
# q-expansion of newform 1183.2.a.m, downloaded from the LMFDB on 17 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 = 1183
weight = 2
poly_data = [-2, -6, 7, 8, -5, -2, 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, 0, 0], 1], [[0, 1, 0, 0, 0, 0], 1], [[-2, -1, 1, 0, 0, 0], 1], [[1, -3, -1, 1, 0, 0], 1], [[2, 2, -4, -1, 1, 0], 1], [[-2, 4, 6, -4, -2, 1], 1]]

hecke_ring_character_values = None
aps_data = [[-1, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0], [-1, 0, -1, 0, 1, 0], [1, 0, 0, 0, 0, 0], [0, -1, 1, 0, 1, 1], [0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, -1], [0, 0, 2, 0, 0, 1], [-2, -2, -2, 0, 0, -2], [0, -2, -1, -1, 1, 2], [3, 0, 1, 1, 0, 2], [-3, 0, -1, -2, -1, -3], [-5, 0, 1, -2, 1, -1], [-1, 2, -3, 1, -1, -2], [-3, 0, -1, 3, -2, -2], [-3, 0, 0, 0, -2, 2], [-1, 2, 3, -1, 0, -2], [-2, -3, 0, -1, -1, -2], [-5, 2, 3, 1, 1, -1], [0, -2, -2, -2, 2, 0], [-5, -2, -1, -2, 1, 1], [-4, 0, 4, 0, 0, 2], [-5, 2, -1, -1, -2, 0], [-6, 0, 0, 0, -2, 3], [-5, 4, 3, 1, -2, 1], [2, -1, 6, 3, 2, 0], [3, 3, 2, -1, -2, 3], [-6, 5, -1, 4, -2, -2], [-3, 6, 1, 1, -1, -3], [-1, 0, 2, -2, 2, 2], [-1, 2, -5, -1, 1, -2], [1, 5, -4, -1, 1, 1], [-5, -3, 0, -4, 4, 6], [1, -5, -4, 5, 1, 3], [-1, 3, 2, 0, -2, -2], [-5, 8, 1, -3, 1, 3], [-1, 4, 2, 6, -4, -1], [4, -4, -8, 0, 0, -2], [-3, -2, -3, -3, 0, -4], [5, 1, -2, -3, 0, -1], [-7, -5, 2, -1, 3, 2], [6, -5, -6, -1, 1, -2], [-4, 3, -5, -2, -2, -2], [-3, -4, -5, 6, -1, 1], [2, 3, 3, 2, -5, 5], [3, -5, 6, -5, 6, 5], [-3, -1, 0, 1, 5, -2], [-3, 2, -1, 1, 4, 5], [11, -2, 3, -3, -2, 0], [2, -4, 2, 10, 2, -2], [-6, 11, 5, 0, 0, 0], [-3, -6, 5, 5, -5, 3], [10, -4, -4, 1, -5, 2], [2, 0, 0, -8, 2, 2], [9, -2, -8, -2, 2, -3], [-3, -1, 2, 7, 3, -4], [11, 5, 6, -3, 1, 7], [5, 4, -1, -7, 2, 2], [7, -6, 6, 0, 2, 6], [-5, 2, 3, 8, -5, -1], [-10, -4, -6, -2, 6, 2], [-5, 10, 7, 0, -7, -1], [3, -2, -3, -5, 2, -3], [5, -11, 0, -1, 3, 1], [10, -6, -6, 4, 8, 2], [-14, -2, 4, 3, -2, 0], [6, -3, -7, 2, 5, 3], [8, -10, -1, -1, 3, 6], [10, -10, -2, 0, 4, -2], [7, -4, 5, -1, 0, 6], [-7, 0, -3, 0, -3, -12], [5, -7, -2, -3, 0, 4], [2, 12, -4, 4, -2, -6], [-3, 1, 1, -10, 0, -6], [-8, 1, -5, -6, 5, 3], [9, 0, 1, -1, 0, 5], [0, 5, 6, 9, -1, 4], [-8, 10, -6, 4, -8, -10], [1, -1, 4, 4, -4, 0], [-11, -8, -7, 8, -5, 2], [17, -5, 8, 5, 0, 3], [13, 6, 1, 2, -3, 3], [-5, 5, 4, 11, -10, 2], [9, 2, -6, 4, -7, -9], [1, 3, 4, -3, 11, 9], [-16, 8, -4, -4, -8, 0], [-18, -6, 0, -6, 4, 0], [1, -16, 3, -4, 3, 1], [-1, 6, -7, 6, -5, -6], [1, 10, 5, 1, -5, -11], [11, -1, 0, 1, -4, 5], [10, -6, 10, 6, -4, 1], [-3, -5, 0, 9, 0, 6], [9, -12, 1, 7, 3, 4], [9, -9, -4, -1, 10, 4], [-2, 4, 4, 8, -9, -2], [-4, -4, -4, 1, 1, 5], [6, 3, 2, -1, -5, 2], [-4, 0, -12, -4, -2, 2], [-15, 5, 2, 0, 4, 2], [11, 5, 4, -9, -1, -4], [-12, 9, 11, 3, 1, -1], [10, -4, -2, -6, 8, -6], [-8, -4, -2, -2, 0, 8], [12, -1, 1, 0, 2, 0], [-12, 10, -2, 1, 1, -7], [-21, 0, -9, 1, -2, -11], [-8, -4, 6, -3, 11, -2], [-9, -2, 3, -5, 11, 4], [12, -1, 6, -5, 0, 8], [5, 5, 6, 9, -2, -7], [6, -3, -3, 5, 3, 1], [-16, -4, 4, -5, 2, 2], [16, 2, -6, -8, 10, -10], [-2, 5, 5, 2, 9, 3], [-1, 1, -13, 4, 2, -4], [3, 0, -3, 11, 0, -2], [14, 4, 8, -4, -2, 4], [-5, -4, 1, -9, -7, -4], [-8, 8, 4, 12, -14, -2], [7, -10, -5, -8, 13, 1], [-1, 8, 4, -4, 0, 8], [2, -6, -12, -2, 7, 0], [-7, 6, -3, -1, -9, -7], [0, 16, 14, -2, -2, 1], [-10, 2, -10, 16, -8, -8], [1, 3, 0, -8, 8, -8], [-13, 17, 2, -3, -6, -9], [-3, -9, 2, -13, 4, 5], [2, 16, 12, 3, -5, 4], [10, -4, 6, -6, 0, -2], [-9, -7, -4, -1, 0, 4], [-3, 1, -4, -5, -3, -18], [20, 10, 16, -2, -6, 6], [-21, 6, -3, 5, -4, -7], [13, -2, 9, 5, -8, 4], [-5, 8, 5, 7, 4, 2], [0, -4, 0, -2, -4, 4], [30, -14, -10, 0, 0, 0], [5, -2, -6, -8, 8, -2], [12, -4, -6, -8, 4, -9], [19, -2, 7, -5, -1, 11], [5, -1, -6, 3, 5, -6], [7, 13, -4, 3, 6, -6], [0, -1, 4, 5, -4, 2], [-8, -10, -12, 0, 8, 5], [0, 0, -16, -3, 1, -5], [3, -8, -12, 0, -2, -3], [-6, 0, 8, 12, -10, 10], [-28, 16, 4, -4, -6, -8], [0, -8, 0, -9, 2, -6], [-1, -16, -6, 6, -1, 1], [-7, 7, 0, 9, 1, -2], [-10, -4, -14, 2, -4, -16], [4, -6, -10, -4, -2, 6], [-12, -13, -11, 0, 8, 6], [-20, 4, 6, -2, 10, 2], [-28, -2, -2, -3, 9, 3], [-9, 12, 0, 4, 5, -1], [-7, -10, -19, -7, 6, 6], [-8, -1, -11, 10, 1, 3], [-11, 17, 10, 1, -5, 0], [5, -2, -3, -1, -9, -7], [17, 11, 12, 5, -8, 7], [-18, 6, 0, 1, 6, 0], [9, -4, -7, 9, -4, 5], [-12, 7, -1, -2, 12, 0], [-13, 2, 16, -10, 11, 1], [-1, -11, -20, -1, 12, 6], [-21, 8, -12, 4, 3, -3], [9, 7, 4, 7, -4, -6], [22, 2, 2, -1, 5, -4], [5, -12, 3, -11, -1, 14], [2, 13, -3, 3, -11, -11], [2, -6, -8, -2, -3, 8], [-1, -8, -12, -4, 3, -11], [6, 16, -10, -4, 6, -4], [-16, -1, 9, -1, 17, -5], [-23, 29, 2, 1, -7, -5], [-5, 7, 8, 9, -11, 1], [19, 0, 5, 5, 3, 1], [0, 6, 8, 14, -13, -8], [-33, 16, 2, -2, -8, -8], [-4, 0, 10, 10, 0, 3], [6, 5, -11, -6, -13, 5], [5, 8, 0, 0, -9, 3], [4, 4, 2, 10, -6, -4], [-5, -10, -7, 3, -4, -1], [7, -16, 17, 2, -3, 13], [6, -8, -4, 12, 0, -11], [-9, 6, -12, -10, -9, -1], [19, -9, 14, -3, 0, 12], [2, 3, 19, -6, 2, 6], [30, -18, -18, 8, -2, 3], [6, 3, -9, -6, -10, -4], [3, -13, -2, -13, 2, -3], [6, -3, -3, -1, -11, -9], [-35, 21, -1, -4, -8, -10], [-5, 4, -7, -4, 11, -1], [-3, 3, 16, 1, -6, -5], [-12, -13, 5, -15, 1, 1], [4, 14, -6, 4, 5, -4], [12, -12, -4, -13, 15, -5], [-14, 15, -20, 5, -14, -14], [5, 4, 7, -13, 22, 8], [-15, 18, 14, 0, -13, -1], [-8, 14, -10, 4, -2, -6], [-3, -15, -16, 7, -5, 6], [-10, 13, -7, -1, -1, -17], [8, 8, 4, -4, 14, 6], [14, 2, -5, -1, -1, 10], [21, -7, 16, -1, 11, 11], [-23, -3, -10, -7, 3, -16], [-3, -6, -3, -9, -8, -4], [-15, -6, -5, -19, 6, -1], [16, -2, -2, 1, -5, 11], [15, 1, 0, -9, 7, 8], [31, 12, -2, 10, -10, -3], [-1, 13, -8, -13, -4, 0], [-14, -7, 5, 6, -13, 5], [-18, 10, -17, -5, -1, -6], [23, -22, 3, -15, -4, 16], [-5, 8, -3, -4, -11, 5], [25, -23, 10, -11, 0, 16], [-2, -24, 8, -12, 5, 10], [0, -4, 9, -9, 7, 12], [-17, 12, 4, -8, 9, 9], [-15, 4, 13, 11, -19, 2], [-11, -1, 6, 3, -9, -13], [-7, -16, -1, 1, -7, -15], [22, -5, 6, 11, -9, -8], [39, -9, 4, -7, 8, 9], [-15, -3, 2, 3, -18, 6], [10, -7, -10, -11, 3, -12], [6, -8, -12, 0, -10, 10], [1, -10, 1, -29, 6, 8], [-7, 10, 15, -5, 6, 9], [18, 0, 14, -11, -2, 2], [-4, -2, 4, 6, 6, 8], [5, 11, 6, 7, -20, 3], [-18, 11, -5, -8, -3, -1], [-23, 7, 20, 13, -10, 7], [-1, -1, 18, 7, 5, 21], [24, -2, 14, 7, 2, 0], [-13, -18, 7, 12, -7, 9], [-14, 0, -6, 2, -18, -10], [-18, 6, 4, 8, -16, -2], [12, 6, 0, 18, -6, -9], [0, -16, -12, 8, -6, -10], [-3, 10, -7, -7, 1, -24], [-5, 13, 12, 11, 6, -2], [-13, -10, 5, 21, -2, 7], [21, -2, 13, 9, -7, -7], [-13, 4, 4, -16, 11, 3], [23, 9, -4, -1, 4, -1], [9, -18, 1, 5, -5, -1], [2, 7, 5, -28, 2, 4], [-29, -12, 13, 3, -2, 6], [-11, 9, 6, 5, -10, -5], [-24, 4, 26, -7, 15, -4], [-3, -13, -22, -9, 17, -14], [5, 7, 6, 15, -11, 14], [20, -12, 24, 9, 3, 13], [15, -22, -8, -2, 13, 11], [-21, 10, -1, 28, 1, -5], [-22, -4, -20, 12, 6, -10], [-21, 6, -5, -9, -13, -15], [-10, 32, 8, -1, -3, -6], [-23, -18, -19, 3, 11, -15], [-1, -6, -6, -4, -6, -10], [7, -6, 28, 6, 0, 3], [-25, 13, 4, -5, -4, 2], [-5, 8, 1, -2, -15, 15], [13, -13, -12, 13, 13, 12], [0, 6, -12, 6, -18, -22], [-20, -2, 8, -8, 0, -9], [-37, 8, -15, -5, -5, -1], [-19, 16, 3, 12, -21, -9], [-9, 2, -9, 10, -23, -11], [7, -9, 6, -9, -8, -1], [17, -27, -6, 9, 1, 5], [-1, -9, 6, -11, -2, -2], [14, -6, 10, 12, -11, -12], [13, 4, 3, -10, -5, -11], [3, -12, -11, -13, 0, 0], [-22, 28, 8, -12, 6, 0], [5, 1, -11, -6, 18, 0], [4, -1, -3, 13, 5, -13], [24, 10, -6, -20, 0, -6], [14, -17, 20, -3, 12, 20], [21, -5, -28, -3, -9, 0], [15, 5, -8, -13, 9, -11], [-37, 6, -1, -4, 3, -1], [13, 0, 21, -23, 14, 11], [38, -12, 8, 12, 0, 2], [19, -12, -8, 24, 2, 5], [-36, -2, -2, -24, 3, -2], [-23, 2, -12, 14, 0, -1], [-2, 18, 22, -4, -17, -2], [6, -24, -12, -2, 4, -17], [-16, -9, -16, 13, 3, -14], [-9, 3, 2, -8, 18, -2], [18, 14, 0, 18, -4, 8], [-3, -11, 12, 3, 7, -6], [-11, 19, -4, -3, -1, -2], [-4, 6, -2, 1, 20, 12], [-15, 17, 4, -1, 13, -4], [27, 2, -12, 18, -5, -1], [-1, -2, -13, 7, -3, -9], [-2, -27, 0, 7, -1, 4], [19, -27, -10, -15, 26, 9], [-3, -6, -7, 1, -15, 9], [11, 14, 0, 2, 4, -2], [12, -15, 15, 4, 6, 12], [11, 3, 6, -7, -10, 2], [-33, 0, 4, 8, -10, 7], [-15, 10, 9, 19, 10, 4], [-22, 12, 2, 0, 0, 6], [17, 10, 5, -11, 2, 0], [20, -15, 8, -21, -7, 8], [1, 7, -4, -11, 10, -7], [-21, -3, 4, 3, 10, -18], [-11, 2, 6, -8, 5, 1], [-28, -1, 21, -12, -1, -3], [-16, -22, -10, -4, 0, -8], [-11, 0, -8, 12, -18, -15], [-42, -13, 7, 12, -9, 5], [26, 26, -6, 8, -4, 1], [0, 15, -15, 4, 8, 2], [2, -13, -4, -3, 15, 8], [8, -10, -26, -13, 4, -6], [-16, -12, 13, 3, 9, 6], [34, -12, 0, 12, 2, 2], [-5, 12, 29, 9, 2, 6], [11, -15, -24, 3, 2, 10], [-18, -6, 4, -20, 2, 4], [14, -16, -2, -11, 18, 8], [-18, -22, 8, -2, 8, -4], [-28, 8, -2, 1, -9, -6], [31, -5, 8, -3, 14, 13], [-13, -3, -8, -13, 12, -10], [-2, -27, -8, 23, -7, -4], [18, 10, 14, -1, -5, 14], [0, -16, -4, -8, 28, -4], [-2, -28, 14, -10, 10, 22], [-34, 10, 8, -6, -12, -6], [12, -33, -4, -19, 18, 4], [36, -10, -2, -4, -14, 2], [13, -6, -11, -5, 0, 14], [7, 1, 8, -5, -2, 9], [-13, 6, 27, -9, -10, 2], [-22, -22, -20, 22, -4, -8], [24, -24, -4, 7, 8, -6], [9, 21, -4, 3, -9, -7], [-5, -10, -22, 8, -4, -4], [-23, -12, -18, 2, 6, -19], [12, 2, 4, -10, 20, -3], [6, -20, -10, -10, 8, 0], [48, -10, 5, -7, -1, 12], [4, -4, 0, 26, -16, -4], [48, -5, -3, -1, 15, -3], [14, 11, 0, 1, -15, -10], [-36, 46, 12, 14, -10, -14], [45, 8, 7, 5, -7, -7], [14, -24, -10, 10, -10, -4], [1, -17, 16, 29, -5, 2], [-24, 4, -20, -8, -14, 2], [2, 1, 25, 6, 15, -7], [-21, 23, -3, 4, -12, -26], [-32, 14, 28, -10, 16, 2], [7, -3, -28, -1, -16, -11], [-12, 19, -15, -2, -13, -19], [-23, 17, 27, 12, -6, 8], [26, 14, 2, -4, 2, -14], [-6, -18, -9, -25, 21, 8], [-13, 20, -15, 5, -14, -4], [-16, -18, -2, -24, 6, -6], [9, 18, -9, 7, -16, -25], [-11, 36, 2, 18, -11, -13], [47, 0, 8, -8, 3, 5], [17, -36, -7, -9, 10, 15], [18, -20, 6, -19, 16, 10], [11, 0, -7, 7, -7, 1], [-9, 6, 23, -9, 23, 21], [1, -26, 11, 3, 2, 7], [0, 10, 6, 14, 14, -6], [-17, 18, -25, -5, 19, -3], [1, -15, 2, -7, -6, -7], [-19, 36, 3, 3, -15, -15], [-2, 8, 20, -16, -16, -2], [1, 9, 2, 18, 0, 0], [-22, -20, -14, -1, 7, 1], [21, -19, -22, -7, 6, -8], [3, -3, 10, 9, 10, 5], [31, -7, 4, 23, 10, 16], [32, -6, 0, 6, 18, 4], [19, -18, -23, 9, 8, 17], [26, 12, 12, -16, 2, 12], [9, 28, 5, -9, -1, -28], [13, -5, -12, 2, 24, -4], [1, -14, -19, 1, -10, 16], [29, 8, 27, -7, 9, 10], [-12, 6, 6, 28, -12, 8], [-44, 2, -24, -3, -5, -12], [-16, -4, -10, 23, 1, -16], [25, -21, -4, -23, 14, 5], [14, -18, 12, -2, -6, -6], [-17, -4, 13, -24, 15, 11], [-26, 10, 12, 22, -4, 8], [-20, 28, -24, 4, -12, -36], [-31, -16, -34, -2, 8, 5], [-22, -11, 8, -9, 15, 4], [-21, 3, 28, -19, -9, -4], [4, -20, -2, -2, 0, 6], [31, -8, -14, -18, -8, -4], [4, 19, -20, -3, -1, -12], [8, 8, -4, -6, 14, 0], [-2, 8, 26, 14, -13, 12], [-7, -6, -7, 17, 16, 8], [-21, 15, -8, -15, 18, 11], [-17, -16, -8, 0, 18, 18], [-8, -14, -22, -1, 27, 3], [13, 30, 33, -3, -5, 5], [-1, 5, -4, 15, 14, 3], [-8, 2, 14, -4, -10, 2], [-34, 46, 10, 7, -23, -10], [36, -8, 4, 8, -20, 0], [17, -5, 0, 12, 16, 24], [-15, -16, -21, -17, 24, 8], [21, 19, -4, 1, -6, 9]]
