
# q-expansion of newform 2312.2.a.m, downloaded from the LMFDB on 03 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, 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 = 2312
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, 1], [-2, 0], [-1, -1], [-1, -1], [0, -2], [0, 0], [-2, 2], [-1, -1], [-2, 0], [1, 1], [2, -4], [-2, 0], [-6, -2], [4, -4], [-2, 0], [10, -2], [2, 4], [-12, 0], [-7, 1], [-6, 4], [-5, 3], [6, 2], [-12, -2], [-2, 0], [0, 6], [-8, 0], [-3, -3], [-10, 0], [-6, 4], [-2, -2], [1, 9], [-12, -2], [9, 1], [6, 0], [10, 2], [6, 8], [11, -5], [13, -3], [-2, 0], [10, -2], [-10, 0], [-4, 4], [-6, 4], [2, 4], [-11, 5], [-1, -1], [10, 2], [-1, -1], [-20, 2], [6, -8], [0, -8], [10, 4], [0, 4], [0, 10], [-2, 6], [10, 4], [0, 8], [-6, -12], [2, -8], [-1, -1], [-10, 0], [-12, -8], [17, 1], [-10, -8], [-2, -8], [-6, -2], [6, 8], [7, -1], [6, 0], [-6, -8], [18, 2], [-17, -1], [8, -2], [17, -7], [-10, 6], [-12, -6], [-22, -4], [-22, 4], [-6, 0], [23, -1], [-16, -10], [-13, 11], [-24, 2], [-15, -7], [-4, 0], [-10, -8], [24, -6], [14, -8], [4, -12], [10, 14], [-27, -3], [-41, -1], [6, 2], [27, -5], [-21, 3], [-2, 0], [6, -8], [-16, 4], [30, 0], [5, -3], [8, 6], [-2, 10], [18, 8], [-1, -1], [-4, -2], [10, -10], [-6, 8], [20, 12], [-30, -4], [13, -3], [6, 0], [-34, 0], [-17, 7], [2, -14], [-10, 8], [-15, -15], [32, 0], [-6, 12], [-8, 12], [22, 8], [-2, -16], [-26, -8], [21, 13], [-13, 11], [-8, -10], [-6, 4], [-7, -7], [12, -4], [-18, 8], [-30, 6], [33, -7], [29, -3], [12, 2], [12, -10], [12, -2], [-4, -14], [9, -7], [-2, -8], [34, -4], [-7, 9], [-50, 0], [27, 11], [23, -1], [-26, 8], [3, -13], [-2, 16], [-10, -8], [10, -18], [-4, 12], [14, 8], [-10, -8], [-20, -8], [9, 9], [39, -9], [-21, 3], [-24, 8], [18, -4], [26, 0], [-10, -16], [1, 17], [-28, -2], [2, 2], [14, 2], [34, 0], [9, 1], [-15, 1], [-38, -4], [14, 0], [30, 8], [44, 8], [-40, 6], [-5, 3], [36, -2], [-46, 2], [-14, 12], [-19, 13], [38, 0], [6, 22], [28, -6], [-24, -8], [-3, -19], [18, 12], [-14, 8], [-38, 10], [-32, -10], [-26, -8], [12, 16], [-22, -12], [5, -27], [-2, -16], [15, -1], [-10, -22], [20, 2], [-7, 1], [34, 12], [-6, -12], [-26, 0], [38, 8], [-16, -8], [-26, 8], [-1, -9], [38, 0], [0, -22], [4, 0], [-36, 10], [-36, -4], [18, 6], [-38, 4], [4, 16], [22, 8], [28, 2], [-21, 3], [-26, 2], [-25, -1], [-10, 8], [-16, -16], [10, -24], [-17, 7], [40, 6], [46, 8], [-15, 17], [18, -16], [-19, 5], [12, -8], [16, 22], [-30, -4], [-5, -5], [-38, 2], [-37, -5], [-18, -24], [-15, 1], [-34, 6], [-8, -22], [40, -4], [14, 6], [-38, 4], [10, -12], [33, -15], [18, 2], [11, 3], [28, -8], [44, 12], [-20, 10], [-6, 4], [-15, 17], [41, 9], [3, 11], [14, 2], [50, -6], [-10, 8], [-30, 12], [-10, -18], [10, 4], [-36, 10], [16, -4], [-6, -4], [1, -7], [-10, 24], [-8, 10], [17, -7], [-60, -8], [6, -8], [-6, 12], [22, -24], [-20, -8], [54, 0], [2, 24], [-21, 3], [-18, -8], [-22, -4], [-32, 4], [10, -16], [22, -2], [-38, -8], [-22, -22], [38, -6], [16, -18], [-4, 6], [18, 14], [12, -28], [1, 1], [-29, -21], [-42, 8], [-23, -7], [40, 8], [-10, -8], [38, -16], [-18, -10], [-8, 18], [-42, -8], [9, -7], [40, 10], [-45, -13], [54, -8], [-26, 0], [20, 4], [-24, -2], [7, 15], [-2, -6], [44, 6], [52, 2], [67, 3], [65, 9], [-7, -15], [-10, -8], [-32, -4], [10, -4], [0, -24], [-18, 8], [-21, 19], [14, -16], [-34, 16], [18, 6], [-20, -12], [-22, 16], [18, -10], [-39, -15], [42, 4], [10, -24], [-25, 23], [30, -16], [30, 0], [0, 32], [-10, -24], [18, 16], [-27, 5], [19, 19], [-27, -27], [-10, -24], [66, -12], [-6, 28], [-15, -15], [28, 16], [-25, -9], [-73, -1], [-26, 0], [66, -4], [30, 0], [-2, 6], [-2, 0], [26, 16], [10, 20], [24, -16], [-32, 14], [51, 11], [22, 8], [36, -24], [53, -11], [14, 8], [42, 6], [-38, 4], [38, 8], [1, 9], [12, -14], [36, -18], [-62, -6], [45, -3], [10, -12], [14, -10], [30, 8], [-10, -8], [8, 16], [27, 11], [22, 10], [-14, 24], [-18, 16], [12, -4], [-22, 4], [-18, -22], [-1, -17], [-17, 7], [16, 22], [16, -16], [-46, 4], [-11, -3], [11, -13], [-56, 2], [26, -24], [-6, 0], [10, 4], [-14, 8], [-11, -27], [2, 12], [27, -13], [59, 3], [18, 18], [28, 2], [29, -3], [24, -8], [42, 20], [-30, 4], [-24, -12], [-16, 4], [-2, -26], [-34, 0], [8, 16], [-30, -8], [-9, -1], [24, -18], [-26, 24], [4, 38], [-28, -28], [22, 8], [-42, 16], [45, 5], [-12, -30], [-4, 30], [-58, 18], [33, 25], [-2, 16], [-12, -6], [32, 28], [41, 17], [2, -24], [6, -8], [11, -37], [21, 13], [-58, 8], [-68, 12], [-26, -16], [-10, -16], [-51, 13], [46, 10], [38, -8], [-42, -8], [41, 25], [-14, 28], [24, -4], [-41, 23]]
