Properties

Label 3640.2.a.x.1.4
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.04948\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.04948 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.29934 q^{9} +O(q^{10})\) \(q+3.04948 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.29934 q^{9} +2.68830 q^{11} +1.00000 q^{13} -3.04948 q^{15} +3.20037 q^{17} +1.48793 q^{19} -3.04948 q^{21} -0.688301 q^{23} +1.00000 q^{25} +10.0613 q^{27} +0.361180 q^{29} +2.07362 q^{31} +8.19792 q^{33} +1.00000 q^{35} -9.88503 q^{37} +3.04948 q^{39} -1.24866 q^{41} +8.78726 q^{43} -6.29934 q^{45} -3.12311 q^{47} +1.00000 q^{49} +9.75948 q^{51} +7.12311 q^{53} -2.68830 q^{55} +4.53741 q^{57} -6.52505 q^{59} +8.24621 q^{61} -6.29934 q^{63} -1.00000 q^{65} +1.10141 q^{67} -2.09896 q^{69} +16.1566 q^{71} +7.07482 q^{73} +3.04948 q^{75} -2.68830 q^{77} -12.9221 q^{79} +11.7836 q^{81} -1.17139 q^{83} -3.20037 q^{85} +1.10141 q^{87} -6.91156 q^{89} -1.00000 q^{91} +6.32348 q^{93} -1.48793 q^{95} +9.57214 q^{97} +16.9345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.04948 1.76062 0.880309 0.474400i \(-0.157335\pi\)
0.880309 + 0.474400i \(0.157335\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 6.29934 2.09978
\(10\) 0 0
\(11\) 2.68830 0.810553 0.405277 0.914194i \(-0.367175\pi\)
0.405277 + 0.914194i \(0.367175\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −3.04948 −0.787373
\(16\) 0 0
\(17\) 3.20037 0.776204 0.388102 0.921616i \(-0.373131\pi\)
0.388102 + 0.921616i \(0.373131\pi\)
\(18\) 0 0
\(19\) 1.48793 0.341354 0.170677 0.985327i \(-0.445405\pi\)
0.170677 + 0.985327i \(0.445405\pi\)
\(20\) 0 0
\(21\) −3.04948 −0.665451
\(22\) 0 0
\(23\) −0.688301 −0.143521 −0.0717604 0.997422i \(-0.522862\pi\)
−0.0717604 + 0.997422i \(0.522862\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 10.0613 1.93629
\(28\) 0 0
\(29\) 0.361180 0.0670694 0.0335347 0.999438i \(-0.489324\pi\)
0.0335347 + 0.999438i \(0.489324\pi\)
\(30\) 0 0
\(31\) 2.07362 0.372434 0.186217 0.982509i \(-0.440377\pi\)
0.186217 + 0.982509i \(0.440377\pi\)
\(32\) 0 0
\(33\) 8.19792 1.42708
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −9.88503 −1.62509 −0.812545 0.582899i \(-0.801918\pi\)
−0.812545 + 0.582899i \(0.801918\pi\)
\(38\) 0 0
\(39\) 3.04948 0.488308
\(40\) 0 0
\(41\) −1.24866 −0.195008 −0.0975039 0.995235i \(-0.531086\pi\)
−0.0975039 + 0.995235i \(0.531086\pi\)
\(42\) 0 0
\(43\) 8.78726 1.34005 0.670023 0.742341i \(-0.266284\pi\)
0.670023 + 0.742341i \(0.266284\pi\)
\(44\) 0 0
\(45\) −6.29934 −0.939049
\(46\) 0 0
\(47\) −3.12311 −0.455552 −0.227776 0.973714i \(-0.573145\pi\)
−0.227776 + 0.973714i \(0.573145\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 9.75948 1.36660
\(52\) 0 0
\(53\) 7.12311 0.978434 0.489217 0.872162i \(-0.337283\pi\)
0.489217 + 0.872162i \(0.337283\pi\)
\(54\) 0 0
\(55\) −2.68830 −0.362490
\(56\) 0 0
\(57\) 4.53741 0.600995
\(58\) 0 0
\(59\) −6.52505 −0.849489 −0.424744 0.905313i \(-0.639636\pi\)
−0.424744 + 0.905313i \(0.639636\pi\)
\(60\) 0 0
\(61\) 8.24621 1.05582 0.527910 0.849301i \(-0.322976\pi\)
0.527910 + 0.849301i \(0.322976\pi\)
\(62\) 0 0
\(63\) −6.29934 −0.793642
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 1.10141 0.134559 0.0672794 0.997734i \(-0.478568\pi\)
0.0672794 + 0.997734i \(0.478568\pi\)
\(68\) 0 0
\(69\) −2.09896 −0.252685
\(70\) 0 0
\(71\) 16.1566 1.91743 0.958717 0.284363i \(-0.0917821\pi\)
0.958717 + 0.284363i \(0.0917821\pi\)
\(72\) 0 0
\(73\) 7.07482 0.828045 0.414022 0.910267i \(-0.364123\pi\)
0.414022 + 0.910267i \(0.364123\pi\)
\(74\) 0 0
\(75\) 3.04948 0.352124
\(76\) 0 0
\(77\) −2.68830 −0.306360
\(78\) 0 0
\(79\) −12.9221 −1.45386 −0.726928 0.686714i \(-0.759052\pi\)
−0.726928 + 0.686714i \(0.759052\pi\)
\(80\) 0 0
\(81\) 11.7836 1.30929
\(82\) 0 0
\(83\) −1.17139 −0.128577 −0.0642885 0.997931i \(-0.520478\pi\)
−0.0642885 + 0.997931i \(0.520478\pi\)
\(84\) 0 0
\(85\) −3.20037 −0.347129
\(86\) 0 0
\(87\) 1.10141 0.118084
\(88\) 0 0
\(89\) −6.91156 −0.732624 −0.366312 0.930492i \(-0.619380\pi\)
−0.366312 + 0.930492i \(0.619380\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 6.32348 0.655714
\(94\) 0 0
\(95\) −1.48793 −0.152658
\(96\) 0 0
\(97\) 9.57214 0.971903 0.485952 0.873986i \(-0.338473\pi\)
0.485952 + 0.873986i \(0.338473\pi\)
\(98\) 0 0
\(99\) 16.9345 1.70198
\(100\) 0 0
\(101\) −8.74592 −0.870252 −0.435126 0.900370i \(-0.643296\pi\)
−0.435126 + 0.900370i \(0.643296\pi\)
\(102\) 0 0
\(103\) −0.562747 −0.0554492 −0.0277246 0.999616i \(-0.508826\pi\)
−0.0277246 + 0.999616i \(0.508826\pi\)
\(104\) 0 0
\(105\) 3.04948 0.297599
\(106\) 0 0
\(107\) −9.28697 −0.897806 −0.448903 0.893581i \(-0.648185\pi\)
−0.448903 + 0.893581i \(0.648185\pi\)
\(108\) 0 0
\(109\) 3.02653 0.289889 0.144945 0.989440i \(-0.453700\pi\)
0.144945 + 0.989440i \(0.453700\pi\)
\(110\) 0 0
\(111\) −30.1442 −2.86116
\(112\) 0 0
\(113\) 17.4756 1.64396 0.821981 0.569514i \(-0.192869\pi\)
0.821981 + 0.569514i \(0.192869\pi\)
\(114\) 0 0
\(115\) 0.688301 0.0641844
\(116\) 0 0
\(117\) 6.29934 0.582374
\(118\) 0 0
\(119\) −3.20037 −0.293378
\(120\) 0 0
\(121\) −3.77304 −0.343003
\(122\) 0 0
\(123\) −3.80776 −0.343335
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.64001 0.589206 0.294603 0.955620i \(-0.404813\pi\)
0.294603 + 0.955620i \(0.404813\pi\)
\(128\) 0 0
\(129\) 26.7966 2.35931
\(130\) 0 0
\(131\) 0.876894 0.0766146 0.0383073 0.999266i \(-0.487803\pi\)
0.0383073 + 0.999266i \(0.487803\pi\)
\(132\) 0 0
\(133\) −1.48793 −0.129020
\(134\) 0 0
\(135\) −10.0613 −0.865936
\(136\) 0 0
\(137\) −1.73778 −0.148469 −0.0742344 0.997241i \(-0.523651\pi\)
−0.0742344 + 0.997241i \(0.523651\pi\)
\(138\) 0 0
\(139\) 12.5504 1.06451 0.532255 0.846584i \(-0.321345\pi\)
0.532255 + 0.846584i \(0.321345\pi\)
\(140\) 0 0
\(141\) −9.52385 −0.802053
\(142\) 0 0
\(143\) 2.68830 0.224807
\(144\) 0 0
\(145\) −0.361180 −0.0299943
\(146\) 0 0
\(147\) 3.04948 0.251517
\(148\) 0 0
\(149\) −7.12311 −0.583548 −0.291774 0.956487i \(-0.594245\pi\)
−0.291774 + 0.956487i \(0.594245\pi\)
\(150\) 0 0
\(151\) −18.1083 −1.47363 −0.736816 0.676093i \(-0.763672\pi\)
−0.736816 + 0.676093i \(0.763672\pi\)
\(152\) 0 0
\(153\) 20.1602 1.62986
\(154\) 0 0
\(155\) −2.07362 −0.166558
\(156\) 0 0
\(157\) 14.5108 1.15809 0.579045 0.815296i \(-0.303426\pi\)
0.579045 + 0.815296i \(0.303426\pi\)
\(158\) 0 0
\(159\) 21.7218 1.72265
\(160\) 0 0
\(161\) 0.688301 0.0542458
\(162\) 0 0
\(163\) −6.76432 −0.529822 −0.264911 0.964273i \(-0.585343\pi\)
−0.264911 + 0.964273i \(0.585343\pi\)
\(164\) 0 0
\(165\) −8.19792 −0.638208
\(166\) 0 0
\(167\) 8.94874 0.692474 0.346237 0.938147i \(-0.387459\pi\)
0.346237 + 0.938147i \(0.387459\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 9.37296 0.716768
\(172\) 0 0
\(173\) 16.0347 1.21909 0.609547 0.792750i \(-0.291351\pi\)
0.609547 + 0.792750i \(0.291351\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −19.8980 −1.49563
\(178\) 0 0
\(179\) 22.1689 1.65698 0.828492 0.560000i \(-0.189199\pi\)
0.828492 + 0.560000i \(0.189199\pi\)
\(180\) 0 0
\(181\) −0.476148 −0.0353918 −0.0176959 0.999843i \(-0.505633\pi\)
−0.0176959 + 0.999843i \(0.505633\pi\)
\(182\) 0 0
\(183\) 25.1467 1.85890
\(184\) 0 0
\(185\) 9.88503 0.726762
\(186\) 0 0
\(187\) 8.60357 0.629155
\(188\) 0 0
\(189\) −10.0613 −0.731849
\(190\) 0 0
\(191\) 2.86089 0.207007 0.103503 0.994629i \(-0.466995\pi\)
0.103503 + 0.994629i \(0.466995\pi\)
\(192\) 0 0
\(193\) −13.7001 −0.986153 −0.493077 0.869986i \(-0.664128\pi\)
−0.493077 + 0.869986i \(0.664128\pi\)
\(194\) 0 0
\(195\) −3.04948 −0.218378
\(196\) 0 0
\(197\) −0.868174 −0.0618549 −0.0309274 0.999522i \(-0.509846\pi\)
−0.0309274 + 0.999522i \(0.509846\pi\)
\(198\) 0 0
\(199\) −15.2963 −1.08433 −0.542163 0.840273i \(-0.682395\pi\)
−0.542163 + 0.840273i \(0.682395\pi\)
\(200\) 0 0
\(201\) 3.35873 0.236907
\(202\) 0 0
\(203\) −0.361180 −0.0253498
\(204\) 0 0
\(205\) 1.24866 0.0872102
\(206\) 0 0
\(207\) −4.33584 −0.301362
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 2.80202 0.192899 0.0964494 0.995338i \(-0.469251\pi\)
0.0964494 + 0.995338i \(0.469251\pi\)
\(212\) 0 0
\(213\) 49.2692 3.37587
\(214\) 0 0
\(215\) −8.78726 −0.599286
\(216\) 0 0
\(217\) −2.07362 −0.140767
\(218\) 0 0
\(219\) 21.5745 1.45787
\(220\) 0 0
\(221\) 3.20037 0.215280
\(222\) 0 0
\(223\) 12.0718 0.808391 0.404195 0.914673i \(-0.367552\pi\)
0.404195 + 0.914673i \(0.367552\pi\)
\(224\) 0 0
\(225\) 6.29934 0.419956
\(226\) 0 0
\(227\) −22.3687 −1.48467 −0.742333 0.670032i \(-0.766281\pi\)
−0.742333 + 0.670032i \(0.766281\pi\)
\(228\) 0 0
\(229\) −15.5220 −1.02572 −0.512861 0.858472i \(-0.671414\pi\)
−0.512861 + 0.858472i \(0.671414\pi\)
\(230\) 0 0
\(231\) −8.19792 −0.539384
\(232\) 0 0
\(233\) −7.17139 −0.469814 −0.234907 0.972018i \(-0.575478\pi\)
−0.234907 + 0.972018i \(0.575478\pi\)
\(234\) 0 0
\(235\) 3.12311 0.203729
\(236\) 0 0
\(237\) −39.4059 −2.55969
\(238\) 0 0
\(239\) 16.8862 1.09228 0.546140 0.837694i \(-0.316097\pi\)
0.546140 + 0.837694i \(0.316097\pi\)
\(240\) 0 0
\(241\) 11.5962 0.746978 0.373489 0.927635i \(-0.378161\pi\)
0.373489 + 0.927635i \(0.378161\pi\)
\(242\) 0 0
\(243\) 5.75015 0.368872
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 1.48793 0.0946746
\(248\) 0 0
\(249\) −3.57214 −0.226375
\(250\) 0 0
\(251\) −18.2003 −1.14879 −0.574397 0.818577i \(-0.694763\pi\)
−0.574397 + 0.818577i \(0.694763\pi\)
\(252\) 0 0
\(253\) −1.85036 −0.116331
\(254\) 0 0
\(255\) −9.75948 −0.611162
\(256\) 0 0
\(257\) −11.6155 −0.724557 −0.362278 0.932070i \(-0.618001\pi\)
−0.362278 + 0.932070i \(0.618001\pi\)
\(258\) 0 0
\(259\) 9.88503 0.614226
\(260\) 0 0
\(261\) 2.27519 0.140831
\(262\) 0 0
\(263\) −21.3105 −1.31406 −0.657032 0.753863i \(-0.728188\pi\)
−0.657032 + 0.753863i \(0.728188\pi\)
\(264\) 0 0
\(265\) −7.12311 −0.437569
\(266\) 0 0
\(267\) −21.0767 −1.28987
\(268\) 0 0
\(269\) 1.50029 0.0914744 0.0457372 0.998954i \(-0.485436\pi\)
0.0457372 + 0.998954i \(0.485436\pi\)
\(270\) 0 0
\(271\) −0.510206 −0.0309928 −0.0154964 0.999880i \(-0.504933\pi\)
−0.0154964 + 0.999880i \(0.504933\pi\)
\(272\) 0 0
\(273\) −3.04948 −0.184563
\(274\) 0 0
\(275\) 2.68830 0.162111
\(276\) 0 0
\(277\) −22.3476 −1.34273 −0.671367 0.741125i \(-0.734293\pi\)
−0.671367 + 0.741125i \(0.734293\pi\)
\(278\) 0 0
\(279\) 13.0625 0.782029
\(280\) 0 0
\(281\) −24.5021 −1.46167 −0.730836 0.682553i \(-0.760870\pi\)
−0.730836 + 0.682553i \(0.760870\pi\)
\(282\) 0 0
\(283\) −8.88809 −0.528342 −0.264171 0.964476i \(-0.585098\pi\)
−0.264171 + 0.964476i \(0.585098\pi\)
\(284\) 0 0
\(285\) −4.53741 −0.268773
\(286\) 0 0
\(287\) 1.24866 0.0737060
\(288\) 0 0
\(289\) −6.75761 −0.397507
\(290\) 0 0
\(291\) 29.1901 1.71115
\(292\) 0 0
\(293\) 18.7768 1.09695 0.548475 0.836167i \(-0.315209\pi\)
0.548475 + 0.836167i \(0.315209\pi\)
\(294\) 0 0
\(295\) 6.52505 0.379903
\(296\) 0 0
\(297\) 27.0477 1.56947
\(298\) 0 0
\(299\) −0.688301 −0.0398055
\(300\) 0 0
\(301\) −8.78726 −0.506489
\(302\) 0 0
\(303\) −26.6705 −1.53218
\(304\) 0 0
\(305\) −8.24621 −0.472177
\(306\) 0 0
\(307\) −12.6976 −0.724692 −0.362346 0.932044i \(-0.618024\pi\)
−0.362346 + 0.932044i \(0.618024\pi\)
\(308\) 0 0
\(309\) −1.71609 −0.0976248
\(310\) 0 0
\(311\) −25.1394 −1.42552 −0.712762 0.701406i \(-0.752556\pi\)
−0.712762 + 0.701406i \(0.752556\pi\)
\(312\) 0 0
\(313\) −11.5151 −0.650874 −0.325437 0.945564i \(-0.605511\pi\)
−0.325437 + 0.945564i \(0.605511\pi\)
\(314\) 0 0
\(315\) 6.29934 0.354927
\(316\) 0 0
\(317\) −24.8546 −1.39597 −0.697986 0.716112i \(-0.745920\pi\)
−0.697986 + 0.716112i \(0.745920\pi\)
\(318\) 0 0
\(319\) 0.970960 0.0543633
\(320\) 0 0
\(321\) −28.3204 −1.58069
\(322\) 0 0
\(323\) 4.76193 0.264961
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 9.22935 0.510384
\(328\) 0 0
\(329\) 3.12311 0.172182
\(330\) 0 0
\(331\) −0.934513 −0.0513655 −0.0256827 0.999670i \(-0.508176\pi\)
−0.0256827 + 0.999670i \(0.508176\pi\)
\(332\) 0 0
\(333\) −62.2691 −3.41233
\(334\) 0 0
\(335\) −1.10141 −0.0601765
\(336\) 0 0
\(337\) −3.15022 −0.171603 −0.0858017 0.996312i \(-0.527345\pi\)
−0.0858017 + 0.996312i \(0.527345\pi\)
\(338\) 0 0
\(339\) 53.2914 2.89439
\(340\) 0 0
\(341\) 5.57453 0.301878
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.09896 0.113004
\(346\) 0 0
\(347\) 4.63273 0.248698 0.124349 0.992239i \(-0.460316\pi\)
0.124349 + 0.992239i \(0.460316\pi\)
\(348\) 0 0
\(349\) −31.8367 −1.70418 −0.852091 0.523394i \(-0.824666\pi\)
−0.852091 + 0.523394i \(0.824666\pi\)
\(350\) 0 0
\(351\) 10.0613 0.537030
\(352\) 0 0
\(353\) 22.1979 1.18148 0.590738 0.806863i \(-0.298837\pi\)
0.590738 + 0.806863i \(0.298837\pi\)
\(354\) 0 0
\(355\) −16.1566 −0.857502
\(356\) 0 0
\(357\) −9.75948 −0.516526
\(358\) 0 0
\(359\) 24.8356 1.31077 0.655385 0.755295i \(-0.272506\pi\)
0.655385 + 0.755295i \(0.272506\pi\)
\(360\) 0 0
\(361\) −16.7861 −0.883477
\(362\) 0 0
\(363\) −11.5058 −0.603898
\(364\) 0 0
\(365\) −7.07482 −0.370313
\(366\) 0 0
\(367\) 20.4560 1.06779 0.533897 0.845550i \(-0.320727\pi\)
0.533897 + 0.845550i \(0.320727\pi\)
\(368\) 0 0
\(369\) −7.86573 −0.409473
\(370\) 0 0
\(371\) −7.12311 −0.369813
\(372\) 0 0
\(373\) −11.8455 −0.613335 −0.306667 0.951817i \(-0.599214\pi\)
−0.306667 + 0.951817i \(0.599214\pi\)
\(374\) 0 0
\(375\) −3.04948 −0.157475
\(376\) 0 0
\(377\) 0.361180 0.0186017
\(378\) 0 0
\(379\) 6.76370 0.347428 0.173714 0.984796i \(-0.444423\pi\)
0.173714 + 0.984796i \(0.444423\pi\)
\(380\) 0 0
\(381\) 20.2486 1.03737
\(382\) 0 0
\(383\) −13.3234 −0.680795 −0.340397 0.940282i \(-0.610562\pi\)
−0.340397 + 0.940282i \(0.610562\pi\)
\(384\) 0 0
\(385\) 2.68830 0.137009
\(386\) 0 0
\(387\) 55.3539 2.81380
\(388\) 0 0
\(389\) 26.3223 1.33459 0.667297 0.744792i \(-0.267451\pi\)
0.667297 + 0.744792i \(0.267451\pi\)
\(390\) 0 0
\(391\) −2.20282 −0.111401
\(392\) 0 0
\(393\) 2.67407 0.134889
\(394\) 0 0
\(395\) 12.9221 0.650184
\(396\) 0 0
\(397\) −23.6179 −1.18535 −0.592675 0.805442i \(-0.701928\pi\)
−0.592675 + 0.805442i \(0.701928\pi\)
\(398\) 0 0
\(399\) −4.53741 −0.227155
\(400\) 0 0
\(401\) −2.79776 −0.139714 −0.0698568 0.997557i \(-0.522254\pi\)
−0.0698568 + 0.997557i \(0.522254\pi\)
\(402\) 0 0
\(403\) 2.07362 0.103295
\(404\) 0 0
\(405\) −11.7836 −0.585533
\(406\) 0 0
\(407\) −26.5739 −1.31722
\(408\) 0 0
\(409\) −20.2239 −1.00001 −0.500003 0.866024i \(-0.666668\pi\)
−0.500003 + 0.866024i \(0.666668\pi\)
\(410\) 0 0
\(411\) −5.29934 −0.261397
\(412\) 0 0
\(413\) 6.52505 0.321077
\(414\) 0 0
\(415\) 1.17139 0.0575014
\(416\) 0 0
\(417\) 38.2722 1.87420
\(418\) 0 0
\(419\) 18.6952 0.913322 0.456661 0.889641i \(-0.349045\pi\)
0.456661 + 0.889641i \(0.349045\pi\)
\(420\) 0 0
\(421\) −39.8135 −1.94039 −0.970194 0.242328i \(-0.922089\pi\)
−0.970194 + 0.242328i \(0.922089\pi\)
\(422\) 0 0
\(423\) −19.6735 −0.956558
\(424\) 0 0
\(425\) 3.20037 0.155241
\(426\) 0 0
\(427\) −8.24621 −0.399062
\(428\) 0 0
\(429\) 8.19792 0.395800
\(430\) 0 0
\(431\) 1.34254 0.0646681 0.0323341 0.999477i \(-0.489706\pi\)
0.0323341 + 0.999477i \(0.489706\pi\)
\(432\) 0 0
\(433\) −37.0447 −1.78025 −0.890127 0.455713i \(-0.849384\pi\)
−0.890127 + 0.455713i \(0.849384\pi\)
\(434\) 0 0
\(435\) −1.10141 −0.0528086
\(436\) 0 0
\(437\) −1.02414 −0.0489914
\(438\) 0 0
\(439\) 11.3283 0.540671 0.270336 0.962766i \(-0.412865\pi\)
0.270336 + 0.962766i \(0.412865\pi\)
\(440\) 0 0
\(441\) 6.29934 0.299968
\(442\) 0 0
\(443\) −22.8789 −1.08701 −0.543506 0.839406i \(-0.682903\pi\)
−0.543506 + 0.839406i \(0.682903\pi\)
\(444\) 0 0
\(445\) 6.91156 0.327640
\(446\) 0 0
\(447\) −21.7218 −1.02740
\(448\) 0 0
\(449\) −25.7701 −1.21616 −0.608082 0.793874i \(-0.708061\pi\)
−0.608082 + 0.793874i \(0.708061\pi\)
\(450\) 0 0
\(451\) −3.35677 −0.158064
\(452\) 0 0
\(453\) −55.2209 −2.59450
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) 1.65865 0.0775884 0.0387942 0.999247i \(-0.487648\pi\)
0.0387942 + 0.999247i \(0.487648\pi\)
\(458\) 0 0
\(459\) 32.1998 1.50296
\(460\) 0 0
\(461\) −15.3123 −0.713165 −0.356583 0.934264i \(-0.616058\pi\)
−0.356583 + 0.934264i \(0.616058\pi\)
\(462\) 0 0
\(463\) 7.82616 0.363712 0.181856 0.983325i \(-0.441789\pi\)
0.181856 + 0.983325i \(0.441789\pi\)
\(464\) 0 0
\(465\) −6.32348 −0.293244
\(466\) 0 0
\(467\) −14.7100 −0.680697 −0.340349 0.940299i \(-0.610545\pi\)
−0.340349 + 0.940299i \(0.610545\pi\)
\(468\) 0 0
\(469\) −1.10141 −0.0508584
\(470\) 0 0
\(471\) 44.2505 2.03895
\(472\) 0 0
\(473\) 23.6228 1.08618
\(474\) 0 0
\(475\) 1.48793 0.0682708
\(476\) 0 0
\(477\) 44.8708 2.05449
\(478\) 0 0
\(479\) −0.776154 −0.0354634 −0.0177317 0.999843i \(-0.505644\pi\)
−0.0177317 + 0.999843i \(0.505644\pi\)
\(480\) 0 0
\(481\) −9.88503 −0.450719
\(482\) 0 0
\(483\) 2.09896 0.0955061
\(484\) 0 0
\(485\) −9.57214 −0.434648
\(486\) 0 0
\(487\) −3.23807 −0.146731 −0.0733656 0.997305i \(-0.523374\pi\)
−0.0733656 + 0.997305i \(0.523374\pi\)
\(488\) 0 0
\(489\) −20.6277 −0.932815
\(490\) 0 0
\(491\) 22.4441 1.01289 0.506445 0.862272i \(-0.330959\pi\)
0.506445 + 0.862272i \(0.330959\pi\)
\(492\) 0 0
\(493\) 1.15591 0.0520596
\(494\) 0 0
\(495\) −16.9345 −0.761150
\(496\) 0 0
\(497\) −16.1566 −0.724722
\(498\) 0 0
\(499\) 31.2821 1.40038 0.700189 0.713958i \(-0.253099\pi\)
0.700189 + 0.713958i \(0.253099\pi\)
\(500\) 0 0
\(501\) 27.2890 1.21918
\(502\) 0 0
\(503\) 1.88681 0.0841287 0.0420643 0.999115i \(-0.486607\pi\)
0.0420643 + 0.999115i \(0.486607\pi\)
\(504\) 0 0
\(505\) 8.74592 0.389188
\(506\) 0 0
\(507\) 3.04948 0.135432
\(508\) 0 0
\(509\) −12.4248 −0.550721 −0.275360 0.961341i \(-0.588797\pi\)
−0.275360 + 0.961341i \(0.588797\pi\)
\(510\) 0 0
\(511\) −7.07482 −0.312972
\(512\) 0 0
\(513\) 14.9704 0.660961
\(514\) 0 0
\(515\) 0.562747 0.0247976
\(516\) 0 0
\(517\) −8.39585 −0.369249
\(518\) 0 0
\(519\) 48.8974 2.14636
\(520\) 0 0
\(521\) 18.9735 0.831243 0.415621 0.909538i \(-0.363564\pi\)
0.415621 + 0.909538i \(0.363564\pi\)
\(522\) 0 0
\(523\) 4.76370 0.208302 0.104151 0.994561i \(-0.466787\pi\)
0.104151 + 0.994561i \(0.466787\pi\)
\(524\) 0 0
\(525\) −3.04948 −0.133090
\(526\) 0 0
\(527\) 6.63637 0.289085
\(528\) 0 0
\(529\) −22.5262 −0.979402
\(530\) 0 0
\(531\) −41.1035 −1.78374
\(532\) 0 0
\(533\) −1.24866 −0.0540855
\(534\) 0 0
\(535\) 9.28697 0.401511
\(536\) 0 0
\(537\) 67.6038 2.91732
\(538\) 0 0
\(539\) 2.68830 0.115793
\(540\) 0 0
\(541\) −17.8726 −0.768402 −0.384201 0.923249i \(-0.625523\pi\)
−0.384201 + 0.923249i \(0.625523\pi\)
\(542\) 0 0
\(543\) −1.45200 −0.0623115
\(544\) 0 0
\(545\) −3.02653 −0.129642
\(546\) 0 0
\(547\) −15.8597 −0.678112 −0.339056 0.940766i \(-0.610108\pi\)
−0.339056 + 0.940766i \(0.610108\pi\)
\(548\) 0 0
\(549\) 51.9457 2.21699
\(550\) 0 0
\(551\) 0.537410 0.0228944
\(552\) 0 0
\(553\) 12.9221 0.549506
\(554\) 0 0
\(555\) 30.1442 1.27955
\(556\) 0 0
\(557\) 33.5788 1.42278 0.711389 0.702798i \(-0.248066\pi\)
0.711389 + 0.702798i \(0.248066\pi\)
\(558\) 0 0
\(559\) 8.78726 0.371662
\(560\) 0 0
\(561\) 26.2364 1.10770
\(562\) 0 0
\(563\) 43.5586 1.83578 0.917888 0.396839i \(-0.129893\pi\)
0.917888 + 0.396839i \(0.129893\pi\)
\(564\) 0 0
\(565\) −17.4756 −0.735203
\(566\) 0 0
\(567\) −11.7836 −0.494866
\(568\) 0 0
\(569\) 0.213931 0.00896845 0.00448422 0.999990i \(-0.498573\pi\)
0.00448422 + 0.999990i \(0.498573\pi\)
\(570\) 0 0
\(571\) 16.0076 0.669895 0.334948 0.942237i \(-0.391281\pi\)
0.334948 + 0.942237i \(0.391281\pi\)
\(572\) 0 0
\(573\) 8.72422 0.364460
\(574\) 0 0
\(575\) −0.688301 −0.0287042
\(576\) 0 0
\(577\) −14.4170 −0.600188 −0.300094 0.953910i \(-0.597018\pi\)
−0.300094 + 0.953910i \(0.597018\pi\)
\(578\) 0 0
\(579\) −41.7781 −1.73624
\(580\) 0 0
\(581\) 1.17139 0.0485975
\(582\) 0 0
\(583\) 19.1491 0.793073
\(584\) 0 0
\(585\) −6.29934 −0.260445
\(586\) 0 0
\(587\) −11.2988 −0.466352 −0.233176 0.972435i \(-0.574912\pi\)
−0.233176 + 0.972435i \(0.574912\pi\)
\(588\) 0 0
\(589\) 3.08540 0.127132
\(590\) 0 0
\(591\) −2.64748 −0.108903
\(592\) 0 0
\(593\) −24.2722 −0.996738 −0.498369 0.866965i \(-0.666068\pi\)
−0.498369 + 0.866965i \(0.666068\pi\)
\(594\) 0 0
\(595\) 3.20037 0.131203
\(596\) 0 0
\(597\) −46.6458 −1.90909
\(598\) 0 0
\(599\) −17.3953 −0.710751 −0.355376 0.934724i \(-0.615647\pi\)
−0.355376 + 0.934724i \(0.615647\pi\)
\(600\) 0 0
\(601\) −11.3259 −0.461994 −0.230997 0.972954i \(-0.574199\pi\)
−0.230997 + 0.972954i \(0.574199\pi\)
\(602\) 0 0
\(603\) 6.93816 0.282544
\(604\) 0 0
\(605\) 3.77304 0.153396
\(606\) 0 0
\(607\) 23.2884 0.945247 0.472624 0.881264i \(-0.343307\pi\)
0.472624 + 0.881264i \(0.343307\pi\)
\(608\) 0 0
\(609\) −1.10141 −0.0446314
\(610\) 0 0
\(611\) −3.12311 −0.126347
\(612\) 0 0
\(613\) −22.8015 −0.920944 −0.460472 0.887674i \(-0.652320\pi\)
−0.460472 + 0.887674i \(0.652320\pi\)
\(614\) 0 0
\(615\) 3.80776 0.153544
\(616\) 0 0
\(617\) −5.03855 −0.202844 −0.101422 0.994843i \(-0.532339\pi\)
−0.101422 + 0.994843i \(0.532339\pi\)
\(618\) 0 0
\(619\) 34.1701 1.37341 0.686706 0.726936i \(-0.259056\pi\)
0.686706 + 0.726936i \(0.259056\pi\)
\(620\) 0 0
\(621\) −6.92518 −0.277898
\(622\) 0 0
\(623\) 6.91156 0.276906
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 12.1979 0.487138
\(628\) 0 0
\(629\) −31.6358 −1.26140
\(630\) 0 0
\(631\) 18.6032 0.740583 0.370291 0.928916i \(-0.379258\pi\)
0.370291 + 0.928916i \(0.379258\pi\)
\(632\) 0 0
\(633\) 8.54470 0.339621
\(634\) 0 0
\(635\) −6.64001 −0.263501
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 101.776 4.02619
\(640\) 0 0
\(641\) 4.14581 0.163750 0.0818749 0.996643i \(-0.473909\pi\)
0.0818749 + 0.996643i \(0.473909\pi\)
\(642\) 0 0
\(643\) 39.0737 1.54091 0.770457 0.637492i \(-0.220028\pi\)
0.770457 + 0.637492i \(0.220028\pi\)
\(644\) 0 0
\(645\) −26.7966 −1.05511
\(646\) 0 0
\(647\) 30.3222 1.19209 0.596045 0.802951i \(-0.296738\pi\)
0.596045 + 0.802951i \(0.296738\pi\)
\(648\) 0 0
\(649\) −17.5413 −0.688556
\(650\) 0 0
\(651\) −6.32348 −0.247837
\(652\) 0 0
\(653\) 19.5721 0.765917 0.382958 0.923766i \(-0.374905\pi\)
0.382958 + 0.923766i \(0.374905\pi\)
\(654\) 0 0
\(655\) −0.876894 −0.0342631
\(656\) 0 0
\(657\) 44.5667 1.73871
\(658\) 0 0
\(659\) 38.1442 1.48589 0.742944 0.669354i \(-0.233429\pi\)
0.742944 + 0.669354i \(0.233429\pi\)
\(660\) 0 0
\(661\) −49.3711 −1.92031 −0.960156 0.279465i \(-0.909843\pi\)
−0.960156 + 0.279465i \(0.909843\pi\)
\(662\) 0 0
\(663\) 9.75948 0.379027
\(664\) 0 0
\(665\) 1.48793 0.0576994
\(666\) 0 0
\(667\) −0.248601 −0.00962585
\(668\) 0 0
\(669\) 36.8129 1.42327
\(670\) 0 0
\(671\) 22.1683 0.855798
\(672\) 0 0
\(673\) 2.66440 0.102705 0.0513525 0.998681i \(-0.483647\pi\)
0.0513525 + 0.998681i \(0.483647\pi\)
\(674\) 0 0
\(675\) 10.0613 0.387258
\(676\) 0 0
\(677\) 43.7863 1.68285 0.841423 0.540377i \(-0.181718\pi\)
0.841423 + 0.540377i \(0.181718\pi\)
\(678\) 0 0
\(679\) −9.57214 −0.367345
\(680\) 0 0
\(681\) −68.2130 −2.61393
\(682\) 0 0
\(683\) −1.42063 −0.0543591 −0.0271795 0.999631i \(-0.508653\pi\)
−0.0271795 + 0.999631i \(0.508653\pi\)
\(684\) 0 0
\(685\) 1.73778 0.0663973
\(686\) 0 0
\(687\) −47.3340 −1.80590
\(688\) 0 0
\(689\) 7.12311 0.271369
\(690\) 0 0
\(691\) 0.641449 0.0244019 0.0122009 0.999926i \(-0.496116\pi\)
0.0122009 + 0.999926i \(0.496116\pi\)
\(692\) 0 0
\(693\) −16.9345 −0.643289
\(694\) 0 0
\(695\) −12.5504 −0.476063
\(696\) 0 0
\(697\) −3.99618 −0.151366
\(698\) 0 0
\(699\) −21.8690 −0.827163
\(700\) 0 0
\(701\) −16.2221 −0.612701 −0.306351 0.951919i \(-0.599108\pi\)
−0.306351 + 0.951919i \(0.599108\pi\)
\(702\) 0 0
\(703\) −14.7082 −0.554731
\(704\) 0 0
\(705\) 9.52385 0.358689
\(706\) 0 0
\(707\) 8.74592 0.328924
\(708\) 0 0
\(709\) 5.99883 0.225291 0.112645 0.993635i \(-0.464068\pi\)
0.112645 + 0.993635i \(0.464068\pi\)
\(710\) 0 0
\(711\) −81.4010 −3.05277
\(712\) 0 0
\(713\) −1.42728 −0.0534520
\(714\) 0 0
\(715\) −2.68830 −0.100537
\(716\) 0 0
\(717\) 51.4942 1.92309
\(718\) 0 0
\(719\) −18.5987 −0.693613 −0.346807 0.937937i \(-0.612734\pi\)
−0.346807 + 0.937937i \(0.612734\pi\)
\(720\) 0 0
\(721\) 0.562747 0.0209578
\(722\) 0 0
\(723\) 35.3625 1.31514
\(724\) 0 0
\(725\) 0.361180 0.0134139
\(726\) 0 0
\(727\) −28.5969 −1.06060 −0.530300 0.847810i \(-0.677921\pi\)
−0.530300 + 0.847810i \(0.677921\pi\)
\(728\) 0 0
\(729\) −17.8159 −0.659848
\(730\) 0 0
\(731\) 28.1225 1.04015
\(732\) 0 0
\(733\) 31.5932 1.16692 0.583461 0.812141i \(-0.301698\pi\)
0.583461 + 0.812141i \(0.301698\pi\)
\(734\) 0 0
\(735\) −3.04948 −0.112482
\(736\) 0 0
\(737\) 2.96092 0.109067
\(738\) 0 0
\(739\) −51.1892 −1.88303 −0.941513 0.336976i \(-0.890596\pi\)
−0.941513 + 0.336976i \(0.890596\pi\)
\(740\) 0 0
\(741\) 4.53741 0.166686
\(742\) 0 0
\(743\) 8.59919 0.315474 0.157737 0.987481i \(-0.449580\pi\)
0.157737 + 0.987481i \(0.449580\pi\)
\(744\) 0 0
\(745\) 7.12311 0.260970
\(746\) 0 0
\(747\) −7.37899 −0.269983
\(748\) 0 0
\(749\) 9.28697 0.339339
\(750\) 0 0
\(751\) −34.8796 −1.27277 −0.636386 0.771370i \(-0.719572\pi\)
−0.636386 + 0.771370i \(0.719572\pi\)
\(752\) 0 0
\(753\) −55.5015 −2.02259
\(754\) 0 0
\(755\) 18.1083 0.659028
\(756\) 0 0
\(757\) 31.1702 1.13290 0.566451 0.824096i \(-0.308316\pi\)
0.566451 + 0.824096i \(0.308316\pi\)
\(758\) 0 0
\(759\) −5.64264 −0.204815
\(760\) 0 0
\(761\) −1.36549 −0.0494991 −0.0247496 0.999694i \(-0.507879\pi\)
−0.0247496 + 0.999694i \(0.507879\pi\)
\(762\) 0 0
\(763\) −3.02653 −0.109568
\(764\) 0 0
\(765\) −20.1602 −0.728894
\(766\) 0 0
\(767\) −6.52505 −0.235606
\(768\) 0 0
\(769\) −27.2245 −0.981739 −0.490870 0.871233i \(-0.663321\pi\)
−0.490870 + 0.871233i \(0.663321\pi\)
\(770\) 0 0
\(771\) −35.4213 −1.27567
\(772\) 0 0
\(773\) 19.5745 0.704047 0.352023 0.935991i \(-0.385494\pi\)
0.352023 + 0.935991i \(0.385494\pi\)
\(774\) 0 0
\(775\) 2.07362 0.0744868
\(776\) 0 0
\(777\) 30.1442 1.08142
\(778\) 0 0
\(779\) −1.85792 −0.0665668
\(780\) 0 0
\(781\) 43.4338 1.55418
\(782\) 0 0
\(783\) 3.63392 0.129866
\(784\) 0 0
\(785\) −14.5108 −0.517913
\(786\) 0 0
\(787\) −44.3132 −1.57959 −0.789797 0.613369i \(-0.789814\pi\)
−0.789797 + 0.613369i \(0.789814\pi\)
\(788\) 0 0
\(789\) −64.9861 −2.31357
\(790\) 0 0
\(791\) −17.4756 −0.621360
\(792\) 0 0
\(793\) 8.24621 0.292832
\(794\) 0 0
\(795\) −21.7218 −0.770392
\(796\) 0 0
\(797\) −45.7244 −1.61964 −0.809821 0.586677i \(-0.800436\pi\)
−0.809821 + 0.586677i \(0.800436\pi\)
\(798\) 0 0
\(799\) −9.99510 −0.353601
\(800\) 0 0
\(801\) −43.5383 −1.53835
\(802\) 0 0
\(803\) 19.0192 0.671175
\(804\) 0 0
\(805\) −0.688301 −0.0242594
\(806\) 0 0
\(807\) 4.57511 0.161052
\(808\) 0 0
\(809\) 25.0531 0.880821 0.440410 0.897797i \(-0.354833\pi\)
0.440410 + 0.897797i \(0.354833\pi\)
\(810\) 0 0
\(811\) −12.0239 −0.422216 −0.211108 0.977463i \(-0.567707\pi\)
−0.211108 + 0.977463i \(0.567707\pi\)
\(812\) 0 0
\(813\) −1.55586 −0.0545665
\(814\) 0 0
\(815\) 6.76432 0.236944
\(816\) 0 0
\(817\) 13.0748 0.457430
\(818\) 0 0
\(819\) −6.29934 −0.220117
\(820\) 0 0
\(821\) −5.22935 −0.182506 −0.0912529 0.995828i \(-0.529087\pi\)
−0.0912529 + 0.995828i \(0.529087\pi\)
\(822\) 0 0
\(823\) −33.5554 −1.16967 −0.584834 0.811153i \(-0.698840\pi\)
−0.584834 + 0.811153i \(0.698840\pi\)
\(824\) 0 0
\(825\) 8.19792 0.285415
\(826\) 0 0
\(827\) −27.6560 −0.961695 −0.480848 0.876804i \(-0.659671\pi\)
−0.480848 + 0.876804i \(0.659671\pi\)
\(828\) 0 0
\(829\) 52.2576 1.81498 0.907491 0.420073i \(-0.137995\pi\)
0.907491 + 0.420073i \(0.137995\pi\)
\(830\) 0 0
\(831\) −68.1485 −2.36404
\(832\) 0 0
\(833\) 3.20037 0.110886
\(834\) 0 0
\(835\) −8.94874 −0.309684
\(836\) 0 0
\(837\) 20.8633 0.721140
\(838\) 0 0
\(839\) −1.79024 −0.0618058 −0.0309029 0.999522i \(-0.509838\pi\)
−0.0309029 + 0.999522i \(0.509838\pi\)
\(840\) 0 0
\(841\) −28.8695 −0.995502
\(842\) 0 0
\(843\) −74.7187 −2.57345
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 3.77304 0.129643
\(848\) 0 0
\(849\) −27.1041 −0.930209
\(850\) 0 0
\(851\) 6.80388 0.233234
\(852\) 0 0
\(853\) 7.40255 0.253459 0.126729 0.991937i \(-0.459552\pi\)
0.126729 + 0.991937i \(0.459552\pi\)
\(854\) 0 0
\(855\) −9.37296 −0.320548
\(856\) 0 0
\(857\) −8.86328 −0.302764 −0.151382 0.988475i \(-0.548372\pi\)
−0.151382 + 0.988475i \(0.548372\pi\)
\(858\) 0 0
\(859\) −46.6319 −1.59106 −0.795530 0.605914i \(-0.792807\pi\)
−0.795530 + 0.605914i \(0.792807\pi\)
\(860\) 0 0
\(861\) 3.80776 0.129768
\(862\) 0 0
\(863\) −49.6173 −1.68899 −0.844496 0.535562i \(-0.820100\pi\)
−0.844496 + 0.535562i \(0.820100\pi\)
\(864\) 0 0
\(865\) −16.0347 −0.545195
\(866\) 0 0
\(867\) −20.6072 −0.699858
\(868\) 0 0
\(869\) −34.7386 −1.17843
\(870\) 0 0
\(871\) 1.10141 0.0373199
\(872\) 0 0
\(873\) 60.2981 2.04078
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 9.27578 0.313221 0.156610 0.987660i \(-0.449943\pi\)
0.156610 + 0.987660i \(0.449943\pi\)
\(878\) 0 0
\(879\) 57.2594 1.93131
\(880\) 0 0
\(881\) 53.3102 1.79607 0.898033 0.439928i \(-0.144996\pi\)
0.898033 + 0.439928i \(0.144996\pi\)
\(882\) 0 0
\(883\) 38.7070 1.30259 0.651297 0.758823i \(-0.274225\pi\)
0.651297 + 0.758823i \(0.274225\pi\)
\(884\) 0 0
\(885\) 19.8980 0.668864
\(886\) 0 0
\(887\) 33.3560 1.11999 0.559993 0.828497i \(-0.310804\pi\)
0.559993 + 0.828497i \(0.310804\pi\)
\(888\) 0 0
\(889\) −6.64001 −0.222699
\(890\) 0 0
\(891\) 31.6779 1.06125
\(892\) 0 0
\(893\) −4.64696 −0.155505
\(894\) 0 0
\(895\) −22.1689 −0.741026
\(896\) 0 0
\(897\) −2.09896 −0.0700823
\(898\) 0 0
\(899\) 0.748951 0.0249789
\(900\) 0 0
\(901\) 22.7966 0.759465
\(902\) 0 0
\(903\) −26.7966 −0.891735
\(904\) 0 0
\(905\) 0.476148 0.0158277
\(906\) 0 0
\(907\) 50.4311 1.67454 0.837269 0.546791i \(-0.184151\pi\)
0.837269 + 0.546791i \(0.184151\pi\)
\(908\) 0 0
\(909\) −55.0935 −1.82734
\(910\) 0 0
\(911\) 1.42914 0.0473497 0.0236748 0.999720i \(-0.492463\pi\)
0.0236748 + 0.999720i \(0.492463\pi\)
\(912\) 0 0
\(913\) −3.14906 −0.104219
\(914\) 0 0
\(915\) −25.1467 −0.831323
\(916\) 0 0
\(917\) −0.876894 −0.0289576
\(918\) 0 0
\(919\) −6.53438 −0.215549 −0.107775 0.994175i \(-0.534372\pi\)
−0.107775 + 0.994175i \(0.534372\pi\)
\(920\) 0 0
\(921\) −38.7212 −1.27591
\(922\) 0 0
\(923\) 16.1566 0.531800
\(924\) 0 0
\(925\) −9.88503 −0.325018
\(926\) 0 0
\(927\) −3.54493 −0.116431
\(928\) 0 0
\(929\) −10.7089 −0.351346 −0.175673 0.984449i \(-0.556210\pi\)
−0.175673 + 0.984449i \(0.556210\pi\)
\(930\) 0 0
\(931\) 1.48793 0.0487649
\(932\) 0 0
\(933\) −76.6621 −2.50980
\(934\) 0 0
\(935\) −8.60357 −0.281367
\(936\) 0 0
\(937\) 20.1527 0.658359 0.329180 0.944267i \(-0.393228\pi\)
0.329180 + 0.944267i \(0.393228\pi\)
\(938\) 0 0
\(939\) −35.1152 −1.14594
\(940\) 0 0
\(941\) 30.1678 0.983441 0.491721 0.870753i \(-0.336368\pi\)
0.491721 + 0.870753i \(0.336368\pi\)
\(942\) 0 0
\(943\) 0.859454 0.0279877
\(944\) 0 0
\(945\) 10.0613 0.327293
\(946\) 0 0
\(947\) 59.7659 1.94213 0.971065 0.238816i \(-0.0767593\pi\)
0.971065 + 0.238816i \(0.0767593\pi\)
\(948\) 0 0
\(949\) 7.07482 0.229658
\(950\) 0 0
\(951\) −75.7935 −2.45777
\(952\) 0 0
\(953\) 22.2191 0.719747 0.359874 0.933001i \(-0.382820\pi\)
0.359874 + 0.933001i \(0.382820\pi\)
\(954\) 0 0
\(955\) −2.86089 −0.0925762
\(956\) 0 0
\(957\) 2.96092 0.0957131
\(958\) 0 0
\(959\) 1.73778 0.0561159
\(960\) 0 0
\(961\) −26.7001 −0.861293
\(962\) 0 0
\(963\) −58.5018 −1.88519
\(964\) 0 0
\(965\) 13.7001 0.441021
\(966\) 0 0
\(967\) −5.81080 −0.186863 −0.0934313 0.995626i \(-0.529784\pi\)
−0.0934313 + 0.995626i \(0.529784\pi\)
\(968\) 0 0
\(969\) 14.5214 0.466495
\(970\) 0 0
\(971\) −14.5692 −0.467547 −0.233773 0.972291i \(-0.575107\pi\)
−0.233773 + 0.972291i \(0.575107\pi\)
\(972\) 0 0
\(973\) −12.5504 −0.402347
\(974\) 0 0
\(975\) 3.04948 0.0976616
\(976\) 0 0
\(977\) 53.1684 1.70101 0.850503 0.525970i \(-0.176297\pi\)
0.850503 + 0.525970i \(0.176297\pi\)
\(978\) 0 0
\(979\) −18.5804 −0.593831
\(980\) 0 0
\(981\) 19.0651 0.608703
\(982\) 0 0
\(983\) 60.1364 1.91805 0.959027 0.283315i \(-0.0914344\pi\)
0.959027 + 0.283315i \(0.0914344\pi\)
\(984\) 0 0
\(985\) 0.868174 0.0276623
\(986\) 0 0
\(987\) 9.52385 0.303148
\(988\) 0 0
\(989\) −6.04829 −0.192324
\(990\) 0 0
\(991\) 15.2263 0.483680 0.241840 0.970316i \(-0.422249\pi\)
0.241840 + 0.970316i \(0.422249\pi\)
\(992\) 0 0
\(993\) −2.84978 −0.0904350
\(994\) 0 0
\(995\) 15.2963 0.484925
\(996\) 0 0
\(997\) −37.4676 −1.18661 −0.593306 0.804977i \(-0.702178\pi\)
−0.593306 + 0.804977i \(0.702178\pi\)
\(998\) 0 0
\(999\) −99.4559 −3.14664
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 3640.2.a.x.1.4 4
4.3 odd 2 7280.2.a.br.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.x.1.4 4 1.1 even 1 trivial
7280.2.a.br.1.1 4 4.3 odd 2