Properties

Label 531.5.c.d.235.16
Level $531$
Weight $5$
Character 531.235
Analytic conductor $54.889$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 235.16
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.25

$q$-expansion

\(f(q)\) \(=\) \(q-2.31914i q^{2} +10.6216 q^{4} +38.6371 q^{5} +9.48964 q^{7} -61.7392i q^{8} +O(q^{10})\) \(q-2.31914i q^{2} +10.6216 q^{4} +38.6371 q^{5} +9.48964 q^{7} -61.7392i q^{8} -89.6048i q^{10} +58.4880i q^{11} +290.914i q^{13} -22.0078i q^{14} +26.7637 q^{16} +467.291 q^{17} -261.109 q^{19} +410.387 q^{20} +135.642 q^{22} -190.226i q^{23} +867.825 q^{25} +674.670 q^{26} +100.795 q^{28} +407.645 q^{29} +1893.18i q^{31} -1049.90i q^{32} -1083.71i q^{34} +366.652 q^{35} -306.386i q^{37} +605.549i q^{38} -2385.42i q^{40} +486.816 q^{41} -1830.69i q^{43} +621.236i q^{44} -441.160 q^{46} +1576.43i q^{47} -2310.95 q^{49} -2012.61i q^{50} +3089.97i q^{52} +2715.01 q^{53} +2259.81i q^{55} -585.883i q^{56} -945.384i q^{58} +(-3310.90 - 1074.84i) q^{59} +3956.15i q^{61} +4390.55 q^{62} -2006.63 q^{64} +11240.1i q^{65} +5269.11i q^{67} +4963.37 q^{68} -850.317i q^{70} -7353.03 q^{71} -8254.16i q^{73} -710.553 q^{74} -2773.40 q^{76} +555.030i q^{77} +8388.54 q^{79} +1034.07 q^{80} -1128.99i q^{82} +8211.40i q^{83} +18054.8 q^{85} -4245.63 q^{86} +3611.00 q^{88} -13635.4i q^{89} +2760.67i q^{91} -2020.50i q^{92} +3655.95 q^{94} -10088.5 q^{95} +5348.42i q^{97} +5359.41i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.31914i 0.579785i −0.957059 0.289892i \(-0.906380\pi\)
0.957059 0.289892i \(-0.0936195\pi\)
\(3\) 0 0
\(4\) 10.6216 0.663850
\(5\) 38.6371 1.54548 0.772742 0.634720i \(-0.218885\pi\)
0.772742 + 0.634720i \(0.218885\pi\)
\(6\) 0 0
\(7\) 9.48964 0.193666 0.0968331 0.995301i \(-0.469129\pi\)
0.0968331 + 0.995301i \(0.469129\pi\)
\(8\) 61.7392i 0.964675i
\(9\) 0 0
\(10\) 89.6048i 0.896048i
\(11\) 58.4880i 0.483372i 0.970355 + 0.241686i \(0.0777004\pi\)
−0.970355 + 0.241686i \(0.922300\pi\)
\(12\) 0 0
\(13\) 290.914i 1.72138i 0.509125 + 0.860692i \(0.329969\pi\)
−0.509125 + 0.860692i \(0.670031\pi\)
\(14\) 22.0078i 0.112285i
\(15\) 0 0
\(16\) 26.7637 0.104546
\(17\) 467.291 1.61692 0.808462 0.588548i \(-0.200300\pi\)
0.808462 + 0.588548i \(0.200300\pi\)
\(18\) 0 0
\(19\) −261.109 −0.723294 −0.361647 0.932315i \(-0.617785\pi\)
−0.361647 + 0.932315i \(0.617785\pi\)
\(20\) 410.387 1.02597
\(21\) 0 0
\(22\) 135.642 0.280252
\(23\) 190.226i 0.359595i −0.983704 0.179798i \(-0.942456\pi\)
0.983704 0.179798i \(-0.0575443\pi\)
\(24\) 0 0
\(25\) 867.825 1.38852
\(26\) 674.670 0.998033
\(27\) 0 0
\(28\) 100.795 0.128565
\(29\) 407.645 0.484714 0.242357 0.970187i \(-0.422079\pi\)
0.242357 + 0.970187i \(0.422079\pi\)
\(30\) 0 0
\(31\) 1893.18i 1.97001i 0.172518 + 0.985006i \(0.444810\pi\)
−0.172518 + 0.985006i \(0.555190\pi\)
\(32\) 1049.90i 1.02529i
\(33\) 0 0
\(34\) 1083.71i 0.937468i
\(35\) 366.652 0.299308
\(36\) 0 0
\(37\) 306.386i 0.223803i −0.993719 0.111902i \(-0.964306\pi\)
0.993719 0.111902i \(-0.0356941\pi\)
\(38\) 605.549i 0.419355i
\(39\) 0 0
\(40\) 2385.42i 1.49089i
\(41\) 486.816 0.289599 0.144799 0.989461i \(-0.453746\pi\)
0.144799 + 0.989461i \(0.453746\pi\)
\(42\) 0 0
\(43\) 1830.69i 0.990098i −0.868865 0.495049i \(-0.835150\pi\)
0.868865 0.495049i \(-0.164850\pi\)
\(44\) 621.236i 0.320886i
\(45\) 0 0
\(46\) −441.160 −0.208488
\(47\) 1576.43i 0.713637i 0.934174 + 0.356819i \(0.116139\pi\)
−0.934174 + 0.356819i \(0.883861\pi\)
\(48\) 0 0
\(49\) −2310.95 −0.962493
\(50\) 2012.61i 0.805043i
\(51\) 0 0
\(52\) 3089.97i 1.14274i
\(53\) 2715.01 0.966541 0.483271 0.875471i \(-0.339449\pi\)
0.483271 + 0.875471i \(0.339449\pi\)
\(54\) 0 0
\(55\) 2259.81i 0.747043i
\(56\) 585.883i 0.186825i
\(57\) 0 0
\(58\) 945.384i 0.281030i
\(59\) −3310.90 1074.84i −0.951135 0.308774i
\(60\) 0 0
\(61\) 3956.15i 1.06320i 0.846997 + 0.531598i \(0.178408\pi\)
−0.846997 + 0.531598i \(0.821592\pi\)
\(62\) 4390.55 1.14218
\(63\) 0 0
\(64\) −2006.63 −0.489901
\(65\) 11240.1i 2.66037i
\(66\) 0 0
\(67\) 5269.11i 1.17378i 0.809666 + 0.586891i \(0.199648\pi\)
−0.809666 + 0.586891i \(0.800352\pi\)
\(68\) 4963.37 1.07339
\(69\) 0 0
\(70\) 850.317i 0.173534i
\(71\) −7353.03 −1.45865 −0.729323 0.684169i \(-0.760165\pi\)
−0.729323 + 0.684169i \(0.760165\pi\)
\(72\) 0 0
\(73\) 8254.16i 1.54891i −0.632627 0.774457i \(-0.718023\pi\)
0.632627 0.774457i \(-0.281977\pi\)
\(74\) −710.553 −0.129758
\(75\) 0 0
\(76\) −2773.40 −0.480158
\(77\) 555.030i 0.0936128i
\(78\) 0 0
\(79\) 8388.54 1.34410 0.672051 0.740505i \(-0.265414\pi\)
0.672051 + 0.740505i \(0.265414\pi\)
\(80\) 1034.07 0.161574
\(81\) 0 0
\(82\) 1128.99i 0.167905i
\(83\) 8211.40i 1.19196i 0.803000 + 0.595979i \(0.203236\pi\)
−0.803000 + 0.595979i \(0.796764\pi\)
\(84\) 0 0
\(85\) 18054.8 2.49893
\(86\) −4245.63 −0.574044
\(87\) 0 0
\(88\) 3611.00 0.466297
\(89\) 13635.4i 1.72142i −0.509092 0.860712i \(-0.670019\pi\)
0.509092 0.860712i \(-0.329981\pi\)
\(90\) 0 0
\(91\) 2760.67i 0.333374i
\(92\) 2020.50i 0.238717i
\(93\) 0 0
\(94\) 3655.95 0.413756
\(95\) −10088.5 −1.11784
\(96\) 0 0
\(97\) 5348.42i 0.568436i 0.958760 + 0.284218i \(0.0917340\pi\)
−0.958760 + 0.284218i \(0.908266\pi\)
\(98\) 5359.41i 0.558039i
\(99\) 0 0
\(100\) 9217.68 0.921768
\(101\) 11578.9i 1.13508i −0.823347 0.567538i \(-0.807896\pi\)
0.823347 0.567538i \(-0.192104\pi\)
\(102\) 0 0
\(103\) 18902.9i 1.78178i −0.454222 0.890888i \(-0.650083\pi\)
0.454222 0.890888i \(-0.349917\pi\)
\(104\) 17960.8 1.66058
\(105\) 0 0
\(106\) 6296.50i 0.560386i
\(107\) −15473.8 −1.35155 −0.675773 0.737110i \(-0.736190\pi\)
−0.675773 + 0.737110i \(0.736190\pi\)
\(108\) 0 0
\(109\) 10396.5i 0.875049i −0.899206 0.437525i \(-0.855855\pi\)
0.899206 0.437525i \(-0.144145\pi\)
\(110\) 5240.80 0.433124
\(111\) 0 0
\(112\) 253.978 0.0202470
\(113\) 3655.52i 0.286281i −0.989702 0.143141i \(-0.954280\pi\)
0.989702 0.143141i \(-0.0457201\pi\)
\(114\) 0 0
\(115\) 7349.77i 0.555748i
\(116\) 4329.83 0.321777
\(117\) 0 0
\(118\) −2492.71 + 7678.44i −0.179022 + 0.551454i
\(119\) 4434.42 0.313143
\(120\) 0 0
\(121\) 11220.2 0.766352
\(122\) 9174.87 0.616425
\(123\) 0 0
\(124\) 20108.6i 1.30779i
\(125\) 9382.04 0.600451
\(126\) 0 0
\(127\) 8960.25 0.555536 0.277768 0.960648i \(-0.410405\pi\)
0.277768 + 0.960648i \(0.410405\pi\)
\(128\) 12144.7i 0.741252i
\(129\) 0 0
\(130\) 26067.3 1.54244
\(131\) 12620.5i 0.735417i −0.929941 0.367708i \(-0.880142\pi\)
0.929941 0.367708i \(-0.119858\pi\)
\(132\) 0 0
\(133\) −2477.83 −0.140078
\(134\) 12219.8 0.680541
\(135\) 0 0
\(136\) 28850.2i 1.55981i
\(137\) 21067.5 1.12246 0.561232 0.827659i \(-0.310328\pi\)
0.561232 + 0.827659i \(0.310328\pi\)
\(138\) 0 0
\(139\) 23705.1 1.22691 0.613455 0.789729i \(-0.289779\pi\)
0.613455 + 0.789729i \(0.289779\pi\)
\(140\) 3894.43 0.198695
\(141\) 0 0
\(142\) 17052.7i 0.845701i
\(143\) −17015.0 −0.832069
\(144\) 0 0
\(145\) 15750.2 0.749118
\(146\) −19142.5 −0.898037
\(147\) 0 0
\(148\) 3254.31i 0.148572i
\(149\) 5346.43i 0.240819i −0.992724 0.120410i \(-0.961579\pi\)
0.992724 0.120410i \(-0.0384208\pi\)
\(150\) 0 0
\(151\) 27531.3i 1.20746i 0.797188 + 0.603731i \(0.206320\pi\)
−0.797188 + 0.603731i \(0.793680\pi\)
\(152\) 16120.7i 0.697744i
\(153\) 0 0
\(154\) 1287.19 0.0542753
\(155\) 73147.1i 3.04462i
\(156\) 0 0
\(157\) 29614.0i 1.20143i −0.799465 0.600713i \(-0.794883\pi\)
0.799465 0.600713i \(-0.205117\pi\)
\(158\) 19454.2i 0.779289i
\(159\) 0 0
\(160\) 40564.9i 1.58457i
\(161\) 1805.17i 0.0696414i
\(162\) 0 0
\(163\) 523.410 0.0197000 0.00985001 0.999951i \(-0.496865\pi\)
0.00985001 + 0.999951i \(0.496865\pi\)
\(164\) 5170.76 0.192250
\(165\) 0 0
\(166\) 19043.4 0.691079
\(167\) −8380.72 −0.300503 −0.150251 0.988648i \(-0.548008\pi\)
−0.150251 + 0.988648i \(0.548008\pi\)
\(168\) 0 0
\(169\) −56070.0 −1.96317
\(170\) 41871.5i 1.44884i
\(171\) 0 0
\(172\) 19444.9i 0.657276i
\(173\) 31624.9i 1.05666i −0.849038 0.528331i \(-0.822818\pi\)
0.849038 0.528331i \(-0.177182\pi\)
\(174\) 0 0
\(175\) 8235.35 0.268909
\(176\) 1565.36i 0.0505345i
\(177\) 0 0
\(178\) −31622.4 −0.998055
\(179\) 14600.1i 0.455668i 0.973700 + 0.227834i \(0.0731644\pi\)
−0.973700 + 0.227834i \(0.926836\pi\)
\(180\) 0 0
\(181\) −48410.5 −1.47769 −0.738843 0.673877i \(-0.764628\pi\)
−0.738843 + 0.673877i \(0.764628\pi\)
\(182\) 6402.38 0.193285
\(183\) 0 0
\(184\) −11744.4 −0.346892
\(185\) 11837.9i 0.345884i
\(186\) 0 0
\(187\) 27330.9i 0.781575i
\(188\) 16744.1i 0.473748i
\(189\) 0 0
\(190\) 23396.6i 0.648106i
\(191\) 11986.4i 0.328566i −0.986413 0.164283i \(-0.947469\pi\)
0.986413 0.164283i \(-0.0525310\pi\)
\(192\) 0 0
\(193\) 19496.9 0.523420 0.261710 0.965147i \(-0.415714\pi\)
0.261710 + 0.965147i \(0.415714\pi\)
\(194\) 12403.7 0.329571
\(195\) 0 0
\(196\) −24545.9 −0.638951
\(197\) 42645.4 1.09885 0.549426 0.835542i \(-0.314846\pi\)
0.549426 + 0.835542i \(0.314846\pi\)
\(198\) 0 0
\(199\) 10669.4 0.269421 0.134711 0.990885i \(-0.456990\pi\)
0.134711 + 0.990885i \(0.456990\pi\)
\(200\) 53578.8i 1.33947i
\(201\) 0 0
\(202\) −26853.1 −0.658100
\(203\) 3868.40 0.0938727
\(204\) 0 0
\(205\) 18809.1 0.447570
\(206\) −43838.4 −1.03305
\(207\) 0 0
\(208\) 7785.94i 0.179963i
\(209\) 15271.8i 0.349620i
\(210\) 0 0
\(211\) 15256.3i 0.342677i −0.985212 0.171339i \(-0.945191\pi\)
0.985212 0.171339i \(-0.0548092\pi\)
\(212\) 28837.8 0.641638
\(213\) 0 0
\(214\) 35886.0i 0.783605i
\(215\) 70732.6i 1.53018i
\(216\) 0 0
\(217\) 17965.6i 0.381525i
\(218\) −24110.8 −0.507340
\(219\) 0 0
\(220\) 24002.7i 0.495924i
\(221\) 135941.i 2.78335i
\(222\) 0 0
\(223\) 17062.9 0.343117 0.171558 0.985174i \(-0.445120\pi\)
0.171558 + 0.985174i \(0.445120\pi\)
\(224\) 9963.13i 0.198564i
\(225\) 0 0
\(226\) −8477.67 −0.165981
\(227\) 15718.2i 0.305037i −0.988301 0.152518i \(-0.951262\pi\)
0.988301 0.152518i \(-0.0487383\pi\)
\(228\) 0 0
\(229\) 10725.1i 0.204518i 0.994758 + 0.102259i \(0.0326071\pi\)
−0.994758 + 0.102259i \(0.967393\pi\)
\(230\) −17045.1 −0.322214
\(231\) 0 0
\(232\) 25167.6i 0.467591i
\(233\) 34953.2i 0.643836i 0.946768 + 0.321918i \(0.104327\pi\)
−0.946768 + 0.321918i \(0.895673\pi\)
\(234\) 0 0
\(235\) 60908.5i 1.10291i
\(236\) −35167.1 11416.5i −0.631411 0.204979i
\(237\) 0 0
\(238\) 10284.0i 0.181556i
\(239\) −43682.3 −0.764732 −0.382366 0.924011i \(-0.624891\pi\)
−0.382366 + 0.924011i \(0.624891\pi\)
\(240\) 0 0
\(241\) −37284.2 −0.641935 −0.320968 0.947090i \(-0.604008\pi\)
−0.320968 + 0.947090i \(0.604008\pi\)
\(242\) 26021.1i 0.444319i
\(243\) 0 0
\(244\) 42020.7i 0.705802i
\(245\) −89288.3 −1.48752
\(246\) 0 0
\(247\) 75960.3i 1.24507i
\(248\) 116884. 1.90042
\(249\) 0 0
\(250\) 21758.3i 0.348132i
\(251\) 42855.1 0.680229 0.340114 0.940384i \(-0.389534\pi\)
0.340114 + 0.940384i \(0.389534\pi\)
\(252\) 0 0
\(253\) 11125.9 0.173818
\(254\) 20780.1i 0.322092i
\(255\) 0 0
\(256\) −60271.3 −0.919668
\(257\) 67944.9 1.02870 0.514352 0.857579i \(-0.328033\pi\)
0.514352 + 0.857579i \(0.328033\pi\)
\(258\) 0 0
\(259\) 2907.50i 0.0433431i
\(260\) 119387.i 1.76609i
\(261\) 0 0
\(262\) −29268.7 −0.426383
\(263\) −6967.85 −0.100737 −0.0503683 0.998731i \(-0.516040\pi\)
−0.0503683 + 0.998731i \(0.516040\pi\)
\(264\) 0 0
\(265\) 104900. 1.49377
\(266\) 5746.44i 0.0812149i
\(267\) 0 0
\(268\) 55966.3i 0.779215i
\(269\) 19627.0i 0.271238i 0.990761 + 0.135619i \(0.0433023\pi\)
−0.990761 + 0.135619i \(0.956698\pi\)
\(270\) 0 0
\(271\) −25914.9 −0.352866 −0.176433 0.984313i \(-0.556456\pi\)
−0.176433 + 0.984313i \(0.556456\pi\)
\(272\) 12506.4 0.169043
\(273\) 0 0
\(274\) 48858.5i 0.650787i
\(275\) 50757.3i 0.671171i
\(276\) 0 0
\(277\) −104861. −1.36664 −0.683318 0.730121i \(-0.739464\pi\)
−0.683318 + 0.730121i \(0.739464\pi\)
\(278\) 54975.5i 0.711344i
\(279\) 0 0
\(280\) 22636.8i 0.288735i
\(281\) 14867.0 0.188283 0.0941417 0.995559i \(-0.469989\pi\)
0.0941417 + 0.995559i \(0.469989\pi\)
\(282\) 0 0
\(283\) 72926.7i 0.910570i 0.890346 + 0.455285i \(0.150463\pi\)
−0.890346 + 0.455285i \(0.849537\pi\)
\(284\) −78100.9 −0.968322
\(285\) 0 0
\(286\) 39460.1i 0.482421i
\(287\) 4619.71 0.0560855
\(288\) 0 0
\(289\) 134840. 1.61444
\(290\) 36526.9i 0.434327i
\(291\) 0 0
\(292\) 87672.3i 1.02825i
\(293\) −69720.2 −0.812125 −0.406063 0.913845i \(-0.633099\pi\)
−0.406063 + 0.913845i \(0.633099\pi\)
\(294\) 0 0
\(295\) −127924. 41528.8i −1.46996 0.477205i
\(296\) −18916.1 −0.215897
\(297\) 0 0
\(298\) −12399.1 −0.139623
\(299\) 55339.4 0.619002
\(300\) 0 0
\(301\) 17372.6i 0.191748i
\(302\) 63849.0 0.700068
\(303\) 0 0
\(304\) −6988.25 −0.0756173
\(305\) 152854.i 1.64315i
\(306\) 0 0
\(307\) −140429. −1.48997 −0.744987 0.667079i \(-0.767544\pi\)
−0.744987 + 0.667079i \(0.767544\pi\)
\(308\) 5895.30i 0.0621448i
\(309\) 0 0
\(310\) 169638. 1.76523
\(311\) 106994. 1.10621 0.553105 0.833112i \(-0.313443\pi\)
0.553105 + 0.833112i \(0.313443\pi\)
\(312\) 0 0
\(313\) 111408.i 1.13717i −0.822623 0.568587i \(-0.807490\pi\)
0.822623 0.568587i \(-0.192510\pi\)
\(314\) −68678.9 −0.696569
\(315\) 0 0
\(316\) 89099.6 0.892281
\(317\) −139639. −1.38960 −0.694799 0.719204i \(-0.744507\pi\)
−0.694799 + 0.719204i \(0.744507\pi\)
\(318\) 0 0
\(319\) 23842.3i 0.234297i
\(320\) −77530.5 −0.757134
\(321\) 0 0
\(322\) −4186.45 −0.0403770
\(323\) −122014. −1.16951
\(324\) 0 0
\(325\) 252462.i 2.39018i
\(326\) 1213.86i 0.0114218i
\(327\) 0 0
\(328\) 30055.6i 0.279369i
\(329\) 14959.7i 0.138207i
\(330\) 0 0
\(331\) 54555.5 0.497947 0.248973 0.968510i \(-0.419907\pi\)
0.248973 + 0.968510i \(0.419907\pi\)
\(332\) 87218.1i 0.791281i
\(333\) 0 0
\(334\) 19436.1i 0.174227i
\(335\) 203583.i 1.81406i
\(336\) 0 0
\(337\) 48646.0i 0.428339i 0.976797 + 0.214169i \(0.0687045\pi\)
−0.976797 + 0.214169i \(0.931296\pi\)
\(338\) 130034.i 1.13821i
\(339\) 0 0
\(340\) 191770. 1.65891
\(341\) −110728. −0.952249
\(342\) 0 0
\(343\) −44714.7 −0.380069
\(344\) −113025. −0.955123
\(345\) 0 0
\(346\) −73342.5 −0.612637
\(347\) 115161.i 0.956413i −0.878248 0.478206i \(-0.841287\pi\)
0.878248 0.478206i \(-0.158713\pi\)
\(348\) 0 0
\(349\) 45753.9i 0.375645i −0.982203 0.187822i \(-0.939857\pi\)
0.982203 0.187822i \(-0.0601429\pi\)
\(350\) 19098.9i 0.155909i
\(351\) 0 0
\(352\) 61406.3 0.495596
\(353\) 118207.i 0.948622i −0.880357 0.474311i \(-0.842697\pi\)
0.880357 0.474311i \(-0.157303\pi\)
\(354\) 0 0
\(355\) −284100. −2.25431
\(356\) 144830.i 1.14277i
\(357\) 0 0
\(358\) 33859.6 0.264189
\(359\) −210839. −1.63592 −0.817961 0.575274i \(-0.804896\pi\)
−0.817961 + 0.575274i \(0.804896\pi\)
\(360\) 0 0
\(361\) −62143.0 −0.476846
\(362\) 112271.i 0.856741i
\(363\) 0 0
\(364\) 29322.7i 0.221310i
\(365\) 318917.i 2.39382i
\(366\) 0 0
\(367\) 29773.6i 0.221054i 0.993873 + 0.110527i \(0.0352539\pi\)
−0.993873 + 0.110527i \(0.964746\pi\)
\(368\) 5091.15i 0.0375942i
\(369\) 0 0
\(370\) −27453.7 −0.200538
\(371\) 25764.5 0.187186
\(372\) 0 0
\(373\) −25947.3 −0.186498 −0.0932492 0.995643i \(-0.529725\pi\)
−0.0932492 + 0.995643i \(0.529725\pi\)
\(374\) 63384.2 0.453146
\(375\) 0 0
\(376\) 97327.2 0.688428
\(377\) 118590.i 0.834379i
\(378\) 0 0
\(379\) −268474. −1.86906 −0.934530 0.355884i \(-0.884180\pi\)
−0.934530 + 0.355884i \(0.884180\pi\)
\(380\) −107156. −0.742077
\(381\) 0 0
\(382\) −27798.2 −0.190498
\(383\) −164228. −1.11956 −0.559782 0.828640i \(-0.689115\pi\)
−0.559782 + 0.828640i \(0.689115\pi\)
\(384\) 0 0
\(385\) 21444.7i 0.144677i
\(386\) 45216.0i 0.303471i
\(387\) 0 0
\(388\) 56808.7i 0.377356i
\(389\) 231786. 1.53175 0.765876 0.642988i \(-0.222305\pi\)
0.765876 + 0.642988i \(0.222305\pi\)
\(390\) 0 0
\(391\) 88890.8i 0.581438i
\(392\) 142676.i 0.928493i
\(393\) 0 0
\(394\) 98900.6i 0.637098i
\(395\) 324109. 2.07729
\(396\) 0 0
\(397\) 66558.9i 0.422304i 0.977453 + 0.211152i \(0.0677215\pi\)
−0.977453 + 0.211152i \(0.932278\pi\)
\(398\) 24743.7i 0.156206i
\(399\) 0 0
\(400\) 23226.2 0.145164
\(401\) 22759.8i 0.141540i −0.997493 0.0707702i \(-0.977454\pi\)
0.997493 0.0707702i \(-0.0225457\pi\)
\(402\) 0 0
\(403\) −550753. −3.39115
\(404\) 122986.i 0.753520i
\(405\) 0 0
\(406\) 8971.36i 0.0544260i
\(407\) 17919.9 0.108180
\(408\) 0 0
\(409\) 312184.i 1.86623i 0.359583 + 0.933113i \(0.382919\pi\)
−0.359583 + 0.933113i \(0.617081\pi\)
\(410\) 43621.0i 0.259494i
\(411\) 0 0
\(412\) 200779.i 1.18283i
\(413\) −31419.3 10199.9i −0.184203 0.0597991i
\(414\) 0 0
\(415\) 317264.i 1.84215i
\(416\) 305429. 1.76492
\(417\) 0 0
\(418\) −35417.3 −0.202704
\(419\) 314728.i 1.79270i −0.443346 0.896351i \(-0.646209\pi\)
0.443346 0.896351i \(-0.353791\pi\)
\(420\) 0 0
\(421\) 25224.4i 0.142317i −0.997465 0.0711585i \(-0.977330\pi\)
0.997465 0.0711585i \(-0.0226696\pi\)
\(422\) −35381.6 −0.198679
\(423\) 0 0
\(424\) 167623.i 0.932398i
\(425\) 405527. 2.24513
\(426\) 0 0
\(427\) 37542.5i 0.205905i
\(428\) −164357. −0.897223
\(429\) 0 0
\(430\) −164039. −0.887175
\(431\) 58420.2i 0.314491i −0.987560 0.157246i \(-0.949739\pi\)
0.987560 0.157246i \(-0.0502614\pi\)
\(432\) 0 0
\(433\) 59683.1 0.318328 0.159164 0.987252i \(-0.449120\pi\)
0.159164 + 0.987252i \(0.449120\pi\)
\(434\) 41664.8 0.221202
\(435\) 0 0
\(436\) 110427.i 0.580901i
\(437\) 49669.7i 0.260093i
\(438\) 0 0
\(439\) 200004. 1.03779 0.518895 0.854838i \(-0.326344\pi\)
0.518895 + 0.854838i \(0.326344\pi\)
\(440\) 139519. 0.720654
\(441\) 0 0
\(442\) 315267. 1.61374
\(443\) 299059.i 1.52388i 0.647649 + 0.761939i \(0.275752\pi\)
−0.647649 + 0.761939i \(0.724248\pi\)
\(444\) 0 0
\(445\) 526832.i 2.66043i
\(446\) 39571.1i 0.198934i
\(447\) 0 0
\(448\) −19042.2 −0.0948773
\(449\) −308677. −1.53113 −0.765565 0.643359i \(-0.777540\pi\)
−0.765565 + 0.643359i \(0.777540\pi\)
\(450\) 0 0
\(451\) 28472.9i 0.139984i
\(452\) 38827.5i 0.190048i
\(453\) 0 0
\(454\) −36452.8 −0.176856
\(455\) 106664.i 0.515224i
\(456\) 0 0
\(457\) 224815.i 1.07645i 0.842803 + 0.538223i \(0.180904\pi\)
−0.842803 + 0.538223i \(0.819096\pi\)
\(458\) 24873.1 0.118577
\(459\) 0 0
\(460\) 78066.3i 0.368933i
\(461\) −186398. −0.877079 −0.438539 0.898712i \(-0.644504\pi\)
−0.438539 + 0.898712i \(0.644504\pi\)
\(462\) 0 0
\(463\) 45013.2i 0.209980i −0.994473 0.104990i \(-0.966519\pi\)
0.994473 0.104990i \(-0.0334810\pi\)
\(464\) 10910.1 0.0506748
\(465\) 0 0
\(466\) 81061.3 0.373286
\(467\) 243089.i 1.11463i −0.830300 0.557316i \(-0.811831\pi\)
0.830300 0.557316i \(-0.188169\pi\)
\(468\) 0 0
\(469\) 50001.9i 0.227322i
\(470\) 141255. 0.639453
\(471\) 0 0
\(472\) −66359.9 + 204412.i −0.297866 + 0.917536i
\(473\) 107073. 0.478586
\(474\) 0 0
\(475\) −226597. −1.00431
\(476\) 47100.6 0.207880
\(477\) 0 0
\(478\) 101305.i 0.443380i
\(479\) −115564. −0.503677 −0.251839 0.967769i \(-0.581035\pi\)
−0.251839 + 0.967769i \(0.581035\pi\)
\(480\) 0 0
\(481\) 89132.1 0.385251
\(482\) 86467.3i 0.372184i
\(483\) 0 0
\(484\) 119176. 0.508742
\(485\) 206647.i 0.878509i
\(486\) 0 0
\(487\) 151048. 0.636880 0.318440 0.947943i \(-0.396841\pi\)
0.318440 + 0.947943i \(0.396841\pi\)
\(488\) 244250. 1.02564
\(489\) 0 0
\(490\) 207072.i 0.862440i
\(491\) 20445.7 0.0848086 0.0424043 0.999101i \(-0.486498\pi\)
0.0424043 + 0.999101i \(0.486498\pi\)
\(492\) 0 0
\(493\) 190489. 0.783746
\(494\) −176163. −0.721871
\(495\) 0 0
\(496\) 50668.6i 0.205957i
\(497\) −69777.7 −0.282490
\(498\) 0 0
\(499\) −59385.2 −0.238494 −0.119247 0.992865i \(-0.538048\pi\)
−0.119247 + 0.992865i \(0.538048\pi\)
\(500\) 99652.2 0.398609
\(501\) 0 0
\(502\) 99386.9i 0.394386i
\(503\) 156531.i 0.618677i 0.950952 + 0.309339i \(0.100108\pi\)
−0.950952 + 0.309339i \(0.899892\pi\)
\(504\) 0 0
\(505\) 447375.i 1.75424i
\(506\) 25802.6i 0.100777i
\(507\) 0 0
\(508\) 95172.1 0.368793
\(509\) 250897.i 0.968410i −0.874954 0.484205i \(-0.839109\pi\)
0.874954 0.484205i \(-0.160891\pi\)
\(510\) 0 0
\(511\) 78329.0i 0.299972i
\(512\) 54537.0i 0.208042i
\(513\) 0 0
\(514\) 157574.i 0.596427i
\(515\) 730352.i 2.75371i
\(516\) 0 0
\(517\) −92201.9 −0.344952
\(518\) −6742.89 −0.0251297
\(519\) 0 0
\(520\) 693953. 2.56639
\(521\) 149931. 0.552351 0.276175 0.961107i \(-0.410933\pi\)
0.276175 + 0.961107i \(0.410933\pi\)
\(522\) 0 0
\(523\) −90744.5 −0.331754 −0.165877 0.986146i \(-0.553046\pi\)
−0.165877 + 0.986146i \(0.553046\pi\)
\(524\) 134050.i 0.488206i
\(525\) 0 0
\(526\) 16159.4i 0.0584055i
\(527\) 884667.i 3.18536i
\(528\) 0 0
\(529\) 243655. 0.870691
\(530\) 243278.i 0.866067i
\(531\) 0 0
\(532\) −26318.5 −0.0929904
\(533\) 141621.i 0.498511i
\(534\) 0 0
\(535\) −597864. −2.08879
\(536\) 325310. 1.13232
\(537\) 0 0
\(538\) 45517.8 0.157259
\(539\) 135163.i 0.465242i
\(540\) 0 0
\(541\) 293061.i 1.00130i −0.865650 0.500650i \(-0.833094\pi\)
0.865650 0.500650i \(-0.166906\pi\)
\(542\) 60100.2i 0.204587i
\(543\) 0 0
\(544\) 490607.i 1.65781i
\(545\) 401689.i 1.35237i
\(546\) 0 0
\(547\) −264316. −0.883384 −0.441692 0.897167i \(-0.645622\pi\)
−0.441692 + 0.897167i \(0.645622\pi\)
\(548\) 223771. 0.745147
\(549\) 0 0
\(550\) 117713. 0.389135
\(551\) −106440. −0.350591
\(552\) 0 0
\(553\) 79604.2 0.260307
\(554\) 243187.i 0.792355i
\(555\) 0 0
\(556\) 251786. 0.814484
\(557\) −349573. −1.12675 −0.563375 0.826202i \(-0.690497\pi\)
−0.563375 + 0.826202i \(0.690497\pi\)
\(558\) 0 0
\(559\) 532574. 1.70434
\(560\) 9812.97 0.0312914
\(561\) 0 0
\(562\) 34478.7i 0.109164i
\(563\) 87644.3i 0.276508i −0.990397 0.138254i \(-0.955851\pi\)
0.990397 0.138254i \(-0.0441490\pi\)
\(564\) 0 0
\(565\) 141239.i 0.442443i
\(566\) 169127. 0.527935
\(567\) 0 0
\(568\) 453970.i 1.40712i
\(569\) 81258.9i 0.250984i 0.992095 + 0.125492i \(0.0400510\pi\)
−0.992095 + 0.125492i \(0.959949\pi\)
\(570\) 0 0
\(571\) 490531.i 1.50451i 0.658875 + 0.752253i \(0.271033\pi\)
−0.658875 + 0.752253i \(0.728967\pi\)
\(572\) −180726. −0.552369
\(573\) 0 0
\(574\) 10713.7i 0.0325175i
\(575\) 165083.i 0.499305i
\(576\) 0 0
\(577\) 458744. 1.37790 0.688951 0.724807i \(-0.258071\pi\)
0.688951 + 0.724807i \(0.258071\pi\)
\(578\) 312712.i 0.936029i
\(579\) 0 0
\(580\) 167292. 0.497301
\(581\) 77923.2i 0.230842i
\(582\) 0 0
\(583\) 158796.i 0.467199i
\(584\) −509605. −1.49420
\(585\) 0 0
\(586\) 161691.i 0.470858i
\(587\) 346247.i 1.00487i 0.864615 + 0.502435i \(0.167562\pi\)
−0.864615 + 0.502435i \(0.832438\pi\)
\(588\) 0 0
\(589\) 494327.i 1.42490i
\(590\) −96311.0 + 296673.i −0.276676 + 0.852263i
\(591\) 0 0
\(592\) 8200.04i 0.0233977i
\(593\) −507006. −1.44180 −0.720898 0.693041i \(-0.756271\pi\)
−0.720898 + 0.693041i \(0.756271\pi\)
\(594\) 0 0
\(595\) 171333. 0.483958
\(596\) 56787.6i 0.159868i
\(597\) 0 0
\(598\) 128340.i 0.358888i
\(599\) −571461. −1.59270 −0.796348 0.604838i \(-0.793238\pi\)
−0.796348 + 0.604838i \(0.793238\pi\)
\(600\) 0 0
\(601\) 393314.i 1.08891i 0.838792 + 0.544453i \(0.183263\pi\)
−0.838792 + 0.544453i \(0.816737\pi\)
\(602\) −40289.5 −0.111173
\(603\) 0 0
\(604\) 292427.i 0.801573i
\(605\) 433514. 1.18438
\(606\) 0 0
\(607\) −130471. −0.354108 −0.177054 0.984201i \(-0.556657\pi\)
−0.177054 + 0.984201i \(0.556657\pi\)
\(608\) 274137.i 0.741585i
\(609\) 0 0
\(610\) 354490. 0.952675
\(611\) −458604. −1.22844
\(612\) 0 0
\(613\) 512655.i 1.36428i 0.731220 + 0.682142i \(0.238951\pi\)
−0.731220 + 0.682142i \(0.761049\pi\)
\(614\) 325673.i 0.863864i
\(615\) 0 0
\(616\) 34267.1 0.0903059
\(617\) −120180. −0.315692 −0.157846 0.987464i \(-0.550455\pi\)
−0.157846 + 0.987464i \(0.550455\pi\)
\(618\) 0 0
\(619\) 460899. 1.20289 0.601443 0.798916i \(-0.294593\pi\)
0.601443 + 0.798916i \(0.294593\pi\)
\(620\) 776938.i 2.02117i
\(621\) 0 0
\(622\) 248133.i 0.641364i
\(623\) 129395.i 0.333382i
\(624\) 0 0
\(625\) −179896. −0.460533
\(626\) −258370. −0.659316
\(627\) 0 0
\(628\) 314547.i 0.797566i
\(629\) 143172.i 0.361873i
\(630\) 0 0
\(631\) 218188. 0.547989 0.273995 0.961731i \(-0.411655\pi\)
0.273995 + 0.961731i \(0.411655\pi\)
\(632\) 517901.i 1.29662i
\(633\) 0 0
\(634\) 323843.i 0.805668i
\(635\) 346198. 0.858572
\(636\) 0 0
\(637\) 672287.i 1.65682i
\(638\) 55293.6 0.135842
\(639\) 0 0
\(640\) 469235.i 1.14559i
\(641\) 761314. 1.85288 0.926441 0.376440i \(-0.122852\pi\)
0.926441 + 0.376440i \(0.122852\pi\)
\(642\) 0 0
\(643\) −131827. −0.318848 −0.159424 0.987210i \(-0.550964\pi\)
−0.159424 + 0.987210i \(0.550964\pi\)
\(644\) 19173.8i 0.0462314i
\(645\) 0 0
\(646\) 282967.i 0.678065i
\(647\) 673769. 1.60954 0.804771 0.593585i \(-0.202288\pi\)
0.804771 + 0.593585i \(0.202288\pi\)
\(648\) 0 0
\(649\) 62865.4 193648.i 0.149253 0.459752i
\(650\) 585495. 1.38579
\(651\) 0 0
\(652\) 5559.45 0.0130779
\(653\) 310717. 0.728684 0.364342 0.931265i \(-0.381294\pi\)
0.364342 + 0.931265i \(0.381294\pi\)
\(654\) 0 0
\(655\) 487619.i 1.13657i
\(656\) 13029.0 0.0302763
\(657\) 0 0
\(658\) 34693.6 0.0801306
\(659\) 459674.i 1.05847i 0.848475 + 0.529235i \(0.177521\pi\)
−0.848475 + 0.529235i \(0.822479\pi\)
\(660\) 0 0
\(661\) −70113.4 −0.160472 −0.0802358 0.996776i \(-0.525567\pi\)
−0.0802358 + 0.996776i \(0.525567\pi\)
\(662\) 126522.i 0.288702i
\(663\) 0 0
\(664\) 506965. 1.14985
\(665\) −95736.2 −0.216488
\(666\) 0 0
\(667\) 77544.5i 0.174301i
\(668\) −89016.6 −0.199489
\(669\) 0 0
\(670\) 472137. 1.05177
\(671\) −231388. −0.513919
\(672\) 0 0
\(673\) 832203.i 1.83738i 0.394979 + 0.918690i \(0.370752\pi\)
−0.394979 + 0.918690i \(0.629248\pi\)
\(674\) 112817. 0.248344
\(675\) 0 0
\(676\) −595552. −1.30325
\(677\) −372770. −0.813323 −0.406661 0.913579i \(-0.633307\pi\)
−0.406661 + 0.913579i \(0.633307\pi\)
\(678\) 0 0
\(679\) 50754.6i 0.110087i
\(680\) 1.11469e6i 2.41065i
\(681\) 0 0
\(682\) 256795.i 0.552099i
\(683\) 231892.i 0.497102i 0.968619 + 0.248551i \(0.0799543\pi\)
−0.968619 + 0.248551i \(0.920046\pi\)
\(684\) 0 0
\(685\) 813988. 1.73475
\(686\) 103700.i 0.220358i
\(687\) 0 0
\(688\) 48996.1i 0.103511i
\(689\) 789836.i 1.66379i
\(690\) 0 0
\(691\) 177978.i 0.372744i −0.982479 0.186372i \(-0.940327\pi\)
0.982479 0.186372i \(-0.0596730\pi\)
\(692\) 335906.i 0.701465i
\(693\) 0 0
\(694\) −267074. −0.554514
\(695\) 915898. 1.89617
\(696\) 0 0
\(697\) 227485. 0.468259
\(698\) −106110. −0.217793
\(699\) 0 0
\(700\) 87472.5 0.178515
\(701\) 316317.i 0.643705i 0.946790 + 0.321852i \(0.104305\pi\)
−0.946790 + 0.321852i \(0.895695\pi\)
\(702\) 0 0
\(703\) 80000.3i 0.161875i
\(704\) 117364.i 0.236804i
\(705\) 0 0
\(706\) −274138. −0.549997
\(707\) 109880.i 0.219826i
\(708\) 0 0
\(709\) 288012. 0.572952 0.286476 0.958087i \(-0.407516\pi\)
0.286476 + 0.958087i \(0.407516\pi\)
\(710\) 658867.i 1.30702i
\(711\) 0 0
\(712\) −841838. −1.66061
\(713\) 360132. 0.708407
\(714\) 0 0
\(715\) −657409. −1.28595
\(716\) 155076.i 0.302495i
\(717\) 0 0
\(718\) 488966.i 0.948483i
\(719\) 622044.i 1.20327i −0.798771 0.601636i \(-0.794516\pi\)
0.798771 0.601636i \(-0.205484\pi\)
\(720\) 0 0
\(721\) 179381.i 0.345070i
\(722\) 144118.i 0.276468i
\(723\) 0 0
\(724\) −514197. −0.980962
\(725\) 353764. 0.673035
\(726\) 0 0
\(727\) −323221. −0.611548 −0.305774 0.952104i \(-0.598915\pi\)
−0.305774 + 0.952104i \(0.598915\pi\)
\(728\) 170441. 0.321597
\(729\) 0 0
\(730\) −739612. −1.38790
\(731\) 855466.i 1.60091i
\(732\) 0 0
\(733\) −314696. −0.585711 −0.292856 0.956157i \(-0.594606\pi\)
−0.292856 + 0.956157i \(0.594606\pi\)
\(734\) 69049.0 0.128164
\(735\) 0 0
\(736\) −199717. −0.368689
\(737\) −308180. −0.567373
\(738\) 0 0
\(739\) 703004.i 1.28727i −0.765334 0.643634i \(-0.777426\pi\)
0.765334 0.643634i \(-0.222574\pi\)
\(740\) 125737.i 0.229615i
\(741\) 0 0
\(742\) 59751.5i 0.108528i
\(743\) 128013. 0.231886 0.115943 0.993256i \(-0.463011\pi\)
0.115943 + 0.993256i \(0.463011\pi\)
\(744\) 0 0
\(745\) 206571.i 0.372182i
\(746\) 60175.5i 0.108129i
\(747\) 0 0
\(748\) 290298.i 0.518849i
\(749\) −146841. −0.261749
\(750\) 0 0
\(751\) 905740.i 1.60592i −0.596034 0.802959i \(-0.703258\pi\)
0.596034 0.802959i \(-0.296742\pi\)
\(752\) 42191.0i 0.0746078i
\(753\) 0 0
\(754\) 275026. 0.483760
\(755\) 1.06373e6i 1.86611i
\(756\) 0 0
\(757\) 302200. 0.527355 0.263678 0.964611i \(-0.415065\pi\)
0.263678 + 0.964611i \(0.415065\pi\)
\(758\) 622628.i 1.08365i
\(759\) 0 0
\(760\) 622856.i 1.07835i
\(761\) −844330. −1.45795 −0.728976 0.684539i \(-0.760003\pi\)
−0.728976 + 0.684539i \(0.760003\pi\)
\(762\) 0 0
\(763\) 98658.6i 0.169467i
\(764\) 127315.i 0.218118i
\(765\) 0 0
\(766\) 380867.i 0.649107i
\(767\) 312687. 963188.i 0.531519 1.63727i
\(768\) 0 0
\(769\) 232294.i 0.392813i 0.980523 + 0.196406i \(0.0629272\pi\)
−0.980523 + 0.196406i \(0.937073\pi\)
\(770\) 49733.4 0.0838815
\(771\) 0 0
\(772\) 207088. 0.347472
\(773\) 340313.i 0.569534i −0.958597 0.284767i \(-0.908084\pi\)
0.958597 0.284767i \(-0.0919162\pi\)
\(774\) 0 0
\(775\) 1.64295e6i 2.73540i
\(776\) 330207. 0.548356
\(777\) 0 0
\(778\) 537545.i 0.888087i
\(779\) −127112. −0.209465
\(780\) 0 0
\(781\) 430064.i 0.705068i
\(782\) −206150. −0.337109
\(783\) 0 0
\(784\) −61849.5 −0.100625
\(785\) 1.14420e6i 1.85679i
\(786\) 0 0
\(787\) −18969.9 −0.0306278 −0.0153139 0.999883i \(-0.504875\pi\)
−0.0153139 + 0.999883i \(0.504875\pi\)
\(788\) 452962. 0.729473
\(789\) 0 0
\(790\) 751653.i 1.20438i
\(791\) 34689.6i 0.0554429i
\(792\) 0 0
\(793\) −1.15090e6 −1.83017
\(794\) 154359. 0.244846
\(795\) 0 0
\(796\) 113326. 0.178855
\(797\) 845431.i