Properties

Label 320.6.d.c.161.11
Level 320
Weight 6
Character 320.161
Analytic conductor 51.323
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + 26901210 x^{4} + 434775816 x^{3} + 1271335685 x^{2} + 1475231592 x + 653157349\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.11
Root \(-8.35048 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.c.161.2

$q$-expansion

\(f(q)\) \(=\) \(q+20.6292i q^{3} +25.0000i q^{5} +93.5254 q^{7} -182.562 q^{9} +O(q^{10})\) \(q+20.6292i q^{3} +25.0000i q^{5} +93.5254 q^{7} -182.562 q^{9} +659.055i q^{11} -183.844i q^{13} -515.729 q^{15} -467.075 q^{17} +225.914i q^{19} +1929.35i q^{21} -1109.57 q^{23} -625.000 q^{25} +1246.78i q^{27} +1738.98i q^{29} +7270.81 q^{31} -13595.7 q^{33} +2338.14i q^{35} +14229.7i q^{37} +3792.54 q^{39} -4929.92 q^{41} +12379.7i q^{43} -4564.05i q^{45} +619.842 q^{47} -8060.00 q^{49} -9635.36i q^{51} -16989.0i q^{53} -16476.4 q^{55} -4660.40 q^{57} -50742.8i q^{59} +20666.3i q^{61} -17074.2 q^{63} +4596.09 q^{65} -41920.3i q^{67} -22889.5i q^{69} -54687.7 q^{71} -25211.5 q^{73} -12893.2i q^{75} +61638.4i q^{77} -29000.4 q^{79} -70082.7 q^{81} -92679.1i q^{83} -11676.9i q^{85} -35873.7 q^{87} +54644.0 q^{89} -17194.1i q^{91} +149991. i q^{93} -5647.84 q^{95} +45542.8 q^{97} -120318. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 268q^{7} - 428q^{9} + O(q^{10}) \) \( 12q - 268q^{7} - 428q^{9} + 900q^{15} + 2400q^{17} - 8108q^{23} - 7500q^{25} - 7976q^{31} - 20776q^{33} - 40984q^{39} - 56408q^{41} - 21172q^{47} - 5540q^{49} - 18400q^{55} + 39992q^{57} - 179516q^{63} + 44000q^{65} - 367704q^{71} + 58736q^{73} - 26192q^{79} + 411692q^{81} - 183200q^{87} + 87672q^{89} - 121000q^{95} - 172336q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 20.6292i 1.32336i 0.749786 + 0.661680i \(0.230156\pi\)
−0.749786 + 0.661680i \(0.769844\pi\)
\(4\) 0 0
\(5\) 25.0000i 0.447214i
\(6\) 0 0
\(7\) 93.5254 0.721414 0.360707 0.932679i \(-0.382535\pi\)
0.360707 + 0.932679i \(0.382535\pi\)
\(8\) 0 0
\(9\) −182.562 −0.751284
\(10\) 0 0
\(11\) 659.055i 1.64225i 0.570747 + 0.821126i \(0.306654\pi\)
−0.570747 + 0.821126i \(0.693346\pi\)
\(12\) 0 0
\(13\) − 183.844i − 0.301711i −0.988556 0.150855i \(-0.951797\pi\)
0.988556 0.150855i \(-0.0482027\pi\)
\(14\) 0 0
\(15\) −515.729 −0.591825
\(16\) 0 0
\(17\) −467.075 −0.391980 −0.195990 0.980606i \(-0.562792\pi\)
−0.195990 + 0.980606i \(0.562792\pi\)
\(18\) 0 0
\(19\) 225.914i 0.143568i 0.997420 + 0.0717841i \(0.0228693\pi\)
−0.997420 + 0.0717841i \(0.977131\pi\)
\(20\) 0 0
\(21\) 1929.35i 0.954691i
\(22\) 0 0
\(23\) −1109.57 −0.437357 −0.218678 0.975797i \(-0.570175\pi\)
−0.218678 + 0.975797i \(0.570175\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) 1246.78i 0.329141i
\(28\) 0 0
\(29\) 1738.98i 0.383972i 0.981398 + 0.191986i \(0.0614929\pi\)
−0.981398 + 0.191986i \(0.938507\pi\)
\(30\) 0 0
\(31\) 7270.81 1.35887 0.679436 0.733735i \(-0.262225\pi\)
0.679436 + 0.733735i \(0.262225\pi\)
\(32\) 0 0
\(33\) −13595.7 −2.17329
\(34\) 0 0
\(35\) 2338.14i 0.322626i
\(36\) 0 0
\(37\) 14229.7i 1.70881i 0.519610 + 0.854403i \(0.326077\pi\)
−0.519610 + 0.854403i \(0.673923\pi\)
\(38\) 0 0
\(39\) 3792.54 0.399272
\(40\) 0 0
\(41\) −4929.92 −0.458015 −0.229008 0.973425i \(-0.573548\pi\)
−0.229008 + 0.973425i \(0.573548\pi\)
\(42\) 0 0
\(43\) 12379.7i 1.02103i 0.859870 + 0.510514i \(0.170545\pi\)
−0.859870 + 0.510514i \(0.829455\pi\)
\(44\) 0 0
\(45\) − 4564.05i − 0.335984i
\(46\) 0 0
\(47\) 619.842 0.0409295 0.0204647 0.999791i \(-0.493485\pi\)
0.0204647 + 0.999791i \(0.493485\pi\)
\(48\) 0 0
\(49\) −8060.00 −0.479562
\(50\) 0 0
\(51\) − 9635.36i − 0.518731i
\(52\) 0 0
\(53\) − 16989.0i − 0.830765i −0.909647 0.415382i \(-0.863648\pi\)
0.909647 0.415382i \(-0.136352\pi\)
\(54\) 0 0
\(55\) −16476.4 −0.734437
\(56\) 0 0
\(57\) −4660.40 −0.189993
\(58\) 0 0
\(59\) − 50742.8i − 1.89777i −0.315615 0.948887i \(-0.602211\pi\)
0.315615 0.948887i \(-0.397789\pi\)
\(60\) 0 0
\(61\) 20666.3i 0.711111i 0.934655 + 0.355555i \(0.115708\pi\)
−0.934655 + 0.355555i \(0.884292\pi\)
\(62\) 0 0
\(63\) −17074.2 −0.541987
\(64\) 0 0
\(65\) 4596.09 0.134929
\(66\) 0 0
\(67\) − 41920.3i − 1.14087i −0.821341 0.570437i \(-0.806774\pi\)
0.821341 0.570437i \(-0.193226\pi\)
\(68\) 0 0
\(69\) − 22889.5i − 0.578781i
\(70\) 0 0
\(71\) −54687.7 −1.28749 −0.643744 0.765241i \(-0.722620\pi\)
−0.643744 + 0.765241i \(0.722620\pi\)
\(72\) 0 0
\(73\) −25211.5 −0.553722 −0.276861 0.960910i \(-0.589294\pi\)
−0.276861 + 0.960910i \(0.589294\pi\)
\(74\) 0 0
\(75\) − 12893.2i − 0.264672i
\(76\) 0 0
\(77\) 61638.4i 1.18474i
\(78\) 0 0
\(79\) −29000.4 −0.522802 −0.261401 0.965230i \(-0.584184\pi\)
−0.261401 + 0.965230i \(0.584184\pi\)
\(80\) 0 0
\(81\) −70082.7 −1.18686
\(82\) 0 0
\(83\) − 92679.1i − 1.47668i −0.674428 0.738341i \(-0.735610\pi\)
0.674428 0.738341i \(-0.264390\pi\)
\(84\) 0 0
\(85\) − 11676.9i − 0.175299i
\(86\) 0 0
\(87\) −35873.7 −0.508134
\(88\) 0 0
\(89\) 54644.0 0.731253 0.365627 0.930762i \(-0.380855\pi\)
0.365627 + 0.930762i \(0.380855\pi\)
\(90\) 0 0
\(91\) − 17194.1i − 0.217658i
\(92\) 0 0
\(93\) 149991.i 1.79828i
\(94\) 0 0
\(95\) −5647.84 −0.0642056
\(96\) 0 0
\(97\) 45542.8 0.491463 0.245731 0.969338i \(-0.420972\pi\)
0.245731 + 0.969338i \(0.420972\pi\)
\(98\) 0 0
\(99\) − 120318.i − 1.23380i
\(100\) 0 0
\(101\) − 195492.i − 1.90689i −0.301571 0.953444i \(-0.597511\pi\)
0.301571 0.953444i \(-0.402489\pi\)
\(102\) 0 0
\(103\) 81681.1 0.758627 0.379314 0.925268i \(-0.376160\pi\)
0.379314 + 0.925268i \(0.376160\pi\)
\(104\) 0 0
\(105\) −48233.8 −0.426951
\(106\) 0 0
\(107\) − 107507.i − 0.907775i −0.891059 0.453888i \(-0.850037\pi\)
0.891059 0.453888i \(-0.149963\pi\)
\(108\) 0 0
\(109\) 170366.i 1.37346i 0.726912 + 0.686731i \(0.240955\pi\)
−0.726912 + 0.686731i \(0.759045\pi\)
\(110\) 0 0
\(111\) −293548. −2.26137
\(112\) 0 0
\(113\) −81363.4 −0.599423 −0.299711 0.954030i \(-0.596890\pi\)
−0.299711 + 0.954030i \(0.596890\pi\)
\(114\) 0 0
\(115\) − 27739.3i − 0.195592i
\(116\) 0 0
\(117\) 33562.9i 0.226670i
\(118\) 0 0
\(119\) −43683.4 −0.282780
\(120\) 0 0
\(121\) −273302. −1.69699
\(122\) 0 0
\(123\) − 101700.i − 0.606119i
\(124\) 0 0
\(125\) − 15625.0i − 0.0894427i
\(126\) 0 0
\(127\) 291463. 1.60352 0.801759 0.597647i \(-0.203898\pi\)
0.801759 + 0.597647i \(0.203898\pi\)
\(128\) 0 0
\(129\) −255382. −1.35119
\(130\) 0 0
\(131\) 276390.i 1.40716i 0.710616 + 0.703580i \(0.248416\pi\)
−0.710616 + 0.703580i \(0.751584\pi\)
\(132\) 0 0
\(133\) 21128.7i 0.103572i
\(134\) 0 0
\(135\) −31169.6 −0.147196
\(136\) 0 0
\(137\) −366347. −1.66760 −0.833798 0.552069i \(-0.813839\pi\)
−0.833798 + 0.552069i \(0.813839\pi\)
\(138\) 0 0
\(139\) 162496.i 0.713353i 0.934228 + 0.356676i \(0.116090\pi\)
−0.934228 + 0.356676i \(0.883910\pi\)
\(140\) 0 0
\(141\) 12786.8i 0.0541645i
\(142\) 0 0
\(143\) 121163. 0.495485
\(144\) 0 0
\(145\) −43474.5 −0.171718
\(146\) 0 0
\(147\) − 166271.i − 0.634633i
\(148\) 0 0
\(149\) 301589.i 1.11288i 0.830887 + 0.556441i \(0.187834\pi\)
−0.830887 + 0.556441i \(0.812166\pi\)
\(150\) 0 0
\(151\) −375519. −1.34026 −0.670131 0.742243i \(-0.733762\pi\)
−0.670131 + 0.742243i \(0.733762\pi\)
\(152\) 0 0
\(153\) 85270.1 0.294488
\(154\) 0 0
\(155\) 181770.i 0.607706i
\(156\) 0 0
\(157\) 548149.i 1.77480i 0.460999 + 0.887400i \(0.347491\pi\)
−0.460999 + 0.887400i \(0.652509\pi\)
\(158\) 0 0
\(159\) 350469. 1.09940
\(160\) 0 0
\(161\) −103773. −0.315515
\(162\) 0 0
\(163\) − 19042.7i − 0.0561383i −0.999606 0.0280692i \(-0.991064\pi\)
0.999606 0.0280692i \(-0.00893587\pi\)
\(164\) 0 0
\(165\) − 339894.i − 0.971926i
\(166\) 0 0
\(167\) 287366. 0.797340 0.398670 0.917094i \(-0.369472\pi\)
0.398670 + 0.917094i \(0.369472\pi\)
\(168\) 0 0
\(169\) 337494. 0.908971
\(170\) 0 0
\(171\) − 41243.2i − 0.107860i
\(172\) 0 0
\(173\) − 352199.i − 0.894692i −0.894361 0.447346i \(-0.852369\pi\)
0.894361 0.447346i \(-0.147631\pi\)
\(174\) 0 0
\(175\) −58453.4 −0.144283
\(176\) 0 0
\(177\) 1.04678e6 2.51144
\(178\) 0 0
\(179\) 87611.7i 0.204376i 0.994765 + 0.102188i \(0.0325843\pi\)
−0.994765 + 0.102188i \(0.967416\pi\)
\(180\) 0 0
\(181\) − 534314.i − 1.21227i −0.795361 0.606136i \(-0.792719\pi\)
0.795361 0.606136i \(-0.207281\pi\)
\(182\) 0 0
\(183\) −426328. −0.941056
\(184\) 0 0
\(185\) −355744. −0.764202
\(186\) 0 0
\(187\) − 307828.i − 0.643730i
\(188\) 0 0
\(189\) 116606.i 0.237447i
\(190\) 0 0
\(191\) −45622.1 −0.0904882 −0.0452441 0.998976i \(-0.514407\pi\)
−0.0452441 + 0.998976i \(0.514407\pi\)
\(192\) 0 0
\(193\) −26392.3 −0.0510017 −0.0255009 0.999675i \(-0.508118\pi\)
−0.0255009 + 0.999675i \(0.508118\pi\)
\(194\) 0 0
\(195\) 94813.5i 0.178560i
\(196\) 0 0
\(197\) 483767.i 0.888119i 0.895997 + 0.444059i \(0.146462\pi\)
−0.895997 + 0.444059i \(0.853538\pi\)
\(198\) 0 0
\(199\) 51497.1 0.0921828 0.0460914 0.998937i \(-0.485323\pi\)
0.0460914 + 0.998937i \(0.485323\pi\)
\(200\) 0 0
\(201\) 864781. 1.50979
\(202\) 0 0
\(203\) 162639.i 0.277003i
\(204\) 0 0
\(205\) − 123248.i − 0.204831i
\(206\) 0 0
\(207\) 202566. 0.328579
\(208\) 0 0
\(209\) −148889. −0.235775
\(210\) 0 0
\(211\) 716759.i 1.10833i 0.832408 + 0.554163i \(0.186961\pi\)
−0.832408 + 0.554163i \(0.813039\pi\)
\(212\) 0 0
\(213\) − 1.12816e6i − 1.70381i
\(214\) 0 0
\(215\) −309491. −0.456617
\(216\) 0 0
\(217\) 680005. 0.980309
\(218\) 0 0
\(219\) − 520092.i − 0.732774i
\(220\) 0 0
\(221\) 85868.8i 0.118265i
\(222\) 0 0
\(223\) 747997. 1.00725 0.503625 0.863922i \(-0.331999\pi\)
0.503625 + 0.863922i \(0.331999\pi\)
\(224\) 0 0
\(225\) 114101. 0.150257
\(226\) 0 0
\(227\) 598186.i 0.770498i 0.922813 + 0.385249i \(0.125884\pi\)
−0.922813 + 0.385249i \(0.874116\pi\)
\(228\) 0 0
\(229\) − 655203.i − 0.825633i −0.910814 0.412816i \(-0.864545\pi\)
0.910814 0.412816i \(-0.135455\pi\)
\(230\) 0 0
\(231\) −1.27155e6 −1.56784
\(232\) 0 0
\(233\) 170801. 0.206111 0.103055 0.994676i \(-0.467138\pi\)
0.103055 + 0.994676i \(0.467138\pi\)
\(234\) 0 0
\(235\) 15496.0i 0.0183042i
\(236\) 0 0
\(237\) − 598255.i − 0.691855i
\(238\) 0 0
\(239\) 1.26920e6 1.43726 0.718629 0.695393i \(-0.244770\pi\)
0.718629 + 0.695393i \(0.244770\pi\)
\(240\) 0 0
\(241\) 274289. 0.304205 0.152103 0.988365i \(-0.451396\pi\)
0.152103 + 0.988365i \(0.451396\pi\)
\(242\) 0 0
\(243\) − 1.14278e6i − 1.24150i
\(244\) 0 0
\(245\) − 201500.i − 0.214467i
\(246\) 0 0
\(247\) 41532.8 0.0433160
\(248\) 0 0
\(249\) 1.91189e6 1.95418
\(250\) 0 0
\(251\) 861371.i 0.862990i 0.902115 + 0.431495i \(0.142014\pi\)
−0.902115 + 0.431495i \(0.857986\pi\)
\(252\) 0 0
\(253\) − 731268.i − 0.718250i
\(254\) 0 0
\(255\) 240884. 0.231984
\(256\) 0 0
\(257\) 524792. 0.495626 0.247813 0.968808i \(-0.420288\pi\)
0.247813 + 0.968808i \(0.420288\pi\)
\(258\) 0 0
\(259\) 1.33084e6i 1.23276i
\(260\) 0 0
\(261\) − 317472.i − 0.288472i
\(262\) 0 0
\(263\) 2.03635e6 1.81536 0.907680 0.419664i \(-0.137852\pi\)
0.907680 + 0.419664i \(0.137852\pi\)
\(264\) 0 0
\(265\) 424725. 0.371529
\(266\) 0 0
\(267\) 1.12726e6i 0.967712i
\(268\) 0 0
\(269\) − 477001.i − 0.401919i −0.979600 0.200960i \(-0.935594\pi\)
0.979600 0.200960i \(-0.0644060\pi\)
\(270\) 0 0
\(271\) 350310. 0.289754 0.144877 0.989450i \(-0.453721\pi\)
0.144877 + 0.989450i \(0.453721\pi\)
\(272\) 0 0
\(273\) 354699. 0.288040
\(274\) 0 0
\(275\) − 411909.i − 0.328450i
\(276\) 0 0
\(277\) 2.08816e6i 1.63518i 0.575804 + 0.817588i \(0.304689\pi\)
−0.575804 + 0.817588i \(0.695311\pi\)
\(278\) 0 0
\(279\) −1.32737e6 −1.02090
\(280\) 0 0
\(281\) −1.24077e6 −0.937399 −0.468699 0.883358i \(-0.655277\pi\)
−0.468699 + 0.883358i \(0.655277\pi\)
\(282\) 0 0
\(283\) 1.85643e6i 1.37788i 0.724818 + 0.688940i \(0.241924\pi\)
−0.724818 + 0.688940i \(0.758076\pi\)
\(284\) 0 0
\(285\) − 116510.i − 0.0849672i
\(286\) 0 0
\(287\) −461072. −0.330419
\(288\) 0 0
\(289\) −1.20170e6 −0.846352
\(290\) 0 0
\(291\) 939510.i 0.650383i
\(292\) 0 0
\(293\) 1.64126e6i 1.11689i 0.829543 + 0.558443i \(0.188601\pi\)
−0.829543 + 0.558443i \(0.811399\pi\)
\(294\) 0 0
\(295\) 1.26857e6 0.848711
\(296\) 0 0
\(297\) −821699. −0.540532
\(298\) 0 0
\(299\) 203988.i 0.131955i
\(300\) 0 0
\(301\) 1.15781e6i 0.736583i
\(302\) 0 0
\(303\) 4.03283e6 2.52350
\(304\) 0 0
\(305\) −516657. −0.318018
\(306\) 0 0
\(307\) 1.40107e6i 0.848423i 0.905563 + 0.424212i \(0.139449\pi\)
−0.905563 + 0.424212i \(0.860551\pi\)
\(308\) 0 0
\(309\) 1.68501e6i 1.00394i
\(310\) 0 0
\(311\) −1.24137e6 −0.727783 −0.363891 0.931441i \(-0.618552\pi\)
−0.363891 + 0.931441i \(0.618552\pi\)
\(312\) 0 0
\(313\) −2.66880e6 −1.53977 −0.769885 0.638183i \(-0.779686\pi\)
−0.769885 + 0.638183i \(0.779686\pi\)
\(314\) 0 0
\(315\) − 426855.i − 0.242384i
\(316\) 0 0
\(317\) − 2.96026e6i − 1.65456i −0.561793 0.827278i \(-0.689888\pi\)
0.561793 0.827278i \(-0.310112\pi\)
\(318\) 0 0
\(319\) −1.14608e6 −0.630579
\(320\) 0 0
\(321\) 2.21778e6 1.20131
\(322\) 0 0
\(323\) − 105518.i − 0.0562759i
\(324\) 0 0
\(325\) 114902.i 0.0603421i
\(326\) 0 0
\(327\) −3.51450e6 −1.81758
\(328\) 0 0
\(329\) 57971.0 0.0295271
\(330\) 0 0
\(331\) 1.50180e6i 0.753430i 0.926329 + 0.376715i \(0.122946\pi\)
−0.926329 + 0.376715i \(0.877054\pi\)
\(332\) 0 0
\(333\) − 2.59781e6i − 1.28380i
\(334\) 0 0
\(335\) 1.04801e6 0.510214
\(336\) 0 0
\(337\) −1.61150e6 −0.772957 −0.386478 0.922298i \(-0.626309\pi\)
−0.386478 + 0.922298i \(0.626309\pi\)
\(338\) 0 0
\(339\) − 1.67846e6i − 0.793253i
\(340\) 0 0
\(341\) 4.79186e6i 2.23161i
\(342\) 0 0
\(343\) −2.32570e6 −1.06738
\(344\) 0 0
\(345\) 572238. 0.258839
\(346\) 0 0
\(347\) − 497865.i − 0.221967i −0.993822 0.110983i \(-0.964600\pi\)
0.993822 0.110983i \(-0.0354001\pi\)
\(348\) 0 0
\(349\) − 3.24174e6i − 1.42467i −0.701839 0.712335i \(-0.747638\pi\)
0.701839 0.712335i \(-0.252362\pi\)
\(350\) 0 0
\(351\) 229214. 0.0993053
\(352\) 0 0
\(353\) −436867. −0.186600 −0.0933001 0.995638i \(-0.529742\pi\)
−0.0933001 + 0.995638i \(0.529742\pi\)
\(354\) 0 0
\(355\) − 1.36719e6i − 0.575783i
\(356\) 0 0
\(357\) − 901151.i − 0.374220i
\(358\) 0 0
\(359\) −1.67096e6 −0.684274 −0.342137 0.939650i \(-0.611151\pi\)
−0.342137 + 0.939650i \(0.611151\pi\)
\(360\) 0 0
\(361\) 2.42506e6 0.979388
\(362\) 0 0
\(363\) − 5.63799e6i − 2.24573i
\(364\) 0 0
\(365\) − 630288.i − 0.247632i
\(366\) 0 0
\(367\) 3.06532e6 1.18799 0.593993 0.804471i \(-0.297551\pi\)
0.593993 + 0.804471i \(0.297551\pi\)
\(368\) 0 0
\(369\) 900015. 0.344099
\(370\) 0 0
\(371\) − 1.58890e6i − 0.599325i
\(372\) 0 0
\(373\) − 3.45643e6i − 1.28634i −0.765724 0.643169i \(-0.777619\pi\)
0.765724 0.643169i \(-0.222381\pi\)
\(374\) 0 0
\(375\) 322331. 0.118365
\(376\) 0 0
\(377\) 319701. 0.115849
\(378\) 0 0
\(379\) 1.61905e6i 0.578979i 0.957181 + 0.289489i \(0.0934855\pi\)
−0.957181 + 0.289489i \(0.906514\pi\)
\(380\) 0 0
\(381\) 6.01263e6i 2.12203i
\(382\) 0 0
\(383\) −151934. −0.0529248 −0.0264624 0.999650i \(-0.508424\pi\)
−0.0264624 + 0.999650i \(0.508424\pi\)
\(384\) 0 0
\(385\) −1.54096e6 −0.529833
\(386\) 0 0
\(387\) − 2.26005e6i − 0.767081i
\(388\) 0 0
\(389\) 2.83414e6i 0.949615i 0.880090 + 0.474807i \(0.157482\pi\)
−0.880090 + 0.474807i \(0.842518\pi\)
\(390\) 0 0
\(391\) 518253. 0.171435
\(392\) 0 0
\(393\) −5.70168e6 −1.86218
\(394\) 0 0
\(395\) − 725011.i − 0.233804i
\(396\) 0 0
\(397\) 1.66354e6i 0.529732i 0.964285 + 0.264866i \(0.0853277\pi\)
−0.964285 + 0.264866i \(0.914672\pi\)
\(398\) 0 0
\(399\) −435866. −0.137063
\(400\) 0 0
\(401\) 3.72720e6 1.15750 0.578751 0.815504i \(-0.303540\pi\)
0.578751 + 0.815504i \(0.303540\pi\)
\(402\) 0 0
\(403\) − 1.33669e6i − 0.409986i
\(404\) 0 0
\(405\) − 1.75207e6i − 0.530778i
\(406\) 0 0
\(407\) −9.37818e6 −2.80629
\(408\) 0 0
\(409\) −2.06547e6 −0.610536 −0.305268 0.952267i \(-0.598746\pi\)
−0.305268 + 0.952267i \(0.598746\pi\)
\(410\) 0 0
\(411\) − 7.55743e6i − 2.20683i
\(412\) 0 0
\(413\) − 4.74574e6i − 1.36908i
\(414\) 0 0
\(415\) 2.31698e6 0.660392
\(416\) 0 0
\(417\) −3.35215e6 −0.944023
\(418\) 0 0
\(419\) − 2.02283e6i − 0.562892i −0.959577 0.281446i \(-0.909186\pi\)
0.959577 0.281446i \(-0.0908141\pi\)
\(420\) 0 0
\(421\) 4.21007e6i 1.15767i 0.815445 + 0.578834i \(0.196492\pi\)
−0.815445 + 0.578834i \(0.803508\pi\)
\(422\) 0 0
\(423\) −113160. −0.0307497
\(424\) 0 0
\(425\) 291922. 0.0783960
\(426\) 0 0
\(427\) 1.93282e6i 0.513005i
\(428\) 0 0
\(429\) 2.49949e6i 0.655705i
\(430\) 0 0
\(431\) 3.35990e6 0.871231 0.435616 0.900133i \(-0.356531\pi\)
0.435616 + 0.900133i \(0.356531\pi\)
\(432\) 0 0
\(433\) 1.98488e6 0.508762 0.254381 0.967104i \(-0.418128\pi\)
0.254381 + 0.967104i \(0.418128\pi\)
\(434\) 0 0
\(435\) − 896843.i − 0.227244i
\(436\) 0 0
\(437\) − 250667.i − 0.0627905i
\(438\) 0 0
\(439\) −5.52991e6 −1.36948 −0.684741 0.728786i \(-0.740085\pi\)
−0.684741 + 0.728786i \(0.740085\pi\)
\(440\) 0 0
\(441\) 1.47145e6 0.360287
\(442\) 0 0
\(443\) − 2.55974e6i − 0.619708i −0.950784 0.309854i \(-0.899720\pi\)
0.950784 0.309854i \(-0.100280\pi\)
\(444\) 0 0
\(445\) 1.36610e6i 0.327026i
\(446\) 0 0
\(447\) −6.22152e6 −1.47275
\(448\) 0 0
\(449\) −6.29746e6 −1.47418 −0.737088 0.675796i \(-0.763800\pi\)
−0.737088 + 0.675796i \(0.763800\pi\)
\(450\) 0 0
\(451\) − 3.24908e6i − 0.752176i
\(452\) 0 0
\(453\) − 7.74664e6i − 1.77365i
\(454\) 0 0
\(455\) 429852. 0.0973397
\(456\) 0 0
\(457\) 4.50464e6 1.00895 0.504475 0.863426i \(-0.331686\pi\)
0.504475 + 0.863426i \(0.331686\pi\)
\(458\) 0 0
\(459\) − 582341.i − 0.129017i
\(460\) 0 0
\(461\) 8.91962e6i 1.95476i 0.211486 + 0.977381i \(0.432170\pi\)
−0.211486 + 0.977381i \(0.567830\pi\)
\(462\) 0 0
\(463\) 7.68162e6 1.66533 0.832665 0.553778i \(-0.186814\pi\)
0.832665 + 0.553778i \(0.186814\pi\)
\(464\) 0 0
\(465\) −3.74977e6 −0.804214
\(466\) 0 0
\(467\) 1.20606e6i 0.255903i 0.991780 + 0.127951i \(0.0408402\pi\)
−0.991780 + 0.127951i \(0.959160\pi\)
\(468\) 0 0
\(469\) − 3.92062e6i − 0.823042i
\(470\) 0 0
\(471\) −1.13079e7 −2.34870
\(472\) 0 0
\(473\) −8.15887e6 −1.67678
\(474\) 0 0
\(475\) − 141196.i − 0.0287136i
\(476\) 0 0
\(477\) 3.10155e6i 0.624140i
\(478\) 0 0
\(479\) 1.89691e6 0.377753 0.188876 0.982001i \(-0.439515\pi\)
0.188876 + 0.982001i \(0.439515\pi\)
\(480\) 0 0
\(481\) 2.61605e6 0.515565
\(482\) 0 0
\(483\) − 2.14075e6i − 0.417540i
\(484\) 0 0
\(485\) 1.13857e6i 0.219789i
\(486\) 0 0
\(487\) 2.24462e6 0.428864 0.214432 0.976739i \(-0.431210\pi\)
0.214432 + 0.976739i \(0.431210\pi\)
\(488\) 0 0
\(489\) 392835. 0.0742913
\(490\) 0 0
\(491\) 9.45333e6i 1.76962i 0.465948 + 0.884812i \(0.345714\pi\)
−0.465948 + 0.884812i \(0.654286\pi\)
\(492\) 0 0
\(493\) − 812234.i − 0.150510i
\(494\) 0 0
\(495\) 3.00796e6 0.551771
\(496\) 0 0
\(497\) −5.11469e6 −0.928812
\(498\) 0 0
\(499\) 9.68957e6i 1.74202i 0.491265 + 0.871010i \(0.336535\pi\)
−0.491265 + 0.871010i \(0.663465\pi\)
\(500\) 0 0
\(501\) 5.92811e6i 1.05517i
\(502\) 0 0
\(503\) 4.98436e6 0.878394 0.439197 0.898391i \(-0.355263\pi\)
0.439197 + 0.898391i \(0.355263\pi\)
\(504\) 0 0
\(505\) 4.88729e6 0.852786
\(506\) 0 0
\(507\) 6.96223e6i 1.20290i
\(508\) 0 0
\(509\) 3.72859e6i 0.637896i 0.947772 + 0.318948i \(0.103330\pi\)
−0.947772 + 0.318948i \(0.896670\pi\)
\(510\) 0 0
\(511\) −2.35792e6 −0.399463
\(512\) 0 0
\(513\) −281665. −0.0472542
\(514\) 0 0
\(515\) 2.04203e6i 0.339268i
\(516\) 0 0
\(517\) 408510.i 0.0672165i
\(518\) 0 0
\(519\) 7.26558e6 1.18400
\(520\) 0 0
\(521\) −7.50570e6 −1.21143 −0.605713 0.795683i \(-0.707112\pi\)
−0.605713 + 0.795683i \(0.707112\pi\)
\(522\) 0 0
\(523\) − 8.28837e6i − 1.32500i −0.749063 0.662498i \(-0.769496\pi\)
0.749063 0.662498i \(-0.230504\pi\)
\(524\) 0 0
\(525\) − 1.20584e6i − 0.190938i
\(526\) 0 0
\(527\) −3.39601e6 −0.532651
\(528\) 0 0
\(529\) −5.20519e6 −0.808719
\(530\) 0 0
\(531\) 9.26371e6i 1.42577i
\(532\) 0 0
\(533\) 906334.i 0.138188i
\(534\) 0 0
\(535\) 2.68768e6 0.405970
\(536\) 0 0
\(537\) −1.80735e6 −0.270463
\(538\) 0 0
\(539\) − 5.31198e6i − 0.787561i
\(540\) 0 0
\(541\) − 4.42308e6i − 0.649728i −0.945761 0.324864i \(-0.894682\pi\)
0.945761 0.324864i \(-0.105318\pi\)
\(542\) 0 0
\(543\) 1.10224e7 1.60427
\(544\) 0 0
\(545\) −4.25915e6 −0.614231
\(546\) 0 0
\(547\) 1.60000e6i 0.228640i 0.993444 + 0.114320i \(0.0364689\pi\)
−0.993444 + 0.114320i \(0.963531\pi\)
\(548\) 0 0
\(549\) − 3.77287e6i − 0.534246i
\(550\) 0 0
\(551\) −392859. −0.0551262
\(552\) 0 0
\(553\) −2.71228e6 −0.377156
\(554\) 0 0
\(555\) − 7.33869e6i − 1.01131i
\(556\) 0 0
\(557\) 5.33213e6i 0.728220i 0.931356 + 0.364110i \(0.118627\pi\)
−0.931356 + 0.364110i \(0.881373\pi\)
\(558\) 0 0
\(559\) 2.27592e6 0.308055
\(560\) 0 0
\(561\) 6.35023e6 0.851887
\(562\) 0 0
\(563\) − 5.39829e6i − 0.717770i −0.933382 0.358885i \(-0.883157\pi\)
0.933382 0.358885i \(-0.116843\pi\)
\(564\) 0 0
\(565\) − 2.03409e6i − 0.268070i
\(566\) 0 0
\(567\) −6.55451e6 −0.856215
\(568\) 0 0
\(569\) 1.34873e7 1.74641 0.873204 0.487355i \(-0.162038\pi\)
0.873204 + 0.487355i \(0.162038\pi\)
\(570\) 0 0
\(571\) 6.16933e6i 0.791859i 0.918281 + 0.395929i \(0.129578\pi\)
−0.918281 + 0.395929i \(0.870422\pi\)
\(572\) 0 0
\(573\) − 941145.i − 0.119748i
\(574\) 0 0
\(575\) 693482. 0.0874713
\(576\) 0 0
\(577\) 1.47140e7 1.83989 0.919943 0.392052i \(-0.128235\pi\)
0.919943 + 0.392052i \(0.128235\pi\)
\(578\) 0 0
\(579\) − 544452.i − 0.0674937i
\(580\) 0 0
\(581\) − 8.66786e6i − 1.06530i
\(582\) 0 0
\(583\) 1.11967e7 1.36432
\(584\) 0 0
\(585\) −839072. −0.101370
\(586\) 0 0
\(587\) 1.19357e6i 0.142972i 0.997442 + 0.0714860i \(0.0227741\pi\)
−0.997442 + 0.0714860i \(0.977226\pi\)
\(588\) 0 0
\(589\) 1.64257e6i 0.195091i
\(590\) 0 0
\(591\) −9.97971e6 −1.17530
\(592\) 0 0
\(593\) 9.29056e6 1.08494 0.542469 0.840076i \(-0.317490\pi\)
0.542469 + 0.840076i \(0.317490\pi\)
\(594\) 0 0
\(595\) − 1.09208e6i − 0.126463i
\(596\) 0 0
\(597\) 1.06234e6i 0.121991i
\(598\) 0 0
\(599\) −1.60680e7 −1.82976 −0.914880 0.403725i \(-0.867715\pi\)
−0.914880 + 0.403725i \(0.867715\pi\)
\(600\) 0 0
\(601\) 1.28035e7 1.44591 0.722957 0.690893i \(-0.242782\pi\)
0.722957 + 0.690893i \(0.242782\pi\)
\(602\) 0 0
\(603\) 7.65306e6i 0.857120i
\(604\) 0 0
\(605\) − 6.83255e6i − 0.758918i
\(606\) 0 0
\(607\) 8.81825e6 0.971428 0.485714 0.874118i \(-0.338560\pi\)
0.485714 + 0.874118i \(0.338560\pi\)
\(608\) 0 0
\(609\) −3.35510e6 −0.366575
\(610\) 0 0
\(611\) − 113954.i − 0.0123489i
\(612\) 0 0
\(613\) − 1.50578e7i − 1.61849i −0.587468 0.809247i \(-0.699875\pi\)
0.587468 0.809247i \(-0.300125\pi\)
\(614\) 0 0
\(615\) 2.54250e6 0.271065
\(616\) 0 0
\(617\) 5.92373e6 0.626445 0.313222 0.949680i \(-0.398592\pi\)
0.313222 + 0.949680i \(0.398592\pi\)
\(618\) 0 0
\(619\) 1.67289e6i 0.175485i 0.996143 + 0.0877426i \(0.0279653\pi\)
−0.996143 + 0.0877426i \(0.972035\pi\)
\(620\) 0 0
\(621\) − 1.38340e6i − 0.143952i
\(622\) 0 0
\(623\) 5.11061e6 0.527536
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 3.07146e6i − 0.312016i
\(628\) 0 0
\(629\) − 6.64636e6i − 0.669818i
\(630\) 0 0
\(631\) −1.34503e7 −1.34480 −0.672402 0.740186i \(-0.734737\pi\)
−0.672402 + 0.740186i \(0.734737\pi\)
\(632\) 0 0
\(633\) −1.47861e7 −1.46671
\(634\) 0 0
\(635\) 7.28657e6i 0.717115i
\(636\) 0 0
\(637\) 1.48178e6i 0.144689i
\(638\) 0 0
\(639\) 9.98389e6 0.967270
\(640\) 0 0
\(641\) 4.66915e6 0.448841 0.224421 0.974492i \(-0.427951\pi\)
0.224421 + 0.974492i \(0.427951\pi\)
\(642\) 0 0
\(643\) 1.53466e7i 1.46381i 0.681405 + 0.731906i \(0.261369\pi\)
−0.681405 + 0.731906i \(0.738631\pi\)
\(644\) 0 0
\(645\) − 6.38454e6i − 0.604269i
\(646\) 0 0
\(647\) 1.28923e7 1.21079 0.605395 0.795925i \(-0.293015\pi\)
0.605395 + 0.795925i \(0.293015\pi\)
\(648\) 0 0
\(649\) 3.34423e7 3.11662
\(650\) 0 0
\(651\) 1.40279e7i 1.29730i
\(652\) 0 0
\(653\) − 6.98908e6i − 0.641412i −0.947179 0.320706i \(-0.896080\pi\)
0.947179 0.320706i \(-0.103920\pi\)
\(654\) 0 0
\(655\) −6.90974e6 −0.629301
\(656\) 0 0
\(657\) 4.60267e6 0.416003
\(658\) 0 0
\(659\) − 5.92616e6i − 0.531569i −0.964032 0.265785i \(-0.914369\pi\)
0.964032 0.265785i \(-0.0856310\pi\)
\(660\) 0 0
\(661\) 6.41040e6i 0.570665i 0.958429 + 0.285333i \(0.0921040\pi\)
−0.958429 + 0.285333i \(0.907896\pi\)
\(662\) 0 0
\(663\) −1.77140e6 −0.156507
\(664\) 0 0
\(665\) −528216. −0.0463188
\(666\) 0 0
\(667\) − 1.92952e6i − 0.167933i
\(668\) 0 0
\(669\) 1.54305e7i 1.33296i
\(670\) 0 0
\(671\) −1.36202e7 −1.16782
\(672\) 0 0
\(673\) 1.44237e7 1.22755 0.613775 0.789481i \(-0.289650\pi\)
0.613775 + 0.789481i \(0.289650\pi\)
\(674\) 0 0
\(675\) − 779240.i − 0.0658282i
\(676\) 0 0
\(677\) − 2.61864e6i − 0.219585i −0.993955 0.109793i \(-0.964981\pi\)
0.993955 0.109793i \(-0.0350187\pi\)
\(678\) 0 0
\(679\) 4.25941e6 0.354548
\(680\) 0 0
\(681\) −1.23401e7 −1.01965
\(682\) 0 0
\(683\) − 1.73254e7i − 1.42112i −0.703635 0.710561i \(-0.748441\pi\)
0.703635 0.710561i \(-0.251559\pi\)
\(684\) 0 0
\(685\) − 9.15867e6i − 0.745772i
\(686\) 0 0
\(687\) 1.35163e7 1.09261
\(688\) 0 0
\(689\) −3.12332e6 −0.250650
\(690\) 0 0
\(691\) 9.93677e6i 0.791681i 0.918319 + 0.395840i \(0.129547\pi\)
−0.918319 + 0.395840i \(0.870453\pi\)
\(692\) 0 0
\(693\) − 1.12528e7i − 0.890079i
\(694\) 0 0
\(695\) −4.06239e6 −0.319021
\(696\) 0 0
\(697\) 2.30264e6 0.179533
\(698\) 0 0
\(699\) 3.52348e6i 0.272759i
\(700\) 0 0
\(701\) 2.70085e6i 0.207590i 0.994599 + 0.103795i \(0.0330985\pi\)
−0.994599 + 0.103795i \(0.966901\pi\)
\(702\) 0 0
\(703\) −3.21469e6 −0.245330
\(704\) 0 0
\(705\) −319670. −0.0242231
\(706\) 0 0
\(707\) − 1.82835e7i − 1.37566i
\(708\) 0 0
\(709\) 1.98847e6i 0.148561i 0.997237 + 0.0742804i \(0.0236660\pi\)
−0.997237 + 0.0742804i \(0.976334\pi\)
\(710\) 0 0
\(711\) 5.29438e6 0.392773
\(712\) 0 0
\(713\) −8.06748e6 −0.594312
\(714\) 0 0
\(715\) 3.02908e6i 0.221587i
\(716\) 0 0
\(717\) 2.61825e7i 1.90201i
\(718\) 0 0
\(719\) −5.01212e6 −0.361576 −0.180788 0.983522i \(-0.557865\pi\)
−0.180788 + 0.983522i \(0.557865\pi\)
\(720\) 0 0
\(721\) 7.63926e6 0.547284
\(722\) 0 0
\(723\) 5.65836e6i 0.402573i
\(724\) 0 0
\(725\) − 1.08686e6i − 0.0767945i
\(726\) 0 0
\(727\) 1.08406e7 0.760709 0.380355 0.924841i \(-0.375802\pi\)
0.380355 + 0.924841i \(0.375802\pi\)
\(728\) 0 0
\(729\) 6.54445e6 0.456094
\(730\) 0 0
\(731\) − 5.78222e6i − 0.400222i
\(732\) 0 0
\(733\) − 9.18857e6i − 0.631667i −0.948815 0.315833i \(-0.897716\pi\)
0.948815 0.315833i \(-0.102284\pi\)
\(734\) 0 0
\(735\) 4.15677e6 0.283817
\(736\) 0 0
\(737\) 2.76278e7 1.87360
\(738\) 0 0
\(739\) 6.19968e6i 0.417598i 0.977959 + 0.208799i \(0.0669554\pi\)
−0.977959 + 0.208799i \(0.933045\pi\)
\(740\) 0 0
\(741\) 856786.i 0.0573227i
\(742\) 0 0
\(743\) 7.11862e6 0.473068 0.236534 0.971623i \(-0.423988\pi\)
0.236534 + 0.971623i \(0.423988\pi\)
\(744\) 0 0
\(745\) −7.53972e6 −0.497696
\(746\) 0 0
\(747\) 1.69197e7i 1.10941i
\(748\) 0 0
\(749\) − 1.00547e7i − 0.654882i
\(750\) 0 0
\(751\) 7.67860e6 0.496801 0.248400 0.968657i \(-0.420095\pi\)
0.248400 + 0.968657i \(0.420095\pi\)
\(752\) 0 0
\(753\) −1.77694e7 −1.14205
\(754\) 0 0
\(755\) − 9.38797e6i − 0.599383i
\(756\) 0 0
\(757\) 5.96101e6i 0.378077i 0.981970 + 0.189039i \(0.0605371\pi\)
−0.981970 + 0.189039i \(0.939463\pi\)
\(758\) 0 0
\(759\) 1.50854e7 0.950504
\(760\) 0 0
\(761\) −2.00820e7 −1.25703 −0.628513 0.777799i \(-0.716336\pi\)
−0.628513 + 0.777799i \(0.716336\pi\)
\(762\) 0 0
\(763\) 1.59335e7i 0.990834i
\(764\) 0 0
\(765\) 2.13175e6i 0.131699i
\(766\) 0 0
\(767\) −9.32875e6 −0.572579
\(768\) 0 0
\(769\) −2.98669e7 −1.82127 −0.910635 0.413212i \(-0.864407\pi\)
−0.910635 + 0.413212i \(0.864407\pi\)
\(770\) 0 0
\(771\) 1.08260e7i 0.655892i
\(772\) 0 0
\(773\) 1.24588e7i 0.749944i 0.927036 + 0.374972i \(0.122348\pi\)
−0.927036 + 0.374972i \(0.877652\pi\)
\(774\) 0 0
\(775\) −4.54425e6 −0.271774
\(776\) 0 0
\(777\) −2.74542e7 −1.63138
\(778\) 0 0
\(779\) − 1.11373e6i − 0.0657564i
\(780\) 0 0
\(781\) − 3.60422e7i − 2.11438i
\(782\) 0 0
\(783\) −2.16813e6 −0.126381
\(784\) 0 0
\(785\) −1.37037e7 −0.793715
\(786\) 0 0
\(787\) 1.73472e7i 0.998372i 0.866495 + 0.499186i \(0.166368\pi\)
−0.866495 + 0.499186i \(0.833632\pi\)
\(788\) 0 0
\(789\) 4.20081e7i 2.40238i
\(790\) 0 0
\(791\) −7.60955e6 −0.432432
\(792\) 0 0
\(793\) 3.79936e6 0.214550
\(794\) 0 0
\(795\) 8.76172e6i 0.491667i
\(796\) 0 0
\(797\) 6.44517e6i 0.359408i 0.983721 + 0.179704i \(0.0575141\pi\)
−0.983721 + 0.179704i \(0.942486\pi\)
\(798\) 0 0
\(799\) −289512. −0.0160435
\(800\) 0 0
\(801\) −9.97592e6 −0.549379
\(802\) 0 0
\(803\) − 1.66158e7i − 0.909351i
\(804\) 0 0
\(805\) − 2.59433e6i − 0.141103i
\(806\) 0 0
\(807\) 9.84013e6 0.531884
\(808\) 0 0
\(809\) 1.85411e6 0.0996010 0.0498005 0.998759i \(-0.484141\pi\)
0.0498005 + 0.998759i \(0.484141\pi\)
\(810\) 0 0
\(811\) 1.23842e7i 0.661176i 0.943775 + 0.330588i \(0.107247\pi\)
−0.943775 + 0.330588i \(0.892753\pi\)
\(812\) 0 0
\(813\) 7.22659e6i 0.383449i
\(814\) 0 0
\(815\) 476067. 0.0251058
\(816\) 0 0
\(817\) −2.79673e6 −0.146587
\(818\) 0 0
\(819\) 3.13898e6i 0.163523i
\(820\) 0 0
\(821\) 1.24225e7i 0.643207i 0.946874 + 0.321603i \(0.104222\pi\)
−0.946874 + 0.321603i \(0.895778\pi\)
\(822\) 0 0
\(823\) 2.32448e7 1.19626 0.598132 0.801398i \(-0.295910\pi\)
0.598132 + 0.801398i \(0.295910\pi\)
\(824\) 0 0
\(825\) 8.49734e6 0.434658
\(826\) 0 0
\(827\) − 2.28900e7i − 1.16381i −0.813257 0.581904i \(-0.802308\pi\)
0.813257 0.581904i \(-0.197692\pi\)
\(828\) 0 0
\(829\) − 3.27454e7i − 1.65487i −0.561562 0.827435i \(-0.689799\pi\)
0.561562 0.827435i \(-0.310201\pi\)
\(830\) 0 0
\(831\) −4.30770e7 −2.16393
\(832\) 0 0
\(833\) 3.76462e6 0.187979
\(834\) 0 0
\(835\) 7.18414e6i 0.356581i
\(836\) 0 0
\(837\) 9.06513e6i 0.447260i
\(838\) 0 0
\(839\) 2.43580e7 1.19464 0.597319 0.802003i \(-0.296232\pi\)
0.597319 + 0.802003i \(0.296232\pi\)
\(840\) 0 0
\(841\) 1.74871e7 0.852565
\(842\) 0 0
\(843\) − 2.55960e7i − 1.24052i
\(844\) 0 0
\(845\) 8.43736e6i 0.406504i
\(846\) 0 0
\(847\) −2.55607e7 −1.22423
\(848\) 0 0
\(849\) −3.82965e7 −1.82343
\(850\) 0 0
\(851\) − 1.57889e7i − 0.747358i
\(852\) 0 0
\(853\) 4.14097e7i 1.94863i 0.225192 + 0.974314i \(0.427699\pi\)
−0.225192 + 0.974314i \(0.572301\pi\)
\(854\) 0 0
\(855\) 1.03108e6 0.0482367
\(856\) 0 0
\(857\) −6.65781e6 −0.309656 −0.154828 0.987941i \(-0.549482\pi\)
−0.154828 + 0.987941i \(0.549482\pi\)
\(858\) 0 0
\(859\) − 1.99635e7i − 0.923111i −0.887111 0.461555i \(-0.847292\pi\)
0.887111 0.461555i \(-0.152708\pi\)
\(860\) 0 0
\(861\) − 9.51153e6i − 0.437263i
\(862\) 0 0
\(863\) −2.53117e7 −1.15690 −0.578448 0.815719i \(-0.696341\pi\)
−0.578448 + 0.815719i \(0.696341\pi\)
\(864\) 0 0
\(865\) 8.80498e6 0.400118
\(866\) 0 0
\(867\) − 2.47900e7i − 1.12003i
\(868\) 0 0
\(869\) − 1.91129e7i − 0.858572i
\(870\) 0 0
\(871\) −7.70679e6 −0.344214
\(872\) 0 0
\(873\) −8.31439e6 −0.369228
\(874\) 0 0
\(875\) − 1.46133e6i − 0.0645252i
\(876\) 0 0
\(877\) − 3.34243e7i − 1.46745i −0.679446 0.733726i \(-0.737780\pi\)
0.679446 0.733726i \(-0.262220\pi\)
\(878\) 0 0
\(879\) −3.38578e7 −1.47804
\(880\) 0 0
\(881\) −1.99218e7 −0.864745 −0.432373 0.901695i \(-0.642323\pi\)
−0.432373 + 0.901695i \(0.642323\pi\)
\(882\) 0 0
\(883\) − 3.47030e7i − 1.49784i −0.662661 0.748920i \(-0.730573\pi\)
0.662661 0.748920i \(-0.269427\pi\)
\(884\) 0 0
\(885\) 2.61695e7i 1.12315i
\(886\) 0 0
\(887\) −1.55960e7 −0.665586 −0.332793 0.943000i \(-0.607991\pi\)
−0.332793 + 0.943000i \(0.607991\pi\)
\(888\) 0 0
\(889\) 2.72592e7 1.15680
\(890\) 0 0
\(891\) − 4.61883e7i − 1.94912i
\(892\) 0 0
\(893\) 140031.i 0.00587617i
\(894\) 0 0
\(895\) −2.19029e6 −0.0913996
\(896\) 0 0
\(897\) −4.20810e6 −0.174624
\(898\) 0 0
\(899\) 1.26438e7i 0.521769i
\(900\) 0 0
\(901\) 7.93513e6i 0.325643i
\(902\) 0 0
\(903\) −2.38847e7 −0.974765
\(904\) 0 0
\(905\) 1.33578e7 0.542144
\(906\) 0 0
\(907\) 1.98603e7i 0.801620i 0.916161 + 0.400810i \(0.131271\pi\)
−0.916161 + 0.400810i \(0.868729\pi\)
\(908\) 0 0
\(909\) 3.56894e7i 1.43261i
\(910\) 0 0
\(911\) 1.26907e7 0.506627 0.253313 0.967384i \(-0.418480\pi\)
0.253313 + 0.967384i \(0.418480\pi\)
\(912\) 0 0
\(913\) 6.10806e7 2.42508
\(914\) 0 0
\(915\) − 1.06582e7i − 0.420853i
\(916\) 0 0
\(917\) 2.58494e7i 1.01514i
\(918\) 0 0
\(919\) 1.26560e7 0.494318 0.247159 0.968975i \(-0.420503\pi\)
0.247159 + 0.968975i \(0.420503\pi\)
\(920\) 0 0
\(921\) −2.89028e7 −1.12277
\(922\) 0 0
\(923\) 1.00540e7i 0.388449i
\(924\) 0 0
\(925\) − 8.89359e6i − 0.341761i
\(926\) 0 0
\(927\) −1.49119e7 −0.569944
\(928\) 0 0
\(929\) −3.77116e7 −1.43363 −0.716813 0.697265i \(-0.754400\pi\)
−0.716813 + 0.697265i \(0.754400\pi\)
\(930\) 0 0
\(931\) − 1.82086e6i − 0.0688498i
\(932\) 0 0
\(933\) − 2.56085e7i − 0.963119i
\(934\) 0 0
\(935\) 7.69569e6 0.287885
\(936\) 0 0
\(937\) −1.13200e7 −0.421208 −0.210604 0.977572i \(-0.567543\pi\)
−0.210604 + 0.977572i \(0.567543\pi\)
\(938\) 0 0
\(939\) − 5.50551e7i − 2.03767i
\(940\) 0 0
\(941\) 1.30012e7i 0.478641i 0.970941 + 0.239320i \(0.0769246\pi\)
−0.970941 + 0.239320i \(0.923075\pi\)
\(942\) 0 0
\(943\) 5.47009e6 0.200316
\(944\) 0 0
\(945\) −2.91515e6 −0.106189
\(946\) 0 0
\(947\) − 2.89411e7i − 1.04867i −0.851511 0.524337i \(-0.824313\pi\)
0.851511 0.524337i \(-0.175687\pi\)
\(948\) 0 0
\(949\) 4.63498e6i 0.167064i
\(950\) 0 0
\(951\) 6.10676e7 2.18957
\(952\) 0 0
\(953\) −2.70255e6 −0.0963922 −0.0481961 0.998838i \(-0.515347\pi\)
−0.0481961 + 0.998838i \(0.515347\pi\)
\(954\) 0 0
\(955\) − 1.14055e6i − 0.0404675i
\(956\) 0 0
\(957\) − 2.36427e7i − 0.834484i
\(958\) 0 0
\(959\) −3.42627e7 −1.20303
\(960\) 0 0
\(961\) 2.42355e7 0.846532
\(962\) 0 0
\(963\) 1.96267e7i 0.681997i
\(964\) 0 0
\(965\) − 659809.i − 0.0228087i
\(966\) 0 0
\(967\) −2.39559e7 −0.823847 −0.411923 0.911218i \(-0.635143\pi\)
−0.411923 + 0.911218i \(0.635143\pi\)
\(968\) 0 0
\(969\) 2.17676e6 0.0744733
\(970\) 0 0
\(971\) − 1.62085e7i − 0.551689i −0.961202 0.275844i \(-0.911043\pi\)
0.961202 0.275844i \(-0.0889574\pi\)
\(972\) 0 0
\(973\) 1.51975e7i 0.514623i
\(974\) 0 0
\(975\) −2.37034e6 −0.0798544
\(976\) 0 0
\(977\) −2.57236e7 −0.862175 −0.431087 0.902310i \(-0.641870\pi\)
−0.431087 + 0.902310i \(0.641870\pi\)
\(978\) 0 0
\(979\) 3.60134e7i 1.20090i
\(980\) 0 0
\(981\) − 3.11023e7i − 1.03186i
\(982\) 0 0
\(983\) −5.83343e6 −0.192548 −0.0962742 0.995355i \(-0.530693\pi\)
−0.0962742 + 0.995355i \(0.530693\pi\)
\(984\) 0 0
\(985\) −1.20942e7 −0.397179
\(986\) 0 0
\(987\) 1.19589e6i 0.0390750i
\(988\) 0 0
\(989\) − 1.37361e7i − 0.446553i
\(990\) 0 0
\(991\) 4.31418e7 1.39545 0.697725 0.716366i \(-0.254196\pi\)
0.697725 + 0.716366i \(0.254196\pi\)
\(992\) 0 0
\(993\) −3.09809e7 −0.997059
\(994\) 0 0
\(995\) 1.28743e6i 0.0412254i
\(996\) 0 0
\(997\) − 4.95272e7i − 1.57800i −0.614396 0.788998i \(-0.710600\pi\)
0.614396 0.788998i \(-0.289400\pi\)
\(998\) 0 0
\(999\) −1.77414e7 −0.562438
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 320.6.d.c.161.11 yes 12
4.3 odd 2 320.6.d.d.161.2 yes 12
8.3 odd 2 320.6.d.d.161.11 yes 12
8.5 even 2 inner 320.6.d.c.161.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.c.161.2 12 8.5 even 2 inner
320.6.d.c.161.11 yes 12 1.1 even 1 trivial
320.6.d.d.161.2 yes 12 4.3 odd 2
320.6.d.d.161.11 yes 12 8.3 odd 2