# q-expansion of newform 588.6.i.b, downloaded from the LMFDB on 21 September 2026.

# 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 = 588
weight = 6
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 = [[295, [1, 0]], [197, [1, 0]], [493, [0, -1]]]
aps_data = [[0, 0], [-9, 9], [0, 6], [0, 0], [108, -108], [346, 0], [-1398, 1398], [0, -1012], [0, 1536], [-3762, 0], [-736, 736], [0, -2054], [15534, 0], [11036, 0], [0, 4560], [7962, -7962], [-7020, 7020], [0, 26870], [-52148, 52148], [-2544, 0], [-9766, 9766], [0, -68672], [61668, 0], [0, -41454], [111262, 0], [-180426, 180426], [0, 35912], [0, 30492], [-82382, 82382], [-160398, 0], [-80896, 0], [0, 173676], [-390426, 390426], [-83204, 0], [0, -140358], [-320360, 320360], [-158266, 158266], [0, -345476], [20568, 0], [0, 732558], [-572220, 572220], [352402, 0], [0, 18456], [-832322, 832322], [612438, 0], [-501352, 501352], [-556588, 0], [1256800, 0], [-700932, 700932], [0, 153374], [0, -154266], [-926376, 0], [-1056622, 1056622], [1459836, 0], [0, 1571418], [1462752, -1462752], [-230850, 230850], [0, -574432], [-510950, 510950], [931146, 0], [2155604, -2155604], [-962070, 0], [-1719884, 0], [-2381208, 2381208], [0, -293494], [0, 1946418], [0, -1251836], [297458, 0], [3406572, -3406572], [420826, 0], [1394346, -1394346], [0, 3550368], [-391696, 391696], [0, 163834], [206156, 0], [0, 484176], [4174794, -4174794], [0, -4399978], [0, 5459166], [2183066, -2183066], [-5445612, 0], [-4830538, 0], [7581816, -7581816], [99838, 0], [0, -7776904], [0, -3154884], [4914498, 0], [0, 5770006], [-1652718, 0], [7691456, 0], [0, 4958796], [-1468992, 1468992], [6466792, -6466792], [8944788, 0], [0, -10505204], [-7979928, 0], [0, 8134254], [1768002, -1768002], [0, -4072108], [0, 4609042], [-5810908, 0], [9599490, -9599490], [-3211524, 3211524], [0, 9235734], [-2816828, 2816828], [4134146, -4134146], [4381548, 0], [0, 12297114], [9209280, -9209280], [16339366, 0], [0, -3730816], [-14650310, 14650310], [-4335270, 0], [-17909452, 17909452], [-7136152, 0], [2470254, -2470254], [-11371724, 0], [-16981176, 16981176], [0, 15672450], [2854236, 0], [15639566, -15639566], [-3012094, 0], [0, 8987334], [-7139316, 7139316], [0, -15510436], [-22294770, 0], [0, -2534], [0, 16672800], [940648, 0], [0, 27678278], [-10035140, 10035140], [16720224, 0], [0, -9817280], [-2729482, 0], [0, 2781762], [-5043314, 0], [-14982378, 14982378], [4828508, -4828508], [22212546, 0], [-13431210, 13431210], [19967308, 0], [0, 29006298], [-23880104, 23880104], [-9587244, 0], [-26683546, 26683546], [-5418696, 0], [20232322, 0], [7160706, -7160706], [0, -12483964], [0, -33273144], [0, -4102478], [2996358, 0], [-10751404, 0], [0, 20496984], [19751956, -19751956], [24851112, 0], [0, -15486392], [0, 17384874], [27289654, 0], [-12040050, 12040050], [0, -38280828], [18365466, 0], [18378056, 0], [0, -19079532], [-33740706, 33740706], [15653544, -15653544], [11012224, -11012224], [23835614, -23835614]]