
# q-expansion of newform 414.2.i.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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 414
weight = 2
poly_data = [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[47, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [235, [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -2, 1, 0, 0, -1, 0, 0, 1], [0, 0, -1, 0, -1, -1, -1, 0, -1, 0], [1, 2, -2, -1, 0, 0, 1, 1, 1, 0], [-2, 2, 0, 0, 0, 1, -2, 1, 0, 0], [-2, -1, -1, 3, -1, -1, -2, 0, 0, 0], [0, -1, -2, 2, 1, 0, 0, 1, 2, 1], [0, 0, 1, -2, -2, -1, 0, -1, -1, 3], [0, -1, -3, 0, -3, -1, 0, 0, -2, 2], [-5, 2, -5, 0, -3, 5, -3, 3, -5, 3], [5, -5, 1, 0, 0, -1, 5, -5, 0, -5], [7, -5, 4, -5, 7, 0, 5, -4, 4, -5], [-1, 3, 0, 3, -4, 4, -3, 0, -3, 1], [-1, 0, 0, -2, 4, 2, -2, -4, 2, 0], [0, 0, 0, 6, -3, 3, -3, 6, 0, 0], [-2, 2, 0, 2, -4, 4, -3, 4, -4, 2], [6, -5, 6, 0, 5, 1, 2, -2, -1, -5], [-2, 5, 0, -2, 1, -2, 0, 5, -2, 0], [-8, 3, -4, 2, -5, 2, -4, 3, -8, 0], [-2, 1, -3, 3, -1, 2, 0, 0, -2, 0], [-1, 1, 0, 2, -2, 2, -5, 2, -2, 2], [6, -7, 9, -5, 5, -9, 7, -6, 0, -5], [3, 1, 4, -4, 0, 0, 4, -4, -1, -3], [-3, 0, 3, 0, -3, 0, -2, -3, 3, 2], [2, -3, -3, 0, 0, 3, 3, -2, 0, -4], [1, -3, 3, -1, 0, -6, 2, -3, 2, -6], [0, -2, 0, 0, -2, 3, -6, 6, -3, 2], [-4, 3, -7, 6, -7, 3, -4, 0, -5, 5], [-2, 3, 3, 1, 2, 1, 3, 3, -2, 0], [3, -1, 1, -3, 0, -2, -1, 3, -1, -2], [0, -3, 2, -2, -2, -1, -2, -2, 2, -3], [2, 0, 3, -3, -2, -3, 3, 2, 3, -3], [-2, 0, 0, -10, 1, -5, 5, -1, 10, 0], [-8, 6, -6, 8, 0, 2, -8, 3, -8, 2], [-2, 2, 0, 6, -6, 5, -8, 5, -6, 6], [-4, 4, 1, -1, -4, 4, 0, -5, 8, -5], [-4, -9, 8, -8, 1, -8, 8, -9, -4, 0], [7, 0, 0, -6, 0, 0, 7, 0, 5, -5], [7, -2, 2, -7, 0, -10, 8, -14, 8, -10], [0, 2, 2, 2, 0, 0, -5, -4, 4, 5], [-6, 6, -2, 10, -6, 6, -10, 2, -6, 6], [0, 3, -6, 9, -4, -6, -4, 9, -6, 3], [-2, -12, 4, -9, 9, -4, 12, 2, 0, -5], [1, -1, 0, 0, -4, -4, -2, -4, -4, 0], [4, 3, 4, 0, -1, -7, -2, 2, 7, 1], [-13, 10, -11, 11, -10, 13, 0, 16, -18, 16], [0, 5, -8, -2, 5, 6, 5, -2, -8, 5], [-6, 1, 0, 6, -2, 2, -6, 0, -1, 6], [13, 0, 1, -5, 11, -9, 9, -11, 5, -1], [-8, -2, -1, -5, -5, 5, 5, 1, 2, 8], [2, 4, -3, -4, 4, 3, -4, -2, 0, 7], [1, -1, 1, -1, 0, -10, 2, -5, 2, -10], [-7, 4, -14, 4, -7, 4, -14, 4, -7, 0], [4, 3, -5, 5, -3, -4, 0, 1, -6, 1], [-8, 8, 0, 10, 1, 2, 5, 2, 1, 10], [0, -3, 7, 3, -2, 5, -2, 3, 7, -3], [-6, -5, 3, -5, -6, 0, 0, 2, -2, 0], [2, 0, -11, 10, -5, 12, -12, 5, -10, 11], [10, -5, 3, -5, 10, 0, -4, -6, 6, 4], [0, 0, 13, 4, -2, -6, -2, 4, 13, 0], [1, 6, 7, 1, 7, 6, 1, 0, 0, 0], [8, 4, 8, 0, 18, -14, 6, -6, 14, -18], [-2, 2, -2, 2, 0, 1, -1, 14, -1, 1], [-1, 0, 3, -3, 3, -3, 0, 1, 0, -7], [-4, -8, 14, -8, -4, 0, 2, 7, -7, -2], [1, 9, -8, -6, 6, 8, -9, -1, 0, 0], [-4, -14, -4, 0, 8, 1, -6, 6, -1, -8], [6, -20, 9, -12, 9, -12, 9, -20, 6, 0], [0, -9, -8, 8, 9, 0, 0, -2, 3, -2], [-3, 1, -3, 0, -6, 2, -6, 6, -2, 6], [-1, -6, -7, 9, -9, 7, 6, 1, 0, -3], [14, 0, 6, -9, -2, 1, -1, 2, 9, -6], [7, -5, 13, -5, 7, 0, -6, -4, 4, 6], [-10, 11, -11, 10, 0, 6, -18, 3, -18, 6], [14, -9, 14, 0, 8, -1, 6, -6, 1, -8], [18, -5, 10, -1, 4, -1, 10, -5, 18, 0], [0, -2, -3, 1, -3, -2, 0, 0, -14, 14], [4, -4, 0, -14, 10, 0, 4, 0, 10, -14], [9, -13, 19, -13, 9, 0, 4, -10, 10, -4], [-3, -8, -4, -8, -3, 0, -5, 10, -10, 5], [0, 5, -14, 12, -14, 5, -14, 12, -14, 5], [-13, -1, -7, -7, -7, -1, -13, 0, 3, -3], [-6, -3, 2, -2, 3, 6, 0, -4, 10, -4], [4, -11, 13, -5, 22, -5, 13, -11, 4, 0], [1, 6, 1, -4, 1, 6, 1, 0, 0, 0], [1, -4, 4, -1, 0, -15, -3, -11, -3, -15], [-12, 10, -8, -11, -5, 5, 11, 8, -10, 12], [-11, 0, -24, 19, -11, 23, -23, 11, -19, 24], [-5, 2, 4, -2, 2, -2, 2, -4, -2, 5], [0, 6, 3, -4, 7, 11, 7, -4, 3, 6], [11, -2, 4, -4, 2, -11, 0, -4, 17, -4], [12, -17, 7, -7, 17, -12, 0, -10, 11, -10], [3, 2, 3, 0, 7, -7, -9, 9, 7, -7], [0, 10, -8, 14, -19, 3, -19, 14, -8, 10], [-14, 6, 7, 3, -5, 5, -3, -7, -6, 14], [-8, 24, -5, 6, -3, 3, -6, 5, -24, 8], [-8, 6, -8, 0, -5, 6, -19, 19, -6, 5], [-14, 10, -12, 4, -12, 10, -14, 0, 3, -3], [7, -1, 1, -7, 0, -8, 8, 1, 8, -8], [-5, -11, 7, -11, -5, 0, 7, 1, -1, -7], [2, -6, -17, -6, 2, 0, -9, 0, 0, 9], [16, -9, 9, -16, 0, -15, 16, -17, 16, -15], [10, -14, 2, -9, 2, -14, 10, 0, 9, -9], [-6, -5, -28, 28, 5, 6, 0, 10, 1, 10], [3, 5, -5, -14, 3, -14, -5, 5, 3, 0], [-12, 8, -8, 12, 0, 9, -10, 4, -10, 9], [-4, 16, 0, 16, -4, 0, 8, -1, 1, -8], [-5, 0, 6, -2, -10, 4, -4, 10, 2, -6], [-11, 3, -15, -3, -3, 3, 3, 15, -3, 11], [22, -4, 6, 7, -7, -6, 4, -22, 0, -8], [17, -9, 17, 0, 4, -9, 3, -3, 9, -4], [4, -10, -4, 4, 10, -4, 0, -16, 1, -16], [-5, 13, -1, 2, 2, 2, -1, 13, -5, 0], [-1, 1, 0, -2, 5, -14, -2, -14, 5, -2], [-4, 12, -11, 7, -23, 23, -7, 11, -12, 4], [29, 0, 5, 0, 12, -10, 10, -12, 0, -5], [-17, -5, -4, 11, 0, 0, -11, 4, 5, 17], [2, 14, 0, 8, -8, 0, -14, -2, 0, -2], [-9, 14, -9, 0, -8, 4, 6, -6, -4, 8], [1, 15, -17, 7, -17, 15, 1, 0, -14, 14], [-4, -4, 0, -23, 0, -4, -4, 0, 1, -1], [-6, 6, 0, -5, 15, -17, -3, -17, 15, -5], [0, 11, 9, 11, 0, -2, 0, 11, 9, 11], [3, 0, 7, -8, -8, -13, 13, 8, 8, -7], [3, -1, 1, -3, 0, 19, -3, 6, -3, 19], [13, -21, 27, -27, 21, -13, 0, -14, 10, -14], [21, -16, 15, -9, 15, -16, 21, 0, 7, -7], [-18, 7, -6, 3, -3, 6, -7, 18, 0, -15], [-18, 12, -1, 14, -6, 6, -14, 1, -12, 18], [-6, -5, -2, 8, -4, 4, -8, 2, 5, 6], [0, 0, -3, 5, 8, -26, 8, 5, -3, 0], [-4, 10, -4, 0, -11, 18, -4, 4, -18, 11], [5, -8, 9, -10, 34, -10, 9, -8, 5, 0], [-18, 21, -6, -1, -3, -1, -6, 21, -18, 0], [-3, 3, 0, -5, -2, -14, 16, -14, -2, -5], [14, -6, -2, -16, 16, 2, 6, -14, 0, -5], [20, -12, 27, -12, 20, 0, -2, -15, 15, 2], [19, -8, 19, 0, 16, -7, 17, -17, 7, -16], [1, -11, 11, -11, 11, -1, 0, -11, -9, -11], [16, 11, -4, -2, -4, 11, 16, 0, -3, 3], [0, 19, -21, 12, -30, 20, -30, 12, -21, 19], [9, 5, -16, 5, 9, 0, 7, -2, 2, -7], [-6, 0, 20, 0, 13, -7, 7, -13, 0, -20], [-11, 0, -14, 11, -8, 25, -25, 8, -11, 14], [-7, 1, -1, 7, 0, -8, 0, -29, 0, -8], [-13, 9, 7, -2, -11, -2, 7, 9, -13, 0], [9, -15, 14, -17, 14, -15, 9, 0, 14, -14], [15, -3, 15, 0, 11, 0, 1, -1, 0, -11], [10, 1, -1, -10, 0, 8, -5, 9, -5, 8], [11, -9, 20, 4, -4, 4, -4, -20, 9, -11], [0, -5, 14, 14, 1, -8, 1, 14, 14, -5], [9, -8, 18, -9, 9, -18, 8, -9, 0, -15], [-5, 5, 0, -2, -9, 11, -6, 11, -9, -2], [-10, 10, 0, -6, 7, 6, 12, 6, 7, -6], [16, -12, 14, -29, 29, -14, 12, -16, 0, -35], [-26, 0, -15, 27, -22, 13, -13, 22, -27, 15], [-23, 12, -28, 5, -5, 28, -12, 23, 0, 13], [-17, -1, -6, -8, -6, -1, -17, 0, -8, 8], [-19, 8, 2, 6, 22, 6, 2, 8, -19, 0], [3, 20, -6, 6, -20, -3, 0, 3, -16, 3], [-7, 7, 0, 18, 0, -1, -4, -1, 0, 18], [-8, 0, 20, -8, 15, -5, 5, -15, 8, -20], [2, 32, 0, 32, 2, 0, -9, 7, -7, 9], [5, -4, 4, -5, 0, 0, -11, -5, -11, 0], [3, 17, -9, -18, -9, 17, 3, 0, 17, -17], [4, 7, -11, -14, -9, -14, -11, 7, 4, 0], [20, -16, 20, 0, 3, -26, 2, -2, 26, -3]]
