Properties

Label 315.2.a.e.1.2
Level $315$
Weight $2$
Character 315.1
Self dual yes
Analytic conductor $2.515$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.51528766367\)
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 \(\chi\) \(=\) 315.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.56155 q^{8} +O(q^{10})\) \(q+2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -1.00000 q^{7} +6.56155 q^{8} -2.56155 q^{10} +1.56155 q^{11} +0.438447 q^{13} -2.56155 q^{14} +7.68466 q^{16} +0.438447 q^{17} -7.12311 q^{19} -4.56155 q^{20} +4.00000 q^{22} -3.12311 q^{23} +1.00000 q^{25} +1.12311 q^{26} -4.56155 q^{28} -6.68466 q^{29} +6.56155 q^{32} +1.12311 q^{34} +1.00000 q^{35} +6.00000 q^{37} -18.2462 q^{38} -6.56155 q^{40} -5.12311 q^{41} +0.876894 q^{43} +7.12311 q^{44} -8.00000 q^{46} +8.68466 q^{47} +1.00000 q^{49} +2.56155 q^{50} +2.00000 q^{52} +5.12311 q^{53} -1.56155 q^{55} -6.56155 q^{56} -17.1231 q^{58} +4.00000 q^{59} +15.3693 q^{61} +1.43845 q^{64} -0.438447 q^{65} +10.2462 q^{67} +2.00000 q^{68} +2.56155 q^{70} -8.00000 q^{71} -12.2462 q^{73} +15.3693 q^{74} -32.4924 q^{76} -1.56155 q^{77} -2.43845 q^{79} -7.68466 q^{80} -13.1231 q^{82} -4.00000 q^{83} -0.438447 q^{85} +2.24621 q^{86} +10.2462 q^{88} +1.12311 q^{89} -0.438447 q^{91} -14.2462 q^{92} +22.2462 q^{94} +7.12311 q^{95} +5.80776 q^{97} +2.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} - q^{10} - q^{11} + 5q^{13} - q^{14} + 3q^{16} + 5q^{17} - 6q^{19} - 5q^{20} + 8q^{22} + 2q^{23} + 2q^{25} - 6q^{26} - 5q^{28} - q^{29} + 9q^{32} - 6q^{34} + 2q^{35} + 12q^{37} - 20q^{38} - 9q^{40} - 2q^{41} + 10q^{43} + 6q^{44} - 16q^{46} + 5q^{47} + 2q^{49} + q^{50} + 4q^{52} + 2q^{53} + q^{55} - 9q^{56} - 26q^{58} + 8q^{59} + 6q^{61} + 7q^{64} - 5q^{65} + 4q^{67} + 4q^{68} + q^{70} - 16q^{71} - 8q^{73} + 6q^{74} - 32q^{76} + q^{77} - 9q^{79} - 3q^{80} - 18q^{82} - 8q^{83} - 5q^{85} - 12q^{86} + 4q^{88} - 6q^{89} - 5q^{91} - 12q^{92} + 28q^{94} + 6q^{95} - 9q^{97} + q^{98} + O(q^{100}) \)

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
\(7\) −1.00000 −0.377964
\(8\) 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 \(-0.480634\pi\)
0.0608017 + 0.998150i \(0.480634\pi\)
\(14\) −2.56155 −0.684604
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 0.438447 0.106339 0.0531695 0.998586i \(-0.483068\pi\)
0.0531695 + 0.998586i \(0.483068\pi\)
\(18\) 0 0
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) −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
\(28\) −4.56155 −0.862052
\(29\) −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
\(35\) 1.00000 0.169031
\(36\) 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 \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 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 \(-0.281661\pi\)
0.633394 + 0.773830i \(0.281661\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 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
\(56\) −6.56155 −0.876824
\(57\) 0 0
\(58\) −17.1231 −2.24837
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 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 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
\(70\) 2.56155 0.306164
\(71\) −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 \(-0.754328\pi\)
−0.716655 + 0.697428i \(0.754328\pi\)
\(74\) 15.3693 1.78665
\(75\) 0 0
\(76\) −32.4924 −3.72714
\(77\) −1.56155 −0.177955
\(78\) 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 \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 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 \(-0.481042\pi\)
0.0595245 + 0.998227i \(0.481042\pi\)
\(90\) 0 0
\(91\) −0.438447 −0.0459618
\(92\) −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 \(-0.404732\pi\)
0.294845 + 0.955545i \(0.404732\pi\)
\(98\) 2.56155 0.258756
\(99\) 0 0
\(100\) 4.56155 0.456155
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) 5.56155 0.547996 0.273998 0.961730i \(-0.411654\pi\)
0.273998 + 0.961730i \(0.411654\pi\)
\(104\) 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
\(112\) −7.68466 −0.726132
\(113\) 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
\(119\) −0.438447 −0.0401924
\(120\) 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 \(-0.487803\pi\)
0.0383073 + 0.999266i \(0.487803\pi\)
\(132\) 0 0
\(133\) 7.12311 0.617652
\(134\) 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 \(-0.721631\pi\)
−0.641363 + 0.767238i \(0.721631\pi\)
\(140\) 4.56155 0.385522
\(141\) 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
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 20.2462 1.61582 0.807912 0.589303i \(-0.200598\pi\)
0.807912 + 0.589303i \(0.200598\pi\)
\(158\) −6.24621 −0.496922
\(159\) 0 0
\(160\) −6.56155 −0.518736
\(161\) 3.12311 0.246135
\(162\) 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 \(-0.413583\pi\)
0.268163 + 0.963373i \(0.413583\pi\)
\(168\) 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 \(-0.446035\pi\)
0.168724 + 0.985663i \(0.446035\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 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 \(-0.727195\pi\)
−0.654676 + 0.755910i \(0.727195\pi\)
\(182\) −1.12311 −0.0832501
\(183\) 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
\(196\) 4.56155 0.325825
\(197\) −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 \(-0.519799\pi\)
−0.0621614 + 0.998066i \(0.519799\pi\)
\(200\) 6.56155 0.463972
\(201\) 0 0
\(202\) 41.6155 2.92806
\(203\) 6.68466 0.469171
\(204\) 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 \(-0.526017\pi\)
−0.0816453 + 0.996661i \(0.526017\pi\)
\(224\) −6.56155 −0.438412
\(225\) 0 0
\(226\) 35.8617 2.38549
\(227\) −11.3153 −0.751026 −0.375513 0.926817i \(-0.622533\pi\)
−0.375513 + 0.926817i \(0.622533\pi\)
\(228\) 0 0
\(229\) 10.8769 0.718765 0.359383 0.933190i \(-0.382987\pi\)
0.359383 + 0.933190i \(0.382987\pi\)
\(230\) 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
\(238\) −1.12311 −0.0728001
\(239\) −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 \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(242\) −21.9309 −1.40977
\(243\) 0 0
\(244\) 70.1080 4.48820
\(245\) −1.00000 −0.0638877
\(246\) 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 \(-0.409615\pi\)
0.280152 + 0.959956i \(0.409615\pi\)
\(252\) 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 \(-0.393878\pi\)
0.327250 + 0.944938i \(0.393878\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −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
\(266\) 18.2462 1.11875
\(267\) 0 0
\(268\) 46.7386 2.85502
\(269\) 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 \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 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
\(280\) 6.56155 0.392128
\(281\) −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 \(-0.609180\pi\)
−0.336314 + 0.941750i \(0.609180\pi\)
\(284\) −36.4924 −2.16543
\(285\) 0 0
\(286\) 1.75379 0.103704
\(287\) 5.12311 0.302407
\(288\) 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 \(-0.475013\pi\)
0.0784197 + 0.996920i \(0.475013\pi\)
\(294\) 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
\(301\) −0.876894 −0.0505434
\(302\) −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 \(-0.685827\pi\)
−0.551192 + 0.834378i \(0.685827\pi\)
\(308\) −7.12311 −0.405877
\(309\) 0 0
\(310\) 0 0
\(311\) −31.6155 −1.79275 −0.896376 0.443294i \(-0.853810\pi\)
−0.896376 + 0.443294i \(0.853810\pi\)
\(312\) 0 0
\(313\) −22.3002 −1.26048 −0.630241 0.776400i \(-0.717044\pi\)
−0.630241 + 0.776400i \(0.717044\pi\)
\(314\) 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
\(322\) 8.00000 0.445823
\(323\) −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
\(329\) −8.68466 −0.478801
\(330\) 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
\(343\) −1.00000 −0.0539949
\(344\) 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 \(-0.409393\pi\)
0.280823 + 0.959760i \(0.409393\pi\)
\(350\) −2.56155 −0.136921
\(351\) 0 0
\(352\) 10.2462 0.546125
\(353\) −5.80776 −0.309116 −0.154558 0.987984i \(-0.549395\pi\)
−0.154558 + 0.987984i \(0.549395\pi\)
\(354\) 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
\(364\) −2.00000 −0.104828
\(365\) 12.2462 0.640996
\(366\) 0 0
\(367\) −8.68466 −0.453335 −0.226668 0.973972i \(-0.572783\pi\)
−0.226668 + 0.973972i \(0.572783\pi\)
\(368\) −24.0000 −1.25109
\(369\) 0 0
\(370\) −15.3693 −0.799013
\(371\) −5.12311 −0.265978
\(372\) 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 \(-0.551015\pi\)
−0.159583 + 0.987184i \(0.551015\pi\)
\(384\) 0 0
\(385\) 1.56155 0.0795841
\(386\) 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
\(392\) 6.56155 0.331408
\(393\) 0 0
\(394\) −2.87689 −0.144936
\(395\) 2.43845 0.122692
\(396\) 0 0
\(397\) 27.5616 1.38327 0.691637 0.722245i \(-0.256890\pi\)
0.691637 + 0.722245i \(0.256890\pi\)
\(398\) −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
\(406\) 17.1231 0.849805
\(407\) 9.36932 0.464420
\(408\) 0 0
\(409\) 6.49242 0.321030 0.160515 0.987033i \(-0.448685\pi\)
0.160515 + 0.987033i \(0.448685\pi\)
\(410\) 13.1231 0.648104
\(411\) 0 0
\(412\) 25.3693 1.24986
\(413\) −4.00000 −0.196827
\(414\) 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 \(-0.721524\pi\)
−0.641106 + 0.767453i \(0.721524\pi\)
\(420\) 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
\(427\) −15.3693 −0.743773
\(428\) −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 \(-0.436509\pi\)
0.198144 + 0.980173i \(0.436509\pi\)
\(434\) 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 \(-0.428223\pi\)
0.223587 + 0.974684i \(0.428223\pi\)
\(440\) −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
\(448\) −1.43845 −0.0679602
\(449\) 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
\(455\) 0.438447 0.0205547
\(456\) 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 \(-0.401142\pi\)
0.305602 + 0.952159i \(0.401142\pi\)
\(462\) 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 \(-0.673757\pi\)
−0.519164 + 0.854675i \(0.673757\pi\)
\(468\) 0 0
\(469\) −10.2462 −0.473126
\(470\) −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
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) −50.7386 −2.32073
\(479\) −4.87689 −0.222831 −0.111415 0.993774i \(-0.535538\pi\)
−0.111415 + 0.993774i \(0.535538\pi\)
\(480\) 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
\(490\) −2.56155 −0.115719
\(491\) 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
\(497\) 8.00000 0.358849
\(498\) 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 \(-0.834544\pi\)
−0.867921 + 0.496703i \(0.834544\pi\)
\(504\) 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 \(-0.416116\pi\)
0.260489 + 0.965477i \(0.416116\pi\)
\(510\) 0 0
\(511\) 12.2462 0.541740
\(512\) −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
\(518\) −15.3693 −0.675289
\(519\) 0 0
\(520\) −2.87689 −0.126160
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 40.4924 1.77061 0.885305 0.465011i \(-0.153950\pi\)
0.885305 + 0.465011i \(0.153950\pi\)
\(524\) 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
\(532\) 32.4924 1.40873
\(533\) −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
\(539\) 1.56155 0.0672608
\(540\) 0 0
\(541\) −37.8078 −1.62548 −0.812741 0.582625i \(-0.802026\pi\)
−0.812741 + 0.582625i \(0.802026\pi\)
\(542\) −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
\(553\) 2.43845 0.103693
\(554\) −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
\(560\) 7.68466 0.324736
\(561\) 0 0
\(562\) −31.8617 −1.34401
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 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
\(574\) 13.1231 0.547748
\(575\) −3.12311 −0.130243
\(576\) 0 0
\(577\) −24.0540 −1.00138 −0.500690 0.865627i \(-0.666920\pi\)
−0.500690 + 0.865627i \(0.666920\pi\)
\(578\) −43.0540 −1.79081
\(579\) 0 0
\(580\) 30.4924 1.26613
\(581\) 4.00000 0.165948
\(582\) 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 \(-0.317800\pi\)
0.541649 + 0.840605i \(0.317800\pi\)
\(588\) 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 \(-0.308525\pi\)
0.565909 + 0.824468i \(0.308525\pi\)
\(594\) 0 0
\(595\) 0.438447 0.0179746
\(596\) −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 \(-0.457727\pi\)
0.132416 + 0.991194i \(0.457727\pi\)
\(602\) −2.24621 −0.0915487
\(603\) 0 0
\(604\) −31.6155 −1.28642
\(605\) 8.56155 0.348077
\(606\) 0 0
\(607\) −42.0540 −1.70692 −0.853459 0.521160i \(-0.825500\pi\)
−0.853459 + 0.521160i \(0.825500\pi\)
\(608\) −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
\(616\) −10.2462 −0.412832
\(617\) −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 \(-0.276747\pi\)
0.645264 + 0.763960i \(0.276747\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −80.9848 −3.24720
\(623\) −1.12311 −0.0449963
\(624\) 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
\(637\) 0.438447 0.0173719
\(638\) −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 \(-0.490197\pi\)
0.0307908 + 0.999526i \(0.490197\pi\)
\(644\) 14.2462 0.561379
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) −36.4924 −1.43467 −0.717333 0.696731i \(-0.754637\pi\)
−0.717333 + 0.696731i \(0.754637\pi\)
\(648\) 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
\(658\) −22.2462 −0.867248
\(659\) −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 \(-0.531767\pi\)
−0.0996329 + 0.995024i \(0.531767\pi\)
\(662\) 30.7386 1.19469
\(663\) 0 0
\(664\) −26.2462 −1.01855
\(665\) −7.12311 −0.276222
\(666\) 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 \(-0.530207\pi\)
−0.0947544 + 0.995501i \(0.530207\pi\)
\(678\) 0 0
\(679\) −5.80776 −0.222882
\(680\) −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
\(686\) −2.56155 −0.0978005
\(687\) 0 0
\(688\) 6.73863 0.256908
\(689\) 2.24621 0.0855738
\(690\) 0 0
\(691\) −24.4924 −0.931736 −0.465868 0.884854i \(-0.654258\pi\)
−0.465868 + 0.884854i \(0.654258\pi\)
\(692\) 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
\(700\) −4.56155 −0.172410
\(701\) −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
\(707\) −16.2462 −0.611002
\(708\) 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 \(-0.549971\pi\)
−0.156344 + 0.987703i \(0.549971\pi\)
\(720\) 0 0
\(721\) −5.56155 −0.207123
\(722\) 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 \(-0.0735572\pi\)
0.973418 + 0.229035i \(0.0735572\pi\)
\(728\) −2.87689 −0.106625
\(729\) 0 0
\(730\) 31.3693 1.16103
\(731\) 0.384472 0.0142202
\(732\) 0 0
\(733\) 6.68466 0.246903 0.123452 0.992351i \(-0.460604\pi\)
0.123452 + 0.992351i \(0.460604\pi\)
\(734\) −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
\(742\) −13.1231 −0.481764
\(743\) 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
\(749\) 13.3693 0.488504
\(750\) 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 \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) 0 0
\(763\) −5.31534 −0.192428
\(764\) 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 \(-0.777834\pi\)
−0.766158 + 0.642652i \(0.777834\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) 88.3542 3.17994
\(773\) −36.9309 −1.32831 −0.664156 0.747594i \(-0.731209\pi\)
−0.664156 + 0.747594i \(0.731209\pi\)
\(774\) 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
\(784\) 7.68466 0.274452
\(785\) −20.2462 −0.722618
\(786\) 0 0
\(787\) −49.1771 −1.75297 −0.876487 0.481426i \(-0.840119\pi\)
−0.876487 + 0.481426i \(0.840119\pi\)
\(788\) −5.12311 −0.182503
\(789\) 0 0
\(790\) 6.24621 0.222230
\(791\) −14.0000 −0.497783
\(792\) 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 \(-0.640084\pi\)
−0.426018 + 0.904715i \(0.640084\pi\)
\(798\) 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
\(805\) −3.12311 −0.110075
\(806\) 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 \(-0.388083\pi\)
0.344397 + 0.938824i \(0.388083\pi\)
\(812\) 30.4924 1.07007
\(813\) 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
\(826\) −10.2462 −0.356511
\(827\) 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 \(-0.292908\pi\)
0.605662 + 0.795722i \(0.292908\pi\)
\(830\) 10.2462 0.355651
\(831\) 0 0
\(832\) 0.630683 0.0218650
\(833\) 0.438447 0.0151913
\(834\) 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 \(-0.333895\pi\)
0.498471 + 0.866907i \(0.333895\pi\)
\(840\) 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
\(847\) 8.56155 0.294178
\(848\) 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 \(-0.539672\pi\)
−0.124312 + 0.992243i \(0.539672\pi\)
\(854\) −39.3693 −1.34719
\(855\) 0 0
\(856\) −87.7235 −2.99833
\(857\) 15.7538 0.538139 0.269070 0.963121i \(-0.413284\pi\)
0.269070 + 0.963121i \(0.413284\pi\)
\(858\) 0 0
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) −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
\(875\) 1.00000 0.0338062
\(876\) 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 \(-0.435966\pi\)
0.199816 + 0.979833i \(0.435966\pi\)
\(882\) 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 \(-0.611794\pi\)
−0.344034 + 0.938957i \(0.611794\pi\)
\(888\) 0 0
\(889\) 6.24621 0.209491
\(890\) −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
\(896\) 9.43845 0.315316
\(897\) 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
\(910\) 1.12311 0.0372306
\(911\) 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
\(917\) −0.876894 −0.0289576
\(918\) 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 \(-0.618134\pi\)
−0.362669 + 0.931918i \(0.618134\pi\)
\(930\) 0 0
\(931\) −7.12311 −0.233450
\(932\) −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 \(-0.863399\pi\)
−0.909322 + 0.416094i \(0.863399\pi\)
\(938\) −26.2462 −0.856969
\(939\) 0 0
\(940\) −39.6155 −1.29212
\(941\) −43.8617 −1.42985 −0.714926 0.699200i \(-0.753540\pi\)
−0.714926 + 0.699200i \(0.753540\pi\)
\(942\) 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
\(952\) −2.87689 −0.0932407
\(953\) 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
\(959\) −17.1231 −0.552934
\(960\) 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 \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) 0 0
\(973\) 15.1231 0.484825
\(974\) −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
\(980\) −4.56155 −0.145713
\(981\) 0 0
\(982\) 105.477 3.36591
\(983\) −51.4233 −1.64015 −0.820074 0.572257i \(-0.806068\pi\)
−0.820074 + 0.572257i \(0.806068\pi\)
\(984\) 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
\(994\) 20.4924 0.649980
\(995\) 1.75379 0.0555988
\(996\) 0 0
\(997\) −2.68466 −0.0850240 −0.0425120 0.999096i \(-0.513536\pi\)
−0.0425120 + 0.999096i \(0.513536\pi\)
\(998\) 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 315.2.a.e.1.2 2
3.2 odd 2 35.2.a.b.1.1 2
4.3 odd 2 5040.2.a.bt.1.1 2
5.2 odd 4 1575.2.d.e.1324.4 4
5.3 odd 4 1575.2.d.e.1324.1 4
5.4 even 2 1575.2.a.p.1.1 2
7.6 odd 2 2205.2.a.x.1.2 2
12.11 even 2 560.2.a.i.1.1 2
15.2 even 4 175.2.b.b.99.1 4
15.8 even 4 175.2.b.b.99.4 4
15.14 odd 2 175.2.a.f.1.2 2
21.2 odd 6 245.2.e.i.116.2 4
21.5 even 6 245.2.e.h.116.2 4
21.11 odd 6 245.2.e.i.226.2 4
21.17 even 6 245.2.e.h.226.2 4
21.20 even 2 245.2.a.d.1.1 2
24.5 odd 2 2240.2.a.bh.1.1 2
24.11 even 2 2240.2.a.bd.1.2 2
33.32 even 2 4235.2.a.m.1.2 2
39.38 odd 2 5915.2.a.l.1.2 2
60.23 odd 4 2800.2.g.t.449.2 4
60.47 odd 4 2800.2.g.t.449.3 4
60.59 even 2 2800.2.a.bi.1.2 2
84.83 odd 2 3920.2.a.bs.1.2 2
105.62 odd 4 1225.2.b.f.99.1 4
105.83 odd 4 1225.2.b.f.99.4 4
105.104 even 2 1225.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 3.2 odd 2
175.2.a.f.1.2 2 15.14 odd 2
175.2.b.b.99.1 4 15.2 even 4
175.2.b.b.99.4 4 15.8 even 4
245.2.a.d.1.1 2 21.20 even 2
245.2.e.h.116.2 4 21.5 even 6
245.2.e.h.226.2 4 21.17 even 6
245.2.e.i.116.2 4 21.2 odd 6
245.2.e.i.226.2 4 21.11 odd 6
315.2.a.e.1.2 2 1.1 even 1 trivial
560.2.a.i.1.1 2 12.11 even 2
1225.2.a.s.1.2 2 105.104 even 2
1225.2.b.f.99.1 4 105.62 odd 4
1225.2.b.f.99.4 4 105.83 odd 4
1575.2.a.p.1.1 2 5.4 even 2
1575.2.d.e.1324.1 4 5.3 odd 4
1575.2.d.e.1324.4 4 5.2 odd 4
2205.2.a.x.1.2 2 7.6 odd 2
2240.2.a.bd.1.2 2 24.11 even 2
2240.2.a.bh.1.1 2 24.5 odd 2
2800.2.a.bi.1.2 2 60.59 even 2
2800.2.g.t.449.2 4 60.23 odd 4
2800.2.g.t.449.3 4 60.47 odd 4
3920.2.a.bs.1.2 2 84.83 odd 2
4235.2.a.m.1.2 2 33.32 even 2
5040.2.a.bt.1.1 2 4.3 odd 2
5915.2.a.l.1.2 2 39.38 odd 2