
# q-expansion of newform 1792.2.m.f, downloaded from the LMFDB on 16 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 = 1792
weight = 2
poly_data = [4, -40, 164, -344, 390, -336, 358, -180, 351, -176, 118, -24, 67, -48, 12, -4, 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], 1], [[-29073536866216359, 117377507327509198, -301042761683687430, 330162633044914154, -298674348575977988, 317962271773663747, -141663658022557777, 316269746406518514, -158494098831050719, 109855023226949419, -16311504998557221, 59571828256046246, -44591424589229034, 11056775295709911, -3722676599840684, 951071408409672], 2586996236383195], [[22416575382206, -121829545946822, 253762122985590, -315437625800966, 247799490169192, -297759021619953, 125536721618373, -285346377635636, 136856641603546, -98359361008081, 16633551650999, -52647402989314, 39378227440746, -9774129145049, 3268059368601, -837450978918], 1646719437545], [[63352765090121984, -285278940866108728, 640181323544103480, -950165080048245094, 567587652935576758, -974932204477243482, 215321576232905127, -856904562132546654, 359150264381483384, -323008880354869584, 27728735279128781, -141252009753288336, 120429826515046084, -29239824225526546, 9848563721439349, -2704100770805192], 5173992472766390], [[-23048604152387462, 194573129558701754, -502632258726641150, 562177999739357012, -515646310273577514, 534273028711390981, -240129108623449166, 546976894209326142, -280841928485519762, 187581053285797687, -24742665676005658, 104476272119691108, -78294768665013192, 19424222999220833, -6572201638917102, 1674708372990976], 2586996236383195], [[-258618081490, 1753345508964, -4768566345236, 6274129258728, -4492931472132, 6516717498206, -1643327598977, 6018321521990, -2719627979686, 2258419134670, -152629602893, 1035106924390, -869648666130, 211530176114, -71829768307, 19482487322], 18265555126], [[-124297662602168868, 849908894353444156, -2095030105848071860, 2566105999528852478, -2042609308803956566, 2518330312414188084, -909797757822592589, 2427731722617841338, -1161231004486048088, 863513038753683928, -101252104556846217, 440818146003185412, -344785787247940018, 84898454002472672, -28658627888228153, 7498893603309034], 5173992472766390], [[-67882571420950212, 447008615232415754, -1152862769202227570, 1283713884161744232, -1173798486360141284, 1224404821394753001, -549310814953693356, 1240943173396670972, -633782556466103087, 428257247412482512, -59330090045209703, 236350861493329618, -176863864075565747, 43873064609843573, -14818624514131492, 3777555552239611], 2586996236383195], [[40980131050682892, -314841413375322204, 917338120456251020, -1263918842903687457, 857235288719333779, -1363528429379028841, 261476413468482641, -1231844713236319142, 540760231449412867, -476344981375704222, 18177688506737658, -205792432068434953, 180471639613998717, -43592489723122903, 14866011666638747, -4111165082614591], 2586996236383195], [[339029565430912, -3001512409125524, 10174933047931800, -15153946647046832, 8947496261644624, -17151627123727186, 1835649209159641, -14919834611631202, 6135862988795822, -6032292554488402, 15018673117033, -2359965597354778, 2216205890956042, -529691956965078, 181410266124727, -51700125227286], 28585593772190], [[-58837164424030928, 435161639642974356, -1206798639331144120, 1612124569733992598, -1143547375046559696, 1698017523089689604, -391651315022393869, 1557941182908479308, -700440973017072543, 591816357963522438, -32882698304956797, 265978766975545407, -226796911018514863, 55026059693644252, -18723422330113773, 5111890261955639], 2586996236383195], [[-69616014115651576, 472716677119123542, -1321246981077163610, 1703428508268658061, -1251632447632723287, 1789141729619286818, -453843017266031758, 1650583194286411736, -751717640243443631, 623516248794297746, -40441701960073524, 285186819270035739, -239788425805503616, 58296781349821544, -19813878646531791, 5374657323581928], 2586996236383195], [[76096265993090204, -493126775820972468, 1351849139305683920, -1760055912567976994, 1269554917909688188, -1847016073682336012, 458715003921674597, -1697590220722951414, 766246241691993109, -641433639158466729, 41301296869441296, -291627635161742131, 246097516841643734, -59803135064160756, 20318406884992724, -5522619438975152], 2586996236383195], [[13545912097023734, -105659607438284102, 308964463100290878, -415638376318740034, 291324658777882928, -443797052249781231, 94218727843551051, -405109873988720704, 180665252133307975, -155431896656409780, 7085686180956252, -68547510138283141, 59261194792100792, -14345307664530663, 4888822756910656, -1342976102346028], 517399247276639], [[1133976629719396, -8262089065435772, 23746847574694880, -30280524171409216, 22789052228712772, -31568902874141438, 8291064774952483, -29521345230669606, 13589173271460526, -11070904840803886, 714420636220699, -5137895187302654, 4295184660226986, -1045387256303994, 355577051279181, -96105262004818], 28585593772190], [[129759186731597904, -988212428548741298, 2966846397725994870, -3971125038086180174, 2775101054643465278, -4254657735688371947, 886977360247710202, -3881717238759491924, 1724470044458942519, -1493003685079757954, 64154006218803931, -654936333530095316, 568446387146979339, -137516440256165211, 46882114975152344, -12901040348769007], 2586996236383195]]

hecke_ring_character_values = [[1023, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1541, [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1025, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, 1, 0, -1], [-1, 0, 1, -1, 1, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [-1, 1, 0, 2, -1, 0, 2, 1, 0, -1, 0, 0, 0, 0, 1, 0], [0, 1, -1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1], [0, 0, 2, -1, 0, 0, 1, 0, 0, -1, -1, 0, 0, -1, -1, 3], [0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1], [-1, -1, 0, 1, -2, 0, 1, 1, 0, 1, -1, 0, -1, 0, -1, 0], [0, 0, 2, 0, -2, 0, 2, 2, -2, -3, -2, 0, 0, 0, 0, 2], [0, 0, 0, -1, 0, 0, 1, 0, 1, 0, -1, -1, 1, 0, 0, 2], [1, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0], [3, 1, 0, 0, 2, 0, 0, -1, 1, 1, 1, 1, 1, 0, -1, 0], [4, 2, 1, 0, 1, 2, -2, -2, 0, 1, 0, 0, 0, -2, 0, 0], [1, 0, 1, 0, -1, 1, 1, 1, -4, -2, 0, 0, 0, 0, 0, 1], [0, -2, 1, 0, -1, -2, 0, 0, 0, 0, -2, 0, -1, -2, 0, 2], [-4, -1, -2, 5, -3, 3, 0, 3, 0, 0, -1, -2, -3, -1, 1, -2], [0, 0, -1, 1, 0, 2, -1, 0, 0, -1, -1, 0, 1, 0, -1, 0], [0, 0, -4, 1, 0, 0, -1, 0, -1, 1, 2, 1, 1, 3, 1, -2], [-3, -3, 0, 0, -1, 0, 0, 1, 2, 2, -1, 2, -1, 0, -2, 0], [-2, 0, 3, -2, 3, 2, 0, -2, 0, 0, 0, 3, 2, 0, -4, 2], [0, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 1, 1, -1, 0], [-7, -3, 0, -1, 1, 0, -1, -1, 3, 3, 0, 3, 0, 0, -3, 0], [-3, -2, 0, 0, -2, -1, 1, 2, 3, 2, 1, 0, 0, 2, 0, 0], [0, 0, 1, 0, 0, 6, 0, 0, -1, 1, 4, 1, -1, 3, 1, -2], [2, 3, 2, 0, 1, -1, 1, 0, 2, -1, -2, 0, 0, -3, 0, 3], [0, -2, 3, -4, 1, -2, 0, 0, 0, 0, -2, -2, 0, -2, -1, 2], [2, 0, 0, -2, 3, 0, -2, -3, 2, 2, 3, 2, 3, 0, -2, 0], [-6, -4, 0, 2, -3, 0, 2, 3, 0, 0, -2, 0, -2, 0, 0, 0], [1, 1, -3, 2, -2, 0, 0, 3, 0, 0, 1, -3, 0, 1, 0, -4], [0, 0, -1, 0, 0, 4, 0, 0, -1, 1, -1, 1, 3, 2, 1, 0], [-1, 0, 2, 0, -2, -1, 3, 5, 0, -2, -2, 0, 0, 0, 0, 5], [0, 2, 2, 0, 0, -2, 0, -3, -2, -3, 2, 0, 0, -2, 0, -1], [0, 0, 0, -1, 0, 0, 1, 0, 2, 1, 3, -2, -4, -1, 1, 1], [-1, -2, 4, -5, 2, -1, 0, -4, 0, 0, -2, 1, 1, -2, -4, 6], [-3, -2, 1, -3, -1, 1, 0, -3, 0, 0, -2, 2, 1, -2, -1, 5], [0, 0, -3, 0, 0, -2, 0, 0, 1, -1, 0, -1, -3, -3, -1, 0], [3, -2, 3, -3, 1, -5, 0, 0, 0, 0, -2, 3, -3, -2, -4, 2], [11, 4, -1, -1, 3, -7, 0, -2, 0, 0, 4, -2, 5, 4, -1, -2], [0, 2, 3, 0, -1, -2, 4, 0, 4, 2, -1, 0, 0, -2, 0, 2], [9, -1, 0, -3, 4, 0, -3, -5, 4, 1, 2, 4, 2, 0, -1, 0], [-4, 0, 0, 2, -2, 0, 2, 3, -3, -2, -2, -3, -2, 0, 2, 0], [2, 2, -3, 0, 5, 0, -4, 0, 2, -1, 0, 0, 0, -2, 0, 2], [0, 0, -3, 1, 0, -8, -1, 0, -2, 1, 2, 2, 1, 3, 1, 0], [-5, 2, -2, -3, 0, 7, 0, 0, 0, 0, 2, 0, 1, 2, -1, -2], [-4, 2, 0, -1, 0, 0, -1, -2, 3, 0, 0, 3, 0, 0, 0, 0], [1, 2, -5, 7, -3, 1, 0, 1, 0, 0, 2, -2, -2, 2, 4, -3], [5, 0, -2, 0, 2, 5, -1, 0, 1, -4, -1, 0, 0, 0, 0, 0], [0, 0, -3, 2, 0, -6, -2, 0, 3, -2, -1, -3, 1, 0, -2, 0], [2, -4, 0, -2, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 7, 0, -2, 3, 0, -2, -3, 4, 1, 4, 4, 4, 0, -1, 0], [-6, -2, -1, 0, -1, -4, 0, 2, 1, -4, -4, 0, 0, 2, 0, 0], [0, 0, 0, -1, 0, 0, -1, 4, -1, 2, -2, -1, -2, 0, -2, 0], [0, 0, 1, 1, 0, -2, -1, 0, -2, 1, 1, 2, -3, -2, 1, -2], [-3, -4, 1, -3, -3, -1, 0, 2, 0, 0, -4, -1, -3, -4, 0, 2], [-2, -4, 0, -5, 4, 0, -5, 0, -1, 2, 0, -1, 0, 0, -2, 0], [-4, 2, 4, 0, -2, -6, 8, 3, -2, -1, -2, 0, 0, -2, 0, 5], [0, 0, 8, -2, 0, -2, 2, 0, -6, -5, -4, 6, 2, -2, -5, 4], [-1, -5, -3, 0, -2, 4, -2, 3, 1, 6, -1, 0, 0, 5, 0, -2], [-5, -2, -1, 0, -1, -3, 5, 2, -5, 2, -3, 0, 0, 2, 0, 0], [1, 1, -2, 6, -1, 0, 0, 3, 0, 0, 1, -3, 4, 1, 6, -4], [0, 0, 6, -1, 0, -6, 1, 0, 1, -2, -4, -1, -6, -10, -2, 2], [0, 0, 2, 1, 0, 0, -1, 0, 1, 0, -1, -1, -5, -6, 0, -2], [10, 4, -4, -2, 0, -6, 0, -4, 0, 0, 4, -2, 8, 4, -1, 0], [-6, -1, 5, 0, -6, -5, 7, 1, -2, -1, 2, 0, 0, 1, 0, 0], [10, 4, 0, -4, 5, 0, -4, -5, 0, 2, 1, 0, 1, 0, -2, 0], [3, -2, 1, 0, -3, 5, 1, 2, -4, -5, 5, 0, 0, 2, 0, 0], [-4, 2, -2, 2, 0, 6, 0, -4, 0, 0, 2, 6, 3, 2, 6, 2], [-6, -2, 0, 1, -4, 0, 1, 4, -1, 0, -4, -1, -4, 0, 0, 0], [0, 0, -4, 3, 0, -8, -3, 0, 2, -3, -1, -2, 2, 1, -3, 1], [4, 0, 0, 0, 2, 0, 0, -2, 0, -4, -2, 0, -2, 0, 4, 0], [2, 0, 1, 0, -1, 2, 0, 3, -4, -3, 2, 0, 0, 0, 0, 3], [-7, 4, -2, 0, 6, -11, -3, -6, 4, -1, 3, 0, 0, -4, 0, -2], [-9, 3, 0, 3, 0, 0, 3, 3, 0, -3, 3, 0, 3, 0, 3, 0], [-4, 2, 1, 0, 1, -6, 4, 2, 2, 3, 2, 0, 0, -2, 0, 4], [-4, -2, 2, 2, 0, 2, 0, 2, 0, 0, -2, 2, 1, -2, -2, 0], [-2, -6, 0, -4, 2, 0, -4, -1, -1, -2, -4, -1, -4, 0, 2, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -3, 0, 3, 0, 0, -4], [1, 1, 4, 2, 5, 0, 0, 1, 0, 0, 1, 5, 2, 1, 0, -2], [4, -2, -6, 0, 4, 6, -4, -1, 6, 1, -2, 0, 0, 2, 0, -3], [-4, -6, 0, -2, -5, 0, -2, -3, 0, 0, -2, 0, -2, 0, 0, 0], [-5, 3, 0, 3, -7, 0, 3, -1, -1, -1, 2, -1, 2, 0, 1, 0], [0, 0, -2, 0, 0, -4, 0, 0, 2, 0, 0, -2, 4, 4, 0, -6], [0, 5, -3, 0, 8, -5, -5, -7, 6, 1, 2, 0, 0, -5, 0, -2], [2, -6, 0, 4, -9, 0, 4, 8, -7, -4, -1, -7, -1, 0, 4, 0], [0, 0, 5, -4, 0, 0, 4, 0, -6, -4, 3, 6, 1, 4, -4, -1], [-2, 0, 6, -10, 6, 2, 0, -6, 0, 0, 0, 6, 3, 0, -2, 6], [-3, 7, 0, 9, -8, 0, 9, 3, -2, -5, 2, -2, 2, 0, 5, 0], [2, 5, 2, -9, 7, 3, 0, -4, 0, 0, 5, 2, 0, 5, -2, -1], [11, 1, 0, -2, -2, 0, -2, -3, 3, -1, 1, 3, 1, 0, 1, 0], [0, 0, -8, 5, 0, -4, -5, 0, 10, 7, -1, -10, -4, -5, 7, -5], [-7, 0, 3, 0, -3, -7, 5, 6, -4, -7, -3, 0, 0, 0, 0, 6], [7, 2, -3, -1, -1, -5, 0, 3, 0, 0, 2, -4, -5, 2, 1, -5], [0, 0, 0, -6, 0, 0, 6, 0, -2, -2, 6, 2, 2, 8, -2, -2], [-4, 2, -6, 2, -4, 6, 0, -2, 0, 0, 2, 2, 1, 2, -2, 0], [0, 0, 4, 1, 0, -4, -1, 0, -1, 2, -7, 1, -1, -8, 2, 8], [8, 7, 2, 0, 5, 1, -1, -6, 2, 4, 3, 0, 0, -7, 0, 1], [-4, -6, 4, 2, -2, -2, 0, 1, 0, 0, -6, -4, -4, -6, -3, 5], [-12, -1, 3, -5, 2, 11, 0, 2, 0, 0, -1, -2, -3, -1, 1, -1], [-8, -4, 4, -8, 0, 4, 0, 2, 0, 0, -4, 4, -8, -4, -3, 2], [10, -5, 5, -3, 0, -15, 0, 2, 0, 0, -5, 0, -2, -5, -2, 3], [0, 0, -11, 8, 0, -8, -8, 0, 4, 2, 1, -4, 7, 8, 2, -7], [-6, 3, 9, 0, -6, -9, 7, 1, 0, 1, 0, 0, 0, -3, 0, 4], [4, -4, 0, 1, -2, 0, 1, -2, 1, -2, -2, 1, -2, 0, 2, 0], [2, -1, -1, 0, 0, 3, -3, 4, -2, 0, -7, 0, 0, 1, 0, 3], [8, -4, 0, 3, -6, 0, 3, 4, -5, 0, -2, -5, -2, 0, 0, 0], [0, 0, -6, 4, 0, -4, -4, 0, 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