
# q-expansion of newform 3344.2.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 = 3344
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], [-2], [4], [1], [2], [2], [-1], [0], [2], [0], [10], [-2], [4], [-8], [2], [-8], [10], [8], [0], [2], [-8], [4], [-6], [2], [10], [8], [12], [-14], [10], [8], [4], [10], [-4], [10], [-8], [14], [12], [-16], [2], [24], [10], [-16], [22], [-6], [24], [-12], [24], [-12], [14], [10], [-20], [-2], [4], [18], [-4], [-30], [-12], [10], [6], [4], [-6], [-4], [0], [-6], [18], [24], [22], [28], [-14], [34], [-36], [16], [10], [-8], [-32], [-34], [-2], [34], [-26], [-12], [-30], [8], [-14], [-8], [-28], [-6], [2], [2], [-16], [-28], [4], [0], [4], [-12], [-36], [-30], [-6], [20], [-38], [-4], [18], [-4], [30], [-12], [2], [-12], [-14], [24], [22], [8], [34], [-6], [-4], [-8], [-6], [4], [8], [38], [-20], [26], [14], [-6], [0], [-4], [-6], [-34], [-24], [-16], [42], [36], [-24], [-24], [-18], [42], [-6], [-6], [4], [-6], [18], [-28], [-22], [8], [-36], [18], [-40], [-14], [-10], [52], [-24], [50], [18], [20], [-24], [-56], [32], [-44], [-30], [-30], [-30], [-36], [30], [-4], [0], [2], [-16], [-8], [42], [-18], [-2], [-28], [10], [-60], [-38], [32], [-54], [-52], [-50], [-64], [-30], [16], [-4], [26], [-18], [32], [30], [2], [64], [14], [-4], [-54], [4], [-32], [-14], [60], [-6], [42], [-34], [2], [-36], [50], [40], [50], [54], [60], [-34], [-56], [12], [66], [40], [10], [-26], [-48], [40], [40], [-46], [68], [-42], [-24], [62], [26], [-56], [50], [16], [20], [-38], [34], [8], [48], [60], [38], [60], [32], [70], [20], [12], [-30], [-70], [20], [24], [-24], [-36], [-56], [42], [46], [44], [72], [24], [52], [20], [66], [50], [24], [-38], [-54], [-60], [-10], [-20], [-30], [-38], [16], [68], [-46], [-30], [18], [48], [2], [26], [-36], [-30], [-38], [56], [-70], [-12], [-66], [56], [-28], [-14], [70], [-28], [0], [-40], [-12], [-38], [-68], [48], [-30], [-6], [0], [-66], [78], [-12], [6], [12], [-46], [-14], [80], [70], [44], [-68], [-70], [18], [-28], [-48], [-52], [58], [-48], [82], [56], [-6], [-60], [-18], [22], [64], [12], [22], [28], [-24], [66], [-6], [-60], [74], [18], [32], [-6], [58], [8], [-56], [-24], [-46], [66], [2], [-28], [-36], [20], [-92], [-62], [-42], [42], [76], [-6], [-22], [42], [-40], [-6], [-36], [-10], [-32], [72], [-18], [4], [2], [-50], [-24], [10], [38], [48], [-20], [-30], [8], [82], [-14], [16], [60], [-76], [66], [-54], [80], [-50], [68], [-48], [-60], [-54], [-100], [78], [-48], [60], [-46], [-70], [-42], [-14], [10], [-44], [-14], [-4], [-24], [-88], [-42], [-52], [24], [-70], [-54], [20], [-20], [-16], [30], [8], [-78], [-40], [-14], [-14], [50], [-48], [-30], [-6], [-68], [94], [30], [-24], [-36], [38], [-22], [-72], [-36], [50], [46], [48], [-32], [-30], [16], [98], [-102], [24], [76], [-30], [-30], [40], [-86], [-44], [84]]
