
# q-expansion of newform 429.2.a.f, downloaded from the LMFDB on 19 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 = 429
weight = 2
poly_data = [1, -3, -1, 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], 1], [[0, 1, 0], 1], [[-2, -1, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 1, 0], [1, 0, 0], [0, 0, -1], [-1, 1, -1], [1, 0, 0], [1, 0, 0], [3, -1, 2], [3, -3, -1], [3, -1, 1], [1, -1, -2], [-2, 0, 1], [-4, 0, -2], [5, -3, -1], [1, -1, 4], [0, 2, -2], [-2, 4, 6], [4, -4, 2], [-2, 2, 2], [0, 2, 3], [2, -4, -2], [-3, 5, 3], [5, -1, 2], [-2, 2, 0], [-6, 8, 3], [-4, 0, -2], [3, 3, 0], [-10, 2, 2], [4, 4, -2], [-5, -5, 5], [4, 2, -4], [11, -13, -6], [2, 6, -2], [2, -4, 7], [-3, 9, 4], [3, -9, 1], [7, 1, 3], [-1, -3, -3], [-10, 2, -3], [2, 6, 2], [13, 1, 2], [5, 5, 5], [-19, 3, 3], [6, -6, -4], [-9, -1, -11], [-5, 3, -9], [2, -6, -6], [15, 1, -4], [6, 0, 7], [9, -9, -1], [10, -10, -10], [5, -5, -4], [7, -3, -7], [6, 0, 14], [2, 2, -4], [2, 4, -4], [0, 8, 10], [-4, -2, 0], [11, 5, 9], [0, 6, -10], [-9, -1, -9], [13, 1, 2], [-13, -5, 3], [5, -5, -3], [17, -7, -1], [-17, 13, 3], [-10, 8, 11], [4, 12, 3], [-18, 8, -2], [-6, 4, -2], [-2, 8, 2], [-8, 0, 13], [-15, -5, -5], [-8, 0, -6], [-18, 0, -2], [-2, 2, -1], [-14, 6, 0], [-14, -8, 2], [4, -16, -2], [10, -2, -13], [9, 1, -13], [9, -19, -3], [-20, -2, 2], [2, -10, 10], [-10, -4, 4], [-3, -1, -16], [8, -12, -12], [-14, -10, 1], [-20, -6, -6], [9, 5, 7], [0, 0, -9], [11, 3, -3], [-3, -1, -1], [-10, -10, 7], [4, 10, 12], [-6, -4, -3], [8, -12, -2], [2, -12, 1], [6, -4, -8], [-9, 17, 8], [10, -4, 16], [7, 11, 12], [-5, -1, 9], [22, 2, 14], [13, 9, -10], [-3, 7, -14], [-28, -2, -2], [-20, 8, 6], [-1, 15, -5], [6, 18, 0], [2, -4, -12], [-9, 15, -2], [-11, -7, -21], [18, -2, -11], [8, -8, -11], [0, 4, -5], [-4, -2, -22], [16, 2, -7], [-2, 2, -4], [2, -16, -20], [0, 2, 6], [8, 18, 0], [-16, 24, 2], [-1, 13, 10], [26, -32, -14], [36, -8, -5], [7, 3, -16], [8, -8, 4], [-20, 4, 12], [-2, 14, 12], [1, 1, 15], [9, 7, -15], [-20, -4, 14], [-4, 24, 6], [11, 13, 17], [-3, 17, 1], [10, 0, 0], [-10, -12, -5], [21, 3, -3], [-20, 18, 6], [-3, -3, 18], [17, -25, -15], [-21, 31, 15], [-18, -6, 0], [-43, -1, -9], [26, 4, 8], [-18, -6, -6], [12, -22, 2], [17, -5, -6], [26, 14, -12], [6, 6, 0], [-2, 4, 2], [26, -8, 0], [-6, 14, 2], [-8, -10, -26], [12, -24, -18], [-18, 6, -12], [-17, 3, 2], [4, 10, 9], [-10, -18, 8], [17, -27, -21], [18, 12, 16], [-1, -7, 4], [-21, -3, 17], [-19, 25, 19], [-2, 12, 1], [-22, -10, -26], [0, 12, -26], [-14, -8, -6]]
