
# q-expansion of newform 2394.2.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 = 2394
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], [4], [-1], [6], [-4], [4], [-1], [-2], [-2], [4], [2], [-6], [0], [8], [14], [4], [-10], [10], [4], [-14], [2], [0], [-6], [0], [-4], [-4], [-12], [-6], [-18], [-2], [20], [18], [20], [18], [-14], [-22], [-24], [-8], [-10], [-4], [16], [14], [26], [-22], [8], [10], [-16], [20], [-2], [-6], [22], [28], [20], [6], [2], [6], [8], [-10], [-10], [-4], [-18], [-8], [4], [-14], [18], [-10], [6], [-6], [-30], [-4], [34], [-32], [14], [-34], [24], [-34], [10], [-30], [24], [12], [-2], [-12], [0], [20], [-18], [30], [-10], [-36], [0], [8], [-4], [-34], [-30], [24], [-28], [-42], [-38], [16], [-14], [10], [10], [-44], [10], [-4], [-38], [0], [-16], [28], [-8], [-24], [-34], [2], [12], [12], [22], [-36], [-12], [14], [36], [0], [22], [-6], [-8], [-4], [6], [10], [-12], [-32], [-30], [4], [32], [22], [46], [-16], [18], [-42], [32], [-2], [-10], [-40], [-30], [24], [28], [-44], [-48], [46], [18], [4], [-48], [-54], [-12], [0], [24], [-26], [-52], [0], [52], [38], [-2], [-30], [22], [4], [12], [18], [-16], [-30], [-14], [-38], [-36], [24], [32], [42], [2], [24], [-40], [-28], [38], [-40], [14], [-44], [12], [-14], [-34], [34], [-28], [38], [-44], [22], [-40], [12], [42], [34], [-54], [58], [-6], [10], [22], [44], [-36], [-42], [40], [-32], [4], [32], [-4], [-24], [24], [6], [48], [14], [-28], [-24], [20], [24], [52], [-32], [-6], [-12], [6], [-30], [32], [18], [24], [-12], [22], [54], [-64], [-68], [-54], [18], [-24], [-2], [-6], [28], [24], [-62], [-14], [30], [-24], [72], [-36], [40], [6], [22], [28], [40], [-52], [46], [-54], [50], [-40], [54], [-20], [16], [-50], [10], [-4], [42], [-18], [14], [56], [10], [-76], [24], [44], [-10], [-40], [-38], [78], [0], [-58], [18], [44], [0], [-48], [-18], [22], [10], [-16], [24], [-40], [52], [28], [-52], [18], [-34], [54], [-64], [-58], [10], [-8], [86], [4], [-18], [-48], [20], [68], [-4], [-60], [54], [-18], [28], [80], [8], [46], [8], [-4], [-34], [-46], [8], [30], [-26], [50], [-90], [-32], [-72], [-52], [-66], [-54], [-52], [70], [66], [-6], [-66], [52], [-58], [24], [4], [-34], [18], [82], [-64], [4], [40], [48], [-30], [-38], [-42], [48], [14], [58], [18], [22], [-46], [84], [-22], [-38], [-16], [12], [36], [34], [90], [-24], [-50], [-38], [48], [60], [-18], [60], [42], [56], [60], [-64], [4], [-10], [8], [-42], [74], [-74], [-68], [-6], [-22], [32], [10], [-36], [-34], [-6], [72], [28], [66], [70], [20], [6], [28], [-48], [-36], [-10], [16], [32], [-18], [2], [2], [68], [56], [38], [60], [-18], [38], [30], [-60], [-10], [-38], [-58], [-10], [-48], [46], [-54], [-40], [64], [56], [70], [-60], [56], [22], [96], [48], [32], [-28], [-40], [50], [48], [82], [84], [96], [46], [-44], [-6], [-28], [0]]
