
# q-expansion of newform 192.4.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
    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 = 192
weight = 4
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[0], [-3], [18], [8], [-36], [10], [18], [100], [72], [234], [-16], [226], [90], [-452], [432], [-414], [684], [-422], [-332], [-360], [26], [512], [1188], [-630], [-1054], [-558], [8], [-1764], [-1622], [-1134], [-592], [1908], [954], [-2564], [738], [-2440], [2554], [820], [1944], [1242], [-1116], [-1070], [-576], [-1342], [-1422], [872], [-1340], [4880], [-2700], [-254], [4410], [-3888], [5138], [-4788], [-5886], [2232], [666], [-5536], [-2126], [-2934], [-2036], [-2286], [-1244], [1224], [1898], [9162], [4348], [7154], [1836], [-5894], [11106], [13176], [-6112], [13618], [-692], [-8064], [-12654], [106], [-4014], [3914], [-4428], [15490], [6768], [1298], [-2248], [9612], [162], [1370], [15354], [-13024], [14436], [12096], [6056], [-7524], [-5276], [4968], [-10998], [-8838], [-22436], [4762], [6004], [5274], [12420], [-21366], [-21140], [3266], [-17028], [9522], [-10296], [-3382], [-20656], [22114], [19962], [604], [152], [4194], [7252], [-6696], [-28422], [19908], [-14318], [30050], [-22158], [3132], [20932], [21834], [-12446], [-12528], [11576], [29338], [-2540], [-18792], [4832], [20818], [12042], [13058], [11826], [-11996], [-6966], [-40806], [17980], [12834], [-37864], [-42516], [-45638], [17496], [-32174], [-38934], [-29780], [-48096], [-21302], [-7470], [-764], [32328], [36316], [-13392], [38072], [-12798], [34874], [-17190], [-40284], [15498], [37160], [-18468], [10386], [44136], [-28432], [39778]]
