
# q-expansion of newform 2304.2.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 = 2304
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 = [[0], [0], [0], [-4], [-4], [4], [2], [-4], [8], [8], [-4], [-4], [-6], [4], [8], [8], [12], [12], [12], [-8], [-6], [-4], [4], [6], [-2], [8], [4], [-4], [4], [14], [4], [-12], [-6], [20], [0], [12], [-4], [-20], [0], [0], [4], [4], [-16], [-18], [24], [4], [-4], [20], [4], [4], [22], [0], [-2], [12], [-2], [16], [-24], [-12], [20], [6], [4], [-24], [12], [-32], [22], [-8], [4], [2], [12], [28], [-34], [24], [-36], [-4], [4], [16], [-32], [-20], [34], [-26], [20], [20], [24], [14], [-12], [-36], [-14], [-10], [0], [20], [-12], [8], [-20], [12], [-4], [24], [24], [-6], [20], [-28], [-36], [-16], [-12], [10], [4], [18], [-4], [-18], [24], [-22], [-28], [-36], [-10], [-28], [44], [18], [-36], [-24], [-16], [36], [-4], [-34], [16], [44], [44], [-8], [20], [24], [28], [-28], [28], [-16], [44], [-12], [10], [46], [-8], [12], [16], [26], [-44], [24], [-4], [-4], [4], [-24], [44], [-22], [-28], [-16], [-20], [14], [28], [48], [4], [0], [-36], [-14], [-22], [32], [-12], [42], [-12], [-20], [-30], [-16], [-20], [44], [-18], [48], [-4], [12], [48], [10], [-4], [-58], [20], [32], [-44], [44], [-4], [4], [36], [-42], [48], [16], [-12], [28], [38], [0], [2], [-20], [-52], [48], [36], [54], [2], [-44], [30], [32], [0], [68], [52], [18], [-36], [24], [4], [68], [-10], [20], [-34], [48], [-12], [60], [-24], [54], [-52], [-30], [0], [-32], [20], [36], [50], [-28], [20], [-20], [6], [40], [4], [-20], [-28], [-68], [20], [-42], [4], [8], [14], [-64], [-4], [-24], [-28], [-60], [60], [-20], [-50], [-24], [28], [-12], [20], [-16], [-12], [62], [-24], [-54], [-24], [-28], [-60], [-76], [-72], [38], [-44], [36], [-60], [44], [18], [-20], [48], [6], [4], [-32], [28], [44], [-38], [60], [50], [36], [-36], [44], [-38], [84], [-24], [-76], [24], [-36], [36], [16], [-34], [48], [60], [30], [-16], [52], [-26], [60], [-36], [-32], [-44], [-24], [-36], [-4], [70], [-48], [-68], [-60], [20], [14], [-68], [4], [72], [-20], [48], [-88], [-78], [28], [-32], [-86], [36], [-40], [-14], [-50], [60], [-10], [72], [-68], [58], [-82], [12], [-12], [64], [0], [-12], [0], [-44], [20], [-12], [12], [-52], [-66], [26], [20], [-68], [42], [40], [-28], [-72], [-60], [36], [-92], [88], [56], [-52], [-74], [-40], [36], [60], [-90], [64], [-84], [-62], [-96], [92], [22], [-48], [-84], [28], [58], [72], [44], [70], [-12], [52], [64], [-24], [-4], [4], [4], [-48], [34], [34], [-42], [-64], [26], [-20], [30], [44], [0], [12], [-76], [52], [-72], [-62], [-16], [60], [-52], [48], [-90], [52], [26], [-28], [32], [92], [-78], [-44], [42], [40], [-100], [-44], [2], [12], [20], [34], [48], [-52], [12], [26], [104], [-48], [-20], [-30], [8], [-32], [28], [-80], [-52], [-58], [-24], [-76], [6], [52], [8]]
