
# q-expansion of newform 6930.2.a.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 = 6930
weight = 2
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[-1], [0], [-1], [1], [1], [2], [-6], [2], [6], [0], [8], [8], [0], [-4], [0], [-12], [-12], [-10], [-4], [12], [14], [-10], [0], [18], [8], [18], [-4], [-12], [8], [6], [-16], [18], [6], [-10], [0], [-10], [2], [-4], [0], [-6], [12], [-10], [0], [14], [-18], [8], [20], [-4], [12], [26], [6], [-18], [8], [12], [24], [24], [-18], [20], [-10], [-6], [-4], [-18], [32], [-24], [-16], [0], [20], [-22], [12], [2], [24], [18], [20], [-22], [-28], [12], [6], [2], [18], [8], [12], [14], [18], [-4], [20], [-12], [18], [26], [30], [2], [-18], [0], [-22], [-12], [20], [0], [42], [-6], [-16], [20], [44], [6], [36], [6], [32], [-16], [-18], [-30], [36], [-28], [-16], [26], [-30], [44], [-16], [30], [14], [-12], [36], [-24], [-10], [2], [-6], [24], [8], [36], [-46], [0], [-16], [14], [20], [24], [44], [-52], [48], [20], [42], [-4], [-42], [-30], [14], [-36], [2], [-36], [2], [48], [14], [-30], [-28], [30], [-22], [30], [-16], [0], [32], [-12], [-34], [18], [2], [18], [48], [-6], [-16], [-60], [-42], [-12], [56], [-10], [-34], [54], [24], [38], [-18], [26], [56], [6], [-28], [-54], [-16], [38], [-10], [-30], [-52], [-42], [18], [-42], [-46], [-10], [-22], [-24], [-52], [-60], [-52], [66], [-36], [-36], [38], [-4], [-18], [24], [-36], [-40], [62], [-40], [24], [-54], [-16], [-60], [6], [8], [-58], [54], [50], [-42], [0], [38], [-52], [24], [-18], [36], [32], [44], [66], [38], [-36], [-28], [60], [-12], [-16], [0], [32], [-34], [62], [-42], [26], [-24], [14], [-54], [-36], [24], [24], [14], [32], [-10], [-66], [60], [20], [-36], [-76], [24], [-58], [-60], [42], [26], [-18], [-60], [2], [-64], [66], [74], [8], [36], [-58], [-34], [-48], [-28], [30], [-6], [68], [42], [-58], [68], [-28], [2], [-46], [-16], [-60], [32], [26], [78], [0], [-4], [-72], [-58], [44], [24], [-58], [66], [-64], [0], [-6], [-6], [6], [-66], [-34], [-18], [8], [-66], [30], [8], [26], [30], [-46], [48], [-28], [26], [48], [-22], [60], [2], [72], [18], [-90], [32], [-12], [-16], [-72], [-66], [44], [-30], [-70], [74], [42], [-46], [42], [-34], [-28], [26], [48], [-66], [44], [48], [68], [0], [-64], [-78], [-70], [54], [14], [56], [-28], [-18], [-18], [-64], [0], [-60], [26], [-52], [12], [-78], [50], [14], [-90], [80], [32], [-30], [-48], [-6], [-30], [18], [-10], [-72], [18], [-36], [-10], [-10], [-18], [-76], [-10], [36], [-16], [-84], [48], [92], [56], [24], [-78], [50], [36], [26], [30], [-30], [-28], [30], [26], [-48], [56], [-58], [-100], [36], [14], [-30], [-36], [2], [96], [14], [44], [-30], [20], [12], [-58], [42], [-40], [-60], [-66], [20], [8], [18], [-34], [60], [-46], [90], [36], [-16], [50], [42], [-54], [-28], [-48], [-24], [-30], [2], [6], [42], [56], [-78], [84], [-30], [-4], [24], [-22], [-48], [-76], [-96], [-34], [54], [86], [38], [80], [32], [54], [30], [44], [66], [44], [-54], [44], [48], [74], [-10], [92], [-24], [84], [-60], [-70], [-78], [-64], [54], [86], [-90], [-52], [-52], [-48], [-10], [-76], [14], [-10], [6], [-66], [68], [-100], [-72], [72], [74], [-12], [2], [54], [-40], [0], [-12], [32], [48]]
