Properties

Label 2304.3.e.o.1025.8
Level $2304$
Weight $3$
Character 2304.1025
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.540942598144.16
Defining polynomial: \(x^{8} + 16 x^{6} + 71 x^{4} + 56 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.8
Root \(-2.78961i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1025
Dual form 2304.3.e.o.1025.2

$q$-expansion

\(f(q)\) \(=\) \(q+7.67101i q^{5} -7.21110 q^{7} +O(q^{10})\) \(q+7.67101i q^{5} -7.21110 q^{7} -6.05164i q^{11} +2.29014 q^{13} -21.8103i q^{17} +34.8355 q^{19} +21.5117i q^{23} -33.8444 q^{25} -10.9098i q^{29} +37.6333 q^{31} -55.3164i q^{35} -34.8355 q^{37} -13.3250i q^{41} +60.5104 q^{43} +3.34701i q^{47} +3.00000 q^{49} -35.1163i q^{53} +46.4222 q^{55} -37.1615i q^{59} +25.6749 q^{61} +17.5677i q^{65} +25.6749 q^{67} +37.2881i q^{71} +77.6888 q^{73} +43.6390i q^{77} -31.3221 q^{79} +55.3164i q^{83} +167.307 q^{85} +5.43682i q^{89} -16.5144 q^{91} +267.223i q^{95} -52.8444 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 40q^{25} + 128q^{31} + 24q^{49} + 256q^{55} + 160q^{73} + 384q^{79} - 192q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 7.67101i 1.53420i 0.641526 + 0.767101i \(0.278302\pi\)
−0.641526 + 0.767101i \(0.721698\pi\)
\(6\) 0 0
\(7\) −7.21110 −1.03016 −0.515079 0.857143i \(-0.672237\pi\)
−0.515079 + 0.857143i \(0.672237\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 6.05164i − 0.550149i −0.961423 0.275075i \(-0.911297\pi\)
0.961423 0.275075i \(-0.0887026\pi\)
\(12\) 0 0
\(13\) 2.29014 0.176164 0.0880821 0.996113i \(-0.471926\pi\)
0.0880821 + 0.996113i \(0.471926\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 21.8103i − 1.28296i −0.767140 0.641479i \(-0.778321\pi\)
0.767140 0.641479i \(-0.221679\pi\)
\(18\) 0 0
\(19\) 34.8355 1.83345 0.916723 0.399523i \(-0.130824\pi\)
0.916723 + 0.399523i \(0.130824\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 21.5117i 0.935293i 0.883915 + 0.467647i \(0.154898\pi\)
−0.883915 + 0.467647i \(0.845102\pi\)
\(24\) 0 0
\(25\) −33.8444 −1.35378
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 10.9098i − 0.376198i −0.982150 0.188099i \(-0.939767\pi\)
0.982150 0.188099i \(-0.0602326\pi\)
\(30\) 0 0
\(31\) 37.6333 1.21398 0.606989 0.794710i \(-0.292377\pi\)
0.606989 + 0.794710i \(0.292377\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 55.3164i − 1.58047i
\(36\) 0 0
\(37\) −34.8355 −0.941499 −0.470750 0.882267i \(-0.656017\pi\)
−0.470750 + 0.882267i \(0.656017\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 13.3250i − 0.325000i −0.986709 0.162500i \(-0.948044\pi\)
0.986709 0.162500i \(-0.0519558\pi\)
\(42\) 0 0
\(43\) 60.5104 1.40722 0.703609 0.710587i \(-0.251570\pi\)
0.703609 + 0.710587i \(0.251570\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.34701i 0.0712129i 0.999366 + 0.0356065i \(0.0113363\pi\)
−0.999366 + 0.0356065i \(0.988664\pi\)
\(48\) 0 0
\(49\) 3.00000 0.0612245
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 35.1163i − 0.662572i −0.943530 0.331286i \(-0.892518\pi\)
0.943530 0.331286i \(-0.107482\pi\)
\(54\) 0 0
\(55\) 46.4222 0.844040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 37.1615i − 0.629856i −0.949116 0.314928i \(-0.898020\pi\)
0.949116 0.314928i \(-0.101980\pi\)
\(60\) 0 0
\(61\) 25.6749 0.420901 0.210450 0.977605i \(-0.432507\pi\)
0.210450 + 0.977605i \(0.432507\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 17.5677i 0.270272i
\(66\) 0 0
\(67\) 25.6749 0.383208 0.191604 0.981472i \(-0.438631\pi\)
0.191604 + 0.981472i \(0.438631\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 37.2881i 0.525185i 0.964907 + 0.262592i \(0.0845775\pi\)
−0.964907 + 0.262592i \(0.915423\pi\)
\(72\) 0 0
\(73\) 77.6888 1.06423 0.532115 0.846672i \(-0.321397\pi\)
0.532115 + 0.846672i \(0.321397\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 43.6390i 0.566740i
\(78\) 0 0
\(79\) −31.3221 −0.396483 −0.198241 0.980153i \(-0.563523\pi\)
−0.198241 + 0.980153i \(0.563523\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 55.3164i 0.666463i 0.942845 + 0.333232i \(0.108139\pi\)
−0.942845 + 0.333232i \(0.891861\pi\)
\(84\) 0 0
\(85\) 167.307 1.96832
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.43682i 0.0610878i 0.999533 + 0.0305439i \(0.00972394\pi\)
−0.999533 + 0.0305439i \(0.990276\pi\)
\(90\) 0 0
\(91\) −16.5144 −0.181477
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 267.223i 2.81288i
\(96\) 0 0
\(97\) −52.8444 −0.544788 −0.272394 0.962186i \(-0.587815\pi\)
−0.272394 + 0.962186i \(0.587815\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 90.0069i 0.891158i 0.895243 + 0.445579i \(0.147002\pi\)
−0.895243 + 0.445579i \(0.852998\pi\)
\(102\) 0 0
\(103\) 119.211 1.15739 0.578695 0.815544i \(-0.303562\pi\)
0.578695 + 0.815544i \(0.303562\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 24.2066i − 0.226230i −0.993582 0.113115i \(-0.963917\pi\)
0.993582 0.113115i \(-0.0360828\pi\)
\(108\) 0 0
\(109\) −132.472 −1.21533 −0.607667 0.794192i \(-0.707895\pi\)
−0.607667 + 0.794192i \(0.707895\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 160.560i 1.42089i 0.703754 + 0.710444i \(0.251506\pi\)
−0.703754 + 0.710444i \(0.748494\pi\)
\(114\) 0 0
\(115\) −165.017 −1.43493
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 157.276i 1.32165i
\(120\) 0 0
\(121\) 84.3776 0.697336
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 67.8456i − 0.542765i
\(126\) 0 0
\(127\) 146.478 1.15337 0.576684 0.816967i \(-0.304346\pi\)
0.576684 + 0.816967i \(0.304346\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 184.956i 1.41188i 0.708273 + 0.705939i \(0.249475\pi\)
−0.708273 + 0.705939i \(0.750525\pi\)
\(132\) 0 0
\(133\) −251.202 −1.88874
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 133.313i 0.973089i 0.873656 + 0.486544i \(0.161743\pi\)
−0.873656 + 0.486544i \(0.838257\pi\)
\(138\) 0 0
\(139\) −165.017 −1.18717 −0.593586 0.804771i \(-0.702288\pi\)
−0.593586 + 0.804771i \(0.702288\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 13.8591i − 0.0969166i
\(144\) 0 0
\(145\) 83.6888 0.577164
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 228.937i − 1.53649i −0.640157 0.768244i \(-0.721131\pi\)
0.640157 0.768244i \(-0.278869\pi\)
\(150\) 0 0
\(151\) 162.478 1.07601 0.538006 0.842941i \(-0.319178\pi\)
0.538006 + 0.842941i \(0.319178\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 288.686i 1.86249i
\(156\) 0 0
\(157\) 113.667 0.723993 0.361997 0.932179i \(-0.382095\pi\)
0.361997 + 0.932179i \(0.382095\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 155.123i − 0.963499i
\(162\) 0 0
\(163\) −111.860 −0.686259 −0.343130 0.939288i \(-0.611487\pi\)
−0.343130 + 0.939288i \(0.611487\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 151.541i 0.907430i 0.891147 + 0.453715i \(0.149902\pi\)
−0.891147 + 0.453715i \(0.850098\pi\)
\(168\) 0 0
\(169\) −163.755 −0.968966
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 98.0198i 0.566588i 0.959033 + 0.283294i \(0.0914273\pi\)
−0.959033 + 0.283294i \(0.908573\pi\)
\(174\) 0 0
\(175\) 244.056 1.39460
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 222.117i − 1.24088i −0.784254 0.620440i \(-0.786954\pi\)
0.784254 0.620440i \(-0.213046\pi\)
\(180\) 0 0
\(181\) 118.731 0.655971 0.327985 0.944683i \(-0.393630\pi\)
0.327985 + 0.944683i \(0.393630\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 267.223i − 1.44445i
\(186\) 0 0
\(187\) −131.988 −0.705818
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 151.541i 0.793408i 0.917947 + 0.396704i \(0.129846\pi\)
−0.917947 + 0.396704i \(0.870154\pi\)
\(192\) 0 0
\(193\) 113.378 0.587449 0.293724 0.955890i \(-0.405105\pi\)
0.293724 + 0.955890i \(0.405105\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 319.454i 1.62159i 0.585329 + 0.810796i \(0.300965\pi\)
−0.585329 + 0.810796i \(0.699035\pi\)
\(198\) 0 0
\(199\) −41.0109 −0.206085 −0.103043 0.994677i \(-0.532858\pi\)
−0.103043 + 0.994677i \(0.532858\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 78.6713i 0.387544i
\(204\) 0 0
\(205\) 102.216 0.498616
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 210.812i − 1.00867i
\(210\) 0 0
\(211\) 95.3459 0.451876 0.225938 0.974142i \(-0.427455\pi\)
0.225938 + 0.974142i \(0.427455\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 464.176i 2.15896i
\(216\) 0 0
\(217\) −271.378 −1.25059
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 49.9485i − 0.226011i
\(222\) 0 0
\(223\) 268.167 1.20254 0.601270 0.799046i \(-0.294662\pi\)
0.601270 + 0.799046i \(0.294662\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 65.7164i 0.289499i 0.989468 + 0.144750i \(0.0462377\pi\)
−0.989468 + 0.144750i \(0.953762\pi\)
\(228\) 0 0
\(229\) −155.373 −0.678484 −0.339242 0.940699i \(-0.610171\pi\)
−0.339242 + 0.940699i \(0.610171\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 285.451i 1.22511i 0.790427 + 0.612556i \(0.209859\pi\)
−0.790427 + 0.612556i \(0.790141\pi\)
\(234\) 0 0
\(235\) −25.6749 −0.109255
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 421.153i 1.76214i 0.472982 + 0.881072i \(0.343178\pi\)
−0.472982 + 0.881072i \(0.656822\pi\)
\(240\) 0 0
\(241\) −191.600 −0.795019 −0.397510 0.917598i \(-0.630126\pi\)
−0.397510 + 0.917598i \(0.630126\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 23.0130i 0.0939307i
\(246\) 0 0
\(247\) 79.7779 0.322988
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 483.190i − 1.92506i −0.271178 0.962529i \(-0.587413\pi\)
0.271178 0.962529i \(-0.412587\pi\)
\(252\) 0 0
\(253\) 130.181 0.514551
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 21.2132i − 0.0825416i −0.999148 0.0412708i \(-0.986859\pi\)
0.999148 0.0412708i \(-0.0131406\pi\)
\(258\) 0 0
\(259\) 251.202 0.969893
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 246.670i 0.937910i 0.883222 + 0.468955i \(0.155369\pi\)
−0.883222 + 0.468955i \(0.844631\pi\)
\(264\) 0 0
\(265\) 269.378 1.01652
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 241.040i − 0.896060i −0.894019 0.448030i \(-0.852126\pi\)
0.894019 0.448030i \(-0.147874\pi\)
\(270\) 0 0
\(271\) −61.7443 −0.227839 −0.113919 0.993490i \(-0.536341\pi\)
−0.113919 + 0.993490i \(0.536341\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 204.814i 0.744779i
\(276\) 0 0
\(277\) 379.093 1.36857 0.684284 0.729215i \(-0.260115\pi\)
0.684284 + 0.729215i \(0.260115\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 301.227i − 1.07198i −0.844223 0.535992i \(-0.819938\pi\)
0.844223 0.535992i \(-0.180062\pi\)
\(282\) 0 0
\(283\) 399.705 1.41238 0.706192 0.708020i \(-0.250412\pi\)
0.706192 + 0.708020i \(0.250412\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 96.0880i 0.334801i
\(288\) 0 0
\(289\) −186.689 −0.645982
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 258.085i − 0.880838i −0.897792 0.440419i \(-0.854830\pi\)
0.897792 0.440419i \(-0.145170\pi\)
\(294\) 0 0
\(295\) 285.066 0.966327
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 49.2648i 0.164765i
\(300\) 0 0
\(301\) −436.347 −1.44966
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 196.953i 0.645747i
\(306\) 0 0
\(307\) 286.038 0.931719 0.465859 0.884859i \(-0.345745\pi\)
0.465859 + 0.884859i \(0.345745\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 464.647i − 1.49404i −0.664800 0.747021i \(-0.731483\pi\)
0.664800 0.747021i \(-0.268517\pi\)
\(312\) 0 0
\(313\) −158.000 −0.504792 −0.252396 0.967624i \(-0.581219\pi\)
−0.252396 + 0.967624i \(0.581219\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.6388i 0.0903433i 0.998979 + 0.0451717i \(0.0143835\pi\)
−0.998979 + 0.0451717i \(0.985617\pi\)
\(318\) 0 0
\(319\) −66.0219 −0.206965
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 759.772i − 2.35224i
\(324\) 0 0
\(325\) −77.5083 −0.238487
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 24.1356i − 0.0733605i
\(330\) 0 0
\(331\) 428.993 1.29605 0.648026 0.761619i \(-0.275595\pi\)
0.648026 + 0.761619i \(0.275595\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 196.953i 0.587919i
\(336\) 0 0
\(337\) 290.755 0.862775 0.431388 0.902167i \(-0.358024\pi\)
0.431388 + 0.902167i \(0.358024\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 227.743i − 0.667869i
\(342\) 0 0
\(343\) 331.711 0.967087
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 179.756i − 0.518029i −0.965873 0.259014i \(-0.916602\pi\)
0.965873 0.259014i \(-0.0833977\pi\)
\(348\) 0 0
\(349\) −225.527 −0.646210 −0.323105 0.946363i \(-0.604727\pi\)
−0.323105 + 0.946363i \(0.604727\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 133.784i 0.378992i 0.981881 + 0.189496i \(0.0606854\pi\)
−0.981881 + 0.189496i \(0.939315\pi\)
\(354\) 0 0
\(355\) −286.038 −0.805740
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 239.018i 0.665787i 0.942964 + 0.332894i \(0.108025\pi\)
−0.942964 + 0.332894i \(0.891975\pi\)
\(360\) 0 0
\(361\) 852.511 2.36153
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 595.952i 1.63274i
\(366\) 0 0
\(367\) 188.389 0.513320 0.256660 0.966502i \(-0.417378\pi\)
0.256660 + 0.966502i \(0.417378\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 253.227i 0.682554i
\(372\) 0 0
\(373\) −561.108 −1.50431 −0.752156 0.658985i \(-0.770986\pi\)
−0.752156 + 0.658985i \(0.770986\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 24.9848i − 0.0662727i
\(378\) 0 0
\(379\) 328.227 0.866034 0.433017 0.901386i \(-0.357449\pi\)
0.433017 + 0.901386i \(0.357449\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 204.118i − 0.532945i −0.963843 0.266472i \(-0.914142\pi\)
0.963843 0.266472i \(-0.0858581\pi\)
\(384\) 0 0
\(385\) −334.755 −0.869494
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 60.1746i 0.154690i 0.997004 + 0.0773452i \(0.0246444\pi\)
−0.997004 + 0.0773452i \(0.975356\pi\)
\(390\) 0 0
\(391\) 469.177 1.19994
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 240.272i − 0.608285i
\(396\) 0 0
\(397\) 592.203 1.49170 0.745848 0.666116i \(-0.232045\pi\)
0.745848 + 0.666116i \(0.232045\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 653.178i 1.62887i 0.580254 + 0.814436i \(0.302953\pi\)
−0.580254 + 0.814436i \(0.697047\pi\)
\(402\) 0 0
\(403\) 86.1854 0.213859
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 210.812i 0.517965i
\(408\) 0 0
\(409\) −355.156 −0.868351 −0.434176 0.900828i \(-0.642960\pi\)
−0.434176 + 0.900828i \(0.642960\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 267.976i 0.648851i
\(414\) 0 0
\(415\) −424.333 −1.02249
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 299.085i 0.713808i 0.934141 + 0.356904i \(0.116168\pi\)
−0.934141 + 0.356904i \(0.883832\pi\)
\(420\) 0 0
\(421\) 769.638 1.82812 0.914059 0.405582i \(-0.132931\pi\)
0.914059 + 0.405582i \(0.132931\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 738.156i 1.73684i
\(426\) 0 0
\(427\) −185.145 −0.433594
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 411.583i 0.954948i 0.878646 + 0.477474i \(0.158448\pi\)
−0.878646 + 0.477474i \(0.841552\pi\)
\(432\) 0 0
\(433\) 516.133 1.19199 0.595996 0.802987i \(-0.296757\pi\)
0.595996 + 0.802987i \(0.296757\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 749.372i 1.71481i
\(438\) 0 0
\(439\) −865.788 −1.97218 −0.986091 0.166205i \(-0.946849\pi\)
−0.986091 + 0.166205i \(0.946849\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 546.261i 1.23310i 0.787318 + 0.616548i \(0.211469\pi\)
−0.787318 + 0.616548i \(0.788531\pi\)
\(444\) 0 0
\(445\) −41.7059 −0.0937211
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 428.978i − 0.955407i −0.878521 0.477704i \(-0.841469\pi\)
0.878521 0.477704i \(-0.158531\pi\)
\(450\) 0 0
\(451\) −80.6382 −0.178799
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 126.682i − 0.278422i
\(456\) 0 0
\(457\) −75.5997 −0.165426 −0.0827130 0.996573i \(-0.526358\pi\)
−0.0827130 + 0.996573i \(0.526358\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 771.365i − 1.67324i −0.547781 0.836622i \(-0.684527\pi\)
0.547781 0.836622i \(-0.315473\pi\)
\(462\) 0 0
\(463\) 468.278 1.01140 0.505699 0.862710i \(-0.331235\pi\)
0.505699 + 0.862710i \(0.331235\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 452.842i − 0.969682i −0.874602 0.484841i \(-0.838877\pi\)
0.874602 0.484841i \(-0.161123\pi\)
\(468\) 0 0
\(469\) −185.145 −0.394765
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 366.187i − 0.774180i
\(474\) 0 0
\(475\) −1178.99 −2.48208
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 467.523i − 0.976040i −0.872832 0.488020i \(-0.837719\pi\)
0.872832 0.488020i \(-0.162281\pi\)
\(480\) 0 0
\(481\) −79.7779 −0.165859
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 405.370i − 0.835815i
\(486\) 0 0
\(487\) −863.766 −1.77365 −0.886824 0.462108i \(-0.847093\pi\)
−0.886824 + 0.462108i \(0.847093\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 149.498i 0.304476i 0.988344 + 0.152238i \(0.0486480\pi\)
−0.988344 + 0.152238i \(0.951352\pi\)
\(492\) 0 0
\(493\) −237.945 −0.482647
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 268.889i − 0.541023i
\(498\) 0 0
\(499\) −205.400 −0.411622 −0.205811 0.978592i \(-0.565983\pi\)
−0.205811 + 0.978592i \(0.565983\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 349.923i − 0.695673i −0.937555 0.347836i \(-0.886916\pi\)
0.937555 0.347836i \(-0.113084\pi\)
\(504\) 0 0
\(505\) −690.444 −1.36722
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 529.468i − 1.04021i −0.854102 0.520106i \(-0.825892\pi\)
0.854102 0.520106i \(-0.174108\pi\)
\(510\) 0 0
\(511\) −560.222 −1.09632
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 914.470i 1.77567i
\(516\) 0 0
\(517\) 20.2549 0.0391777
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 367.318i 0.705026i 0.935807 + 0.352513i \(0.114673\pi\)
−0.935807 + 0.352513i \(0.885327\pi\)
\(522\) 0 0
\(523\) −577.495 −1.10420 −0.552099 0.833779i \(-0.686173\pi\)
−0.552099 + 0.833779i \(0.686173\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 820.793i − 1.55748i
\(528\) 0 0
\(529\) 66.2447 0.125226
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 30.5161i − 0.0572534i
\(534\) 0 0
\(535\) 185.689 0.347082
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 18.1549i − 0.0336826i
\(540\) 0 0
\(541\) −326.777 −0.604023 −0.302012 0.953304i \(-0.597658\pi\)
−0.302012 + 0.953304i \(0.597658\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1016.19i − 1.86457i
\(546\) 0 0
\(547\) 273.137 0.499336 0.249668 0.968332i \(-0.419679\pi\)
0.249668 + 0.968332i \(0.419679\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 380.046i − 0.689739i
\(552\) 0 0
\(553\) 225.867 0.408440
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 93.2457i − 0.167407i −0.996491 0.0837035i \(-0.973325\pi\)
0.996491 0.0837035i \(-0.0266749\pi\)
\(558\) 0 0
\(559\) 138.577 0.247902
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 42.3615i 0.0752424i 0.999292 + 0.0376212i \(0.0119780\pi\)
−0.999292 + 0.0376212i \(0.988022\pi\)
\(564\) 0 0
\(565\) −1231.66 −2.17993
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 1055.57i − 1.85513i −0.373663 0.927564i \(-0.621898\pi\)
0.373663 0.927564i \(-0.378102\pi\)
\(570\) 0 0
\(571\) 21.9344 0.0384141 0.0192070 0.999816i \(-0.493886\pi\)
0.0192070 + 0.999816i \(0.493886\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 728.052i − 1.26618i
\(576\) 0 0
\(577\) 161.378 0.279684 0.139842 0.990174i \(-0.455341\pi\)
0.139842 + 0.990174i \(0.455341\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 398.893i − 0.686562i
\(582\) 0 0
\(583\) −212.511 −0.364513
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 481.396i 0.820096i 0.912064 + 0.410048i \(0.134488\pi\)
−0.912064 + 0.410048i \(0.865512\pi\)
\(588\) 0 0
\(589\) 1310.97 2.22576
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 321.781i 0.542632i 0.962490 + 0.271316i \(0.0874588\pi\)
−0.962490 + 0.271316i \(0.912541\pi\)
\(594\) 0 0
\(595\) −1206.47 −2.02768
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 35.8419i 0.0598362i 0.999552 + 0.0299181i \(0.00952464\pi\)
−0.999552 + 0.0299181i \(0.990475\pi\)
\(600\) 0 0
\(601\) −147.867 −0.246035 −0.123018 0.992404i \(-0.539257\pi\)
−0.123018 + 0.992404i \(0.539257\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 647.262i 1.06985i
\(606\) 0 0
\(607\) 636.611 1.04878 0.524391 0.851478i \(-0.324293\pi\)
0.524391 + 0.851478i \(0.324293\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.66510i 0.0125452i
\(612\) 0 0
\(613\) 870.887 1.42070 0.710348 0.703850i \(-0.248537\pi\)
0.710348 + 0.703850i \(0.248537\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1126.69i 1.82607i 0.407876 + 0.913037i \(0.366270\pi\)
−0.407876 + 0.913037i \(0.633730\pi\)
\(618\) 0 0
\(619\) 311.713 0.503575 0.251787 0.967783i \(-0.418982\pi\)
0.251787 + 0.967783i \(0.418982\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 39.2054i − 0.0629301i
\(624\) 0 0
\(625\) −325.666 −0.521066
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 759.772i 1.20790i
\(630\) 0 0
\(631\) −203.278 −0.322153 −0.161076 0.986942i \(-0.551497\pi\)
−0.161076 + 0.986942i \(0.551497\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1123.63i 1.76950i
\(636\) 0 0
\(637\) 6.87041 0.0107856
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 586.490i 0.914960i 0.889220 + 0.457480i \(0.151248\pi\)
−0.889220 + 0.457480i \(0.848752\pi\)
\(642\) 0 0
\(643\) −9.16054 −0.0142466 −0.00712328 0.999975i \(-0.502267\pi\)
−0.00712328 + 0.999975i \(0.502267\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 859.040i − 1.32773i −0.747853 0.663864i \(-0.768915\pi\)
0.747853 0.663864i \(-0.231085\pi\)
\(648\) 0 0
\(649\) −224.888 −0.346515
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 131.775i 0.201799i 0.994897 + 0.100899i \(0.0321720\pi\)
−0.994897 + 0.100899i \(0.967828\pi\)
\(654\) 0 0
\(655\) −1418.80 −2.16611
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 305.137i − 0.463030i −0.972831 0.231515i \(-0.925632\pi\)
0.972831 0.231515i \(-0.0743683\pi\)
\(660\) 0 0
\(661\) 623.298 0.942962 0.471481 0.881876i \(-0.343720\pi\)
0.471481 + 0.881876i \(0.343720\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1926.97i − 2.89771i
\(666\) 0 0
\(667\) 234.688 0.351856
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 155.376i − 0.231558i
\(672\) 0 0
\(673\) −263.867 −0.392076 −0.196038 0.980596i \(-0.562808\pi\)
−0.196038 + 0.980596i \(0.562808\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 788.411i − 1.16457i −0.812986 0.582283i \(-0.802160\pi\)
0.812986 0.582283i \(-0.197840\pi\)
\(678\) 0 0
\(679\) 381.066 0.561217
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 905.773i − 1.32617i −0.748545 0.663084i \(-0.769247\pi\)
0.748545 0.663084i \(-0.230753\pi\)
\(684\) 0 0
\(685\) −1022.65 −1.49291
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 80.4211i − 0.116721i
\(690\) 0 0
\(691\) −45.8027 −0.0662847 −0.0331423 0.999451i \(-0.510551\pi\)
−0.0331423 + 0.999451i \(0.510551\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1265.85i − 1.82136i
\(696\) 0 0
\(697\) −290.622 −0.416962
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1019.73i 1.45468i 0.686279 + 0.727339i \(0.259243\pi\)
−0.686279 + 0.727339i \(0.740757\pi\)
\(702\) 0 0
\(703\) −1213.51 −1.72619
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 649.049i − 0.918033i
\(708\) 0 0
\(709\) 84.7350 0.119513 0.0597567 0.998213i \(-0.480968\pi\)
0.0597567 + 0.998213i \(0.480968\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 809.558i 1.13543i
\(714\) 0 0
\(715\) 106.313 0.148690
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 646.311i − 0.898903i −0.893305 0.449451i \(-0.851619\pi\)
0.893305 0.449451i \(-0.148381\pi\)
\(720\) 0 0
\(721\) −859.643 −1.19229
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 369.234i 0.509288i
\(726\) 0 0
\(727\) −648.789 −0.892419 −0.446210 0.894928i \(-0.647226\pi\)
−0.446210 + 0.894928i \(0.647226\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 1319.75i − 1.80540i
\(732\) 0 0
\(733\) −696.353 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 155.376i − 0.210822i
\(738\) 0 0
\(739\) 740.833 1.00248 0.501240 0.865308i \(-0.332877\pi\)
0.501240 + 0.865308i \(0.332877\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 111.377i 0.149901i 0.997187 + 0.0749507i \(0.0238800\pi\)
−0.997187 + 0.0749507i \(0.976120\pi\)
\(744\) 0 0
\(745\) 1756.18 2.35728
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 174.556i 0.233052i
\(750\) 0 0
\(751\) −923.699 −1.22996 −0.614979 0.788543i \(-0.710836\pi\)
−0.614979 + 0.788543i \(0.710836\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1246.37i 1.65082i
\(756\) 0 0
\(757\) −1233.47 −1.62941 −0.814707 0.579873i \(-0.803102\pi\)
−0.814707 + 0.579873i \(0.803102\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 179.070i 0.235309i 0.993055 + 0.117654i \(0.0375375\pi\)
−0.993055 + 0.117654i \(0.962463\pi\)
\(762\) 0 0
\(763\) 955.266 1.25199
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 85.1049i − 0.110958i
\(768\) 0 0
\(769\) −251.511 −0.327062 −0.163531 0.986538i \(-0.552288\pi\)
−0.163531 + 0.986538i \(0.552288\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 184.110i 0.238176i 0.992884 + 0.119088i \(0.0379971\pi\)
−0.992884 + 0.119088i \(0.962003\pi\)
\(774\) 0 0
\(775\) −1273.68 −1.64345
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 464.183i − 0.595870i
\(780\) 0 0
\(781\) 225.654 0.288930
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 871.941i 1.11075i
\(786\) 0 0
\(787\) 1351.36 1.71710 0.858550 0.512731i \(-0.171366\pi\)
0.858550 + 0.512731i \(0.171366\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1157.82i − 1.46374i
\(792\) 0 0
\(793\) 58.7991 0.0741476
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1407.20i − 1.76562i −0.469727 0.882812i \(-0.655648\pi\)
0.469727 0.882812i \(-0.344352\pi\)
\(798\) 0 0
\(799\) 72.9992 0.0913632
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 470.145i − 0.585485i
\(804\) 0 0
\(805\) 1189.95 1.47820
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 741.991i 0.917171i 0.888650 + 0.458585i \(0.151644\pi\)
−0.888650 + 0.458585i \(0.848356\pi\)
\(810\) 0 0
\(811\) 716.837 0.883893 0.441947 0.897041i \(-0.354288\pi\)
0.441947 + 0.897041i \(0.354288\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 858.082i − 1.05286i
\(816\) 0 0
\(817\) 2107.91 2.58006
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 244.447i − 0.297743i −0.988857 0.148871i \(-0.952436\pi\)
0.988857 0.148871i \(-0.0475640\pi\)
\(822\) 0 0
\(823\) 331.500 0.402795 0.201398 0.979510i \(-0.435452\pi\)
0.201398 + 0.979510i \(0.435452\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 724.314i − 0.875833i −0.899016 0.437916i \(-0.855717\pi\)
0.899016 0.437916i \(-0.144283\pi\)
\(828\) 0 0
\(829\) −77.5083 −0.0934961 −0.0467481 0.998907i \(-0.514886\pi\)
−0.0467481 + 0.998907i \(0.514886\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 65.4309i − 0.0785485i
\(834\) 0 0
\(835\) −1162.47 −1.39218
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 952.252i − 1.13498i −0.823379 0.567492i \(-0.807914\pi\)
0.823379 0.567492i \(-0.192086\pi\)
\(840\) 0 0
\(841\) 721.977 0.858475
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1256.17i − 1.48659i
\(846\) 0 0
\(847\) −608.456 −0.718366
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 749.372i − 0.880578i
\(852\) 0 0
\(853\) 282.424 0.331095 0.165548 0.986202i \(-0.447061\pi\)
0.165548 + 0.986202i \(0.447061\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 539.665i − 0.629714i −0.949139 0.314857i \(-0.898044\pi\)
0.949139 0.314857i \(-0.101956\pi\)
\(858\) 0 0
\(859\) 53.2837 0.0620299 0.0310150 0.999519i \(-0.490126\pi\)
0.0310150 + 0.999519i \(0.490126\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 632.435i 0.732834i 0.930451 + 0.366417i \(0.119416\pi\)
−0.930451 + 0.366417i \(0.880584\pi\)
\(864\) 0 0
\(865\) −751.911 −0.869261
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 189.550i 0.218125i
\(870\) 0 0
\(871\) 58.7991 0.0675075
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 489.241i 0.559133i
\(876\) 0 0
\(877\) −346.548 −0.395152 −0.197576 0.980288i \(-0.563307\pi\)
−0.197576 + 0.980288i \(0.563307\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1394.26i 1.58258i 0.611438 + 0.791292i \(0.290591\pi\)
−0.611438 + 0.791292i \(0.709409\pi\)
\(882\) 0 0
\(883\) 1714.29 1.94144 0.970720 0.240212i \(-0.0772170\pi\)
0.970720 + 0.240212i \(0.0772170\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1149.19i 1.29559i 0.761815 + 0.647795i \(0.224309\pi\)
−0.761815 + 0.647795i \(0.775691\pi\)
\(888\) 0 0
\(889\) −1056.27 −1.18815
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 116.595i 0.130565i
\(894\) 0 0
\(895\) 1703.87 1.90376
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 410.570i − 0.456696i
\(900\) 0 0
\(901\) −765.897 −0.850052
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 910.785i 1.00639i
\(906\) 0 0
\(907\) 507.952 0.560035 0.280017 0.959995i \(-0.409660\pi\)
0.280017 + 0.959995i \(0.409660\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1488.13i − 1.63351i −0.576984 0.816756i \(-0.695770\pi\)
0.576984 0.816756i \(-0.304230\pi\)
\(912\) 0 0
\(913\) 334.755 0.366654
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1333.74i − 1.45446i
\(918\) 0 0
\(919\) −335.790 −0.365386 −0.182693 0.983170i \(-0.558481\pi\)
−0.182693 + 0.983170i \(0.558481\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 85.3949i 0.0925188i
\(924\) 0 0
\(925\) 1178.99 1.27458
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1685.87i − 1.81471i −0.420363 0.907356i \(-0.638097\pi\)
0.420363 0.907356i \(-0.361903\pi\)
\(930\) 0 0
\(931\) 104.506 0.112252
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 1012.48i − 1.08287i
\(936\) 0 0
\(937\) 49.7342 0.0530781 0.0265390 0.999648i \(-0.491551\pi\)
0.0265390 + 0.999648i \(0.491551\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1689.50i 1.79543i 0.440577 + 0.897715i \(0.354774\pi\)
−0.440577 + 0.897715i \(0.645226\pi\)
\(942\) 0 0
\(943\) 286.644 0.303971
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1036.44i − 1.09445i −0.836986 0.547225i \(-0.815684\pi\)
0.836986 0.547225i \(-0.184316\pi\)
\(948\) 0 0
\(949\) 177.918 0.187479
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 388.814i 0.407989i 0.978972 + 0.203995i \(0.0653925\pi\)
−0.978972 + 0.203995i \(0.934608\pi\)
\(954\) 0 0
\(955\) −1162.47 −1.21725
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 961.335i − 1.00243i
\(960\) 0 0
\(961\) 455.266 0.473742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 869.721i 0.901265i
\(966\) 0 0
\(967\) −201.899 −0.208789 −0.104395 0.994536i \(-0.533290\pi\)
−0.104395 + 0.994536i \(0.533290\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1500.54i 1.54535i 0.634800 + 0.772676i \(0.281082\pi\)
−0.634800 + 0.772676i \(0.718918\pi\)
\(972\) 0 0
\(973\) 1189.95 1.22297
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 866.378i − 0.886774i −0.896330 0.443387i \(-0.853777\pi\)
0.896330 0.443387i \(-0.146223\pi\)
\(978\) 0 0
\(979\) 32.9017 0.0336074
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 613.800i 0.624415i 0.950014 + 0.312207i \(0.101068\pi\)
−0.950014 + 0.312207i \(0.898932\pi\)
\(984\) 0 0
\(985\) −2450.53 −2.48785
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1301.68i 1.31616i
\(990\) 0 0
\(991\) −1460.50 −1.47376 −0.736882 0.676022i \(-0.763703\pi\)
−0.736882 + 0.676022i \(0.763703\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 314.595i − 0.316176i
\(996\) 0 0
\(997\) −1378.84 −1.38299 −0.691494 0.722382i \(-0.743047\pi\)
−0.691494 + 0.722382i \(0.743047\pi\)
\(998\) 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 2304.3.e.o.1025.8 8
3.2 odd 2 inner 2304.3.e.o.1025.2 8
4.3 odd 2 2304.3.e.n.1025.8 8
8.3 odd 2 2304.3.e.n.1025.1 8
8.5 even 2 inner 2304.3.e.o.1025.1 8
12.11 even 2 2304.3.e.n.1025.2 8
16.3 odd 4 72.3.h.a.53.6 yes 8
16.5 even 4 288.3.h.a.17.1 8
16.11 odd 4 72.3.h.a.53.4 yes 8
16.13 even 4 288.3.h.a.17.8 8
24.5 odd 2 inner 2304.3.e.o.1025.7 8
24.11 even 2 2304.3.e.n.1025.7 8
48.5 odd 4 288.3.h.a.17.7 8
48.11 even 4 72.3.h.a.53.5 yes 8
48.29 odd 4 288.3.h.a.17.2 8
48.35 even 4 72.3.h.a.53.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.h.a.53.3 8 48.35 even 4
72.3.h.a.53.4 yes 8 16.11 odd 4
72.3.h.a.53.5 yes 8 48.11 even 4
72.3.h.a.53.6 yes 8 16.3 odd 4
288.3.h.a.17.1 8 16.5 even 4
288.3.h.a.17.2 8 48.29 odd 4
288.3.h.a.17.7 8 48.5 odd 4
288.3.h.a.17.8 8 16.13 even 4
2304.3.e.n.1025.1 8 8.3 odd 2
2304.3.e.n.1025.2 8 12.11 even 2
2304.3.e.n.1025.7 8 24.11 even 2
2304.3.e.n.1025.8 8 4.3 odd 2
2304.3.e.o.1025.1 8 8.5 even 2 inner
2304.3.e.o.1025.2 8 3.2 odd 2 inner
2304.3.e.o.1025.7 8 24.5 odd 2 inner
2304.3.e.o.1025.8 8 1.1 even 1 trivial