# q-expansion of newform 605.6.a.b, downloaded from the LMFDB on 24 June 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, 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 = 605
weight = 6
poly_data = [40, -30, -1, 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], 1], [[0, 1, 0], 1], [[-20, 1, 1], 2]]

hecke_ring_character_values = None
aps_data = [[2, 0, 1], [-11, -3, 0], [25, 0, 0], [37, 1, -10], [0, 0, 0], [566, 4, -56], [557, -37, 108], [175, -521, 6], [-1366, 338, -116], [-1155, 711, 784], [1947, 55, 700], [-7397, -591, 750], [1658, 2340, -100], [-10004, 1152, -784], [-2142, 1298, -3304], [-12601, 4861, 1286], [-8170, -7166, 236], [15063, 4585, -400], [-36712, -1844, 88], [-19023, 3325, -4700], [646, 1856, -9348], [-24190, 1178, -4688], [-11534, -9690, -552], [16445, -11385, -12490], [-57632, 11030, 11384], [-39782, -28160, 3940], [43384, -16056, 5788], [88772, 6392, -11596], [-61050, 14500, -2980], [-85176, -15138, -1860], [-40728, 37808, 21168], [-67957, 35875, 3430], [-171852, -5278, -1588], [-52900, -42024, 29384], [217455, -39943, -11552], [33538, -1070, 7480], [55883, -29527, -47874], [-152751, 18761, -28224], [-142163, 78665, 27006], [144126, -37732, -3060], [1780, -22144, 47084], [-60658, 31800, 35860], [-121108, -110900, -23640], [88761, 52615, -31092], [-318488, -59306, -3928], [510745, 22165, 6540], [-300517, 131555, -27490], [-566026, -47818, 64360], [-469878, -121874, 71020], [-818850, 88952, -45652], [-9379, -92061, -47476], [453830, -87426, -135004], [-28602, -46680, 150380], [-167998, 60430, 88420], [-115182, 62944, 135260], [999311, -22733, 74306], [98060, -153002, 69892], [-6792, 107320, 110320], [455712, -65090, -210144], [830678, 266400, -137380], [370746, -88682, -183120], [102306, -148184, 274452], [-1699758, -124346, 143252], [683867, 308415, -226180], [84754, 410360, -184568], [-668567, -69905, -356486], [344132, 206040, 163280], [1431907, -15799, 223380], [29222, -375502, 258068], [-1480650, -25276, -31004], [-248666, 423108, -136316], [-2161040, 24776, -93436], [357228, -370236, 319960], [2255466, -24128, -276644], [397450, 380838, 92892], [206094, -39154, -226824], [3227110, -218976, 337756], [-1481482, 13016, -402712], [-1383793, 457305, -539310], [1125040, 656498, 175072], [1758540, 523560, 284660], [-1306788, -171010, -523800], [-2000392, 255000, -347760], [-4146956, -66646, 287688], [1267320, -188800, -97740], [-1111156, 321832, -353080], [-222230, -789528, 573768], [-1483153, -1073507, 476668], [2361003, -85955, 694400], [-3162606, -488382, 612604], [3167653, -1214003, -483928], [2266620, 11132, -149732], [2390588, -299332, 297216], [-895097, 512535, 202310], [-2204000, -652036, -284644], [2070176, -970512, -439000], [1962360, 26706, 721144], [54422, 1952700, -203320], [3950836, -822120, 188748], [-2401287, -742125, 378360], [-4815788, 1448032, 246004], [136862, -1057204, -259480], [2477686, -90166, -1409956], [7552780, -322946, 447976], [1293393, -1257975, 101930], [2950218, -846728, 1549612], [-5795777, -681641, -241320], [2681126, -1323768, 729356], [-10546365, 503223, 803732], [-6757542, 817300, -1035260], [-7813713, 1354715, 1010586], [-8441974, -211232, -357520], [-10475092, 728266, -853592], [-3575240, 878236, 1427044], [3240577, -712195, 636620], [-10909933, 1114685, 1125650], [-1693951, 1305625, 1455792], [-9479362, 1719190, 647004], [7085449, -940321, -1419262], [3443205, -1854955, 1830970], [16904602, -405820, 783520], [298121, 1883799, -847596], [-13716128, -1010162, 747288], [1876219, -2344077, -1021696], [-3731538, 333090, 1713800], [-2572137, 75625, 1026160], [-1291330, -3512632, 479432], [-5140905, -1318621, 1452356], [-13445262, 32906, 1187268], [-18448904, 1354926, -9708], [-7728440, -2962292, 1040152], [-9164019, -1274055, -40842], [-5450833, 214955, 1303380], [-9201762, -1541480, 819304], [6840198, 370800, -1979700], [-8111970, 993840, -3152180], [9526179, -1908743, 3041030], [-7285608, 4597828, 804548], [-7002782, -585628, -664328], [2381990, -342768, 531208], [16688933, 1630045, -355230], [11257918, 349060, 484620], [3835404, -1949124, -1856804], [-6076668, 1012680, -3322684], [14246860, -948826, -256124], [-8746360, -2632704, -861736], [7173306, -1998080, 1305968], [4965197, 3437859, -517108], [24885870, -908390, 1308300], [8443854, 901230, 1130912], [-7184058, 2992132, -660516], [-9896718, -1768640, -880240], [-19438581, 1160507, -702920], [-4050008, 1323176, -2241280], [9934533, -3574443, -326248], [3365437, 7370545, -3911980], [36547440, -1362168, -464072], [-7598825, 1783633, -161078], [20746342, 1665760, 1271396], [-922367, -4367605, 470520], [-4394057, 5735591, -2996112], [4189411, 709521, -1334148], [-9284213, 667119, 5378162], [-287188, -2716080, 3622020], [-12528232, -3688930, -558416], [10624054, -2542546, -2989772], [-16749448, 8810280, 3501520], [6025392, -3169874, 1761940], [23689990, 2019704, -1475304], [21722979, 8184873, -1862786], [17833215, 678671, -463266], [-4326478, 1294180, 1725300], [-17010382, 6555330, 313900], [30791281, -2238713, 3348836], [-39122320, -2927552, 1688432], [27588735, 3431969, -3071734], [39367228, -80840, 2066160], [26723472, -5951390, 958920], [-46219169, -156829, -296558], [-713805, 7539173, 917712], [24241118, 3884030, 5017004], [-11797357, -2603125, 2517830], [-8132611, 5419499, -7883262], [-10286383, -11503833, 3084564], [12250674, -426878, -54260], [27966230, -6435008, -1365932], [4922452, -2135142, -4169332], [-26566796, -2965432, 1343584], [-46242190, -2693684, 1062664], [5028308, -6825660, 1306900], [-5208096, -1710394, -8089364], [1666196, 7341288, 3628840], [-22857538, -5741870, 3181560], [41352692, 5511630, -2424620], [-37837968, -5417076, 2745652], [-6673006, 11581480, 2631592], [-27739712, -6118550, -1125540], [6916834, 3893572, 9427720], [48373282, 1107236, -4712420], [-30914849, -9597501, -723926], [6306325, -2183921, -766424], [717898, -4585630, -2261420], [-2612707, 2137947, -1638358], [8910900, -7507218, 3539928], [72579760, -33932, 1190092], [-15141397, 13649801, 4998758], [18526000, 4622256, -6817336], [-20158464, -13491204, 623632], [34900350, -11594232, -7026448], [2060692, -17189320, -305220], [18045527, 5894669, -2767668], [-25413108, -13895570, 1002780], [-1128136, 9694840, -5300948], [70371823, -1471945, 6968584], [-18343620, 5516580, 1388180], [-2146793, 6329865, -42470], [-30297313, -1342629, 8250010], [1067498, -459540, -11725760], [36694888, 15938248, -543624], [28307869, -12819149, -3695694], [-25401922, 9294580, -1134260], [37526870, 9772118, 1677252], [41567505, -14752445, -3807130], [-55875596, -7199556, 4223448], [-101840408, -1436188, -2890096], [-8689185, 19593545, -1922200], [-10234206, 18724072, 1052640], [-59568395, 5490857, 11784468], [12639552, -2045392, 11883128], [-12847052, -7762560, -8547240], [51997034, -10308452, -7888256], [73228445, 6077261, 5744394], [12128818, -7079270, -11875500], [30161258, 2648100, 13941040], [-1979956, -5139304, -7059224], [19184407, 13656435, -1815294], [-48182065, 12761209, 4613946], [72368306, 1644728, 8748120], [-45106040, 14765916, 4752604], [93416127, 9134595, -1956020], [-77888481, 9443455, 3422512], [10348188, -15190600, -7414840], [16868904, -15865768, 4180860], [65136070, 429512, -11149872], [-20080254, 5033684, 11196524], [6300890, -9489846, -3262404], [-41257272, 6480272, 4914772], [-25560738, -6308830, -3316680], [44097155, -9807853, 5492518], [-5145249, 23634187, 1404066], [-71521358, 20607520, 254856], [-35581312, 6134090, 6937720], [34693578, -21641854, 7129728], [91966975, -6750799, -1902966], [92161476, 14534826, -5878208], [-53093205, -2155837, 9120222], [-58379288, 10523690, -6982600], [-96324738, 4773514, 5975612], [-70428802, -10120600, 10467104], [73198027, 15544281, -3589380], [-20433624, 31358144, -5364856], [-117542748, -11722272, -4262352], [-26314925, -16792451, -8060064], [-57416874, -12407228, 4097516], [25317308, -20212942, 9472836], [-31636380, 9669408, 2369212], [40812340, -22295826, 9787136], [69068717, 880055, -15749970], [59043316, -11801304, -6788308], [34719646, -9271420, -1789152], [-48169358, -10632780, -1715460], [-73297797, -5752645, 3918624], [-44515796, 9143514, -295352], [-127215490, -4824170, 1925080], [105639317, -10741117, 7445748], [-149397686, 527298, 6249384], [29174928, -28078868, -11811528], [40647370, -14228184, -4033476], [58601778, 6886220, 3898000], [66265593, 28867465, -8172910], [86608781, 15997873, 6386430], [21247252, -8110100, -9106200], [-58834328, -12810616, -8820728], [-54246857, 29888465, -21124360], [139895012, -2462328, 2489304], [113165227, 11241335, -15307260], [136001854, 8155132, 2475680], [-26940918, -30795984, -6601880], [-22182715, 33086657, 12410108], [61249010, 26348884, 11311156], [-1050368, -3195510, 18299680], [-11064172, 18893384, 14238960], [-130802639, -10130633, -5078844], [-122389787, 18082365, 13656770], [-86567334, -15818712, 6197940], [77263610, -1609032, 34681812], [-73096488, 23383400, -27803960], [74318949, -36623581, -263822], [-175825060, -9764760, -6289320], [31828522, 18123406, 10712460], [-204661049, 8520789, -11312596], [104674752, -27395082, 3471768], [-206416480, 28600384, 6719716], [-45096956, 32673928, -31950296], [21983542, -35176510, 20107520], [-124640432, -42415306, 6502140], [207477268, 18684208, -14566384], [-131247880, -39112622, 4166332], [25064175, -6818221, -4894764], [-214378304, 10831326, 2070012], [-71609044, 7065700, 21011648], [-56256250, 21326320, 17365540], [78792468, -11763010, 19331340], [-146492471, 21278049, 2692408], [56619317, 30279857, -15132546], [-102799010, 11127480, -38419740], [79036410, -7565298, -1973552], [-36499212, -16407100, -10570340], [99159634, 11794848, 1195484], [240554260, 36321582, -12737572], [-42207387, 8087045, -2998510], [224158038, 8898988, -386904], [125551738, -38408040, -21680100], [122090974, -33610754, -10416264], [21386331, 33886393, -13781260], [-145939833, 21925105, -37597270], [-145268830, 39162790, -10220120], [30286124, 246792, 6377120], [-204015783, -5328259, 11814270], [224759376, -20690066, -14200596], [90787553, 2812235, -28184840], [-131946102, -25782676, 13226040], [-115066340, -17169508, 17561188], [80616206, 47139922, -26773364], [245551223, 7283655, 20905070], [227207173, -1286939, 20847448], [-120858950, -4903364, 10906904], [80112301, 10538411, 2952292], [48947017, 26436595, 20847990], [-63193863, 16778029, 1963142], [-40558286, -71261060, -2052088], [-7902082, 30719460, -165596], [149204585, -25023925, 5802760], [113815217, -1247235, -28659180], [-236467921, 18412093, 12107574], [217331785, 27272601, -10436326], [-164370368, 19486610, -17637040], [90631303, -7217441, -9525520], [-159631492, -17919060, 5507000], [57807393, -7948177, -4600734], [-129287997, 32621435, -32273690], [210364538, 42158816, -17447092], [-222227988, -46119690, 14766180], [21621871, -48095429, 4474122], [18073575, -57456847, 22728592], [164120851, 24642625, -27799092], [-65947625, 37997875, -1770900], [-30065827, 23924885, 42777490], [173768422, -36167288, 28238604], [-193446956, 38313012, -1851040], [266050982, -32956612, -3354252], [313923148, -32828050, -11460800], [90262188, 43410212, 26409512], [-214810075, 34955997, 39413858], [171962073, -36883647, -15990664], [76242544, -56192114, -42732804], [170694842, -5356920, 45839556], [-68020719, 8458445, -50755082], [242901188, -47535370, -977720], [-294585768, -6165980, 23346000], [-201364840, -78892708, 24834828], [-473323544, -9157936, -16742376], [152920095, -22934711, 81483136], [70497698, -22016830, -3734420], [-49541962, -36712088, 41538392], [286346340, -21262992, -15417668], [-215879962, -26073250, -2560540], [107832861, -19288997, -49928060], [-171656300, 49130558, -38422048], [102194727, -2374811, 6795852], [165382292, 26762110, 33346000], [-85775346, -1152396, 3432128], [228491572, 69491076, -40842240], [-71246493, 53399513, -55258012], [-132771520, 46636484, -38042024], [97457024, 11397248, 935744], [-69415368, 98976120, -25495200], [-181218067, -4468661, -13795790], [-182257184, 10294052, 42841236], [-215514682, 28227238, 55224316], [147135545, 30145483, -8119818], [10095129, 46417911, -10072134], [38365240, 89580844, -7672064], [74587048, -9008084, -28341112], [6386592, -25675240, -13144600], [-218020214, 89506916, 52901492], [-40845550, 22308218, 2308512], [-121950405, -70275395, 2852490], [-120633298, 53680010, 34638640], [8554463, -6044695, 9261480], [78889545, -12147965, -40660080], [-45554546, -38521588, -23516780], [121095692, 61501908, -40689072], [-109852772, 50795402, -42199328], [218749670, -84901668, -56541212], [-168671142, 4323610, 62073480], [-153139447, 93259423, 2059226], [88215058, 5594580, 37228960], [1998259, 135966691, -6276704], [170893510, -88758374, -44869696], [-20603974, 79291340, -15272492], [124372312, 25915174, 8404572], [164942244, -83005120, 6998192], [-281149772, 78213560, 40345960], [523085822, -43103368, 21811964], [227320292, -37955010, -1955880], [210350420, 68075044, 43916816], [-185104752, 33076832, -33147448], [413173638, 5239684, 31030920], [-170919319, 57242997, -32904954], [56024180, 79815438, -38821948], [13329812, 52299090, -4165144], [-209415932, 49719676, -119246432], [-123625385, 14869063, -16951258], [8490674, 132987800, 28592132], [55221353, -3704385, -45157206], [-314187081, 27539159, 26505928], [148929700, -5384970, -27629840], [-379071728, -32969900, -23418020], [72943390, 56593966, -7868976]]