Properties

Label 1143.3.b.a.890.15
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.15
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.70

$q$-expansion

\(f(q)\) \(=\) \(q-2.89430i q^{2} -4.37696 q^{4} +0.368899i q^{5} -11.4242 q^{7} +1.09103i q^{8} +O(q^{10})\) \(q-2.89430i q^{2} -4.37696 q^{4} +0.368899i q^{5} -11.4242 q^{7} +1.09103i q^{8} +1.06770 q^{10} +12.4660i q^{11} -14.3568 q^{13} +33.0651i q^{14} -14.3501 q^{16} +4.03686i q^{17} +4.50638 q^{19} -1.61466i q^{20} +36.0802 q^{22} -24.7854i q^{23} +24.8639 q^{25} +41.5529i q^{26} +50.0034 q^{28} -40.2521i q^{29} -0.142872 q^{31} +45.8975i q^{32} +11.6839 q^{34} -4.21439i q^{35} +70.5507 q^{37} -13.0428i q^{38} -0.402480 q^{40} +72.7348i q^{41} +32.2116 q^{43} -54.5630i q^{44} -71.7363 q^{46} +29.8941i q^{47} +81.5130 q^{49} -71.9636i q^{50} +62.8392 q^{52} -46.0232i q^{53} -4.59868 q^{55} -12.4642i q^{56} -116.501 q^{58} +67.6830i q^{59} -40.9491 q^{61} +0.413515i q^{62} +75.4407 q^{64} -5.29621i q^{65} -13.7399 q^{67} -17.6692i q^{68} -12.1977 q^{70} -14.7337i q^{71} +131.028 q^{73} -204.195i q^{74} -19.7242 q^{76} -142.414i q^{77} +86.0968 q^{79} -5.29373i q^{80} +210.516 q^{82} +141.334i q^{83} -1.48920 q^{85} -93.2301i q^{86} -13.6007 q^{88} +136.262i q^{89} +164.016 q^{91} +108.485i q^{92} +86.5224 q^{94} +1.66240i q^{95} +22.3537 q^{97} -235.923i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.89430i − 1.44715i −0.690246 0.723574i \(-0.742498\pi\)
0.690246 0.723574i \(-0.257502\pi\)
\(3\) 0 0
\(4\) −4.37696 −1.09424
\(5\) 0.368899i 0.0737798i 0.999319 + 0.0368899i \(0.0117451\pi\)
−0.999319 + 0.0368899i \(0.988255\pi\)
\(6\) 0 0
\(7\) −11.4242 −1.63203 −0.816016 0.578029i \(-0.803822\pi\)
−0.816016 + 0.578029i \(0.803822\pi\)
\(8\) 1.09103i 0.136379i
\(9\) 0 0
\(10\) 1.06770 0.106770
\(11\) 12.4660i 1.13327i 0.823969 + 0.566635i \(0.191755\pi\)
−0.823969 + 0.566635i \(0.808245\pi\)
\(12\) 0 0
\(13\) −14.3568 −1.10437 −0.552185 0.833722i \(-0.686206\pi\)
−0.552185 + 0.833722i \(0.686206\pi\)
\(14\) 33.0651i 2.36179i
\(15\) 0 0
\(16\) −14.3501 −0.896879
\(17\) 4.03686i 0.237463i 0.992926 + 0.118731i \(0.0378827\pi\)
−0.992926 + 0.118731i \(0.962117\pi\)
\(18\) 0 0
\(19\) 4.50638 0.237178 0.118589 0.992943i \(-0.462163\pi\)
0.118589 + 0.992943i \(0.462163\pi\)
\(20\) − 1.61466i − 0.0807328i
\(21\) 0 0
\(22\) 36.0802 1.64001
\(23\) − 24.7854i − 1.07763i −0.842425 0.538813i \(-0.818873\pi\)
0.842425 0.538813i \(-0.181127\pi\)
\(24\) 0 0
\(25\) 24.8639 0.994557
\(26\) 41.5529i 1.59819i
\(27\) 0 0
\(28\) 50.0034 1.78583
\(29\) − 40.2521i − 1.38800i −0.719974 0.694001i \(-0.755846\pi\)
0.719974 0.694001i \(-0.244154\pi\)
\(30\) 0 0
\(31\) −0.142872 −0.00460879 −0.00230439 0.999997i \(-0.500734\pi\)
−0.00230439 + 0.999997i \(0.500734\pi\)
\(32\) 45.8975i 1.43430i
\(33\) 0 0
\(34\) 11.6839 0.343644
\(35\) − 4.21439i − 0.120411i
\(36\) 0 0
\(37\) 70.5507 1.90678 0.953388 0.301747i \(-0.0975699\pi\)
0.953388 + 0.301747i \(0.0975699\pi\)
\(38\) − 13.0428i − 0.343232i
\(39\) 0 0
\(40\) −0.402480 −0.0100620
\(41\) 72.7348i 1.77402i 0.461751 + 0.887009i \(0.347221\pi\)
−0.461751 + 0.887009i \(0.652779\pi\)
\(42\) 0 0
\(43\) 32.2116 0.749108 0.374554 0.927205i \(-0.377796\pi\)
0.374554 + 0.927205i \(0.377796\pi\)
\(44\) − 54.5630i − 1.24007i
\(45\) 0 0
\(46\) −71.7363 −1.55949
\(47\) 29.8941i 0.636044i 0.948083 + 0.318022i \(0.103019\pi\)
−0.948083 + 0.318022i \(0.896981\pi\)
\(48\) 0 0
\(49\) 81.5130 1.66353
\(50\) − 71.9636i − 1.43927i
\(51\) 0 0
\(52\) 62.8392 1.20845
\(53\) − 46.0232i − 0.868362i −0.900826 0.434181i \(-0.857038\pi\)
0.900826 0.434181i \(-0.142962\pi\)
\(54\) 0 0
\(55\) −4.59868 −0.0836124
\(56\) − 12.4642i − 0.222575i
\(57\) 0 0
\(58\) −116.501 −2.00865
\(59\) 67.6830i 1.14717i 0.819146 + 0.573585i \(0.194448\pi\)
−0.819146 + 0.573585i \(0.805552\pi\)
\(60\) 0 0
\(61\) −40.9491 −0.671297 −0.335648 0.941987i \(-0.608955\pi\)
−0.335648 + 0.941987i \(0.608955\pi\)
\(62\) 0.413515i 0.00666960i
\(63\) 0 0
\(64\) 75.4407 1.17876
\(65\) − 5.29621i − 0.0814802i
\(66\) 0 0
\(67\) −13.7399 −0.205073 −0.102536 0.994729i \(-0.532696\pi\)
−0.102536 + 0.994729i \(0.532696\pi\)
\(68\) − 17.6692i − 0.259841i
\(69\) 0 0
\(70\) −12.1977 −0.174253
\(71\) − 14.7337i − 0.207517i −0.994603 0.103759i \(-0.966913\pi\)
0.994603 0.103759i \(-0.0330870\pi\)
\(72\) 0 0
\(73\) 131.028 1.79491 0.897454 0.441109i \(-0.145415\pi\)
0.897454 + 0.441109i \(0.145415\pi\)
\(74\) − 204.195i − 2.75939i
\(75\) 0 0
\(76\) −19.7242 −0.259530
\(77\) − 142.414i − 1.84953i
\(78\) 0 0
\(79\) 86.0968 1.08983 0.544916 0.838490i \(-0.316561\pi\)
0.544916 + 0.838490i \(0.316561\pi\)
\(80\) − 5.29373i − 0.0661716i
\(81\) 0 0
\(82\) 210.516 2.56727
\(83\) 141.334i 1.70281i 0.524505 + 0.851407i \(0.324250\pi\)
−0.524505 + 0.851407i \(0.675750\pi\)
\(84\) 0 0
\(85\) −1.48920 −0.0175199
\(86\) − 93.2301i − 1.08407i
\(87\) 0 0
\(88\) −13.6007 −0.154554
\(89\) 136.262i 1.53104i 0.643413 + 0.765519i \(0.277518\pi\)
−0.643413 + 0.765519i \(0.722482\pi\)
\(90\) 0 0
\(91\) 164.016 1.80237
\(92\) 108.485i 1.17918i
\(93\) 0 0
\(94\) 86.5224 0.920451
\(95\) 1.66240i 0.0174989i
\(96\) 0 0
\(97\) 22.3537 0.230450 0.115225 0.993339i \(-0.463241\pi\)
0.115225 + 0.993339i \(0.463241\pi\)
\(98\) − 235.923i − 2.40738i
\(99\) 0 0
\(100\) −108.828 −1.08828
\(101\) − 143.434i − 1.42014i −0.704131 0.710070i \(-0.748663\pi\)
0.704131 0.710070i \(-0.251337\pi\)
\(102\) 0 0
\(103\) 48.6766 0.472588 0.236294 0.971682i \(-0.424067\pi\)
0.236294 + 0.971682i \(0.424067\pi\)
\(104\) − 15.6637i − 0.150613i
\(105\) 0 0
\(106\) −133.205 −1.25665
\(107\) − 59.3338i − 0.554522i −0.960795 0.277261i \(-0.910573\pi\)
0.960795 0.277261i \(-0.0894266\pi\)
\(108\) 0 0
\(109\) −170.051 −1.56010 −0.780050 0.625717i \(-0.784806\pi\)
−0.780050 + 0.625717i \(0.784806\pi\)
\(110\) 13.3100i 0.121000i
\(111\) 0 0
\(112\) 163.938 1.46374
\(113\) 75.6100i 0.669115i 0.942375 + 0.334558i \(0.108587\pi\)
−0.942375 + 0.334558i \(0.891413\pi\)
\(114\) 0 0
\(115\) 9.14331 0.0795071
\(116\) 176.182i 1.51881i
\(117\) 0 0
\(118\) 195.895 1.66012
\(119\) − 46.1181i − 0.387547i
\(120\) 0 0
\(121\) −34.4002 −0.284299
\(122\) 118.519i 0.971466i
\(123\) 0 0
\(124\) 0.625347 0.00504312
\(125\) 18.3948i 0.147158i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) − 34.7579i − 0.271546i
\(129\) 0 0
\(130\) −15.3288 −0.117914
\(131\) − 155.996i − 1.19081i −0.803425 0.595406i \(-0.796991\pi\)
0.803425 0.595406i \(-0.203009\pi\)
\(132\) 0 0
\(133\) −51.4819 −0.387082
\(134\) 39.7672i 0.296770i
\(135\) 0 0
\(136\) −4.40434 −0.0323848
\(137\) 73.9267i 0.539611i 0.962915 + 0.269806i \(0.0869594\pi\)
−0.962915 + 0.269806i \(0.913041\pi\)
\(138\) 0 0
\(139\) −264.466 −1.90264 −0.951318 0.308212i \(-0.900269\pi\)
−0.951318 + 0.308212i \(0.900269\pi\)
\(140\) 18.4462i 0.131759i
\(141\) 0 0
\(142\) −42.6438 −0.300308
\(143\) − 178.971i − 1.25155i
\(144\) 0 0
\(145\) 14.8489 0.102407
\(146\) − 379.235i − 2.59750i
\(147\) 0 0
\(148\) −308.798 −2.08647
\(149\) 81.6878i 0.548240i 0.961695 + 0.274120i \(0.0883866\pi\)
−0.961695 + 0.274120i \(0.911613\pi\)
\(150\) 0 0
\(151\) −245.937 −1.62872 −0.814360 0.580360i \(-0.802912\pi\)
−0.814360 + 0.580360i \(0.802912\pi\)
\(152\) 4.91660i 0.0323460i
\(153\) 0 0
\(154\) −412.188 −2.67655
\(155\) − 0.0527055i 0 0.000340035i
\(156\) 0 0
\(157\) 231.778 1.47629 0.738147 0.674640i \(-0.235701\pi\)
0.738147 + 0.674640i \(0.235701\pi\)
\(158\) − 249.190i − 1.57715i
\(159\) 0 0
\(160\) −16.9315 −0.105822
\(161\) 283.154i 1.75872i
\(162\) 0 0
\(163\) 124.426 0.763347 0.381673 0.924297i \(-0.375348\pi\)
0.381673 + 0.924297i \(0.375348\pi\)
\(164\) − 318.357i − 1.94120i
\(165\) 0 0
\(166\) 409.062 2.46423
\(167\) 47.4894i 0.284368i 0.989840 + 0.142184i \(0.0454124\pi\)
−0.989840 + 0.142184i \(0.954588\pi\)
\(168\) 0 0
\(169\) 37.1181 0.219634
\(170\) 4.31017i 0.0253540i
\(171\) 0 0
\(172\) −140.989 −0.819704
\(173\) 220.614i 1.27522i 0.770357 + 0.637612i \(0.220078\pi\)
−0.770357 + 0.637612i \(0.779922\pi\)
\(174\) 0 0
\(175\) −284.051 −1.62315
\(176\) − 178.887i − 1.01641i
\(177\) 0 0
\(178\) 394.384 2.21564
\(179\) 94.0577i 0.525462i 0.964869 + 0.262731i \(0.0846232\pi\)
−0.964869 + 0.262731i \(0.915377\pi\)
\(180\) 0 0
\(181\) 115.413 0.637642 0.318821 0.947815i \(-0.396713\pi\)
0.318821 + 0.947815i \(0.396713\pi\)
\(182\) − 474.710i − 2.60830i
\(183\) 0 0
\(184\) 27.0416 0.146965
\(185\) 26.0261i 0.140682i
\(186\) 0 0
\(187\) −50.3234 −0.269109
\(188\) − 130.845i − 0.695985i
\(189\) 0 0
\(190\) 4.81148 0.0253236
\(191\) 74.0418i 0.387654i 0.981036 + 0.193827i \(0.0620900\pi\)
−0.981036 + 0.193827i \(0.937910\pi\)
\(192\) 0 0
\(193\) −14.0989 −0.0730514 −0.0365257 0.999333i \(-0.511629\pi\)
−0.0365257 + 0.999333i \(0.511629\pi\)
\(194\) − 64.6983i − 0.333496i
\(195\) 0 0
\(196\) −356.779 −1.82030
\(197\) − 312.804i − 1.58784i −0.608024 0.793919i \(-0.708038\pi\)
0.608024 0.793919i \(-0.291962\pi\)
\(198\) 0 0
\(199\) 69.5241 0.349367 0.174684 0.984625i \(-0.444110\pi\)
0.174684 + 0.984625i \(0.444110\pi\)
\(200\) 27.1273i 0.135636i
\(201\) 0 0
\(202\) −415.141 −2.05515
\(203\) 459.849i 2.26526i
\(204\) 0 0
\(205\) −26.8318 −0.130887
\(206\) − 140.885i − 0.683906i
\(207\) 0 0
\(208\) 206.021 0.990487
\(209\) 56.1764i 0.268787i
\(210\) 0 0
\(211\) 211.824 1.00390 0.501952 0.864896i \(-0.332615\pi\)
0.501952 + 0.864896i \(0.332615\pi\)
\(212\) 201.442i 0.950196i
\(213\) 0 0
\(214\) −171.730 −0.802476
\(215\) 11.8828i 0.0552691i
\(216\) 0 0
\(217\) 1.63221 0.00752169
\(218\) 492.178i 2.25770i
\(219\) 0 0
\(220\) 20.1282 0.0914920
\(221\) − 57.9565i − 0.262247i
\(222\) 0 0
\(223\) −264.549 −1.18632 −0.593159 0.805085i \(-0.702120\pi\)
−0.593159 + 0.805085i \(0.702120\pi\)
\(224\) − 524.343i − 2.34082i
\(225\) 0 0
\(226\) 218.838 0.968309
\(227\) 390.263i 1.71922i 0.510949 + 0.859611i \(0.329294\pi\)
−0.510949 + 0.859611i \(0.670706\pi\)
\(228\) 0 0
\(229\) −156.112 −0.681714 −0.340857 0.940115i \(-0.610717\pi\)
−0.340857 + 0.940115i \(0.610717\pi\)
\(230\) − 26.4635i − 0.115059i
\(231\) 0 0
\(232\) 43.9162 0.189294
\(233\) − 327.852i − 1.40709i −0.710650 0.703545i \(-0.751599\pi\)
0.710650 0.703545i \(-0.248401\pi\)
\(234\) 0 0
\(235\) −11.0279 −0.0469272
\(236\) − 296.246i − 1.25528i
\(237\) 0 0
\(238\) −133.479 −0.560838
\(239\) 218.750i 0.915274i 0.889139 + 0.457637i \(0.151304\pi\)
−0.889139 + 0.457637i \(0.848696\pi\)
\(240\) 0 0
\(241\) −145.126 −0.602182 −0.301091 0.953595i \(-0.597351\pi\)
−0.301091 + 0.953595i \(0.597351\pi\)
\(242\) 99.5643i 0.411423i
\(243\) 0 0
\(244\) 179.232 0.734559
\(245\) 30.0701i 0.122735i
\(246\) 0 0
\(247\) −64.6973 −0.261932
\(248\) − 0.155878i 0 0.000628540i
\(249\) 0 0
\(250\) 53.2399 0.212960
\(251\) − 347.366i − 1.38393i −0.721933 0.691963i \(-0.756746\pi\)
0.721933 0.691963i \(-0.243254\pi\)
\(252\) 0 0
\(253\) 308.974 1.22124
\(254\) − 32.6171i − 0.128414i
\(255\) 0 0
\(256\) 201.163 0.785793
\(257\) 121.233i 0.471722i 0.971787 + 0.235861i \(0.0757910\pi\)
−0.971787 + 0.235861i \(0.924209\pi\)
\(258\) 0 0
\(259\) −805.987 −3.11192
\(260\) 23.1813i 0.0891589i
\(261\) 0 0
\(262\) −451.500 −1.72328
\(263\) 288.631i 1.09746i 0.836000 + 0.548729i \(0.184888\pi\)
−0.836000 + 0.548729i \(0.815112\pi\)
\(264\) 0 0
\(265\) 16.9779 0.0640676
\(266\) 149.004i 0.560166i
\(267\) 0 0
\(268\) 60.1388 0.224398
\(269\) 67.7121i 0.251718i 0.992048 + 0.125859i \(0.0401687\pi\)
−0.992048 + 0.125859i \(0.959831\pi\)
\(270\) 0 0
\(271\) 396.016 1.46131 0.730657 0.682745i \(-0.239214\pi\)
0.730657 + 0.682745i \(0.239214\pi\)
\(272\) − 57.9293i − 0.212975i
\(273\) 0 0
\(274\) 213.966 0.780898
\(275\) 309.953i 1.12710i
\(276\) 0 0
\(277\) −215.982 −0.779718 −0.389859 0.920875i \(-0.627476\pi\)
−0.389859 + 0.920875i \(0.627476\pi\)
\(278\) 765.444i 2.75340i
\(279\) 0 0
\(280\) 4.59802 0.0164215
\(281\) − 170.535i − 0.606886i −0.952850 0.303443i \(-0.901864\pi\)
0.952850 0.303443i \(-0.0981362\pi\)
\(282\) 0 0
\(283\) −0.946273 −0.00334372 −0.00167186 0.999999i \(-0.500532\pi\)
−0.00167186 + 0.999999i \(0.500532\pi\)
\(284\) 64.4889i 0.227074i
\(285\) 0 0
\(286\) −517.997 −1.81118
\(287\) − 830.939i − 2.89526i
\(288\) 0 0
\(289\) 272.704 0.943612
\(290\) − 42.9773i − 0.148197i
\(291\) 0 0
\(292\) −573.505 −1.96406
\(293\) 308.048i 1.05136i 0.850683 + 0.525679i \(0.176189\pi\)
−0.850683 + 0.525679i \(0.823811\pi\)
\(294\) 0 0
\(295\) −24.9682 −0.0846380
\(296\) 76.9729i 0.260044i
\(297\) 0 0
\(298\) 236.429 0.793385
\(299\) 355.839i 1.19010i
\(300\) 0 0
\(301\) −367.993 −1.22257
\(302\) 711.814i 2.35700i
\(303\) 0 0
\(304\) −64.6669 −0.212720
\(305\) − 15.1061i − 0.0495281i
\(306\) 0 0
\(307\) −226.372 −0.737369 −0.368685 0.929555i \(-0.620192\pi\)
−0.368685 + 0.929555i \(0.620192\pi\)
\(308\) 623.340i 2.02383i
\(309\) 0 0
\(310\) −0.152545 −0.000492082 0
\(311\) 286.254i 0.920431i 0.887807 + 0.460215i \(0.152228\pi\)
−0.887807 + 0.460215i \(0.847772\pi\)
\(312\) 0 0
\(313\) 403.280 1.28844 0.644218 0.764842i \(-0.277183\pi\)
0.644218 + 0.764842i \(0.277183\pi\)
\(314\) − 670.835i − 2.13642i
\(315\) 0 0
\(316\) −376.842 −1.19254
\(317\) 252.786i 0.797431i 0.917075 + 0.398715i \(0.130544\pi\)
−0.917075 + 0.398715i \(0.869456\pi\)
\(318\) 0 0
\(319\) 501.780 1.57298
\(320\) 27.8300i 0.0869688i
\(321\) 0 0
\(322\) 819.532 2.54513
\(323\) 18.1917i 0.0563209i
\(324\) 0 0
\(325\) −356.967 −1.09836
\(326\) − 360.125i − 1.10468i
\(327\) 0 0
\(328\) −79.3558 −0.241938
\(329\) − 341.517i − 1.03804i
\(330\) 0 0
\(331\) 493.284 1.49028 0.745142 0.666906i \(-0.232382\pi\)
0.745142 + 0.666906i \(0.232382\pi\)
\(332\) − 618.611i − 1.86329i
\(333\) 0 0
\(334\) 137.448 0.411522
\(335\) − 5.06862i − 0.0151302i
\(336\) 0 0
\(337\) 437.426 1.29800 0.649000 0.760789i \(-0.275188\pi\)
0.649000 + 0.760789i \(0.275188\pi\)
\(338\) − 107.431i − 0.317843i
\(339\) 0 0
\(340\) 6.51815 0.0191710
\(341\) − 1.78104i − 0.00522300i
\(342\) 0 0
\(343\) −371.436 −1.08290
\(344\) 35.1439i 0.102162i
\(345\) 0 0
\(346\) 638.522 1.84544
\(347\) 585.153i 1.68632i 0.537663 + 0.843160i \(0.319307\pi\)
−0.537663 + 0.843160i \(0.680693\pi\)
\(348\) 0 0
\(349\) 526.031 1.50725 0.753626 0.657304i \(-0.228303\pi\)
0.753626 + 0.657304i \(0.228303\pi\)
\(350\) 822.128i 2.34894i
\(351\) 0 0
\(352\) −572.156 −1.62544
\(353\) 50.5781i 0.143281i 0.997431 + 0.0716404i \(0.0228234\pi\)
−0.997431 + 0.0716404i \(0.977177\pi\)
\(354\) 0 0
\(355\) 5.43526 0.0153106
\(356\) − 596.415i − 1.67532i
\(357\) 0 0
\(358\) 272.231 0.760422
\(359\) 141.833i 0.395078i 0.980295 + 0.197539i \(0.0632949\pi\)
−0.980295 + 0.197539i \(0.936705\pi\)
\(360\) 0 0
\(361\) −340.693 −0.943747
\(362\) − 334.040i − 0.922763i
\(363\) 0 0
\(364\) −717.889 −1.97222
\(365\) 48.3362i 0.132428i
\(366\) 0 0
\(367\) −158.951 −0.433110 −0.216555 0.976270i \(-0.569482\pi\)
−0.216555 + 0.976270i \(0.569482\pi\)
\(368\) 355.672i 0.966501i
\(369\) 0 0
\(370\) 75.3272 0.203587
\(371\) 525.779i 1.41719i
\(372\) 0 0
\(373\) 530.611 1.42255 0.711275 0.702914i \(-0.248118\pi\)
0.711275 + 0.702914i \(0.248118\pi\)
\(374\) 145.651i 0.389441i
\(375\) 0 0
\(376\) −32.6153 −0.0867429
\(377\) 577.891i 1.53287i
\(378\) 0 0
\(379\) 244.663 0.645548 0.322774 0.946476i \(-0.395385\pi\)
0.322774 + 0.946476i \(0.395385\pi\)
\(380\) − 7.27626i − 0.0191480i
\(381\) 0 0
\(382\) 214.299 0.560992
\(383\) − 167.289i − 0.436786i −0.975861 0.218393i \(-0.929919\pi\)
0.975861 0.218393i \(-0.0700814\pi\)
\(384\) 0 0
\(385\) 52.5364 0.136458
\(386\) 40.8065i 0.105716i
\(387\) 0 0
\(388\) −97.8412 −0.252168
\(389\) − 412.946i − 1.06156i −0.847511 0.530779i \(-0.821900\pi\)
0.847511 0.530779i \(-0.178100\pi\)
\(390\) 0 0
\(391\) 100.055 0.255896
\(392\) 88.9331i 0.226870i
\(393\) 0 0
\(394\) −905.348 −2.29784
\(395\) 31.7610i 0.0804077i
\(396\) 0 0
\(397\) 536.080 1.35033 0.675164 0.737668i \(-0.264073\pi\)
0.675164 + 0.737668i \(0.264073\pi\)
\(398\) − 201.223i − 0.505587i
\(399\) 0 0
\(400\) −356.799 −0.891997
\(401\) − 535.852i − 1.33629i −0.744032 0.668144i \(-0.767089\pi\)
0.744032 0.668144i \(-0.232911\pi\)
\(402\) 0 0
\(403\) 2.05119 0.00508981
\(404\) 627.805i 1.55397i
\(405\) 0 0
\(406\) 1330.94 3.27817
\(407\) 879.482i 2.16089i
\(408\) 0 0
\(409\) −357.786 −0.874783 −0.437391 0.899271i \(-0.644098\pi\)
−0.437391 + 0.899271i \(0.644098\pi\)
\(410\) 77.6592i 0.189413i
\(411\) 0 0
\(412\) −213.056 −0.517125
\(413\) − 773.226i − 1.87222i
\(414\) 0 0
\(415\) −52.1378 −0.125633
\(416\) − 658.942i − 1.58399i
\(417\) 0 0
\(418\) 162.591 0.388974
\(419\) 613.475i 1.46414i 0.681229 + 0.732070i \(0.261446\pi\)
−0.681229 + 0.732070i \(0.738554\pi\)
\(420\) 0 0
\(421\) 569.239 1.35211 0.676056 0.736850i \(-0.263688\pi\)
0.676056 + 0.736850i \(0.263688\pi\)
\(422\) − 613.081i − 1.45280i
\(423\) 0 0
\(424\) 50.2127 0.118426
\(425\) 100.372i 0.236170i
\(426\) 0 0
\(427\) 467.812 1.09558
\(428\) 259.702i 0.606780i
\(429\) 0 0
\(430\) 34.3925 0.0799826
\(431\) 269.812i 0.626014i 0.949751 + 0.313007i \(0.101336\pi\)
−0.949751 + 0.313007i \(0.898664\pi\)
\(432\) 0 0
\(433\) 151.221 0.349241 0.174620 0.984636i \(-0.444130\pi\)
0.174620 + 0.984636i \(0.444130\pi\)
\(434\) − 4.72409i − 0.0108850i
\(435\) 0 0
\(436\) 744.306 1.70712
\(437\) − 111.692i − 0.255589i
\(438\) 0 0
\(439\) 272.925 0.621698 0.310849 0.950459i \(-0.399387\pi\)
0.310849 + 0.950459i \(0.399387\pi\)
\(440\) − 5.01730i − 0.0114029i
\(441\) 0 0
\(442\) −167.743 −0.379510
\(443\) − 173.360i − 0.391331i −0.980671 0.195666i \(-0.937313\pi\)
0.980671 0.195666i \(-0.0626867\pi\)
\(444\) 0 0
\(445\) −50.2671 −0.112960
\(446\) 765.684i 1.71678i
\(447\) 0 0
\(448\) −861.852 −1.92378
\(449\) 97.2936i 0.216690i 0.994113 + 0.108345i \(0.0345550\pi\)
−0.994113 + 0.108345i \(0.965445\pi\)
\(450\) 0 0
\(451\) −906.709 −2.01044
\(452\) − 330.942i − 0.732172i
\(453\) 0 0
\(454\) 1129.54 2.48797
\(455\) 60.5052i 0.132978i
\(456\) 0 0
\(457\) 30.5417 0.0668310 0.0334155 0.999442i \(-0.489362\pi\)
0.0334155 + 0.999442i \(0.489362\pi\)
\(458\) 451.836i 0.986541i
\(459\) 0 0
\(460\) −40.0199 −0.0869998
\(461\) 88.1669i 0.191251i 0.995417 + 0.0956257i \(0.0304852\pi\)
−0.995417 + 0.0956257i \(0.969515\pi\)
\(462\) 0 0
\(463\) −794.001 −1.71491 −0.857453 0.514563i \(-0.827954\pi\)
−0.857453 + 0.514563i \(0.827954\pi\)
\(464\) 577.620i 1.24487i
\(465\) 0 0
\(466\) −948.902 −2.03627
\(467\) − 142.933i − 0.306066i −0.988221 0.153033i \(-0.951096\pi\)
0.988221 0.153033i \(-0.0489041\pi\)
\(468\) 0 0
\(469\) 156.967 0.334685
\(470\) 31.9180i 0.0679107i
\(471\) 0 0
\(472\) −73.8442 −0.156450
\(473\) 401.549i 0.848941i
\(474\) 0 0
\(475\) 112.046 0.235887
\(476\) 201.857i 0.424069i
\(477\) 0 0
\(478\) 633.129 1.32454
\(479\) − 555.490i − 1.15969i −0.814728 0.579843i \(-0.803114\pi\)
0.814728 0.579843i \(-0.196886\pi\)
\(480\) 0 0
\(481\) −1012.88 −2.10579
\(482\) 420.037i 0.871447i
\(483\) 0 0
\(484\) 150.568 0.311091
\(485\) 8.24626i 0.0170026i
\(486\) 0 0
\(487\) 456.287 0.936934 0.468467 0.883481i \(-0.344807\pi\)
0.468467 + 0.883481i \(0.344807\pi\)
\(488\) − 44.6767i − 0.0915506i
\(489\) 0 0
\(490\) 87.0317 0.177616
\(491\) − 428.089i − 0.871872i −0.899978 0.435936i \(-0.856417\pi\)
0.899978 0.435936i \(-0.143583\pi\)
\(492\) 0 0
\(493\) 162.492 0.329598
\(494\) 187.253i 0.379055i
\(495\) 0 0
\(496\) 2.05023 0.00413353
\(497\) 168.321i 0.338675i
\(498\) 0 0
\(499\) −216.198 −0.433263 −0.216632 0.976253i \(-0.569507\pi\)
−0.216632 + 0.976253i \(0.569507\pi\)
\(500\) − 80.5131i − 0.161026i
\(501\) 0 0
\(502\) −1005.38 −2.00275
\(503\) 484.500i 0.963220i 0.876386 + 0.481610i \(0.159948\pi\)
−0.876386 + 0.481610i \(0.840052\pi\)
\(504\) 0 0
\(505\) 52.9127 0.104778
\(506\) − 894.262i − 1.76732i
\(507\) 0 0
\(508\) −49.3258 −0.0970981
\(509\) 618.428i 1.21499i 0.794325 + 0.607493i \(0.207825\pi\)
−0.794325 + 0.607493i \(0.792175\pi\)
\(510\) 0 0
\(511\) −1496.90 −2.92935
\(512\) − 721.258i − 1.40871i
\(513\) 0 0
\(514\) 350.883 0.682652
\(515\) 17.9568i 0.0348675i
\(516\) 0 0
\(517\) −372.658 −0.720809
\(518\) 2332.77i 4.50341i
\(519\) 0 0
\(520\) 5.77833 0.0111122
\(521\) 15.6085i 0.0299587i 0.999888 + 0.0149793i \(0.00476825\pi\)
−0.999888 + 0.0149793i \(0.995232\pi\)
\(522\) 0 0
\(523\) −422.072 −0.807022 −0.403511 0.914975i \(-0.632210\pi\)
−0.403511 + 0.914975i \(0.632210\pi\)
\(524\) 682.789i 1.30303i
\(525\) 0 0
\(526\) 835.385 1.58818
\(527\) − 0.576756i − 0.00109441i
\(528\) 0 0
\(529\) −85.3161 −0.161278
\(530\) − 49.1391i − 0.0927153i
\(531\) 0 0
\(532\) 225.334 0.423561
\(533\) − 1044.24i − 1.95917i
\(534\) 0 0
\(535\) 21.8882 0.0409125
\(536\) − 14.9906i − 0.0279675i
\(537\) 0 0
\(538\) 195.979 0.364273
\(539\) 1016.14i 1.88523i
\(540\) 0 0
\(541\) −797.374 −1.47389 −0.736944 0.675953i \(-0.763732\pi\)
−0.736944 + 0.675953i \(0.763732\pi\)
\(542\) − 1146.19i − 2.11474i
\(543\) 0 0
\(544\) −185.282 −0.340592
\(545\) − 62.7316i − 0.115104i
\(546\) 0 0
\(547\) 607.308 1.11025 0.555126 0.831766i \(-0.312670\pi\)
0.555126 + 0.831766i \(0.312670\pi\)
\(548\) − 323.574i − 0.590464i
\(549\) 0 0
\(550\) 897.095 1.63108
\(551\) − 181.391i − 0.329204i
\(552\) 0 0
\(553\) −983.589 −1.77864
\(554\) 625.116i 1.12837i
\(555\) 0 0
\(556\) 1157.56 2.08194
\(557\) 763.344i 1.37046i 0.728329 + 0.685228i \(0.240297\pi\)
−0.728329 + 0.685228i \(0.759703\pi\)
\(558\) 0 0
\(559\) −462.457 −0.827293
\(560\) 60.4767i 0.107994i
\(561\) 0 0
\(562\) −493.579 −0.878254
\(563\) 911.688i 1.61934i 0.586886 + 0.809670i \(0.300354\pi\)
−0.586886 + 0.809670i \(0.699646\pi\)
\(564\) 0 0
\(565\) −27.8925 −0.0493672
\(566\) 2.73880i 0.00483886i
\(567\) 0 0
\(568\) 16.0749 0.0283009
\(569\) 680.350i 1.19569i 0.801610 + 0.597847i \(0.203977\pi\)
−0.801610 + 0.597847i \(0.796023\pi\)
\(570\) 0 0
\(571\) 74.3734 0.130251 0.0651256 0.997877i \(-0.479255\pi\)
0.0651256 + 0.997877i \(0.479255\pi\)
\(572\) 783.351i 1.36949i
\(573\) 0 0
\(574\) −2404.98 −4.18987
\(575\) − 616.262i − 1.07176i
\(576\) 0 0
\(577\) 234.061 0.405652 0.202826 0.979215i \(-0.434987\pi\)
0.202826 + 0.979215i \(0.434987\pi\)
\(578\) − 789.286i − 1.36555i
\(579\) 0 0
\(580\) −64.9932 −0.112057
\(581\) − 1614.63i − 2.77905i
\(582\) 0 0
\(583\) 573.723 0.984088
\(584\) 142.956i 0.244787i
\(585\) 0 0
\(586\) 891.583 1.52147
\(587\) − 547.257i − 0.932295i −0.884707 0.466147i \(-0.845642\pi\)
0.884707 0.466147i \(-0.154358\pi\)
\(588\) 0 0
\(589\) −0.643838 −0.00109310
\(590\) 72.2654i 0.122484i
\(591\) 0 0
\(592\) −1012.41 −1.71015
\(593\) − 453.958i − 0.765528i −0.923846 0.382764i \(-0.874972\pi\)
0.923846 0.382764i \(-0.125028\pi\)
\(594\) 0 0
\(595\) 17.0129 0.0285931
\(596\) − 357.544i − 0.599906i
\(597\) 0 0
\(598\) 1029.90 1.72225
\(599\) 852.190i 1.42269i 0.702844 + 0.711344i \(0.251913\pi\)
−0.702844 + 0.711344i \(0.748087\pi\)
\(600\) 0 0
\(601\) 70.0421 0.116543 0.0582713 0.998301i \(-0.481441\pi\)
0.0582713 + 0.998301i \(0.481441\pi\)
\(602\) 1065.08i 1.76924i
\(603\) 0 0
\(604\) 1076.45 1.78221
\(605\) − 12.6902i − 0.0209755i
\(606\) 0 0
\(607\) 569.835 0.938773 0.469387 0.882993i \(-0.344475\pi\)
0.469387 + 0.882993i \(0.344475\pi\)
\(608\) 206.832i 0.340184i
\(609\) 0 0
\(610\) −43.7215 −0.0716746
\(611\) − 429.184i − 0.702428i
\(612\) 0 0
\(613\) 477.707 0.779294 0.389647 0.920964i \(-0.372597\pi\)
0.389647 + 0.920964i \(0.372597\pi\)
\(614\) 655.189i 1.06708i
\(615\) 0 0
\(616\) 155.378 0.252237
\(617\) − 1014.71i − 1.64459i −0.569061 0.822295i \(-0.692693\pi\)
0.569061 0.822295i \(-0.307307\pi\)
\(618\) 0 0
\(619\) −401.464 −0.648569 −0.324284 0.945960i \(-0.605123\pi\)
−0.324284 + 0.945960i \(0.605123\pi\)
\(620\) 0.230690i 0 0.000372080i
\(621\) 0 0
\(622\) 828.504 1.33200
\(623\) − 1556.69i − 2.49871i
\(624\) 0 0
\(625\) 614.812 0.983699
\(626\) − 1167.21i − 1.86456i
\(627\) 0 0
\(628\) −1014.48 −1.61542
\(629\) 284.804i 0.452788i
\(630\) 0 0
\(631\) −807.950 −1.28043 −0.640214 0.768197i \(-0.721154\pi\)
−0.640214 + 0.768197i \(0.721154\pi\)
\(632\) 93.9342i 0.148630i
\(633\) 0 0
\(634\) 731.637 1.15400
\(635\) 4.15728i 0.00654690i
\(636\) 0 0
\(637\) −1170.27 −1.83715
\(638\) − 1452.30i − 2.27634i
\(639\) 0 0
\(640\) 12.8222 0.0200346
\(641\) 225.405i 0.351646i 0.984422 + 0.175823i \(0.0562587\pi\)
−0.984422 + 0.175823i \(0.943741\pi\)
\(642\) 0 0
\(643\) −282.242 −0.438946 −0.219473 0.975619i \(-0.570434\pi\)
−0.219473 + 0.975619i \(0.570434\pi\)
\(644\) − 1239.35i − 1.92446i
\(645\) 0 0
\(646\) 52.6521 0.0815047
\(647\) − 927.159i − 1.43301i −0.697581 0.716506i \(-0.745740\pi\)
0.697581 0.716506i \(-0.254260\pi\)
\(648\) 0 0
\(649\) −843.734 −1.30005
\(650\) 1033.17i 1.58949i
\(651\) 0 0
\(652\) −544.605 −0.835284
\(653\) 91.8009i 0.140583i 0.997526 + 0.0702916i \(0.0223930\pi\)
−0.997526 + 0.0702916i \(0.977607\pi\)
\(654\) 0 0
\(655\) 57.5469 0.0878578
\(656\) − 1043.75i − 1.59108i
\(657\) 0 0
\(658\) −988.451 −1.50221
\(659\) 201.283i 0.305437i 0.988270 + 0.152718i \(0.0488027\pi\)
−0.988270 + 0.152718i \(0.951197\pi\)
\(660\) 0 0
\(661\) −1000.56 −1.51371 −0.756856 0.653582i \(-0.773266\pi\)
−0.756856 + 0.653582i \(0.773266\pi\)
\(662\) − 1427.71i − 2.15666i
\(663\) 0 0
\(664\) −154.199 −0.232228
\(665\) − 18.9916i − 0.0285589i
\(666\) 0 0
\(667\) −997.663 −1.49575
\(668\) − 207.859i − 0.311166i
\(669\) 0 0
\(670\) −14.6701 −0.0218957
\(671\) − 510.470i − 0.760760i
\(672\) 0 0
\(673\) −787.143 −1.16960 −0.584801 0.811177i \(-0.698828\pi\)
−0.584801 + 0.811177i \(0.698828\pi\)
\(674\) − 1266.04i − 1.87840i
\(675\) 0 0
\(676\) −162.464 −0.240332
\(677\) − 846.014i − 1.24965i −0.780764 0.624825i \(-0.785170\pi\)
0.780764 0.624825i \(-0.214830\pi\)
\(678\) 0 0
\(679\) −255.374 −0.376103
\(680\) − 1.62476i − 0.00238935i
\(681\) 0 0
\(682\) −5.15486 −0.00755845
\(683\) 24.7020i 0.0361669i 0.999836 + 0.0180834i \(0.00575645\pi\)
−0.999836 + 0.0180834i \(0.994244\pi\)
\(684\) 0 0
\(685\) −27.2715 −0.0398124
\(686\) 1075.05i 1.56712i
\(687\) 0 0
\(688\) −462.239 −0.671860
\(689\) 660.746i 0.958993i
\(690\) 0 0
\(691\) 1031.11 1.49219 0.746097 0.665838i \(-0.231926\pi\)
0.746097 + 0.665838i \(0.231926\pi\)
\(692\) − 965.618i − 1.39540i
\(693\) 0 0
\(694\) 1693.61 2.44036
\(695\) − 97.5614i − 0.140376i
\(696\) 0 0
\(697\) −293.620 −0.421263
\(698\) − 1522.49i − 2.18122i
\(699\) 0 0
\(700\) 1243.28 1.77611
\(701\) − 131.146i − 0.187085i −0.995615 0.0935424i \(-0.970181\pi\)
0.995615 0.0935424i \(-0.0298191\pi\)
\(702\) 0 0
\(703\) 317.928 0.452245
\(704\) 940.441i 1.33585i
\(705\) 0 0
\(706\) 146.388 0.207349
\(707\) 1638.62i 2.31772i
\(708\) 0 0
\(709\) 488.183 0.688552 0.344276 0.938869i \(-0.388124\pi\)
0.344276 + 0.938869i \(0.388124\pi\)
\(710\) − 15.7313i − 0.0221567i
\(711\) 0 0
\(712\) −148.666 −0.208801
\(713\) 3.54115i 0.00496655i
\(714\) 0 0
\(715\) 66.0224 0.0923390
\(716\) − 411.687i − 0.574982i
\(717\) 0 0
\(718\) 410.507 0.571737
\(719\) − 14.8164i − 0.0206069i −0.999947 0.0103035i \(-0.996720\pi\)
0.999947 0.0103035i \(-0.00327975\pi\)
\(720\) 0 0
\(721\) −556.093 −0.771280
\(722\) 986.066i 1.36574i
\(723\) 0 0
\(724\) −505.159 −0.697733
\(725\) − 1000.82i − 1.38045i
\(726\) 0 0
\(727\) 288.303 0.396565 0.198282 0.980145i \(-0.436464\pi\)
0.198282 + 0.980145i \(0.436464\pi\)
\(728\) 178.946i 0.245805i
\(729\) 0 0
\(730\) 139.899 0.191643
\(731\) 130.034i 0.177885i
\(732\) 0 0
\(733\) 1127.93 1.53879 0.769393 0.638776i \(-0.220559\pi\)
0.769393 + 0.638776i \(0.220559\pi\)
\(734\) 460.053i 0.626775i
\(735\) 0 0
\(736\) 1137.59 1.54564
\(737\) − 171.281i − 0.232402i
\(738\) 0 0
\(739\) 289.034 0.391115 0.195557 0.980692i \(-0.437348\pi\)
0.195557 + 0.980692i \(0.437348\pi\)
\(740\) − 113.915i − 0.153939i
\(741\) 0 0
\(742\) 1521.76 2.05089
\(743\) − 1057.13i − 1.42279i −0.702793 0.711395i \(-0.748064\pi\)
0.702793 0.711395i \(-0.251936\pi\)
\(744\) 0 0
\(745\) −30.1345 −0.0404491
\(746\) − 1535.75i − 2.05864i
\(747\) 0 0
\(748\) 220.263 0.294470
\(749\) 677.843i 0.904998i
\(750\) 0 0
\(751\) 528.864 0.704214 0.352107 0.935960i \(-0.385465\pi\)
0.352107 + 0.935960i \(0.385465\pi\)
\(752\) − 428.982i − 0.570455i
\(753\) 0 0
\(754\) 1672.59 2.21829
\(755\) − 90.7258i − 0.120167i
\(756\) 0 0
\(757\) 1351.92 1.78589 0.892943 0.450170i \(-0.148637\pi\)
0.892943 + 0.450170i \(0.148637\pi\)
\(758\) − 708.126i − 0.934203i
\(759\) 0 0
\(760\) −1.81373 −0.00238648
\(761\) − 615.698i − 0.809064i −0.914524 0.404532i \(-0.867434\pi\)
0.914524 0.404532i \(-0.132566\pi\)
\(762\) 0 0
\(763\) 1942.70 2.54613
\(764\) − 324.078i − 0.424186i
\(765\) 0 0
\(766\) −484.184 −0.632094
\(767\) − 971.712i − 1.26690i
\(768\) 0 0
\(769\) −1291.47 −1.67942 −0.839711 0.543034i \(-0.817275\pi\)
−0.839711 + 0.543034i \(0.817275\pi\)
\(770\) − 152.056i − 0.197475i
\(771\) 0 0
\(772\) 61.7104 0.0799357
\(773\) − 354.437i − 0.458521i −0.973365 0.229261i \(-0.926369\pi\)
0.973365 0.229261i \(-0.0736308\pi\)
\(774\) 0 0
\(775\) −3.55237 −0.00458370
\(776\) 24.3885i 0.0314285i
\(777\) 0 0
\(778\) −1195.19 −1.53623
\(779\) 327.771i 0.420758i
\(780\) 0 0
\(781\) 183.670 0.235173
\(782\) − 289.590i − 0.370319i
\(783\) 0 0
\(784\) −1169.72 −1.49199
\(785\) 85.5027i 0.108921i
\(786\) 0 0
\(787\) −1040.99 −1.32273 −0.661363 0.750066i \(-0.730022\pi\)
−0.661363 + 0.750066i \(0.730022\pi\)
\(788\) 1369.13i 1.73747i
\(789\) 0 0
\(790\) 91.9259 0.116362
\(791\) − 863.786i − 1.09202i
\(792\) 0 0
\(793\) 587.898 0.741360
\(794\) − 1551.58i − 1.95413i
\(795\) 0 0
\(796\) −304.304 −0.382292
\(797\) − 423.595i − 0.531486i −0.964044 0.265743i \(-0.914383\pi\)
0.964044 0.265743i \(-0.0856173\pi\)
\(798\) 0 0
\(799\) −120.678 −0.151037
\(800\) 1141.19i 1.42649i
\(801\) 0 0
\(802\) −1550.91 −1.93381
\(803\) 1633.39i 2.03411i
\(804\) 0 0
\(805\) −104.455 −0.129758
\(806\) − 5.93676i − 0.00736571i
\(807\) 0 0
\(808\) 156.491 0.193677
\(809\) 643.837i 0.795843i 0.917419 + 0.397922i \(0.130268\pi\)
−0.917419 + 0.397922i \(0.869732\pi\)
\(810\) 0 0
\(811\) 42.7687 0.0527357 0.0263679 0.999652i \(-0.491606\pi\)
0.0263679 + 0.999652i \(0.491606\pi\)
\(812\) − 2012.74i − 2.47874i
\(813\) 0 0
\(814\) 2545.48 3.12713
\(815\) 45.9005i 0.0563196i
\(816\) 0 0
\(817\) 145.158 0.177672
\(818\) 1035.54i 1.26594i
\(819\) 0 0
\(820\) 117.442 0.143221
\(821\) − 155.646i − 0.189581i −0.995497 0.0947906i \(-0.969782\pi\)
0.995497 0.0947906i \(-0.0302182\pi\)
\(822\) 0 0
\(823\) −835.537 −1.01523 −0.507617 0.861583i \(-0.669473\pi\)
−0.507617 + 0.861583i \(0.669473\pi\)
\(824\) 53.1076i 0.0644510i
\(825\) 0 0
\(826\) −2237.95 −2.70938
\(827\) − 319.635i − 0.386499i −0.981150 0.193250i \(-0.938097\pi\)
0.981150 0.193250i \(-0.0619027\pi\)
\(828\) 0 0
\(829\) −745.509 −0.899288 −0.449644 0.893208i \(-0.648449\pi\)
−0.449644 + 0.893208i \(0.648449\pi\)
\(830\) 150.902i 0.181810i
\(831\) 0 0
\(832\) −1083.09 −1.30179
\(833\) 329.057i 0.395026i
\(834\) 0 0
\(835\) −17.5188 −0.0209806
\(836\) − 245.882i − 0.294117i
\(837\) 0 0
\(838\) 1775.58 2.11883
\(839\) 1295.12i 1.54365i 0.635838 + 0.771823i \(0.280655\pi\)
−0.635838 + 0.771823i \(0.719345\pi\)
\(840\) 0 0
\(841\) −779.228 −0.926549
\(842\) − 1647.55i − 1.95671i
\(843\) 0 0
\(844\) −927.143 −1.09851
\(845\) 13.6928i 0.0162045i
\(846\) 0 0
\(847\) 392.995 0.463985
\(848\) 660.436i 0.778816i
\(849\) 0 0
\(850\) 290.507 0.341773
\(851\) − 1748.63i − 2.05479i
\(852\) 0 0
\(853\) 30.6295 0.0359080 0.0179540 0.999839i \(-0.494285\pi\)
0.0179540 + 0.999839i \(0.494285\pi\)
\(854\) − 1353.99i − 1.58546i
\(855\) 0 0
\(856\) 64.7350 0.0756250
\(857\) − 446.511i − 0.521016i −0.965472 0.260508i \(-0.916110\pi\)
0.965472 0.260508i \(-0.0838901\pi\)
\(858\) 0 0
\(859\) −1409.65 −1.64103 −0.820516 0.571624i \(-0.806314\pi\)
−0.820516 + 0.571624i \(0.806314\pi\)
\(860\) − 52.0107i − 0.0604776i
\(861\) 0 0
\(862\) 780.916 0.905935
\(863\) 1037.85i 1.20260i 0.799022 + 0.601301i \(0.205351\pi\)
−0.799022 + 0.601301i \(0.794649\pi\)
\(864\) 0 0
\(865\) −81.3842 −0.0940858
\(866\) − 437.679i − 0.505403i
\(867\) 0 0
\(868\) −7.14410 −0.00823053
\(869\) 1073.28i 1.23507i
\(870\) 0 0
\(871\) 197.261 0.226476
\(872\) − 185.531i − 0.212764i
\(873\) 0 0
\(874\) −323.271 −0.369876
\(875\) − 210.146i − 0.240167i
\(876\) 0 0
\(877\) 74.7317 0.0852129 0.0426065 0.999092i \(-0.486434\pi\)
0.0426065 + 0.999092i \(0.486434\pi\)
\(878\) − 789.927i − 0.899689i
\(879\) 0 0
\(880\) 65.9914 0.0749902
\(881\) 278.915i 0.316589i 0.987392 + 0.158294i \(0.0505995\pi\)
−0.987392 + 0.158294i \(0.949400\pi\)
\(882\) 0 0
\(883\) 172.659 0.195537 0.0977686 0.995209i \(-0.468829\pi\)
0.0977686 + 0.995209i \(0.468829\pi\)
\(884\) 253.673i 0.286961i
\(885\) 0 0
\(886\) −501.755 −0.566314
\(887\) 221.858i 0.250122i 0.992149 + 0.125061i \(0.0399127\pi\)
−0.992149 + 0.125061i \(0.960087\pi\)
\(888\) 0 0
\(889\) −128.745 −0.144819
\(890\) 145.488i 0.163470i
\(891\) 0 0
\(892\) 1157.92 1.29812
\(893\) 134.714i 0.150856i
\(894\) 0 0
\(895\) −34.6978 −0.0387685
\(896\) 397.082i 0.443172i
\(897\) 0 0
\(898\) 281.597 0.313582
\(899\) 5.75091i 0.00639700i
\(900\) 0 0
\(901\) 185.789 0.206203
\(902\) 2624.28i 2.90941i
\(903\) 0 0
\(904\) −82.4928 −0.0912531
\(905\) 42.5758i 0.0470451i
\(906\) 0 0
\(907\) 680.121 0.749858 0.374929 0.927054i \(-0.377667\pi\)
0.374929 + 0.927054i \(0.377667\pi\)
\(908\) − 1708.17i − 1.88124i
\(909\) 0 0
\(910\) 175.120 0.192440
\(911\) 1046.68i 1.14894i 0.818527 + 0.574469i \(0.194791\pi\)
−0.818527 + 0.574469i \(0.805209\pi\)
\(912\) 0 0
\(913\) −1761.86 −1.92975
\(914\) − 88.3969i − 0.0967143i
\(915\) 0 0
\(916\) 683.298 0.745958
\(917\) 1782.14i 1.94344i
\(918\) 0 0
\(919\) −244.736 −0.266306 −0.133153 0.991095i \(-0.542510\pi\)
−0.133153 + 0.991095i \(0.542510\pi\)
\(920\) 9.97562i 0.0108431i
\(921\) 0 0
\(922\) 255.181 0.276769
\(923\) 211.529i 0.229176i
\(924\) 0 0
\(925\) 1754.17 1.89640
\(926\) 2298.08i 2.48172i
\(927\) 0 0
\(928\) 1847.47 1.99081
\(929\) 120.321i 0.129516i 0.997901 + 0.0647582i \(0.0206276\pi\)
−0.997901 + 0.0647582i \(0.979372\pi\)
\(930\) 0 0
\(931\) 367.329 0.394553
\(932\) 1435.00i 1.53969i
\(933\) 0 0
\(934\) −413.690 −0.442923
\(935\) − 18.5642i − 0.0198548i
\(936\) 0 0
\(937\) 412.610 0.440353 0.220176 0.975460i \(-0.429337\pi\)
0.220176 + 0.975460i \(0.429337\pi\)
\(938\) − 454.310i − 0.484339i
\(939\) 0 0
\(940\) 48.2687 0.0513496
\(941\) 288.520i 0.306610i 0.988179 + 0.153305i \(0.0489917\pi\)
−0.988179 + 0.153305i \(0.951008\pi\)
\(942\) 0 0
\(943\) 1802.76 1.91173
\(944\) − 971.256i − 1.02887i
\(945\) 0 0
\(946\) 1162.20 1.22854
\(947\) 301.186i 0.318042i 0.987275 + 0.159021i \(0.0508338\pi\)
−0.987275 + 0.159021i \(0.949166\pi\)
\(948\) 0 0
\(949\) −1881.15 −1.98224
\(950\) − 324.295i − 0.341364i
\(951\) 0 0
\(952\) 50.3162 0.0528531
\(953\) − 791.241i − 0.830263i −0.909761 0.415132i \(-0.863735\pi\)
0.909761 0.415132i \(-0.136265\pi\)
\(954\) 0 0
\(955\) −27.3140 −0.0286010
\(956\) − 957.462i − 1.00153i
\(957\) 0 0
\(958\) −1607.75 −1.67824
\(959\) − 844.556i − 0.880663i
\(960\) 0 0
\(961\) −960.980 −0.999979
\(962\) 2931.59i 3.04739i
\(963\) 0 0
\(964\) 635.210 0.658931
\(965\) − 5.20108i − 0.00538972i
\(966\) 0 0
\(967\) 110.779 0.114560 0.0572798 0.998358i \(-0.481757\pi\)
0.0572798 + 0.998358i \(0.481757\pi\)
\(968\) − 37.5316i − 0.0387723i
\(969\) 0 0
\(970\) 23.8671 0.0246053
\(971\) − 1479.69i − 1.52388i −0.647648 0.761940i \(-0.724247\pi\)
0.647648 0.761940i \(-0.275753\pi\)
\(972\) 0 0
\(973\) 3021.32 3.10516
\(974\) − 1320.63i − 1.35588i
\(975\) 0 0
\(976\) 587.622 0.602072
\(977\) − 565.276i − 0.578584i −0.957241 0.289292i \(-0.906580\pi\)
0.957241 0.289292i \(-0.0934198\pi\)
\(978\) 0 0
\(979\) −1698.64 −1.73508
\(980\) − 131.615i − 0.134301i
\(981\) 0 0
\(982\) −1239.02 −1.26173
\(983\) 1189.08i 1.20965i 0.796360 + 0.604823i \(0.206756\pi\)
−0.796360 + 0.604823i \(0.793244\pi\)
\(984\) 0 0
\(985\) 115.393 0.117150
\(986\) − 470.300i − 0.476978i
\(987\) 0 0
\(988\) 283.177 0.286617
\(989\) − 798.379i − 0.807258i
\(990\) 0 0
\(991\) −1580.49 −1.59484 −0.797421 0.603424i \(-0.793803\pi\)
−0.797421 + 0.603424i \(0.793803\pi\)
\(992\) − 6.55748i − 0.00661037i
\(993\) 0 0
\(994\) 487.172 0.490113
\(995\) 25.6474i 0.0257763i
\(996\) 0 0
\(997\) −1820.34 −1.82582 −0.912908 0.408166i \(-0.866168\pi\)
−0.912908 + 0.408166i \(0.866168\pi\)
\(998\) 625.742i 0.626996i
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1143.3.b.a.890.15 84
3.2 odd 2 inner 1143.3.b.a.890.70 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.15 84 1.1 even 1 trivial
1143.3.b.a.890.70 yes 84 3.2 odd 2 inner