Properties

Label 864.3.e.f.161.7
Level $864$
Weight $3$
Character 864.161
Analytic conductor $23.542$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 864.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.5422948407\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.7
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 864.161
Dual form 864.3.e.f.161.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.61037i q^{5} -9.43879 q^{7} +O(q^{10})\) \(q+6.61037i q^{5} -9.43879 q^{7} +17.6969i q^{11} +20.6969 q^{13} -16.0492i q^{17} +12.2993 q^{19} +26.6969i q^{23} -18.6969 q^{25} +0.921404i q^{29} -53.7722 q^{31} -62.3939i q^{35} -64.0908 q^{37} -19.7990i q^{41} -69.7893 q^{43} +0.606123i q^{47} +40.0908 q^{49} +45.3512i q^{53} -116.983 q^{55} -71.3939i q^{59} -4.00000 q^{61} +136.814i q^{65} -7.43545 q^{67} +27.3031i q^{71} +6.30306 q^{73} -167.038i q^{77} +42.6191 q^{79} -72.3031i q^{83} +106.091 q^{85} -126.486i q^{89} -195.354 q^{91} +81.3031i q^{95} +27.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 48 q^{13} - 32 q^{25} - 160 q^{37} - 32 q^{49} - 32 q^{61} + 168 q^{73} + 496 q^{85} + 216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 6.61037i 1.32207i 0.750354 + 0.661037i \(0.229883\pi\)
−0.750354 + 0.661037i \(0.770117\pi\)
\(6\) 0 0
\(7\) −9.43879 −1.34840 −0.674200 0.738549i \(-0.735511\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 17.6969i 1.60881i 0.594080 + 0.804406i \(0.297516\pi\)
−0.594080 + 0.804406i \(0.702484\pi\)
\(12\) 0 0
\(13\) 20.6969 1.59207 0.796036 0.605249i \(-0.206927\pi\)
0.796036 + 0.605249i \(0.206927\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 16.0492i − 0.944068i −0.881580 0.472034i \(-0.843520\pi\)
0.881580 0.472034i \(-0.156480\pi\)
\(18\) 0 0
\(19\) 12.2993 0.647333 0.323667 0.946171i \(-0.395084\pi\)
0.323667 + 0.946171i \(0.395084\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 26.6969i 1.16074i 0.814354 + 0.580368i \(0.197091\pi\)
−0.814354 + 0.580368i \(0.802909\pi\)
\(24\) 0 0
\(25\) −18.6969 −0.747878
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.921404i 0.0317725i 0.999874 + 0.0158863i \(0.00505697\pi\)
−0.999874 + 0.0158863i \(0.994943\pi\)
\(30\) 0 0
\(31\) −53.7722 −1.73459 −0.867294 0.497796i \(-0.834143\pi\)
−0.867294 + 0.497796i \(0.834143\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 62.3939i − 1.78268i
\(36\) 0 0
\(37\) −64.0908 −1.73218 −0.866092 0.499884i \(-0.833376\pi\)
−0.866092 + 0.499884i \(0.833376\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 19.7990i − 0.482902i −0.970413 0.241451i \(-0.922377\pi\)
0.970413 0.241451i \(-0.0776233\pi\)
\(42\) 0 0
\(43\) −69.7893 −1.62301 −0.811503 0.584348i \(-0.801350\pi\)
−0.811503 + 0.584348i \(0.801350\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.606123i 0.0128962i 0.999979 + 0.00644812i \(0.00205251\pi\)
−0.999979 + 0.00644812i \(0.997947\pi\)
\(48\) 0 0
\(49\) 40.0908 0.818180
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 45.3512i 0.855682i 0.903854 + 0.427841i \(0.140726\pi\)
−0.903854 + 0.427841i \(0.859274\pi\)
\(54\) 0 0
\(55\) −116.983 −2.12697
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 71.3939i − 1.21007i −0.796201 0.605033i \(-0.793160\pi\)
0.796201 0.605033i \(-0.206840\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.0655738 −0.0327869 0.999462i \(-0.510438\pi\)
−0.0327869 + 0.999462i \(0.510438\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 136.814i 2.10484i
\(66\) 0 0
\(67\) −7.43545 −0.110977 −0.0554884 0.998459i \(-0.517672\pi\)
−0.0554884 + 0.998459i \(0.517672\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 27.3031i 0.384550i 0.981341 + 0.192275i \(0.0615866\pi\)
−0.981341 + 0.192275i \(0.938413\pi\)
\(72\) 0 0
\(73\) 6.30306 0.0863433 0.0431717 0.999068i \(-0.486254\pi\)
0.0431717 + 0.999068i \(0.486254\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 167.038i − 2.16932i
\(78\) 0 0
\(79\) 42.6191 0.539482 0.269741 0.962933i \(-0.413062\pi\)
0.269741 + 0.962933i \(0.413062\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 72.3031i − 0.871121i −0.900159 0.435561i \(-0.856550\pi\)
0.900159 0.435561i \(-0.143450\pi\)
\(84\) 0 0
\(85\) 106.091 1.24813
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 126.486i − 1.42119i −0.703599 0.710597i \(-0.748425\pi\)
0.703599 0.710597i \(-0.251575\pi\)
\(90\) 0 0
\(91\) −195.354 −2.14675
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 81.3031i 0.855822i
\(96\) 0 0
\(97\) 27.0000 0.278351 0.139175 0.990268i \(-0.455555\pi\)
0.139175 + 0.990268i \(0.455555\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 33.7806i 0.334461i 0.985918 + 0.167231i \(0.0534824\pi\)
−0.985918 + 0.167231i \(0.946518\pi\)
\(102\) 0 0
\(103\) −95.2451 −0.924710 −0.462355 0.886695i \(-0.652995\pi\)
−0.462355 + 0.886695i \(0.652995\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 96.5755i 0.902575i 0.892379 + 0.451287i \(0.149035\pi\)
−0.892379 + 0.451287i \(0.850965\pi\)
\(108\) 0 0
\(109\) 90.8786 0.833748 0.416874 0.908964i \(-0.363126\pi\)
0.416874 + 0.908964i \(0.363126\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 50.1829i 0.444097i 0.975036 + 0.222048i \(0.0712743\pi\)
−0.975036 + 0.222048i \(0.928726\pi\)
\(114\) 0 0
\(115\) −176.477 −1.53458
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 151.485i 1.27298i
\(120\) 0 0
\(121\) −192.182 −1.58828
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 41.6655i 0.333324i
\(126\) 0 0
\(127\) 134.146 1.05627 0.528136 0.849160i \(-0.322891\pi\)
0.528136 + 0.849160i \(0.322891\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 46.8184i − 0.357392i −0.983904 0.178696i \(-0.942812\pi\)
0.983904 0.178696i \(-0.0571879\pi\)
\(132\) 0 0
\(133\) −116.091 −0.872863
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 182.926i 1.33523i 0.744507 + 0.667614i \(0.232685\pi\)
−0.744507 + 0.667614i \(0.767315\pi\)
\(138\) 0 0
\(139\) −137.864 −0.991829 −0.495914 0.868371i \(-0.665167\pi\)
−0.495914 + 0.868371i \(0.665167\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 366.272i 2.56135i
\(144\) 0 0
\(145\) −6.09082 −0.0420056
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 43.6368i 0.292864i 0.989221 + 0.146432i \(0.0467790\pi\)
−0.989221 + 0.146432i \(0.953221\pi\)
\(150\) 0 0
\(151\) −84.0920 −0.556900 −0.278450 0.960451i \(-0.589821\pi\)
−0.278450 + 0.960451i \(0.589821\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 355.454i − 2.29325i
\(156\) 0 0
\(157\) −236.545 −1.50666 −0.753328 0.657645i \(-0.771553\pi\)
−0.753328 + 0.657645i \(0.771553\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 251.987i − 1.56514i
\(162\) 0 0
\(163\) 282.307 1.73194 0.865971 0.500094i \(-0.166701\pi\)
0.865971 + 0.500094i \(0.166701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 70.1816i − 0.420249i −0.977675 0.210125i \(-0.932613\pi\)
0.977675 0.210125i \(-0.0673870\pi\)
\(168\) 0 0
\(169\) 259.363 1.53469
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 338.586i 1.95715i 0.205901 + 0.978573i \(0.433987\pi\)
−0.205901 + 0.978573i \(0.566013\pi\)
\(174\) 0 0
\(175\) 176.477 1.00844
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 348.576i − 1.94735i −0.227942 0.973675i \(-0.573200\pi\)
0.227942 0.973675i \(-0.426800\pi\)
\(180\) 0 0
\(181\) 16.1816 0.0894013 0.0447006 0.999000i \(-0.485767\pi\)
0.0447006 + 0.999000i \(0.485767\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 423.664i − 2.29007i
\(186\) 0 0
\(187\) 284.021 1.51883
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 171.303i 0.896875i 0.893814 + 0.448437i \(0.148019\pi\)
−0.893814 + 0.448437i \(0.851981\pi\)
\(192\) 0 0
\(193\) −241.091 −1.24918 −0.624588 0.780955i \(-0.714733\pi\)
−0.624588 + 0.780955i \(0.714733\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 182.165i − 0.924698i −0.886698 0.462349i \(-0.847007\pi\)
0.886698 0.462349i \(-0.152993\pi\)
\(198\) 0 0
\(199\) −70.0782 −0.352152 −0.176076 0.984377i \(-0.556340\pi\)
−0.176076 + 0.984377i \(0.556340\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 8.69694i − 0.0428421i
\(204\) 0 0
\(205\) 130.879 0.638432
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 217.660i 1.04144i
\(210\) 0 0
\(211\) 102.681 0.486638 0.243319 0.969946i \(-0.421764\pi\)
0.243319 + 0.969946i \(0.421764\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 461.333i − 2.14573i
\(216\) 0 0
\(217\) 507.545 2.33892
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 332.168i − 1.50302i
\(222\) 0 0
\(223\) −205.082 −0.919650 −0.459825 0.888010i \(-0.652088\pi\)
−0.459825 + 0.888010i \(0.652088\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 103.757i 0.457080i 0.973535 + 0.228540i \(0.0733952\pi\)
−0.973535 + 0.228540i \(0.926605\pi\)
\(228\) 0 0
\(229\) 435.576 1.90208 0.951038 0.309073i \(-0.100019\pi\)
0.951038 + 0.309073i \(0.100019\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 387.344i 1.66242i 0.555959 + 0.831210i \(0.312351\pi\)
−0.555959 + 0.831210i \(0.687649\pi\)
\(234\) 0 0
\(235\) −4.00670 −0.0170498
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 225.303i 0.942691i 0.881949 + 0.471345i \(0.156231\pi\)
−0.881949 + 0.471345i \(0.843769\pi\)
\(240\) 0 0
\(241\) −127.757 −0.530113 −0.265056 0.964233i \(-0.585391\pi\)
−0.265056 + 0.964233i \(0.585391\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 265.015i 1.08169i
\(246\) 0 0
\(247\) 254.558 1.03060
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 54.0000i 0.215139i 0.994198 + 0.107570i \(0.0343069\pi\)
−0.994198 + 0.107570i \(0.965693\pi\)
\(252\) 0 0
\(253\) −472.454 −1.86741
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 181.941i − 0.707941i −0.935257 0.353970i \(-0.884831\pi\)
0.935257 0.353970i \(-0.115169\pi\)
\(258\) 0 0
\(259\) 604.940 2.33568
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 42.8786i 0.163036i 0.996672 + 0.0815182i \(0.0259769\pi\)
−0.996672 + 0.0815182i \(0.974023\pi\)
\(264\) 0 0
\(265\) −299.788 −1.13127
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 140.500i − 0.522305i −0.965298 0.261152i \(-0.915898\pi\)
0.965298 0.261152i \(-0.0841025\pi\)
\(270\) 0 0
\(271\) 264.854 0.977323 0.488661 0.872474i \(-0.337485\pi\)
0.488661 + 0.872474i \(0.337485\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 330.879i − 1.20319i
\(276\) 0 0
\(277\) −91.8184 −0.331474 −0.165737 0.986170i \(-0.553000\pi\)
−0.165737 + 0.986170i \(0.553000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 417.814i 1.48688i 0.668801 + 0.743442i \(0.266808\pi\)
−0.668801 + 0.743442i \(0.733192\pi\)
\(282\) 0 0
\(283\) 389.851 1.37757 0.688783 0.724968i \(-0.258145\pi\)
0.688783 + 0.724968i \(0.258145\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 186.879i 0.651145i
\(288\) 0 0
\(289\) 31.4245 0.108735
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 108.145i − 0.369095i −0.982824 0.184547i \(-0.940918\pi\)
0.982824 0.184547i \(-0.0590819\pi\)
\(294\) 0 0
\(295\) 471.940 1.59980
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 552.545i 1.84798i
\(300\) 0 0
\(301\) 658.727 2.18846
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 26.4415i − 0.0866933i
\(306\) 0 0
\(307\) −52.3468 −0.170511 −0.0852554 0.996359i \(-0.527171\pi\)
−0.0852554 + 0.996359i \(0.527171\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 412.999i 1.32797i 0.747745 + 0.663985i \(0.231136\pi\)
−0.747745 + 0.663985i \(0.768864\pi\)
\(312\) 0 0
\(313\) −447.636 −1.43015 −0.715073 0.699050i \(-0.753607\pi\)
−0.715073 + 0.699050i \(0.753607\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 422.582i 1.33307i 0.745476 + 0.666533i \(0.232222\pi\)
−0.745476 + 0.666533i \(0.767778\pi\)
\(318\) 0 0
\(319\) −16.3060 −0.0511161
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 197.394i − 0.611127i
\(324\) 0 0
\(325\) −386.969 −1.19068
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 5.72107i − 0.0173893i
\(330\) 0 0
\(331\) 211.082 0.637711 0.318855 0.947803i \(-0.396702\pi\)
0.318855 + 0.947803i \(0.396702\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 49.1510i − 0.146719i
\(336\) 0 0
\(337\) −113.455 −0.336662 −0.168331 0.985731i \(-0.553838\pi\)
−0.168331 + 0.985731i \(0.553838\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 951.604i − 2.79063i
\(342\) 0 0
\(343\) 84.0920 0.245166
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 448.485i 1.29246i 0.763141 + 0.646232i \(0.223656\pi\)
−0.763141 + 0.646232i \(0.776344\pi\)
\(348\) 0 0
\(349\) −260.454 −0.746287 −0.373143 0.927774i \(-0.621720\pi\)
−0.373143 + 0.927774i \(0.621720\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 541.129i − 1.53294i −0.642279 0.766471i \(-0.722011\pi\)
0.642279 0.766471i \(-0.277989\pi\)
\(354\) 0 0
\(355\) −180.483 −0.508403
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 314.908i 0.877181i 0.898687 + 0.438591i \(0.144522\pi\)
−0.898687 + 0.438591i \(0.855478\pi\)
\(360\) 0 0
\(361\) −209.727 −0.580960
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 41.6655i 0.114152i
\(366\) 0 0
\(367\) −227.677 −0.620374 −0.310187 0.950676i \(-0.600392\pi\)
−0.310187 + 0.950676i \(0.600392\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 428.060i − 1.15380i
\(372\) 0 0
\(373\) 165.546 0.443823 0.221911 0.975067i \(-0.428770\pi\)
0.221911 + 0.975067i \(0.428770\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 19.0702i 0.0505842i
\(378\) 0 0
\(379\) 124.708 0.329044 0.164522 0.986373i \(-0.447392\pi\)
0.164522 + 0.986373i \(0.447392\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 382.454i − 0.998575i −0.866436 0.499287i \(-0.833595\pi\)
0.866436 0.499287i \(-0.166405\pi\)
\(384\) 0 0
\(385\) 1104.18 2.86800
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 215.057i 0.552845i 0.961036 + 0.276423i \(0.0891489\pi\)
−0.961036 + 0.276423i \(0.910851\pi\)
\(390\) 0 0
\(391\) 428.463 1.09581
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 281.728i 0.713234i
\(396\) 0 0
\(397\) −241.909 −0.609343 −0.304672 0.952457i \(-0.598547\pi\)
−0.304672 + 0.952457i \(0.598547\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 325.183i 0.810929i 0.914111 + 0.405464i \(0.132890\pi\)
−0.914111 + 0.405464i \(0.867110\pi\)
\(402\) 0 0
\(403\) −1112.92 −2.76159
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1134.21i − 2.78676i
\(408\) 0 0
\(409\) 419.302 1.02519 0.512594 0.858631i \(-0.328685\pi\)
0.512594 + 0.858631i \(0.328685\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 673.872i 1.63165i
\(414\) 0 0
\(415\) 477.950 1.15169
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 337.151i − 0.804656i −0.915496 0.402328i \(-0.868201\pi\)
0.915496 0.402328i \(-0.131799\pi\)
\(420\) 0 0
\(421\) −422.969 −1.00468 −0.502339 0.864671i \(-0.667527\pi\)
−0.502339 + 0.864671i \(0.667527\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 300.070i 0.706047i
\(426\) 0 0
\(427\) 37.7552 0.0884196
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 137.333i − 0.318637i −0.987227 0.159319i \(-0.949070\pi\)
0.987227 0.159319i \(-0.0509297\pi\)
\(432\) 0 0
\(433\) 523.545 1.20911 0.604555 0.796563i \(-0.293351\pi\)
0.604555 + 0.796563i \(0.293351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 328.354i 0.751383i
\(438\) 0 0
\(439\) 566.617 1.29070 0.645349 0.763888i \(-0.276712\pi\)
0.645349 + 0.763888i \(0.276712\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 592.788i 1.33812i 0.743208 + 0.669061i \(0.233303\pi\)
−0.743208 + 0.669061i \(0.766697\pi\)
\(444\) 0 0
\(445\) 836.120 1.87892
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 127.151i − 0.283187i −0.989925 0.141593i \(-0.954777\pi\)
0.989925 0.141593i \(-0.0452225\pi\)
\(450\) 0 0
\(451\) 350.382 0.776899
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1291.36i − 2.83816i
\(456\) 0 0
\(457\) −192.576 −0.421391 −0.210695 0.977552i \(-0.567573\pi\)
−0.210695 + 0.977552i \(0.567573\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 227.077i 0.492575i 0.969197 + 0.246287i \(0.0792107\pi\)
−0.969197 + 0.246287i \(0.920789\pi\)
\(462\) 0 0
\(463\) −638.399 −1.37883 −0.689416 0.724365i \(-0.742133\pi\)
−0.689416 + 0.724365i \(0.742133\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 604.423i 1.29427i 0.762376 + 0.647134i \(0.224033\pi\)
−0.762376 + 0.647134i \(0.775967\pi\)
\(468\) 0 0
\(469\) 70.1816 0.149641
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 1235.06i − 2.61111i
\(474\) 0 0
\(475\) −229.960 −0.484126
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 368.302i 0.768898i 0.923146 + 0.384449i \(0.125609\pi\)
−0.923146 + 0.384449i \(0.874391\pi\)
\(480\) 0 0
\(481\) −1326.48 −2.75776
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 178.480i 0.368000i
\(486\) 0 0
\(487\) −536.865 −1.10239 −0.551196 0.834376i \(-0.685828\pi\)
−0.551196 + 0.834376i \(0.685828\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 405.606i 0.826082i 0.910712 + 0.413041i \(0.135533\pi\)
−0.910712 + 0.413041i \(0.864467\pi\)
\(492\) 0 0
\(493\) 14.7878 0.0299954
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 257.708i − 0.518527i
\(498\) 0 0
\(499\) −874.090 −1.75168 −0.875842 0.482598i \(-0.839693\pi\)
−0.875842 + 0.482598i \(0.839693\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 586.515i − 1.16603i −0.812460 0.583017i \(-0.801872\pi\)
0.812460 0.583017i \(-0.198128\pi\)
\(504\) 0 0
\(505\) −223.302 −0.442182
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 541.096i − 1.06306i −0.847040 0.531529i \(-0.821618\pi\)
0.847040 0.531529i \(-0.178382\pi\)
\(510\) 0 0
\(511\) −59.4933 −0.116425
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 629.605i − 1.22253i
\(516\) 0 0
\(517\) −10.7265 −0.0207476
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 563.050i 1.08071i 0.841437 + 0.540355i \(0.181710\pi\)
−0.841437 + 0.540355i \(0.818290\pi\)
\(522\) 0 0
\(523\) −353.232 −0.675396 −0.337698 0.941254i \(-0.609648\pi\)
−0.337698 + 0.941254i \(0.609648\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 862.999i 1.63757i
\(528\) 0 0
\(529\) −183.727 −0.347309
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 409.778i − 0.768815i
\(534\) 0 0
\(535\) −638.399 −1.19327
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 709.485i 1.31630i
\(540\) 0 0
\(541\) 122.302 0.226067 0.113033 0.993591i \(-0.463943\pi\)
0.113033 + 0.993591i \(0.463943\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 600.741i 1.10228i
\(546\) 0 0
\(547\) −189.055 −0.345622 −0.172811 0.984955i \(-0.555285\pi\)
−0.172811 + 0.984955i \(0.555285\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.3326i 0.0205674i
\(552\) 0 0
\(553\) −402.272 −0.727437
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 522.883i − 0.938749i −0.882999 0.469375i \(-0.844479\pi\)
0.882999 0.469375i \(-0.155521\pi\)
\(558\) 0 0
\(559\) −1444.42 −2.58394
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 743.574i 1.32074i 0.750942 + 0.660368i \(0.229600\pi\)
−0.750942 + 0.660368i \(0.770400\pi\)
\(564\) 0 0
\(565\) −331.728 −0.587128
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 317.984i − 0.558848i −0.960168 0.279424i \(-0.909857\pi\)
0.960168 0.279424i \(-0.0901435\pi\)
\(570\) 0 0
\(571\) 1032.27 1.80782 0.903911 0.427720i \(-0.140683\pi\)
0.903911 + 0.427720i \(0.140683\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 499.151i − 0.868089i
\(576\) 0 0
\(577\) 374.545 0.649125 0.324562 0.945864i \(-0.394783\pi\)
0.324562 + 0.945864i \(0.394783\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 682.454i 1.17462i
\(582\) 0 0
\(583\) −802.577 −1.37663
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 338.455i 0.576585i 0.957542 + 0.288292i \(0.0930875\pi\)
−0.957542 + 0.288292i \(0.906913\pi\)
\(588\) 0 0
\(589\) −661.362 −1.12286
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 134.371i − 0.226596i −0.993561 0.113298i \(-0.963859\pi\)
0.993561 0.113298i \(-0.0361414\pi\)
\(594\) 0 0
\(595\) −1001.37 −1.68297
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 615.031i − 1.02676i −0.858161 0.513381i \(-0.828393\pi\)
0.858161 0.513381i \(-0.171607\pi\)
\(600\) 0 0
\(601\) 373.817 0.621992 0.310996 0.950411i \(-0.399337\pi\)
0.310996 + 0.950411i \(0.399337\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 1270.39i − 2.09982i
\(606\) 0 0
\(607\) −471.082 −0.776083 −0.388042 0.921642i \(-0.626848\pi\)
−0.388042 + 0.921642i \(0.626848\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.5449i 0.0205317i
\(612\) 0 0
\(613\) 210.727 0.343763 0.171881 0.985118i \(-0.445015\pi\)
0.171881 + 0.985118i \(0.445015\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1037.73i 1.68190i 0.541114 + 0.840949i \(0.318003\pi\)
−0.541114 + 0.840949i \(0.681997\pi\)
\(618\) 0 0
\(619\) −75.7896 −0.122439 −0.0612194 0.998124i \(-0.519499\pi\)
−0.0612194 + 0.998124i \(0.519499\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1193.88i 1.91634i
\(624\) 0 0
\(625\) −742.848 −1.18856
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1028.60i 1.63530i
\(630\) 0 0
\(631\) 289.174 0.458279 0.229139 0.973394i \(-0.426409\pi\)
0.229139 + 0.973394i \(0.426409\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 886.757i 1.39647i
\(636\) 0 0
\(637\) 829.757 1.30260
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 985.234i 1.53703i 0.639834 + 0.768513i \(0.279003\pi\)
−0.639834 + 0.768513i \(0.720997\pi\)
\(642\) 0 0
\(643\) 743.103 1.15568 0.577840 0.816150i \(-0.303896\pi\)
0.577840 + 0.816150i \(0.303896\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 328.849i − 0.508267i −0.967169 0.254134i \(-0.918210\pi\)
0.967169 0.254134i \(-0.0817903\pi\)
\(648\) 0 0
\(649\) 1263.45 1.94677
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 205.865i 0.315261i 0.987498 + 0.157630i \(0.0503854\pi\)
−0.987498 + 0.157630i \(0.949615\pi\)
\(654\) 0 0
\(655\) 309.487 0.472499
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 364.151i 0.552581i 0.961074 + 0.276291i \(0.0891052\pi\)
−0.961074 + 0.276291i \(0.910895\pi\)
\(660\) 0 0
\(661\) 507.728 0.768120 0.384060 0.923308i \(-0.374526\pi\)
0.384060 + 0.923308i \(0.374526\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 767.403i − 1.15399i
\(666\) 0 0
\(667\) −24.5987 −0.0368795
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 70.7878i − 0.105496i
\(672\) 0 0
\(673\) 847.363 1.25908 0.629542 0.776967i \(-0.283243\pi\)
0.629542 + 0.776967i \(0.283243\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 902.076i − 1.33246i −0.745746 0.666230i \(-0.767907\pi\)
0.745746 0.666230i \(-0.232093\pi\)
\(678\) 0 0
\(679\) −254.847 −0.375328
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 754.604i − 1.10484i −0.833567 0.552419i \(-0.813705\pi\)
0.833567 0.552419i \(-0.186295\pi\)
\(684\) 0 0
\(685\) −1209.21 −1.76527
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 938.630i 1.36231i
\(690\) 0 0
\(691\) 420.450 0.608466 0.304233 0.952598i \(-0.401600\pi\)
0.304233 + 0.952598i \(0.401600\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 911.333i − 1.31127i
\(696\) 0 0
\(697\) −317.757 −0.455893
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 423.953i 0.604783i 0.953184 + 0.302391i \(0.0977850\pi\)
−0.953184 + 0.302391i \(0.902215\pi\)
\(702\) 0 0
\(703\) −788.274 −1.12130
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 318.848i − 0.450987i
\(708\) 0 0
\(709\) 14.3020 0.0201721 0.0100861 0.999949i \(-0.496789\pi\)
0.0100861 + 0.999949i \(0.496789\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 1435.55i − 2.01340i
\(714\) 0 0
\(715\) −2421.19 −3.38629
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 453.453i − 0.630672i −0.948980 0.315336i \(-0.897883\pi\)
0.948980 0.315336i \(-0.102117\pi\)
\(720\) 0 0
\(721\) 898.999 1.24688
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 17.2274i − 0.0237620i
\(726\) 0 0
\(727\) −818.324 −1.12562 −0.562809 0.826587i \(-0.690279\pi\)
−0.562809 + 0.826587i \(0.690279\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1120.06i 1.53223i
\(732\) 0 0
\(733\) 164.636 0.224605 0.112303 0.993674i \(-0.464177\pi\)
0.112303 + 0.993674i \(0.464177\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 131.585i − 0.178541i
\(738\) 0 0
\(739\) 392.143 0.530641 0.265320 0.964160i \(-0.414522\pi\)
0.265320 + 0.964160i \(0.414522\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1341.12i − 1.80501i −0.430684 0.902503i \(-0.641728\pi\)
0.430684 0.902503i \(-0.358272\pi\)
\(744\) 0 0
\(745\) −288.455 −0.387188
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 911.556i − 1.21703i
\(750\) 0 0
\(751\) 330.637 0.440262 0.220131 0.975470i \(-0.429351\pi\)
0.220131 + 0.975470i \(0.429351\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 555.879i − 0.736263i
\(756\) 0 0
\(757\) −70.1816 −0.0927102 −0.0463551 0.998925i \(-0.514761\pi\)
−0.0463551 + 0.998925i \(0.514761\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 761.704i 1.00093i 0.865758 + 0.500463i \(0.166837\pi\)
−0.865758 + 0.500463i \(0.833163\pi\)
\(762\) 0 0
\(763\) −857.784 −1.12423
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1477.63i − 1.92651i
\(768\) 0 0
\(769\) −853.272 −1.10959 −0.554794 0.831988i \(-0.687203\pi\)
−0.554794 + 0.831988i \(0.687203\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 487.925i 0.631209i 0.948891 + 0.315605i \(0.102207\pi\)
−0.948891 + 0.315605i \(0.897793\pi\)
\(774\) 0 0
\(775\) 1005.38 1.29726
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 243.514i − 0.312599i
\(780\) 0 0
\(781\) −483.181 −0.618669
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1563.65i − 1.99191i
\(786\) 0 0
\(787\) 492.532 0.625834 0.312917 0.949780i \(-0.398694\pi\)
0.312917 + 0.949780i \(0.398694\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 473.666i − 0.598820i
\(792\) 0 0
\(793\) −82.7878 −0.104398
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 361.085i 0.453055i 0.974005 + 0.226528i \(0.0727374\pi\)
−0.974005 + 0.226528i \(0.927263\pi\)
\(798\) 0 0
\(799\) 9.72777 0.0121749
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 111.545i 0.138910i
\(804\) 0 0
\(805\) 1665.73 2.06922
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 579.548i − 0.716376i −0.933649 0.358188i \(-0.883395\pi\)
0.933649 0.358188i \(-0.116605\pi\)
\(810\) 0 0
\(811\) 1406.37 1.73412 0.867059 0.498205i \(-0.166007\pi\)
0.867059 + 0.498205i \(0.166007\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1866.15i 2.28975i
\(816\) 0 0
\(817\) −858.361 −1.05063
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 111.702i − 0.136056i −0.997683 0.0680280i \(-0.978329\pi\)
0.997683 0.0680280i \(-0.0216707\pi\)
\(822\) 0 0
\(823\) 161.895 0.196713 0.0983564 0.995151i \(-0.468641\pi\)
0.0983564 + 0.995151i \(0.468641\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 592.788i 0.716793i 0.933569 + 0.358396i \(0.116676\pi\)
−0.933569 + 0.358396i \(0.883324\pi\)
\(828\) 0 0
\(829\) −950.363 −1.14640 −0.573199 0.819416i \(-0.694298\pi\)
−0.573199 + 0.819416i \(0.694298\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 643.424i − 0.772418i
\(834\) 0 0
\(835\) 463.926 0.555600
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 326.636i − 0.389316i −0.980871 0.194658i \(-0.937640\pi\)
0.980871 0.194658i \(-0.0623596\pi\)
\(840\) 0 0
\(841\) 840.151 0.998991
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1714.49i 2.02898i
\(846\) 0 0
\(847\) 1813.96 2.14163
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1711.03i − 2.01061i
\(852\) 0 0
\(853\) −906.727 −1.06299 −0.531493 0.847063i \(-0.678369\pi\)
−0.531493 + 0.847063i \(0.678369\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 715.483i 0.834869i 0.908707 + 0.417435i \(0.137071\pi\)
−0.908707 + 0.417435i \(0.862929\pi\)
\(858\) 0 0
\(859\) 49.7558 0.0579229 0.0289615 0.999581i \(-0.490780\pi\)
0.0289615 + 0.999581i \(0.490780\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1595.76i 1.84908i 0.381086 + 0.924539i \(0.375550\pi\)
−0.381086 + 0.924539i \(0.624450\pi\)
\(864\) 0 0
\(865\) −2238.18 −2.58749
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 754.227i 0.867925i
\(870\) 0 0
\(871\) −153.891 −0.176683
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 393.272i − 0.449454i
\(876\) 0 0
\(877\) −494.182 −0.563491 −0.281746 0.959489i \(-0.590913\pi\)
−0.281746 + 0.959489i \(0.590913\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 728.383i 0.826768i 0.910557 + 0.413384i \(0.135653\pi\)
−0.910557 + 0.413384i \(0.864347\pi\)
\(882\) 0 0
\(883\) −1378.34 −1.56098 −0.780489 0.625170i \(-0.785030\pi\)
−0.780489 + 0.625170i \(0.785030\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1329.00i 1.49831i 0.662397 + 0.749153i \(0.269539\pi\)
−0.662397 + 0.749153i \(0.730461\pi\)
\(888\) 0 0
\(889\) −1266.18 −1.42428
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.45491i 0.00834816i
\(894\) 0 0
\(895\) 2304.21 2.57454
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 49.5459i − 0.0551123i
\(900\) 0 0
\(901\) 727.848 0.807822
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 106.967i 0.118195i
\(906\) 0 0
\(907\) 940.152 1.03655 0.518276 0.855214i \(-0.326574\pi\)
0.518276 + 0.855214i \(0.326574\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 962.486i 1.05652i 0.849084 + 0.528258i \(0.177155\pi\)
−0.849084 + 0.528258i \(0.822845\pi\)
\(912\) 0 0
\(913\) 1279.54 1.40147
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 441.909i 0.481907i
\(918\) 0 0
\(919\) 441.610 0.480533 0.240267 0.970707i \(-0.422765\pi\)
0.240267 + 0.970707i \(0.422765\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 565.090i 0.612232i
\(924\) 0 0
\(925\) 1198.30 1.29546
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 401.078i 0.431731i 0.976423 + 0.215866i \(0.0692573\pi\)
−0.976423 + 0.215866i \(0.930743\pi\)
\(930\) 0 0
\(931\) 493.090 0.529635
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1877.48i 2.00800i
\(936\) 0 0
\(937\) 493.000 0.526147 0.263074 0.964776i \(-0.415264\pi\)
0.263074 + 0.964776i \(0.415264\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 556.439i − 0.591328i −0.955292 0.295664i \(-0.904459\pi\)
0.955292 0.295664i \(-0.0955408\pi\)
\(942\) 0 0
\(943\) 528.572 0.560522
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1295.91i − 1.36844i −0.729278 0.684218i \(-0.760144\pi\)
0.729278 0.684218i \(-0.239856\pi\)
\(948\) 0 0
\(949\) 130.454 0.137465
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 838.672i 0.880034i 0.897990 + 0.440017i \(0.145028\pi\)
−0.897990 + 0.440017i \(0.854972\pi\)
\(954\) 0 0
\(955\) −1132.38 −1.18573
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1726.60i − 1.80042i
\(960\) 0 0
\(961\) 1930.45 2.00880
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1593.70i − 1.65150i
\(966\) 0 0
\(967\) 96.6705 0.0999695 0.0499848 0.998750i \(-0.484083\pi\)
0.0499848 + 0.998750i \(0.484083\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1218.66i 1.25506i 0.778592 + 0.627531i \(0.215934\pi\)
−0.778592 + 0.627531i \(0.784066\pi\)
\(972\) 0 0
\(973\) 1301.27 1.33738
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 315.692i 0.323124i 0.986863 + 0.161562i \(0.0516532\pi\)
−0.986863 + 0.161562i \(0.948347\pi\)
\(978\) 0 0
\(979\) 2238.42 2.28643
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 40.2429i − 0.0409388i −0.999790 0.0204694i \(-0.993484\pi\)
0.999790 0.0204694i \(-0.00651607\pi\)
\(984\) 0 0
\(985\) 1204.18 1.22252
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1863.16i − 1.88388i
\(990\) 0 0
\(991\) −173.635 −0.175212 −0.0876062 0.996155i \(-0.527922\pi\)
−0.0876062 + 0.996155i \(0.527922\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 463.243i − 0.465571i
\(996\) 0 0
\(997\) −818.454 −0.820917 −0.410458 0.911879i \(-0.634631\pi\)
−0.410458 + 0.911879i \(0.634631\pi\)
\(998\) 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 864.3.e.f.161.7 yes 8
3.2 odd 2 inner 864.3.e.f.161.1 8
4.3 odd 2 inner 864.3.e.f.161.8 yes 8
8.3 odd 2 1728.3.e.u.1025.2 8
8.5 even 2 1728.3.e.u.1025.1 8
12.11 even 2 inner 864.3.e.f.161.2 yes 8
24.5 odd 2 1728.3.e.u.1025.7 8
24.11 even 2 1728.3.e.u.1025.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.f.161.1 8 3.2 odd 2 inner
864.3.e.f.161.2 yes 8 12.11 even 2 inner
864.3.e.f.161.7 yes 8 1.1 even 1 trivial
864.3.e.f.161.8 yes 8 4.3 odd 2 inner
1728.3.e.u.1025.1 8 8.5 even 2
1728.3.e.u.1025.2 8 8.3 odd 2
1728.3.e.u.1025.7 8 24.5 odd 2
1728.3.e.u.1025.8 8 24.11 even 2