
# q-expansion of newform 128.4.a.f, downloaded from the LMFDB on 18 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 = 128
weight = 4
poly_data = [-3, 0, 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], 1], [[0, 4], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0], [-2, 1], [2, 2], [4, 2], [-46, -1], [50, -6], [46, 12], [-2, -7], [-4, -18], [42, 10], [192, 16], [-86, -14], [-150, 8], [-150, -5], [8, -44], [-6, -14], [322, -33], [146, -70], [86, -83], [-204, 42], [206, -52], [200, 36], [-474, 13], [286, -116], [1102, 92], [-1062, -14], [1732, -14], [722, -145], [714, 42], [98, -96], [1328, -120], [-474, -171], [-710, 40], [-1094, 179], [882, 386], [-772, 302], [2082, -6], [-386, 401], [-364, -246], [-2622, 26], [678, 253], [1250, 98], [-2480, -232], [-242, -132], [-2894, 34], [-380, 162], [2806, 149], [4416, -144], [-2130, 233], [1138, 66], [2286, 428], [-936, 428], [622, -756], [3042, -49], [2322, -192], [2484, 714], [1882, 202], [1912, -244], [3642, -366], [-1458, -132], [-4566, 291], [3442, -478], [942, -127], [5596, -722], [-2150, 840], [4074, -534], [7042, -169], [3154, -304], [3042, 127], [-4358, -1046], [5010, -944], [8644, 306], [1928, 1268], [-7094, -302], [7058, 1031], [-6144, -512], [7786, -206], [3210, 1418], [5070, 236], [2266, 1256], [-8954, -683], [-1694, -990], [-9832, -340], [-10098, 172], [3228, 206], [-1494, -1229], [-3346, -4], [-4566, -1304], [-1190, -342], [-2872, 1572], [-7474, 1377], [11328, -1104], [-10732, -22], [-326, 2147], [-2954, -795], [-1940, 422], [16066, -230], [-7270, -504], [-638, 311], [-12302, -1478], [13382, -459], [-8206, 154], [-9810, -1407], [2090, -1144], [12682, -413], [2114, -240], [-12686, -1745], [7986, -1760], [5580, -1082], [-18114, -180], [6928, -1880], [-10526, -478], [17086, -740], [2018, 919], [-11860, -250], [-17874, -36], [-1098, 3773], [-11340, 2506], [-3078, -1590], [-10458, -979], [8002, 1762], [1102, 828], [2570, -398], [13146, 1491], [-11610, 765], [2834, -4006], [-3382, 2578], [-9432, 436], [3276, -3450], [13866, -1174], [-11346, -3607], [9172, -694], [2488, -2692], [16370, -1726], [13594, 2120], [-8338, -964], [8106, 82], [-7178, -2379], [37026, -230], [-16278, 328], [-9438, -1897], [7882, 1938], [4828, 1262], [7898, -3925], [12610, 2330], [-15324, 2130], [11738, 3314], [4938, 4680], [7298, 3215], [-10528, -160], [20474, -86], [-25486, -1584], [-802, -3943], [-35556, -914], [-9782, 3227], [-9192, -4420], [10332, -194], [21102, -3012], [-14162, 4588], [-7766, -3286], [39758, 337], [-52022, 488], [-10780, -6094], [-3638, -3269], [-19506, -1588], [-7892, 6486], [17264, 4408], [3690, -2478]]
