
# q-expansion of newform 8048.2.a.m, downloaded from the LMFDB on 19 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 = 8048
weight = 2
poly_data = [3, -4, -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, 0], 1], [[0, 1, 0], 1], [[-3, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0], [0, -1, 0], [0, 0, 0], [0, -1, 1], [-4, 1, 0], [2, -1, -1], [4, 0, 2], [-4, 0, 0], [4, 0, 0], [0, -2, -2], [4, -2, 2], [2, -4, 0], [0, 2, 4], [-1, -2, 3], [-7, 2, 1], [6, -4, -2], [1, 2, 1], [2, 0, 0], [-11, -2, 1], [-4, 0, 2], [12, -1, 2], [4, -5, -3], [0, 1, -1], [0, 4, 2], [-6, -5, 3], [-8, 4, -2], [2, -8, 4], [4, -4, 0], [4, 2, 6], [3, -8, 3], [-6, 2, -8], [1, 2, -3], [12, 0, 2], [-4, 8, 0], [6, 0, -2], [2, 10, -4], [4, 2, 2], [4, -2, 0], [6, -4, 0], [0, 7, -6], [16, 2, -2], [6, -4, -6], [8, -2, -4], [12, -4, -6], [-2, -8, 4], [-4, -7, -4], [10, -4, -2], [16, -3, 0], [-2, 4, 4], [1, 0, 9], [0, -5, -6], [10, 2, -4], [0, 4, -2], [0, 0, 4], [-2, 9, 5], [0, 1, 7], [-6, 8, -4], [3, 6, -9], [4, 0, 4], [-2, 4, 8], [1, 2, -3], [-1, 0, -1], [4, 10, -6], [-18, -2, 6], [-14, 4, -6], [-19, 4, -3], [-6, 0, -4], [10, -4, 14], [10, -8, 0], [-8, -6, 4], [-6, 0, -2], [6, 6, -4], [8, -5, 4], [16, -3, 6], [-24, 1, -5], [15, -2, 15], [-10, 1, -3], [3, 4, 11], [18, -9, -5], [10, 6, -10], [-2, 12, -6], [23, -4, -1], [6, 10, 6], [-3, 12, -11], [-6, -2, 8], [12, 11, 1], [-12, 6, 0], [10, 4, -8], [-14, -2, 2], [-8, -8, -4], [-6, 6, 0], [16, 6, 2], [28, 0, 8], [-6, -14, 6], [2, 4, 12], [-1, 0, 0], [8, 13, -10], [-14, 1, 5], [6, 4, -12], [-4, 4, 8], [3, 10, 7], [12, -11, 2], [-8, 12, 2], [32, -3, 6], [0, -10, 2], [-16, 8, 6], [-8, 9, 8], [-16, 2, 4], [12, -11, -1], [11, 8, -5], [32, 5, -5], [-2, 6, 18], [-28, 4, 8], [2, -4, -10], [5, 10, 1], [19, 0, -5], [-14, 16, -6], [0, -8, -16], [-6, 19, -9], [12, -5, 17], [-6, -5, 15], [11, -12, -13], [-24, 8, -2], [-20, 10, 8], [8, 7, 1], [12, -11, -2], [-6, -10, 2], [12, 1, -12], [-16, 3, 4], [10, 12, -2], [16, -9, 0], [4, 12, 4], [4, -4, 10], [20, 10, 6], [-16, 19, -2], [-6, 4, -8], [-2, -4, 4], [20, 10, -4], [-4, 13, 2], [-8, -6, 6], [4, 8, 8], [-24, -10, 4], [20, -16, -14], [-20, -8, 4], [-6, 0, -8], [-32, -7, 3], [-2, -1, -1], [16, 9, 6], [18, -12, -2], [18, 16, -8], [4, 20, -2], [7, -4, -9], [-16, -15, 8], [9, 18, -15], [-28, 4, 8], [-18, -2, -12], [-8, -7, 3], [-18, 0, 4], [-2, 12, 2], [18, -5, -1], [-16, -8, 2], [23, -4, 7], [20, 10, 6], [32, -4, 4], [5, -4, -23], [-4, -2, 10], [-12, -1, -4], [12, -20, -12], [16, 3, -6], [-2, 3, -9], [-12, -4, 0], [2, -4, 14], [12, 17, -8], [2, -13, -1], [23, -6, -5], [2, 8, -8], [2, -4, 18], [0, -4, -10], [-20, -6, -2], [-8, -5, -22], [-20, 13, -13], [4, -3, -9], [14, -4, 12], [-9, 4, 15], [-16, -1, -7], [-42, -10, 2], [-16, 20, 18], [-7, -18, -7], [4, 10, -16], [12, 11, -15], [5, -12, 13], [-8, 0, -28], [8, 10, 14], [-32, 5, 2], [-26, -4, 14], [16, -14, -12], [22, -16, -8], [-20, -3, -6], [4, -4, 26], [-32, -6, 8], [-2, 3, 11], [-40, -7, 0], [22, 4, -4], [38, 0, -8], [9, -6, 13], [-34, 5, -19], [4, -15, 15], [6, 2, -20], [24, -1, 14], [-19, 6, 13], [-2, 28, -4], [8, -4, 10], [-4, -27, 0], [-20, -7, 12], [0, -12, 14], [-16, 10, -16], [6, 8, -12], [14, 5, -3], [20, -17, -19], [2, 9, -7], [-4, 12, -12], [-37, 14, 15], [28, -20, -8], [6, 6, 16], [28, -1, 4], [-30, 13, -15], [8, 15, -22], [28, 23, -11], [28, -16, 8], [-39, 2, 5], [-14, -2, -22], [-28, 16, -10], [16, -13, 12], [-48, 8, -8], [26, 28, -8], [0, 22, 0], [0, -19, -6], [24, -12, 2], [0, 21, -13], [-23, -6, -3], [0, 17, 11], [40, -3, 15], [4, -22, -4], [-6, 0, -30], [-11, 4, 17], [-7, -26, -3], [-28, 22, -8], [-42, -4, 12], [-20, 6, 22], [18, -18, 6], [13, 24, -15], [-16, 9, 10], [28, 1, 0], [-10, 11, 3], [-62, -3, 1], [34, -2, -8], [18, -7, -19], [-28, -19, 8], [14, 12, 12], [12, 0, -18], [-40, 3, -4], [-9, -22, 3], [-20, -18, 2], [-17, -32, 15], [9, 16, -7], [13, 6, -11], [12, -5, 14], [-32, 2, -4], [-8, 2, 20], [-9, 16, -13], [-10, 16, 0], [-22, 18, -16], [54, -3, -7], [2, -16, -2], [8, -1, 22], [-12, -6, -30], [-28, 20, 10], [-10, -14, 4], [-8, 31, -2], [12, -4, 10], [-10, -40, 10], [-20, 1, 0], [12, 7, 20], [-18, 1, 25], [12, -12, -2], [-14, 16, 2], [-18, 15, -17], [-6, -5, -17], [-4, 12, 12], [-18, -4, 24], [-8, 19, 14], [20, -17, 25], [-30, 12, 2], [24, -4, -14], [2, 24, 2], [10, -4, 2], [32, -12, 6], [0, 2, 20], [28, 4, -22], [16, 14, -18], [40, 1, 22], [8, 19, 18], [6, -22, 12], [-6, 14, 10], [-34, 16, 14], [20, -2, -14], [18, 20, 14], [-6, 14, -22], [-3, -18, 1], [6, -14, 12], [48, 10, -10], [-24, 8, -8], [-37, 12, -5], [4, 0, 30], [-8, -31, 3], [-8, -13, 10], [-12, -24, -2], [-29, 2, -5], [30, 14, -2], [-26, 7, -1], [-46, 10, -24], [40, -14, -4], [-15, 24, 9], [-20, 13, 8], [-17, 28, 11], [22, -26, 8], [-38, 14, -34], [-20, -6, -10], [-12, 2, 14], [10, 3, 7], [-34, -20, 2], [-22, 4, -32], [-8, 8, 12], [4, 11, 9], [-8, -16, -28], [32, 12, 12], [-50, 1, 5], [-18, -18, 16], [8, 20, -10], [8, -7, 7], [20, -3, -6], [-14, 21, 21], [-23, 20, 21], [20, 15, 4], [40, -2, -16], [30, -16, -10], [16, -21, 2], [24, 2, 10], [-8, 35, -28], [-24, 26, 4], [-2, 40, -2], [8, 20, -10], [-30, 0, -18], [14, -10, -20], [-18, 6, 8], [-36, 22, -22], [-24, 15, 24], [12, 18, 24], [-14, -16, 14], [-66, 0, 6], [72, -12, -10], [-9, 40, -17], [16, -13, 24], [-48, -12, -6], [42, -18, 6], [30, -8, 30], [6, -7, -23], [-14, 8, -40], [29, -12, 13], [-32, -9, -3], [44, 19, -4], [33, 2, -11], [-44, -14, 12], [32, 16, 8], [21, -28, -23], [-16, 16, 0], [-18, -10, 14], [-21, -4, -17], [-6, -9, 27], [2, 24, -6], [36, -14, 38], [0, 11, -14], [-8, 11, 0], [-12, 12, 32], [20, -15, 7], [-22, 10, 28], [4, -8, 4], [83, 0, -5], [-40, 17, -8], [24, -3, 4], [-16, 2, 2], [-54, 4, 18], [72, -9, 12], [28, 21, -21], [24, -9, 0], [34, -5, -25], [10, -30, 20], [0, 26, -28], [-32, -9, -11], [42, -16, -24], [-24, 7, -22], [18, -10, -12], [15, 14, -5], [20, 13, 10], [-14, -28, 8], [35, -34, -21], [56, -3, -14], [3, -20, 31], [52, -15, 8], [40, 10, 4], [32, -6, 10], [-15, 12, -19], [-26, 4, -32], [-20, 21, -13], [22, 30, -16], [90, 3, 3], [-29, -22, 7], [20, 0, -30], [-50, 16, -22], [-37, 26, -17], [54, -7, 1], [13, 28, 1], [18, -20, 16], [-34, 24, 26], [54, 24, -8], [18, -26, 22], [16, 27, 4], [-8, -22, 6], [46, -6, 12], [8, 5, 15]]
