
# q-expansion of newform 1288.2.a.f, downloaded from the LMFDB on 27 September 2025.

# 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 = 1288
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], [2], [0], [-1], [4], [6], [0], [0], [1], [2], [2], [2], [-6], [4], [-2], [14], [14], [12], [-4], [0], [-2], [-8], [-4], [-16], [4], [6], [-8], [-12], [2], [10], [-12], [-10], [-10], [-10], [14], [-20], [12], [-12], [-14], [-22], [0], [-24], [8], [-14], [22], [-8], [-24], [-14], [0], [16], [6], [-20], [24], [0], [-6], [0], [10], [-2], [10], [6], [16], [0], [18], [18], [8], [2], [-8], [-34], [8], [-14], [22], [-16], [-12], [14], [-20], [28], [-30], [-34], [22], [14], [8], [-10], [32], [-20], [34], [-12], [-10], [-22], [-6], [-4], [8], [24], [-32], [36], [-40], [16], [42], [4], [40], [18], [16], [30], [-24], [14], [-4], [22], [-2], [42], [12], [6], [22], [-6], [-26], [36], [-8], [-50], [-16], [-14], [14], [-12], [-4], [14], [12], [44], [-30], [18], [-38], [2], [32], [32], [-36], [-40], [24], [34], [-18], [40], [-36], [-20], [8], [-22], [-34], [-14], [-16], [12], [18], [28], [-34], [10], [-34], [-32], [-14], [4], [-16], [-30], [-4], [-40], [-24], [42], [-28], [-24], [48], [-14], [-40], [36], [-10], [12], [56], [26], [26], [-10], [-36], [2], [8], [54], [-26], [-24], [20], [-42], [16], [24], [-20], [64], [14], [-2], [-24], [56], [58], [36], [-10], [-2], [-38], [20], [-12], [-26], [60], [-32], [-10], [-62], [12], [-2], [54], [10], [-62], [36], [24], [-6], [-28], [8], [30], [-58], [46], [54], [16], [56], [-54], [-16], [20], [-2], [40], [14], [-26], [-40], [-30], [24], [54], [58], [8], [-20], [68], [-52], [-10], [56], [-8], [38], [-16], [-4], [-64], [38], [56], [18], [-44], [-34], [38], [46], [-6], [-58], [38], [-12], [-52], [-40], [62], [-8], [24], [4], [6], [-72], [-22], [-24], [18], [-34], [16], [-4], [66], [12], [-14], [16], [-18], [-56], [76], [-2], [-46], [52], [16], [-16], [54], [14], [-12], [42], [-46], [0], [-14], [-32], [12], [-12], [-14], [60], [34], [-42], [-42], [-58], [54], [72], [6], [-84], [62], [-60], [48], [-10], [-42], [58], [76], [-6], [-16], [88], [52], [2], [60], [-52], [-72], [70], [14], [-78], [46], [36], [-72], [32], [-18], [-48], [24], [34], [52], [14], [78], [20], [26], [-32], [-4], [66], [72], [-14], [-62], [78], [74], [18], [84], [42], [26], [-24], [-54], [4], [-46], [-26], [30], [8], [-54], [-84], [-74], [-80], [-64], [-64], [50], [-70], [-46], [4], [-42], [38], [-92], [64], [-82], [-16], [22], [-90], [64], [36], [-18], [50], [-18], [-56], [2], [68], [46], [-36], [6], [-28], [94], [-86], [-24], [12], [-100], [42], [16], [34], [32], [-30], [0], [-30], [-48], [42], [20], [-20], [-10], [50], [-88], [62], [8], [46], [60], [-36], [-92], [74], [-50], [-66], [-72], [-36], [-34], [46], [38], [82], [-20], [54], [-38], [-10], [12], [-60], [-10], [-70], [60], [-2], [-12], [-96], [-46], [-56], [-84], [58], [38], [-18], [-52], [-6], [-42], [-2]]
