# q-expansion of newform 588.8.a.g, downloaded from the LMFDB on 28 July 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 = 588 weight = 8 poly_data = [1071280, -36766, -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], [[-1, 2, 0], 1], [[-98095, 168, 4], 15]] hecke_ring_character_values = None aps_data = [[0, 0, 0], [-27, 0, 0], [-85, -1, 0], [0, 0, 0], [-438, 5, 1], [3118, 0, 2], [-48, 45, 3], [13627, 48, -1], [25077, -39, 12], [-61785, -280, -11], [-35491, 108, 25], [-143463, -720, -9], [294114, 663, 75], [-156188, -2112, 24], [121106, -1060, 78], [-207549, 1892, -95], [384974, 380, -378], [-181288, 4146, 82], [-1348616, 5478, 390], [-2265465, -10165, 244], [259952, -1770, 298], [1358686, -930, 1560], [317946, 1860, 1146], [2200564, -53, 849], [-2258032, -3870, 214], [-1831885, 8861, 1386], [3186823, -12180, -1837], [6764352, 23653, -961], [-10884080, 1680, 678], [-1724172, 56284, 518], [-110698, 57966, 2712], [9072708, 6708, -2028], [-8856270, 9800, 1060], [-5418102, -104688, 1686], [6752523, 46580, 7129], [7423252, -114600, -492], [-1471232, -187590, -2470], [20995812, 26274, -930], [-26628458, 166960, -7506], [12767539, -185555, -11694], [15208074, -72165, 3387], [-76952950, -233832, 3958], [-43023099, -110805, -12342], [37573085, 286356, -19929], [72515643, -59688, 13941], [66341708, -200760, 22444], [-59422024, -554820, -336], [-65249614, -443040, -18398], [86957938, 106960, -33402], [234265030, 254772, -3898], [-51517458, -328036, 23500], [8953257, -629295, -18768], [186472268, -589602, -45098], [-142522108, -196600, -1224], [335858380, -36185, 7533], [17382603, -700207, 7762], [362491943, -134869, 15720], [348865681, -544404, -27451], [20997115, 589128, 106293], [54185976, 1412700, -9114], [377359913, 55080, 33013], [-209064879, 121005, 53928], [-418653403, -104520, 56113], [431584906, 391156, 16854], [487397172, -519960, -18204], [-238539459, 94740, -143277], [237874316, 1197900, 51012], [-393979702, 235656, -197100], [-118459416, -481773, -8511], [27303480, 1394202, 10722], [908873088, -556335, 88599], [892146273, -325205, 93206], [682874192, 3496560, -124112], [-182727218, -2416728, -63672], [-823621524, -885660, 147228], [810585000, -1773060, -134556], [-635442717, 834000, 246657], [194148004, -3984870, 259550], [91180530, 76080, 304116], [643064388, 2280678, 100422], [-204595696, -179452, 242520], [116345919, -3372480, 185265], [221573133, 3903235, 24386], [-1122996344, 1282620, -163804], [-76111086, 732912, -522846], [388234536, -3284409, -251691], [1634126706, -3693300, 223620], [-1232840434, 1461528, -318564], [-648564183, -835407, 329364], [185759498, 3314994, 144768], [-449514140, 24400, 373752], [-550757160, 713892, 327612], [2401067332, 3578700, 221136], [3053633286, 5847615, -239301], [58782156, 2921100, 486660], [2305302358, -3882620, -145758], [-1584228395, 197275, 254790], [-557772848, -963059, -993693], [-59845712, 6676920, 990380], [-2107686881, 5650200, 575217], [-907976708, 9658902, -695262], [-1495771017, -11260452, 323157], [-1598660198, 6533140, 483378], [4358441484, 10255380, -173886], [1347327120, 5860290, 310146], [-1476490456, -10067220, 997996], [3985643762, 10531040, 132222], [-5195602722, 206565, -407007], [-1953272325, -148185, -1495908], [1371178108, -5668092, 1335992], [-1229015022, 22252440, 892890], [-833869342, -8092512, -363888], [3107962896, -18480416, -1520758], [-8170841974, 5378880, -492722], [-2641291878, -7912410, 127572], [-3874298556, 21144040, 1039838], [3366463275, -9027480, 179439], [-4450243926, -1812840, 1442490], [2939102763, -23678020, 86389], [4269725484, -5194475, 2074907], [-3661151912, -26554614, 55610], [-9647457721, -14052252, 561765], [-11332294665, 5439765, 567198], [-6632466564, 7590619, 574517], [-4676478460, 4880016, -2689064], [4209286395, 2051980, -1269499], [-871631696, -29487840, 1376862], [346811052, -3922956, 1777920], [6254981623, -26412540, -511957], [-11023565278, -14219040, 392494], [-3572605040, -11325870, -1607238], [13465830201, -8657625, -1394142], [-7262775468, -2443200, -2922060], [-5795642408, 41483448, 487134], [-4030581772, 8027507, 996561], [-7291471172, 5906508, -611848], [3422355423, 33486765, 212454], [-1498033642, 47555880, 2660818], [-3064395153, -22258215, 1625970], [5598474396, 47827620, 1377630], [10269303388, 26533968, -288688], [16675765407, -32987860, -496715], [509220298, -19566486, 1082448], [14813755530, 16592313, -1166187], [-135621814, 8523864, 2829742], [6201291864, 44569824, 824016], [-7256291168, 50512290, 1081634], [5540760204, -36724485, 2874897], [11380474707, -4593696, 869055], [12597315219, 22101201, -519102], [-7526154474, -38234232, 4604436], [-22698825574, -35735191, 221097], [-2732278520, -14937804, -693144], [11283916926, -16971240, 3162558], [18801849976, 14373744, 4511172], [32190860745, -7641365, 2277926], [-17273478384, 951420, 3342948], [16125231126, 117479055, -2998377], [-11860324020, 58481400, -1948812], [21560388993, -49870917, -3217878], [4700327178, -62940391, -5772587], [26819904606, -28730544, 2406084], [-10483597322, -37280658, -75264], [22272268550, -74791768, -5592558], [5551254966, 67465436, -6438356], [8341524482, -47287840, 3678810], [-2836177108, 39412140, -3883128], [22459719916, 2700930, -2604394], [-28998700073, -55876020, -766695], [11010059493, 20724165, -2174820], [-14544159396, 74123985, 3791823], [-29526025128, 22376562, 3258858], [28928121135, -30318705, 222120], [-1898987943, 105535164, 1892535], [3312487676, -26965248, -2700860], [-27034904678, -2841623, 5408793], [-24785370148, 72165150, 7716186], [-17170119249, -47057580, 8118729], [18216290377, 50731500, 8500301], [17414135828, -85338510, -2433026], [22526616786, 1407042, 5697360], [30754619572, 99423040, -2667984], [-17425917087, 35119680, -10153041], [-17624458938, -126945075, 2517633], [-22595378007, 93869861, 5383180], [46873751473, -1427003, -1592664], [3175731596, -117117432, -3246354], [19854969732, -73264080, 8120904], [6320404465, 118310820, -8746809], [12058478470, 161889172, -1230678], [-24807241304, 124825230, -3556102], [462295134, 162604957, -7809931], [-39851387800, -9531720, -5305644], [6528211473, 105573909, 8921448], [-25796826234, 88972453, -10241539], [26737570646, -51277075, -6211215], [-18429157373, 51681660, -716775], [-36546298811, -67625952, -9838245], [1534652114, 254937935, 3335403], [60870884946, -161186220, -1993506], [-44351180463, -78927280, 197851], [36074599222, -67699632, -437986], [-732707174, 201718860, 6003770], [12632806036, -2400150, 17972858], [12538999848, -161216256, 16536828], [1588732569, 75145845, 13989576], [-7633303666, -221234040, -15972578], [-6232978818, -118411971, -3995355], [24603142116, -110737160, -19554766], [-59490383873, -204637200, 8269691], [27980167770, 47164992, 23696652], [-56264686659, 3514191, -21804366], [31769104468, 6968652, -14548704], [-6857322780, 157353120, -24905832], [10511724680, 191287628, 22438428], [-20982564320, 204377976, 15410576], [47109991708, 211522860, -4966056], [-54539785686, -61835715, -7172403], [72456192531, -175802427, 11141766], [28526518659, -21489592, 21486385], [-58253916764, -60027000, 17509578], [66956172039, -176678460, -20872413], [-90591499194, -187559580, -13632000], [4122195692, -124349136, -22187460], [114224193894, -63552540, 16683510], [20185905473, 20001000, -12535401], [-127809948742, 197877515, 590103], [-31805987745, 21224545, -8805082], [-27625760847, -318743100, 8884989], [-9363963180, 96143355, 4544469], [27674107668, 44950104, 8766018], [-99139670947, -55710408, -26056799], [85876459828, 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[20887570226, 242231256, 93052584], [-257512095339, 75967176, -23201841], [105948717389, 316857900, -3483623], [80652867888, -407701341, 35947653], [251444308761, 729323280, -26680221], [168060366860, 554139260, 30257316], [-217252850184, -203892048, 10991250], [-233524784494, -133093036, -40931598], [275536005426, 62231400, -46333692], [45573988135, -257851175, 46619850], [29738412056, -112348392, -38153984], [111129875315, 893725715, 21405996], [153806062278, -932152416, 42016338], [-73318091650, -124243560, -43880606], [283089574028, 217310940, 53048008], [-80921013831, 676175688, -71130825], [-143123275130, 651553890, -52159092], [-276512085222, 265874073, 64593237], [14744556960, 962132010, 38179578], [-34489005046, 709316184, -16421484], [148709022474, -258854365, 7312147], [-81782350334, -966032772, -27717454], [-111956672979, 1306836675, 73452330], [180491125135, 177124872, -84903303], [148030541238, 497513040, 17424822], [274743180753, 64332780, 4873251], [-357566451168, 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