Properties

Label 1728.3.e.v.1025.3
Level $1728$
Weight $3$
Character 1728.1025
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2441150464.4
Defining polynomial: \( x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.3
Root \(2.27249 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1025
Dual form 1728.3.e.v.1025.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.88259i q^{5} -10.9726 q^{7} +O(q^{10})\) \(q-1.88259i q^{5} -10.9726 q^{7} +0.171573i q^{11} -6.48528 q^{13} -27.2699i q^{17} +5.32478 q^{19} +3.51472i q^{23} +21.4558 q^{25} -34.8003i q^{29} +26.9469 q^{31} +20.6569i q^{35} -46.4264 q^{37} +23.5047i q^{41} -55.1858 q^{43} +57.5980i q^{47} +71.3970 q^{49} +51.7436i q^{53} +0.323002 q^{55} +82.2843i q^{59} -79.8823 q^{61} +12.2091i q^{65} +44.5362 q^{67} +41.2304i q^{71} +66.3381 q^{73} -1.88259i q^{77} -115.050 q^{79} -36.3381i q^{83} -51.3381 q^{85} +158.027i q^{89} +71.1601 q^{91} -10.0244i q^{95} +62.8823 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{13} - 32 q^{25} - 32 q^{37} + 96 q^{49} - 96 q^{61} - 216 q^{73} + 336 q^{85} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.88259i − 0.376519i −0.982119 0.188259i \(-0.939715\pi\)
0.982119 0.188259i \(-0.0602845\pi\)
\(6\) 0 0
\(7\) −10.9726 −1.56751 −0.783754 0.621071i \(-0.786698\pi\)
−0.783754 + 0.621071i \(0.786698\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.171573i 0.0155975i 0.999970 + 0.00779877i \(0.00248245\pi\)
−0.999970 + 0.00779877i \(0.997518\pi\)
\(12\) 0 0
\(13\) −6.48528 −0.498868 −0.249434 0.968392i \(-0.580245\pi\)
−0.249434 + 0.968392i \(0.580245\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 27.2699i − 1.60411i −0.597249 0.802056i \(-0.703740\pi\)
0.597249 0.802056i \(-0.296260\pi\)
\(18\) 0 0
\(19\) 5.32478 0.280251 0.140126 0.990134i \(-0.455249\pi\)
0.140126 + 0.990134i \(0.455249\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.51472i 0.152814i 0.997077 + 0.0764069i \(0.0243448\pi\)
−0.997077 + 0.0764069i \(0.975655\pi\)
\(24\) 0 0
\(25\) 21.4558 0.858234
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 34.8003i − 1.20001i −0.799997 0.600004i \(-0.795165\pi\)
0.799997 0.600004i \(-0.204835\pi\)
\(30\) 0 0
\(31\) 26.9469 0.869254 0.434627 0.900610i \(-0.356880\pi\)
0.434627 + 0.900610i \(0.356880\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 20.6569i 0.590196i
\(36\) 0 0
\(37\) −46.4264 −1.25477 −0.627384 0.778710i \(-0.715874\pi\)
−0.627384 + 0.778710i \(0.715874\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 23.5047i 0.573285i 0.958038 + 0.286643i \(0.0925393\pi\)
−0.958038 + 0.286643i \(0.907461\pi\)
\(42\) 0 0
\(43\) −55.1858 −1.28339 −0.641695 0.766960i \(-0.721769\pi\)
−0.641695 + 0.766960i \(0.721769\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 57.5980i 1.22549i 0.790281 + 0.612744i \(0.209935\pi\)
−0.790281 + 0.612744i \(0.790065\pi\)
\(48\) 0 0
\(49\) 71.3970 1.45708
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 51.7436i 0.976294i 0.872761 + 0.488147i \(0.162327\pi\)
−0.872761 + 0.488147i \(0.837673\pi\)
\(54\) 0 0
\(55\) 0.323002 0.00587276
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 82.2843i 1.39465i 0.716756 + 0.697324i \(0.245626\pi\)
−0.716756 + 0.697324i \(0.754374\pi\)
\(60\) 0 0
\(61\) −79.8823 −1.30955 −0.654773 0.755826i \(-0.727236\pi\)
−0.654773 + 0.755826i \(0.727236\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.2091i 0.187833i
\(66\) 0 0
\(67\) 44.5362 0.664720 0.332360 0.943153i \(-0.392155\pi\)
0.332360 + 0.943153i \(0.392155\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 41.2304i 0.580711i 0.956919 + 0.290355i \(0.0937735\pi\)
−0.956919 + 0.290355i \(0.906227\pi\)
\(72\) 0 0
\(73\) 66.3381 0.908741 0.454371 0.890813i \(-0.349864\pi\)
0.454371 + 0.890813i \(0.349864\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.88259i − 0.0244493i
\(78\) 0 0
\(79\) −115.050 −1.45633 −0.728167 0.685400i \(-0.759627\pi\)
−0.728167 + 0.685400i \(0.759627\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 36.3381i − 0.437808i −0.975746 0.218904i \(-0.929752\pi\)
0.975746 0.218904i \(-0.0702482\pi\)
\(84\) 0 0
\(85\) −51.3381 −0.603978
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 158.027i 1.77558i 0.460245 + 0.887792i \(0.347762\pi\)
−0.460245 + 0.887792i \(0.652238\pi\)
\(90\) 0 0
\(91\) 71.1601 0.781979
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 10.0244i − 0.105520i
\(96\) 0 0
\(97\) 62.8823 0.648271 0.324135 0.946011i \(-0.394927\pi\)
0.324135 + 0.946011i \(0.394927\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 130.702i 1.29408i 0.762458 + 0.647038i \(0.223992\pi\)
−0.762458 + 0.647038i \(0.776008\pi\)
\(102\) 0 0
\(103\) 72.4521 0.703419 0.351709 0.936109i \(-0.385601\pi\)
0.351709 + 0.936109i \(0.385601\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 168.255i 1.57248i 0.617924 + 0.786238i \(0.287974\pi\)
−0.617924 + 0.786238i \(0.712026\pi\)
\(108\) 0 0
\(109\) −103.397 −0.948596 −0.474298 0.880364i \(-0.657298\pi\)
−0.474298 + 0.880364i \(0.657298\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 140.115i − 1.23995i −0.784621 0.619976i \(-0.787142\pi\)
0.784621 0.619976i \(-0.212858\pi\)
\(114\) 0 0
\(115\) 6.61678 0.0575373
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 299.220i 2.51446i
\(120\) 0 0
\(121\) 120.971 0.999757
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 87.4574i − 0.699660i
\(126\) 0 0
\(127\) 120.052 0.945292 0.472646 0.881252i \(-0.343299\pi\)
0.472646 + 0.881252i \(0.343299\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 198.421i 1.51467i 0.653028 + 0.757333i \(0.273498\pi\)
−0.653028 + 0.757333i \(0.726502\pi\)
\(132\) 0 0
\(133\) −58.4264 −0.439296
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 120.375i − 0.878650i −0.898328 0.439325i \(-0.855218\pi\)
0.898328 0.439325i \(-0.144782\pi\)
\(138\) 0 0
\(139\) −11.9416 −0.0859105 −0.0429553 0.999077i \(-0.513677\pi\)
−0.0429553 + 0.999077i \(0.513677\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 1.11270i − 0.00778111i
\(144\) 0 0
\(145\) −65.5147 −0.451826
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 55.5088i − 0.372542i −0.982498 0.186271i \(-0.940360\pi\)
0.982498 0.186271i \(-0.0596403\pi\)
\(150\) 0 0
\(151\) 203.154 1.34539 0.672695 0.739920i \(-0.265137\pi\)
0.672695 + 0.739920i \(0.265137\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 50.7300i − 0.327290i
\(156\) 0 0
\(157\) 164.971 1.05077 0.525384 0.850865i \(-0.323922\pi\)
0.525384 + 0.850865i \(0.323922\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 38.5654i − 0.239537i
\(162\) 0 0
\(163\) −178.948 −1.09784 −0.548919 0.835875i \(-0.684961\pi\)
−0.548919 + 0.835875i \(0.684961\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 43.8924i 0.262828i 0.991328 + 0.131414i \(0.0419518\pi\)
−0.991328 + 0.131414i \(0.958048\pi\)
\(168\) 0 0
\(169\) −126.941 −0.751131
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 191.858i − 1.10901i −0.832181 0.554504i \(-0.812908\pi\)
0.832181 0.554504i \(-0.187092\pi\)
\(174\) 0 0
\(175\) −235.425 −1.34529
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 144.765i 0.808740i 0.914595 + 0.404370i \(0.132509\pi\)
−0.914595 + 0.404370i \(0.867491\pi\)
\(180\) 0 0
\(181\) 0.735065 0.00406113 0.00203057 0.999998i \(-0.499354\pi\)
0.00203057 + 0.999998i \(0.499354\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 87.4020i 0.472443i
\(186\) 0 0
\(187\) 4.67877 0.0250202
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 290.132i 1.51902i 0.650498 + 0.759508i \(0.274560\pi\)
−0.650498 + 0.759508i \(0.725440\pi\)
\(192\) 0 0
\(193\) 63.6030 0.329549 0.164775 0.986331i \(-0.447310\pi\)
0.164775 + 0.986331i \(0.447310\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 88.3710i − 0.448584i −0.974522 0.224292i \(-0.927993\pi\)
0.974522 0.224292i \(-0.0720069\pi\)
\(198\) 0 0
\(199\) −318.204 −1.59902 −0.799508 0.600656i \(-0.794906\pi\)
−0.799508 + 0.600656i \(0.794906\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 381.848i 1.88102i
\(204\) 0 0
\(205\) 44.2498 0.215853
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.913587i 0.00437123i
\(210\) 0 0
\(211\) 46.6310 0.221000 0.110500 0.993876i \(-0.464755\pi\)
0.110500 + 0.993876i \(0.464755\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 103.892i 0.483220i
\(216\) 0 0
\(217\) −295.676 −1.36256
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 176.853i 0.800239i
\(222\) 0 0
\(223\) 47.2770 0.212004 0.106002 0.994366i \(-0.466195\pi\)
0.106002 + 0.994366i \(0.466195\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 342.500i − 1.50881i −0.656410 0.754404i \(-0.727926\pi\)
0.656410 0.754404i \(-0.272074\pi\)
\(228\) 0 0
\(229\) 381.647 1.66658 0.833290 0.552836i \(-0.186455\pi\)
0.833290 + 0.552836i \(0.186455\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 383.716i − 1.64685i −0.567424 0.823426i \(-0.692060\pi\)
0.567424 0.823426i \(-0.307940\pi\)
\(234\) 0 0
\(235\) 108.434 0.461419
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 315.161i − 1.31867i −0.751850 0.659334i \(-0.770838\pi\)
0.751850 0.659334i \(-0.229162\pi\)
\(240\) 0 0
\(241\) −256.558 −1.06456 −0.532279 0.846569i \(-0.678664\pi\)
−0.532279 + 0.846569i \(0.678664\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 134.411i − 0.548618i
\(246\) 0 0
\(247\) −34.5327 −0.139808
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 246.000i 0.980080i 0.871700 + 0.490040i \(0.163018\pi\)
−0.871700 + 0.490040i \(0.836982\pi\)
\(252\) 0 0
\(253\) −0.603030 −0.00238352
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 443.046i 1.72391i 0.506982 + 0.861957i \(0.330761\pi\)
−0.506982 + 0.861957i \(0.669239\pi\)
\(258\) 0 0
\(259\) 509.416 1.96686
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 479.897i 1.82470i 0.409409 + 0.912351i \(0.365735\pi\)
−0.409409 + 0.912351i \(0.634265\pi\)
\(264\) 0 0
\(265\) 97.4121 0.367593
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 418.406i 1.55541i 0.628628 + 0.777706i \(0.283617\pi\)
−0.628628 + 0.777706i \(0.716383\pi\)
\(270\) 0 0
\(271\) 278.347 1.02711 0.513555 0.858057i \(-0.328328\pi\)
0.513555 + 0.858057i \(0.328328\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.68124i 0.0133863i
\(276\) 0 0
\(277\) 192.617 0.695369 0.347685 0.937612i \(-0.386968\pi\)
0.347685 + 0.937612i \(0.386968\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 435.515i 1.54988i 0.632037 + 0.774939i \(0.282219\pi\)
−0.632037 + 0.774939i \(0.717781\pi\)
\(282\) 0 0
\(283\) −518.774 −1.83312 −0.916562 0.399894i \(-0.869047\pi\)
−0.916562 + 0.399894i \(0.869047\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 257.907i − 0.898629i
\(288\) 0 0
\(289\) −454.647 −1.57317
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 117.634i − 0.401482i −0.979644 0.200741i \(-0.935665\pi\)
0.979644 0.200741i \(-0.0643350\pi\)
\(294\) 0 0
\(295\) 154.908 0.525111
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 22.7939i − 0.0762339i
\(300\) 0 0
\(301\) 605.529 2.01172
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 150.386i 0.493068i
\(306\) 0 0
\(307\) −371.618 −1.21048 −0.605241 0.796042i \(-0.706923\pi\)
−0.605241 + 0.796042i \(0.706923\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.8680i 0.0896076i 0.998996 + 0.0448038i \(0.0142663\pi\)
−0.998996 + 0.0448038i \(0.985734\pi\)
\(312\) 0 0
\(313\) −279.368 −0.892548 −0.446274 0.894896i \(-0.647249\pi\)
−0.446274 + 0.894896i \(0.647249\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 546.367i 1.72355i 0.507287 + 0.861777i \(0.330648\pi\)
−0.507287 + 0.861777i \(0.669352\pi\)
\(318\) 0 0
\(319\) 5.97078 0.0187172
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 145.206i − 0.449554i
\(324\) 0 0
\(325\) −139.147 −0.428145
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 631.997i − 1.92096i
\(330\) 0 0
\(331\) −305.294 −0.922337 −0.461168 0.887313i \(-0.652570\pi\)
−0.461168 + 0.887313i \(0.652570\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 83.8436i − 0.250279i
\(336\) 0 0
\(337\) 159.206 0.472422 0.236211 0.971702i \(-0.424094\pi\)
0.236211 + 0.971702i \(0.424094\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.62335i 0.0135582i
\(342\) 0 0
\(343\) −245.752 −0.716478
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 296.387i 0.854141i 0.904218 + 0.427070i \(0.140454\pi\)
−0.904218 + 0.427070i \(0.859546\pi\)
\(348\) 0 0
\(349\) −108.839 −0.311858 −0.155929 0.987768i \(-0.549837\pi\)
−0.155929 + 0.987768i \(0.549837\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.3200i 0.0349008i 0.999848 + 0.0174504i \(0.00555492\pi\)
−0.999848 + 0.0174504i \(0.994445\pi\)
\(354\) 0 0
\(355\) 77.6201 0.218648
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 92.2254i 0.256895i 0.991716 + 0.128448i \(0.0409994\pi\)
−0.991716 + 0.128448i \(0.959001\pi\)
\(360\) 0 0
\(361\) −332.647 −0.921459
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 124.888i − 0.342158i
\(366\) 0 0
\(367\) −13.0673 −0.0356058 −0.0178029 0.999842i \(-0.505667\pi\)
−0.0178029 + 0.999842i \(0.505667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 567.759i − 1.53035i
\(372\) 0 0
\(373\) 233.632 0.626361 0.313180 0.949694i \(-0.398606\pi\)
0.313180 + 0.949694i \(0.398606\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 225.689i 0.598646i
\(378\) 0 0
\(379\) 683.528 1.80351 0.901753 0.432253i \(-0.142281\pi\)
0.901753 + 0.432253i \(0.142281\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 279.858i − 0.730699i −0.930870 0.365350i \(-0.880949\pi\)
0.930870 0.365350i \(-0.119051\pi\)
\(384\) 0 0
\(385\) −3.54416 −0.00920560
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 260.434i − 0.669497i −0.942308 0.334748i \(-0.891349\pi\)
0.942308 0.334748i \(-0.108651\pi\)
\(390\) 0 0
\(391\) 95.8460 0.245130
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 216.593i 0.548337i
\(396\) 0 0
\(397\) −167.809 −0.422693 −0.211346 0.977411i \(-0.567785\pi\)
−0.211346 + 0.977411i \(0.567785\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 60.2430i 0.150232i 0.997175 + 0.0751159i \(0.0239327\pi\)
−0.997175 + 0.0751159i \(0.976067\pi\)
\(402\) 0 0
\(403\) −174.758 −0.433643
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 7.96551i − 0.0195713i
\(408\) 0 0
\(409\) 532.765 1.30260 0.651301 0.758819i \(-0.274223\pi\)
0.651301 + 0.758819i \(0.274223\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 902.869i − 2.18612i
\(414\) 0 0
\(415\) −68.4098 −0.164843
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 483.255i − 1.15335i −0.816973 0.576676i \(-0.804349\pi\)
0.816973 0.576676i \(-0.195651\pi\)
\(420\) 0 0
\(421\) 29.2061 0.0693731 0.0346865 0.999398i \(-0.488957\pi\)
0.0346865 + 0.999398i \(0.488957\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 585.098i − 1.37670i
\(426\) 0 0
\(427\) 876.512 2.05272
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 634.971i − 1.47325i −0.676302 0.736625i \(-0.736418\pi\)
0.676302 0.736625i \(-0.263582\pi\)
\(432\) 0 0
\(433\) −28.5004 −0.0658209 −0.0329104 0.999458i \(-0.510478\pi\)
−0.0329104 + 0.999458i \(0.510478\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 18.7151i 0.0428263i
\(438\) 0 0
\(439\) −287.547 −0.655006 −0.327503 0.944850i \(-0.606207\pi\)
−0.327503 + 0.944850i \(0.606207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.176624i 0 0.000398699i 1.00000 0.000199349i \(6.34549e-5\pi\)
−1.00000 0.000199349i \(0.999937\pi\)
\(444\) 0 0
\(445\) 297.500 0.668540
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 36.5166i 0.0813287i 0.999173 + 0.0406644i \(0.0129474\pi\)
−0.999173 + 0.0406644i \(0.987053\pi\)
\(450\) 0 0
\(451\) −4.03277 −0.00894184
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 133.966i − 0.294430i
\(456\) 0 0
\(457\) −243.235 −0.532244 −0.266122 0.963939i \(-0.585742\pi\)
−0.266122 + 0.963939i \(0.585742\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 155.231i 0.336726i 0.985725 + 0.168363i \(0.0538481\pi\)
−0.985725 + 0.168363i \(0.946152\pi\)
\(462\) 0 0
\(463\) 444.550 0.960151 0.480076 0.877227i \(-0.340609\pi\)
0.480076 + 0.877227i \(0.340609\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 853.357i − 1.82732i −0.406482 0.913659i \(-0.633245\pi\)
0.406482 0.913659i \(-0.366755\pi\)
\(468\) 0 0
\(469\) −488.676 −1.04195
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 9.46838i − 0.0200177i
\(474\) 0 0
\(475\) 114.248 0.240521
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 445.882i − 0.930861i −0.885085 0.465430i \(-0.845900\pi\)
0.885085 0.465430i \(-0.154100\pi\)
\(480\) 0 0
\(481\) 301.088 0.625963
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 118.382i − 0.244086i
\(486\) 0 0
\(487\) 438.413 0.900232 0.450116 0.892970i \(-0.351383\pi\)
0.450116 + 0.892970i \(0.351383\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 186.754i − 0.380355i −0.981750 0.190178i \(-0.939094\pi\)
0.981750 0.190178i \(-0.0609064\pi\)
\(492\) 0 0
\(493\) −948.999 −1.92495
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 452.403i − 0.910268i
\(498\) 0 0
\(499\) −432.775 −0.867284 −0.433642 0.901085i \(-0.642772\pi\)
−0.433642 + 0.901085i \(0.642772\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 45.6913i 0.0908377i 0.998968 + 0.0454188i \(0.0144622\pi\)
−0.998968 + 0.0454188i \(0.985538\pi\)
\(504\) 0 0
\(505\) 246.058 0.487244
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 267.051i 0.524658i 0.964978 + 0.262329i \(0.0844906\pi\)
−0.964978 + 0.262329i \(0.915509\pi\)
\(510\) 0 0
\(511\) −727.898 −1.42446
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 136.398i − 0.264850i
\(516\) 0 0
\(517\) −9.88225 −0.0191146
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 144.101i 0.276586i 0.990391 + 0.138293i \(0.0441616\pi\)
−0.990391 + 0.138293i \(0.955838\pi\)
\(522\) 0 0
\(523\) −631.083 −1.20666 −0.603330 0.797491i \(-0.706160\pi\)
−0.603330 + 0.797491i \(0.706160\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 734.839i − 1.39438i
\(528\) 0 0
\(529\) 516.647 0.976648
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 152.435i − 0.285994i
\(534\) 0 0
\(535\) 316.755 0.592066
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.2498i 0.0227269i
\(540\) 0 0
\(541\) −235.765 −0.435794 −0.217897 0.975972i \(-0.569920\pi\)
−0.217897 + 0.975972i \(0.569920\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 194.654i 0.357164i
\(546\) 0 0
\(547\) −813.418 −1.48705 −0.743526 0.668707i \(-0.766848\pi\)
−0.743526 + 0.668707i \(0.766848\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 185.304i − 0.336304i
\(552\) 0 0
\(553\) 1262.40 2.28281
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 175.773i 0.315571i 0.987473 + 0.157786i \(0.0504355\pi\)
−0.987473 + 0.157786i \(0.949565\pi\)
\(558\) 0 0
\(559\) 357.895 0.640242
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 572.927i − 1.01763i −0.860875 0.508816i \(-0.830083\pi\)
0.860875 0.508816i \(-0.169917\pi\)
\(564\) 0 0
\(565\) −263.779 −0.466865
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 814.664i 1.43175i 0.698230 + 0.715873i \(0.253971\pi\)
−0.698230 + 0.715873i \(0.746029\pi\)
\(570\) 0 0
\(571\) −128.284 −0.224665 −0.112333 0.993671i \(-0.535832\pi\)
−0.112333 + 0.993671i \(0.535832\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 75.4113i 0.131150i
\(576\) 0 0
\(577\) 30.4996 0.0528589 0.0264294 0.999651i \(-0.491586\pi\)
0.0264294 + 0.999651i \(0.491586\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 398.722i 0.686268i
\(582\) 0 0
\(583\) −8.87780 −0.0152278
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 193.794i 0.330143i 0.986282 + 0.165071i \(0.0527855\pi\)
−0.986282 + 0.165071i \(0.947215\pi\)
\(588\) 0 0
\(589\) 143.486 0.243610
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 684.488i − 1.15428i −0.816645 0.577140i \(-0.804169\pi\)
0.816645 0.577140i \(-0.195831\pi\)
\(594\) 0 0
\(595\) 563.310 0.946740
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 814.244i 1.35934i 0.733519 + 0.679669i \(0.237877\pi\)
−0.733519 + 0.679669i \(0.762123\pi\)
\(600\) 0 0
\(601\) −531.368 −0.884139 −0.442069 0.896981i \(-0.645756\pi\)
−0.442069 + 0.896981i \(0.645756\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 227.738i − 0.376427i
\(606\) 0 0
\(607\) −464.880 −0.765865 −0.382933 0.923776i \(-0.625086\pi\)
−0.382933 + 0.923776i \(0.625086\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 373.539i − 0.611357i
\(612\) 0 0
\(613\) −1120.59 −1.82804 −0.914019 0.405672i \(-0.867038\pi\)
−0.914019 + 0.405672i \(0.867038\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 561.372i 0.909841i 0.890532 + 0.454921i \(0.150332\pi\)
−0.890532 + 0.454921i \(0.849668\pi\)
\(618\) 0 0
\(619\) −471.340 −0.761454 −0.380727 0.924687i \(-0.624326\pi\)
−0.380727 + 0.924687i \(0.624326\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1733.96i − 2.78324i
\(624\) 0 0
\(625\) 371.749 0.594799
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1266.04i 2.01279i
\(630\) 0 0
\(631\) −431.160 −0.683296 −0.341648 0.939828i \(-0.610985\pi\)
−0.341648 + 0.939828i \(0.610985\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 226.009i − 0.355920i
\(636\) 0 0
\(637\) −463.029 −0.726891
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 399.580i 0.623370i 0.950186 + 0.311685i \(0.100893\pi\)
−0.950186 + 0.311685i \(0.899107\pi\)
\(642\) 0 0
\(643\) 1013.66 1.57646 0.788231 0.615380i \(-0.210997\pi\)
0.788231 + 0.615380i \(0.210997\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 28.4710i − 0.0440046i −0.999758 0.0220023i \(-0.992996\pi\)
0.999758 0.0220023i \(-0.00700412\pi\)
\(648\) 0 0
\(649\) −14.1177 −0.0217531
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 274.692i − 0.420662i −0.977630 0.210331i \(-0.932546\pi\)
0.977630 0.210331i \(-0.0674542\pi\)
\(654\) 0 0
\(655\) 373.547 0.570300
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 87.7065i − 0.133090i −0.997783 0.0665451i \(-0.978802\pi\)
0.997783 0.0665451i \(-0.0211976\pi\)
\(660\) 0 0
\(661\) 212.368 0.321282 0.160641 0.987013i \(-0.448644\pi\)
0.160641 + 0.987013i \(0.448644\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 109.993i 0.165403i
\(666\) 0 0
\(667\) 122.313 0.183378
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 13.7056i − 0.0204257i
\(672\) 0 0
\(673\) 1099.17 1.63325 0.816623 0.577171i \(-0.195843\pi\)
0.816623 + 0.577171i \(0.195843\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 715.745i 1.05723i 0.848862 + 0.528615i \(0.177288\pi\)
−0.848862 + 0.528615i \(0.822712\pi\)
\(678\) 0 0
\(679\) −689.979 −1.01617
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 588.823i − 0.862112i −0.902325 0.431056i \(-0.858141\pi\)
0.902325 0.431056i \(-0.141859\pi\)
\(684\) 0 0
\(685\) −226.617 −0.330828
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 335.572i − 0.487042i
\(690\) 0 0
\(691\) 466.975 0.675796 0.337898 0.941183i \(-0.390284\pi\)
0.337898 + 0.941183i \(0.390284\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 22.4811i 0.0323469i
\(696\) 0 0
\(697\) 640.971 0.919613
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 374.525i 0.534273i 0.963659 + 0.267136i \(0.0860774\pi\)
−0.963659 + 0.267136i \(0.913923\pi\)
\(702\) 0 0
\(703\) −247.210 −0.351650
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1434.13i − 2.02847i
\(708\) 0 0
\(709\) 259.647 0.366215 0.183108 0.983093i \(-0.441384\pi\)
0.183108 + 0.983093i \(0.441384\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 94.7107i 0.132834i
\(714\) 0 0
\(715\) −2.09476 −0.00292973
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 1124.90i − 1.56454i −0.622942 0.782268i \(-0.714063\pi\)
0.622942 0.782268i \(-0.285937\pi\)
\(720\) 0 0
\(721\) −794.985 −1.10261
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 746.669i − 1.02989i
\(726\) 0 0
\(727\) −185.241 −0.254803 −0.127401 0.991851i \(-0.540664\pi\)
−0.127401 + 0.991851i \(0.540664\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1504.91i 2.05870i
\(732\) 0 0
\(733\) −62.0732 −0.0846837 −0.0423419 0.999103i \(-0.513482\pi\)
−0.0423419 + 0.999103i \(0.513482\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.64121i 0.0103680i
\(738\) 0 0
\(739\) −156.357 −0.211579 −0.105789 0.994389i \(-0.533737\pi\)
−0.105789 + 0.994389i \(0.533737\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 371.220i 0.499624i 0.968294 + 0.249812i \(0.0803688\pi\)
−0.968294 + 0.249812i \(0.919631\pi\)
\(744\) 0 0
\(745\) −104.500 −0.140269
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1846.19i − 2.46487i
\(750\) 0 0
\(751\) 380.653 0.506861 0.253431 0.967354i \(-0.418441\pi\)
0.253431 + 0.967354i \(0.418441\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 382.456i − 0.506564i
\(756\) 0 0
\(757\) −351.088 −0.463789 −0.231895 0.972741i \(-0.574492\pi\)
−0.231895 + 0.972741i \(0.574492\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 300.772i 0.395232i 0.980280 + 0.197616i \(0.0633199\pi\)
−0.980280 + 0.197616i \(0.936680\pi\)
\(762\) 0 0
\(763\) 1134.53 1.48693
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 533.637i − 0.695745i
\(768\) 0 0
\(769\) −378.955 −0.492790 −0.246395 0.969170i \(-0.579246\pi\)
−0.246395 + 0.969170i \(0.579246\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 249.305i − 0.322516i −0.986912 0.161258i \(-0.948445\pi\)
0.986912 0.161258i \(-0.0515552\pi\)
\(774\) 0 0
\(775\) 578.168 0.746023
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 125.157i 0.160664i
\(780\) 0 0
\(781\) −7.07403 −0.00905765
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 310.572i − 0.395634i
\(786\) 0 0
\(787\) −1126.62 −1.43154 −0.715769 0.698337i \(-0.753924\pi\)
−0.715769 + 0.698337i \(0.753924\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1537.42i 1.94364i
\(792\) 0 0
\(793\) 518.059 0.653290
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 451.324i − 0.566278i −0.959079 0.283139i \(-0.908624\pi\)
0.959079 0.283139i \(-0.0913758\pi\)
\(798\) 0 0
\(799\) 1570.69 1.96582
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.3818i 0.0141741i
\(804\) 0 0
\(805\) −72.6030 −0.0901901
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 684.710i 0.846365i 0.906044 + 0.423183i \(0.139087\pi\)
−0.906044 + 0.423183i \(0.860913\pi\)
\(810\) 0 0
\(811\) −1324.62 −1.63331 −0.816656 0.577125i \(-0.804174\pi\)
−0.816656 + 0.577125i \(0.804174\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 336.886i 0.413357i
\(816\) 0 0
\(817\) −293.852 −0.359672
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 837.836i 1.02051i 0.860024 + 0.510253i \(0.170448\pi\)
−0.860024 + 0.510253i \(0.829552\pi\)
\(822\) 0 0
\(823\) −217.033 −0.263710 −0.131855 0.991269i \(-0.542093\pi\)
−0.131855 + 0.991269i \(0.542093\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 458.215i 0.554069i 0.960860 + 0.277035i \(0.0893517\pi\)
−0.960860 + 0.277035i \(0.910648\pi\)
\(828\) 0 0
\(829\) 335.706 0.404953 0.202476 0.979287i \(-0.435101\pi\)
0.202476 + 0.979287i \(0.435101\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1946.99i − 2.33732i
\(834\) 0 0
\(835\) 82.6314 0.0989598
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1202.59i − 1.43336i −0.697401 0.716681i \(-0.745660\pi\)
0.697401 0.716681i \(-0.254340\pi\)
\(840\) 0 0
\(841\) −370.058 −0.440021
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 238.978i 0.282815i
\(846\) 0 0
\(847\) −1327.36 −1.56713
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 163.176i − 0.191746i
\(852\) 0 0
\(853\) 730.471 0.856355 0.428178 0.903695i \(-0.359156\pi\)
0.428178 + 0.903695i \(0.359156\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1244.03i − 1.45161i −0.687899 0.725807i \(-0.741467\pi\)
0.687899 0.725807i \(-0.258533\pi\)
\(858\) 0 0
\(859\) −1584.24 −1.84428 −0.922141 0.386855i \(-0.873561\pi\)
−0.922141 + 0.386855i \(0.873561\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 746.392i 0.864881i 0.901663 + 0.432440i \(0.142347\pi\)
−0.901663 + 0.432440i \(0.857653\pi\)
\(864\) 0 0
\(865\) −361.191 −0.417562
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 19.7395i − 0.0227152i
\(870\) 0 0
\(871\) −288.830 −0.331607
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 959.632i 1.09672i
\(876\) 0 0
\(877\) 1351.97 1.54158 0.770792 0.637087i \(-0.219861\pi\)
0.770792 + 0.637087i \(0.219861\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 649.799i 0.737569i 0.929515 + 0.368785i \(0.120226\pi\)
−0.929515 + 0.368785i \(0.879774\pi\)
\(882\) 0 0
\(883\) 829.881 0.939843 0.469922 0.882708i \(-0.344282\pi\)
0.469922 + 0.882708i \(0.344282\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 296.662i 0.334455i 0.985918 + 0.167228i \(0.0534815\pi\)
−0.985918 + 0.167228i \(0.946519\pi\)
\(888\) 0 0
\(889\) −1317.28 −1.48175
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 306.696i 0.343445i
\(894\) 0 0
\(895\) 272.533 0.304506
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 937.759i − 1.04311i
\(900\) 0 0
\(901\) 1411.04 1.56608
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1.38383i − 0.00152909i
\(906\) 0 0
\(907\) 964.450 1.06334 0.531670 0.846952i \(-0.321565\pi\)
0.531670 + 0.846952i \(0.321565\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 399.941i − 0.439013i −0.975611 0.219507i \(-0.929555\pi\)
0.975611 0.219507i \(-0.0704448\pi\)
\(912\) 0 0
\(913\) 6.23463 0.00682873
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2177.19i − 2.37425i
\(918\) 0 0
\(919\) 1363.99 1.48421 0.742107 0.670281i \(-0.233826\pi\)
0.742107 + 0.670281i \(0.233826\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 267.391i − 0.289698i
\(924\) 0 0
\(925\) −996.118 −1.07688
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 78.7364i − 0.0847539i −0.999102 0.0423770i \(-0.986507\pi\)
0.999102 0.0423770i \(-0.0134930\pi\)
\(930\) 0 0
\(931\) 380.173 0.408349
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 8.80822i − 0.00942056i
\(936\) 0 0
\(937\) −861.705 −0.919642 −0.459821 0.888012i \(-0.652086\pi\)
−0.459821 + 0.888012i \(0.652086\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 641.991i 0.682243i 0.940019 + 0.341122i \(0.110807\pi\)
−0.940019 + 0.341122i \(0.889193\pi\)
\(942\) 0 0
\(943\) −82.6124 −0.0876060
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1259.50i 1.32999i 0.746849 + 0.664994i \(0.231566\pi\)
−0.746849 + 0.664994i \(0.768434\pi\)
\(948\) 0 0
\(949\) −430.221 −0.453342
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 867.155i − 0.909921i −0.890512 0.454961i \(-0.849653\pi\)
0.890512 0.454961i \(-0.150347\pi\)
\(954\) 0 0
\(955\) 546.200 0.571938
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1320.82i 1.37729i
\(960\) 0 0
\(961\) −234.865 −0.244397
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 119.739i − 0.124081i
\(966\) 0 0
\(967\) 157.972 0.163363 0.0816813 0.996659i \(-0.473971\pi\)
0.0816813 + 0.996659i \(0.473971\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 826.466i 0.851149i 0.904923 + 0.425575i \(0.139928\pi\)
−0.904923 + 0.425575i \(0.860072\pi\)
\(972\) 0 0
\(973\) 131.029 0.134665
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1777.33i 1.81917i 0.415514 + 0.909587i \(0.363602\pi\)
−0.415514 + 0.909587i \(0.636398\pi\)
\(978\) 0 0
\(979\) −27.1131 −0.0276947
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 587.324i 0.597481i 0.954334 + 0.298740i \(0.0965665\pi\)
−0.954334 + 0.298740i \(0.903434\pi\)
\(984\) 0 0
\(985\) −166.367 −0.168900
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 193.962i − 0.196120i
\(990\) 0 0
\(991\) −1132.76 −1.14304 −0.571522 0.820587i \(-0.693647\pi\)
−0.571522 + 0.820587i \(0.693647\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 599.049i 0.602059i
\(996\) 0 0
\(997\) 1296.22 1.30012 0.650060 0.759883i \(-0.274744\pi\)
0.650060 + 0.759883i \(0.274744\pi\)
\(998\) 0 0
\(999\) 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 1728.3.e.v.1025.3 8
3.2 odd 2 inner 1728.3.e.v.1025.5 8
4.3 odd 2 inner 1728.3.e.v.1025.4 8
8.3 odd 2 864.3.e.e.161.6 yes 8
8.5 even 2 864.3.e.e.161.5 yes 8
12.11 even 2 inner 1728.3.e.v.1025.6 8
24.5 odd 2 864.3.e.e.161.3 8
24.11 even 2 864.3.e.e.161.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.e.161.3 8 24.5 odd 2
864.3.e.e.161.4 yes 8 24.11 even 2
864.3.e.e.161.5 yes 8 8.5 even 2
864.3.e.e.161.6 yes 8 8.3 odd 2
1728.3.e.v.1025.3 8 1.1 even 1 trivial
1728.3.e.v.1025.4 8 4.3 odd 2 inner
1728.3.e.v.1025.5 8 3.2 odd 2 inner
1728.3.e.v.1025.6 8 12.11 even 2 inner