
# q-expansion of newform 3024.2.r.f, downloaded from the LMFDB on 20 January 2026.

# 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 = 3024
weight = 2
poly_data = [9, -3, -2, -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, 0], 1], [[-3, -2, 2, 1], 6], [[0, 5, 1, -1], 3], [[-9, 2, 1, 2], 3]]

hecke_ring_character_values = [[1135, [1, 0, 0, 0]], [757, [1, 0, 0, 0]], [785, [0, -1, 0, 0]], [2593, [1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 1], [1, -1, 0, 0], [-1, 2, 1, -1], [0, -2, 0, 0], [1, 0, -1, 0], [-5, 0, 0, 0], [0, 5, 0, -1], [-4, 2, -2, 2], [0, 2, 0, 0], [2, 0, 0, 0], [0, 8, 0, -1], [-1, -2, -3, 3], [0, 0, 0, 0], [4, 0, 2, 0], [0, 0, 0, 3], [4, -7, -3, 3], [0, 8, 0, -3], [0, 0, -3, 0], [5, 0, 3, 0], [2, -5, -3, 3], [-8, 4, -4, 4], [-10, 0, -2, 0], [1, 2, 3, -3], [4, -5, -1, 1], [0, -10, 0, 0], [-5, 0, -1, 0], [14, 0, 0, 0], [0, -1, 0, -1], [4, 0, -3, 0], [0, 5, 0, -1], [13, -14, -1, 1], [0, -7, 0, 6], [0, 10, 0, -2], [2, 1, 3, -3], [0, 1, 0, -3], [4, 0, 6, 0], [0, -8, 0, 4], [-6, 6, 0, 0], [8, 0, -2, 0], [8, 0, -3, 0], [12, -9, 3, -3], [0, 7, 0, 0], [6, 0, 0, 0], [10, 0, 0, 0], [0, -16, 0, 0], [-4, 4, 0, 0], [17, -19, -2, 2], [0, 13, 0, -3], [5, 0, -2, 0], [0, 7, 0, -5], [-11, 8, -3, 3], [-9, 0, 0, 0], [0, 10, 0, -5], [-10, 11, 1, -1], [4, 0, -1, 0], [-2, 0, -6, 0], [-2, 8, 6, -6], [14, -13, 1, -1], [0, 17, 0, 3], [0, -7, 0, -1], [13, 0, 0, 0], [0, 12, 0, -6], [13, -10, 3, -3], [-6, 6, 0, 0], [8, -2, 6, -6], [0, -2, 0, 3], [0, 0, 0, 3], [22, -22, 0, 0], [-11, 10, -1, 1], [10, 0, 5, 0], [2, -8, -6, 6], [0, 10, 0, 0], [1, 0, 3, 0], [0, 26, 0, 2], [-8, 16, 8, -8], [-22, 0, 0, 0], [0, -5, 0, -2], [0, -8, 0, 9], [0, -11, 0, 7], [-20, 14, -6, 6], [-16, 0, 4, 0], [-13, 0, -3, 0], [8, -8, 0, 0], [33, -30, 3, -3], [-33, 0, 0, 0], [7, 5, 12, -12], [14, -19, -5, 5], [0, -13, 0, -3], [7, 0, 11, 0], [-28, 26, -2, 2], [-14, 0, -9, 0], [0, 22, 0, 1], [0, 8, 0, 3], [-18, 0, -6, 0], [0, -2, 0, -8], [13, 0, 5, 0], [28, 0, 3, 0], [8, 0, -6, 0], [11, -8, 3, -3], [-20, 0, -4, 0], [0, -3, 0, 0], [9, -6, 3, -3], [0, 8, 0, -9], [17, 0, -3, 0], [-5, 1, -4, 4], [-10, 0, -2, 0], [0, -24, 0, 0], [-17, 14, -3, 3], [0, 2, 0, 6], [-22, 0, -6, 0], [0, 2, 0, 5], [17, -5, 12, -12], [16, 0, 9, 0], [1, 13, 14, -14], [0, -4, 0, 9], [-4, 0, 4, 0], [0, -20, 0, -2], [16, -14, 2, -2], [0, -17, 0, -3], [10, -13, -3, 3], [6, 6, 12, -12], [-1, 0, 13, 0], [-16, 13, -3, 3], [24, 0, 6, 0], [-44, 44, 0, 0], [-2, 0, 2, 0], [-22, 16, -6, 6], [0, 31, 0, 3], [1, 0, 3, 0], [0, -20, 0, 10], [0, -7, 0, -3], [32, 0, 6, 0], [0, 22, 0, 4], [0, 10, 0, 0], [28, 0, 5, 0], [0, -4, 0, 0], [0, -25, 0, 5], [-5, 0, 5, 0], [-35, 0, -3, 0], [-18, 24, 6, -6], [0, 8, 0, -6], [12, 0, 0, 0], [-28, 0, 6, 0], [4, -8, -4, 4], [28, -25, 3, -3], [-2, -14, -16, 16], [0, 44, 0, -3], [-2, 0, 5, 0], [0, -2, 0, 18], [6, 0, 6, 0], [19, 0, 9, 0], [0, 8, 0, -10], [-19, 16, -3, 3], [40, -41, -1, 1], [40, 0, 3, 0], [30, -24, 6, -6], [2, 0, 12, 0], [0, 9, 0, 3], [-27, 24, -3, 3], [-21, 0, -3, 0], [0, 41, 0, -3], [4, 0, -1, 0], [0, 30, 0, 3], [44, -46, -2, 2], [-20, 0, -12, 0], [-2, 5, 3, -3], [26, 0, 0, 0], [0, -5, 0, 13], [-49, 50, 1, -1], [0, -11, 0, 9], [0, -13, 0, -13], [1, -19, -18, 18], [0, -40, 0, 0], [0, 20, 0, 2], [-13, 10, -3, 3], [22, 0, 2, 0], [28, 0, 6, 0], [-38, 41, 3, -3], [20, -11, 9, -9], [-5, 16, 11, -11], [0, 58, 0, 0], [-37, 0, 1, 0], [0, -15, 0, -9], [10, -2, 8, -8], [-4, 0, -6, 0], [-52, 49, -3, 3], [0, 37, 0, -6], [-16, 0, 4, 0], [17, 0, -3, 0], [-29, 22, -7, 7], [-23, 0, -3, 0], [42, -39, 3, -3], [-7, 0, -5, 0], [0, -12, 0, 9], [0, -17, 0, -6], [34, -40, -6, 6], [29, -22, 7, -7], [10, 0, 2, 0], [0, -6, 0, 12], [26, -38, -12, 12], [0, -17, 0, -15], [-5, -4, -9, 9], [39, 0, 12, 0], [-2, 0, -1, 0], [-8, 0, -18, 0], [0, -4, 0, -1], [-37, 26, -11, 11], [0, 5, 0, 18], [41, 0, 6, 0], [0, 14, 0, -22], [38, -44, -6, 6], [0, 3, 3, -3], [0, -36, 0, 6], [13, -22, -9, 9], [0, 29, 0, -3], [-17, 28, 11, -11], [-14, 0, -22, 0], [0, -28, 0, 2], [0, 4, 0, 6], [0, -10, 0, 12], [0, -47, 0, -8], [22, 0, 6, 0], [0, 16, 0, -5], [10, -16, -6, 6], [6, -24, -18, 18], [32, 0, -5, 0], [56, -56, 0, 0], [-12, 12, 0, 0], [0, 16, 0, 6], [-20, 0, -15, 0], [0, 29, 0, 9], [0, -30, 0, -3], [-4, -11, -15, 15], [-46, 50, 4, -4], [0, -20, 0, 6], [-10, 0, -20, 0], [0, 16, 0, 1], [12, 0, 6, 0], [-55, 56, 1, -1], [-44, 0, 0, 0], [0, -28, 0, 12], [8, 0, -18, 0], [0, 50, 0, -1], [-18, 0, -18, 18], [-2, 0, 0, 0], [0, 21, 0, 12], [0, -16, 0, -12], [18, 0, 3, 0], [0, -20, 0, 12], [-13, 0, -5, 0], [0, 27, 0, -15], [-26, 32, 6, -6], [-2, 10, 8, -8], [25, 0, 5, 0], [-10, 0, -12, 0], [20, -11, 9, -9], [-6, 0, 3, 0], [35, 0, -9, 0], [2, 16, 18, -18], [43, -44, -1, 1], [0, 10, 0, 12], [-10, 0, -3, 0], [0, 60, 0, -6], [-13, 4, -9, 9], [30, 0, -12, 0], [-19, 8, -11, 11], [0, 20, 0, 9], [0, 0, 0, 6], [0, -14, 0, 0], [16, 0, 12, 0], [7, -16, -9, 9], [0, -16, 0, 12], [0, 22, 0, -3], [-8, 0, 0, 0], [0, -4, 0, -16], [64, -58, 6, -6], [-61, 0, -3, 0], [21, -15, 6, -6], [0, 48, 0, 0], [0, -16, 0, 18], [-26, 34, 8, -8], [-50, 53, 3, -3], [0, 23, 0, -6], [-20, 0, -1, 0], [2, 0, -3, 0], [0, -52, 0, 2], [20, -2, 18, -18], [-45, 0, -3, 0], [-2, 4, 2, -2], [-19, 0, 10, 0], [0, 13, 0, 10], [0, 52, 0, -5], [10, -10, 0, 0], [0, -57, 0, 9], [56, -50, 6, -6], [-22, 5, -17, 17], [-32, 0, -4, 0], [-4, 4, 0, 0], [0, -56, 0, 9], [32, 0, 16, 0], [-2, 0, -15, 0]]
