
# q-expansion of newform 2320.2.a.m, downloaded from the LMFDB on 23 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 = 2320
weight = 2
poly_data = [3, -5, -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], [[-1, 2, 0], 1], [[-4, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0], [-1, 0, -1], [-1, 0, 0], [1, 0, -1], [-3, 0, -1], [-1, -1, 0], [0, -1, -1], [-2, 1, 1], [0, 1, -1], [-1, 0, 0], [-2, -1, -1], [-4, 1, -3], [3, -1, 0], [-5, -2, 1], [3, 0, -1], [3, 1, -2], [-3, -3, 2], [-7, -1, 2], [4, 1, -1], [3, -1, 2], [-1, -2, 1], [1, 2, 1], [-3, 0, 3], [6, -2, 2], [-13, 0, -1], [6, -2, 6], [10, -1, -1], [3, 0, 1], [-1, -1, -4], [9, 0, -3], [7, -2, 1], [0, 1, -1], [12, 3, -1], [4, 2, 2], [18, 0, 0], [1, -3, 2], [5, 2, -1], [-5, 2, -3], [6, -3, 5], [3, 3, -4], [-6, -2, 0], [5, -3, 0], [-3, -2, 5], [-10, 3, 1], [0, 0, -2], [-11, -3, 2], [1, -2, 5], [4, -3, 3], [6, -1, -5], [2, 0, 0], [0, -2, 0], [-15, 1, -2], [8, -2, 8], [9, 0, -1], [3, -5, -4], [-9, -4, -1], [-15, 1, 0], [-11, -4, 3], [5, 3, -6], [3, -1, -4], [-8, -1, 5], [3, 0, 7], [-17, -2, 1], [-6, 3, -9], [-19, 1, 0], [18, 3, -3], [-8, 5, -1], [-7, -2, 3], [-3, 2, 1], [2, 4, -4], [-3, 3, 2], [0, 1, -1], [19, 0, 3], [-4, 0, 6], [-17, -6, 3], [3, 4, 1], [-15, -3, -4], [-16, -4, 2], [9, 1, 4], [-13, 1, 6], [3, -3, 4], [11, 5, -2], [12, 0, 4], [-16, 3, -9], [19, 1, 4], [-15, -2, -1], [-3, 3, 2], [11, -3, -6], [12, 4, 2], [16, 5, -5], [-3, 0, 5], [-9, 2, -1], [-17, -6, 3], [6, -1, 3], [-2, -2, -8], [-15, 2, 3], [-3, 1, -4], [-15, 1, -4], [16, -5, 1], [2, -6, 10], [-11, 2, -15], [18, 0, 12], [-9, -2, -7], [-15, -1, -2], [4, 0, -8], [-4, -3, 1], [-18, 3, 3], [15, 1, -2], [21, -2, -3], [5, -3, 0], [-5, -4, -5], [8, -2, 4], [-3, 6, -9], [-5, 0, -7], [-32, 2, -6], [-21, 3, -12], [-8, 3, 9], [-9, 6, -5], [-9, -6, 5], [18, 5, 1], [29, -3, 4], [8, 0, -2], [-9, -8, 11], [6, 1, 13], [13, -5, -8], [-15, -3, 8], [23, 3, 0], [-9, -9, 6], [7, 0, 3], [-22, 1, -5], [4, 7, -11], [15, -4, 15], [-2, 5, 13], [11, -2, 17], [6, -8, -8], [2, -10, -6], [-12, 1, -7], [-17, 4, 1], [3, 2, -11], [18, 2, 10], [-8, 2, 2], [36, 2, -4], [13, 6, 3], [-3, -2, -5], [-19, -7, 4], [-3, -4, -5], [2, 5, -9], [3, -5, 4], [19, -4, 5], [3, 8, -11], [23, -3, -2], [-3, 1, 8], [-11, 0, 7], [21, 0, 5], [-5, 0, 3], [-18, -11, 5], [-29, -1, -10], [12, -2, -4], [23, -3, -6], [6, -4, 4], [3, -2, -7], [12, -6, -12], [7, 10, -7], [0, 7, -11], [15, 3, 8], [21, 2, -5], [-11, 5, -2], [-13, 2, -7], [20, 6, -4], [24, -1, 3], [3, -11, 4], [38, -4, -4], [27, 1, -4], [-46, 1, -9], [-11, -7, 6], [-27, 5, -20], [7, -3, 12], [0, -8, -10], [-17, 4, 3], [8, -2, 4], [19, -2, 9], [6, 7, -1], [2, 2, 14], [-9, 7, 8], [30, -1, 7], [6, 8, -8], [14, -1, -11], [19, -2, -7], [-25, -3, -6], [0, -4, 8], [-22, 2, 2], [15, -6, -3], [-26, 1, -11], [3, -11, 6], [33, 4, -7], [12, -4, 6], [-19, 1, -16], [-4, 0, -2], [-12, 4, -10], [15, -4, 9], [-6, -8, -4], [46, -4, -2], [8, 3, -5], [8, 4, -10], [18, 7, 7], [12, 12, -6], [-26, 7, -5], [30, 1, 5], [33, -7, 4], [1, -10, 1], [-22, -7, -5], [-6, 4, 12], [25, 8, 1], [33, 2, -9], [9, -6, 9], [14, -8, 8], [-41, -2, -5], [-24, 2, -20], [-3, 4, -9], [45, 4, 1], [17, -5, 2], [-14, 0, -6], [36, 0, 6], [19, 8, -1], [-27, -2, -15], [5, 5, 0], [6, 13, -1], [-15, 8, 15], [1, -10, -5], [-15, -1, -8], [8, 9, 9], [1, -5, 4], [7, 4, -11], [-15, -3, 0], [1, 0, -1], [15, -6, -3], [35, 1, -10], [-15, -10, 7], [0, 10, -2], [6, 3, 3], [-33, 8, 3], [-11, 7, -4], [37, 2, 1], [-16, 2, -8], [-12, -14, 8], [6, -6, -12], [-23, 4, -5], [21, -3, 22], [7, -11, 16], [9, -6, -9], [26, -7, 11], [-6, 8, 4], [-15, 0, -3], [-13, -7, -6], [-18, 9, 11], [-12, -2, 2], [-16, 0, -10], [13, 2, 11], [6, -4, -4], [-10, 10, -2], [-11, 0, -19], [-27, 10, -5], [26, -8, -12], [20, -3, 5], [-3, 2, 19], [49, -4, 11], [24, 2, 4], [-12, 6, -4], [7, 16, -5], [6, -8, -16], [8, -6, 8], [-38, -1, -13], [-40, 0, -6], [-35, -6, 5], [14, 15, -3], [-5, -4, -5], [-27, 0, 9], [-13, -9, 0], [-13, -1, 8], [-21, 3, 10], [-39, 2, 1], [34, 8, 2], [21, 8, -13], [11, -1, 8], [37, -4, -3], [30, -3, 21], [-25, 0, 3], [-60, -1, 3], [-44, 6, -6], [-36, -10, 4], [12, 14, -16], [39, 12, -3], [48, 0, -2], [-3, 0, -5], [11, 0, -9], [9, 5, 8], [34, 9, 1], [9, -1, -14], [9, 1, -18], [1, -2, -1], [-31, 10, -9], [3, -11, 6], [-32, 3, -3], [-9, -6, -19], [-8, 5, 19], [17, 13, -8], [-21, -4, -17], [14, 4, 0], [-21, -3, 4], [20, 4, 6], [-12, 5, 3], [-30, -4, -8], [-9, -9, 0], [-23, -12, 3], [-21, -6, -9], [11, -9, -12], [3, -2, 3], [24, 16, -4], [-7, 5, -16], [-15, -1, 10], [49, -4, -9], [-31, 5, -24], [9, -7, -16], [73, 0, -3], [0, 10, -4], [5, 13, 0], [-2, 10, 4], [31, 6, -9], [-39, 0, -11], [-42, 0, -12], [-25, -7, -6], [-33, 11, 0], [-2, -2, 0], [27, -4, 15], [-38, 3, -9], [-27, -6, 5], [32, -6, 0], [6, -5, 1], [11, 19, -8], [-41, -2, -17], [-7, -14, -9], [-6, 2, 6], [-9, 3, -4], [-14, -6, 12], [45, -7, 4], [15, -10, -1], [41, 19, -14], [1, 10, 7], [33, 2, 9], [-33, 0, -25], [13, 5, -14], [11, -15, 10], [33, 7, 2], [40, 1, 19], [14, 0, 24], [21, -4, 21], [-54, -13, 7], [-6, -4, 22], [6, 7, 1], [15, 0, -11], [56, 0, 6], [18, 0, 0], [3, 4, -1], [6, -8, 14], [-29, 0, -13], [-22, 5, -1], [30, 3, -15], [28, -11, -1], [-49, 3, -12], [-12, -11, 11], [16, -6, 22], [12, -3, 27], [-33, -17, 8], [13, 9, 14], [26, 8, 8], [-60, 1, 7], [51, 6, 3], [17, -4, 13], [-15, 1, 16], [-13, 3, 8], [-36, -12, 14], [33, -11, 8], [-17, 10, 5], [18, -5, 5], [19, -1, -18], [-24, 9, -33], [1, 6, 9], [-7, -11, 8], [13, -4, 9], [57, 4, -3], [-13, 5, 2], [3, -3, 14], [-18, 5, 17], [1, 16, 9], [-9, 16, -15], [-25, 3, 8], [10, -2, 0], [0, 0, 6], [37, -3, -6], [-9, 5, 10], [-16, -6, 0], [42, 9, -9], [-23, 0, 9], [-12, 12, -6], [-27, -15, 0], [-38, -2, -20], [-28, 6, -20], [30, -12, 0], [-5, 10, -3], [-30, 2, 4], [5, -7, 8], [9, -9, 18], [-27, -8, -9], [31, -5, -14], [38, -9, 17], [48, -8, -10], [-69, -2, 3], [-14, -15, 9], [33, 18, -5], [-9, 10, -11], [27, -9, -8], [-25, -6, -3], [9, -10, -1], [-9, -12, 21], [80, 2, 8], [6, -16, 4], [-63, 0, 7], [-21, 1, -2], [31, -1, -10], [12, 7, 13]]
