
# q-expansion of newform 6026.2.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, 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 = 6026
weight = 2
poly_data = [-18, -78, 1159, 576, -9137, 2042, 25315, -14682, -28084, 21886, 14837, -14776, -3723, 5359, 248, -1083, 78, 115, -17, -5, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-5029558767, -6210157485, 80734359230, 266688463738, -154167082409, -1142544646218, -50250467098, 1839263430391, 330166097514, -1492751196985, -311582977642, 686826922760, 136061253901, -187493827325, -31119918252, 30194840133, 3599346675, -2663674000, -165884442, 99716990], 1711992393], [[-1903874094, 19091308350, -239014121539, -514352194880, 1484823148105, 1927564243203, -3564239814652, -2433986096297, 4014750146469, 1387964422985, -2433681484336, -353315039944, 845435323468, 12131055271, -168614176551, 12991637169, 17942676336, -2581287385, -788021802, 152897921], 1711992393], [[10210385484, 12667611897, -655658120918, -868959764302, 3828596607116, 3181938636582, -8681537826470, -3483690211072, 9252549939678, 1465883080024, -5302318876388, -79154083532, 1740985774982, -134696224735, -328492457904, 46406228355, 33136669317, -6233624999, -1383955827, 309851458], 1711992393], [[11277017919, -13818490434, -444981502009, -479591138117, 2669559331153, 2022451755138, -6497668918675, -2230153018763, 7345735584573, 822808635896, -4429190235145, 94977561293, 1522754372488, -160904820680, -299751293436, 48925034667, 31439546223, -6450050938, -1360457148, 322669970], 1711992393], [[7853033133, -6970520862, -443269509616, -481303130510, 2669559331153, 2022451755138, -6497668918675, -2230153018763, 7345735584573, 822808635896, -4429190235145, 94977561293, 1522754372488, -160904820680, -299751293436, 48925034667, 31439546223, -6450050938, -1360457148, 322669970], 1711992393], [[3805064997, 13425493694, -141470821499, -263557449692, 880564478145, 917906878673, -2228055002861, -1049940919284, 2647408333697, 474253792532, -1685927227003, -27928299023, 611952043660, -50992377522, -126771603270, 19013317515, 13927919237, -2750855046, -627942667, 146343330], 570664131], [[-21496095405, -123020445429, 862543526110, 2213572376246, -5106166101469, -7746619103055, 12597173558332, 9136296553166, -14170725778413, -4771815292994, 8410243049992, 1050361806526, -2832716706445, 14470325336, 545581112568, -50354083845, -56024761614, 8657638597, 2377631685, -477486794], 1711992393], [[18210231207, 129553986273, -827450554963, -1822112773976, 5021707486708, 5657252453124, -12398618288461, -5708901648527, 13944905779887, 2096214081833, -8265873305065, 96329231267, 2775395086510, -296476727960, -531636343350, 89385080193, 54163977555, -11409071752, -2275865718, 550477613], 1711992393], [[12725619606, 7395908349, -741450867083, -1130822268817, 4609864588088, 4417753791675, -11746626266969, -4598705287915, 13445110060605, 1516003317697, -8055235042100, 284298537844, 2725240933223, -334836535351, -525192157785, 94239617940, 53784538836, -11766008126, -2270260821, 562516600], 1711992393], [[-15043298778, -98272405995, 1077516056294, 2217972391426, -6931059756389, -7487287629810, 17620952974694, 7662576443257, -20128396990494, -2722915606537, 12065066105960, -252098034391, -4091073927500, 460823919703, 791153125932, -135968571153, -81372015369, 17340713522, 3451389795, -840747436], 1711992393], [[-6583539255, -25266898338, 334887564545, 583750175422, -2191104403268, -1963791412764, 5660449894751, 1860096940774, -6556627302303, -438853391215, 3983590226324, -271585338031, -1368544326728, 200567866756, 267919145877, -52656595317, -27863463012, 6440353895, 1193388237, -306455260], 570664131], [[3636815862, 28438633600, -316169845004, -494972035686, 2119661708299, 1522471869796, -5441991275711, -1212682461482, 6281885236696, -5754095250, -3812158841505, 436085177527, 1309458273684, -234380734790, -256415042700, 56303390241, 26675284060, -6603875713, -1142759006, 306841820], 570664131], [[21903959223, 110179494222, -1351112921560, -2585383086992, 8535466217248, 9087254168172, -21643300706941, -9712589542436, 24888060042309, 3787701103535, -15082153681294, 90802281113, 5180055874786, -521211788708, -1015296823950, 165046457061, 105828203667, -21817310038, -4546314924, 1084667243], 1711992393], [[26342084538, 167597701761, -1588589428406, -3344500909942, 9847996099748, 11437902014430, -24392018763977, -12573673222303, 27580460910462, 5529256958347, -16492312061762, -493304121437, 5600687059850, -412087129414, -1087065350931, 154602262989, 112365319416, -21455653031, -4793379879, 1089425851], 1711992393], [[133089489, 740541705, -6391055815, -13420672217, 39427171861, 44859980994, -98477264995, -47391522458, 111562121574, 18907936892, -66542626897, -294234073, 22488696304, -2109921236, -4338879945, 685127538, 445558719, -90366943, -18879387, 4456799], 6661449], [[44758625463, 204713127345, -2180302356475, -4131055705850, 13409940203227, 14054638663002, -33358718318455, -14792189063120, 37854401980263, 5678248146920, -22682866537195, 160415769779, 7710757987687, -779896124435, -1496966561592, 243686192967, 154671482871, -31890712168, -6592015260, 1571933387], 1711992393], [[16459377834, 81393303662, -859091649656, -1558745920940, 5245158178278, 5219898632999, -12843632042951, -5460601016910, 14459887772984, 2074169438390, -8637298145164, 81393002008, 2934376722508, -298853019066, -570148648821, 92842904574, 59005753619, -12156366042, -2519966401, 600429471], 570664131]]

hecke_ring_character_values = None
aps_data = [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, -1, -1, -1, -1, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0], [-1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, -1, 0], [-2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 1, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 1, -1, -1], [0, 0, -1, -1, 0, 0, 0, 0, 1, -1, 1, 1, 0, 1, 2, 0, 0, 0, 0, -2], [1, -2, 1, 0, 1, 1, 1, -2, 0, -1, 1, 1, 2, 0, 1, -1, 0, 0, 0, -1], [-3, 0, 1, -1, -1, 2, -1, 2, 0, 1, 0, -1, -1, 0, -2, 0, -2, -1, 1, 1], [-1, 1, 1, 0, -2, 1, 0, 1, -2, 1, 0, -1, 0, 1, -1, 1, 0, 1, 0, 1], [-2, 0, 2, 2, 1, 0, 1, 1, 0, 0, 1, -1, 1, -1, -1, 0, 0, 1, 1, 0], [-2, 2, -2, 0, 1, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 1, 0, 1, -1], [-2, 2, 2, 0, -1, 0, 1, 1, -1, 1, -1, -1, 1, 0, -2, 1, 1, 1, -1, 2], [1, 2, -2, -3, 1, -3, 1, -2, -1, -1, -2, 0, -1, 1, 2, -2, 2, -2, 1, 0], [-1, 0, 0, 0, 0, 2, -2, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1], [-3, -1, 0, -3, 4, 0, 0, -6, 2, -2, -2, 0, -2, -1, 0, -2, 0, -3, 2, 0], [-5, -2, 3, -1, 0, 2, 1, 1, 1, 0, 1, -2, 0, -3, -2, 0, -1, 1, -1, -1], [-1, -3, 0, 0, 2, 0, -1, -3, 3, -1, -1, 3, 0, 0, 1, -1, -1, -2, 1, 1], [-3, -3, 2, 0, 3, 2, -1, -1, 1, 0, 0, -1, -1, -2, -2, -1, -1, 0, 1, -1], [-1, 0, 1, -1, 0, -2, 0, -1, 0, 0, 0, 0, 1, -1, 2, -2, 1, 0, -1, 0], [-2, -2, 2, 1, 1, -1, 1, 0, 2, -1, 0, 0, 1, -2, 0, 0, 1, 3, -2, 0], [-1, 2, -3, -2, -2, -1, -3, 1, -1, 1, -1, 0, -4, 1, -2, 0, -1, -3, 3, 0], [2, 1, -1, 1, 3, -2, 1, -4, -1, 0, -3, 2, 0, 2, 2, 0, 2, 0, 2, 0], [-1, 0, 0, -1, -1, -2, -2, -1, 2, 1, 0, 1, -2, 0, 0, -1, 0, -1, 1, 1], [-11, -2, 3, 2, 0, 2, -1, 5, 0, 4, 0, -6, -2, -3, -7, 2, -3, 2, 2, 1], [-3, 1, 1, 4, -3, 1, 1, 7, -2, 2, 3, -3, 0, -1, -4, 4, -3, 1, -1, -1], [-2, -5, 2, 5, 2, 3, 3, -2, 1, 1, 4, 0, 2, 0, 1, -1, -2, 3, 0, -2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 5, 0, -2, 2, 1, 2, -1, -1, 1, -3, -1, -1, -3, 1, -2, 2, -1, 3], [0, -1, 0, 1, -1, 0, 1, 0, -2, 0, -1, 2, 0, 0, 1, 0, -1, 1, -1, 2], [-3, -6, 4, 3, 4, 1, 1, -2, 2, -1, 1, 1, 1, -2, 1, -1, -1, 3, 1, -2], [-4, 2, 0, 0, -3, 2, -1, 1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 2, -2, 0], [2, -5, -1, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, -1, 2, -3, 0, -2, 0, -3], [-4, 0, 3, 2, -1, 1, -2, 5, 0, 2, -1, -1, 1, -2, -2, -1, 1, 2, -1, 1], [2, -1, -5, 3, 0, -1, 2, -2, -4, -1, 0, 0, 0, -1, 2, -1, 2, -1, -3, -2], [-1, -2, -1, -1, 5, 1, 1, -6, -1, -1, -4, 3, 3, -1, 2, -3, 4, 2, 1, -2], [2, -2, 1, 0, 4, 2, 1, -7, 2, -2, 1, 4, 3, 0, 2, 0, 4, 3, -1, -1], [4, 1, -1, -1, 1, -2, -3, -3, -2, -2, -2, 2, -2, 1, 1, -1, 2, -2, 4, 1], [-1, -1, 2, 3, -1, 3, -4, 3, 0, 1, 2, 1, 3, -2, -2, 1, -2, 1, 2, 1], [-5, 5, -5, -3, -4, -2, -4, 5, 1, 1, -1, -3, -4, 3, -2, 3, -3, -6, 0, 1], [-3, 0, -4, 0, 1, -1, 2, 2, -1, 0, 1, 0, 0, 0, 2, -3, 1, 0, -1, -1], [2, 0, -2, 2, 1, 0, 1, 2, -4, -1, 0, -1, 0, 0, 0, 0, 2, 0, -1, 0], [-1, 3, 3, 3, -1, 0, 2, 1, -2, 1, 0, -1, 2, -1, -2, 4, 3, 3, -1, 0], [6, 4, -7, -4, -1, -5, -3, -3, 1, -2, -3, 7, 1, 3, 7, -2, 4, -2, -4, -2], [-1, -3, 4, 0, 1, 2, 1, -1, 2, 3, 1, -1, 2, -2, -3, -1, 0, 0, 0, -1], [4, -3, -3, 0, 2, -3, 1, -7, 0, 1, -4, 3, 1, 0, 4, -5, 0, -3, 0, 1], [8, -2, -5, -5, 2, -4, 3, -9, -4, -4, -5, 7, 2, 4, 7, -4, 3, -3, -1, -1], [-1, 1, -1, 1, -2, 0, 4, 3, -3, 0, 2, -2, 2, 2, -1, 2, -1, 1, 0, 0], [-3, -3, 0, -1, 3, 2, -1, -2, 1, 0, 2, 0, 0, 0, 1, -2, 0, 1, 0, -2], [-2, -4, 4, 2, 4, 3, 0, 0, 1, -1, 2, 2, 3, -2, -2, -1, -1, 3, 0, 0], [1, 3, -3, 2, -1, -3, -2, 4, 0, 1, 0, -2, -3, 2, -1, 3, -4, -7, 3, 2], [-3, -1, -2, -3, -3, -1, -2, -1, 1, 1, -2, -2, -2, -3, 0, -1, -2, -3, -2, 2], [-5, 4, 0, -2, 4, -3, -3, -1, 0, -2, -5, -1, -1, -1, -1, 1, 3, 1, 3, 2], [-1, 3, 0, 2, -6, 0, -1, 5, -2, 1, 2, -4, 1, 0, -3, 3, -1, -1, -1, 1], [-4, -3, 0, 1, -3, 2, -1, 3, 2, 0, 2, -2, -5, -1, -3, 3, -6, -3, 2, 1], [0, 0, -2, 4, -1, -1, 0, 7, -1, 3, 2, -4, 0, 1, -3, 3, -1, -3, 3, -2], [9, 2, -8, -1, -4, -1, 1, 2, -4, 1, 0, 0, 0, 6, 5, 2, 3, 1, -3, -2], [-10, -3, 2, -1, 0, 3, 1, 2, 2, 0, 1, 0, -1, -3, -4, 2, -4, 0, -2, 1], [4, -5, -2, -3, -3, 0, -1, 1, 0, 1, -1, 0, -3, 2, -1, -2, -3, -5, 4, -2], [4, -3, 0, -1, 9, 0, 6, -10, 0, -4, -1, 1, 3, 0, 5, -3, 5, 4, 0, -3], [6, -2, 3, 0, 5, 2, 0, -4, 0, 1, -2, 3, 3, 2, 0, -1, 6, 4, 1, -2], [-1, -4, 7, -1, 6, 2, 7, -8, 2, -5, 3, 1, 6, -2, 3, -1, 2, 4, -5, 1], [-3, 0, -3, 0, -2, 1, -1, 4, 0, 6, 1, -4, -2, 3, -3, -1, -2, -5, 3, -1], [-1, 2, -1, -2, -3, -1, -3, 0, -2, 1, -3, 4, -1, 3, -3, 1, 0, -1, 0, 6], [-3, -1, 4, 0, 2, 2, -4, -2, 0, 0, -4, 2, 2, -2, 0, 0, 3, 7, 1, -1], [-9, -5, 6, -1, -3, 6, -3, 3, 3, 4, 3, -3, -1, -4, -5, 1, -6, 0, 1, -1], [-3, 4, -1, -3, -9, 1, -4, 10, -2, 5, 1, -4, -3, 2, -5, 5, -2, -3, 0, 4], [-2, 1, 0, 2, -3, 0, -3, 3, 1, 2, -2, -2, -1, -4, -2, 1, 0, 0, -3, 5], [1, 2, -3, 3, 6, -1, -2, -7, 1, -3, -3, 3, 1, 2, 1, -2, 2, 4, 1, -1], [4, 2, -2, -6, 2, -3, 5, -5, -4, -3, -6, -2, 1, 2, 3, -3, 2, -4, -2, 6], [-1, -2, 3, -4, 8, 1, 1, -13, 4, -4, -5, 5, 6, 0, 4, -5, 3, 1, 0, 1], [3, -1, 4, 3, -6, -1, 2, 2, 2, 1, 3, 1, 0, -1, 1, 0, -1, 5, -2, 2], [4, 4, 1, -3, 0, -5, 1, -9, 3, -4, -4, 6, 3, 3, 9, -2, 4, 1, -5, 5], [0, 2, -1, 2, 1, -4, 0, 5, -3, -1, -3, -3, -2, 1, -4, 3, 3, 1, 2, -1], [0, -7, -1, 1, 1, 3, -1, -1, 3, 1, 2, 1, -1, 0, -2, -3, -3, -5, 2, -1], [-8, 4, -1, 2, -6, -2, -5, 9, 2, 6, 3, -2, -5, -1, -6, 3, -3, -2, 1, 4], [5, 4, -5, 2, 1, -6, -1, -2, -4, -1, -6, 2, 0, 1, -1, 0, 3, 1, 2, 1], [4, 2, -9, -3, -4, -4, 4, 0, -4, -2, 1, 3, 1, 3, 7, 0, 4, 5, -7, 0], [-4, 2, -3, -1, -4, -1, 2, 2, -4, 1, 2, -2, 0, -2, 3, -2, -1, -1, -2, 3], [7, 3, -1, 2, -1, -2, 3, 2, -2, -3, 3, 0, 1, 2, 2, 5, 2, 2, -1, -3], [-4, -4, 9, 6, 4, 5, 4, -3, 1, -1, 3, -1, 8, -3, -3, 0, 3, 10, -1, -1], [2, -1, -4, -7, 4, -4, 1, -7, 3, -3, -5, 0, -4, 2, 6, -4, -2, -12, 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