Properties

Label 2205.2.a.x.1.2
Level $2205$
Weight $2$
Character 2205.1
Self dual yes
Analytic conductor $17.607$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{5} +6.56155 q^{8} +O(q^{10})\) \(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{5} +6.56155 q^{8} +2.56155 q^{10} +1.56155 q^{11} -0.438447 q^{13} +7.68466 q^{16} -0.438447 q^{17} +7.12311 q^{19} +4.56155 q^{20} +4.00000 q^{22} -3.12311 q^{23} +1.00000 q^{25} -1.12311 q^{26} -6.68466 q^{29} +6.56155 q^{32} -1.12311 q^{34} +6.00000 q^{37} +18.2462 q^{38} +6.56155 q^{40} +5.12311 q^{41} +0.876894 q^{43} +7.12311 q^{44} -8.00000 q^{46} -8.68466 q^{47} +2.56155 q^{50} -2.00000 q^{52} +5.12311 q^{53} +1.56155 q^{55} -17.1231 q^{58} -4.00000 q^{59} -15.3693 q^{61} +1.43845 q^{64} -0.438447 q^{65} +10.2462 q^{67} -2.00000 q^{68} -8.00000 q^{71} +12.2462 q^{73} +15.3693 q^{74} +32.4924 q^{76} -2.43845 q^{79} +7.68466 q^{80} +13.1231 q^{82} +4.00000 q^{83} -0.438447 q^{85} +2.24621 q^{86} +10.2462 q^{88} -1.12311 q^{89} -14.2462 q^{92} -22.2462 q^{94} +7.12311 q^{95} -5.80776 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + q^{10} - q^{11} - 5q^{13} + 3q^{16} - 5q^{17} + 6q^{19} + 5q^{20} + 8q^{22} + 2q^{23} + 2q^{25} + 6q^{26} - q^{29} + 9q^{32} + 6q^{34} + 12q^{37} + 20q^{38} + 9q^{40} + 2q^{41} + 10q^{43} + 6q^{44} - 16q^{46} - 5q^{47} + q^{50} - 4q^{52} + 2q^{53} - q^{55} - 26q^{58} - 8q^{59} - 6q^{61} + 7q^{64} - 5q^{65} + 4q^{67} - 4q^{68} - 16q^{71} + 8q^{73} + 6q^{74} + 32q^{76} - 9q^{79} + 3q^{80} + 18q^{82} + 8q^{83} - 5q^{85} - 12q^{86} + 4q^{88} + 6q^{89} - 12q^{92} - 28q^{94} + 6q^{95} + 9q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 0 0
\(4\) 4.56155 2.28078
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 6.56155 2.31986
\(9\) 0 0
\(10\) 2.56155 0.810034
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) −0.438447 −0.121603 −0.0608017 0.998150i \(-0.519366\pi\)
−0.0608017 + 0.998150i \(0.519366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) −0.438447 −0.106339 −0.0531695 0.998586i \(-0.516932\pi\)
−0.0531695 + 0.998586i \(0.516932\pi\)
\(18\) 0 0
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) 4.56155 1.01999
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −3.12311 −0.651213 −0.325606 0.945505i \(-0.605568\pi\)
−0.325606 + 0.945505i \(0.605568\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.12311 −0.220259
\(27\) 0 0
\(28\) 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 \(0.5\pi\)
\(32\) 6.56155 1.15993
\(33\) 0 0
\(34\) −1.12311 −0.192611
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 18.2462 2.95993
\(39\) 0 0
\(40\) 6.56155 1.03747
\(41\) 5.12311 0.800095 0.400047 0.916494i \(-0.368994\pi\)
0.400047 + 0.916494i \(0.368994\pi\)
\(42\) 0 0
\(43\) 0.876894 0.133725 0.0668626 0.997762i \(-0.478701\pi\)
0.0668626 + 0.997762i \(0.478701\pi\)
\(44\) 7.12311 1.07385
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −8.68466 −1.26679 −0.633394 0.773830i \(-0.718339\pi\)
−0.633394 + 0.773830i \(0.718339\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.56155 0.362258
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 5.12311 0.703713 0.351856 0.936054i \(-0.385551\pi\)
0.351856 + 0.936054i \(0.385551\pi\)
\(54\) 0 0
\(55\) 1.56155 0.210560
\(56\) 0 0
\(57\) 0 0
\(58\) −17.1231 −2.24837
\(59\) −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 \(-0.942839\pi\)
−0.983920 + 0.178611i \(0.942839\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) −0.438447 −0.0543827
\(66\) 0 0
\(67\) 10.2462 1.25177 0.625887 0.779914i \(-0.284737\pi\)
0.625887 + 0.779914i \(0.284737\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) 15.3693 1.78665
\(75\) 0 0
\(76\) 32.4924 3.72714
\(77\) 0 0
\(78\) 0 0
\(79\) −2.43845 −0.274347 −0.137173 0.990547i \(-0.543802\pi\)
−0.137173 + 0.990547i \(0.543802\pi\)
\(80\) 7.68466 0.859171
\(81\) 0 0
\(82\) 13.1231 1.44920
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −0.438447 −0.0475563
\(86\) 2.24621 0.242215
\(87\) 0 0
\(88\) 10.2462 1.09225
\(89\) −1.12311 −0.119049 −0.0595245 0.998227i \(-0.518958\pi\)
−0.0595245 + 0.998227i \(0.518958\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −14.2462 −1.48527
\(93\) 0 0
\(94\) −22.2462 −2.29452
\(95\) 7.12311 0.730815
\(96\) 0 0
\(97\) −5.80776 −0.589689 −0.294845 0.955545i \(-0.595268\pi\)
−0.294845 + 0.955545i \(0.595268\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4.56155 0.456155
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 0 0
\(103\) −5.56155 −0.547996 −0.273998 0.961730i \(-0.588346\pi\)
−0.273998 + 0.961730i \(0.588346\pi\)
\(104\) −2.87689 −0.282103
\(105\) 0 0
\(106\) 13.1231 1.27463
\(107\) −13.3693 −1.29246 −0.646230 0.763142i \(-0.723655\pi\)
−0.646230 + 0.763142i \(0.723655\pi\)
\(108\) 0 0
\(109\) 5.31534 0.509117 0.254559 0.967057i \(-0.418070\pi\)
0.254559 + 0.967057i \(0.418070\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −3.12311 −0.291231
\(116\) −30.4924 −2.83115
\(117\) 0 0
\(118\) −10.2462 −0.943240
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) −39.3693 −3.56433
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.24621 −0.554262 −0.277131 0.960832i \(-0.589384\pi\)
−0.277131 + 0.960832i \(0.589384\pi\)
\(128\) −9.43845 −0.834249
\(129\) 0 0
\(130\) −1.12311 −0.0985029
\(131\) −0.876894 −0.0766146 −0.0383073 0.999266i \(-0.512197\pi\)
−0.0383073 + 0.999266i \(0.512197\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 26.2462 2.26733
\(135\) 0 0
\(136\) −2.87689 −0.246692
\(137\) 17.1231 1.46293 0.731463 0.681881i \(-0.238838\pi\)
0.731463 + 0.681881i \(0.238838\pi\)
\(138\) 0 0
\(139\) 15.1231 1.28273 0.641363 0.767238i \(-0.278369\pi\)
0.641363 + 0.767238i \(0.278369\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −20.4924 −1.71969
\(143\) −0.684658 −0.0572540
\(144\) 0 0
\(145\) −6.68466 −0.555131
\(146\) 31.3693 2.59614
\(147\) 0 0
\(148\) 27.3693 2.24974
\(149\) −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 \(-0.591002\pi\)
−0.282013 + 0.959411i \(0.591002\pi\)
\(152\) 46.7386 3.79100
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.2462 −1.61582 −0.807912 0.589303i \(-0.799402\pi\)
−0.807912 + 0.589303i \(0.799402\pi\)
\(158\) −6.24621 −0.496922
\(159\) 0 0
\(160\) 6.56155 0.518736
\(161\) 0 0
\(162\) 0 0
\(163\) −7.12311 −0.557925 −0.278962 0.960302i \(-0.589990\pi\)
−0.278962 + 0.960302i \(0.589990\pi\)
\(164\) 23.3693 1.82484
\(165\) 0 0
\(166\) 10.2462 0.795260
\(167\) −6.93087 −0.536327 −0.268163 0.963373i \(-0.586417\pi\)
−0.268163 + 0.963373i \(0.586417\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) −1.12311 −0.0861383
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −4.43845 −0.337449 −0.168724 0.985663i \(-0.553965\pi\)
−0.168724 + 0.985663i \(0.553965\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) −2.87689 −0.215632
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 17.6155 1.30935 0.654676 0.755910i \(-0.272805\pi\)
0.654676 + 0.755910i \(0.272805\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −20.4924 −1.51072
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) −0.684658 −0.0500672
\(188\) −39.6155 −2.88926
\(189\) 0 0
\(190\) 18.2462 1.32372
\(191\) 13.5616 0.981280 0.490640 0.871363i \(-0.336763\pi\)
0.490640 + 0.871363i \(0.336763\pi\)
\(192\) 0 0
\(193\) 19.3693 1.39423 0.697117 0.716957i \(-0.254466\pi\)
0.697117 + 0.716957i \(0.254466\pi\)
\(194\) −14.8769 −1.06810
\(195\) 0 0
\(196\) 0 0
\(197\) −1.12311 −0.0800180 −0.0400090 0.999199i \(-0.512739\pi\)
−0.0400090 + 0.999199i \(0.512739\pi\)
\(198\) 0 0
\(199\) 1.75379 0.124323 0.0621614 0.998066i \(-0.480201\pi\)
0.0621614 + 0.998066i \(0.480201\pi\)
\(200\) 6.56155 0.463972
\(201\) 0 0
\(202\) −41.6155 −2.92806
\(203\) 0 0
\(204\) 0 0
\(205\) 5.12311 0.357813
\(206\) −14.2462 −0.992581
\(207\) 0 0
\(208\) −3.36932 −0.233620
\(209\) 11.1231 0.769401
\(210\) 0 0
\(211\) 14.0540 0.967516 0.483758 0.875202i \(-0.339272\pi\)
0.483758 + 0.875202i \(0.339272\pi\)
\(212\) 23.3693 1.60501
\(213\) 0 0
\(214\) −34.2462 −2.34102
\(215\) 0.876894 0.0598037
\(216\) 0 0
\(217\) 0 0
\(218\) 13.6155 0.922160
\(219\) 0 0
\(220\) 7.12311 0.480240
\(221\) 0.192236 0.0129312
\(222\) 0 0
\(223\) 2.43845 0.163291 0.0816453 0.996661i \(-0.473983\pi\)
0.0816453 + 0.996661i \(0.473983\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 35.8617 2.38549
\(227\) 11.3153 0.751026 0.375513 0.926817i \(-0.377467\pi\)
0.375513 + 0.926817i \(0.377467\pi\)
\(228\) 0 0
\(229\) −10.8769 −0.718765 −0.359383 0.933190i \(-0.617013\pi\)
−0.359383 + 0.933190i \(0.617013\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −43.8617 −2.87966
\(233\) −5.12311 −0.335626 −0.167813 0.985819i \(-0.553670\pi\)
−0.167813 + 0.985819i \(0.553670\pi\)
\(234\) 0 0
\(235\) −8.68466 −0.566525
\(236\) −18.2462 −1.18773
\(237\) 0 0
\(238\) 0 0
\(239\) −19.8078 −1.28126 −0.640629 0.767851i \(-0.721326\pi\)
−0.640629 + 0.767851i \(0.721326\pi\)
\(240\) 0 0
\(241\) 4.24621 0.273523 0.136761 0.990604i \(-0.456331\pi\)
0.136761 + 0.990604i \(0.456331\pi\)
\(242\) −21.9309 −1.40977
\(243\) 0 0
\(244\) −70.1080 −4.48820
\(245\) 0 0
\(246\) 0 0
\(247\) −3.12311 −0.198718
\(248\) 0 0
\(249\) 0 0
\(250\) 2.56155 0.162007
\(251\) −8.87689 −0.560305 −0.280152 0.959956i \(-0.590385\pi\)
−0.280152 + 0.959956i \(0.590385\pi\)
\(252\) 0 0
\(253\) −4.87689 −0.306608
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) −10.4924 −0.654499 −0.327250 0.944938i \(-0.606122\pi\)
−0.327250 + 0.944938i \(0.606122\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −2.24621 −0.138771
\(263\) 12.8769 0.794023 0.397012 0.917814i \(-0.370047\pi\)
0.397012 + 0.917814i \(0.370047\pi\)
\(264\) 0 0
\(265\) 5.12311 0.314710
\(266\) 0 0
\(267\) 0 0
\(268\) 46.7386 2.85502
\(269\) −20.7386 −1.26446 −0.632228 0.774782i \(-0.717860\pi\)
−0.632228 + 0.774782i \(0.717860\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −3.36932 −0.204295
\(273\) 0 0
\(274\) 43.8617 2.64978
\(275\) 1.56155 0.0941652
\(276\) 0 0
\(277\) −0.246211 −0.0147934 −0.00739670 0.999973i \(-0.502354\pi\)
−0.00739670 + 0.999973i \(0.502354\pi\)
\(278\) 38.7386 2.32339
\(279\) 0 0
\(280\) 0 0
\(281\) −12.4384 −0.742016 −0.371008 0.928630i \(-0.620988\pi\)
−0.371008 + 0.928630i \(0.620988\pi\)
\(282\) 0 0
\(283\) 11.3153 0.672627 0.336314 0.941750i \(-0.390820\pi\)
0.336314 + 0.941750i \(0.390820\pi\)
\(284\) −36.4924 −2.16543
\(285\) 0 0
\(286\) −1.75379 −0.103704
\(287\) 0 0
\(288\) 0 0
\(289\) −16.8078 −0.988692
\(290\) −17.1231 −1.00550
\(291\) 0 0
\(292\) 55.8617 3.26906
\(293\) −2.68466 −0.156839 −0.0784197 0.996920i \(-0.524987\pi\)
−0.0784197 + 0.996920i \(0.524987\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 39.3693 2.28830
\(297\) 0 0
\(298\) −31.3693 −1.81718
\(299\) 1.36932 0.0791896
\(300\) 0 0
\(301\) 0 0
\(302\) −17.7538 −1.02162
\(303\) 0 0
\(304\) 54.7386 3.13948
\(305\) −15.3693 −0.880045
\(306\) 0 0
\(307\) 19.3153 1.10238 0.551192 0.834378i \(-0.314173\pi\)
0.551192 + 0.834378i \(0.314173\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 31.6155 1.79275 0.896376 0.443294i \(-0.146190\pi\)
0.896376 + 0.443294i \(0.146190\pi\)
\(312\) 0 0
\(313\) 22.3002 1.26048 0.630241 0.776400i \(-0.282956\pi\)
0.630241 + 0.776400i \(0.282956\pi\)
\(314\) −51.8617 −2.92673
\(315\) 0 0
\(316\) −11.1231 −0.625724
\(317\) −10.4924 −0.589313 −0.294657 0.955603i \(-0.595205\pi\)
−0.294657 + 0.955603i \(0.595205\pi\)
\(318\) 0 0
\(319\) −10.4384 −0.584441
\(320\) 1.43845 0.0804116
\(321\) 0 0
\(322\) 0 0
\(323\) −3.12311 −0.173774
\(324\) 0 0
\(325\) −0.438447 −0.0243207
\(326\) −18.2462 −1.01056
\(327\) 0 0
\(328\) 33.6155 1.85611
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 18.2462 1.00139
\(333\) 0 0
\(334\) −17.7538 −0.971444
\(335\) 10.2462 0.559810
\(336\) 0 0
\(337\) −1.50758 −0.0821230 −0.0410615 0.999157i \(-0.513074\pi\)
−0.0410615 + 0.999157i \(0.513074\pi\)
\(338\) −32.8078 −1.78451
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 5.75379 0.310223
\(345\) 0 0
\(346\) −11.3693 −0.611218
\(347\) −7.12311 −0.382388 −0.191194 0.981552i \(-0.561236\pi\)
−0.191194 + 0.981552i \(0.561236\pi\)
\(348\) 0 0
\(349\) −10.4924 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.2462 0.546125
\(353\) 5.80776 0.309116 0.154558 0.987984i \(-0.450605\pi\)
0.154558 + 0.987984i \(0.450605\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) −5.12311 −0.271524
\(357\) 0 0
\(358\) −51.2311 −2.70765
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 45.1231 2.37162
\(363\) 0 0
\(364\) 0 0
\(365\) 12.2462 0.640996
\(366\) 0 0
\(367\) 8.68466 0.453335 0.226668 0.973972i \(-0.427217\pi\)
0.226668 + 0.973972i \(0.427217\pi\)
\(368\) −24.0000 −1.25109
\(369\) 0 0
\(370\) 15.3693 0.799013
\(371\) 0 0
\(372\) 0 0
\(373\) 4.63068 0.239768 0.119884 0.992788i \(-0.461748\pi\)
0.119884 + 0.992788i \(0.461748\pi\)
\(374\) −1.75379 −0.0906863
\(375\) 0 0
\(376\) −56.9848 −2.93877
\(377\) 2.93087 0.150947
\(378\) 0 0
\(379\) −16.4924 −0.847159 −0.423579 0.905859i \(-0.639227\pi\)
−0.423579 + 0.905859i \(0.639227\pi\)
\(380\) 32.4924 1.66683
\(381\) 0 0
\(382\) 34.7386 1.77738
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 49.6155 2.52536
\(387\) 0 0
\(388\) −26.4924 −1.34495
\(389\) 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
\(393\) 0 0
\(394\) −2.87689 −0.144936
\(395\) −2.43845 −0.122692
\(396\) 0 0
\(397\) −27.5616 −1.38327 −0.691637 0.722245i \(-0.743110\pi\)
−0.691637 + 0.722245i \(0.743110\pi\)
\(398\) 4.49242 0.225185
\(399\) 0 0
\(400\) 7.68466 0.384233
\(401\) −31.5616 −1.57611 −0.788054 0.615606i \(-0.788911\pi\)
−0.788054 + 0.615606i \(0.788911\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −74.1080 −3.68701
\(405\) 0 0
\(406\) 0 0
\(407\) 9.36932 0.464420
\(408\) 0 0
\(409\) −6.49242 −0.321030 −0.160515 0.987033i \(-0.551315\pi\)
−0.160515 + 0.987033i \(0.551315\pi\)
\(410\) 13.1231 0.648104
\(411\) 0 0
\(412\) −25.3693 −1.24986
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −2.87689 −0.141051
\(417\) 0 0
\(418\) 28.4924 1.39361
\(419\) 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 \(0.520839\pi\)
\(422\) 36.0000 1.75245
\(423\) 0 0
\(424\) 33.6155 1.63251
\(425\) −0.438447 −0.0212678
\(426\) 0 0
\(427\) 0 0
\(428\) −60.9848 −2.94781
\(429\) 0 0
\(430\) 2.24621 0.108322
\(431\) 19.8078 0.954106 0.477053 0.878874i \(-0.341705\pi\)
0.477053 + 0.878874i \(0.341705\pi\)
\(432\) 0 0
\(433\) −8.24621 −0.396288 −0.198144 0.980173i \(-0.563491\pi\)
−0.198144 + 0.980173i \(0.563491\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 24.2462 1.16118
\(437\) −22.2462 −1.06418
\(438\) 0 0
\(439\) −9.36932 −0.447173 −0.223587 0.974684i \(-0.571777\pi\)
−0.223587 + 0.974684i \(0.571777\pi\)
\(440\) 10.2462 0.488469
\(441\) 0 0
\(442\) 0.492423 0.0234221
\(443\) 2.63068 0.124988 0.0624938 0.998045i \(-0.480095\pi\)
0.0624938 + 0.998045i \(0.480095\pi\)
\(444\) 0 0
\(445\) −1.12311 −0.0532403
\(446\) 6.24621 0.295767
\(447\) 0 0
\(448\) 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
\(452\) 63.8617 3.00380
\(453\) 0 0
\(454\) 28.9848 1.36033
\(455\) 0 0
\(456\) 0 0
\(457\) −17.1231 −0.800985 −0.400493 0.916300i \(-0.631161\pi\)
−0.400493 + 0.916300i \(0.631161\pi\)
\(458\) −27.8617 −1.30189
\(459\) 0 0
\(460\) −14.2462 −0.664233
\(461\) −13.1231 −0.611204 −0.305602 0.952159i \(-0.598858\pi\)
−0.305602 + 0.952159i \(0.598858\pi\)
\(462\) 0 0
\(463\) 12.4924 0.580572 0.290286 0.956940i \(-0.406250\pi\)
0.290286 + 0.956940i \(0.406250\pi\)
\(464\) −51.3693 −2.38476
\(465\) 0 0
\(466\) −13.1231 −0.607916
\(467\) 22.4384 1.03833 0.519164 0.854675i \(-0.326243\pi\)
0.519164 + 0.854675i \(0.326243\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −22.2462 −1.02614
\(471\) 0 0
\(472\) −26.2462 −1.20808
\(473\) 1.36932 0.0629613
\(474\) 0 0
\(475\) 7.12311 0.326831
\(476\) 0 0
\(477\) 0 0
\(478\) −50.7386 −2.32073
\(479\) 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
\(482\) 10.8769 0.495429
\(483\) 0 0
\(484\) −39.0540 −1.77518
\(485\) −5.80776 −0.263717
\(486\) 0 0
\(487\) −3.12311 −0.141521 −0.0707607 0.997493i \(-0.522543\pi\)
−0.0707607 + 0.997493i \(0.522543\pi\)
\(488\) −100.847 −4.56511
\(489\) 0 0
\(490\) 0 0
\(491\) 41.1771 1.85830 0.929148 0.369708i \(-0.120542\pi\)
0.929148 + 0.369708i \(0.120542\pi\)
\(492\) 0 0
\(493\) 2.93087 0.132000
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 41.1771 1.84334 0.921670 0.387976i \(-0.126826\pi\)
0.921670 + 0.387976i \(0.126826\pi\)
\(500\) 4.56155 0.203999
\(501\) 0 0
\(502\) −22.7386 −1.01487
\(503\) 38.9309 1.73584 0.867921 0.496703i \(-0.165456\pi\)
0.867921 + 0.496703i \(0.165456\pi\)
\(504\) 0 0
\(505\) −16.2462 −0.722947
\(506\) −12.4924 −0.555356
\(507\) 0 0
\(508\) −28.4924 −1.26415
\(509\) −11.7538 −0.520978 −0.260489 0.965477i \(-0.583884\pi\)
−0.260489 + 0.965477i \(0.583884\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −50.4233 −2.22842
\(513\) 0 0
\(514\) −26.8769 −1.18549
\(515\) −5.56155 −0.245071
\(516\) 0 0
\(517\) −13.5616 −0.596436
\(518\) 0 0
\(519\) 0 0
\(520\) −2.87689 −0.126160
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) −40.4924 −1.77061 −0.885305 0.465011i \(-0.846050\pi\)
−0.885305 + 0.465011i \(0.846050\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 32.9848 1.43821
\(527\) 0 0
\(528\) 0 0
\(529\) −13.2462 −0.575922
\(530\) 13.1231 0.570031
\(531\) 0 0
\(532\) 0 0
\(533\) −2.24621 −0.0972942
\(534\) 0 0
\(535\) −13.3693 −0.578006
\(536\) 67.2311 2.90394
\(537\) 0 0
\(538\) −53.1231 −2.29030
\(539\) 0 0
\(540\) 0 0
\(541\) −37.8078 −1.62548 −0.812741 0.582625i \(-0.802026\pi\)
−0.812741 + 0.582625i \(0.802026\pi\)
\(542\) 40.9848 1.76045
\(543\) 0 0
\(544\) −2.87689 −0.123346
\(545\) 5.31534 0.227684
\(546\) 0 0
\(547\) −2.24621 −0.0960411 −0.0480205 0.998846i \(-0.515291\pi\)
−0.0480205 + 0.998846i \(0.515291\pi\)
\(548\) 78.1080 3.33661
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) −47.6155 −2.02849
\(552\) 0 0
\(553\) 0 0
\(554\) −0.630683 −0.0267952
\(555\) 0 0
\(556\) 68.9848 2.92561
\(557\) 13.1231 0.556044 0.278022 0.960575i \(-0.410321\pi\)
0.278022 + 0.960575i \(0.410321\pi\)
\(558\) 0 0
\(559\) −0.384472 −0.0162614
\(560\) 0 0
\(561\) 0 0
\(562\) −31.8617 −1.34401
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 28.9848 1.21832
\(567\) 0 0
\(568\) −52.4924 −2.20253
\(569\) 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 \(-0.178240\pi\)
0.847278 + 0.531150i \(0.178240\pi\)
\(572\) −3.12311 −0.130584
\(573\) 0 0
\(574\) 0 0
\(575\) −3.12311 −0.130243
\(576\) 0 0
\(577\) 24.0540 1.00138 0.500690 0.865627i \(-0.333080\pi\)
0.500690 + 0.865627i \(0.333080\pi\)
\(578\) −43.0540 −1.79081
\(579\) 0 0
\(580\) −30.4924 −1.26613
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 80.3542 3.32508
\(585\) 0 0
\(586\) −6.87689 −0.284082
\(587\) −26.2462 −1.08330 −0.541649 0.840605i \(-0.682200\pi\)
−0.541649 + 0.840605i \(0.682200\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −10.2462 −0.421830
\(591\) 0 0
\(592\) 46.1080 1.89503
\(593\) −27.5616 −1.13182 −0.565909 0.824468i \(-0.691475\pi\)
−0.565909 + 0.824468i \(0.691475\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −55.8617 −2.28819
\(597\) 0 0
\(598\) 3.50758 0.143436
\(599\) 11.8078 0.482452 0.241226 0.970469i \(-0.422450\pi\)
0.241226 + 0.970469i \(0.422450\pi\)
\(600\) 0 0
\(601\) −6.49242 −0.264831 −0.132416 0.991194i \(-0.542273\pi\)
−0.132416 + 0.991194i \(0.542273\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −31.6155 −1.28642
\(605\) −8.56155 −0.348077
\(606\) 0 0
\(607\) 42.0540 1.70692 0.853459 0.521160i \(-0.174500\pi\)
0.853459 + 0.521160i \(0.174500\pi\)
\(608\) 46.7386 1.89550
\(609\) 0 0
\(610\) −39.3693 −1.59402
\(611\) 3.80776 0.154046
\(612\) 0 0
\(613\) 40.7386 1.64542 0.822709 0.568463i \(-0.192462\pi\)
0.822709 + 0.568463i \(0.192462\pi\)
\(614\) 49.4773 1.99674
\(615\) 0 0
\(616\) 0 0
\(617\) −32.2462 −1.29818 −0.649092 0.760710i \(-0.724851\pi\)
−0.649092 + 0.760710i \(0.724851\pi\)
\(618\) 0 0
\(619\) −32.1080 −1.29053 −0.645264 0.763960i \(-0.723253\pi\)
−0.645264 + 0.763960i \(0.723253\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 80.9848 3.24720
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 57.1231 2.28310
\(627\) 0 0
\(628\) −92.3542 −3.68533
\(629\) −2.63068 −0.104892
\(630\) 0 0
\(631\) −11.8078 −0.470060 −0.235030 0.971988i \(-0.575519\pi\)
−0.235030 + 0.971988i \(0.575519\pi\)
\(632\) −16.0000 −0.636446
\(633\) 0 0
\(634\) −26.8769 −1.06742
\(635\) −6.24621 −0.247873
\(636\) 0 0
\(637\) 0 0
\(638\) −26.7386 −1.05859
\(639\) 0 0
\(640\) −9.43845 −0.373087
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −1.56155 −0.0615816 −0.0307908 0.999526i \(-0.509803\pi\)
−0.0307908 + 0.999526i \(0.509803\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 36.4924 1.43467 0.717333 0.696731i \(-0.245363\pi\)
0.717333 + 0.696731i \(0.245363\pi\)
\(648\) 0 0
\(649\) −6.24621 −0.245185
\(650\) −1.12311 −0.0440518
\(651\) 0 0
\(652\) −32.4924 −1.27250
\(653\) 33.2311 1.30043 0.650216 0.759750i \(-0.274678\pi\)
0.650216 + 0.759750i \(0.274678\pi\)
\(654\) 0 0
\(655\) −0.876894 −0.0342631
\(656\) 39.3693 1.53711
\(657\) 0 0
\(658\) 0 0
\(659\) −9.17708 −0.357488 −0.178744 0.983896i \(-0.557203\pi\)
−0.178744 + 0.983896i \(0.557203\pi\)
\(660\) 0 0
\(661\) 5.12311 0.199266 0.0996329 0.995024i \(-0.468233\pi\)
0.0996329 + 0.995024i \(0.468233\pi\)
\(662\) 30.7386 1.19469
\(663\) 0 0
\(664\) 26.2462 1.01855
\(665\) 0 0
\(666\) 0 0
\(667\) 20.8769 0.808357
\(668\) −31.6155 −1.22324
\(669\) 0 0
\(670\) 26.2462 1.01398
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 31.8617 1.22818 0.614090 0.789236i \(-0.289523\pi\)
0.614090 + 0.789236i \(0.289523\pi\)
\(674\) −3.86174 −0.148749
\(675\) 0 0
\(676\) −58.4233 −2.24705
\(677\) 4.93087 0.189509 0.0947544 0.995501i \(-0.469793\pi\)
0.0947544 + 0.995501i \(0.469793\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.87689 −0.110324
\(681\) 0 0
\(682\) 0 0
\(683\) 6.73863 0.257847 0.128923 0.991655i \(-0.458848\pi\)
0.128923 + 0.991655i \(0.458848\pi\)
\(684\) 0 0
\(685\) 17.1231 0.654240
\(686\) 0 0
\(687\) 0 0
\(688\) 6.73863 0.256908
\(689\) −2.24621 −0.0855738
\(690\) 0 0
\(691\) 24.4924 0.931736 0.465868 0.884854i \(-0.345742\pi\)
0.465868 + 0.884854i \(0.345742\pi\)
\(692\) −20.2462 −0.769645
\(693\) 0 0
\(694\) −18.2462 −0.692617
\(695\) 15.1231 0.573652
\(696\) 0 0
\(697\) −2.24621 −0.0850813
\(698\) −26.8769 −1.01731
\(699\) 0 0
\(700\) 0 0
\(701\) −28.9309 −1.09270 −0.546352 0.837556i \(-0.683984\pi\)
−0.546352 + 0.837556i \(0.683984\pi\)
\(702\) 0 0
\(703\) 42.7386 1.61192
\(704\) 2.24621 0.0846573
\(705\) 0 0
\(706\) 14.8769 0.559899
\(707\) 0 0
\(708\) 0 0
\(709\) 27.1771 1.02066 0.510328 0.859980i \(-0.329524\pi\)
0.510328 + 0.859980i \(0.329524\pi\)
\(710\) −20.4924 −0.769067
\(711\) 0 0
\(712\) −7.36932 −0.276177
\(713\) 0 0
\(714\) 0 0
\(715\) −0.684658 −0.0256048
\(716\) −91.2311 −3.40946
\(717\) 0 0
\(718\) −20.4924 −0.764770
\(719\) 8.38447 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 81.3002 3.02568
\(723\) 0 0
\(724\) 80.3542 2.98634
\(725\) −6.68466 −0.248262
\(726\) 0 0
\(727\) −52.4924 −1.94684 −0.973418 0.229035i \(-0.926443\pi\)
−0.973418 + 0.229035i \(0.926443\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 31.3693 1.16103
\(731\) −0.384472 −0.0142202
\(732\) 0 0
\(733\) −6.68466 −0.246903 −0.123452 0.992351i \(-0.539396\pi\)
−0.123452 + 0.992351i \(0.539396\pi\)
\(734\) 22.2462 0.821123
\(735\) 0 0
\(736\) −20.4924 −0.755361
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 34.9309 1.28495 0.642476 0.766305i \(-0.277907\pi\)
0.642476 + 0.766305i \(0.277907\pi\)
\(740\) 27.3693 1.00612
\(741\) 0 0
\(742\) 0 0
\(743\) 32.9848 1.21010 0.605048 0.796189i \(-0.293154\pi\)
0.605048 + 0.796189i \(0.293154\pi\)
\(744\) 0 0
\(745\) −12.2462 −0.448666
\(746\) 11.8617 0.434289
\(747\) 0 0
\(748\) −3.12311 −0.114192
\(749\) 0 0
\(750\) 0 0
\(751\) 17.0691 0.622861 0.311431 0.950269i \(-0.399192\pi\)
0.311431 + 0.950269i \(0.399192\pi\)
\(752\) −66.7386 −2.43371
\(753\) 0 0
\(754\) 7.50758 0.273410
\(755\) −6.93087 −0.252240
\(756\) 0 0
\(757\) 39.3693 1.43090 0.715451 0.698663i \(-0.246221\pi\)
0.715451 + 0.698663i \(0.246221\pi\)
\(758\) −42.2462 −1.53445
\(759\) 0 0
\(760\) 46.7386 1.69539
\(761\) 48.2462 1.74892 0.874462 0.485094i \(-0.161215\pi\)
0.874462 + 0.485094i \(0.161215\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 61.8617 2.23808
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 1.75379 0.0633256
\(768\) 0 0
\(769\) 42.4924 1.53232 0.766158 0.642652i \(-0.222166\pi\)
0.766158 + 0.642652i \(0.222166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 88.3542 3.17994
\(773\) 36.9309 1.32831 0.664156 0.747594i \(-0.268791\pi\)
0.664156 + 0.747594i \(0.268791\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −38.1080 −1.36800
\(777\) 0 0
\(778\) 63.8617 2.28955
\(779\) 36.4924 1.30748
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) 3.50758 0.125431
\(783\) 0 0
\(784\) 0 0
\(785\) −20.2462 −0.722618
\(786\) 0 0
\(787\) 49.1771 1.75297 0.876487 0.481426i \(-0.159881\pi\)
0.876487 + 0.481426i \(0.159881\pi\)
\(788\) −5.12311 −0.182503
\(789\) 0 0
\(790\) −6.24621 −0.222230
\(791\) 0 0
\(792\) 0 0
\(793\) 6.73863 0.239296
\(794\) −70.6004 −2.50551
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 24.0540 0.852036 0.426018 0.904715i \(-0.359916\pi\)
0.426018 + 0.904715i \(0.359916\pi\)
\(798\) 0 0
\(799\) 3.80776 0.134709
\(800\) 6.56155 0.231986
\(801\) 0 0
\(802\) −80.8466 −2.85479
\(803\) 19.1231 0.674840
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −106.600 −3.75019
\(809\) −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
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) −7.12311 −0.249512
\(816\) 0 0
\(817\) 6.24621 0.218527
\(818\) −16.6307 −0.581478
\(819\) 0 0
\(820\) 23.3693 0.816092
\(821\) 21.4233 0.747678 0.373839 0.927494i \(-0.378041\pi\)
0.373839 + 0.927494i \(0.378041\pi\)
\(822\) 0 0
\(823\) −36.4924 −1.27205 −0.636023 0.771670i \(-0.719422\pi\)
−0.636023 + 0.771670i \(0.719422\pi\)
\(824\) −36.4924 −1.27127
\(825\) 0 0
\(826\) 0 0
\(827\) 5.36932 0.186709 0.0933547 0.995633i \(-0.470241\pi\)
0.0933547 + 0.995633i \(0.470241\pi\)
\(828\) 0 0
\(829\) −34.8769 −1.21132 −0.605662 0.795722i \(-0.707092\pi\)
−0.605662 + 0.795722i \(0.707092\pi\)
\(830\) 10.2462 0.355651
\(831\) 0 0
\(832\) −0.630683 −0.0218650
\(833\) 0 0
\(834\) 0 0
\(835\) −6.93087 −0.239853
\(836\) 50.7386 1.75483
\(837\) 0 0
\(838\) 67.2311 2.32246
\(839\) −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
\(842\) −6.87689 −0.236993
\(843\) 0 0
\(844\) 64.1080 2.20669
\(845\) −12.8078 −0.440600
\(846\) 0 0
\(847\) 0 0
\(848\) 39.3693 1.35195
\(849\) 0 0
\(850\) −1.12311 −0.0385222
\(851\) −18.7386 −0.642352
\(852\) 0 0
\(853\) 7.26137 0.248624 0.124312 0.992243i \(-0.460328\pi\)
0.124312 + 0.992243i \(0.460328\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −87.7235 −2.99833
\(857\) −15.7538 −0.538139 −0.269070 0.963121i \(-0.586716\pi\)
−0.269070 + 0.963121i \(0.586716\pi\)
\(858\) 0 0
\(859\) 16.4924 0.562714 0.281357 0.959603i \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 50.7386 1.72816
\(863\) 25.7538 0.876669 0.438335 0.898812i \(-0.355569\pi\)
0.438335 + 0.898812i \(0.355569\pi\)
\(864\) 0 0
\(865\) −4.43845 −0.150912
\(866\) −21.1231 −0.717792
\(867\) 0 0
\(868\) 0 0
\(869\) −3.80776 −0.129170
\(870\) 0 0
\(871\) −4.49242 −0.152220
\(872\) 34.8769 1.18108
\(873\) 0 0
\(874\) −56.9848 −1.92754
\(875\) 0 0
\(876\) 0 0
\(877\) −40.2462 −1.35902 −0.679509 0.733667i \(-0.737807\pi\)
−0.679509 + 0.733667i \(0.737807\pi\)
\(878\) −24.0000 −0.809961
\(879\) 0 0
\(880\) 12.0000 0.404520
\(881\) −11.8617 −0.399632 −0.199816 0.979833i \(-0.564034\pi\)
−0.199816 + 0.979833i \(0.564034\pi\)
\(882\) 0 0
\(883\) −8.49242 −0.285793 −0.142896 0.989738i \(-0.545642\pi\)
−0.142896 + 0.989738i \(0.545642\pi\)
\(884\) 0.876894 0.0294931
\(885\) 0 0
\(886\) 6.73863 0.226389
\(887\) 20.4924 0.688068 0.344034 0.938957i \(-0.388206\pi\)
0.344034 + 0.938957i \(0.388206\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.87689 −0.0964337
\(891\) 0 0
\(892\) 11.1231 0.372429
\(893\) −61.8617 −2.07012
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 4.63068 0.154528
\(899\) 0 0
\(900\) 0 0
\(901\) −2.24621 −0.0748321
\(902\) 20.4924 0.682323
\(903\) 0 0
\(904\) 91.8617 3.05528
\(905\) 17.6155 0.585560
\(906\) 0 0
\(907\) −24.1080 −0.800491 −0.400246 0.916408i \(-0.631075\pi\)
−0.400246 + 0.916408i \(0.631075\pi\)
\(908\) 51.6155 1.71292
\(909\) 0 0
\(910\) 0 0
\(911\) 28.4924 0.943996 0.471998 0.881600i \(-0.343533\pi\)
0.471998 + 0.881600i \(0.343533\pi\)
\(912\) 0 0
\(913\) 6.24621 0.206719
\(914\) −43.8617 −1.45082
\(915\) 0 0
\(916\) −49.6155 −1.63934
\(917\) 0 0
\(918\) 0 0
\(919\) 40.3002 1.32938 0.664690 0.747119i \(-0.268564\pi\)
0.664690 + 0.747119i \(0.268564\pi\)
\(920\) −20.4924 −0.675615
\(921\) 0 0
\(922\) −33.6155 −1.10707
\(923\) 3.50758 0.115453
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) −43.8617 −1.43983
\(929\) 22.1080 0.725338 0.362669 0.931918i \(-0.381866\pi\)
0.362669 + 0.931918i \(0.381866\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −23.3693 −0.765487
\(933\) 0 0
\(934\) 57.4773 1.88071
\(935\) −0.684658 −0.0223907
\(936\) 0 0
\(937\) 55.6695 1.81864 0.909322 0.416094i \(-0.136601\pi\)
0.909322 + 0.416094i \(0.136601\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −39.6155 −1.29212
\(941\) 43.8617 1.42985 0.714926 0.699200i \(-0.246460\pi\)
0.714926 + 0.699200i \(0.246460\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) −30.7386 −1.00046
\(945\) 0 0
\(946\) 3.50758 0.114041
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) −5.36932 −0.174295
\(950\) 18.2462 0.591985
\(951\) 0 0
\(952\) 0 0
\(953\) 33.1231 1.07296 0.536481 0.843912i \(-0.319753\pi\)
0.536481 + 0.843912i \(0.319753\pi\)
\(954\) 0 0
\(955\) 13.5616 0.438842
\(956\) −90.3542 −2.92226
\(957\) 0 0
\(958\) 12.4924 0.403612
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −6.73863 −0.217262
\(963\) 0 0
\(964\) 19.3693 0.623844
\(965\) 19.3693 0.623520
\(966\) 0 0
\(967\) −35.1231 −1.12948 −0.564741 0.825268i \(-0.691024\pi\)
−0.564741 + 0.825268i \(0.691024\pi\)
\(968\) −56.1771 −1.80560
\(969\) 0 0
\(970\) −14.8769 −0.477668
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −118.108 −3.78054
\(977\) −33.2311 −1.06316 −0.531578 0.847009i \(-0.678401\pi\)
−0.531578 + 0.847009i \(0.678401\pi\)
\(978\) 0 0
\(979\) −1.75379 −0.0560513
\(980\) 0 0
\(981\) 0 0
\(982\) 105.477 3.36591
\(983\) 51.4233 1.64015 0.820074 0.572257i \(-0.193932\pi\)
0.820074 + 0.572257i \(0.193932\pi\)
\(984\) 0 0
\(985\) −1.12311 −0.0357851
\(986\) 7.50758 0.239090
\(987\) 0 0
\(988\) −14.2462 −0.453232
\(989\) −2.73863 −0.0870835
\(990\) 0 0
\(991\) 12.4924 0.396835 0.198417 0.980118i \(-0.436420\pi\)
0.198417 + 0.980118i \(0.436420\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.75379 0.0555988
\(996\) 0 0
\(997\) 2.68466 0.0850240 0.0425120 0.999096i \(-0.486464\pi\)
0.0425120 + 0.999096i \(0.486464\pi\)
\(998\) 105.477 3.33882
\(999\) 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 2205.2.a.x.1.2 2
3.2 odd 2 245.2.a.d.1.1 2
7.6 odd 2 315.2.a.e.1.2 2
12.11 even 2 3920.2.a.bs.1.2 2
15.2 even 4 1225.2.b.f.99.1 4
15.8 even 4 1225.2.b.f.99.4 4
15.14 odd 2 1225.2.a.s.1.2 2
21.2 odd 6 245.2.e.h.116.2 4
21.5 even 6 245.2.e.i.116.2 4
21.11 odd 6 245.2.e.h.226.2 4
21.17 even 6 245.2.e.i.226.2 4
21.20 even 2 35.2.a.b.1.1 2
28.27 even 2 5040.2.a.bt.1.1 2
35.13 even 4 1575.2.d.e.1324.1 4
35.27 even 4 1575.2.d.e.1324.4 4
35.34 odd 2 1575.2.a.p.1.1 2
84.83 odd 2 560.2.a.i.1.1 2
105.62 odd 4 175.2.b.b.99.1 4
105.83 odd 4 175.2.b.b.99.4 4
105.104 even 2 175.2.a.f.1.2 2
168.83 odd 2 2240.2.a.bd.1.2 2
168.125 even 2 2240.2.a.bh.1.1 2
231.230 odd 2 4235.2.a.m.1.2 2
273.272 even 2 5915.2.a.l.1.2 2
420.83 even 4 2800.2.g.t.449.2 4
420.167 even 4 2800.2.g.t.449.3 4
420.419 odd 2 2800.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 21.20 even 2
175.2.a.f.1.2 2 105.104 even 2
175.2.b.b.99.1 4 105.62 odd 4
175.2.b.b.99.4 4 105.83 odd 4
245.2.a.d.1.1 2 3.2 odd 2
245.2.e.h.116.2 4 21.2 odd 6
245.2.e.h.226.2 4 21.11 odd 6
245.2.e.i.116.2 4 21.5 even 6
245.2.e.i.226.2 4 21.17 even 6
315.2.a.e.1.2 2 7.6 odd 2
560.2.a.i.1.1 2 84.83 odd 2
1225.2.a.s.1.2 2 15.14 odd 2
1225.2.b.f.99.1 4 15.2 even 4
1225.2.b.f.99.4 4 15.8 even 4
1575.2.a.p.1.1 2 35.34 odd 2
1575.2.d.e.1324.1 4 35.13 even 4
1575.2.d.e.1324.4 4 35.27 even 4
2205.2.a.x.1.2 2 1.1 even 1 trivial
2240.2.a.bd.1.2 2 168.83 odd 2
2240.2.a.bh.1.1 2 168.125 even 2
2800.2.a.bi.1.2 2 420.419 odd 2
2800.2.g.t.449.2 4 420.83 even 4
2800.2.g.t.449.3 4 420.167 even 4
3920.2.a.bs.1.2 2 12.11 even 2
4235.2.a.m.1.2 2 231.230 odd 2
5040.2.a.bt.1.1 2 28.27 even 2
5915.2.a.l.1.2 2 273.272 even 2