0, 6, 7, 6], [0, 11, -4, 11], [4, -12, 0, 12], [0, 8, -1, 8], [0, 5, -3, 5], [8, 8, 0, -8], [0, -8, -13, -8], [-6, 1, 0, -1], [0, 8, 8, 8], [0, 8, -6, 8], [13, -7, 0, 7], [0, 4, 8, 4], [-20, 1, 0, -1], [0, -4, 8, -4], [19, -4, 0, 4], [-11, -2, 0, 2], [4, 2, 0, -2], [-14, 1, 0, -1], [26, 5, 0, -5], [0, 0, -23, 0], [0, -9, 0, 9], [0, -5, -9, -5], [-9, -4, 0, 4], [0, -2, -13, -2], [0, -1, 0, 1], [0, 0, -24, 0], [0, -3, -3, -3], [-2, -6, 0, 6], [0, 2, -30, 2], [21, -1, 0, 1], [9, 5, 0, -5], [0, -1, -12, -1], [-12, -8, 0, 8], [7, 6, 0, -6], [0, -4, 20, -4], [-20, 8, 0, -8], [0, 5, 5, 5], [0, 11, -12, 11], [0, 7, -19, 7], [7, 10, 0, -10], [25, 5, 0, -5], [0, -6, -1, -6], [0, -11, 3, -11], [0, 12, -12, 12], [-24, -4, 0, 4], [29, 4, 0, -4], [0, -8, -8, -8], [0, 11, 10, 11], [0, -6, 23, -6], [31, -1, 0, 1], [-4, -3, 0, 3], [-37, 0, 0, 0], [0, -5, -3, -5], [0, 2, 30, 2], [0, 0, 34, 0], [26, 4, 0, -4], [-4, 4, 0, -4], [0, 2, 24, 2], [0, 8, 20, 8], [0, 1, 43, 1], [-2, 8, 0, -8], [-10, -12, 0, 12], [-30, -3, 0, 3], [-32, -4, 0, 4], [0, -3, -18, -3], [0, -7, -4, -7], [5, 5, 0, -5], [0, 2, 18, 2], [-4, -20, 0, 20], [0, 0, -10, 0], [-24, 10, 0, -10], [0, -10, 0, 10], [0, -12, -6, -12], [0, -12, -22, -12], [0, 8, -16, 8], [-3, 18, 0, -18], [-5, 5, 0, -5], [-20, 9, 0, -9], [0, 8, 12, 8], [0, 4, -4, 4], [24, 1, 0, -1], [-28, -5, 0, 5], [0, -7, 8, -7], [0, 12, -24, 12], [1, 15, 0, -15], [0, -1, -24, -1], [0, 7, -38, 7], [-30, 0, 0, 0], [0, -1, 9, -1], [0, 3, 12, 3], [0, -15, -3, -15], [12, 5, 0, -5], [-6, 2, 0, -2], [36, -8, 0, 8], [0, 8, -20, 8], [-40, 4, 0, -4], [0, 5, -9, 5], [0, -4, 41, -4], [0, 6, -14, 6], [6, 12, 0, -12], [0, -2, 33, -2], [0, -12, 8, -12], [4, 0, 0, 0], [0, 11, -6, 11]]
