Properties

Label 2800.2.g.t.449.3
Level $2800$
Weight $2$
Character 2800.449
Analytic conductor $22.358$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2800.449
Dual form 2800.2.g.t.449.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +O(q^{10})\) \(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +1.56155 q^{11} -0.438447i q^{13} -0.438447i q^{17} -7.12311 q^{19} -1.56155 q^{21} +3.12311i q^{23} +5.56155i q^{27} -6.68466 q^{29} +2.43845i q^{33} +6.00000i q^{37} +0.684658 q^{39} +5.12311 q^{41} +0.876894i q^{43} +8.68466i q^{47} -1.00000 q^{49} +0.684658 q^{51} +5.12311i q^{53} -11.1231i q^{57} -4.00000 q^{59} +15.3693 q^{61} +0.561553i q^{63} -10.2462i q^{67} -4.87689 q^{69} -8.00000 q^{71} +12.2462i q^{73} +1.56155i q^{77} -2.43845 q^{79} -7.00000 q^{81} +4.00000i q^{83} -10.4384i q^{87} +1.12311 q^{89} +0.438447 q^{91} +5.80776i q^{97} +0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} - 2q^{11} - 12q^{19} + 2q^{21} - 2q^{29} - 22q^{39} + 4q^{41} - 4q^{49} - 22q^{51} - 16q^{59} + 12q^{61} - 36q^{69} - 32q^{71} - 18q^{79} - 28q^{81} - 12q^{89} + 10q^{91} + 20q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.56155i 0.901563i 0.892634 + 0.450781i \(0.148855\pi\)
−0.892634 + 0.450781i \(0.851145\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) − 0.438447i − 0.121603i −0.998150 0.0608017i \(-0.980634\pi\)
0.998150 0.0608017i \(-0.0193657\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.438447i − 0.106339i −0.998586 0.0531695i \(-0.983068\pi\)
0.998586 0.0531695i \(-0.0169324\pi\)
\(18\) 0 0
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) 0 0
\(21\) −1.56155 −0.340759
\(22\) 0 0
\(23\) 3.12311i 0.651213i 0.945505 + 0.325606i \(0.105568\pi\)
−0.945505 + 0.325606i \(0.894432\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.56155i 1.07032i
\(28\) 0 0
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.43845i 0.424479i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0.684658 0.109633
\(40\) 0 0
\(41\) 5.12311 0.800095 0.400047 0.916494i \(-0.368994\pi\)
0.400047 + 0.916494i \(0.368994\pi\)
\(42\) 0 0
\(43\) 0.876894i 0.133725i 0.997762 + 0.0668626i \(0.0212989\pi\)
−0.997762 + 0.0668626i \(0.978701\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.68466i 1.26679i 0.773830 + 0.633394i \(0.218339\pi\)
−0.773830 + 0.633394i \(0.781661\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0.684658 0.0958714
\(52\) 0 0
\(53\) 5.12311i 0.703713i 0.936054 + 0.351856i \(0.114449\pi\)
−0.936054 + 0.351856i \(0.885551\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 11.1231i − 1.47329i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 15.3693 1.96784 0.983920 0.178611i \(-0.0571605\pi\)
0.983920 + 0.178611i \(0.0571605\pi\)
\(62\) 0 0
\(63\) 0.561553i 0.0707490i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.2462i − 1.25177i −0.779914 0.625887i \(-0.784737\pi\)
0.779914 0.625887i \(-0.215263\pi\)
\(68\) 0 0
\(69\) −4.87689 −0.587109
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 12.2462i 1.43331i 0.697428 + 0.716655i \(0.254328\pi\)
−0.697428 + 0.716655i \(0.745672\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.56155i 0.177955i
\(78\) 0 0
\(79\) −2.43845 −0.274347 −0.137173 0.990547i \(-0.543802\pi\)
−0.137173 + 0.990547i \(0.543802\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 10.4384i − 1.11912i
\(88\) 0 0
\(89\) 1.12311 0.119049 0.0595245 0.998227i \(-0.481042\pi\)
0.0595245 + 0.998227i \(0.481042\pi\)
\(90\) 0 0
\(91\) 0.438447 0.0459618
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.80776i 0.589689i 0.955545 + 0.294845i \(0.0952679\pi\)
−0.955545 + 0.294845i \(0.904732\pi\)
\(98\) 0 0
\(99\) 0.876894 0.0881312
\(100\) 0 0
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 0 0
\(103\) 5.56155i 0.547996i 0.961730 + 0.273998i \(0.0883462\pi\)
−0.961730 + 0.273998i \(0.911654\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 13.3693i − 1.29246i −0.763142 0.646230i \(-0.776345\pi\)
0.763142 0.646230i \(-0.223655\pi\)
\(108\) 0 0
\(109\) −5.31534 −0.509117 −0.254559 0.967057i \(-0.581930\pi\)
−0.254559 + 0.967057i \(0.581930\pi\)
\(110\) 0 0
\(111\) −9.36932 −0.889296
\(112\) 0 0
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 0.246211i − 0.0227622i
\(118\) 0 0
\(119\) 0.438447 0.0401924
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.24621i 0.554262i 0.960832 + 0.277131i \(0.0893835\pi\)
−0.960832 + 0.277131i \(0.910616\pi\)
\(128\) 0 0
\(129\) −1.36932 −0.120562
\(130\) 0 0
\(131\) 0.876894 0.0766146 0.0383073 0.999266i \(-0.487803\pi\)
0.0383073 + 0.999266i \(0.487803\pi\)
\(132\) 0 0
\(133\) − 7.12311i − 0.617652i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 17.1231i − 1.46293i −0.681881 0.731463i \(-0.738838\pi\)
0.681881 0.731463i \(-0.261162\pi\)
\(138\) 0 0
\(139\) −15.1231 −1.28273 −0.641363 0.767238i \(-0.721631\pi\)
−0.641363 + 0.767238i \(0.721631\pi\)
\(140\) 0 0
\(141\) −13.5616 −1.14209
\(142\) 0 0
\(143\) − 0.684658i − 0.0572540i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.56155i − 0.128795i
\(148\) 0 0
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 0 0
\(151\) 6.93087 0.564026 0.282013 0.959411i \(-0.408998\pi\)
0.282013 + 0.959411i \(0.408998\pi\)
\(152\) 0 0
\(153\) − 0.246211i − 0.0199050i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.2462i 1.61582i 0.589303 + 0.807912i \(0.299402\pi\)
−0.589303 + 0.807912i \(0.700598\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) −3.12311 −0.246135
\(162\) 0 0
\(163\) − 7.12311i − 0.557925i −0.960302 0.278962i \(-0.910010\pi\)
0.960302 0.278962i \(-0.0899905\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.93087i 0.536327i 0.963373 + 0.268163i \(0.0864167\pi\)
−0.963373 + 0.268163i \(0.913583\pi\)
\(168\) 0 0
\(169\) 12.8078 0.985213
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 4.43845i 0.337449i 0.985663 + 0.168724i \(0.0539648\pi\)
−0.985663 + 0.168724i \(0.946035\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 6.24621i − 0.469494i
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −17.6155 −1.30935 −0.654676 0.755910i \(-0.727195\pi\)
−0.654676 + 0.755910i \(0.727195\pi\)
\(182\) 0 0
\(183\) 24.0000i 1.77413i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 0.684658i − 0.0500672i
\(188\) 0 0
\(189\) −5.56155 −0.404543
\(190\) 0 0
\(191\) 13.5616 0.981280 0.490640 0.871363i \(-0.336763\pi\)
0.490640 + 0.871363i \(0.336763\pi\)
\(192\) 0 0
\(193\) − 19.3693i − 1.39423i −0.716957 0.697117i \(-0.754466\pi\)
0.716957 0.697117i \(-0.245534\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.12311i 0.0800180i 0.999199 + 0.0400090i \(0.0127387\pi\)
−0.999199 + 0.0400090i \(0.987261\pi\)
\(198\) 0 0
\(199\) −1.75379 −0.124323 −0.0621614 0.998066i \(-0.519799\pi\)
−0.0621614 + 0.998066i \(0.519799\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) − 6.68466i − 0.469171i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.75379i 0.121897i
\(208\) 0 0
\(209\) −11.1231 −0.769401
\(210\) 0 0
\(211\) −14.0540 −0.967516 −0.483758 0.875202i \(-0.660728\pi\)
−0.483758 + 0.875202i \(0.660728\pi\)
\(212\) 0 0
\(213\) − 12.4924i − 0.855967i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −19.1231 −1.29222
\(220\) 0 0
\(221\) −0.192236 −0.0129312
\(222\) 0 0
\(223\) − 2.43845i − 0.163291i −0.996661 0.0816453i \(-0.973983\pi\)
0.996661 0.0816453i \(-0.0260175\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 11.3153i − 0.751026i −0.926817 0.375513i \(-0.877467\pi\)
0.926817 0.375513i \(-0.122533\pi\)
\(228\) 0 0
\(229\) −10.8769 −0.718765 −0.359383 0.933190i \(-0.617013\pi\)
−0.359383 + 0.933190i \(0.617013\pi\)
\(230\) 0 0
\(231\) −2.43845 −0.160438
\(232\) 0 0
\(233\) − 5.12311i − 0.335626i −0.985819 0.167813i \(-0.946330\pi\)
0.985819 0.167813i \(-0.0536704\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 3.80776i − 0.247341i
\(238\) 0 0
\(239\) 19.8078 1.28126 0.640629 0.767851i \(-0.278674\pi\)
0.640629 + 0.767851i \(0.278674\pi\)
\(240\) 0 0
\(241\) −4.24621 −0.273523 −0.136761 0.990604i \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(242\) 0 0
\(243\) 5.75379i 0.369106i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.12311i 0.198718i
\(248\) 0 0
\(249\) −6.24621 −0.395838
\(250\) 0 0
\(251\) 8.87689 0.560305 0.280152 0.959956i \(-0.409615\pi\)
0.280152 + 0.959956i \(0.409615\pi\)
\(252\) 0 0
\(253\) 4.87689i 0.306608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.4924i − 0.654499i −0.944938 0.327250i \(-0.893878\pi\)
0.944938 0.327250i \(-0.106122\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −3.75379 −0.232354
\(262\) 0 0
\(263\) − 12.8769i − 0.794023i −0.917814 0.397012i \(-0.870047\pi\)
0.917814 0.397012i \(-0.129953\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.75379i 0.107330i
\(268\) 0 0
\(269\) 20.7386 1.26446 0.632228 0.774782i \(-0.282140\pi\)
0.632228 + 0.774782i \(0.282140\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0.684658i 0.0414374i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 0.246211i − 0.0147934i −0.999973 0.00739670i \(-0.997646\pi\)
0.999973 0.00739670i \(-0.00235446\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.4384 0.742016 0.371008 0.928630i \(-0.379012\pi\)
0.371008 + 0.928630i \(0.379012\pi\)
\(282\) 0 0
\(283\) − 11.3153i − 0.672627i −0.941750 0.336314i \(-0.890820\pi\)
0.941750 0.336314i \(-0.109180\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.12311i 0.302407i
\(288\) 0 0
\(289\) 16.8078 0.988692
\(290\) 0 0
\(291\) −9.06913 −0.531642
\(292\) 0 0
\(293\) 2.68466i 0.156839i 0.996920 + 0.0784197i \(0.0249874\pi\)
−0.996920 + 0.0784197i \(0.975013\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 8.68466i 0.503935i
\(298\) 0 0
\(299\) 1.36932 0.0791896
\(300\) 0 0
\(301\) −0.876894 −0.0505434
\(302\) 0 0
\(303\) − 25.3693i − 1.45743i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.3153i 1.10238i 0.834378 + 0.551192i \(0.185827\pi\)
−0.834378 + 0.551192i \(0.814173\pi\)
\(308\) 0 0
\(309\) −8.68466 −0.494053
\(310\) 0 0
\(311\) −31.6155 −1.79275 −0.896376 0.443294i \(-0.853810\pi\)
−0.896376 + 0.443294i \(0.853810\pi\)
\(312\) 0 0
\(313\) 22.3002i 1.26048i 0.776400 + 0.630241i \(0.217044\pi\)
−0.776400 + 0.630241i \(0.782956\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.4924i 0.589313i 0.955603 + 0.294657i \(0.0952053\pi\)
−0.955603 + 0.294657i \(0.904795\pi\)
\(318\) 0 0
\(319\) −10.4384 −0.584441
\(320\) 0 0
\(321\) 20.8769 1.16523
\(322\) 0 0
\(323\) 3.12311i 0.173774i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 8.30019i − 0.459001i
\(328\) 0 0
\(329\) −8.68466 −0.478801
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 3.36932i 0.184637i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.50758i − 0.0821230i −0.999157 0.0410615i \(-0.986926\pi\)
0.999157 0.0410615i \(-0.0130740\pi\)
\(338\) 0 0
\(339\) −21.8617 −1.18737
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 7.12311i − 0.382388i −0.981552 0.191194i \(-0.938764\pi\)
0.981552 0.191194i \(-0.0612360\pi\)
\(348\) 0 0
\(349\) −10.4924 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(350\) 0 0
\(351\) 2.43845 0.130155
\(352\) 0 0
\(353\) − 5.80776i − 0.309116i −0.987984 0.154558i \(-0.950605\pi\)
0.987984 0.154558i \(-0.0493954\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.684658i 0.0362360i
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) − 13.3693i − 0.701707i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.68466i 0.453335i 0.973972 + 0.226668i \(0.0727831\pi\)
−0.973972 + 0.226668i \(0.927217\pi\)
\(368\) 0 0
\(369\) 2.87689 0.149765
\(370\) 0 0
\(371\) −5.12311 −0.265978
\(372\) 0 0
\(373\) − 4.63068i − 0.239768i −0.992788 0.119884i \(-0.961748\pi\)
0.992788 0.119884i \(-0.0382522\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.93087i 0.150947i
\(378\) 0 0
\(379\) −16.4924 −0.847159 −0.423579 0.905859i \(-0.639227\pi\)
−0.423579 + 0.905859i \(0.639227\pi\)
\(380\) 0 0
\(381\) −9.75379 −0.499702
\(382\) 0 0
\(383\) 6.24621i 0.319166i 0.987184 + 0.159583i \(0.0510150\pi\)
−0.987184 + 0.159583i \(0.948985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.492423i 0.0250312i
\(388\) 0 0
\(389\) 24.9309 1.26405 0.632023 0.774950i \(-0.282225\pi\)
0.632023 + 0.774950i \(0.282225\pi\)
\(390\) 0 0
\(391\) 1.36932 0.0692493
\(392\) 0 0
\(393\) 1.36932i 0.0690729i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 27.5616i 1.38327i 0.722245 + 0.691637i \(0.243110\pi\)
−0.722245 + 0.691637i \(0.756890\pi\)
\(398\) 0 0
\(399\) 11.1231 0.556852
\(400\) 0 0
\(401\) 31.5616 1.57611 0.788054 0.615606i \(-0.211089\pi\)
0.788054 + 0.615606i \(0.211089\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.36932i 0.464420i
\(408\) 0 0
\(409\) −6.49242 −0.321030 −0.160515 0.987033i \(-0.551315\pi\)
−0.160515 + 0.987033i \(0.551315\pi\)
\(410\) 0 0
\(411\) 26.7386 1.31892
\(412\) 0 0
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 23.6155i − 1.15646i
\(418\) 0 0
\(419\) 26.2462 1.28221 0.641106 0.767453i \(-0.278476\pi\)
0.641106 + 0.767453i \(0.278476\pi\)
\(420\) 0 0
\(421\) −2.68466 −0.130842 −0.0654211 0.997858i \(-0.520839\pi\)
−0.0654211 + 0.997858i \(0.520839\pi\)
\(422\) 0 0
\(423\) 4.87689i 0.237123i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15.3693i 0.743773i
\(428\) 0 0
\(429\) 1.06913 0.0516181
\(430\) 0 0
\(431\) 19.8078 0.954106 0.477053 0.878874i \(-0.341705\pi\)
0.477053 + 0.878874i \(0.341705\pi\)
\(432\) 0 0
\(433\) − 8.24621i − 0.396288i −0.980173 0.198144i \(-0.936509\pi\)
0.980173 0.198144i \(-0.0634913\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 22.2462i − 1.06418i
\(438\) 0 0
\(439\) 9.36932 0.447173 0.223587 0.974684i \(-0.428223\pi\)
0.223587 + 0.974684i \(0.428223\pi\)
\(440\) 0 0
\(441\) −0.561553 −0.0267406
\(442\) 0 0
\(443\) − 2.63068i − 0.124988i −0.998045 0.0624938i \(-0.980095\pi\)
0.998045 0.0624938i \(-0.0199054\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 19.1231i − 0.904492i
\(448\) 0 0
\(449\) 1.80776 0.0853137 0.0426568 0.999090i \(-0.486418\pi\)
0.0426568 + 0.999090i \(0.486418\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 10.8229i 0.508505i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 17.1231i − 0.800985i −0.916300 0.400493i \(-0.868839\pi\)
0.916300 0.400493i \(-0.131161\pi\)
\(458\) 0 0
\(459\) 2.43845 0.113817
\(460\) 0 0
\(461\) −13.1231 −0.611204 −0.305602 0.952159i \(-0.598858\pi\)
−0.305602 + 0.952159i \(0.598858\pi\)
\(462\) 0 0
\(463\) 12.4924i 0.580572i 0.956940 + 0.290286i \(0.0937505\pi\)
−0.956940 + 0.290286i \(0.906250\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 22.4384i − 1.03833i −0.854675 0.519164i \(-0.826243\pi\)
0.854675 0.519164i \(-0.173757\pi\)
\(468\) 0 0
\(469\) 10.2462 0.473126
\(470\) 0 0
\(471\) −31.6155 −1.45677
\(472\) 0 0
\(473\) 1.36932i 0.0629613i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.87689i 0.131724i
\(478\) 0 0
\(479\) 4.87689 0.222831 0.111415 0.993774i \(-0.464462\pi\)
0.111415 + 0.993774i \(0.464462\pi\)
\(480\) 0 0
\(481\) 2.63068 0.119949
\(482\) 0 0
\(483\) − 4.87689i − 0.221906i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.12311i 0.141521i 0.997493 + 0.0707607i \(0.0225427\pi\)
−0.997493 + 0.0707607i \(0.977457\pi\)
\(488\) 0 0
\(489\) 11.1231 0.503004
\(490\) 0 0
\(491\) 41.1771 1.85830 0.929148 0.369708i \(-0.120542\pi\)
0.929148 + 0.369708i \(0.120542\pi\)
\(492\) 0 0
\(493\) 2.93087i 0.132000i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.00000i − 0.358849i
\(498\) 0 0
\(499\) 41.1771 1.84334 0.921670 0.387976i \(-0.126826\pi\)
0.921670 + 0.387976i \(0.126826\pi\)
\(500\) 0 0
\(501\) −10.8229 −0.483532
\(502\) 0 0
\(503\) 38.9309i 1.73584i 0.496703 + 0.867921i \(0.334544\pi\)
−0.496703 + 0.867921i \(0.665456\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 20.0000i 0.888231i
\(508\) 0 0
\(509\) 11.7538 0.520978 0.260489 0.965477i \(-0.416116\pi\)
0.260489 + 0.965477i \(0.416116\pi\)
\(510\) 0 0
\(511\) −12.2462 −0.541740
\(512\) 0 0
\(513\) − 39.6155i − 1.74907i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13.5616i 0.596436i
\(518\) 0 0
\(519\) −6.93087 −0.304231
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 40.4924i 1.77061i 0.465011 + 0.885305i \(0.346050\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.2462 0.575922
\(530\) 0 0
\(531\) −2.24621 −0.0974773
\(532\) 0 0
\(533\) − 2.24621i − 0.0972942i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 31.2311i 1.34772i
\(538\) 0 0
\(539\) −1.56155 −0.0672608
\(540\) 0 0
\(541\) −37.8078 −1.62548 −0.812741 0.582625i \(-0.802026\pi\)
−0.812741 + 0.582625i \(0.802026\pi\)
\(542\) 0 0
\(543\) − 27.5076i − 1.18046i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.24621i 0.0960411i 0.998846 + 0.0480205i \(0.0152913\pi\)
−0.998846 + 0.0480205i \(0.984709\pi\)
\(548\) 0 0
\(549\) 8.63068 0.368349
\(550\) 0 0
\(551\) 47.6155 2.02849
\(552\) 0 0
\(553\) − 2.43845i − 0.103693i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 13.1231i − 0.556044i −0.960575 0.278022i \(-0.910321\pi\)
0.960575 0.278022i \(-0.0896788\pi\)
\(558\) 0 0
\(559\) 0.384472 0.0162614
\(560\) 0 0
\(561\) 1.06913 0.0451387
\(562\) 0 0
\(563\) − 28.0000i − 1.18006i −0.807382 0.590030i \(-0.799116\pi\)
0.807382 0.590030i \(-0.200884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 7.00000i − 0.293972i
\(568\) 0 0
\(569\) 30.9848 1.29895 0.649476 0.760382i \(-0.274988\pi\)
0.649476 + 0.760382i \(0.274988\pi\)
\(570\) 0 0
\(571\) −40.4924 −1.69456 −0.847278 0.531150i \(-0.821760\pi\)
−0.847278 + 0.531150i \(0.821760\pi\)
\(572\) 0 0
\(573\) 21.1771i 0.884685i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 24.0540i − 1.00138i −0.865627 0.500690i \(-0.833080\pi\)
0.865627 0.500690i \(-0.166920\pi\)
\(578\) 0 0
\(579\) 30.2462 1.25699
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.2462i 1.08330i 0.840605 + 0.541649i \(0.182200\pi\)
−0.840605 + 0.541649i \(0.817800\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −1.75379 −0.0721412
\(592\) 0 0
\(593\) 27.5616i 1.13182i 0.824468 + 0.565909i \(0.191475\pi\)
−0.824468 + 0.565909i \(0.808525\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 2.73863i − 0.112085i
\(598\) 0 0
\(599\) −11.8078 −0.482452 −0.241226 0.970469i \(-0.577550\pi\)
−0.241226 + 0.970469i \(0.577550\pi\)
\(600\) 0 0
\(601\) 6.49242 0.264831 0.132416 0.991194i \(-0.457727\pi\)
0.132416 + 0.991194i \(0.457727\pi\)
\(602\) 0 0
\(603\) − 5.75379i − 0.234312i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 42.0540i 1.70692i 0.521160 + 0.853459i \(0.325500\pi\)
−0.521160 + 0.853459i \(0.674500\pi\)
\(608\) 0 0
\(609\) 10.4384 0.422987
\(610\) 0 0
\(611\) 3.80776 0.154046
\(612\) 0 0
\(613\) − 40.7386i − 1.64542i −0.568463 0.822709i \(-0.692462\pi\)
0.568463 0.822709i \(-0.307538\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.2462i 1.29818i 0.760710 + 0.649092i \(0.224851\pi\)
−0.760710 + 0.649092i \(0.775149\pi\)
\(618\) 0 0
\(619\) 32.1080 1.29053 0.645264 0.763960i \(-0.276747\pi\)
0.645264 + 0.763960i \(0.276747\pi\)
\(620\) 0 0
\(621\) −17.3693 −0.697007
\(622\) 0 0
\(623\) 1.12311i 0.0449963i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 17.3693i − 0.693664i
\(628\) 0 0
\(629\) 2.63068 0.104892
\(630\) 0 0
\(631\) 11.8078 0.470060 0.235030 0.971988i \(-0.424481\pi\)
0.235030 + 0.971988i \(0.424481\pi\)
\(632\) 0 0
\(633\) − 21.9460i − 0.872276i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.438447i 0.0173719i
\(638\) 0 0
\(639\) −4.49242 −0.177717
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 1.56155i 0.0615816i 0.999526 + 0.0307908i \(0.00980257\pi\)
−0.999526 + 0.0307908i \(0.990197\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 36.4924i − 1.43467i −0.696731 0.717333i \(-0.745363\pi\)
0.696731 0.717333i \(-0.254637\pi\)
\(648\) 0 0
\(649\) −6.24621 −0.245185
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.2311i 1.30043i 0.759750 + 0.650216i \(0.225322\pi\)
−0.759750 + 0.650216i \(0.774678\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.87689i 0.268293i
\(658\) 0 0
\(659\) 9.17708 0.357488 0.178744 0.983896i \(-0.442797\pi\)
0.178744 + 0.983896i \(0.442797\pi\)
\(660\) 0 0
\(661\) −5.12311 −0.199266 −0.0996329 0.995024i \(-0.531767\pi\)
−0.0996329 + 0.995024i \(0.531767\pi\)
\(662\) 0 0
\(663\) − 0.300187i − 0.0116583i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 20.8769i − 0.808357i
\(668\) 0 0
\(669\) 3.80776 0.147217
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) − 31.8617i − 1.22818i −0.789236 0.614090i \(-0.789523\pi\)
0.789236 0.614090i \(-0.210477\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.93087i 0.189509i 0.995501 + 0.0947544i \(0.0302066\pi\)
−0.995501 + 0.0947544i \(0.969793\pi\)
\(678\) 0 0
\(679\) −5.80776 −0.222882
\(680\) 0 0
\(681\) 17.6695 0.677097
\(682\) 0 0
\(683\) − 6.73863i − 0.257847i −0.991655 0.128923i \(-0.958848\pi\)
0.991655 0.128923i \(-0.0411521\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 16.9848i − 0.648012i
\(688\) 0 0
\(689\) 2.24621 0.0855738
\(690\) 0 0
\(691\) 24.4924 0.931736 0.465868 0.884854i \(-0.345742\pi\)
0.465868 + 0.884854i \(0.345742\pi\)
\(692\) 0 0
\(693\) 0.876894i 0.0333105i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 2.24621i − 0.0850813i
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 28.9309 1.09270 0.546352 0.837556i \(-0.316016\pi\)
0.546352 + 0.837556i \(0.316016\pi\)
\(702\) 0 0
\(703\) − 42.7386i − 1.61192i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 16.2462i − 0.611002i
\(708\) 0 0
\(709\) −27.1771 −1.02066 −0.510328 0.859980i \(-0.670476\pi\)
−0.510328 + 0.859980i \(0.670476\pi\)
\(710\) 0 0
\(711\) −1.36932 −0.0513534
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 30.9309i 1.15513i
\(718\) 0 0
\(719\) 8.38447 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(720\) 0 0
\(721\) −5.56155 −0.207123
\(722\) 0 0
\(723\) − 6.63068i − 0.246598i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 52.4924i − 1.94684i −0.229035 0.973418i \(-0.573557\pi\)
0.229035 0.973418i \(-0.426443\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) 0.384472 0.0142202
\(732\) 0 0
\(733\) − 6.68466i − 0.246903i −0.992351 0.123452i \(-0.960604\pi\)
0.992351 0.123452i \(-0.0393964\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 16.0000i − 0.589368i
\(738\) 0 0
\(739\) 34.9309 1.28495 0.642476 0.766305i \(-0.277907\pi\)
0.642476 + 0.766305i \(0.277907\pi\)
\(740\) 0 0
\(741\) −4.87689 −0.179157
\(742\) 0 0
\(743\) − 32.9848i − 1.21010i −0.796189 0.605048i \(-0.793154\pi\)
0.796189 0.605048i \(-0.206846\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.24621i 0.0821846i
\(748\) 0 0
\(749\) 13.3693 0.488504
\(750\) 0 0
\(751\) −17.0691 −0.622861 −0.311431 0.950269i \(-0.600808\pi\)
−0.311431 + 0.950269i \(0.600808\pi\)
\(752\) 0 0
\(753\) 13.8617i 0.505150i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 39.3693i 1.43090i 0.698663 + 0.715451i \(0.253779\pi\)
−0.698663 + 0.715451i \(0.746221\pi\)
\(758\) 0 0
\(759\) −7.61553 −0.276426
\(760\) 0 0
\(761\) 48.2462 1.74892 0.874462 0.485094i \(-0.161215\pi\)
0.874462 + 0.485094i \(0.161215\pi\)
\(762\) 0 0
\(763\) − 5.31534i − 0.192428i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.75379i 0.0633256i
\(768\) 0 0
\(769\) 42.4924 1.53232 0.766158 0.642652i \(-0.222166\pi\)
0.766158 + 0.642652i \(0.222166\pi\)
\(770\) 0 0
\(771\) 16.3845 0.590072
\(772\) 0 0
\(773\) − 36.9309i − 1.32831i −0.747594 0.664156i \(-0.768791\pi\)
0.747594 0.664156i \(-0.231209\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.36932i − 0.336122i
\(778\) 0 0
\(779\) −36.4924 −1.30748
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) 0 0
\(783\) − 37.1771i − 1.32860i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 49.1771i 1.75297i 0.481426 + 0.876487i \(0.340119\pi\)
−0.481426 + 0.876487i \(0.659881\pi\)
\(788\) 0 0
\(789\) 20.1080 0.715862
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) − 6.73863i − 0.239296i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.0540i 0.852036i 0.904715 + 0.426018i \(0.140084\pi\)
−0.904715 + 0.426018i \(0.859916\pi\)
\(798\) 0 0
\(799\) 3.80776 0.134709
\(800\) 0 0
\(801\) 0.630683 0.0222841
\(802\) 0 0
\(803\) 19.1231i 0.674840i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 32.3845i 1.13999i
\(808\) 0 0
\(809\) −16.5464 −0.581740 −0.290870 0.956763i \(-0.593945\pi\)
−0.290870 + 0.956763i \(0.593945\pi\)
\(810\) 0 0
\(811\) −19.6155 −0.688794 −0.344397 0.938824i \(-0.611917\pi\)
−0.344397 + 0.938824i \(0.611917\pi\)
\(812\) 0 0
\(813\) 24.9848i 0.876257i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 6.24621i − 0.218527i
\(818\) 0 0
\(819\) 0.246211 0.00860332
\(820\) 0 0
\(821\) −21.4233 −0.747678 −0.373839 0.927494i \(-0.621959\pi\)
−0.373839 + 0.927494i \(0.621959\pi\)
\(822\) 0 0
\(823\) − 36.4924i − 1.27205i −0.771670 0.636023i \(-0.780578\pi\)
0.771670 0.636023i \(-0.219422\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.36932i 0.186709i 0.995633 + 0.0933547i \(0.0297591\pi\)
−0.995633 + 0.0933547i \(0.970241\pi\)
\(828\) 0 0
\(829\) −34.8769 −1.21132 −0.605662 0.795722i \(-0.707092\pi\)
−0.605662 + 0.795722i \(0.707092\pi\)
\(830\) 0 0
\(831\) 0.384472 0.0133372
\(832\) 0 0
\(833\) 0.438447i 0.0151913i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.8769 −0.996941 −0.498471 0.866907i \(-0.666105\pi\)
−0.498471 + 0.866907i \(0.666105\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 0 0
\(843\) 19.4233i 0.668974i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 8.56155i − 0.294178i
\(848\) 0 0
\(849\) 17.6695 0.606416
\(850\) 0 0
\(851\) −18.7386 −0.642352
\(852\) 0 0
\(853\) 7.26137i 0.248624i 0.992243 + 0.124312i \(0.0396724\pi\)
−0.992243 + 0.124312i \(0.960328\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 15.7538i − 0.538139i −0.963121 0.269070i \(-0.913284\pi\)
0.963121 0.269070i \(-0.0867162\pi\)
\(858\) 0 0
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) − 25.7538i − 0.876669i −0.898812 0.438335i \(-0.855569\pi\)
0.898812 0.438335i \(-0.144431\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 26.2462i 0.891368i
\(868\) 0 0
\(869\) −3.80776 −0.129170
\(870\) 0 0
\(871\) −4.49242 −0.152220
\(872\) 0 0
\(873\) 3.26137i 0.110381i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 40.2462i − 1.35902i −0.733667 0.679509i \(-0.762193\pi\)
0.733667 0.679509i \(-0.237807\pi\)
\(878\) 0 0
\(879\) −4.19224 −0.141401
\(880\) 0 0
\(881\) −11.8617 −0.399632 −0.199816 0.979833i \(-0.564034\pi\)
−0.199816 + 0.979833i \(0.564034\pi\)
\(882\) 0 0
\(883\) − 8.49242i − 0.285793i −0.989738 0.142896i \(-0.954358\pi\)
0.989738 0.142896i \(-0.0456416\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 20.4924i − 0.688068i −0.938957 0.344034i \(-0.888206\pi\)
0.938957 0.344034i \(-0.111794\pi\)
\(888\) 0 0
\(889\) −6.24621 −0.209491
\(890\) 0 0
\(891\) −10.9309 −0.366198
\(892\) 0 0
\(893\) − 61.8617i − 2.07012i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.13826i 0.0713944i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 2.24621 0.0748321
\(902\) 0 0
\(903\) − 1.36932i − 0.0455680i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 24.1080i 0.800491i 0.916408 + 0.400246i \(0.131075\pi\)
−0.916408 + 0.400246i \(0.868925\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) 28.4924 0.943996 0.471998 0.881600i \(-0.343533\pi\)
0.471998 + 0.881600i \(0.343533\pi\)
\(912\) 0 0
\(913\) 6.24621i 0.206719i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.876894i 0.0289576i
\(918\) 0 0
\(919\) 40.3002 1.32938 0.664690 0.747119i \(-0.268564\pi\)
0.664690 + 0.747119i \(0.268564\pi\)
\(920\) 0 0
\(921\) −30.1619 −0.993869
\(922\) 0 0
\(923\) 3.50758i 0.115453i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 3.12311i 0.102576i
\(928\) 0 0
\(929\) −22.1080 −0.725338 −0.362669 0.931918i \(-0.618134\pi\)
−0.362669 + 0.931918i \(0.618134\pi\)
\(930\) 0 0
\(931\) 7.12311 0.233450
\(932\) 0 0
\(933\) − 49.3693i − 1.61628i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 55.6695i − 1.81864i −0.416094 0.909322i \(-0.636601\pi\)
0.416094 0.909322i \(-0.363399\pi\)
\(938\) 0 0
\(939\) −34.8229 −1.13640
\(940\) 0 0
\(941\) 43.8617 1.42985 0.714926 0.699200i \(-0.246460\pi\)
0.714926 + 0.699200i \(0.246460\pi\)
\(942\) 0 0
\(943\) 16.0000i 0.521032i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) 5.36932 0.174295
\(950\) 0 0
\(951\) −16.3845 −0.531303
\(952\) 0 0
\(953\) 33.1231i 1.07296i 0.843912 + 0.536481i \(0.180247\pi\)
−0.843912 + 0.536481i \(0.819753\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 16.3002i − 0.526910i
\(958\) 0 0
\(959\) 17.1231 0.552934
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 7.50758i − 0.241928i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 35.1231i 1.12948i 0.825268 + 0.564741i \(0.191024\pi\)
−0.825268 + 0.564741i \(0.808976\pi\)
\(968\) 0 0
\(969\) −4.87689 −0.156668
\(970\) 0 0
\(971\) −49.4773 −1.58780 −0.793901 0.608048i \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) 0 0
\(973\) − 15.1231i − 0.484825i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.2311i 1.06316i 0.847009 + 0.531578i \(0.178401\pi\)
−0.847009 + 0.531578i \(0.821599\pi\)
\(978\) 0 0
\(979\) 1.75379 0.0560513
\(980\) 0 0
\(981\) −2.98485 −0.0952988
\(982\) 0 0
\(983\) 51.4233i 1.64015i 0.572257 + 0.820074i \(0.306068\pi\)
−0.572257 + 0.820074i \(0.693932\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 13.5616i − 0.431669i
\(988\) 0 0
\(989\) −2.73863 −0.0870835
\(990\) 0 0
\(991\) −12.4924 −0.396835 −0.198417 0.980118i \(-0.563580\pi\)
−0.198417 + 0.980118i \(0.563580\pi\)
\(992\) 0 0
\(993\) − 18.7386i − 0.594653i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 2.68466i − 0.0850240i −0.999096 0.0425120i \(-0.986464\pi\)
0.999096 0.0425120i \(-0.0135361\pi\)
\(998\) 0 0
\(999\) −33.3693 −1.05576
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 2800.2.g.t.449.3 4
4.3 odd 2 175.2.b.b.99.1 4
5.2 odd 4 2800.2.a.bi.1.2 2
5.3 odd 4 560.2.a.i.1.1 2
5.4 even 2 inner 2800.2.g.t.449.2 4
12.11 even 2 1575.2.d.e.1324.4 4
15.8 even 4 5040.2.a.bt.1.1 2
20.3 even 4 35.2.a.b.1.1 2
20.7 even 4 175.2.a.f.1.2 2
20.19 odd 2 175.2.b.b.99.4 4
28.27 even 2 1225.2.b.f.99.1 4
35.13 even 4 3920.2.a.bs.1.2 2
40.3 even 4 2240.2.a.bh.1.1 2
40.13 odd 4 2240.2.a.bd.1.2 2
60.23 odd 4 315.2.a.e.1.2 2
60.47 odd 4 1575.2.a.p.1.1 2
60.59 even 2 1575.2.d.e.1324.1 4
140.3 odd 12 245.2.e.h.226.2 4
140.23 even 12 245.2.e.i.116.2 4
140.27 odd 4 1225.2.a.s.1.2 2
140.83 odd 4 245.2.a.d.1.1 2
140.103 odd 12 245.2.e.h.116.2 4
140.123 even 12 245.2.e.i.226.2 4
140.139 even 2 1225.2.b.f.99.4 4
220.43 odd 4 4235.2.a.m.1.2 2
260.103 even 4 5915.2.a.l.1.2 2
420.83 even 4 2205.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 20.3 even 4
175.2.a.f.1.2 2 20.7 even 4
175.2.b.b.99.1 4 4.3 odd 2
175.2.b.b.99.4 4 20.19 odd 2
245.2.a.d.1.1 2 140.83 odd 4
245.2.e.h.116.2 4 140.103 odd 12
245.2.e.h.226.2 4 140.3 odd 12
245.2.e.i.116.2 4 140.23 even 12
245.2.e.i.226.2 4 140.123 even 12
315.2.a.e.1.2 2 60.23 odd 4
560.2.a.i.1.1 2 5.3 odd 4
1225.2.a.s.1.2 2 140.27 odd 4
1225.2.b.f.99.1 4 28.27 even 2
1225.2.b.f.99.4 4 140.139 even 2
1575.2.a.p.1.1 2 60.47 odd 4
1575.2.d.e.1324.1 4 60.59 even 2
1575.2.d.e.1324.4 4 12.11 even 2
2205.2.a.x.1.2 2 420.83 even 4
2240.2.a.bd.1.2 2 40.13 odd 4
2240.2.a.bh.1.1 2 40.3 even 4
2800.2.a.bi.1.2 2 5.2 odd 4
2800.2.g.t.449.2 4 5.4 even 2 inner
2800.2.g.t.449.3 4 1.1 even 1 trivial
3920.2.a.bs.1.2 2 35.13 even 4
4235.2.a.m.1.2 2 220.43 odd 4
5040.2.a.bt.1.1 2 15.8 even 4
5915.2.a.l.1.2 2 260.103 even 4