
# q-expansion of newform 2013.2.a.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 = 2013
weight = 2
poly_data = [-11, -16, 118, 98, -404, -141, 534, 13, -266, 32, 55, -11, -4, 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, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-3, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[2, -4, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-47, 14, 228, -495, -255, 770, 46, -374, 29, 73, -11, -5, 1], 17], [[8, -184, 201, 441, -695, -320, 506, 108, -150, -17, 20, 1, -1], 17], [[66, 176, -386, -142, 749, -516, -292, 411, -8, -101, 18, 8, -2], 17], [[186, 164, -992, -788, 1350, 1126, -764, -540, 221, 104, -33, -7, 2], 17], [[-21, -427, 129, 1740, -411, -2043, 591, 960, -312, -194, 67, 14, -5], 17], [[267, 122, -1696, 66, 2700, -833, -1384, 586, 263, -140, -11, 11, -1], 17], [[-12, -305, -463, 1675, 1362, -2912, -489, 1604, -91, -348, 58, 26, -6], 17], [[-141, 152, 905, -1556, -1532, 2895, 506, -1604, 91, 348, -58, -26, 6], 17], [[-233, 149, 1747, -1551, -3112, 3278, 1321, -1924, -51, 432, -56, -33, 7], 17]]

hecke_ring_character_values = None
aps_data = [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1], [1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, 1, 0], [0, 0, 0, -1, 0, 1, 1, 1, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1], [0, -1, 0, -1, -1, 1, 1, 1, -1, -2, 1, -1, 0], [1, 0, -1, 0, -1, -1, -2, 0, 0, 1, 0, 0, 0], [2, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1], [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0], [1, -1, -1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0], [2, -1, 1, 0, -2, 0, -1, -1, -1, -2, 0, -1, -2], [3, -2, 0, -1, 0, 0, -1, -2, 0, -2, -2, 0, -2], [2, 0, 1, -2, 2, 2, 3, 1, 0, 0, 1, -2, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 2, -1, -1, -1, 0, 2, 2, 1, 1, 0], [5, -1, -1, -1, 1, -2, -1, -2, 0, 0, -2, 0, -4], [-4, 1, 0, -1, 2, 1, 4, 0, 2, 0, 1, -2, 1], [1, -1, 1, 2, 0, 0, -2, -1, -1, 0, 0, 1, -2], [3, -1, 1, 2, -2, 0, 1, 3, -1, 0, 1, 2, -1], [2, 1, 0, 1, 3, -1, 1, -2, 2, 2, -2, -1, 2], [1, -1, -1, 0, 1, -2, 0, 0, 0, 3, 1, 3, 2], [0, 0, 2, 1, -2, 0, 0, -2, 2, -3, -2, -2, -3], [1, 1, 1, 1, -3, 0, -2, -1, -1, 0, 0, -2, -1], [0, 1, 0, -1, 2, -1, 2, -2, 2, 1, 1, -2, 1], [-3, -2, -2, 1, 1, -1, 0, -3, 1, 1, 0, -1, 1], [2, -3, 2, 0, 0, 2, 1, 4, -2, -1, 2, 0, 1], [-3, 1, -3, -2, 2, -1, -2, -3, 1, 1, -2, -2, 1], [6, -1, 1, 3, 1, -1, 2, 1, -1, 0, -1, 3, 1], [-1, -4, -4, 0, -3, 2, 0, 0, 0, -1, -1, 0, 1], [-4, -4, 0, 1, -3, 3, 1, 0, 0, -1, 3, -3, 0], [-2, 1, -1, 3, -2, 0, -2, -1, 0, 2, 1, 0, 3], [1, -1, 0, -4, 4, 1, 1, 1, -1, 1, 2, -1, 0], [1, -2, 0, -1, 3, -1, 2, 3, -1, -2, 0, -1, 1], [2, 3, -3, -1, 1, -3, 0, -1, 2, 3, -1, 1, 3], [2, 0, -3, -1, 5, -3, -2, 0, -1, 1, -3, 1, 1], [2, 5, 0, 0, -2, 0, 2, 1, 1, 4, 0, 2, 2], [5, -3, -1, 3, -2, -3, -5, -1, 0, 1, -1, 3, -2], [-2, 1, 3, -1, 0, 0, 1, 2, 2, -1, 3, -2, 1], [4, 1, 1, 1, -1, 2, 1, 4, 0, 1, 3, 3, 3], [-2, -3, 5, -1, -4, 5, 4, 4, -1, -4, 2, -1, -1], [1, -1, 1, 1, 3, 0, 3, -2, 1, -1, -2, -3, -2], [4, 0, 2, 0, -3, 3, 0, -1, 0, -3, -3, -1, -4], [-1, 3, 3, 2, -1, -4, -4, -1, 0, 4, 0, 2, 0], [5, -2, 3, 4, 1, 0, 2, 2, 1, 4, 2, 2, 3], [5, -1, -4, -1, 2, 0, 1, -2, 3, 2, -1, 0, 2], [3, -1, 1, 0, 0, 1, -2, -3, -1, 0, 0, -1, 3], [0, -3, 4, 0, 1, 2, 3, 2, 0, -1, 0, 1, -1], [5, 5, 3, -1, 1, 2, 3, 5, -3, -1, 1, 1, 0], [-4, -2, 1, 1, 1, -3, -1, -4, 1, 0, 0, 0, -2], [-1, -3, 1, 0, -1, 3, -1, -3, 2, 2, 1, -2, 3], [-4, 4, 2, -3, -1, 1, 3, 1, 0, -3, 0, -3, -1], [2, -4, 1, 1, 4, -1, -3, -5, 2, 4, 1, -3, -1], [5, -4, -5, 1, 0, 0, -3, 4, -3, 3, 1, 5, 1], [0, 3, -1, 0, 3, -2, 1, -3, 1, 4, -4, 4, -1], [-4, 4, 3, -2, 0, 5, 4, 0, 1, -1, 1, -2, 4], [3, 4, -2, -1, 6, -7, -4, -4, -1, 6, -3, 1, -1], [1, 5, -2, -1, 1, -2, -2, -3, 0, 1, -5, -2, -1], [1, -2, -1, 3, -1, -4, -4, 1, -3, 4, 3, 2, 1], [4, -2, 2, 1, 0, 2, 2, 5, -3, -1, -1, 1, 3], [-2, -2, -3, 1, -4, 1, -4, -1, -1, -2, 0, -1, -1], [-1, 4, 2, 3, -2, -1, 0, -2, 1, 3, -1, 1, -5], [1, 4, 0, 0, 4, -4, 3, 3, 1, 6, 1, 4, 1], [5, 1, 4, 4, 0, -1, -3, 0, -1, 4, 5, 2, 3], [-11, 4, -5, -4, 5, -3, 0, -5, 1, 2, -3, -4, 0], [2, 3, 4, 1, 0, 0, 0, 2, 0, 1, 1, 2, -4], [0, -1, 0, 3, 4, -1, 1, -5, 2, -3, -3, -5, -1], [-2, 0, -3, -1, -1, -1, 3, 1, -2, -2, 1, 1, 2], [6, 0, -4, -2, 8, -3, 1, 0, -2, -1, -3, -1, 0], [3, 8, -2, 1, 3, -2, -2, -6, 1, 4, -3, -3, -3], [-1, -1, 0, 0, 4, -1, 3, 1, 2, 1, -2, -3, 0], [-7, 3, -1, 0, 3, 1, 8, 1, 1, 0, 2, 0, 1], [0, 0, 0, -1, -1, 4, 4, 7, -4, 0, 1, 4, 7], [8, 5, -6, -1, 0, -4, -3, 1, -2, 2, -3, 7, 1], [6, 3, -3, -1, 0, -2, -3, 4, -2, 4, -3, 7, 4], [2, -1, 0, 5, -2, 1, 2, -2, 1, 1, 1, -2, 2], [-4, 1, 1, 0, -1, -1, -1, -6, 2, 0, 1, -2, -4], [7, 0, -2, -2, -5, -2, -10, -3, -2, -4, -2, 1, -1], [-8, -3, -3, -3, -2, 3, -1, -1, 0, -6, -2, -4, 1], [-4, 6, 1, -3, 1, -3, -2, 2, 1, 3, 1, 0, 2], [-3, 3, -3, 5, -4, 0, -1, 4, -1, 4, 0, 4, 3], [-1, -4, 0, 2, 4, -4, -3, -3, 2, -2, -3, 0, -2], [4, -5, 1, -2, 1, -1, 1, -3, 2, -4, 2, -2, 3], [-1, -3, 1, 0, -4, 5, 6, 3, -3, -6, -2, -1, -5], [-8, -1, 4, -5, -8, 8, 7, 7, -6, -11, 1, -3, -2], [-5, -1, 4, 1, -6, 4, 2, -3, 2, -1, 3, -3, -1], [2, -5, 5, -2, 5, 0, 5, 1, 4, -2, 2, -1, -3], [4, -2, 2, 1, -2, 2, 3, 1, 3, -4, 0, -3, -3], [-8, 7, -2, -1, 0, 3, -2, -4, 1, -2, -3, -3, 0], [-4, -1, -1, 1, -1, 2, 5, -1, 1, 1, 0, -1, -2], [0, 10, -4, -4, 8, -3, 4, -1, 2, 7, -1, 1, 5], [-3, 0, -7, -5, 8, 1, 0, -6, 0, 2, 2, -4, 2], [-6, 1, 7, -1, -8, 3, 0, 7, -4, -8, 3, -3, -2], [6, 4, -4, 1, 10, -5, 0, -9, 2, 6, -5, -3, -1], [2, 4, 2, -1, -12, 2, -5, 0, -1, -10, 1, 3, -5], [-8, 2, 6, 1, 0, 2, 7, 1, 2, -1, 4, -2, 4], [-12, 0, 2, 0, -9, 6, 0, 5, 1, -3, 5, -1, -1], [-1, -2, -4, -1, -6, 2, -2, 4, -2, 5, 5, 6, 5], [-4, 10, 1, -1, -5, 0, -4, -7, 6, 0, -2, -3, -5], [0, 2, 0, 0, -2, 2, 2, 6, -4, -2, 0, -1, -1], [-1, 5, -1, 0, 2, 3, 2, 4, 4, 6, 5, -1, 7], [0, -5, -5, -1, 0, -2, -1, 2, 0, -1, -5, 3, 1], [-1, 6, 5, 2, 3, -3, 1, -3, 1, 6, 0, 2, 0], [8, 10, -1, 2, 5, 0, 2, 0, -3, 4, -1, 3, 5], [5, -7, 7, -2, -3, 3, 1, 6, -5, 0, 6, 2, -2], [4, 1, -5, 4, -1, -1, -2, -1, 0, 8, -3, 5, 4], [-4, 5, 5, -1, 3, -1, 4, 2, 0, -2, -2, 3, -4], [-8, -4, 8, -3, -6, 9, 5, 8, -4, -9, 7, -2, 3], [5, -2, -1, 0, 1, -1, -2, -1, 2, -8, -1, 1, -3], [-2, 5, -3, 0, -5, 2, -1, -3, 2, 2, -2, -2, -1], [2, 2, -4, -2, 4, -2, 7, -1, 1, 1, -4, 3, 4], [-7, -2, -3, -2, -6, 1, -1, -4, 6, 1, 5, -3, 4], [10, -1, 6, 0, -1, 6, 2, 8, -1, 1, 5, 2, 1], [18, 1, -2, -1, 3, -7, -2, 2, -4, 8, 3, 5, 2], [-2, 3, 3, 3, -7, -3, -9, 2, -1, 0, 1, 2, -1], [4, 6, 5, -1, -1, 1, 5, 4, -1, -2, 2, -2, -3], [8, -2, -1, -3, -1, -3, -3, 0, -1, -1, 0, 6, -4], [-1, 5, 6, -1, -4, 3, 2, 2, 0, -1, 0, -3, -2], [3, -7, 3, 4, -4, 4, -8, 0, 1, 3, 1, 1, 1], [-8, -3, -6, -1, 3, -6, -4, -5, 3, 6, 2, 0, 4], [6, 0, -1, 6, -1, -5, -8, -1, -2, -1, -4, 4, 0], [-1, 1, 3, 1, -6, 3, 1, 7, -5, -2, 1, 1, 2], [0, 8, -4, -2, 14, -7, 5, -3, 3, 5, -2, 3, 3], [-1, -4, 7, 1, -10, 3, 3, 8, -4, -13, 3, 2, -5], [5, 1, -1, -4, 0, -2, -5, 1, -4, -2, 1, 4, -3], [3, 3, 5, -5, -2, 3, -2, -3, 1, -6, 2, -4, -5], [2, 0, 3, 2, -10, 5, 2, 7, -4, -4, 0, 4, -3], [-5, -2, -5, 0, 0, -6, 1, -3, 5, 2, -2, -2, -8], [-5, 2, 4, -4, 5, 2, 1, 1, -6, -7, 1, -4, -2], [0, 1, -1, -2, 4, -2, -3, -7, -3, -4, -8, -7, -4], [3, 5, 2, 1, 2, -4, -3, -4, 4, 1, -1, -4, -4], [1, 4, 3, 2, 6, 2, -1, 1, 4, 3, -4, -3, 5], [1, -11, 8, 0, -4, 6, 4, 5, 2, -4, 3, -5, -3], [8, 2, 0, 1, -2, 3, 2, 7, -1, -3, -1, 6, -5], [10, -9, 0, 5, 0, -6, -6, 1, -3, 1, -2, 8, -5], [1, 3, 5, -6, -6, 5, 6, 4, -4, -8, 1, -3, -4], [1, -1, -1, 0, -1, -3, 5, 5, 4, 4, 3, 3, 3], [5, -2, -7, -5, 2, 1, 1, 6, -6, -2, -1, 4, 3], [6, 2, 12, 4, -6, 8, 3, 6, -6, 2, 7, -1, -1], [6, -3, -8, 5, 1, -8, -8, -3, -1, -1, -10, 4, -7], [2, -1, 5, 0, 2, 3, 6, 1, 3, 2, -1, -2, 4], [5, 5, -8, 2, 0, -7, -1, 2, -1, -1, -3, 3, 3], [9, -1, -5, 4, -4, -2, -3, -8, 1, 4, -2, 3, -3], [1, -10, -4, 0, 9, 0, 5, 1, -3, 1, -1, -4, 10], [5, 1, -8, -6, -3, 3, -2, -2, 0, 2, 0, -4, -2], [-5, -6, 1, 3, 3, 4, 10, 5, 3, -5, 3, -4, 4], [5, -12, -2, -5, -7, 3, 1, 7, -3, -5, 5, 6, 2], [-7, -5, -6, 0, -5, 1, -8, 1, -4, -4, 3, 5, 3], [-8, 9, 0, 4, 2, 1, 3, 1, 0, 4, -1, -1, 2], [0, -4, 3, 0, -3, 3, 13, 4, 2, -6, 4, -3, -1], [10, 10, -9, -3, 1, -4, -10, -8, 5, 8, -3, 3, 4], [10, 3, 6, 4, 4, 0, 2, -1, -1, 1, 0, -2, -5], [3, 3, 2, 3, 0, -8, -9, -3, 2, -2, -10, 3, -8], [-7, 7, -3, -3, 12, -2, 10, 7, -3, 8, 4, 1, 6], [2, -5, -5, -3, 5, -4, 1, -8, 7, 7, 3, -4, -2], [2, 1, -4, -1, 8, -3, -2, -6, -3, 4, -1, -7, -4], [-4, 4, 6, 11, -5, 1, 1, 1, 1, 3, 6, 5, 3], [6, 1, -6, 2, -4, 3, 0, 6, -3, 3, 3, 5, 1], [2, 11, 6, -1, 6, -2, 3, -13, 7, 2, -1, -8, 1], [0, -9, 5, 3, -2, 4, 2, 5, 4, -5, -3, 0, 3], [-10, 9, 4, -2, -9, 10, 4, -2, 5, -5, 5, -7, 3], [1, -4, 1, 2, -8, 8, -2, 7, -3, -4, 3, -4, 2], [4, -1, -2, 7, 0, -7, -5, -4, 6, 2, -6, 4, -5], [-1, 16, 1, 2, 6, -4, -4, -6, -1, 9, 2, 0, -3], [9, -3, -7, -2, 0, -5, -5, 3, -9, -2, -1, 7, -3], [13, -2, -6, -2, -7, 2, -7, 8, -6, 0, 2, 1, -2], [-4, -2, -1, -10, 2, 6, 2, 2, -7, -8, -3, -1, -3], [7, 6, -7, 4, -3, -3, -1, -6, 4, 3, -5, 4, -1], [-16, -1, -5, -3, -3, 7, 4, 3, -2, -8, -3, -3, 1], [3, 4, -4, 8, 1, -10, -10, -3, 6, 10, -4, 8, 4], [-5, -8, -2, 5, 1, -1, 1, 4, 4, 7, 6, 5, 8], [6, -12, -8, -5, -8, 0, 0, 6, -3, -6, -2, 6, 2], [20, -1, 3, 1, -2, 4, -3, 6, 3, 5, -1, 1, 6], [-16, -10, -5, 3, -8, -2, -4, -1, 3, 0, 4, 3, 1], [1, -4, -8, -6, 7, -7, -2, -1, -3, -2, -6, -3, 1], [9, 0, 0, -2, -6, 0, -13, -6, -7, -9, -6, -4, -4], [-10, -9, -6, -4, 7, 5, 0, -2, -2, -2, 1, -1, 2], [7, 2, -3, -3, -3, 2, -2, -1, 0, 4, 1, 2, 5], [1, 10, 1, -1, 3, 2, 10, 0, 6, 8, 2, -4, 3], [-4, 6, 8, 1, -4, 10, 7, 5, 5, 2, 4, -4, 6], [-4, -3, 4, 2, -5, 0, -3, 1, -5, 5, 3, -5, 1], [5, -4, -8, 1, 4, -5, -5, -5, 1, 3, -1, -2, -5], [-5, 8, -5, 9, -5, -2, 1, 2, 2, 3, -5, 8, 2], [-12, 6, 5, 8, -1, 0, 6, 0, 5, 3, -3, -1, -1], [20, 1, -9, -1, 3, -11, -7, -11, 2, 7, -2, 4, -2], [-11, -1, 2, -5, 1, 5, 8, 0, 9, 2, 8, -1, -1], [15, -5, 7, 3, -5, 0, -1, 6, -2, 2, 2, 2, -6], [-6, 2, 1, -4, 8, 3, 5, 5, 5, -3, 3, -3, 4], [7, -8, 1, -10, -10, 2, -4, 0, -5, -11, -2, -1, -6], [0, 2, 6, -3, -10, 7, 4, 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