
# q-expansion of newform 2352.2.bl.f, downloaded from the LMFDB on 25 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 2352
weight = 2
poly_data = [1, -1, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[1471, [-1, 0]], [1765, [1, 0]], [785, [1, 0]], [2257, [0, 1]]]
aps_data = [[0, 0], [-1, 1], [4, -2], [0, 0], [2, 2], [3, -6], [4, 4], [0, -7], [0, 0], [0, 0], [-5, 5], [0, -1], [6, -12], [1, -2], [0, 6], [0, 0], [0, 0], [0, 0], [-1, -1], [-2, 4], [-5, -5], [18, -9], [-6, 0], [-8, 4], [-4, 8], [2, 2], [0, -5], [8, -4], [-5, 5], [6, 0], [-9, 18], [0, -18], [12, -12], [5, 0], [0, 12], [6, 6], [8, 8], [4, -2], [6, 0], [0, 0], [10, 10], [9, -18], [4, -2], [-13, 13], [12, 0], [-16, 16], [10, -20], [-8, 0], [30, -30], [-2, 1], [0, 18], [-10, 20], [8, 8], [-24, 0], [12, -6], [-16, -16], [18, 18], [0, 16], [17, -17], [12, 0], [5, -5], [-12, 24], [25, 0], [-18, 18], [-18, 9], [0, 12], [-10, 5], [-19, 0], [-20, -20], [-8, 16], [-10, -10], [-8, 4], [-37, 37], [0, -11], [-7, 14], [0, -36], [-30, 30], [-6, 3], [0, -24], [-11, -11], [-30, 0], [-29, 0], [-10, -10], [-17, 34], [0, -16], [-8, 4], [-6, 0], [0, -5], [-8, 16], [7, -14], [0, 6], [-24, 24], [-7, -7], [-12, 24], [14, -7], [-30, 0], [4, -2], [8, 8], [0, 1], [0, 41], [6, -12], [-18, 18], [-18, 18], [0, 6], [15, 15], [1, 1], [12, 0], [44, -22], [-4, -4], [-15, 30], [0, -37], [22, -22], [-42, 0], [-35, 35], [2, -4], [36, -36], [43, 0], [18, -18], [0, 18], [12, -24], [-11, -11], [1, 0], [-40, 20], [24, 24], [0, 13], [0, 0], [0, 10], [0, 18], [-1, 0], [42, -21], [-21, -21], [2, -4], [54, -27], [-22, 0], [-40, 20], [-17, 34], [18, 18], [-8, 8], [24, -48], [-54, 54], [-16, 0], [0, 6], [10, 10], [10, -20], [13, 13], [-24, 0], [1, -2], [0, 0], [0, -40], [36, -18], [0, 22], [-16, 32], [-7, 14], [0, 30], [9, 9], [-12, 24], [-2, 1], [-4, 2], [17, -34], [-28, -28], [20, -10], [36, 0], [-13, 26], [0, -12], [-54, 54], [0, 0], [-7, -7], [-25, -25], [43, 0], [32, -16], [-4, -4], [-12, 24], [0, 0], [29, -29], [41, -41], [24, -48], [-33, 66], [42, -42], [-16, 0], [-58, 29], [42, -21], [-12, 0], [-29, 0], [24, -12], [2, 2], [20, 20], [-14, 14], [25, -25], [0, -31], [-18, 18], [-50, 25], [6, -12], [-28, 14], [-48, 24], [-24, -24], [-32, -32], [-13, 13], [0, -10], [2, -4], [0, 36], [30, -30], [-11, 0], [16, -8], [36, 36], [66, 0], [2, 2], [0, 56], [20, -10], [0, 0], [-4, 4], [0, -47], [-24, 48], [-30, 60], [0, 48], [-60, 60], [-2, 1], [38, 38], [4, 4], [48, -24], [-6, 0], [0, -17], [41, 0], [0, -42], [20, -10], [30, 0], [-26, 0], [16, -8], [-22, -22], [0, 61], [-12, 6], [31, -31], [-28, 28], [-7, 14], [6, -6], [55, 0], [18, -18], [-70, 35], [0, -66], [32, -64], [-42, 0], [22, 22], [0, 7], [41, -41], [0, 14], [-8, 16], [0, 24], [-49, 0], [6, -6], [-22, -22], [32, -64], [41, 0], [64, -32], [26, 26], [5, -10], [-16, -16], [-32, 16], [53, -53], [47, -47], [-2, 4], [-8, 4], [19, 19], [-20, 40], [31, 31], [-41, 82], [-14, -14], [0, 7], [-30, 0], [-10, 20], [38, -76], [-6, 6], [66, -33], [2, 2], [-45, -45], [46, -23], [-20, 40], [0, 4], [76, -38], [-61, 61], [0, -37], [0, 60], [36, -36], [-9, -9], [-42, 0], [35, -70], [0, 28], [-68, 34], [7, -7], [78, 0], [55, -55], [-46, 92], [-48, 48], [-48, 48], [0, -66], [-24, 0], [-1, 0], [-2, -2], [0, -19], [12, -24], [0, -54], [-73, 0], [40, -20], [0, 42], [-26, -26]]
