
# q-expansion of newform 1088.2.a.f, downloaded from the LMFDB on 06 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 = 1088
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], [-2], [4], [-2], [-1], [4], [-6], [-8], [2], [-4], [-2], [-4], [-12], [6], [4], [-4], [4], [6], [-6], [10], [-12], [-10], [-10], [-2], [16], [-4], [0], [10], [-4], [12], [2], [-20], [18], [4], [-14], [-24], [2], [16], [20], [20], [-12], [14], [12], [-18], [20], [8], [16], [-6], [-6], [4], [6], [28], [18], [-20], [0], [-28], [8], [-26], [-20], [18], [-12], [2], [14], [-12], [28], [-30], [24], [-26], [6], [-24], [26], [-18], [-16], [4], [-14], [36], [30], [-26], [0], [-10], [-30], [-34], [38], [-36], [-34], [-26], [22], [8], [-12], [38], [-22], [-20], [28], [26], [-22], [6], [20], [-36], [-20], [6], [-4], [2], [-8], [-46], [-12], [-14], [-8], [34], [-22], [-38], [-6], [-24], [-8], [-26], [-8], [44], [24], [28], [-10], [-26], [-28], [-16], [-20], [-34], [-32], [-26], [-28], [14], [44], [10], [-26], [-38], [18], [-6], [-46], [4], [-2], [6], [40], [52], [-14], [4], [26], [38], [-8], [14], [20], [32], [32], [26], [20], [6], [4], [6], [-36], [30], [22], [60], [4], [54], [-48], [-12], [-38], [-18], [6], [-20], [14], [-4], [-12], [46], [14], [22], [20], [-14], [-8], [-36], [56], [2], [24], [-12], [-52], [-38], [-4], [-42], [-28], [-12], [46], [26], [-14], [-24], [44], [42], [48], [34], [2], [28], [-22], [-16], [-36], [18], [58], [-14], [12], [-54], [-32], [-44], [-62], [-52], [14], [18], [54], [-36], [-58], [14], [-16], [-46], [70], [-42], [-22], [-54], [-2], [-38], [-60], [-50], [-30], [-54], [-52], [48], [-2], [-16], [64], [-18], [-68], [32], [-42], [44], [76], [24], [32], [68], [12], [-54], [10], [26], [-70], [8], [68], [-48], [-38], [50], [-20], [-14], [42], [44], [-4], [-8], [28], [58], [-46], [20], [4], [32], [38], [-36], [-74], [30], [8], [34], [48], [4], [-38], [8], [30], [-56], [-12], [-10], [-10], [-4], [20], [-30], [-18], [-26], [36], [-40], [-66], [-76], [52], [54], [-40], [0], [10], [28], [36], [-56], [72], [-74], [-40], [52], [54], [-30], [38], [64], [-88], [-2], [-28], [28], [0], [10], [66], [40], [-18], [-36], [48], [-26], [20], [-46], [2], [6], [-80], [30], [10], [-56], [-2], [90], [60], [-44], [-26], [28], [-8], [52], [6], [4], [-16], [-84], [86], [66], [50], [-48], [30], [-78], [92], [64], [-34], [48], [-60], [20], [-78], [36], [60], [-82], [50], [-38], [74], [-38], [-48], [0], [62], [-84], [-12], [10], [-4], [64], [12], [6], [-24], [-36], [-50], [20], [-4], [-78], [-18], [-36], [64], [0], [-66], [34], [14], [58], [40], [70], [-2], [58], [-80], [-58], [92], [-22], [24], [64], [-14], [-60], [28], [44], [20], [-50], [52], [-42], [-68], [22], [92], [-6], [-48], [-66], [42], [-38], [26], [-54], [60], [24], [18], [10], [84], [52], [46], [-84], [-54], [26], [-74], [-20], [-18], [-56], [22], [4], [62], [-38], [36], [-86], [44], [-42]]
