
# q-expansion of newform 1690.2.e.m, downloaded from the LMFDB on 12 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 = 1690
weight = 2
poly_data = [100, 0, 10, 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], 10], [[0, 0, 0, 1], 10]]

hecke_ring_character_values = [[677, [1, 0, 0, 0]], [171, [0, 0, 1, 0]]]
aps_data = [[1, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [-1, -1, -1, 0], [0, 0, 0, 0], [0, 1, -2, 1], [0, 1, 5, 1], [-6, 0, -6, 0], [-4, 2, -4, 0], [-4, 0, 0, -1], [3, 1, 3, 0], [-4, 2, -4, 0], [0, 0, 2, 0], [3, 0, 0, 0], [1, 0, 0, -1], [0, -2, 4, -2], [0, 3, 2, 3], [4, 2, 4, 0], [0, -2, -2, -2], [-6, 0, 0, 1], [-4, 0, 0, -3], [0, 0, 0, -3], [-1, 2, -1, 0], [0, -3, -2, -3], [8, -1, 8, 0], [7, 0, 0, -2], [-6, -3, -6, 0], [2, 0, 0, -4], [0, 4, 4, 4], [3, 2, 3, 0], [5, 0, 0, -5], [0, 3, -6, 3], [0, 1, -9, 1], [0, 2, -10, 2], [8, 0, 0, -1], [-9, 0, 0, -1], [0, 3, -8, 3], [7, -2, 7, 0], [0, -1, -1, -1], [-16, 2, -16, 0], [-6, 0, 0, 5], [0, -3, 0, -3], [-4, 0, -4, 0], [-9, 3, -9, 0], [0, -4, -14, -4], [-15, -1, -15, 0], [-3, -2, -3, 0], [0, 2, -4, 2], [-6, 0, 0, 1], [0, 0, 0, 3], [-8, 0, 0, -1], [0, -6, 7, -6], [0, -3, 3, -3], [0, 0, 0, 0], [-1, -4, -1, 0], [0, 3, 12, 3], [0, -2, 0, 0], [0, 3, 11, 3], [0, 0, 0, 6], [22, 0, 22, 0], [0, 1, 25, 1], [-22, 0, 0, 2], [6, 0, 0, -3], [-2, 0, 0, -2], [-3, 0, 0, -3], [0, -4, 12, -4], [8, 0, 0, 3], [0, 3, 12, 3], [8, -3, 8, 0], [2, -1, 2, 0], [6, 0, 0, 0], [8, 4, 8, 0], [0, 4, -18, 4], [-23, 5, -23, 0], [0, 0, -30, 0], [16, 0, 0, -1], [0, 3, -29, 3], [3, -6, 3, 0], [0, -4, 9, -4], [-2, 4, -2, 0], [0, 0, 0, 1], [6, -3, 6, 0], [0, -2, 0, -2], [12, 5, 12, 0], [14, 0, 0, 4], [0, 6, 21, 6], [-4, -9, -4, 0], [0, 3, -12, 3], [-6, 0, 0, -2], [20, 0, 0, 4], [10, 7, 10, 0], [0, 0, 1, 0], [-27, -3, -27, 0], [22, 0, 0, 0], [0, 8, 11, 8], [18, 6, 18, 0], [29, 0, 0, -2], [14, -8, 14, 0], [-30, 0, 0, 1], [6, 0, 0, -4], [19, 1, 19, 0], [0, 0, -24, 0], [15, -6, 15, 0], [-3, 0, 0, -7], [-4, 0, 0, -1], [-6, -6, -6, 0], [16, 0, 0, -4], [6, 0, 0, 12], [-9, 8, -9, 0], [0, 0, 5, 0], [11, -3, 11, 0], [0, -12, -6, -12], [1, 0, 0, 9], [0, -2, 8, -2], [0, 6, 9, 6], [0, 9, 16, 9], [-3, -6, -3, 0], [-27, -3, -27, 0], [0, -8, -2, -8], [-14, -1, -14, 0], [4, 6, 4, 0], [6, 0, 0, -6], [0, -2, -26, -2], [-15, 7, -15, 0], [-20, 0, 0, -1], [0, 3, -32, 3], [0, 0, 12, 0], [-27, 0, 0, 2], [1, 0, 0, -3], [-29, 5, -29, 0], [-18, 0, -18, 0], [16, -6, 16, 0], [-31, 1, -31, 0], [0, -8, -17, -8], [0, -2, 0, 0], [0, 11, -13, 11], [0, -7, 24, -7], [0, -2, -20, -2], [-4, 2, -4, 0], [3, 0, 0, 1], [-2, 1, -2, 0], [0, 0, 47, 0], [42, 0, 0, 3], [-32, 0, -32, 0], [0, -1, 20, -1], [-12, 0, 0, -14], [8, 0, 0, 7], [25, 0, 0, 7], [18, 0, 0, 0], [0, 2, 30, 2], [39, -6, 39, 0], [26, 0, 0, 3], [-5, 10, -5, 0], [-6, -7, -6, 0], [8, 0, 0, -8], [0, -2, -18, -2], [0, 0, -54, 0], [34, 0, 0, -2], [-16, 0, 0, -2], [14, 5, 14, 0], [0, -3, -12, -3], [-15, 0, 0, 10], [0, -11, 25, -11], [0, -6, 0, 0], [-43, 0, 0, 4], [14, -8, 14, 0], [0, 1, 15, 1], [33, 0, 0, -8], [24, 0, 0, -6], [8, 0, 0, -2], [0, -6, -2, -6], [0, -7, -40, -7], [0, -4, 6, -4], [-34, 0, 0, 0], [0, -10, 5, -10], [0, 10, 0, 0], [36, 0, 0, -3], [-26, 12, -26, 0], [34, 0, 34, 0], [6, 0, 0, 16], [29, 0, 0, -5], [-5, 0, 0, 1], [-28, 0, 0, 10], [-1, 20, -1, 0], [0, -9, 6, -9], [-37, 0, 0, 3], [-20, 0, 0, -12], [15, 10, 15, 0], [0, -13, -4, -13], [0, -5, -48, -5], [0, 14, -4, 14], [-27, 0, 0, -7], [26, -1, 26, 0], [0, 2, -34, 2], [-52, -1, -52, 0], [-22, 0, 0, 8], [0, 5, 13, 5], [-16, 0, 0, -2], [32, 0, 0, -8], [0, 10, 34, 10], [0, -1, -56, -1], [22, 8, 22, 0], [23, 0, 0, -12], [21, -15, 21, 0], [42, -6, 42, 0], [48, 0, 0, 4], [0, 0, 36, 0], [7, -8, 7, 0], [0, -3, 35, -3], [46, 3, 46, 0], [18, 0, 0, -3], [-55, 4, -55, 0], [0, 2, -34, 2], [0, -2, 46, -2], [-5, 0, 0, -12], [10, 0, 0, 4], [0, -12, 9, -12], [-27, 12, -27, 0], [-5, 0, 0, -19], [24, 11, 24, 0], [-22, 0, 0, -10], [-3, 0, 0, -12], [0, -4, -30, -4], [-18, 15, -18, 0], [4, 0, 0, 10], [8, 5, 8, 0], [0, 7, 52, 7], [0, 10, 18, 10], [-35, 0, 0, -1], [27, 5, 27, 0], [-9, -1, -9, 0], [38, -3, 38, 0], [19, 0, 0, 14], [-8, 0, 0, 16], [-17, 0, 0, 8], [0, 0, 4, 0], [-18, 12, -18, 0], [0, 5, 23, 5], [-4, -1, -4, 0], [10, 10, 10, 0], [-20, -12, -20, 0], [0, 16, -21, 16], [-70, 0, -70, 0], [0, 7, -8, 7], [-6, 0, 0, -3], [0, -10, -30, -10], [33, -3, 33, 0], [0, 9, 1, 9], [-9, 12, -9, 0], [-18, 16, -18, 0], [15, 6, 15, 0], [9, 0, 0, 12], [-28, -14, -28, 0], [-9, 0, 0, 15], [0, -21, -9, -21], [0, -13, -26, -13], [-38, 8, -38, 0], [-17, 0, 0, 11], [0, 1, 8, 1], [-39, 0, 0, -4], [-4, -1, -4, 0], [28, 0, 0, 0], [40, 6, 40, 0], [0, -8, -38, -8], [0, -20, -6, -20], [0, -6, -30, -6], [45, 0, 0, -6], [0, 1, 32, 1], [0, 16, -14, 16], [-10, 0, 0, -21], [24, 0, 0, 7], [34, -1, 34, 0], [0, 1, -6, 1], [0, 2, 52, 2], [-14, 8, -14, 0], [0, -12, -18, -12], [-6, 0, 0, 7], [0, 6, -8, 6], [0, -4, -40, -4], [9, -12, 9, 0], [0, -14, 0, 0], [6, 0, 0, 12], [12, 22, 12, 0], [22, 0, 0, -15], [12, 0, 0, -15], [-42, 0, 0, -10], [27, 0, 0, 15], [0, 0, 34, 0], [0, 6, 3, 6], [-24, 9, -24, 0], [0, -3, -24, -3], [-12, -12, -12, 0], [0, 5, 35, 5], [0, 2, 10, 2], [54, 0, 0, 6], [4, 0, 0, -11], [33, 3, 33, 0], [10, 4, 10, 0], [-16, 9, -16, 0], [0, 2, 16, 2], [-20, 0, 0, -10], [-20, 18, -20, 0]]
