
# q-expansion of newform 9075.2.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 = 9075
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], [1], [0], [-4], [0], [-2], [-2], [8], [-4], [4], [-8], [-8], [12], [8], [4], [4], [-8], [0], [4], [-12], [-2], [8], [-4], [6], [8], [-4], [12], [-12], [-8], [-12], [4], [8], [4], [16], [20], [0], [-8], [20], [-24], [-14], [0], [-10], [12], [6], [-6], [-16], [-16], [4], [12], [-22], [-6], [-24], [8], [0], [-4], [0], [-26], [16], [-10], [12], [16], [6], [-24], [28], [-16], [-20], [4], [-26], [-12], [16], [-12], [24], [-36], [-22], [-4], [36], [18], [-8], [-2], [-40], [8], [-10], [-24], [-16], [0], [-12], [-30], [2], [-12], [-28], [-12], [-24], [-20], [0], [-12], [-24], [26], [22], [-24], [8], [8], [-14], [-36], [44], [-40], [0], [-12], [14], [-12], [-32], [-20], [42], [28], [-36], [8], [-2], [-12], [28], [-28], [24], [6], [38], [38], [-4], [-20], [12], [26], [36], [-28], [-14], [16], [24], [-16], [0], [-36], [16], [36], [-32], [-12], [-44], [-8], [20], [-20], [20], [-14], [20], [-6], [38], [4], [36], [2], [-30], [4], [16], [52], [-36], [0], [-18], [-14], [-4], [-28], [-54], [-4], [16], [-36], [4], [0], [42], [-24], [-36], [-24], [-14], [-48], [34], [-40], [-22], [8], [-18], [-28], [-16], [20], [48], [-48], [-38], [12], [50], [-30], [28], [-40], [-40], [-40], [44], [52], [-42], [28], [12], [16], [-16], [-30], [-48], [-20], [40], [16], [24], [-40], [-12], [-56], [36], [-60], [-28], [-22], [50], [12], [-12], [8], [-42], [-44], [20], [12], [36], [8], [24], [2], [44], [36], [-32], [-12], [-20], [44], [-40], [32], [64], [-8], [4], [-20], [-24], [-2], [-42], [48], [-36], [52], [0], [60], [-2], [-14], [-8], [44], [-40], [-16], [-24], [-18], [4], [44], [-6], [50], [8], [-54], [-8], [-52], [34], [52], [28], [40], [-34], [60], [-68], [58], [54], [16], [-26], [-14], [-28], [-56], [32], [26], [20], [52], [80], [64], [8], [-40], [-32], [48], [-64], [0], [60], [-8], [86], [24], [36], [6], [-36], [-70], [72], [-66], [20], [-8], [12], [8], [16], [-14], [-78], [64], [-20], [-28], [-88], [84], [30], [36], [10], [-24], [62], [-12], [28], [24], [-56], [0], [32], [40], [12], [0], [32], [12], [-44], [42], [-34], [-36], [84], [16], [-6], [-8], [12], [-40], [44], [48], [-68], [-78], [-78], [-26], [-52], [40], [-12], [36], [-64], [84], [80], [-6], [28], [-40], [-68], [40], [88], [-38], [12], [48], [22], [-20], [-72], [2], [-12], [-26], [36], [-60], [80], [-68], [-16], [82], [44], [-16], [-40], [100], [-40], [-4], [40], [-64], [-24], [0], [-38], [-20], [82], [74], [-52], [44], [-66], [-56], [84], [-48], [16], [-32], [28], [62], [-60], [-48], [-44], [60], [18], [-80], [-26], [-52], [84], [16], [20], [-76], [92], [36], [64], [16], [-60], [60], [-72], [10], [22], [4], [-8], [-98], [-38], [0], [4], [-84], [64], [70], [38], [60], [-16], [32], [-4], [84], [-12], [60], [-40]]
