
# q-expansion of newform 1248.2.a.m, 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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 1248
weight = 2
poly_data = [-1, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0], 1], [[-1, 2], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0], [1, 0], [-1, -1], [-3, 1], [0, 2], [-1, 0], [0, -2], [-1, -1], [-2, -2], [-4, 2], [-7, 1], [0, 2], [-7, 1], [-2, 2], [-4, -2], [-4, -2], [-2, -4], [8, 2], [1, -3], [-10, 0], [0, 2], [0, 4], [-6, -4], [-9, 3], [12, -2], [10, 0], [-2, -6], [-2, 2], [6, -4], [2, 4], [2, 6], [14, 2], [-7, -7], [8, -4], [1, 5], [3, -5], [-8, -2], [9, 1], [-8, 2], [-2, 4], [10, 2], [12, -6], [10, 6], [-6, 0], [-1, 7], [-6, -2], [0, 4], [5, 5], [-12, -6], [-6, 0], [-24, 2], [6, -8], [12, 6], [16, 0], [12, 6], [-28, 0], [4, -6], [-3, -7], [10, 4], [-11, -3], [-26, 2], [9, -11], [15, 3], [22, -2], [14, 8], [-1, 7], [-11, 1], [-14, 4], [-20, 0], [-12, -10], [-19, -3], [-4, 6], [-24, -4], [-2, 4], [3, -13], [8, 6], [14, 8], [8, 2], [15, 3], [0, 10], [-10, 2], [-2, 4], [8, -2], [-12, -6], [6, -6], [-14, -2], [-21, 7], [32, 2], [-1, -1], [25, 5], [-28, 4], [-30, 0], [1, -7], [-8, -8], [-27, 1], [16, -4], [27, -5], [-18, -8], [-20, -8], [2, -16], [4, -16], [1, 13], [-2, -2], [4, 6], [2, 6], [-2, -12], [-14, 4], [11, 7], [12, 0], [-18, 8], [8, -4], [-4, -2], [-11, 13], [15, 11], [-5, 3], [0, -2], [-31, 5], [-16, 0], [16, -2], [-16, -12], [-24, 2], [-40, 2], [12, 6], [16, -6], [13, 9], [-10, 8], [-22, 8], [-2, -2], [6, 10], [10, -8], [11, -1], [-30, -4], [30, 2], [-8, -6], [13, 5], [6, -4], [33, 5], [-37, -1], [6, -16], [-42, 0], [27, 11], [13, 9], [-2, 2], [24, -6], [12, -2], [-14, -8], [-28, 6], [12, -2], [8, -4], [-24, 14], [16, 2], [24, 2], [-10, 2], [-34, -6], [-18, -14], [38, 2], [-12, -8], [9, -7], [8, 22], [-43, 1], [-2, -4], [0, -6], [-3, -3], [16, 12], [-9, 19], [-34, -4], [-12, 0], [-10, -8], [0, 2], [26, 0], [-4, 26], [4, 22], [18, -6], [36, -10], [30, -6], [-10, -4], [7, -5], [-35, -7], [42, -2], [-42, 4], [17, 5], [18, -14], [-8, -22], [-9, -5], [-28, 6], [-10, 0], [-20, 2], [3, -21], [-28, 6], [-12, 6], [8, 10], [34, -4], [36, 0], [-15, -3], [2, 2], [36, -2], [2, 16], [-6, -24], [-25, -5], [6, 18], [-25, 7], [-4, 0], [18, -8], [0, 18], [42, 0], [-4, 14], [-19, -11], [6, 26], [7, -21], [54, 2], [-4, -14], [32, 10], [-8, -4], [4, -2], [-24, -2], [10, -24], [4, 8], [-22, 0], [-14, -12], [13, 25], [-22, -4], [-51, -7], [15, -5], [-15, -23], [-48, 0], [-46, 4], [38, 12], [-16, -12], [14, -14], [14, 12], [-4, 2], [-16, 4], [-25, -13], [20, 10], [28, 8], [-22, -4], [-32, -14], [9, -11], [-14, 18], [2, -10], [30, 16], [-38, 6], [-36, 16], [64, 2], [11, -9], [6, 6], [-15, 5], [50, 0], [15, -17], [6, 30], [-22, -24], [-35, -3], [-46, 0], [26, 4], [0, -2], [-40, 10], [34, 8], [-33, 7], [-6, -28], [-6, -24], [-34, 2], [-56, -4], [54, 4], [-2, 8], [-13, 7], [-4, 32], [-13, 19], [45, -3], [-13, 15], [14, -12], [32, 10], [-3, 17], [-36, -10], [-26, 2], [62, 0], [-5, -25], [-52, -6], [24, -14], [70, -4], [-48, 16], [-18, 26], [49, 5], [-2, 34], [20, 6], [15, -29], [-12, 4], [-28, -6], [-39, -3], [-37, 11], [20, -10], [4, -26], [-52, 0], [11, -17], [26, -24], [16, 6], [18, 12], [-28, 0], [-46, -4], [36, -16], [39, -5], [2, -24], [-9, -17], [30, 2], [4, 0], [-8, 12], [56, 2], [34, -14], [10, 4], [10, -24], [-22, -8], [-48, 12], [-1, -17], [-22, 12], [44, 8], [56, -14], [30, 8], [40, 6], [0, -18], [4, -2], [48, -18], [-38, -18], [22, -12], [-64, -6], [3, -21], [-21, 7], [-20, -14], [-71, 1], [1, 5], [16, -24], [4, 10], [14, 12], [28, 10], [-28, -8], [-20, 22], [67, -5], [42, -20], [8, 2], [-59, 13], [24, 18], [42, -10], [12, -10], [-26, -24], [-7, -27], [22, 2], [-29, 11], [-22, 18], [66, 8], [-27, 13], [-24, 18], [-38, 12], [-7, 37], [-12, 6], [-39, -19], [2, -18], [0, -30], [30, -24], [16, 26], [-4, 18], [-20, 14], [-8, 14], [-8, 18], [72, -10], [60, 8], [-40, 14], [-6, 6], [44, 6], [47, 15], [15, -9], [-24, -22], [26, -10], [-22, 22], [52, -18], [-32, -30], [4, -8], [18, 28], [26, -4], [-16, 0], [-26, -12], [10, 16], [64, 10], [-23, -11], [-3, 13], [9, 5], [-9, -37], [33, 17], [14, 4], [55, 3], [-52, 6], [-21, 3], [-44, 8], [8, -22], [1, 5], [52, 14], [4, 8], [4, -2], [12, 10], [-33, 3], [-20, -6], [8, -4], [43, -21], [-44, -2], [-20, 26], [-11, -15], [-33, 11], [-17, 23], [56, -4], [-30, 8], [7, 19], [7, 35], [12, -26], [-46, 0], [12, 22], [-44, 8], [-50, 2], [70, 4], [16, 18], [46, 8], [8, -20], [-73, 3], [-54, 2], [10, 8], [8, 10], [-2, -16], [-26, -14], [-14, -32], [57, 5], [-40, -24], [51, 15], [5, -27], [26, -10]]
