
# q-expansion of newform 2352.3.f.f, downloaded from the LMFDB on 19 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 = 2352
weight = 3
poly_data = [256, 16, 17, -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], [[120, -17, 17, -1], 136], [[8, 289, -17, 9], 136], [[-65, 0, 0, -3], 17]]

hecke_ring_character_values = [[1471, [1, 0, 0, 0]], [1765, [1, 0, 0, 0]], [785, [1, 0, 0, 0]], [2257, [-1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, 1, 0, 0], [0, 2, 1, 0], [0, 0, 0, 0], [8, 0, 0, -1], [0, 1, 1, 0], [0, -4, -2, 0], [0, -13, 1, 0], [24, 0, 0, 0], [2, 0, 0, -1], [0, -11, -6, 0], [-37, 0, 0, 1], [0, -14, -4, 0], [-19, 0, 0, 3], [0, -14, 2, 0], [-50, 0, 0, 1], [0, -40, 1, 0], [0, 36, 8, 0], [-1, 0, 0, -5], [54, 0, 0, -6], [0, -11, -7, 0], [29, 0, 0, -2], [0, 2, -17, 0], [0, -36, -6, 0], [0, 12, 7, 0], [0, 6, 18, 0], [0, -35, 13, 0], [46, 0, 0, -5], [19, 0, 0, 3], [-106, 0, 0, -4], [25, 0, 0, 0], [0, 90, -9, 0], [-108, 0, 0, 12], [0, 1, 19, 0], [152, 0, 0, 2], [-120, 0, 0, -5], [0, -36, 24, 0], [-18, 0, 0, -10], [0, 22, 14, 0], [0, -120, 12, 0], [-138, 0, 0, 0], [0, 11, 13, 0], [-10, 0, 0, 14], [113, 0, 0, -14], [8, 0, 0, 14], [0, 36, -16, 0], [186, 0, 0, -14], [0, -132, -33, 0], [0, -78, -9, 0], [0, 35, 9, 0], [214, 0, 0, 16], [94, 0, 0, -14], [0, 120, -1, 0], [0, -56, 35, 0], [0, -66, -24, 0], [-228, 0, 0, -18], [0, -98, 5, 0], [0, 12, -13, 0], [-7, 0, 0, -3], [-124, 0, 0, -16], [0, -97, -27, 0], [0, -188, -13, 0], [0, 25, -3, 0], [0, -110, -28, 0], [0, 107, -8, 0], [-254, 0, 0, -23], [329, 0, 0, -15], [85, 0, 0, 18], [64, 0, 0, -14], [0, -108, -4, 0], [0, -46, -2, 0], [-68, 0, 0, -32], [0, 145, 32, 0], [-91, 0, 0, 3], [-83, 0, 0, -49], [0, -268, -26, 0], [-146, 0, 0, -20], [0, -227, 9, 0], [-232, 0, 0, 20], [0, 251, -4, 0], [0, -206, -34, 0], [-31, 0, 0, 33], [-302, 0, 0, -20], [0, 181, -15, 0], [0, -204, -33, 0], [-30, 0, 0, 33], [270, 0, 0, -30], [-11, 0, 0, 0], [0, 424, -16, 0], [-293, 0, 0, 9], [0, 90, -18, 0], [0, -256, 46, 0], [287, 0, 0, 30], [-86, 0, 0, 37], [-715, 0, 0, 15], [0, -22, 100, 0], [0, 106, -49, 0], [0, 88, 2, 0], [0, 167, 35, 0], [-751, 0, 0, 15], [-150, 0, 0, -4], [424, 0, 0, 1], [0, 138, -57, 0], [178, 0, 0, 28], [235, 0, 0, 51], [0, 155, 56, 0], [0, -108, -27, 0], [0, 142, 14, 0], [-340, 0, 0, 56], [0, 263, 48, 0], [0, -131, -136, 0], [546, 0, 0, -26], [-70, 0, 0, 50], [0, -49, -73, 0], [948, 0, 0, 11], [212, 0, 0, 20], [0, 599, 3, 0], [0, 222, 66, 0], [-172, 0, 0, -67], [-776, 0, 0, -38], [0, -25, -95, 0], [-563, 0, 0, 48], [0, -300, -33, 0], [-298, 0, 0, -43], [0, 71, -111, 0], [-554, 0, 0, -35], [-30, 0, 0, -64], [0, -34, 88, 0], [0, 527, 6, 0], [0, 121, 39, 0], [247, 0, 0, -15], [-294, 0, 0, 54], [-97, 0, 0, -36], [354, 0, 0, 2], [0, 100, 56, 0], [0, -239, -22, 0], [0, -274, 58, 0], [0, 48, 152, 0], [0, -288, -93, 0], [-574, 0, 0, -22], [0, -168, -12, 0], [-700, 0, 0, 59], [-462, 0, 0, -16], [-436, 0, 0, 89], [0, -385, -95, 0], [0, 568, 50, 0], [0, 311, -65, 0], [0, 80, 4, 0], [0, -480, 86, 0], [42, 0, 0, 36], [690, 0, 0, -62], [0, 768, 12, 0], [-727, 0, 0, -69], [0, 298, -118, 0], [433, 0, 0, -67], [308, 0, 0, -88], [941, 0, 0, 39], [0, 350, -44, 0], [0, -61, 26, 0], [0, -116, -175, 0], [122, 0, 0, -52], [-784, 0, 0, -70], [-1303, 0, 0, -2], [0, 212, -95, 0], [-714, 0, 0, -24], [0, -64, -50, 0], [919, 0, 0, 26], [0, -611, 57, 0], [115, 0, 0, -22], [0, 416, -71, 0], [1318, 0, 0, 31], [0, 576, -6, 0], [164, 0, 0, 86], [-1117, 0, 0, -66], [0, 119, -112, 0], [0, -112, 64, 0], [151, 0, 0, 81], [76, 0, 0, -59], [0, 12, 80, 0], [0, 167, 87, 0], [145, 0, 0, -60], [0, -52, 169, 0], [125, 0, 0, 21], [0, -92, -256, 0], [934, 0, 0, 52], [0, -476, -43, 0], [-582, 0, 0, -28], [0, -409, 79, 0], [775, 0, 0, 14], [0, -614, -4, 0], [0, -673, 106, 0], [144, 0, 0, -39], [1026, 0, 0, 28], [0, -904, -104, 0], [262, 0, 0, -71], [0, -616, -110, 0], [-185, 0, 0, 33], [714, 0, 0, 46], [0, -642, -108, 0], [0, -364, -152, 0], [1148, 0, 0, 71], [0, -227, 145, 0], [0, 756, 104, 0], [0, -348, -200, 0], [0, 226, 74, 0], [0, 382, 35, 0], [0, 684, -92, 0], [312, 0, 0, -87], [108, 0, 0, 114], [0, -372, -170, 0], [169, 0, 0, -49], [0, 456, 33, 0], [492, 0, 0, -5], [0, -112, -251, 0], [0, -1036, -26, 0], [0, 647, -133, 0], [-448, 0, 0, 99], [0, 1044, -96, 0], [-2264, 0, 0, -32], [-398, 0, 0, -8], [947, 0, 0, -125], [0, 241, -27, 0], [470, 0, 0, -166], [1940, 0, 0, -63], [0, 950, -83, 0], [1470, 0, 0, 50], [0, -496, -200, 0], [-886, 0, 0, -34], [0, 1103, -20, 0], [1002, 0, 0, -48], [211, 0, 0, 51], [0, -204, -126, 0], [239, 0, 0, -138], [362, 0, 0, 80], [0, -289, -355, 0], [0, -282, -78, 0], [0, -527, -172, 0], [468, 0, 0, 51], [852, 0, 0, 162], [0, -754, -38, 0], [26, 0, 0, 200], [0, -155, -193, 0], [0, 371, 64, 0], [-1062, 0, 0, 86], [0, -1440, 48, 0], [0, 664, -154, 0], [0, -61, 146, 0], [0, -158, -406, 0], [178, 0, 0, -126], [-796, 0, 0, -22], [439, 0, 0, -21], [0, 504, 168, 0], [18, 0, 0, 6], [0, 95, -13, 0], [0, 416, -131, 0], [-634, 0, 0, -7], [-1235, 0, 0, 131], [0, -443, -45, 0], [0, 474, -75, 0], [0, -1308, -57, 0], [-281, 0, 0, -147], [498, 0, 0, -123], [0, -563, -119, 0], [0, 361, 379, 0], [0, 318, 174, 0], [0, -215, -3, 0], [636, 0, 0, 183], [0, 978, 258, 0], [-806, 0, 0, -120], [-1220, 0, 0, -107], [0, 541, -127, 0], [-90, 0, 0, 40], [0, 1417, -18, 0], [-185, 0, 0, -40], [0, -876, 255, 0], [0, 792, -164, 0], [1152, 0, 0, -141], [-449, 0, 0, -129], [-1493, 0, 0, 57], [0, 508, 152, 0], [0, 1708, -16, 0], [-407, 0, 0, 96], [0, -442, -254, 0], [0, 577, 235, 0], [0, -1188, -44, 0], [1374, 0, 0, 108], [1073, 0, 0, 76], [2704, 0, 0, -29], [0, 253, -122, 0], [0, -370, -494, 0], [2740, 0, 0, 10], [0, 104, 259, 0], [534, 0, 0, 114], [0, -2024, 20, 0], [-847, 0, 0, -159], [0, 986, 1, 0], [0, 1271, 203, 0], [0, 700, 80, 0], [0, -866, -286, 0], [0, 1355, 161, 0], [0, 180, -128, 0], [224, 0, 0, -13], [2668, 0, 0, 33]]
