Properties

Label 2280.2.a.q.1.2
Level $2280$
Weight $2$
Character 2280.1
Self dual yes
Analytic conductor $18.206$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.2058916609\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2280.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} -6.19615 q^{11} -4.73205 q^{13} +1.00000 q^{15} +6.92820 q^{17} +1.00000 q^{19} -1.26795 q^{21} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.196152 q^{29} -2.00000 q^{31} -6.19615 q^{33} -1.26795 q^{35} -0.732051 q^{37} -4.73205 q^{39} -5.66025 q^{41} -1.26795 q^{43} +1.00000 q^{45} -10.0000 q^{47} -5.39230 q^{49} +6.92820 q^{51} -6.19615 q^{55} +1.00000 q^{57} -2.53590 q^{59} -1.46410 q^{61} -1.26795 q^{63} -4.73205 q^{65} -1.07180 q^{67} -6.00000 q^{69} -8.39230 q^{71} -12.9282 q^{73} +1.00000 q^{75} +7.85641 q^{77} -8.00000 q^{79} +1.00000 q^{81} +15.4641 q^{83} +6.92820 q^{85} +0.196152 q^{87} +9.66025 q^{89} +6.00000 q^{91} -2.00000 q^{93} +1.00000 q^{95} -15.6603 q^{97} -6.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{15} + 2 q^{19} - 6 q^{21} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 2 q^{33} - 6 q^{35} + 2 q^{37} - 6 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} - 20 q^{47} + 10 q^{49} - 2 q^{55} + 2 q^{57} - 12 q^{59} + 4 q^{61} - 6 q^{63} - 6 q^{65} - 16 q^{67} - 12 q^{69} + 4 q^{71} - 12 q^{73} + 2 q^{75} - 12 q^{77} - 16 q^{79} + 2 q^{81} + 24 q^{83} - 10 q^{87} + 2 q^{89} + 12 q^{91} - 4 q^{93} + 2 q^{95} - 14 q^{97} - 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.26795 −0.479240 −0.239620 0.970867i \(-0.577023\pi\)
−0.239620 + 0.970867i \(0.577023\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.19615 −1.86821 −0.934105 0.356998i \(-0.883800\pi\)
−0.934105 + 0.356998i \(0.883800\pi\)
\(12\) 0 0
\(13\) −4.73205 −1.31243 −0.656217 0.754572i \(-0.727845\pi\)
−0.656217 + 0.754572i \(0.727845\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 6.92820 1.68034 0.840168 0.542326i \(-0.182456\pi\)
0.840168 + 0.542326i \(0.182456\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.26795 −0.276689
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.196152 0.0364246 0.0182123 0.999834i \(-0.494203\pi\)
0.0182123 + 0.999834i \(0.494203\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −6.19615 −1.07861
\(34\) 0 0
\(35\) −1.26795 −0.214323
\(36\) 0 0
\(37\) −0.732051 −0.120348 −0.0601742 0.998188i \(-0.519166\pi\)
−0.0601742 + 0.998188i \(0.519166\pi\)
\(38\) 0 0
\(39\) −4.73205 −0.757735
\(40\) 0 0
\(41\) −5.66025 −0.883983 −0.441992 0.897019i \(-0.645728\pi\)
−0.441992 + 0.897019i \(0.645728\pi\)
\(42\) 0 0
\(43\) −1.26795 −0.193360 −0.0966802 0.995315i \(-0.530822\pi\)
−0.0966802 + 0.995315i \(0.530822\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) 6.92820 0.970143
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −6.19615 −0.835489
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −2.53590 −0.330146 −0.165073 0.986281i \(-0.552786\pi\)
−0.165073 + 0.986281i \(0.552786\pi\)
\(60\) 0 0
\(61\) −1.46410 −0.187459 −0.0937295 0.995598i \(-0.529879\pi\)
−0.0937295 + 0.995598i \(0.529879\pi\)
\(62\) 0 0
\(63\) −1.26795 −0.159747
\(64\) 0 0
\(65\) −4.73205 −0.586939
\(66\) 0 0
\(67\) −1.07180 −0.130941 −0.0654704 0.997855i \(-0.520855\pi\)
−0.0654704 + 0.997855i \(0.520855\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −8.39230 −0.995983 −0.497992 0.867182i \(-0.665929\pi\)
−0.497992 + 0.867182i \(0.665929\pi\)
\(72\) 0 0
\(73\) −12.9282 −1.51313 −0.756566 0.653917i \(-0.773124\pi\)
−0.756566 + 0.653917i \(0.773124\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 7.85641 0.895321
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.4641 1.69741 0.848703 0.528870i \(-0.177384\pi\)
0.848703 + 0.528870i \(0.177384\pi\)
\(84\) 0 0
\(85\) 6.92820 0.751469
\(86\) 0 0
\(87\) 0.196152 0.0210297
\(88\) 0 0
\(89\) 9.66025 1.02398 0.511992 0.858990i \(-0.328908\pi\)
0.511992 + 0.858990i \(0.328908\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −15.6603 −1.59006 −0.795029 0.606571i \(-0.792544\pi\)
−0.795029 + 0.606571i \(0.792544\pi\)
\(98\) 0 0
\(99\) −6.19615 −0.622737
\(100\) 0 0
\(101\) 3.46410 0.344691 0.172345 0.985037i \(-0.444865\pi\)
0.172345 + 0.985037i \(0.444865\pi\)
\(102\) 0 0
\(103\) 13.8564 1.36531 0.682656 0.730740i \(-0.260825\pi\)
0.682656 + 0.730740i \(0.260825\pi\)
\(104\) 0 0
\(105\) −1.26795 −0.123739
\(106\) 0 0
\(107\) 2.92820 0.283080 0.141540 0.989933i \(-0.454795\pi\)
0.141540 + 0.989933i \(0.454795\pi\)
\(108\) 0 0
\(109\) 10.3923 0.995402 0.497701 0.867349i \(-0.334178\pi\)
0.497701 + 0.867349i \(0.334178\pi\)
\(110\) 0 0
\(111\) −0.732051 −0.0694832
\(112\) 0 0
\(113\) 9.46410 0.890308 0.445154 0.895454i \(-0.353149\pi\)
0.445154 + 0.895454i \(0.353149\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −4.73205 −0.437478
\(118\) 0 0
\(119\) −8.78461 −0.805284
\(120\) 0 0
\(121\) 27.3923 2.49021
\(122\) 0 0
\(123\) −5.66025 −0.510368
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.9282 −1.32466 −0.662332 0.749211i \(-0.730433\pi\)
−0.662332 + 0.749211i \(0.730433\pi\)
\(128\) 0 0
\(129\) −1.26795 −0.111637
\(130\) 0 0
\(131\) 10.5885 0.925118 0.462559 0.886589i \(-0.346931\pi\)
0.462559 + 0.886589i \(0.346931\pi\)
\(132\) 0 0
\(133\) −1.26795 −0.109945
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −18.7846 −1.60488 −0.802439 0.596734i \(-0.796465\pi\)
−0.802439 + 0.596734i \(0.796465\pi\)
\(138\) 0 0
\(139\) −10.5359 −0.893643 −0.446822 0.894623i \(-0.647444\pi\)
−0.446822 + 0.894623i \(0.647444\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 0 0
\(143\) 29.3205 2.45190
\(144\) 0 0
\(145\) 0.196152 0.0162896
\(146\) 0 0
\(147\) −5.39230 −0.444750
\(148\) 0 0
\(149\) −4.92820 −0.403734 −0.201867 0.979413i \(-0.564701\pi\)
−0.201867 + 0.979413i \(0.564701\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) 6.92820 0.560112
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 3.46410 0.276465 0.138233 0.990400i \(-0.455858\pi\)
0.138233 + 0.990400i \(0.455858\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.60770 0.599570
\(162\) 0 0
\(163\) 0.875644 0.0685858 0.0342929 0.999412i \(-0.489082\pi\)
0.0342929 + 0.999412i \(0.489082\pi\)
\(164\) 0 0
\(165\) −6.19615 −0.482370
\(166\) 0 0
\(167\) −6.39230 −0.494651 −0.247326 0.968932i \(-0.579552\pi\)
−0.247326 + 0.968932i \(0.579552\pi\)
\(168\) 0 0
\(169\) 9.39230 0.722485
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −8.39230 −0.638055 −0.319028 0.947745i \(-0.603356\pi\)
−0.319028 + 0.947745i \(0.603356\pi\)
\(174\) 0 0
\(175\) −1.26795 −0.0958479
\(176\) 0 0
\(177\) −2.53590 −0.190610
\(178\) 0 0
\(179\) 14.5359 1.08646 0.543232 0.839583i \(-0.317200\pi\)
0.543232 + 0.839583i \(0.317200\pi\)
\(180\) 0 0
\(181\) 11.4641 0.852120 0.426060 0.904695i \(-0.359901\pi\)
0.426060 + 0.904695i \(0.359901\pi\)
\(182\) 0 0
\(183\) −1.46410 −0.108230
\(184\) 0 0
\(185\) −0.732051 −0.0538214
\(186\) 0 0
\(187\) −42.9282 −3.13922
\(188\) 0 0
\(189\) −1.26795 −0.0922297
\(190\) 0 0
\(191\) −16.7321 −1.21069 −0.605344 0.795964i \(-0.706965\pi\)
−0.605344 + 0.795964i \(0.706965\pi\)
\(192\) 0 0
\(193\) −22.1962 −1.59771 −0.798857 0.601521i \(-0.794562\pi\)
−0.798857 + 0.601521i \(0.794562\pi\)
\(194\) 0 0
\(195\) −4.73205 −0.338869
\(196\) 0 0
\(197\) 14.9282 1.06359 0.531795 0.846873i \(-0.321518\pi\)
0.531795 + 0.846873i \(0.321518\pi\)
\(198\) 0 0
\(199\) −9.46410 −0.670892 −0.335446 0.942059i \(-0.608887\pi\)
−0.335446 + 0.942059i \(0.608887\pi\)
\(200\) 0 0
\(201\) −1.07180 −0.0755987
\(202\) 0 0
\(203\) −0.248711 −0.0174561
\(204\) 0 0
\(205\) −5.66025 −0.395329
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −6.19615 −0.428597
\(210\) 0 0
\(211\) 6.92820 0.476957 0.238479 0.971148i \(-0.423351\pi\)
0.238479 + 0.971148i \(0.423351\pi\)
\(212\) 0 0
\(213\) −8.39230 −0.575031
\(214\) 0 0
\(215\) −1.26795 −0.0864734
\(216\) 0 0
\(217\) 2.53590 0.172148
\(218\) 0 0
\(219\) −12.9282 −0.873607
\(220\) 0 0
\(221\) −32.7846 −2.20533
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 16.5359 1.09753 0.548763 0.835978i \(-0.315099\pi\)
0.548763 + 0.835978i \(0.315099\pi\)
\(228\) 0 0
\(229\) 10.5359 0.696232 0.348116 0.937452i \(-0.386822\pi\)
0.348116 + 0.937452i \(0.386822\pi\)
\(230\) 0 0
\(231\) 7.85641 0.516914
\(232\) 0 0
\(233\) 27.8564 1.82493 0.912467 0.409150i \(-0.134175\pi\)
0.912467 + 0.409150i \(0.134175\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 16.0526 1.03835 0.519177 0.854667i \(-0.326239\pi\)
0.519177 + 0.854667i \(0.326239\pi\)
\(240\) 0 0
\(241\) 4.92820 0.317453 0.158727 0.987323i \(-0.449261\pi\)
0.158727 + 0.987323i \(0.449261\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −5.39230 −0.344502
\(246\) 0 0
\(247\) −4.73205 −0.301093
\(248\) 0 0
\(249\) 15.4641 0.979998
\(250\) 0 0
\(251\) −17.8038 −1.12377 −0.561884 0.827216i \(-0.689923\pi\)
−0.561884 + 0.827216i \(0.689923\pi\)
\(252\) 0 0
\(253\) 37.1769 2.33729
\(254\) 0 0
\(255\) 6.92820 0.433861
\(256\) 0 0
\(257\) −7.32051 −0.456641 −0.228320 0.973586i \(-0.573323\pi\)
−0.228320 + 0.973586i \(0.573323\pi\)
\(258\) 0 0
\(259\) 0.928203 0.0576757
\(260\) 0 0
\(261\) 0.196152 0.0121415
\(262\) 0 0
\(263\) −3.46410 −0.213606 −0.106803 0.994280i \(-0.534061\pi\)
−0.106803 + 0.994280i \(0.534061\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.66025 0.591198
\(268\) 0 0
\(269\) 19.5167 1.18995 0.594976 0.803744i \(-0.297162\pi\)
0.594976 + 0.803744i \(0.297162\pi\)
\(270\) 0 0
\(271\) −11.3205 −0.687672 −0.343836 0.939030i \(-0.611726\pi\)
−0.343836 + 0.939030i \(0.611726\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) −6.19615 −0.373642
\(276\) 0 0
\(277\) −15.8564 −0.952719 −0.476360 0.879251i \(-0.658044\pi\)
−0.476360 + 0.879251i \(0.658044\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 30.0526 1.79279 0.896393 0.443261i \(-0.146178\pi\)
0.896393 + 0.443261i \(0.146178\pi\)
\(282\) 0 0
\(283\) −24.1962 −1.43831 −0.719156 0.694849i \(-0.755471\pi\)
−0.719156 + 0.694849i \(0.755471\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 7.17691 0.423640
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 0 0
\(291\) −15.6603 −0.918020
\(292\) 0 0
\(293\) −3.60770 −0.210764 −0.105382 0.994432i \(-0.533606\pi\)
−0.105382 + 0.994432i \(0.533606\pi\)
\(294\) 0 0
\(295\) −2.53590 −0.147646
\(296\) 0 0
\(297\) −6.19615 −0.359537
\(298\) 0 0
\(299\) 28.3923 1.64197
\(300\) 0 0
\(301\) 1.60770 0.0926660
\(302\) 0 0
\(303\) 3.46410 0.199007
\(304\) 0 0
\(305\) −1.46410 −0.0838342
\(306\) 0 0
\(307\) −27.3205 −1.55926 −0.779632 0.626238i \(-0.784594\pi\)
−0.779632 + 0.626238i \(0.784594\pi\)
\(308\) 0 0
\(309\) 13.8564 0.788263
\(310\) 0 0
\(311\) 24.7321 1.40243 0.701213 0.712952i \(-0.252642\pi\)
0.701213 + 0.712952i \(0.252642\pi\)
\(312\) 0 0
\(313\) 21.3205 1.20511 0.602553 0.798079i \(-0.294150\pi\)
0.602553 + 0.798079i \(0.294150\pi\)
\(314\) 0 0
\(315\) −1.26795 −0.0714408
\(316\) 0 0
\(317\) 14.9282 0.838451 0.419226 0.907882i \(-0.362302\pi\)
0.419226 + 0.907882i \(0.362302\pi\)
\(318\) 0 0
\(319\) −1.21539 −0.0680488
\(320\) 0 0
\(321\) 2.92820 0.163436
\(322\) 0 0
\(323\) 6.92820 0.385496
\(324\) 0 0
\(325\) −4.73205 −0.262487
\(326\) 0 0
\(327\) 10.3923 0.574696
\(328\) 0 0
\(329\) 12.6795 0.699043
\(330\) 0 0
\(331\) −3.07180 −0.168841 −0.0844206 0.996430i \(-0.526904\pi\)
−0.0844206 + 0.996430i \(0.526904\pi\)
\(332\) 0 0
\(333\) −0.732051 −0.0401161
\(334\) 0 0
\(335\) −1.07180 −0.0585585
\(336\) 0 0
\(337\) 0.732051 0.0398773 0.0199387 0.999801i \(-0.493653\pi\)
0.0199387 + 0.999801i \(0.493653\pi\)
\(338\) 0 0
\(339\) 9.46410 0.514019
\(340\) 0 0
\(341\) 12.3923 0.671081
\(342\) 0 0
\(343\) 15.7128 0.848412
\(344\) 0 0
\(345\) −6.00000 −0.323029
\(346\) 0 0
\(347\) 3.46410 0.185963 0.0929814 0.995668i \(-0.470360\pi\)
0.0929814 + 0.995668i \(0.470360\pi\)
\(348\) 0 0
\(349\) −25.7128 −1.37638 −0.688188 0.725533i \(-0.741593\pi\)
−0.688188 + 0.725533i \(0.741593\pi\)
\(350\) 0 0
\(351\) −4.73205 −0.252578
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) −8.39230 −0.445417
\(356\) 0 0
\(357\) −8.78461 −0.464931
\(358\) 0 0
\(359\) −30.5885 −1.61440 −0.807199 0.590280i \(-0.799017\pi\)
−0.807199 + 0.590280i \(0.799017\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 27.3923 1.43772
\(364\) 0 0
\(365\) −12.9282 −0.676693
\(366\) 0 0
\(367\) −9.26795 −0.483783 −0.241892 0.970303i \(-0.577768\pi\)
−0.241892 + 0.970303i \(0.577768\pi\)
\(368\) 0 0
\(369\) −5.66025 −0.294661
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −14.5885 −0.755362 −0.377681 0.925936i \(-0.623278\pi\)
−0.377681 + 0.925936i \(0.623278\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −0.928203 −0.0478049
\(378\) 0 0
\(379\) −3.85641 −0.198090 −0.0990451 0.995083i \(-0.531579\pi\)
−0.0990451 + 0.995083i \(0.531579\pi\)
\(380\) 0 0
\(381\) −14.9282 −0.764795
\(382\) 0 0
\(383\) −34.6410 −1.77007 −0.885037 0.465521i \(-0.845867\pi\)
−0.885037 + 0.465521i \(0.845867\pi\)
\(384\) 0 0
\(385\) 7.85641 0.400400
\(386\) 0 0
\(387\) −1.26795 −0.0644535
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −41.5692 −2.10225
\(392\) 0 0
\(393\) 10.5885 0.534117
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −3.07180 −0.154169 −0.0770845 0.997025i \(-0.524561\pi\)
−0.0770845 + 0.997025i \(0.524561\pi\)
\(398\) 0 0
\(399\) −1.26795 −0.0634769
\(400\) 0 0
\(401\) 21.6603 1.08166 0.540831 0.841131i \(-0.318110\pi\)
0.540831 + 0.841131i \(0.318110\pi\)
\(402\) 0 0
\(403\) 9.46410 0.471440
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 4.53590 0.224836
\(408\) 0 0
\(409\) −7.46410 −0.369076 −0.184538 0.982825i \(-0.559079\pi\)
−0.184538 + 0.982825i \(0.559079\pi\)
\(410\) 0 0
\(411\) −18.7846 −0.926576
\(412\) 0 0
\(413\) 3.21539 0.158219
\(414\) 0 0
\(415\) 15.4641 0.759103
\(416\) 0 0
\(417\) −10.5359 −0.515945
\(418\) 0 0
\(419\) 4.73205 0.231176 0.115588 0.993297i \(-0.463125\pi\)
0.115588 + 0.993297i \(0.463125\pi\)
\(420\) 0 0
\(421\) −28.9282 −1.40987 −0.704937 0.709270i \(-0.749025\pi\)
−0.704937 + 0.709270i \(0.749025\pi\)
\(422\) 0 0
\(423\) −10.0000 −0.486217
\(424\) 0 0
\(425\) 6.92820 0.336067
\(426\) 0 0
\(427\) 1.85641 0.0898378
\(428\) 0 0
\(429\) 29.3205 1.41561
\(430\) 0 0
\(431\) 14.2487 0.686336 0.343168 0.939274i \(-0.388500\pi\)
0.343168 + 0.939274i \(0.388500\pi\)
\(432\) 0 0
\(433\) −28.0526 −1.34812 −0.674060 0.738677i \(-0.735451\pi\)
−0.674060 + 0.738677i \(0.735451\pi\)
\(434\) 0 0
\(435\) 0.196152 0.00940479
\(436\) 0 0
\(437\) −6.00000 −0.287019
\(438\) 0 0
\(439\) −21.8564 −1.04315 −0.521575 0.853206i \(-0.674655\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(440\) 0 0
\(441\) −5.39230 −0.256776
\(442\) 0 0
\(443\) 20.9282 0.994329 0.497164 0.867656i \(-0.334375\pi\)
0.497164 + 0.867656i \(0.334375\pi\)
\(444\) 0 0
\(445\) 9.66025 0.457940
\(446\) 0 0
\(447\) −4.92820 −0.233096
\(448\) 0 0
\(449\) 20.1962 0.953115 0.476558 0.879143i \(-0.341884\pi\)
0.476558 + 0.879143i \(0.341884\pi\)
\(450\) 0 0
\(451\) 35.0718 1.65147
\(452\) 0 0
\(453\) −2.00000 −0.0939682
\(454\) 0 0
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −12.5359 −0.586405 −0.293202 0.956050i \(-0.594721\pi\)
−0.293202 + 0.956050i \(0.594721\pi\)
\(458\) 0 0
\(459\) 6.92820 0.323381
\(460\) 0 0
\(461\) −30.7846 −1.43378 −0.716891 0.697185i \(-0.754436\pi\)
−0.716891 + 0.697185i \(0.754436\pi\)
\(462\) 0 0
\(463\) −7.12436 −0.331097 −0.165548 0.986202i \(-0.552939\pi\)
−0.165548 + 0.986202i \(0.552939\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) 32.9282 1.52374 0.761868 0.647733i \(-0.224283\pi\)
0.761868 + 0.647733i \(0.224283\pi\)
\(468\) 0 0
\(469\) 1.35898 0.0627520
\(470\) 0 0
\(471\) 3.46410 0.159617
\(472\) 0 0
\(473\) 7.85641 0.361238
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 34.5885 1.58039 0.790193 0.612857i \(-0.209980\pi\)
0.790193 + 0.612857i \(0.209980\pi\)
\(480\) 0 0
\(481\) 3.46410 0.157949
\(482\) 0 0
\(483\) 7.60770 0.346162
\(484\) 0 0
\(485\) −15.6603 −0.711096
\(486\) 0 0
\(487\) −0.392305 −0.0177770 −0.00888851 0.999960i \(-0.502829\pi\)
−0.00888851 + 0.999960i \(0.502829\pi\)
\(488\) 0 0
\(489\) 0.875644 0.0395980
\(490\) 0 0
\(491\) 2.19615 0.0991110 0.0495555 0.998771i \(-0.484220\pi\)
0.0495555 + 0.998771i \(0.484220\pi\)
\(492\) 0 0
\(493\) 1.35898 0.0612056
\(494\) 0 0
\(495\) −6.19615 −0.278496
\(496\) 0 0
\(497\) 10.6410 0.477315
\(498\) 0 0
\(499\) 27.3205 1.22303 0.611517 0.791231i \(-0.290560\pi\)
0.611517 + 0.791231i \(0.290560\pi\)
\(500\) 0 0
\(501\) −6.39230 −0.285587
\(502\) 0 0
\(503\) −23.1769 −1.03341 −0.516704 0.856164i \(-0.672841\pi\)
−0.516704 + 0.856164i \(0.672841\pi\)
\(504\) 0 0
\(505\) 3.46410 0.154150
\(506\) 0 0
\(507\) 9.39230 0.417127
\(508\) 0 0
\(509\) −4.19615 −0.185991 −0.0929956 0.995667i \(-0.529644\pi\)
−0.0929956 + 0.995667i \(0.529644\pi\)
\(510\) 0 0
\(511\) 16.3923 0.725153
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 13.8564 0.610586
\(516\) 0 0
\(517\) 61.9615 2.72506
\(518\) 0 0
\(519\) −8.39230 −0.368381
\(520\) 0 0
\(521\) 39.1244 1.71407 0.857035 0.515259i \(-0.172304\pi\)
0.857035 + 0.515259i \(0.172304\pi\)
\(522\) 0 0
\(523\) −13.4641 −0.588744 −0.294372 0.955691i \(-0.595110\pi\)
−0.294372 + 0.955691i \(0.595110\pi\)
\(524\) 0 0
\(525\) −1.26795 −0.0553378
\(526\) 0 0
\(527\) −13.8564 −0.603595
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −2.53590 −0.110049
\(532\) 0 0
\(533\) 26.7846 1.16017
\(534\) 0 0
\(535\) 2.92820 0.126597
\(536\) 0 0
\(537\) 14.5359 0.627270
\(538\) 0 0
\(539\) 33.4115 1.43914
\(540\) 0 0
\(541\) 33.7128 1.44943 0.724714 0.689050i \(-0.241972\pi\)
0.724714 + 0.689050i \(0.241972\pi\)
\(542\) 0 0
\(543\) 11.4641 0.491972
\(544\) 0 0
\(545\) 10.3923 0.445157
\(546\) 0 0
\(547\) 30.2487 1.29334 0.646671 0.762769i \(-0.276161\pi\)
0.646671 + 0.762769i \(0.276161\pi\)
\(548\) 0 0
\(549\) −1.46410 −0.0624863
\(550\) 0 0
\(551\) 0.196152 0.00835637
\(552\) 0 0
\(553\) 10.1436 0.431349
\(554\) 0 0
\(555\) −0.732051 −0.0310738
\(556\) 0 0
\(557\) −7.85641 −0.332887 −0.166443 0.986051i \(-0.553228\pi\)
−0.166443 + 0.986051i \(0.553228\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) −42.9282 −1.81243
\(562\) 0 0
\(563\) 38.3923 1.61804 0.809021 0.587779i \(-0.199998\pi\)
0.809021 + 0.587779i \(0.199998\pi\)
\(564\) 0 0
\(565\) 9.46410 0.398158
\(566\) 0 0
\(567\) −1.26795 −0.0532489
\(568\) 0 0
\(569\) 20.1962 0.846667 0.423333 0.905974i \(-0.360860\pi\)
0.423333 + 0.905974i \(0.360860\pi\)
\(570\) 0 0
\(571\) 28.3923 1.18818 0.594090 0.804398i \(-0.297512\pi\)
0.594090 + 0.804398i \(0.297512\pi\)
\(572\) 0 0
\(573\) −16.7321 −0.698991
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −9.32051 −0.388018 −0.194009 0.981000i \(-0.562149\pi\)
−0.194009 + 0.981000i \(0.562149\pi\)
\(578\) 0 0
\(579\) −22.1962 −0.922441
\(580\) 0 0
\(581\) −19.6077 −0.813464
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.73205 −0.195646
\(586\) 0 0
\(587\) −18.7846 −0.775324 −0.387662 0.921802i \(-0.626717\pi\)
−0.387662 + 0.921802i \(0.626717\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 14.9282 0.614064
\(592\) 0 0
\(593\) −38.7846 −1.59269 −0.796347 0.604841i \(-0.793237\pi\)
−0.796347 + 0.604841i \(0.793237\pi\)
\(594\) 0 0
\(595\) −8.78461 −0.360134
\(596\) 0 0
\(597\) −9.46410 −0.387340
\(598\) 0 0
\(599\) −28.0000 −1.14405 −0.572024 0.820237i \(-0.693842\pi\)
−0.572024 + 0.820237i \(0.693842\pi\)
\(600\) 0 0
\(601\) −5.60770 −0.228743 −0.114371 0.993438i \(-0.536485\pi\)
−0.114371 + 0.993438i \(0.536485\pi\)
\(602\) 0 0
\(603\) −1.07180 −0.0436469
\(604\) 0 0
\(605\) 27.3923 1.11366
\(606\) 0 0
\(607\) −17.4641 −0.708846 −0.354423 0.935085i \(-0.615323\pi\)
−0.354423 + 0.935085i \(0.615323\pi\)
\(608\) 0 0
\(609\) −0.248711 −0.0100783
\(610\) 0 0
\(611\) 47.3205 1.91438
\(612\) 0 0
\(613\) 35.1769 1.42078 0.710391 0.703807i \(-0.248518\pi\)
0.710391 + 0.703807i \(0.248518\pi\)
\(614\) 0 0
\(615\) −5.66025 −0.228243
\(616\) 0 0
\(617\) −36.7846 −1.48089 −0.740446 0.672116i \(-0.765386\pi\)
−0.740446 + 0.672116i \(0.765386\pi\)
\(618\) 0 0
\(619\) 21.1769 0.851172 0.425586 0.904918i \(-0.360068\pi\)
0.425586 + 0.904918i \(0.360068\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −12.2487 −0.490734
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.19615 −0.247450
\(628\) 0 0
\(629\) −5.07180 −0.202226
\(630\) 0 0
\(631\) 35.7128 1.42170 0.710852 0.703341i \(-0.248309\pi\)
0.710852 + 0.703341i \(0.248309\pi\)
\(632\) 0 0
\(633\) 6.92820 0.275371
\(634\) 0 0
\(635\) −14.9282 −0.592408
\(636\) 0 0
\(637\) 25.5167 1.01101
\(638\) 0 0
\(639\) −8.39230 −0.331994
\(640\) 0 0
\(641\) 7.51666 0.296890 0.148445 0.988921i \(-0.452573\pi\)
0.148445 + 0.988921i \(0.452573\pi\)
\(642\) 0 0
\(643\) 20.9808 0.827400 0.413700 0.910413i \(-0.364236\pi\)
0.413700 + 0.910413i \(0.364236\pi\)
\(644\) 0 0
\(645\) −1.26795 −0.0499255
\(646\) 0 0
\(647\) −41.3205 −1.62448 −0.812238 0.583326i \(-0.801751\pi\)
−0.812238 + 0.583326i \(0.801751\pi\)
\(648\) 0 0
\(649\) 15.7128 0.616782
\(650\) 0 0
\(651\) 2.53590 0.0993897
\(652\) 0 0
\(653\) −48.7846 −1.90909 −0.954545 0.298068i \(-0.903658\pi\)
−0.954545 + 0.298068i \(0.903658\pi\)
\(654\) 0 0
\(655\) 10.5885 0.413725
\(656\) 0 0
\(657\) −12.9282 −0.504377
\(658\) 0 0
\(659\) −11.7128 −0.456266 −0.228133 0.973630i \(-0.573262\pi\)
−0.228133 + 0.973630i \(0.573262\pi\)
\(660\) 0 0
\(661\) 13.6077 0.529278 0.264639 0.964348i \(-0.414747\pi\)
0.264639 + 0.964348i \(0.414747\pi\)
\(662\) 0 0
\(663\) −32.7846 −1.27325
\(664\) 0 0
\(665\) −1.26795 −0.0491690
\(666\) 0 0
\(667\) −1.17691 −0.0455703
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 9.07180 0.350213
\(672\) 0 0
\(673\) 5.51666 0.212652 0.106326 0.994331i \(-0.466091\pi\)
0.106326 + 0.994331i \(0.466091\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −40.0000 −1.53732 −0.768662 0.639655i \(-0.779077\pi\)
−0.768662 + 0.639655i \(0.779077\pi\)
\(678\) 0 0
\(679\) 19.8564 0.762019
\(680\) 0 0
\(681\) 16.5359 0.633657
\(682\) 0 0
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) −18.7846 −0.717723
\(686\) 0 0
\(687\) 10.5359 0.401970
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −12.3923 −0.471425 −0.235713 0.971823i \(-0.575742\pi\)
−0.235713 + 0.971823i \(0.575742\pi\)
\(692\) 0 0
\(693\) 7.85641 0.298440
\(694\) 0 0
\(695\) −10.5359 −0.399649
\(696\) 0 0
\(697\) −39.2154 −1.48539
\(698\) 0 0
\(699\) 27.8564 1.05363
\(700\) 0 0
\(701\) 11.0718 0.418176 0.209088 0.977897i \(-0.432950\pi\)
0.209088 + 0.977897i \(0.432950\pi\)
\(702\) 0 0
\(703\) −0.732051 −0.0276098
\(704\) 0 0
\(705\) −10.0000 −0.376622
\(706\) 0 0
\(707\) −4.39230 −0.165190
\(708\) 0 0
\(709\) −10.2487 −0.384898 −0.192449 0.981307i \(-0.561643\pi\)
−0.192449 + 0.981307i \(0.561643\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 29.3205 1.09652
\(716\) 0 0
\(717\) 16.0526 0.599494
\(718\) 0 0
\(719\) 22.9808 0.857038 0.428519 0.903533i \(-0.359036\pi\)
0.428519 + 0.903533i \(0.359036\pi\)
\(720\) 0 0
\(721\) −17.5692 −0.654312
\(722\) 0 0
\(723\) 4.92820 0.183282
\(724\) 0 0
\(725\) 0.196152 0.00728492
\(726\) 0 0
\(727\) −32.5885 −1.20864 −0.604319 0.796742i \(-0.706555\pi\)
−0.604319 + 0.796742i \(0.706555\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.78461 −0.324911
\(732\) 0 0
\(733\) 6.78461 0.250595 0.125298 0.992119i \(-0.460011\pi\)
0.125298 + 0.992119i \(0.460011\pi\)
\(734\) 0 0
\(735\) −5.39230 −0.198898
\(736\) 0 0
\(737\) 6.64102 0.244625
\(738\) 0 0
\(739\) −9.07180 −0.333711 −0.166856 0.985981i \(-0.553361\pi\)
−0.166856 + 0.985981i \(0.553361\pi\)
\(740\) 0 0
\(741\) −4.73205 −0.173836
\(742\) 0 0
\(743\) 42.6410 1.56435 0.782174 0.623061i \(-0.214111\pi\)
0.782174 + 0.623061i \(0.214111\pi\)
\(744\) 0 0
\(745\) −4.92820 −0.180555
\(746\) 0 0
\(747\) 15.4641 0.565802
\(748\) 0 0
\(749\) −3.71281 −0.135663
\(750\) 0 0
\(751\) −35.8564 −1.30842 −0.654209 0.756313i \(-0.726999\pi\)
−0.654209 + 0.756313i \(0.726999\pi\)
\(752\) 0 0
\(753\) −17.8038 −0.648808
\(754\) 0 0
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) −4.53590 −0.164860 −0.0824300 0.996597i \(-0.526268\pi\)
−0.0824300 + 0.996597i \(0.526268\pi\)
\(758\) 0 0
\(759\) 37.1769 1.34944
\(760\) 0 0
\(761\) −40.9282 −1.48365 −0.741823 0.670596i \(-0.766039\pi\)
−0.741823 + 0.670596i \(0.766039\pi\)
\(762\) 0 0
\(763\) −13.1769 −0.477036
\(764\) 0 0
\(765\) 6.92820 0.250490
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 2.78461 0.100416 0.0502078 0.998739i \(-0.484012\pi\)
0.0502078 + 0.998739i \(0.484012\pi\)
\(770\) 0 0
\(771\) −7.32051 −0.263642
\(772\) 0 0
\(773\) −25.8564 −0.929990 −0.464995 0.885313i \(-0.653944\pi\)
−0.464995 + 0.885313i \(0.653944\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 0.928203 0.0332991
\(778\) 0 0
\(779\) −5.66025 −0.202800
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) 0.196152 0.00700992
\(784\) 0 0
\(785\) 3.46410 0.123639
\(786\) 0 0
\(787\) −5.46410 −0.194774 −0.0973871 0.995247i \(-0.531048\pi\)
−0.0973871 + 0.995247i \(0.531048\pi\)
\(788\) 0 0
\(789\) −3.46410 −0.123325
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 6.92820 0.246028
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 46.2487 1.63821 0.819107 0.573641i \(-0.194470\pi\)
0.819107 + 0.573641i \(0.194470\pi\)
\(798\) 0 0
\(799\) −69.2820 −2.45102
\(800\) 0 0
\(801\) 9.66025 0.341328
\(802\) 0 0
\(803\) 80.1051 2.82685
\(804\) 0 0
\(805\) 7.60770 0.268136
\(806\) 0 0
\(807\) 19.5167 0.687019
\(808\) 0 0
\(809\) −54.4974 −1.91603 −0.958014 0.286723i \(-0.907434\pi\)
−0.958014 + 0.286723i \(0.907434\pi\)
\(810\) 0 0
\(811\) 42.6410 1.49733 0.748664 0.662949i \(-0.230696\pi\)
0.748664 + 0.662949i \(0.230696\pi\)
\(812\) 0 0
\(813\) −11.3205 −0.397028
\(814\) 0 0
\(815\) 0.875644 0.0306725
\(816\) 0 0
\(817\) −1.26795 −0.0443599
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 13.3205 0.464889 0.232444 0.972610i \(-0.425328\pi\)
0.232444 + 0.972610i \(0.425328\pi\)
\(822\) 0 0
\(823\) −40.9808 −1.42850 −0.714250 0.699891i \(-0.753232\pi\)
−0.714250 + 0.699891i \(0.753232\pi\)
\(824\) 0 0
\(825\) −6.19615 −0.215722
\(826\) 0 0
\(827\) −50.1051 −1.74233 −0.871163 0.490994i \(-0.836634\pi\)
−0.871163 + 0.490994i \(0.836634\pi\)
\(828\) 0 0
\(829\) 38.3923 1.33342 0.666710 0.745317i \(-0.267702\pi\)
0.666710 + 0.745317i \(0.267702\pi\)
\(830\) 0 0
\(831\) −15.8564 −0.550053
\(832\) 0 0
\(833\) −37.3590 −1.29441
\(834\) 0 0
\(835\) −6.39230 −0.221215
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 18.5359 0.639930 0.319965 0.947429i \(-0.396329\pi\)
0.319965 + 0.947429i \(0.396329\pi\)
\(840\) 0 0
\(841\) −28.9615 −0.998673
\(842\) 0 0
\(843\) 30.0526 1.03507
\(844\) 0 0
\(845\) 9.39230 0.323105
\(846\) 0 0
\(847\) −34.7321 −1.19341
\(848\) 0 0
\(849\) −24.1962 −0.830410
\(850\) 0 0
\(851\) 4.39230 0.150566
\(852\) 0 0
\(853\) −31.4641 −1.07731 −0.538655 0.842526i \(-0.681067\pi\)
−0.538655 + 0.842526i \(0.681067\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −34.9282 −1.19312 −0.596562 0.802567i \(-0.703467\pi\)
−0.596562 + 0.802567i \(0.703467\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 7.17691 0.244589
\(862\) 0 0
\(863\) 33.8564 1.15249 0.576243 0.817279i \(-0.304518\pi\)
0.576243 + 0.817279i \(0.304518\pi\)
\(864\) 0 0
\(865\) −8.39230 −0.285347
\(866\) 0 0
\(867\) 31.0000 1.05282
\(868\) 0 0
\(869\) 49.5692 1.68152
\(870\) 0 0
\(871\) 5.07180 0.171851
\(872\) 0 0
\(873\) −15.6603 −0.530019
\(874\) 0 0
\(875\) −1.26795 −0.0428645
\(876\) 0 0
\(877\) −46.1962 −1.55993 −0.779967 0.625821i \(-0.784764\pi\)
−0.779967 + 0.625821i \(0.784764\pi\)
\(878\) 0 0
\(879\) −3.60770 −0.121685
\(880\) 0 0
\(881\) 44.2487 1.49078 0.745388 0.666630i \(-0.232264\pi\)
0.745388 + 0.666630i \(0.232264\pi\)
\(882\) 0 0
\(883\) 38.7321 1.30344 0.651719 0.758461i \(-0.274048\pi\)
0.651719 + 0.758461i \(0.274048\pi\)
\(884\) 0 0
\(885\) −2.53590 −0.0852433
\(886\) 0 0
\(887\) −18.9282 −0.635547 −0.317773 0.948167i \(-0.602935\pi\)
−0.317773 + 0.948167i \(0.602935\pi\)
\(888\) 0 0
\(889\) 18.9282 0.634832
\(890\) 0 0
\(891\) −6.19615 −0.207579
\(892\) 0 0
\(893\) −10.0000 −0.334637
\(894\) 0 0
\(895\) 14.5359 0.485881
\(896\) 0 0
\(897\) 28.3923 0.947991
\(898\) 0 0
\(899\) −0.392305 −0.0130841
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.60770 0.0535007
\(904\) 0 0
\(905\) 11.4641 0.381080
\(906\) 0 0
\(907\) 43.0333 1.42890 0.714449 0.699688i \(-0.246677\pi\)
0.714449 + 0.699688i \(0.246677\pi\)
\(908\) 0 0
\(909\) 3.46410 0.114897
\(910\) 0 0
\(911\) 30.9282 1.02470 0.512349 0.858778i \(-0.328776\pi\)
0.512349 + 0.858778i \(0.328776\pi\)
\(912\) 0 0
\(913\) −95.8179 −3.17111
\(914\) 0 0
\(915\) −1.46410 −0.0484017
\(916\) 0 0
\(917\) −13.4256 −0.443353
\(918\) 0 0
\(919\) −9.07180 −0.299251 −0.149625 0.988743i \(-0.547807\pi\)
−0.149625 + 0.988743i \(0.547807\pi\)
\(920\) 0 0
\(921\) −27.3205 −0.900241
\(922\) 0 0
\(923\) 39.7128 1.30716
\(924\) 0 0
\(925\) −0.732051 −0.0240697
\(926\) 0 0
\(927\) 13.8564 0.455104
\(928\) 0 0
\(929\) 11.7513 0.385547 0.192774 0.981243i \(-0.438252\pi\)
0.192774 + 0.981243i \(0.438252\pi\)
\(930\) 0 0
\(931\) −5.39230 −0.176726
\(932\) 0 0
\(933\) 24.7321 0.809691
\(934\) 0 0
\(935\) −42.9282 −1.40390
\(936\) 0 0
\(937\) −19.1769 −0.626482 −0.313241 0.949674i \(-0.601415\pi\)
−0.313241 + 0.949674i \(0.601415\pi\)
\(938\) 0 0
\(939\) 21.3205 0.695768
\(940\) 0 0
\(941\) −54.0526 −1.76206 −0.881032 0.473058i \(-0.843150\pi\)
−0.881032 + 0.473058i \(0.843150\pi\)
\(942\) 0 0
\(943\) 33.9615 1.10594
\(944\) 0 0
\(945\) −1.26795 −0.0412464
\(946\) 0 0
\(947\) −53.3205 −1.73268 −0.866342 0.499451i \(-0.833535\pi\)
−0.866342 + 0.499451i \(0.833535\pi\)
\(948\) 0 0
\(949\) 61.1769 1.98589
\(950\) 0 0
\(951\) 14.9282 0.484080
\(952\) 0 0
\(953\) −45.1769 −1.46342 −0.731712 0.681614i \(-0.761278\pi\)
−0.731712 + 0.681614i \(0.761278\pi\)
\(954\) 0 0
\(955\) −16.7321 −0.541436
\(956\) 0 0
\(957\) −1.21539 −0.0392880
\(958\) 0 0
\(959\) 23.8179 0.769121
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 2.92820 0.0943600
\(964\) 0 0
\(965\) −22.1962 −0.714519
\(966\) 0 0
\(967\) 4.98076 0.160171 0.0800853 0.996788i \(-0.474481\pi\)
0.0800853 + 0.996788i \(0.474481\pi\)
\(968\) 0 0
\(969\) 6.92820 0.222566
\(970\) 0 0
\(971\) −36.4974 −1.17126 −0.585629 0.810579i \(-0.699152\pi\)
−0.585629 + 0.810579i \(0.699152\pi\)
\(972\) 0 0
\(973\) 13.3590 0.428269
\(974\) 0 0
\(975\) −4.73205 −0.151547
\(976\) 0 0
\(977\) −9.85641 −0.315334 −0.157667 0.987492i \(-0.550397\pi\)
−0.157667 + 0.987492i \(0.550397\pi\)
\(978\) 0 0
\(979\) −59.8564 −1.91302
\(980\) 0 0
\(981\) 10.3923 0.331801
\(982\) 0 0
\(983\) 54.3923 1.73485 0.867423 0.497572i \(-0.165775\pi\)
0.867423 + 0.497572i \(0.165775\pi\)
\(984\) 0 0
\(985\) 14.9282 0.475652
\(986\) 0 0
\(987\) 12.6795 0.403593
\(988\) 0 0
\(989\) 7.60770 0.241911
\(990\) 0 0
\(991\) 24.7846 0.787309 0.393655 0.919258i \(-0.371211\pi\)
0.393655 + 0.919258i \(0.371211\pi\)
\(992\) 0 0
\(993\) −3.07180 −0.0974805
\(994\) 0 0
\(995\) −9.46410 −0.300032
\(996\) 0 0
\(997\) −58.3923 −1.84930 −0.924651 0.380815i \(-0.875644\pi\)
−0.924651 + 0.380815i \(0.875644\pi\)
\(998\) 0 0
\(999\) −0.732051 −0.0231611
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 2280.2.a.q.1.2 2
3.2 odd 2 6840.2.a.v.1.2 2
4.3 odd 2 4560.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.q.1.2 2 1.1 even 1 trivial
4560.2.a.bi.1.1 2 4.3 odd 2
6840.2.a.v.1.2 2 3.2 odd 2