
# q-expansion of newform 1003.2.a.f, downloaded from the LMFDB on 24 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 = 1003
weight = 2
poly_data = [4, 2, -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, 0], 1], [[0, 1, 0, 0], 1], [[-3, -1, 1, 0], 1], [[1, -3, -1, 1], 1]]

hecke_ring_character_values = None
aps_data = [[-1, 1, 0, 0], [0, 0, 1, 0], [0, -1, -1, 0], [2, 0, 0, 1], [-3, 1, 0, -1], [-2, -1, 0, 0], [-1, 0, 0, 0], [2, 2, 0, -1], [-2, 1, -2, 0], [-2, -1, 0, -1], [-1, -2, -1, 0], [-8, -2, -1, 1], [1, 4, 2, 0], [4, -1, 1, 1], [-8, 1, 1, -1], [3, 2, 4, 0], [-1, 0, 0, 0], [-3, 0, 3, 0], [-2, -2, -1, -1], [-1, -3, -1, 2], [6, 0, 3, 1], [7, 5, 3, -5], [-6, -4, -3, 1], [-4, 5, -1, -1], [-8, 3, 1, 1], [4, -6, -3, -1], [3, 2, 0, 3], [-1, 1, 3, -1], [-4, 2, -2, 0], [5, -2, -3, -2], [-2, -4, -2, 5], [-6, 2, -2, -2], [-1, -2, -2, -6], [14, -2, 3, 3], [-1, 6, 7, -2], [-9, 5, 1, 0], [-8, 1, 4, 0], [3, 5, 3, -2], [-9, -5, -3, 3], [-11, 2, -1, -6], [2, -3, 0, -6], [3, 0, -1, 3], [-7, -6, -2, 5], [1, 10, 4, -6], [-5, 7, 2, 1], [14, -2, -1, 2], [3, 2, 5, -2], [-1, -3, -11, 0], [-15, -1, 1, 4], [-3, -5, 3, -2], [0, 2, 8, 0], [-3, 5, 5, -3], [14, -7, -3, -2], [-7, -1, 1, -3], [7, -6, 0, 6], [-13, 5, 0, 4], [1, -3, 5, 4], [2, -14, -2, 5], [4, 3, -6, 5], [6, -11, 0, 1], [-4, 1, -3, -1], [-16, -1, -1, 6], [10, 2, 8, 5], [10, -8, -2, -1], [-16, 1, -6, -6], [3, -1, -4, -1], [-5, -7, -10, 0], [1, -1, -10, 3], [-2, -6, 8, -6], [-2, -8, -2, 10], [7, 5, -2, 3], [-14, -4, -9, -4], [-3, -1, 0, -5], [0, -2, -1, -1], [-17, 1, -9, -3], [10, -12, -2, 4], [8, 2, 0, -2], [-4, -5, 2, -10], [13, 2, -1, 2], [-1, -16, 2, 1], [9, 4, 6, -5], [-19, 2, -2, -5], [-2, -6, 3, -1], [-16, 7, -2, -3], [3, -3, -10, -5], [-11, 2, 1, 8], [31, -4, -4, -2], [-10, 20, -1, -5], [-4, 8, 11, -5], [17, -8, 7, 4], [11, -11, 2, 1], [-8, 2, -3, -5], [20, 0, -2, 5], [-4, 14, -3, -2], [8, 0, 5, 12], [3, -3, 1, 4], [1, -2, 4, -9], [6, -8, 8, 4], [25, -3, -1, -5], [10, 11, 9, 1], [14, 8, 5, -11], [-2, -17, -3, 12], [2, -12, 3, 5], [18, 8, -3, 3], [3, -11, 2, -1], [-3, -12, -5, 13], [23, 6, 11, -4], [3, 0, 9, -11], [2, -4, 14, 1], [-3, 7, 3, -8], [-1, 3, -2, 6], [2, 7, 7, 1], [5, -4, 7, 5], [-3, 11, 2, 8], [-3, 21, -3, -2], [3, -3, -6, 5], [-24, 0, -12, 1], [9, 9, 14, -4], [-25, 0, -8, 4], [4, -2, -6, 2], [-7, 18, 0, 2], [-5, 4, -5, -8], [-15, -13, -20, 9], [5, -2, -7, 8], [13, -22, -3, 2], [-11, -4, -16, 1], [-10, 11, 9, -6], [3, 10, -4, 7], [-9, 1, 6, -5], [-3, -7, -12, 1], [-3, 0, -2, 7], [-2, 0, 5, 9], [-2, -8, -18, 2], [1, -8, 1, -5], [-19, -16, -21, 9], [-2, 19, 4, -6], [5, 5, 9, -4], [9, 5, 11, -8], [5, -4, 15, 2], [-2, 22, 0, -12], [5, -1, 3, 0], [5, -7, 1, 16], [15, -2, -11, -4], [0, -2, 11, 5], [-13, 20, -1, -9], [-6, 18, 4, 4], [-17, -10, -8, 6], [-6, 3, 5, 13], [-7, -12, 5, 2], [29, 11, 4, -7], [10, -11, 0, 7], [-13, 9, -3, 8], [15, 17, 7, -19], [26, -8, -7, -5], [7, -3, 3, 3], [5, -15, -22, 4], [15, 17, 7, -10], [12, -8, -8, 0], [-28, -12, -8, 10], [12, 1, -4, 6], [-4, 14, 11, 6], [1, -1, -4, 5], [-3, 1, -11, 4], [13, -1, 4, 2], [8, 1, -15, -1], [31, -1, 1, -2], [10, 10, -13, 3], [-22, -11, -20, 3], [14, 0, 6, 12], [-17, -2, -9, -2], [-11, -3, 5, -14], [-5, -14, -5, 18], [20, -14, 3, 0], [-29, 13, -5, 0], [-4, 6, -3, -8], [-7, 22, 7, -3], [-11, -5, -15, -7], [9, 6, -9, -6], [-1, 5, -2, 3], [-17, 8, -10, -14], [-16, 18, -1, -10], [-10, 2, -7, -10], [11, -13, 2, 7], [6, 1, -6, -9], [-2, -6, -7, -7], [7, 4, -12, 11], [-5, 12, -13, -4], [-17, -5, 13, -2], [21, -4, 11, 6], [-35, 0, -11, -8], [2, 7, 18, 8], [-7, 6, 4, -7], [-3, 8, -8, -5], [-20, -10, -13, 11], [14, -18, 6, 17], [-2, 26, 3, -11], [-4, 8, 5, 9], [-2, 1, 12, -14], [31, -4, -2, 7], [-20, -5, 0, 2], [30, 10, 17, -1], [6, 16, -4, 3], [-20, 22, -5, -9], [-1, 0, 9, -10], [-33, 23, 4, -4], [-31, 13, 13, 2], [-17, -10, -7, -6], [-7, 2, -17, 6], [24, 4, -2, -4], [1, -10, 17, -6], [17, 3, 12, -13], [6, -1, -5, 10], [-2, 2, -19, 2], [25, 9, 1, -15], [-22, -10, 6, -5], [-25, -2, -20, -3], [-1, 3, 14, -17], [-5, -7, 10, 9], [-2, 0, -17, 13], [-6, 24, 12, 0], [26, -15, -17, -3], [16, -16, 3, -3], [19, -38, 3, 10], [-4, -12, 17, -10], [9, 18, 5, 6], [-10, 25, 0, 6], [-6, -5, -4, 1], [-12, 4, -7, 19], [-13, -30, 0, 7], [1, -27, -4, 16], [-24, 19, 7, 1], [14, 6, 11, -17], [21, 18, 9, -10], [-2, 28, 5, -3], [20, -9, -9, 15], [-14, -6, -12, -7], [14, 6, 7, -15], [-10, 8, 21, 7], [15, 20, -1, -2], [-13, -1, -7, 4], [2, -12, 21, 2], [5, -27, -1, 12], [-13, -13, -15, 15], [-32, -7, -10, 7], [31, -20, 17, 13], [-30, 4, -12, -6], [-29, -10, 13, 0], [9, 6, 20, -13], [16, -30, -26, -3], [-13, -7, 12, -14], [22, -23, -5, 0], [7, 8, 0, 13], [-10, 11, -12, -12], [2, 1, 11, 4], [-21, -24, -28, 6], [-5, 7, 0, -23], [-23, -10, -8, 8], [11, 15, 3, -8], [-27, 7, -2, 9], [-7, 9, 10, -21], [24, -1, -9, -9], [4, 2, -11, -12], [-16, -19, -11, 18], [-18, -14, -3, 3], [-25, 6, 2, -8], [13, -20, 6, 1], [-28, -17, -8, 9], [4, 15, 8, 12], [-12, -24, -23, 12], [3, -4, -18, 4], [-12, 33, 28, 0], [-31, 15, -16, -9], [-18, 8, -11, 9], [16, -14, -5, 20], [-8, -7, -4, -5], [-5, 3, -4, -1], [-7, -17, -26, 1], [11, 10, 5, -29], [-24, 0, -17, -15], [14, 0, -2, 27], [-44, 18, 8, -4], [-14, 11, -11, -13], [-30, -12, -5, -3], [-2, -14, 1, -11], [8, 2, 24, 8], [-41, 20, 18, 3], [21, -26, 12, 13], [-19, 2, -11, -15], [11, 6, 15, -8], [-10, 15, 6, -21], [12, -40, -6, 4], [5, 35, 15, -16], [-19, -21, -23, 0], [-16, 13, 8, -4], [-33, 4, 13, 9], [17, -20, 6, 11], [-24, 2, -2, 22], [12, 5, 1, -16], [9, 5, 28, -9], [9, -8, 7, -2], [-6, -1, -7, 10], [-21, 24, 4, 5], [4, 18, 3, 7], [-20, 18, -3, -1], [-10, 6, 10, -3], [-3, 4, 14, -13], [-25, 15, -5, 5], [6, -2, 0, 12], [-24, 12, 12, 0], [-28, -31, -16, 18], [17, -27, -16, 0], [41, 4, -8, 2], [39, -4, 10, 12], [-9, 23, -12, -17], [-21, -9, -12, -8], [38, -29, 1, 13], [-13, 24, 8, -5], [18, 4, 9, 24], [3, -3, -14, 1], [-18, 8, -4, 16], [-19, 23, -3, 2], [-22, -11, -12, 4], [-2, -13, 24, -3], [38, 4, 20, -1], [-2, 13, 7, -16], [-7, -12, 15, -2], [9, -8, 9, 14], [2, 18, -5, -9], [-29, 6, 22, -5], [5, 21, -15, 8], [43, -2, -15, 4], [3, 1, 20, 3], [45, 12, 19, -12], [3, 13, -11, -5], [11, -23, 14, 7], [-33, -5, -14, 2], [-5, 14, 14, 4], [-19, -3, 21, -12], [4, -4, -21, -9], [-8, -18, -11, 0], [7, 3, 16, -5], [14, -11, -12, -16], [-24, 11, 20, 2], [-2, -10, -23, 17], [-27, -14, -34, 7], [-11, -19, -15, 26], [-20, 3, -8, 14], [-10, 34, 1, -20], [-5, -38, -22, 15], [-23, -12, -7, -10], [-49, 5, -18, 1], [35, -14, 8, -2], [30, -21, 6, 17], [17, 4, -9, -10], [-2, -26, -39, -5], [-3, -28, -19, 12], [-16, 25, -3, 7], [-6, -10, 0, 3], [16, 5, 16, -11], [21, 39, 16, -22], [37, 21, 22, -13], [3, 2, -9, 9], [-9, 27, 5, -11], [-9, -1, -12, -1], [38, 24, 17, -23], [-34, 0, 9, -5], [-24, -16, 13, -15], [-30, -2, -8, -3], [36, 16, 1, -13], [-8, 24, 6, 2], [-62, 1, -7, -9], [36, -12, -10, 12], [14, 4, -6, 30], [12, -21, 12, 12], [-2, -7, 6, -13], [11, 24, 10, -19], [-18, 23, 3, -17], [23, 23, 19, -4], [-18, 18, -5, -31], [51, 2, -2, 8], [21, 2, 4, 20], [-13, 28, 13, -2], [-6, 0, 5, -19], [-2, -10, -10, -12], [10, -14, -7, -8], [22, -11, -9, -7], [-8, 0, 8, -5], [-21, 25, 17, -2], [-18, 4, -3, -27], [30, -19, 22, 22], [-45, 1, 8, -5], [-36, 3, -8, -2], [-2, -13, -14, 12], [20, 12, 13, 1], [-41, -23, -12, 29], [51, -9, 5, -2], [-17, 1, -1, -27], [-4, 3, -7, -2], [-55, -3, -1, 3], [-14, -1, -21, -20], [-28, -18, -6, 28], [-8, -2, 10, -20], [-35, -9, 11, -3], [11, -11, 25, -2], [-5, -12, 19, 11], [-15, -31, -4, -5], [58, 15, 10, -16], [-31, -20, 9, 5], [25, 45, 12, -23], [32, 34, 30, -14], [-1, -14, -23, -13], [14, 18, 19, 9], [21, 12, 45, 2], [-17, -15, -8, 12], [-11, -14, -2, 32], [-25, -18, -6, -12], [-6, 36, -7, -19], [23, -12, -7, 4], [-19, 19, -5, -12], [40, 6, 27, -2], [37, 8, -17, 14], [-12, 22, 7, 19], [-52, 18, -17, -19], [20, -4, -14, 15], [11, 16, 42, -6], [-75, 0, 2, -2], [3, -11, -9, -14], [-19, -11, -3, 20], [35, 13, 37, -1], [7, -5, 14, 7]]
