Properties

Label 588.3.d.c.97.4
Level $588$
Weight $3$
Character 588.97
Analytic conductor $16.022$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.4
Root \(1.60021 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 588.97
Dual form 588.3.d.c.97.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} +5.37964i q^{5} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +5.37964i q^{5} -3.00000 q^{9} -8.59159 q^{11} -21.0158i q^{13} +9.31781 q^{15} -5.48622i q^{17} -7.24098i q^{19} +28.0556 q^{23} -3.94054 q^{25} +5.19615i q^{27} +40.3447 q^{29} -40.5101i q^{31} +14.8811i q^{33} +66.6370 q^{37} -36.4005 q^{39} +33.6357i q^{41} +0.932907 q^{43} -16.1389i q^{45} -85.6544i q^{47} -9.50241 q^{51} -44.5954 q^{53} -46.2197i q^{55} -12.5418 q^{57} -63.6950i q^{59} +32.0084i q^{61} +113.058 q^{65} -47.7341 q^{67} -48.5938i q^{69} +14.9676 q^{71} -140.298i q^{73} +6.82521i q^{75} -122.307 q^{79} +9.00000 q^{81} +33.1852i q^{83} +29.5139 q^{85} -69.8791i q^{87} +36.1246i q^{89} -70.1655 q^{93} +38.9539 q^{95} +16.2175i q^{97} +25.7748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9} + O(q^{10}) \) \( 8 q - 24 q^{9} - 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} - 112 q^{43} - 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} - 64 q^{67} + 224 q^{71} - 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} - 272 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) 5.37964i 1.07593i 0.842968 + 0.537964i \(0.180806\pi\)
−0.842968 + 0.537964i \(0.819194\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) −8.59159 −0.781054 −0.390527 0.920592i \(-0.627707\pi\)
−0.390527 + 0.920592i \(0.627707\pi\)
\(12\) 0 0
\(13\) − 21.0158i − 1.61660i −0.588769 0.808301i \(-0.700387\pi\)
0.588769 0.808301i \(-0.299613\pi\)
\(14\) 0 0
\(15\) 9.31781 0.621187
\(16\) 0 0
\(17\) − 5.48622i − 0.322719i −0.986896 0.161359i \(-0.948412\pi\)
0.986896 0.161359i \(-0.0515878\pi\)
\(18\) 0 0
\(19\) − 7.24098i − 0.381104i −0.981677 0.190552i \(-0.938972\pi\)
0.981677 0.190552i \(-0.0610278\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 28.0556 1.21981 0.609905 0.792474i \(-0.291208\pi\)
0.609905 + 0.792474i \(0.291208\pi\)
\(24\) 0 0
\(25\) −3.94054 −0.157621
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 40.3447 1.39120 0.695599 0.718431i \(-0.255139\pi\)
0.695599 + 0.718431i \(0.255139\pi\)
\(30\) 0 0
\(31\) − 40.5101i − 1.30678i −0.757023 0.653388i \(-0.773347\pi\)
0.757023 0.653388i \(-0.226653\pi\)
\(32\) 0 0
\(33\) 14.8811i 0.450941i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 66.6370 1.80100 0.900500 0.434856i \(-0.143201\pi\)
0.900500 + 0.434856i \(0.143201\pi\)
\(38\) 0 0
\(39\) −36.4005 −0.933346
\(40\) 0 0
\(41\) 33.6357i 0.820383i 0.911999 + 0.410191i \(0.134538\pi\)
−0.911999 + 0.410191i \(0.865462\pi\)
\(42\) 0 0
\(43\) 0.932907 0.0216955 0.0108478 0.999941i \(-0.496547\pi\)
0.0108478 + 0.999941i \(0.496547\pi\)
\(44\) 0 0
\(45\) − 16.1389i − 0.358643i
\(46\) 0 0
\(47\) − 85.6544i − 1.82243i −0.411927 0.911217i \(-0.635144\pi\)
0.411927 0.911217i \(-0.364856\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −9.50241 −0.186322
\(52\) 0 0
\(53\) −44.5954 −0.841422 −0.420711 0.907195i \(-0.638219\pi\)
−0.420711 + 0.907195i \(0.638219\pi\)
\(54\) 0 0
\(55\) − 46.2197i − 0.840358i
\(56\) 0 0
\(57\) −12.5418 −0.220031
\(58\) 0 0
\(59\) − 63.6950i − 1.07958i −0.841801 0.539788i \(-0.818504\pi\)
0.841801 0.539788i \(-0.181496\pi\)
\(60\) 0 0
\(61\) 32.0084i 0.524727i 0.964969 + 0.262364i \(0.0845020\pi\)
−0.964969 + 0.262364i \(0.915498\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 113.058 1.73935
\(66\) 0 0
\(67\) −47.7341 −0.712450 −0.356225 0.934400i \(-0.615936\pi\)
−0.356225 + 0.934400i \(0.615936\pi\)
\(68\) 0 0
\(69\) − 48.5938i − 0.704258i
\(70\) 0 0
\(71\) 14.9676 0.210811 0.105405 0.994429i \(-0.466386\pi\)
0.105405 + 0.994429i \(0.466386\pi\)
\(72\) 0 0
\(73\) − 140.298i − 1.92188i −0.276750 0.960942i \(-0.589258\pi\)
0.276750 0.960942i \(-0.410742\pi\)
\(74\) 0 0
\(75\) 6.82521i 0.0910028i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −122.307 −1.54820 −0.774098 0.633066i \(-0.781796\pi\)
−0.774098 + 0.633066i \(0.781796\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 33.1852i 0.399822i 0.979814 + 0.199911i \(0.0640652\pi\)
−0.979814 + 0.199911i \(0.935935\pi\)
\(84\) 0 0
\(85\) 29.5139 0.347222
\(86\) 0 0
\(87\) − 69.8791i − 0.803208i
\(88\) 0 0
\(89\) 36.1246i 0.405894i 0.979190 + 0.202947i \(0.0650519\pi\)
−0.979190 + 0.202947i \(0.934948\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −70.1655 −0.754468
\(94\) 0 0
\(95\) 38.9539 0.410041
\(96\) 0 0
\(97\) 16.2175i 0.167191i 0.996500 + 0.0835956i \(0.0266404\pi\)
−0.996500 + 0.0835956i \(0.973360\pi\)
\(98\) 0 0
\(99\) 25.7748 0.260351
\(100\) 0 0
\(101\) 119.464i 1.18281i 0.806373 + 0.591407i \(0.201427\pi\)
−0.806373 + 0.591407i \(0.798573\pi\)
\(102\) 0 0
\(103\) − 8.93188i − 0.0867173i −0.999060 0.0433586i \(-0.986194\pi\)
0.999060 0.0433586i \(-0.0138058\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 173.856 1.62482 0.812410 0.583086i \(-0.198155\pi\)
0.812410 + 0.583086i \(0.198155\pi\)
\(108\) 0 0
\(109\) −160.315 −1.47078 −0.735388 0.677646i \(-0.763000\pi\)
−0.735388 + 0.677646i \(0.763000\pi\)
\(110\) 0 0
\(111\) − 115.419i − 1.03981i
\(112\) 0 0
\(113\) −81.4420 −0.720725 −0.360363 0.932812i \(-0.617347\pi\)
−0.360363 + 0.932812i \(0.617347\pi\)
\(114\) 0 0
\(115\) 150.929i 1.31243i
\(116\) 0 0
\(117\) 63.0475i 0.538867i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −47.1846 −0.389955
\(122\) 0 0
\(123\) 58.2587 0.473648
\(124\) 0 0
\(125\) 113.292i 0.906339i
\(126\) 0 0
\(127\) 117.172 0.922613 0.461307 0.887241i \(-0.347381\pi\)
0.461307 + 0.887241i \(0.347381\pi\)
\(128\) 0 0
\(129\) − 1.61584i − 0.0125259i
\(130\) 0 0
\(131\) − 244.150i − 1.86374i −0.362795 0.931869i \(-0.618177\pi\)
0.362795 0.931869i \(-0.381823\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −27.9534 −0.207062
\(136\) 0 0
\(137\) 245.412 1.79133 0.895663 0.444733i \(-0.146701\pi\)
0.895663 + 0.444733i \(0.146701\pi\)
\(138\) 0 0
\(139\) − 17.1371i − 0.123288i −0.998098 0.0616441i \(-0.980366\pi\)
0.998098 0.0616441i \(-0.0196344\pi\)
\(140\) 0 0
\(141\) −148.358 −1.05218
\(142\) 0 0
\(143\) 180.559i 1.26265i
\(144\) 0 0
\(145\) 217.040i 1.49683i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 134.242 0.900952 0.450476 0.892789i \(-0.351254\pi\)
0.450476 + 0.892789i \(0.351254\pi\)
\(150\) 0 0
\(151\) 199.009 1.31794 0.658971 0.752168i \(-0.270992\pi\)
0.658971 + 0.752168i \(0.270992\pi\)
\(152\) 0 0
\(153\) 16.4587i 0.107573i
\(154\) 0 0
\(155\) 217.930 1.40600
\(156\) 0 0
\(157\) 77.0189i 0.490566i 0.969451 + 0.245283i \(0.0788810\pi\)
−0.969451 + 0.245283i \(0.921119\pi\)
\(158\) 0 0
\(159\) 77.2415i 0.485795i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −230.379 −1.41337 −0.706684 0.707529i \(-0.749810\pi\)
−0.706684 + 0.707529i \(0.749810\pi\)
\(164\) 0 0
\(165\) −80.0548 −0.485181
\(166\) 0 0
\(167\) 229.231i 1.37264i 0.727298 + 0.686321i \(0.240776\pi\)
−0.727298 + 0.686321i \(0.759224\pi\)
\(168\) 0 0
\(169\) −272.665 −1.61340
\(170\) 0 0
\(171\) 21.7230i 0.127035i
\(172\) 0 0
\(173\) − 112.473i − 0.650133i −0.945691 0.325066i \(-0.894613\pi\)
0.945691 0.325066i \(-0.105387\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −110.323 −0.623293
\(178\) 0 0
\(179\) 105.494 0.589352 0.294676 0.955597i \(-0.404788\pi\)
0.294676 + 0.955597i \(0.404788\pi\)
\(180\) 0 0
\(181\) 15.2683i 0.0843553i 0.999110 + 0.0421776i \(0.0134295\pi\)
−0.999110 + 0.0421776i \(0.986570\pi\)
\(182\) 0 0
\(183\) 55.4401 0.302951
\(184\) 0 0
\(185\) 358.483i 1.93775i
\(186\) 0 0
\(187\) 47.1353i 0.252061i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −306.186 −1.60307 −0.801534 0.597950i \(-0.795982\pi\)
−0.801534 + 0.597950i \(0.795982\pi\)
\(192\) 0 0
\(193\) −1.84100 −0.00953885 −0.00476943 0.999989i \(-0.501518\pi\)
−0.00476943 + 0.999989i \(0.501518\pi\)
\(194\) 0 0
\(195\) − 195.822i − 1.00421i
\(196\) 0 0
\(197\) −255.334 −1.29611 −0.648056 0.761593i \(-0.724418\pi\)
−0.648056 + 0.761593i \(0.724418\pi\)
\(198\) 0 0
\(199\) − 36.6483i − 0.184162i −0.995752 0.0920812i \(-0.970648\pi\)
0.995752 0.0920812i \(-0.0293519\pi\)
\(200\) 0 0
\(201\) 82.6779i 0.411333i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −180.948 −0.882673
\(206\) 0 0
\(207\) −84.1669 −0.406603
\(208\) 0 0
\(209\) 62.2116i 0.297663i
\(210\) 0 0
\(211\) −126.571 −0.599862 −0.299931 0.953961i \(-0.596964\pi\)
−0.299931 + 0.953961i \(0.596964\pi\)
\(212\) 0 0
\(213\) − 25.9246i − 0.121712i
\(214\) 0 0
\(215\) 5.01871i 0.0233428i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −243.002 −1.10960
\(220\) 0 0
\(221\) −115.297 −0.521708
\(222\) 0 0
\(223\) − 212.193i − 0.951536i −0.879571 0.475768i \(-0.842170\pi\)
0.879571 0.475768i \(-0.157830\pi\)
\(224\) 0 0
\(225\) 11.8216 0.0525405
\(226\) 0 0
\(227\) − 105.489i − 0.464708i −0.972631 0.232354i \(-0.925357\pi\)
0.972631 0.232354i \(-0.0746428\pi\)
\(228\) 0 0
\(229\) 6.99086i 0.0305278i 0.999884 + 0.0152639i \(0.00485883\pi\)
−0.999884 + 0.0152639i \(0.995141\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 336.046 1.44226 0.721128 0.692801i \(-0.243624\pi\)
0.721128 + 0.692801i \(0.243624\pi\)
\(234\) 0 0
\(235\) 460.790 1.96081
\(236\) 0 0
\(237\) 211.843i 0.893851i
\(238\) 0 0
\(239\) −232.382 −0.972309 −0.486155 0.873873i \(-0.661601\pi\)
−0.486155 + 0.873873i \(0.661601\pi\)
\(240\) 0 0
\(241\) − 63.7346i − 0.264459i −0.991219 0.132229i \(-0.957786\pi\)
0.991219 0.132229i \(-0.0422136\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −152.175 −0.616094
\(248\) 0 0
\(249\) 57.4784 0.230837
\(250\) 0 0
\(251\) 415.450i 1.65518i 0.561334 + 0.827589i \(0.310288\pi\)
−0.561334 + 0.827589i \(0.689712\pi\)
\(252\) 0 0
\(253\) −241.043 −0.952737
\(254\) 0 0
\(255\) − 51.1196i − 0.200469i
\(256\) 0 0
\(257\) 265.598i 1.03345i 0.856150 + 0.516727i \(0.172850\pi\)
−0.856150 + 0.516727i \(0.827150\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −121.034 −0.463732
\(262\) 0 0
\(263\) 156.373 0.594575 0.297288 0.954788i \(-0.403918\pi\)
0.297288 + 0.954788i \(0.403918\pi\)
\(264\) 0 0
\(265\) − 239.907i − 0.905310i
\(266\) 0 0
\(267\) 62.5696 0.234343
\(268\) 0 0
\(269\) 332.861i 1.23740i 0.785626 + 0.618701i \(0.212341\pi\)
−0.785626 + 0.618701i \(0.787659\pi\)
\(270\) 0 0
\(271\) − 209.089i − 0.771546i −0.922594 0.385773i \(-0.873935\pi\)
0.922594 0.385773i \(-0.126065\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 33.8555 0.123111
\(276\) 0 0
\(277\) −263.231 −0.950293 −0.475146 0.879907i \(-0.657605\pi\)
−0.475146 + 0.879907i \(0.657605\pi\)
\(278\) 0 0
\(279\) 121.530i 0.435592i
\(280\) 0 0
\(281\) 391.519 1.39331 0.696653 0.717408i \(-0.254672\pi\)
0.696653 + 0.717408i \(0.254672\pi\)
\(282\) 0 0
\(283\) 109.394i 0.386550i 0.981145 + 0.193275i \(0.0619109\pi\)
−0.981145 + 0.193275i \(0.938089\pi\)
\(284\) 0 0
\(285\) − 67.4701i − 0.236737i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 258.901 0.895853
\(290\) 0 0
\(291\) 28.0896 0.0965278
\(292\) 0 0
\(293\) 35.3685i 0.120712i 0.998177 + 0.0603558i \(0.0192235\pi\)
−0.998177 + 0.0603558i \(0.980776\pi\)
\(294\) 0 0
\(295\) 342.656 1.16155
\(296\) 0 0
\(297\) − 44.6432i − 0.150314i
\(298\) 0 0
\(299\) − 589.613i − 1.97195i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 206.918 0.682898
\(304\) 0 0
\(305\) −172.194 −0.564569
\(306\) 0 0
\(307\) 125.621i 0.409189i 0.978847 + 0.204594i \(0.0655875\pi\)
−0.978847 + 0.204594i \(0.934412\pi\)
\(308\) 0 0
\(309\) −15.4705 −0.0500662
\(310\) 0 0
\(311\) − 315.735i − 1.01523i −0.861585 0.507613i \(-0.830528\pi\)
0.861585 0.507613i \(-0.169472\pi\)
\(312\) 0 0
\(313\) − 51.6046i − 0.164871i −0.996596 0.0824355i \(-0.973730\pi\)
0.996596 0.0824355i \(-0.0262698\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.99391 −0.0157537 −0.00787683 0.999969i \(-0.502507\pi\)
−0.00787683 + 0.999969i \(0.502507\pi\)
\(318\) 0 0
\(319\) −346.625 −1.08660
\(320\) 0 0
\(321\) − 301.127i − 0.938091i
\(322\) 0 0
\(323\) −39.7256 −0.122990
\(324\) 0 0
\(325\) 82.8137i 0.254811i
\(326\) 0 0
\(327\) 277.673i 0.849153i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 454.780 1.37396 0.686980 0.726677i \(-0.258936\pi\)
0.686980 + 0.726677i \(0.258936\pi\)
\(332\) 0 0
\(333\) −199.911 −0.600333
\(334\) 0 0
\(335\) − 256.792i − 0.766545i
\(336\) 0 0
\(337\) −183.824 −0.545471 −0.272736 0.962089i \(-0.587928\pi\)
−0.272736 + 0.962089i \(0.587928\pi\)
\(338\) 0 0
\(339\) 141.062i 0.416111i
\(340\) 0 0
\(341\) 348.046i 1.02066i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 261.417 0.757731
\(346\) 0 0
\(347\) 144.776 0.417223 0.208611 0.977999i \(-0.433106\pi\)
0.208611 + 0.977999i \(0.433106\pi\)
\(348\) 0 0
\(349\) 187.069i 0.536015i 0.963417 + 0.268007i \(0.0863651\pi\)
−0.963417 + 0.268007i \(0.913635\pi\)
\(350\) 0 0
\(351\) 109.201 0.311115
\(352\) 0 0
\(353\) − 222.601i − 0.630597i −0.948993 0.315299i \(-0.897895\pi\)
0.948993 0.315299i \(-0.102105\pi\)
\(354\) 0 0
\(355\) 80.5201i 0.226817i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 62.9856 0.175447 0.0877237 0.996145i \(-0.472041\pi\)
0.0877237 + 0.996145i \(0.472041\pi\)
\(360\) 0 0
\(361\) 308.568 0.854759
\(362\) 0 0
\(363\) 81.7261i 0.225141i
\(364\) 0 0
\(365\) 754.750 2.06781
\(366\) 0 0
\(367\) − 547.100i − 1.49074i −0.666653 0.745368i \(-0.732274\pi\)
0.666653 0.745368i \(-0.267726\pi\)
\(368\) 0 0
\(369\) − 100.907i − 0.273461i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −32.4417 −0.0869750 −0.0434875 0.999054i \(-0.513847\pi\)
−0.0434875 + 0.999054i \(0.513847\pi\)
\(374\) 0 0
\(375\) 196.228 0.523275
\(376\) 0 0
\(377\) − 847.878i − 2.24901i
\(378\) 0 0
\(379\) 508.859 1.34263 0.671317 0.741170i \(-0.265729\pi\)
0.671317 + 0.741170i \(0.265729\pi\)
\(380\) 0 0
\(381\) − 202.948i − 0.532671i
\(382\) 0 0
\(383\) 26.2839i 0.0686263i 0.999411 + 0.0343131i \(0.0109244\pi\)
−0.999411 + 0.0343131i \(0.989076\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.79872 −0.00723184
\(388\) 0 0
\(389\) −360.128 −0.925779 −0.462890 0.886416i \(-0.653187\pi\)
−0.462890 + 0.886416i \(0.653187\pi\)
\(390\) 0 0
\(391\) − 153.919i − 0.393656i
\(392\) 0 0
\(393\) −422.880 −1.07603
\(394\) 0 0
\(395\) − 657.970i − 1.66575i
\(396\) 0 0
\(397\) 570.992i 1.43827i 0.694872 + 0.719134i \(0.255461\pi\)
−0.694872 + 0.719134i \(0.744539\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 68.4014 0.170577 0.0852885 0.996356i \(-0.472819\pi\)
0.0852885 + 0.996356i \(0.472819\pi\)
\(402\) 0 0
\(403\) −851.353 −2.11254
\(404\) 0 0
\(405\) 48.4168i 0.119548i
\(406\) 0 0
\(407\) −572.518 −1.40668
\(408\) 0 0
\(409\) − 15.3649i − 0.0375671i −0.999824 0.0187835i \(-0.994021\pi\)
0.999824 0.0187835i \(-0.00597934\pi\)
\(410\) 0 0
\(411\) − 425.066i − 1.03422i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −178.524 −0.430179
\(416\) 0 0
\(417\) −29.6823 −0.0711805
\(418\) 0 0
\(419\) − 366.079i − 0.873696i −0.899535 0.436848i \(-0.856095\pi\)
0.899535 0.436848i \(-0.143905\pi\)
\(420\) 0 0
\(421\) 607.135 1.44213 0.721063 0.692870i \(-0.243654\pi\)
0.721063 + 0.692870i \(0.243654\pi\)
\(422\) 0 0
\(423\) 256.963i 0.607478i
\(424\) 0 0
\(425\) 21.6186i 0.0508674i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 312.738 0.728993
\(430\) 0 0
\(431\) −586.353 −1.36045 −0.680224 0.733004i \(-0.738118\pi\)
−0.680224 + 0.733004i \(0.738118\pi\)
\(432\) 0 0
\(433\) 518.769i 1.19808i 0.800719 + 0.599040i \(0.204451\pi\)
−0.800719 + 0.599040i \(0.795549\pi\)
\(434\) 0 0
\(435\) 375.924 0.864194
\(436\) 0 0
\(437\) − 203.150i − 0.464875i
\(438\) 0 0
\(439\) − 221.524i − 0.504610i −0.967648 0.252305i \(-0.918811\pi\)
0.967648 0.252305i \(-0.0811886\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −226.415 −0.511095 −0.255548 0.966796i \(-0.582256\pi\)
−0.255548 + 0.966796i \(0.582256\pi\)
\(444\) 0 0
\(445\) −194.337 −0.436713
\(446\) 0 0
\(447\) − 232.514i − 0.520165i
\(448\) 0 0
\(449\) −378.422 −0.842810 −0.421405 0.906873i \(-0.638463\pi\)
−0.421405 + 0.906873i \(0.638463\pi\)
\(450\) 0 0
\(451\) − 288.984i − 0.640763i
\(452\) 0 0
\(453\) − 344.694i − 0.760914i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 438.153 0.958760 0.479380 0.877607i \(-0.340862\pi\)
0.479380 + 0.877607i \(0.340862\pi\)
\(458\) 0 0
\(459\) 28.5072 0.0621073
\(460\) 0 0
\(461\) 46.3981i 0.100647i 0.998733 + 0.0503234i \(0.0160252\pi\)
−0.998733 + 0.0503234i \(0.983975\pi\)
\(462\) 0 0
\(463\) 367.455 0.793639 0.396820 0.917897i \(-0.370114\pi\)
0.396820 + 0.917897i \(0.370114\pi\)
\(464\) 0 0
\(465\) − 377.465i − 0.811753i
\(466\) 0 0
\(467\) 529.465i 1.13376i 0.823801 + 0.566879i \(0.191849\pi\)
−0.823801 + 0.566879i \(0.808151\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 133.401 0.283229
\(472\) 0 0
\(473\) −8.01516 −0.0169454
\(474\) 0 0
\(475\) 28.5334i 0.0600702i
\(476\) 0 0
\(477\) 133.786 0.280474
\(478\) 0 0
\(479\) − 289.761i − 0.604928i −0.953161 0.302464i \(-0.902191\pi\)
0.953161 0.302464i \(-0.0978093\pi\)
\(480\) 0 0
\(481\) − 1400.43i − 2.91150i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −87.2445 −0.179886
\(486\) 0 0
\(487\) 439.636 0.902744 0.451372 0.892336i \(-0.350935\pi\)
0.451372 + 0.892336i \(0.350935\pi\)
\(488\) 0 0
\(489\) 399.028i 0.816009i
\(490\) 0 0
\(491\) −320.561 −0.652874 −0.326437 0.945219i \(-0.605848\pi\)
−0.326437 + 0.945219i \(0.605848\pi\)
\(492\) 0 0
\(493\) − 221.340i − 0.448965i
\(494\) 0 0
\(495\) 138.659i 0.280119i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 771.973 1.54704 0.773520 0.633772i \(-0.218494\pi\)
0.773520 + 0.633772i \(0.218494\pi\)
\(500\) 0 0
\(501\) 397.040 0.792496
\(502\) 0 0
\(503\) 101.632i 0.202052i 0.994884 + 0.101026i \(0.0322126\pi\)
−0.994884 + 0.101026i \(0.967787\pi\)
\(504\) 0 0
\(505\) −642.675 −1.27262
\(506\) 0 0
\(507\) 472.270i 0.931499i
\(508\) 0 0
\(509\) 605.407i 1.18941i 0.803946 + 0.594703i \(0.202730\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 37.6253 0.0733436
\(514\) 0 0
\(515\) 48.0503 0.0933016
\(516\) 0 0
\(517\) 735.907i 1.42342i
\(518\) 0 0
\(519\) −194.809 −0.375354
\(520\) 0 0
\(521\) − 383.293i − 0.735688i −0.929888 0.367844i \(-0.880096\pi\)
0.929888 0.367844i \(-0.119904\pi\)
\(522\) 0 0
\(523\) 448.095i 0.856777i 0.903595 + 0.428389i \(0.140919\pi\)
−0.903595 + 0.428389i \(0.859081\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −222.247 −0.421721
\(528\) 0 0
\(529\) 258.119 0.487937
\(530\) 0 0
\(531\) 191.085i 0.359859i
\(532\) 0 0
\(533\) 706.882 1.32623
\(534\) 0 0
\(535\) 935.282i 1.74819i
\(536\) 0 0
\(537\) − 182.721i − 0.340262i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −271.258 −0.501400 −0.250700 0.968065i \(-0.580661\pi\)
−0.250700 + 0.968065i \(0.580661\pi\)
\(542\) 0 0
\(543\) 26.4455 0.0487025
\(544\) 0 0
\(545\) − 862.435i − 1.58245i
\(546\) 0 0
\(547\) −590.544 −1.07961 −0.539803 0.841791i \(-0.681501\pi\)
−0.539803 + 0.841791i \(0.681501\pi\)
\(548\) 0 0
\(549\) − 96.0251i − 0.174909i
\(550\) 0 0
\(551\) − 292.135i − 0.530191i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 620.911 1.11876
\(556\) 0 0
\(557\) 188.295 0.338051 0.169026 0.985612i \(-0.445938\pi\)
0.169026 + 0.985612i \(0.445938\pi\)
\(558\) 0 0
\(559\) − 19.6058i − 0.0350730i
\(560\) 0 0
\(561\) 81.6408 0.145527
\(562\) 0 0
\(563\) 204.157i 0.362623i 0.983426 + 0.181312i \(0.0580342\pi\)
−0.983426 + 0.181312i \(0.941966\pi\)
\(564\) 0 0
\(565\) − 438.129i − 0.775449i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −660.913 −1.16153 −0.580767 0.814070i \(-0.697247\pi\)
−0.580767 + 0.814070i \(0.697247\pi\)
\(570\) 0 0
\(571\) −533.978 −0.935162 −0.467581 0.883950i \(-0.654874\pi\)
−0.467581 + 0.883950i \(0.654874\pi\)
\(572\) 0 0
\(573\) 530.329i 0.925531i
\(574\) 0 0
\(575\) −110.554 −0.192268
\(576\) 0 0
\(577\) − 53.4337i − 0.0926062i −0.998927 0.0463031i \(-0.985256\pi\)
0.998927 0.0463031i \(-0.0147440\pi\)
\(578\) 0 0
\(579\) 3.18870i 0.00550726i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 383.145 0.657196
\(584\) 0 0
\(585\) −339.173 −0.579783
\(586\) 0 0
\(587\) 413.063i 0.703684i 0.936059 + 0.351842i \(0.114445\pi\)
−0.936059 + 0.351842i \(0.885555\pi\)
\(588\) 0 0
\(589\) −293.333 −0.498018
\(590\) 0 0
\(591\) 442.252i 0.748311i
\(592\) 0 0
\(593\) − 122.452i − 0.206496i −0.994656 0.103248i \(-0.967076\pi\)
0.994656 0.103248i \(-0.0329235\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −63.4768 −0.106326
\(598\) 0 0
\(599\) −645.136 −1.07702 −0.538511 0.842619i \(-0.681013\pi\)
−0.538511 + 0.842619i \(0.681013\pi\)
\(600\) 0 0
\(601\) − 683.488i − 1.13725i −0.822596 0.568626i \(-0.807475\pi\)
0.822596 0.568626i \(-0.192525\pi\)
\(602\) 0 0
\(603\) 143.202 0.237483
\(604\) 0 0
\(605\) − 253.836i − 0.419564i
\(606\) 0 0
\(607\) 257.253i 0.423811i 0.977290 + 0.211905i \(0.0679669\pi\)
−0.977290 + 0.211905i \(0.932033\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1800.10 −2.94615
\(612\) 0 0
\(613\) 885.713 1.44488 0.722441 0.691433i \(-0.243020\pi\)
0.722441 + 0.691433i \(0.243020\pi\)
\(614\) 0 0
\(615\) 313.411i 0.509612i
\(616\) 0 0
\(617\) 46.0724 0.0746717 0.0373358 0.999303i \(-0.488113\pi\)
0.0373358 + 0.999303i \(0.488113\pi\)
\(618\) 0 0
\(619\) 1096.33i 1.77112i 0.464522 + 0.885562i \(0.346226\pi\)
−0.464522 + 0.885562i \(0.653774\pi\)
\(620\) 0 0
\(621\) 145.781i 0.234753i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −707.986 −1.13278
\(626\) 0 0
\(627\) 107.754 0.171856
\(628\) 0 0
\(629\) − 365.585i − 0.581217i
\(630\) 0 0
\(631\) 606.319 0.960886 0.480443 0.877026i \(-0.340476\pi\)
0.480443 + 0.877026i \(0.340476\pi\)
\(632\) 0 0
\(633\) 219.227i 0.346330i
\(634\) 0 0
\(635\) 630.343i 0.992666i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −44.9027 −0.0702702
\(640\) 0 0
\(641\) 939.184 1.46519 0.732593 0.680667i \(-0.238310\pi\)
0.732593 + 0.680667i \(0.238310\pi\)
\(642\) 0 0
\(643\) − 992.960i − 1.54426i −0.635464 0.772131i \(-0.719191\pi\)
0.635464 0.772131i \(-0.280809\pi\)
\(644\) 0 0
\(645\) 8.69266 0.0134770
\(646\) 0 0
\(647\) 265.184i 0.409867i 0.978776 + 0.204933i \(0.0656978\pi\)
−0.978776 + 0.204933i \(0.934302\pi\)
\(648\) 0 0
\(649\) 547.241i 0.843206i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1134.14 −1.73682 −0.868409 0.495849i \(-0.834857\pi\)
−0.868409 + 0.495849i \(0.834857\pi\)
\(654\) 0 0
\(655\) 1313.44 2.00525
\(656\) 0 0
\(657\) 420.893i 0.640628i
\(658\) 0 0
\(659\) −924.147 −1.40235 −0.701174 0.712990i \(-0.747340\pi\)
−0.701174 + 0.712990i \(0.747340\pi\)
\(660\) 0 0
\(661\) 234.367i 0.354564i 0.984160 + 0.177282i \(0.0567305\pi\)
−0.984160 + 0.177282i \(0.943269\pi\)
\(662\) 0 0
\(663\) 199.701i 0.301208i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1131.90 1.69700
\(668\) 0 0
\(669\) −367.528 −0.549370
\(670\) 0 0
\(671\) − 275.003i − 0.409840i
\(672\) 0 0
\(673\) −465.127 −0.691125 −0.345563 0.938396i \(-0.612312\pi\)
−0.345563 + 0.938396i \(0.612312\pi\)
\(674\) 0 0
\(675\) − 20.4756i − 0.0303343i
\(676\) 0 0
\(677\) 1295.93i 1.91422i 0.289724 + 0.957110i \(0.406437\pi\)
−0.289724 + 0.957110i \(0.593563\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −182.712 −0.268300
\(682\) 0 0
\(683\) 16.2612 0.0238085 0.0119042 0.999929i \(-0.496211\pi\)
0.0119042 + 0.999929i \(0.496211\pi\)
\(684\) 0 0
\(685\) 1320.23i 1.92734i
\(686\) 0 0
\(687\) 12.1085 0.0176252
\(688\) 0 0
\(689\) 937.209i 1.36025i
\(690\) 0 0
\(691\) − 68.5364i − 0.0991843i −0.998770 0.0495922i \(-0.984208\pi\)
0.998770 0.0495922i \(-0.0157922\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 92.1912 0.132649
\(696\) 0 0
\(697\) 184.533 0.264753
\(698\) 0 0
\(699\) − 582.048i − 0.832687i
\(700\) 0 0
\(701\) 923.360 1.31720 0.658602 0.752491i \(-0.271148\pi\)
0.658602 + 0.752491i \(0.271148\pi\)
\(702\) 0 0
\(703\) − 482.518i − 0.686369i
\(704\) 0 0
\(705\) − 798.111i − 1.13207i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −912.863 −1.28754 −0.643768 0.765221i \(-0.722630\pi\)
−0.643768 + 0.765221i \(0.722630\pi\)
\(710\) 0 0
\(711\) 366.922 0.516065
\(712\) 0 0
\(713\) − 1136.54i − 1.59402i
\(714\) 0 0
\(715\) −971.345 −1.35852
\(716\) 0 0
\(717\) 402.497i 0.561363i
\(718\) 0 0
\(719\) 545.020i 0.758025i 0.925391 + 0.379013i \(0.123736\pi\)
−0.925391 + 0.379013i \(0.876264\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −110.392 −0.152685
\(724\) 0 0
\(725\) −158.980 −0.219283
\(726\) 0 0
\(727\) 750.292i 1.03204i 0.856577 + 0.516019i \(0.172587\pi\)
−0.856577 + 0.516019i \(0.827413\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 5.11813i − 0.00700155i
\(732\) 0 0
\(733\) − 926.599i − 1.26412i −0.774920 0.632059i \(-0.782210\pi\)
0.774920 0.632059i \(-0.217790\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 410.112 0.556461
\(738\) 0 0
\(739\) 488.147 0.660551 0.330275 0.943885i \(-0.392858\pi\)
0.330275 + 0.943885i \(0.392858\pi\)
\(740\) 0 0
\(741\) 263.575i 0.355702i
\(742\) 0 0
\(743\) −1091.72 −1.46935 −0.734674 0.678421i \(-0.762665\pi\)
−0.734674 + 0.678421i \(0.762665\pi\)
\(744\) 0 0
\(745\) 722.173i 0.969360i
\(746\) 0 0
\(747\) − 99.5556i − 0.133274i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −480.443 −0.639738 −0.319869 0.947462i \(-0.603639\pi\)
−0.319869 + 0.947462i \(0.603639\pi\)
\(752\) 0 0
\(753\) 719.580 0.955618
\(754\) 0 0
\(755\) 1070.60i 1.41801i
\(756\) 0 0
\(757\) 941.400 1.24359 0.621796 0.783179i \(-0.286403\pi\)
0.621796 + 0.783179i \(0.286403\pi\)
\(758\) 0 0
\(759\) 417.498i 0.550063i
\(760\) 0 0
\(761\) − 1192.34i − 1.56681i −0.621509 0.783407i \(-0.713480\pi\)
0.621509 0.783407i \(-0.286520\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −88.5417 −0.115741
\(766\) 0 0
\(767\) −1338.60 −1.74524
\(768\) 0 0
\(769\) − 908.294i − 1.18114i −0.806988 0.590568i \(-0.798904\pi\)
0.806988 0.590568i \(-0.201096\pi\)
\(770\) 0 0
\(771\) 460.029 0.596665
\(772\) 0 0
\(773\) 1189.38i 1.53866i 0.638854 + 0.769328i \(0.279409\pi\)
−0.638854 + 0.769328i \(0.720591\pi\)
\(774\) 0 0
\(775\) 159.632i 0.205976i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 243.556 0.312652
\(780\) 0 0
\(781\) −128.595 −0.164654
\(782\) 0 0
\(783\) 209.637i 0.267736i
\(784\) 0 0
\(785\) −414.334 −0.527814
\(786\) 0 0
\(787\) 1057.12i 1.34323i 0.740902 + 0.671613i \(0.234398\pi\)
−0.740902 + 0.671613i \(0.765602\pi\)
\(788\) 0 0
\(789\) − 270.846i − 0.343278i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 672.682 0.848275
\(794\) 0 0
\(795\) −415.531 −0.522681
\(796\) 0 0
\(797\) 1037.94i 1.30231i 0.758945 + 0.651154i \(0.225715\pi\)
−0.758945 + 0.651154i \(0.774285\pi\)
\(798\) 0 0
\(799\) −469.919 −0.588133
\(800\) 0 0
\(801\) − 108.374i − 0.135298i
\(802\) 0 0
\(803\) 1205.38i 1.50109i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 576.533 0.714415
\(808\) 0 0
\(809\) 1087.36 1.34408 0.672041 0.740514i \(-0.265418\pi\)
0.672041 + 0.740514i \(0.265418\pi\)
\(810\) 0 0
\(811\) 926.995i 1.14303i 0.820593 + 0.571514i \(0.193644\pi\)
−0.820593 + 0.571514i \(0.806356\pi\)
\(812\) 0 0
\(813\) −362.153 −0.445452
\(814\) 0 0
\(815\) − 1239.36i − 1.52068i
\(816\) 0 0
\(817\) − 6.75517i − 0.00826826i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −430.300 −0.524117 −0.262058 0.965052i \(-0.584401\pi\)
−0.262058 + 0.965052i \(0.584401\pi\)
\(822\) 0 0
\(823\) −1044.74 −1.26942 −0.634712 0.772749i \(-0.718881\pi\)
−0.634712 + 0.772749i \(0.718881\pi\)
\(824\) 0 0
\(825\) − 58.6394i − 0.0710781i
\(826\) 0 0
\(827\) −610.313 −0.737984 −0.368992 0.929433i \(-0.620297\pi\)
−0.368992 + 0.929433i \(0.620297\pi\)
\(828\) 0 0
\(829\) − 1413.90i − 1.70555i −0.522279 0.852775i \(-0.674918\pi\)
0.522279 0.852775i \(-0.325082\pi\)
\(830\) 0 0
\(831\) 455.930i 0.548652i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1233.18 −1.47687
\(836\) 0 0
\(837\) 210.497 0.251489
\(838\) 0 0
\(839\) 46.3303i 0.0552209i 0.999619 + 0.0276105i \(0.00878980\pi\)
−0.999619 + 0.0276105i \(0.991210\pi\)
\(840\) 0 0
\(841\) 786.696 0.935430
\(842\) 0 0
\(843\) − 678.131i − 0.804426i
\(844\) 0 0
\(845\) − 1466.84i − 1.73591i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 189.475 0.223175
\(850\) 0 0
\(851\) 1869.54 2.19688
\(852\) 0 0
\(853\) 773.111i 0.906343i 0.891423 + 0.453172i \(0.149708\pi\)
−0.891423 + 0.453172i \(0.850292\pi\)
\(854\) 0 0
\(855\) −116.862 −0.136680
\(856\) 0 0
\(857\) 438.159i 0.511271i 0.966773 + 0.255636i \(0.0822847\pi\)
−0.966773 + 0.255636i \(0.917715\pi\)
\(858\) 0 0
\(859\) 72.3419i 0.0842164i 0.999113 + 0.0421082i \(0.0134074\pi\)
−0.999113 + 0.0421082i \(0.986593\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 426.495 0.494200 0.247100 0.968990i \(-0.420522\pi\)
0.247100 + 0.968990i \(0.420522\pi\)
\(864\) 0 0
\(865\) 605.064 0.699496
\(866\) 0 0
\(867\) − 448.430i − 0.517221i
\(868\) 0 0
\(869\) 1050.82 1.20922
\(870\) 0 0
\(871\) 1003.17i 1.15175i
\(872\) 0 0
\(873\) − 48.6526i − 0.0557304i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 416.485 0.474898 0.237449 0.971400i \(-0.423689\pi\)
0.237449 + 0.971400i \(0.423689\pi\)
\(878\) 0 0
\(879\) 61.2600 0.0696929
\(880\) 0 0
\(881\) − 1249.01i − 1.41771i −0.705353 0.708857i \(-0.749211\pi\)
0.705353 0.708857i \(-0.250789\pi\)
\(882\) 0 0
\(883\) 81.5906 0.0924016 0.0462008 0.998932i \(-0.485289\pi\)
0.0462008 + 0.998932i \(0.485289\pi\)
\(884\) 0 0
\(885\) − 593.498i − 0.670619i
\(886\) 0 0
\(887\) − 917.730i − 1.03464i −0.855791 0.517322i \(-0.826929\pi\)
0.855791 0.517322i \(-0.173071\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −77.3243 −0.0867837
\(892\) 0 0
\(893\) −620.222 −0.694537
\(894\) 0 0
\(895\) 567.520i 0.634100i
\(896\) 0 0
\(897\) −1021.24 −1.13850
\(898\) 0 0
\(899\) − 1634.37i − 1.81798i
\(900\) 0 0
\(901\) 244.660i 0.271543i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −82.1380 −0.0907602
\(906\) 0 0
\(907\) −1451.40 −1.60022 −0.800110 0.599853i \(-0.795226\pi\)
−0.800110 + 0.599853i \(0.795226\pi\)
\(908\) 0 0
\(909\) − 358.393i − 0.394272i
\(910\) 0 0
\(911\) −562.064 −0.616975 −0.308487 0.951228i \(-0.599823\pi\)
−0.308487 + 0.951228i \(0.599823\pi\)
\(912\) 0 0
\(913\) − 285.114i − 0.312282i
\(914\) 0 0
\(915\) 298.248i 0.325954i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 538.158 0.585591 0.292795 0.956175i \(-0.405414\pi\)
0.292795 + 0.956175i \(0.405414\pi\)
\(920\) 0 0
\(921\) 217.582 0.236245
\(922\) 0 0
\(923\) − 314.556i − 0.340797i
\(924\) 0 0
\(925\) −262.586 −0.283876
\(926\) 0 0
\(927\) 26.7956i 0.0289058i
\(928\) 0 0
\(929\) 827.080i 0.890291i 0.895458 + 0.445145i \(0.146848\pi\)
−0.895458 + 0.445145i \(0.853152\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −546.869 −0.586141
\(934\) 0 0
\(935\) −253.571 −0.271199
\(936\) 0 0
\(937\) − 1501.10i − 1.60203i −0.598644 0.801016i \(-0.704293\pi\)
0.598644 0.801016i \(-0.295707\pi\)
\(938\) 0 0
\(939\) −89.3818 −0.0951883
\(940\) 0 0
\(941\) − 766.672i − 0.814742i −0.913263 0.407371i \(-0.866446\pi\)
0.913263 0.407371i \(-0.133554\pi\)
\(942\) 0 0
\(943\) 943.671i 1.00071i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1362.23 1.43847 0.719237 0.694765i \(-0.244492\pi\)
0.719237 + 0.694765i \(0.244492\pi\)
\(948\) 0 0
\(949\) −2948.47 −3.10692
\(950\) 0 0
\(951\) 8.64971i 0.00909538i
\(952\) 0 0
\(953\) 1431.37 1.50196 0.750980 0.660325i \(-0.229581\pi\)
0.750980 + 0.660325i \(0.229581\pi\)
\(954\) 0 0
\(955\) − 1647.17i − 1.72479i
\(956\) 0 0
\(957\) 600.373i 0.627348i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −680.067 −0.707666
\(962\) 0 0
\(963\) −521.567 −0.541607
\(964\) 0 0
\(965\) − 9.90391i − 0.0102631i
\(966\) 0 0
\(967\) −426.276 −0.440823 −0.220411 0.975407i \(-0.570740\pi\)
−0.220411 + 0.975407i \(0.570740\pi\)
\(968\) 0 0
\(969\) 68.8068i 0.0710080i
\(970\) 0 0
\(971\) − 1728.41i − 1.78003i −0.455934 0.890013i \(-0.650695\pi\)
0.455934 0.890013i \(-0.349305\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 143.437 0.147115
\(976\) 0 0
\(977\) −712.320 −0.729089 −0.364545 0.931186i \(-0.618775\pi\)
−0.364545 + 0.931186i \(0.618775\pi\)
\(978\) 0 0
\(979\) − 310.368i − 0.317025i
\(980\) 0 0
\(981\) 480.944 0.490259
\(982\) 0 0
\(983\) 1042.56i 1.06059i 0.847812 + 0.530297i \(0.177920\pi\)
−0.847812 + 0.530297i \(0.822080\pi\)
\(984\) 0 0
\(985\) − 1373.61i − 1.39452i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 26.1733 0.0264644
\(990\) 0 0
\(991\) 1397.41 1.41010 0.705052 0.709156i \(-0.250924\pi\)
0.705052 + 0.709156i \(0.250924\pi\)
\(992\) 0 0
\(993\) − 787.703i − 0.793256i
\(994\) 0 0
\(995\) 197.155 0.198146
\(996\) 0 0
\(997\) − 717.962i − 0.720122i −0.932929 0.360061i \(-0.882756\pi\)
0.932929 0.360061i \(-0.117244\pi\)
\(998\) 0 0
\(999\) 346.256i 0.346603i
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 588.3.d.c.97.4 8
3.2 odd 2 1764.3.d.h.685.2 8
4.3 odd 2 2352.3.f.j.97.8 8
7.2 even 3 588.3.m.f.325.4 8
7.3 odd 6 588.3.m.f.313.4 8
7.4 even 3 588.3.m.e.313.1 8
7.5 odd 6 588.3.m.e.325.1 8
7.6 odd 2 inner 588.3.d.c.97.5 yes 8
21.2 odd 6 1764.3.z.l.325.1 8
21.5 even 6 1764.3.z.m.325.4 8
21.11 odd 6 1764.3.z.m.901.4 8
21.17 even 6 1764.3.z.l.901.1 8
21.20 even 2 1764.3.d.h.685.7 8
28.27 even 2 2352.3.f.j.97.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.3.d.c.97.4 8 1.1 even 1 trivial
588.3.d.c.97.5 yes 8 7.6 odd 2 inner
588.3.m.e.313.1 8 7.4 even 3
588.3.m.e.325.1 8 7.5 odd 6
588.3.m.f.313.4 8 7.3 odd 6
588.3.m.f.325.4 8 7.2 even 3
1764.3.d.h.685.2 8 3.2 odd 2
1764.3.d.h.685.7 8 21.20 even 2
1764.3.z.l.325.1 8 21.2 odd 6
1764.3.z.l.901.1 8 21.17 even 6
1764.3.z.m.325.4 8 21.5 even 6
1764.3.z.m.901.4 8 21.11 odd 6
2352.3.f.j.97.1 8 28.27 even 2
2352.3.f.j.97.8 8 4.3 odd 2