Properties

Label 1225.2.a.s.1.2
Level $1225$
Weight $2$
Character 1225.1
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.78167424761\)
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 \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} +4.00000 q^{6} +6.56155 q^{8} -0.561553 q^{9} +O(q^{10})\) \(q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} +4.00000 q^{6} +6.56155 q^{8} -0.561553 q^{9} -1.56155 q^{11} +7.12311 q^{12} +0.438447 q^{13} +7.68466 q^{16} -0.438447 q^{17} -1.43845 q^{18} +7.12311 q^{19} -4.00000 q^{22} -3.12311 q^{23} +10.2462 q^{24} +1.12311 q^{26} -5.56155 q^{27} +6.68466 q^{29} +6.56155 q^{32} -2.43845 q^{33} -1.12311 q^{34} -2.56155 q^{36} -6.00000 q^{37} +18.2462 q^{38} +0.684658 q^{39} -5.12311 q^{41} -0.876894 q^{43} -7.12311 q^{44} -8.00000 q^{46} -8.68466 q^{47} +12.0000 q^{48} -0.684658 q^{51} +2.00000 q^{52} +5.12311 q^{53} -14.2462 q^{54} +11.1231 q^{57} +17.1231 q^{58} +4.00000 q^{59} -15.3693 q^{61} +1.43845 q^{64} -6.24621 q^{66} -10.2462 q^{67} -2.00000 q^{68} -4.87689 q^{69} +8.00000 q^{71} -3.68466 q^{72} -12.2462 q^{73} -15.3693 q^{74} +32.4924 q^{76} +1.75379 q^{78} -2.43845 q^{79} -7.00000 q^{81} -13.1231 q^{82} +4.00000 q^{83} -2.24621 q^{86} +10.4384 q^{87} -10.2462 q^{88} +1.12311 q^{89} -14.2462 q^{92} -22.2462 q^{94} +10.2462 q^{96} +5.80776 q^{97} +0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 3q^{9} + q^{11} + 6q^{12} + 5q^{13} + 3q^{16} - 5q^{17} - 7q^{18} + 6q^{19} - 8q^{22} + 2q^{23} + 4q^{24} - 6q^{26} - 7q^{27} + q^{29} + 9q^{32} - 9q^{33} + 6q^{34} - q^{36} - 12q^{37} + 20q^{38} - 11q^{39} - 2q^{41} - 10q^{43} - 6q^{44} - 16q^{46} - 5q^{47} + 24q^{48} + 11q^{51} + 4q^{52} + 2q^{53} - 12q^{54} + 14q^{57} + 26q^{58} + 8q^{59} - 6q^{61} + 7q^{64} + 4q^{66} - 4q^{67} - 4q^{68} - 18q^{69} + 16q^{71} + 5q^{72} - 8q^{73} - 6q^{74} + 32q^{76} + 20q^{78} - 9q^{79} - 14q^{81} - 18q^{82} + 8q^{83} + 12q^{86} + 25q^{87} - 4q^{88} - 6q^{89} - 12q^{92} - 28q^{94} + 4q^{96} - 9q^{97} + 10q^{99} + O(q^{100}) \)

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 \(0.139387\pi\)
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 4.56155 2.28078
\(5\) 0 0
\(6\) 4.00000 1.63299
\(7\) 0 0
\(8\) 6.56155 2.31986
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 7.12311 2.05626
\(13\) 0.438447 0.121603 0.0608017 0.998150i \(-0.480634\pi\)
0.0608017 + 0.998150i \(0.480634\pi\)
\(14\) 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 \(0.516932\pi\)
\(18\) −1.43845 −0.339045
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −3.12311 −0.651213 −0.325606 0.945505i \(-0.605568\pi\)
−0.325606 + 0.945505i \(0.605568\pi\)
\(24\) 10.2462 2.09150
\(25\) 0 0
\(26\) 1.12311 0.220259
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 6.56155 1.15993
\(33\) −2.43845 −0.424479
\(34\) −1.12311 −0.192611
\(35\) 0 0
\(36\) −2.56155 −0.426925
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 18.2462 2.95993
\(39\) 0.684658 0.109633
\(40\) 0 0
\(41\) −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 \(-0.521299\pi\)
−0.0668626 + 0.997762i \(0.521299\pi\)
\(44\) −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 \(0.718339\pi\)
\(48\) 12.0000 1.73205
\(49\) 0 0
\(50\) 0 0
\(51\) −0.684658 −0.0958714
\(52\) 2.00000 0.277350
\(53\) 5.12311 0.703713 0.351856 0.936054i \(-0.385551\pi\)
0.351856 + 0.936054i \(0.385551\pi\)
\(54\) −14.2462 −1.93866
\(55\) 0 0
\(56\) 0 0
\(57\) 11.1231 1.47329
\(58\) 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 \(-0.942839\pi\)
−0.983920 + 0.178611i \(0.942839\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) 0 0
\(66\) −6.24621 −0.768855
\(67\) −10.2462 −1.25177 −0.625887 0.779914i \(-0.715263\pi\)
−0.625887 + 0.779914i \(0.715263\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.87689 −0.587109
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −3.68466 −0.434241
\(73\) −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
\(77\) 0 0
\(78\) 1.75379 0.198577
\(79\) −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
\(82\) −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 0
\(86\) −2.24621 −0.242215
\(87\) 10.4384 1.11912
\(88\) −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 0
\(92\) −14.2462 −1.48527
\(93\) 0 0
\(94\) −22.2462 −2.29452
\(95\) 0 0
\(96\) 10.2462 1.04575
\(97\) 5.80776 0.589689 0.294845 0.955545i \(-0.404732\pi\)
0.294845 + 0.955545i \(0.404732\pi\)
\(98\) 0 0
\(99\) 0.876894 0.0881312
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) −1.75379 −0.173651
\(103\) 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 \(0.723655\pi\)
\(108\) −25.3693 −2.44116
\(109\) 5.31534 0.509117 0.254559 0.967057i \(-0.418070\pi\)
0.254559 + 0.967057i \(0.418070\pi\)
\(110\) 0 0
\(111\) −9.36932 −0.889296
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 28.4924 2.66856
\(115\) 0 0
\(116\) 30.4924 2.83115
\(117\) −0.246211 −0.0227622
\(118\) 10.2462 0.943240
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) −39.3693 −3.56433
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.24621 0.554262 0.277131 0.960832i \(-0.410616\pi\)
0.277131 + 0.960832i \(0.410616\pi\)
\(128\) −9.43845 −0.834249
\(129\) −1.36932 −0.120562
\(130\) 0 0
\(131\) 0.876894 0.0766146 0.0383073 0.999266i \(-0.487803\pi\)
0.0383073 + 0.999266i \(0.487803\pi\)
\(132\) −11.1231 −0.968142
\(133\) 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 \(0.238838\pi\)
\(138\) −12.4924 −1.06343
\(139\) 15.1231 1.28273 0.641363 0.767238i \(-0.278369\pi\)
0.641363 + 0.767238i \(0.278369\pi\)
\(140\) 0 0
\(141\) −13.5616 −1.14209
\(142\) 20.4924 1.71969
\(143\) −0.684658 −0.0572540
\(144\) −4.31534 −0.359612
\(145\) 0 0
\(146\) −31.3693 −2.59614
\(147\) 0 0
\(148\) −27.3693 −2.24974
\(149\) 12.2462 1.00325 0.501624 0.865086i \(-0.332736\pi\)
0.501624 + 0.865086i \(0.332736\pi\)
\(150\) 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
\(153\) 0.246211 0.0199050
\(154\) 0 0
\(155\) 0 0
\(156\) 3.12311 0.250049
\(157\) 20.2462 1.61582 0.807912 0.589303i \(-0.200598\pi\)
0.807912 + 0.589303i \(0.200598\pi\)
\(158\) −6.24621 −0.496922
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) 0 0
\(162\) −17.9309 −1.40878
\(163\) 7.12311 0.557925 0.278962 0.960302i \(-0.410010\pi\)
0.278962 + 0.960302i \(0.410010\pi\)
\(164\) −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
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −4.43845 −0.337449 −0.168724 0.985663i \(-0.553965\pi\)
−0.168724 + 0.985663i \(0.553965\pi\)
\(174\) 26.7386 2.02705
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) 6.24621 0.469494
\(178\) 2.87689 0.215632
\(179\) 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 \(-0.272805\pi\)
0.654676 + 0.755910i \(0.272805\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) −20.4924 −1.51072
\(185\) 0 0
\(186\) 0 0
\(187\) 0.684658 0.0500672
\(188\) −39.6155 −2.88926
\(189\) 0 0
\(190\) 0 0
\(191\) −13.5616 −0.981280 −0.490640 0.871363i \(-0.663237\pi\)
−0.490640 + 0.871363i \(0.663237\pi\)
\(192\) 2.24621 0.162106
\(193\) −19.3693 −1.39423 −0.697117 0.716957i \(-0.745534\pi\)
−0.697117 + 0.716957i \(0.745534\pi\)
\(194\) 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 \(0.512739\pi\)
\(198\) 2.24621 0.159631
\(199\) 1.75379 0.124323 0.0621614 0.998066i \(-0.480201\pi\)
0.0621614 + 0.998066i \(0.480201\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 41.6155 2.92806
\(203\) 0 0
\(204\) −3.12311 −0.218661
\(205\) 0 0
\(206\) 14.2462 0.992581
\(207\) 1.75379 0.121897
\(208\) 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
\(213\) 12.4924 0.855967
\(214\) −34.2462 −2.34102
\(215\) 0 0
\(216\) −36.4924 −2.48299
\(217\) 0 0
\(218\) 13.6155 0.922160
\(219\) −19.1231 −1.29222
\(220\) 0 0
\(221\) −0.192236 −0.0129312
\(222\) −24.0000 −1.61077
\(223\) −2.43845 −0.163291 −0.0816453 0.996661i \(-0.526017\pi\)
−0.0816453 + 0.996661i \(0.526017\pi\)
\(224\) 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 \(0.377467\pi\)
\(228\) 50.7386 3.36025
\(229\) −10.8769 −0.718765 −0.359383 0.933190i \(-0.617013\pi\)
−0.359383 + 0.933190i \(0.617013\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 43.8617 2.87966
\(233\) −5.12311 −0.335626 −0.167813 0.985819i \(-0.553670\pi\)
−0.167813 + 0.985819i \(0.553670\pi\)
\(234\) −0.630683 −0.0412290
\(235\) 0 0
\(236\) 18.2462 1.18773
\(237\) −3.80776 −0.247341
\(238\) 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 \(-0.456331\pi\)
0.136761 + 0.990604i \(0.456331\pi\)
\(242\) −21.9309 −1.40977
\(243\) 5.75379 0.369106
\(244\) −70.1080 −4.48820
\(245\) 0 0
\(246\) −20.4924 −1.30655
\(247\) 3.12311 0.198718
\(248\) 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
\(253\) 4.87689 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 \(0.606122\pi\)
\(258\) −3.50758 −0.218372
\(259\) 0 0
\(260\) 0 0
\(261\) −3.75379 −0.232354
\(262\) 2.24621 0.138771
\(263\) 12.8769 0.794023 0.397012 0.917814i \(-0.370047\pi\)
0.397012 + 0.917814i \(0.370047\pi\)
\(264\) −16.0000 −0.984732
\(265\) 0 0
\(266\) 0 0
\(267\) 1.75379 0.107330
\(268\) −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 \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −3.36932 −0.204295
\(273\) 0 0
\(274\) 43.8617 2.64978
\(275\) 0 0
\(276\) −22.2462 −1.33906
\(277\) 0.246211 0.0147934 0.00739670 0.999973i \(-0.497646\pi\)
0.00739670 + 0.999973i \(0.497646\pi\)
\(278\) 38.7386 2.32339
\(279\) 0 0
\(280\) 0 0
\(281\) 12.4384 0.742016 0.371008 0.928630i \(-0.379012\pi\)
0.371008 + 0.928630i \(0.379012\pi\)
\(282\) −34.7386 −2.06866
\(283\) −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
\(287\) 0 0
\(288\) −3.68466 −0.217121
\(289\) −16.8078 −0.988692
\(290\) 0 0
\(291\) 9.06913 0.531642
\(292\) −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
\(295\) 0 0
\(296\) −39.3693 −2.28830
\(297\) 8.68466 0.503935
\(298\) 31.3693 1.81718
\(299\) −1.36932 −0.0791896
\(300\) 0 0
\(301\) 0 0
\(302\) −17.7538 −1.02162
\(303\) 25.3693 1.45743
\(304\) 54.7386 3.13948
\(305\) 0 0
\(306\) 0.630683 0.0360538
\(307\) −19.3153 −1.10238 −0.551192 0.834378i \(-0.685827\pi\)
−0.551192 + 0.834378i \(0.685827\pi\)
\(308\) 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 \(0.853810\pi\)
\(312\) 4.49242 0.254333
\(313\) −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 \(0.595205\pi\)
\(318\) 20.4924 1.14916
\(319\) −10.4384 −0.584441
\(320\) 0 0
\(321\) −20.8769 −1.16523
\(322\) 0 0
\(323\) −3.12311 −0.173774
\(324\) −31.9309 −1.77394
\(325\) 0 0
\(326\) 18.2462 1.01056
\(327\) 8.30019 0.459001
\(328\) −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
\(333\) 3.36932 0.184637
\(334\) −17.7538 −0.971444
\(335\) 0 0
\(336\) 0 0
\(337\) 1.50758 0.0821230 0.0410615 0.999157i \(-0.486926\pi\)
0.0410615 + 0.999157i \(0.486926\pi\)
\(338\) −32.8078 −1.78451
\(339\) 21.8617 1.18737
\(340\) 0 0
\(341\) 0 0
\(342\) −10.2462 −0.554052
\(343\) 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 \(0.561236\pi\)
\(348\) 47.6155 2.55246
\(349\) −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
\(352\) −10.2462 −0.546125
\(353\) 5.80776 0.309116 0.154558 0.987984i \(-0.450605\pi\)
0.154558 + 0.987984i \(0.450605\pi\)
\(354\) 16.0000 0.850390
\(355\) 0 0
\(356\) 5.12311 0.271524
\(357\) 0 0
\(358\) 51.2311 2.70765
\(359\) 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
\(362\) 45.1231 2.37162
\(363\) −13.3693 −0.701707
\(364\) 0 0
\(365\) 0 0
\(366\) −61.4773 −3.21347
\(367\) −8.68466 −0.453335 −0.226668 0.973972i \(-0.572783\pi\)
−0.226668 + 0.973972i \(0.572783\pi\)
\(368\) −24.0000 −1.25109
\(369\) 2.87689 0.149765
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.63068 −0.239768 −0.119884 0.992788i \(-0.538252\pi\)
−0.119884 + 0.992788i \(0.538252\pi\)
\(374\) 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 \(0.639227\pi\)
\(380\) 0 0
\(381\) 9.75379 0.499702
\(382\) −34.7386 −1.77738
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) −14.7386 −0.752128
\(385\) 0 0
\(386\) −49.6155 −2.52536
\(387\) 0.492423 0.0250312
\(388\) 26.4924 1.34495
\(389\) −24.9309 −1.26405 −0.632023 0.774950i \(-0.717775\pi\)
−0.632023 + 0.774950i \(0.717775\pi\)
\(390\) 0 0
\(391\) 1.36932 0.0692493
\(392\) 0 0
\(393\) 1.36932 0.0690729
\(394\) −2.87689 −0.144936
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 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
\(400\) 0 0
\(401\) 31.5616 1.57611 0.788054 0.615606i \(-0.211089\pi\)
0.788054 + 0.615606i \(0.211089\pi\)
\(402\) −40.9848 −2.04414
\(403\) 0 0
\(404\) 74.1080 3.68701
\(405\) 0 0
\(406\) 0 0
\(407\) 9.36932 0.464420
\(408\) −4.49242 −0.222408
\(409\) −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
\(412\) 25.3693 1.24986
\(413\) 0 0
\(414\) 4.49242 0.220791
\(415\) 0 0
\(416\) 2.87689 0.141051
\(417\) 23.6155 1.15646
\(418\) −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
\(423\) 4.87689 0.237123
\(424\) 33.6155 1.63251
\(425\) 0 0
\(426\) 32.0000 1.55041
\(427\) 0 0
\(428\) −60.9848 −2.94781
\(429\) −1.06913 −0.0516181
\(430\) 0 0
\(431\) −19.8078 −0.954106 −0.477053 0.878874i \(-0.658295\pi\)
−0.477053 + 0.878874i \(0.658295\pi\)
\(432\) −42.7386 −2.05626
\(433\) 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
\(438\) −48.9848 −2.34059
\(439\) −9.36932 −0.447173 −0.223587 0.974684i \(-0.571777\pi\)
−0.223587 + 0.974684i \(0.571777\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.492423 −0.0234221
\(443\) 2.63068 0.124988 0.0624938 0.998045i \(-0.480095\pi\)
0.0624938 + 0.998045i \(0.480095\pi\)
\(444\) −42.7386 −2.02829
\(445\) 0 0
\(446\) −6.24621 −0.295767
\(447\) 19.1231 0.904492
\(448\) 0 0
\(449\) −1.80776 −0.0853137 −0.0426568 0.999090i \(-0.513582\pi\)
−0.0426568 + 0.999090i \(0.513582\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 63.8617 3.00380
\(453\) −10.8229 −0.508505
\(454\) 28.9848 1.36033
\(455\) 0 0
\(456\) 72.9848 3.41783
\(457\) 17.1231 0.800985 0.400493 0.916300i \(-0.368839\pi\)
0.400493 + 0.916300i \(0.368839\pi\)
\(458\) −27.8617 −1.30189
\(459\) 2.43845 0.113817
\(460\) 0 0
\(461\) 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 \(-0.593750\pi\)
−0.290286 + 0.956940i \(0.593750\pi\)
\(464\) 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 \(0.326243\pi\)
\(468\) −1.12311 −0.0519156
\(469\) 0 0
\(470\) 0 0
\(471\) 31.6155 1.45677
\(472\) 26.2462 1.20808
\(473\) 1.36932 0.0629613
\(474\) −9.75379 −0.448006
\(475\) 0 0
\(476\) 0 0
\(477\) −2.87689 −0.131724
\(478\) 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
\(485\) 0 0
\(486\) 14.7386 0.668558
\(487\) 3.12311 0.141521 0.0707607 0.997493i \(-0.477457\pi\)
0.0707607 + 0.997493i \(0.477457\pi\)
\(488\) −100.847 −4.56511
\(489\) 11.1231 0.503004
\(490\) 0 0
\(491\) −41.1771 −1.85830 −0.929148 0.369708i \(-0.879458\pi\)
−0.929148 + 0.369708i \(0.879458\pi\)
\(492\) −36.4924 −1.64521
\(493\) −2.93087 −0.132000
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 16.0000 0.716977
\(499\) 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
\(502\) 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
\(505\) 0 0
\(506\) 12.4924 0.555356
\(507\) −20.0000 −0.888231
\(508\) 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
\(511\) 0 0
\(512\) −50.4233 −2.22842
\(513\) −39.6155 −1.74907
\(514\) −26.8769 −1.18549
\(515\) 0 0
\(516\) −6.24621 −0.274974
\(517\) 13.5616 0.596436
\(518\) 0 0
\(519\) −6.93087 −0.304231
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −9.61553 −0.420860
\(523\) 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
\(528\) −18.7386 −0.815494
\(529\) −13.2462 −0.575922
\(530\) 0 0
\(531\) −2.24621 −0.0974773
\(532\) 0 0
\(533\) −2.24621 −0.0972942
\(534\) 4.49242 0.194406
\(535\) 0 0
\(536\) −67.2311 −2.90394
\(537\) 31.2311 1.34772
\(538\) 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
\(543\) 27.5076 1.18046
\(544\) −2.87689 −0.123346
\(545\) 0 0
\(546\) 0 0
\(547\) 2.24621 0.0960411 0.0480205 0.998846i \(-0.484709\pi\)
0.0480205 + 0.998846i \(0.484709\pi\)
\(548\) 78.1080 3.33661
\(549\) 8.63068 0.368349
\(550\) 0 0
\(551\) 47.6155 2.02849
\(552\) −32.0000 −1.36201
\(553\) 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
\(561\) 1.06913 0.0451387
\(562\) 31.8617 1.34401
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) −61.8617 −2.60485
\(565\) 0 0
\(566\) −28.9848 −1.21832
\(567\) 0 0
\(568\) 52.4924 2.20253
\(569\) −30.9848 −1.29895 −0.649476 0.760382i \(-0.725012\pi\)
−0.649476 + 0.760382i \(0.725012\pi\)
\(570\) 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
\(573\) −21.1771 −0.884685
\(574\) 0 0
\(575\) 0 0
\(576\) −0.807764 −0.0336568
\(577\) −24.0540 −1.00138 −0.500690 0.865627i \(-0.666920\pi\)
−0.500690 + 0.865627i \(0.666920\pi\)
\(578\) −43.0540 −1.79081
\(579\) −30.2462 −1.25699
\(580\) 0 0
\(581\) 0 0
\(582\) 23.2311 0.962958
\(583\) −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
\(590\) 0 0
\(591\) −1.75379 −0.0721412
\(592\) −46.1080 −1.89503
\(593\) −27.5616 −1.13182 −0.565909 0.824468i \(-0.691475\pi\)
−0.565909 + 0.824468i \(0.691475\pi\)
\(594\) 22.2462 0.912773
\(595\) 0 0
\(596\) 55.8617 2.28819
\(597\) 2.73863 0.112085
\(598\) −3.50758 −0.143436
\(599\) −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 \(-0.542273\pi\)
−0.132416 + 0.991194i \(0.542273\pi\)
\(602\) 0 0
\(603\) 5.75379 0.234312
\(604\) −31.6155 −1.28642
\(605\) 0 0
\(606\) 64.9848 2.63983
\(607\) −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
\(610\) 0 0
\(611\) −3.80776 −0.154046
\(612\) 1.12311 0.0453989
\(613\) −40.7386 −1.64542 −0.822709 0.568463i \(-0.807538\pi\)
−0.822709 + 0.568463i \(0.807538\pi\)
\(614\) −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 \(0.724851\pi\)
\(618\) 22.2462 0.894874
\(619\) −32.1080 −1.29053 −0.645264 0.763960i \(-0.723253\pi\)
−0.645264 + 0.763960i \(0.723253\pi\)
\(620\) 0 0
\(621\) 17.3693 0.697007
\(622\) −80.9848 −3.24720
\(623\) 0 0
\(624\) 5.26137 0.210623
\(625\) 0 0
\(626\) −57.1231 −2.28310
\(627\) −17.3693 −0.693664
\(628\) 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
\(633\) 21.9460 0.872276
\(634\) −26.8769 −1.06742
\(635\) 0 0
\(636\) 36.4924 1.44702
\(637\) 0 0
\(638\) −26.7386 −1.05859
\(639\) −4.49242 −0.177717
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −53.4773 −2.11058
\(643\) 1.56155 0.0615816 0.0307908 0.999526i \(-0.490197\pi\)
0.0307908 + 0.999526i \(0.490197\pi\)
\(644\) 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 \(0.245363\pi\)
\(648\) −45.9309 −1.80433
\(649\) −6.24621 −0.245185
\(650\) 0 0
\(651\) 0 0
\(652\) 32.4924 1.27250
\(653\) 33.2311 1.30043 0.650216 0.759750i \(-0.274678\pi\)
0.650216 + 0.759750i \(0.274678\pi\)
\(654\) 21.2614 0.831385
\(655\) 0 0
\(656\) −39.3693 −1.53711
\(657\) 6.87689 0.268293
\(658\) 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 \(-0.468233\pi\)
0.0996329 + 0.995024i \(0.468233\pi\)
\(662\) 30.7386 1.19469
\(663\) −0.300187 −0.0116583
\(664\) 26.2462 1.01855
\(665\) 0 0
\(666\) 8.63068 0.334432
\(667\) −20.8769 −0.808357
\(668\) −31.6155 −1.22324
\(669\) −3.80776 −0.147217
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −31.8617 −1.22818 −0.614090 0.789236i \(-0.710477\pi\)
−0.614090 + 0.789236i \(0.710477\pi\)
\(674\) 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 \(0.469793\pi\)
\(678\) 56.0000 2.15067
\(679\) 0 0
\(680\) 0 0
\(681\) 17.6695 0.677097
\(682\) 0 0
\(683\) 6.73863 0.257847 0.128923 0.991655i \(-0.458848\pi\)
0.128923 + 0.991655i \(0.458848\pi\)
\(684\) −18.2462 −0.697661
\(685\) 0 0
\(686\) 0 0
\(687\) −16.9848 −0.648012
\(688\) −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
\(695\) 0 0
\(696\) 68.4924 2.59620
\(697\) 2.24621 0.0850813
\(698\) −26.8769 −1.01731
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 28.9309 1.09270 0.546352 0.837556i \(-0.316016\pi\)
0.546352 + 0.837556i \(0.316016\pi\)
\(702\) −6.24621 −0.235748
\(703\) −42.7386 −1.61192
\(704\) −2.24621 −0.0846573
\(705\) 0 0
\(706\) 14.8769 0.559899
\(707\) 0 0
\(708\) 28.4924 1.07081
\(709\) 27.1771 1.02066 0.510328 0.859980i \(-0.329524\pi\)
0.510328 + 0.859980i \(0.329524\pi\)
\(710\) 0 0
\(711\) 1.36932 0.0513534
\(712\) 7.36932 0.276177
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 91.2311 3.40946
\(717\) 30.9309 1.15513
\(718\) 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
\(721\) 0 0
\(722\) 81.3002 3.02568
\(723\) 6.63068 0.246598
\(724\) 80.3542 2.98634
\(725\) 0 0
\(726\) −34.2462 −1.27100
\(727\) 52.4924 1.94684 0.973418 0.229035i \(-0.0735572\pi\)
0.973418 + 0.229035i \(0.0735572\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 0.384472 0.0142202
\(732\) −109.477 −4.04640
\(733\) 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
\(738\) 7.36932 0.271268
\(739\) 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
\(743\) 32.9848 1.21010 0.605048 0.796189i \(-0.293154\pi\)
0.605048 + 0.796189i \(0.293154\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −11.8617 −0.434289
\(747\) −2.24621 −0.0821846
\(748\) 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
\(753\) 13.8617 0.505150
\(754\) 7.50758 0.273410
\(755\) 0 0
\(756\) 0 0
\(757\) −39.3693 −1.43090 −0.715451 0.698663i \(-0.753779\pi\)
−0.715451 + 0.698663i \(0.753779\pi\)
\(758\) −42.2462 −1.53445
\(759\) 7.61553 0.276426
\(760\) 0 0
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) 24.9848 0.905105
\(763\) 0 0
\(764\) −61.8617 −2.23808
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 1.75379 0.0633256
\(768\) −42.2462 −1.52443
\(769\) 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
\(772\) −88.3542 −3.17994
\(773\) 36.9309 1.32831 0.664156 0.747594i \(-0.268791\pi\)
0.664156 + 0.747594i \(0.268791\pi\)
\(774\) 1.26137 0.0453389
\(775\) 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
\(783\) −37.1771 −1.32860
\(784\) 0 0
\(785\) 0 0
\(786\) 3.50758 0.125111
\(787\) −49.1771 −1.75297 −0.876487 0.481426i \(-0.840119\pi\)
−0.876487 + 0.481426i \(0.840119\pi\)
\(788\) −5.12311 −0.182503
\(789\) 20.1080 0.715862
\(790\) 0 0
\(791\) 0 0
\(792\) 5.75379 0.204452
\(793\) −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
\(800\) 0 0
\(801\) −0.630683 −0.0222841
\(802\) 80.8466 2.85479
\(803\) 19.1231 0.674840
\(804\) −72.9848 −2.57398
\(805\) 0 0
\(806\) 0 0
\(807\) 32.3845 1.13999
\(808\) 106.600 3.75019
\(809\) 16.5464 0.581740 0.290870 0.956763i \(-0.406055\pi\)
0.290870 + 0.956763i \(0.406055\pi\)
\(810\) 0 0
\(811\) −19.6155 −0.688794 −0.344397 0.938824i \(-0.611917\pi\)
−0.344397 + 0.938824i \(0.611917\pi\)
\(812\) 0 0
\(813\) 24.9848 0.876257
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −5.26137 −0.184185
\(817\) −6.24621 −0.218527
\(818\) −16.6307 −0.581478
\(819\) 0 0
\(820\) 0 0
\(821\) −21.4233 −0.747678 −0.373839 0.927494i \(-0.621959\pi\)
−0.373839 + 0.927494i \(0.621959\pi\)
\(822\) 68.4924 2.38895
\(823\) 36.4924 1.27205 0.636023 0.771670i \(-0.280578\pi\)
0.636023 + 0.771670i \(0.280578\pi\)
\(824\) 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 \(0.470241\pi\)
\(828\) 8.00000 0.278019
\(829\) −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
\(832\) 0.630683 0.0218650
\(833\) 0 0
\(834\) 60.4924 2.09468
\(835\) 0 0
\(836\) −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
\(843\) 19.4233 0.668974
\(844\) 64.1080 2.20669
\(845\) 0 0
\(846\) 12.4924 0.429498
\(847\) 0 0
\(848\) 39.3693 1.35195
\(849\) −17.6695 −0.606416
\(850\) 0 0
\(851\) 18.7386 0.642352
\(852\) 56.9848 1.95227
\(853\) −7.26137 −0.248624 −0.124312 0.992243i \(-0.539672\pi\)
−0.124312 + 0.992243i \(0.539672\pi\)
\(854\) 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 \(0.586716\pi\)
\(858\) −2.73863 −0.0934954
\(859\) 16.4924 0.562714 0.281357 0.959603i \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −50.7386 −1.72816
\(863\) 25.7538 0.876669 0.438335 0.898812i \(-0.355569\pi\)
0.438335 + 0.898812i \(0.355569\pi\)
\(864\) −36.4924 −1.24150
\(865\) 0 0
\(866\) 21.1231 0.717792
\(867\) −26.2462 −0.891368
\(868\) 0 0
\(869\) 3.80776 0.129170
\(870\) 0 0
\(871\) −4.49242 −0.152220
\(872\) 34.8769 1.18108
\(873\) −3.26137 −0.110381
\(874\) −56.9848 −1.92754
\(875\) 0 0
\(876\) −87.2311 −2.94726
\(877\) 40.2462 1.35902 0.679509 0.733667i \(-0.262193\pi\)
0.679509 + 0.733667i \(0.262193\pi\)
\(878\) −24.0000 −0.809961
\(879\) −4.19224 −0.141401
\(880\) 0 0
\(881\) 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 \(-0.454358\pi\)
0.142896 + 0.989738i \(0.454358\pi\)
\(884\) −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 \(0.388206\pi\)
\(888\) −61.4773 −2.06304
\(889\) 0 0
\(890\) 0 0
\(891\) 10.9309 0.366198
\(892\) −11.1231 −0.372429
\(893\) −61.8617 −2.07012
\(894\) 48.9848 1.63830
\(895\) 0 0
\(896\) 0 0
\(897\) −2.13826 −0.0713944
\(898\) −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
\(905\) 0 0
\(906\) −27.7235 −0.921051
\(907\) 24.1080 0.800491 0.400246 0.916408i \(-0.368925\pi\)
0.400246 + 0.916408i \(0.368925\pi\)
\(908\) 51.6155 1.71292
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) −28.4924 −0.943996 −0.471998 0.881600i \(-0.656467\pi\)
−0.471998 + 0.881600i \(0.656467\pi\)
\(912\) 85.4773 2.83044
\(913\) −6.24621 −0.206719
\(914\) 43.8617 1.45082
\(915\) 0 0
\(916\) −49.6155 −1.63934
\(917\) 0 0
\(918\) 6.24621 0.206156
\(919\) 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
\(922\) 33.6155 1.10707
\(923\) 3.50758 0.115453
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) −3.12311 −0.102576
\(928\) 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
\(931\) 0 0
\(932\) −23.3693 −0.765487
\(933\) −49.3693 −1.61628
\(934\) 57.4773 1.88071
\(935\) 0 0
\(936\) −1.61553 −0.0528052
\(937\) −55.6695 −1.81864 −0.909322 0.416094i \(-0.863399\pi\)
−0.909322 + 0.416094i \(0.863399\pi\)
\(938\) 0 0
\(939\) −34.8229 −1.13640
\(940\) 0 0
\(941\) −43.8617 −1.42985 −0.714926 0.699200i \(-0.753540\pi\)
−0.714926 + 0.699200i \(0.753540\pi\)
\(942\) 80.9848 2.63863
\(943\) 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 \(0.520702\pi\)
\(948\) −17.3693 −0.564129
\(949\) −5.36932 −0.174295
\(950\) 0 0
\(951\) −16.3845 −0.531303
\(952\) 0 0
\(953\) 33.1231 1.07296 0.536481 0.843912i \(-0.319753\pi\)
0.536481 + 0.843912i \(0.319753\pi\)
\(954\) −7.36932 −0.238590
\(955\) 0 0
\(956\) 90.3542 2.92226
\(957\) −16.3002 −0.526910
\(958\) −12.4924 −0.403612
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −6.73863 −0.217262
\(963\) 7.50758 0.241928
\(964\) 19.3693 0.623844
\(965\) 0 0
\(966\) 0 0
\(967\) 35.1231 1.12948 0.564741 0.825268i \(-0.308976\pi\)
0.564741 + 0.825268i \(0.308976\pi\)
\(968\) −56.1771 −1.80560
\(969\) −4.87689 −0.156668
\(970\) 0 0
\(971\) −49.4773 −1.58780 −0.793901 0.608048i \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) 26.2462 0.841848
\(973\) 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 \(0.678401\pi\)
\(978\) 28.4924 0.911087
\(979\) −1.75379 −0.0560513
\(980\) 0 0
\(981\) −2.98485 −0.0952988
\(982\) −105.477 −3.36591
\(983\) 51.4233 1.64015 0.820074 0.572257i \(-0.193932\pi\)
0.820074 + 0.572257i \(0.193932\pi\)
\(984\) −52.4924 −1.67340
\(985\) 0 0
\(986\) −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
\(993\) 18.7386 0.594653
\(994\) 0 0
\(995\) 0 0
\(996\) 28.4924 0.902817
\(997\) −2.68466 −0.0850240 −0.0425120 0.999096i \(-0.513536\pi\)
−0.0425120 + 0.999096i \(0.513536\pi\)
\(998\) 105.477 3.33882
\(999\) 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 1225.2.a.s.1.2 2
5.2 odd 4 1225.2.b.f.99.4 4
5.3 odd 4 1225.2.b.f.99.1 4
5.4 even 2 245.2.a.d.1.1 2
7.6 odd 2 175.2.a.f.1.2 2
15.14 odd 2 2205.2.a.x.1.2 2
20.19 odd 2 3920.2.a.bs.1.2 2
21.20 even 2 1575.2.a.p.1.1 2
28.27 even 2 2800.2.a.bi.1.2 2
35.4 even 6 245.2.e.h.226.2 4
35.9 even 6 245.2.e.h.116.2 4
35.13 even 4 175.2.b.b.99.1 4
35.19 odd 6 245.2.e.i.116.2 4
35.24 odd 6 245.2.e.i.226.2 4
35.27 even 4 175.2.b.b.99.4 4
35.34 odd 2 35.2.a.b.1.1 2
105.62 odd 4 1575.2.d.e.1324.1 4
105.83 odd 4 1575.2.d.e.1324.4 4
105.104 even 2 315.2.a.e.1.2 2
140.27 odd 4 2800.2.g.t.449.2 4
140.83 odd 4 2800.2.g.t.449.3 4
140.139 even 2 560.2.a.i.1.1 2
280.69 odd 2 2240.2.a.bh.1.1 2
280.139 even 2 2240.2.a.bd.1.2 2
385.384 even 2 4235.2.a.m.1.2 2
420.419 odd 2 5040.2.a.bt.1.1 2
455.454 odd 2 5915.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 35.34 odd 2
175.2.a.f.1.2 2 7.6 odd 2
175.2.b.b.99.1 4 35.13 even 4
175.2.b.b.99.4 4 35.27 even 4
245.2.a.d.1.1 2 5.4 even 2
245.2.e.h.116.2 4 35.9 even 6
245.2.e.h.226.2 4 35.4 even 6
245.2.e.i.116.2 4 35.19 odd 6
245.2.e.i.226.2 4 35.24 odd 6
315.2.a.e.1.2 2 105.104 even 2
560.2.a.i.1.1 2 140.139 even 2
1225.2.a.s.1.2 2 1.1 even 1 trivial
1225.2.b.f.99.1 4 5.3 odd 4
1225.2.b.f.99.4 4 5.2 odd 4
1575.2.a.p.1.1 2 21.20 even 2
1575.2.d.e.1324.1 4 105.62 odd 4
1575.2.d.e.1324.4 4 105.83 odd 4
2205.2.a.x.1.2 2 15.14 odd 2
2240.2.a.bd.1.2 2 280.139 even 2
2240.2.a.bh.1.1 2 280.69 odd 2
2800.2.a.bi.1.2 2 28.27 even 2
2800.2.g.t.449.2 4 140.27 odd 4
2800.2.g.t.449.3 4 140.83 odd 4
3920.2.a.bs.1.2 2 20.19 odd 2
4235.2.a.m.1.2 2 385.384 even 2
5040.2.a.bt.1.1 2 420.419 odd 2
5915.2.a.l.1.2 2 455.454 odd 2