Properties

Label 3072.2.a.t.1.2
Level $3072$
Weight $2$
Character 3072.1
Self dual yes
Analytic conductor $24.530$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.334904\) of defining polynomial
Character \(\chi\) \(=\) 3072.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.473626 q^{5} +4.55765 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.473626 q^{5} +4.55765 q^{7} +1.00000 q^{9} +3.49824 q^{11} -0.0840215 q^{13} -0.473626 q^{15} -3.61706 q^{17} +3.61706 q^{19} +4.55765 q^{21} +2.82843 q^{23} -4.77568 q^{25} +1.00000 q^{27} +7.30205 q^{29} -0.557647 q^{31} +3.49824 q^{33} -2.15862 q^{35} +6.20285 q^{37} -0.0840215 q^{39} -9.27391 q^{41} +2.27744 q^{43} -0.473626 q^{45} -2.82843 q^{47} +13.7721 q^{49} -3.61706 q^{51} +0.697947 q^{53} -1.65685 q^{55} +3.61706 q^{57} +5.65685 q^{59} -3.85970 q^{61} +4.55765 q^{63} +0.0397948 q^{65} -5.33962 q^{67} +2.82843 q^{69} -9.11529 q^{71} -0.541560 q^{73} -4.77568 q^{75} +15.9437 q^{77} +10.9937 q^{79} +1.00000 q^{81} -15.0496 q^{83} +1.71313 q^{85} +7.30205 q^{87} -14.6533 q^{89} -0.382941 q^{91} -0.557647 q^{93} -1.71313 q^{95} +4.31724 q^{97} +3.49824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{5} + 4q^{7} + 4q^{9} + 8q^{13} + 4q^{15} + 4q^{21} + 4q^{25} + 4q^{27} + 12q^{29} + 12q^{31} + 16q^{37} + 8q^{39} + 4q^{45} + 4q^{49} + 20q^{53} + 16q^{55} + 16q^{61} + 4q^{63} - 8q^{65} - 16q^{67} - 8q^{71} - 8q^{73} + 4q^{75} + 24q^{77} + 12q^{79} + 4q^{81} + 24q^{85} + 12q^{87} - 8q^{89} - 16q^{91} + 12q^{93} - 24q^{95} + O(q^{100}) \)

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
\(5\) −0.473626 −0.211812 −0.105906 0.994376i \(-0.533774\pi\)
−0.105906 + 0.994376i \(0.533774\pi\)
\(6\) 0 0
\(7\) 4.55765 1.72263 0.861314 0.508072i \(-0.169642\pi\)
0.861314 + 0.508072i \(0.169642\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.49824 1.05476 0.527379 0.849630i \(-0.323175\pi\)
0.527379 + 0.849630i \(0.323175\pi\)
\(12\) 0 0
\(13\) −0.0840215 −0.0233034 −0.0116517 0.999932i \(-0.503709\pi\)
−0.0116517 + 0.999932i \(0.503709\pi\)
\(14\) 0 0
\(15\) −0.473626 −0.122290
\(16\) 0 0
\(17\) −3.61706 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(18\) 0 0
\(19\) 3.61706 0.829810 0.414905 0.909865i \(-0.363815\pi\)
0.414905 + 0.909865i \(0.363815\pi\)
\(20\) 0 0
\(21\) 4.55765 0.994560
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −4.77568 −0.955136
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.30205 1.35596 0.677979 0.735082i \(-0.262856\pi\)
0.677979 + 0.735082i \(0.262856\pi\)
\(30\) 0 0
\(31\) −0.557647 −0.100156 −0.0500782 0.998745i \(-0.515947\pi\)
−0.0500782 + 0.998745i \(0.515947\pi\)
\(32\) 0 0
\(33\) 3.49824 0.608965
\(34\) 0 0
\(35\) −2.15862 −0.364873
\(36\) 0 0
\(37\) 6.20285 1.01974 0.509871 0.860251i \(-0.329693\pi\)
0.509871 + 0.860251i \(0.329693\pi\)
\(38\) 0 0
\(39\) −0.0840215 −0.0134542
\(40\) 0 0
\(41\) −9.27391 −1.44834 −0.724171 0.689620i \(-0.757777\pi\)
−0.724171 + 0.689620i \(0.757777\pi\)
\(42\) 0 0
\(43\) 2.27744 0.347307 0.173653 0.984807i \(-0.444443\pi\)
0.173653 + 0.984807i \(0.444443\pi\)
\(44\) 0 0
\(45\) −0.473626 −0.0706040
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 13.7721 1.96745
\(50\) 0 0
\(51\) −3.61706 −0.506490
\(52\) 0 0
\(53\) 0.697947 0.0958704 0.0479352 0.998850i \(-0.484736\pi\)
0.0479352 + 0.998850i \(0.484736\pi\)
\(54\) 0 0
\(55\) −1.65685 −0.223410
\(56\) 0 0
\(57\) 3.61706 0.479091
\(58\) 0 0
\(59\) 5.65685 0.736460 0.368230 0.929735i \(-0.379964\pi\)
0.368230 + 0.929735i \(0.379964\pi\)
\(60\) 0 0
\(61\) −3.85970 −0.494184 −0.247092 0.968992i \(-0.579475\pi\)
−0.247092 + 0.968992i \(0.579475\pi\)
\(62\) 0 0
\(63\) 4.55765 0.574210
\(64\) 0 0
\(65\) 0.0397948 0.00493593
\(66\) 0 0
\(67\) −5.33962 −0.652338 −0.326169 0.945311i \(-0.605758\pi\)
−0.326169 + 0.945311i \(0.605758\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) −9.11529 −1.08179 −0.540893 0.841091i \(-0.681914\pi\)
−0.540893 + 0.841091i \(0.681914\pi\)
\(72\) 0 0
\(73\) −0.541560 −0.0633848 −0.0316924 0.999498i \(-0.510090\pi\)
−0.0316924 + 0.999498i \(0.510090\pi\)
\(74\) 0 0
\(75\) −4.77568 −0.551448
\(76\) 0 0
\(77\) 15.9437 1.81696
\(78\) 0 0
\(79\) 10.9937 1.23689 0.618445 0.785828i \(-0.287763\pi\)
0.618445 + 0.785828i \(0.287763\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.0496 −1.65191 −0.825954 0.563738i \(-0.809363\pi\)
−0.825954 + 0.563738i \(0.809363\pi\)
\(84\) 0 0
\(85\) 1.71313 0.185815
\(86\) 0 0
\(87\) 7.30205 0.782862
\(88\) 0 0
\(89\) −14.6533 −1.55325 −0.776625 0.629964i \(-0.783070\pi\)
−0.776625 + 0.629964i \(0.783070\pi\)
\(90\) 0 0
\(91\) −0.382941 −0.0401431
\(92\) 0 0
\(93\) −0.557647 −0.0578253
\(94\) 0 0
\(95\) −1.71313 −0.175764
\(96\) 0 0
\(97\) 4.31724 0.438349 0.219175 0.975686i \(-0.429664\pi\)
0.219175 + 0.975686i \(0.429664\pi\)
\(98\) 0 0
\(99\) 3.49824 0.351586
\(100\) 0 0
\(101\) −0.641669 −0.0638484 −0.0319242 0.999490i \(-0.510164\pi\)
−0.0319242 + 0.999490i \(0.510164\pi\)
\(102\) 0 0
\(103\) −1.33686 −0.131724 −0.0658622 0.997829i \(-0.520980\pi\)
−0.0658622 + 0.997829i \(0.520980\pi\)
\(104\) 0 0
\(105\) −2.15862 −0.210660
\(106\) 0 0
\(107\) −8.57373 −0.828854 −0.414427 0.910083i \(-0.636018\pi\)
−0.414427 + 0.910083i \(0.636018\pi\)
\(108\) 0 0
\(109\) 8.08402 0.774309 0.387154 0.922015i \(-0.373458\pi\)
0.387154 + 0.922015i \(0.373458\pi\)
\(110\) 0 0
\(111\) 6.20285 0.588748
\(112\) 0 0
\(113\) 9.55136 0.898516 0.449258 0.893402i \(-0.351688\pi\)
0.449258 + 0.893402i \(0.351688\pi\)
\(114\) 0 0
\(115\) −1.33962 −0.124920
\(116\) 0 0
\(117\) −0.0840215 −0.00776779
\(118\) 0 0
\(119\) −16.4853 −1.51120
\(120\) 0 0
\(121\) 1.23765 0.112514
\(122\) 0 0
\(123\) −9.27391 −0.836201
\(124\) 0 0
\(125\) 4.63001 0.414121
\(126\) 0 0
\(127\) 5.09921 0.452481 0.226241 0.974071i \(-0.427356\pi\)
0.226241 + 0.974071i \(0.427356\pi\)
\(128\) 0 0
\(129\) 2.27744 0.200518
\(130\) 0 0
\(131\) −2.99647 −0.261803 −0.130901 0.991395i \(-0.541787\pi\)
−0.130901 + 0.991395i \(0.541787\pi\)
\(132\) 0 0
\(133\) 16.4853 1.42946
\(134\) 0 0
\(135\) −0.473626 −0.0407632
\(136\) 0 0
\(137\) −3.37941 −0.288723 −0.144361 0.989525i \(-0.546113\pi\)
−0.144361 + 0.989525i \(0.546113\pi\)
\(138\) 0 0
\(139\) −8.31724 −0.705459 −0.352729 0.935725i \(-0.614746\pi\)
−0.352729 + 0.935725i \(0.614746\pi\)
\(140\) 0 0
\(141\) −2.82843 −0.238197
\(142\) 0 0
\(143\) −0.293927 −0.0245794
\(144\) 0 0
\(145\) −3.45844 −0.287208
\(146\) 0 0
\(147\) 13.7721 1.13591
\(148\) 0 0
\(149\) 14.1305 1.15761 0.578807 0.815465i \(-0.303518\pi\)
0.578807 + 0.815465i \(0.303518\pi\)
\(150\) 0 0
\(151\) 9.97685 0.811905 0.405952 0.913894i \(-0.366940\pi\)
0.405952 + 0.913894i \(0.366940\pi\)
\(152\) 0 0
\(153\) −3.61706 −0.292422
\(154\) 0 0
\(155\) 0.264116 0.0212143
\(156\) 0 0
\(157\) 22.8562 1.82412 0.912060 0.410056i \(-0.134491\pi\)
0.912060 + 0.410056i \(0.134491\pi\)
\(158\) 0 0
\(159\) 0.697947 0.0553508
\(160\) 0 0
\(161\) 12.8910 1.01595
\(162\) 0 0
\(163\) 10.6135 0.831316 0.415658 0.909521i \(-0.363551\pi\)
0.415658 + 0.909521i \(0.363551\pi\)
\(164\) 0 0
\(165\) −1.65685 −0.128986
\(166\) 0 0
\(167\) 5.83822 0.451775 0.225888 0.974153i \(-0.427472\pi\)
0.225888 + 0.974153i \(0.427472\pi\)
\(168\) 0 0
\(169\) −12.9929 −0.999457
\(170\) 0 0
\(171\) 3.61706 0.276603
\(172\) 0 0
\(173\) −5.12695 −0.389795 −0.194897 0.980824i \(-0.562437\pi\)
−0.194897 + 0.980824i \(0.562437\pi\)
\(174\) 0 0
\(175\) −21.7659 −1.64534
\(176\) 0 0
\(177\) 5.65685 0.425195
\(178\) 0 0
\(179\) 13.1286 0.981279 0.490640 0.871363i \(-0.336763\pi\)
0.490640 + 0.871363i \(0.336763\pi\)
\(180\) 0 0
\(181\) 15.3181 1.13859 0.569294 0.822134i \(-0.307217\pi\)
0.569294 + 0.822134i \(0.307217\pi\)
\(182\) 0 0
\(183\) −3.85970 −0.285317
\(184\) 0 0
\(185\) −2.93783 −0.215993
\(186\) 0 0
\(187\) −12.6533 −0.925303
\(188\) 0 0
\(189\) 4.55765 0.331520
\(190\) 0 0
\(191\) −8.63001 −0.624446 −0.312223 0.950009i \(-0.601074\pi\)
−0.312223 + 0.950009i \(0.601074\pi\)
\(192\) 0 0
\(193\) 11.4514 0.824288 0.412144 0.911119i \(-0.364780\pi\)
0.412144 + 0.911119i \(0.364780\pi\)
\(194\) 0 0
\(195\) 0.0397948 0.00284976
\(196\) 0 0
\(197\) 10.5925 0.754681 0.377340 0.926075i \(-0.376839\pi\)
0.377340 + 0.926075i \(0.376839\pi\)
\(198\) 0 0
\(199\) 3.68000 0.260868 0.130434 0.991457i \(-0.458363\pi\)
0.130434 + 0.991457i \(0.458363\pi\)
\(200\) 0 0
\(201\) −5.33962 −0.376627
\(202\) 0 0
\(203\) 33.2802 2.33581
\(204\) 0 0
\(205\) 4.39236 0.306776
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) 12.6533 0.875249
\(210\) 0 0
\(211\) 14.3102 0.985153 0.492577 0.870269i \(-0.336055\pi\)
0.492577 + 0.870269i \(0.336055\pi\)
\(212\) 0 0
\(213\) −9.11529 −0.624570
\(214\) 0 0
\(215\) −1.07866 −0.0735637
\(216\) 0 0
\(217\) −2.54156 −0.172532
\(218\) 0 0
\(219\) −0.541560 −0.0365952
\(220\) 0 0
\(221\) 0.303911 0.0204433
\(222\) 0 0
\(223\) −4.86156 −0.325554 −0.162777 0.986663i \(-0.552045\pi\)
−0.162777 + 0.986663i \(0.552045\pi\)
\(224\) 0 0
\(225\) −4.77568 −0.318379
\(226\) 0 0
\(227\) −15.0496 −0.998877 −0.499438 0.866349i \(-0.666460\pi\)
−0.499438 + 0.866349i \(0.666460\pi\)
\(228\) 0 0
\(229\) −28.5264 −1.88507 −0.942537 0.334101i \(-0.891567\pi\)
−0.942537 + 0.334101i \(0.891567\pi\)
\(230\) 0 0
\(231\) 15.9437 1.04902
\(232\) 0 0
\(233\) 13.5702 0.889014 0.444507 0.895775i \(-0.353379\pi\)
0.444507 + 0.895775i \(0.353379\pi\)
\(234\) 0 0
\(235\) 1.33962 0.0873869
\(236\) 0 0
\(237\) 10.9937 0.714118
\(238\) 0 0
\(239\) −29.3629 −1.89933 −0.949665 0.313267i \(-0.898576\pi\)
−0.949665 + 0.313267i \(0.898576\pi\)
\(240\) 0 0
\(241\) −24.0063 −1.54638 −0.773190 0.634175i \(-0.781340\pi\)
−0.773190 + 0.634175i \(0.781340\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.52284 −0.416729
\(246\) 0 0
\(247\) −0.303911 −0.0193374
\(248\) 0 0
\(249\) −15.0496 −0.953729
\(250\) 0 0
\(251\) 22.2837 1.40654 0.703268 0.710925i \(-0.251724\pi\)
0.703268 + 0.710925i \(0.251724\pi\)
\(252\) 0 0
\(253\) 9.89450 0.622062
\(254\) 0 0
\(255\) 1.71313 0.107281
\(256\) 0 0
\(257\) 8.66038 0.540220 0.270110 0.962829i \(-0.412940\pi\)
0.270110 + 0.962829i \(0.412940\pi\)
\(258\) 0 0
\(259\) 28.2704 1.75664
\(260\) 0 0
\(261\) 7.30205 0.451986
\(262\) 0 0
\(263\) −13.3208 −0.821394 −0.410697 0.911772i \(-0.634715\pi\)
−0.410697 + 0.911772i \(0.634715\pi\)
\(264\) 0 0
\(265\) −0.330566 −0.0203065
\(266\) 0 0
\(267\) −14.6533 −0.896769
\(268\) 0 0
\(269\) −16.5058 −1.00638 −0.503188 0.864177i \(-0.667840\pi\)
−0.503188 + 0.864177i \(0.667840\pi\)
\(270\) 0 0
\(271\) 21.9769 1.33500 0.667499 0.744610i \(-0.267365\pi\)
0.667499 + 0.744610i \(0.267365\pi\)
\(272\) 0 0
\(273\) −0.382941 −0.0231766
\(274\) 0 0
\(275\) −16.7064 −1.00744
\(276\) 0 0
\(277\) 15.4862 0.930475 0.465237 0.885186i \(-0.345969\pi\)
0.465237 + 0.885186i \(0.345969\pi\)
\(278\) 0 0
\(279\) −0.557647 −0.0333855
\(280\) 0 0
\(281\) 22.8910 1.36556 0.682780 0.730624i \(-0.260771\pi\)
0.682780 + 0.730624i \(0.260771\pi\)
\(282\) 0 0
\(283\) 6.34315 0.377061 0.188530 0.982067i \(-0.439628\pi\)
0.188530 + 0.982067i \(0.439628\pi\)
\(284\) 0 0
\(285\) −1.71313 −0.101477
\(286\) 0 0
\(287\) −42.2672 −2.49496
\(288\) 0 0
\(289\) −3.91688 −0.230405
\(290\) 0 0
\(291\) 4.31724 0.253081
\(292\) 0 0
\(293\) 30.5783 1.78641 0.893203 0.449654i \(-0.148453\pi\)
0.893203 + 0.449654i \(0.148453\pi\)
\(294\) 0 0
\(295\) −2.67923 −0.155991
\(296\) 0 0
\(297\) 3.49824 0.202988
\(298\) 0 0
\(299\) −0.237649 −0.0137436
\(300\) 0 0
\(301\) 10.3798 0.598281
\(302\) 0 0
\(303\) −0.641669 −0.0368629
\(304\) 0 0
\(305\) 1.82805 0.104674
\(306\) 0 0
\(307\) −17.1286 −0.977582 −0.488791 0.872401i \(-0.662562\pi\)
−0.488791 + 0.872401i \(0.662562\pi\)
\(308\) 0 0
\(309\) −1.33686 −0.0760511
\(310\) 0 0
\(311\) −26.8651 −1.52338 −0.761689 0.647943i \(-0.775630\pi\)
−0.761689 + 0.647943i \(0.775630\pi\)
\(312\) 0 0
\(313\) −19.6890 −1.11289 −0.556445 0.830885i \(-0.687835\pi\)
−0.556445 + 0.830885i \(0.687835\pi\)
\(314\) 0 0
\(315\) −2.15862 −0.121624
\(316\) 0 0
\(317\) −30.1860 −1.69541 −0.847706 0.530466i \(-0.822017\pi\)
−0.847706 + 0.530466i \(0.822017\pi\)
\(318\) 0 0
\(319\) 25.5443 1.43021
\(320\) 0 0
\(321\) −8.57373 −0.478539
\(322\) 0 0
\(323\) −13.0831 −0.727964
\(324\) 0 0
\(325\) 0.401260 0.0222579
\(326\) 0 0
\(327\) 8.08402 0.447047
\(328\) 0 0
\(329\) −12.8910 −0.710702
\(330\) 0 0
\(331\) −20.7784 −1.14209 −0.571043 0.820920i \(-0.693461\pi\)
−0.571043 + 0.820920i \(0.693461\pi\)
\(332\) 0 0
\(333\) 6.20285 0.339914
\(334\) 0 0
\(335\) 2.52898 0.138173
\(336\) 0 0
\(337\) 23.0098 1.25342 0.626712 0.779251i \(-0.284400\pi\)
0.626712 + 0.779251i \(0.284400\pi\)
\(338\) 0 0
\(339\) 9.55136 0.518759
\(340\) 0 0
\(341\) −1.95078 −0.105641
\(342\) 0 0
\(343\) 30.8651 1.66656
\(344\) 0 0
\(345\) −1.33962 −0.0721225
\(346\) 0 0
\(347\) −15.4186 −0.827716 −0.413858 0.910341i \(-0.635819\pi\)
−0.413858 + 0.910341i \(0.635819\pi\)
\(348\) 0 0
\(349\) −28.3638 −1.51828 −0.759140 0.650927i \(-0.774380\pi\)
−0.759140 + 0.650927i \(0.774380\pi\)
\(350\) 0 0
\(351\) −0.0840215 −0.00448474
\(352\) 0 0
\(353\) −12.2117 −0.649965 −0.324983 0.945720i \(-0.605358\pi\)
−0.324983 + 0.945720i \(0.605358\pi\)
\(354\) 0 0
\(355\) 4.31724 0.229135
\(356\) 0 0
\(357\) −16.4853 −0.872494
\(358\) 0 0
\(359\) −33.4780 −1.76690 −0.883452 0.468522i \(-0.844786\pi\)
−0.883452 + 0.468522i \(0.844786\pi\)
\(360\) 0 0
\(361\) −5.91688 −0.311415
\(362\) 0 0
\(363\) 1.23765 0.0649597
\(364\) 0 0
\(365\) 0.256497 0.0134256
\(366\) 0 0
\(367\) 0.702379 0.0366639 0.0183319 0.999832i \(-0.494164\pi\)
0.0183319 + 0.999832i \(0.494164\pi\)
\(368\) 0 0
\(369\) −9.27391 −0.482781
\(370\) 0 0
\(371\) 3.18100 0.165149
\(372\) 0 0
\(373\) −26.8132 −1.38834 −0.694168 0.719813i \(-0.744227\pi\)
−0.694168 + 0.719813i \(0.744227\pi\)
\(374\) 0 0
\(375\) 4.63001 0.239093
\(376\) 0 0
\(377\) −0.613530 −0.0315984
\(378\) 0 0
\(379\) −2.51509 −0.129192 −0.0645958 0.997912i \(-0.520576\pi\)
−0.0645958 + 0.997912i \(0.520576\pi\)
\(380\) 0 0
\(381\) 5.09921 0.261240
\(382\) 0 0
\(383\) 25.4880 1.30238 0.651188 0.758916i \(-0.274271\pi\)
0.651188 + 0.758916i \(0.274271\pi\)
\(384\) 0 0
\(385\) −7.55136 −0.384853
\(386\) 0 0
\(387\) 2.27744 0.115769
\(388\) 0 0
\(389\) −16.5532 −0.839281 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(390\) 0 0
\(391\) −10.2306 −0.517383
\(392\) 0 0
\(393\) −2.99647 −0.151152
\(394\) 0 0
\(395\) −5.20690 −0.261988
\(396\) 0 0
\(397\) −12.7936 −0.642094 −0.321047 0.947063i \(-0.604035\pi\)
−0.321047 + 0.947063i \(0.604035\pi\)
\(398\) 0 0
\(399\) 16.4853 0.825296
\(400\) 0 0
\(401\) 18.0853 0.903137 0.451568 0.892237i \(-0.350865\pi\)
0.451568 + 0.892237i \(0.350865\pi\)
\(402\) 0 0
\(403\) 0.0468544 0.00233398
\(404\) 0 0
\(405\) −0.473626 −0.0235347
\(406\) 0 0
\(407\) 21.6990 1.07558
\(408\) 0 0
\(409\) 25.2271 1.24740 0.623699 0.781665i \(-0.285629\pi\)
0.623699 + 0.781665i \(0.285629\pi\)
\(410\) 0 0
\(411\) −3.37941 −0.166694
\(412\) 0 0
\(413\) 25.7819 1.26865
\(414\) 0 0
\(415\) 7.12787 0.349894
\(416\) 0 0
\(417\) −8.31724 −0.407297
\(418\) 0 0
\(419\) −10.2571 −0.501090 −0.250545 0.968105i \(-0.580610\pi\)
−0.250545 + 0.968105i \(0.580610\pi\)
\(420\) 0 0
\(421\) 3.38775 0.165109 0.0825543 0.996587i \(-0.473692\pi\)
0.0825543 + 0.996587i \(0.473692\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) 17.2739 0.837908
\(426\) 0 0
\(427\) −17.5912 −0.851296
\(428\) 0 0
\(429\) −0.293927 −0.0141909
\(430\) 0 0
\(431\) 4.42454 0.213123 0.106561 0.994306i \(-0.466016\pi\)
0.106561 + 0.994306i \(0.466016\pi\)
\(432\) 0 0
\(433\) −7.31371 −0.351474 −0.175737 0.984437i \(-0.556231\pi\)
−0.175737 + 0.984437i \(0.556231\pi\)
\(434\) 0 0
\(435\) −3.45844 −0.165820
\(436\) 0 0
\(437\) 10.2306 0.489395
\(438\) 0 0
\(439\) −29.6533 −1.41527 −0.707637 0.706576i \(-0.750239\pi\)
−0.707637 + 0.706576i \(0.750239\pi\)
\(440\) 0 0
\(441\) 13.7721 0.655817
\(442\) 0 0
\(443\) 14.5743 0.692446 0.346223 0.938152i \(-0.387464\pi\)
0.346223 + 0.938152i \(0.387464\pi\)
\(444\) 0 0
\(445\) 6.94019 0.328997
\(446\) 0 0
\(447\) 14.1305 0.668349
\(448\) 0 0
\(449\) −6.48844 −0.306208 −0.153104 0.988210i \(-0.548927\pi\)
−0.153104 + 0.988210i \(0.548927\pi\)
\(450\) 0 0
\(451\) −32.4423 −1.52765
\(452\) 0 0
\(453\) 9.97685 0.468753
\(454\) 0 0
\(455\) 0.181370 0.00850278
\(456\) 0 0
\(457\) −9.00353 −0.421167 −0.210584 0.977576i \(-0.567536\pi\)
−0.210584 + 0.977576i \(0.567536\pi\)
\(458\) 0 0
\(459\) −3.61706 −0.168830
\(460\) 0 0
\(461\) 20.6783 0.963085 0.481542 0.876423i \(-0.340077\pi\)
0.481542 + 0.876423i \(0.340077\pi\)
\(462\) 0 0
\(463\) 18.6435 0.866437 0.433219 0.901289i \(-0.357378\pi\)
0.433219 + 0.901289i \(0.357378\pi\)
\(464\) 0 0
\(465\) 0.264116 0.0122481
\(466\) 0 0
\(467\) −33.2535 −1.53879 −0.769395 0.638773i \(-0.779442\pi\)
−0.769395 + 0.638773i \(0.779442\pi\)
\(468\) 0 0
\(469\) −24.3361 −1.12374
\(470\) 0 0
\(471\) 22.8562 1.05316
\(472\) 0 0
\(473\) 7.96703 0.366325
\(474\) 0 0
\(475\) −17.2739 −0.792582
\(476\) 0 0
\(477\) 0.697947 0.0319568
\(478\) 0 0
\(479\) −1.08864 −0.0497412 −0.0248706 0.999691i \(-0.507917\pi\)
−0.0248706 + 0.999691i \(0.507917\pi\)
\(480\) 0 0
\(481\) −0.521173 −0.0237634
\(482\) 0 0
\(483\) 12.8910 0.586560
\(484\) 0 0
\(485\) −2.04476 −0.0928476
\(486\) 0 0
\(487\) 35.3298 1.60095 0.800473 0.599369i \(-0.204582\pi\)
0.800473 + 0.599369i \(0.204582\pi\)
\(488\) 0 0
\(489\) 10.6135 0.479960
\(490\) 0 0
\(491\) 18.2306 0.822735 0.411367 0.911470i \(-0.365051\pi\)
0.411367 + 0.911470i \(0.365051\pi\)
\(492\) 0 0
\(493\) −26.4120 −1.18953
\(494\) 0 0
\(495\) −1.65685 −0.0744701
\(496\) 0 0
\(497\) −41.5443 −1.86352
\(498\) 0 0
\(499\) 20.3361 0.910368 0.455184 0.890397i \(-0.349573\pi\)
0.455184 + 0.890397i \(0.349573\pi\)
\(500\) 0 0
\(501\) 5.83822 0.260833
\(502\) 0 0
\(503\) −30.2969 −1.35087 −0.675435 0.737420i \(-0.736044\pi\)
−0.675435 + 0.737420i \(0.736044\pi\)
\(504\) 0 0
\(505\) 0.303911 0.0135239
\(506\) 0 0
\(507\) −12.9929 −0.577037
\(508\) 0 0
\(509\) −14.9660 −0.663355 −0.331677 0.943393i \(-0.607615\pi\)
−0.331677 + 0.943393i \(0.607615\pi\)
\(510\) 0 0
\(511\) −2.46824 −0.109188
\(512\) 0 0
\(513\) 3.61706 0.159697
\(514\) 0 0
\(515\) 0.633169 0.0279008
\(516\) 0 0
\(517\) −9.89450 −0.435160
\(518\) 0 0
\(519\) −5.12695 −0.225048
\(520\) 0 0
\(521\) 24.9049 1.09110 0.545551 0.838078i \(-0.316320\pi\)
0.545551 + 0.838078i \(0.316320\pi\)
\(522\) 0 0
\(523\) 18.2445 0.797775 0.398888 0.917000i \(-0.369396\pi\)
0.398888 + 0.917000i \(0.369396\pi\)
\(524\) 0 0
\(525\) −21.7659 −0.949940
\(526\) 0 0
\(527\) 2.01704 0.0878638
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 5.65685 0.245487
\(532\) 0 0
\(533\) 0.779208 0.0337513
\(534\) 0 0
\(535\) 4.06074 0.175561
\(536\) 0 0
\(537\) 13.1286 0.566542
\(538\) 0 0
\(539\) 48.1782 2.07518
\(540\) 0 0
\(541\) −25.8471 −1.11125 −0.555627 0.831432i \(-0.687522\pi\)
−0.555627 + 0.831432i \(0.687522\pi\)
\(542\) 0 0
\(543\) 15.3181 0.657364
\(544\) 0 0
\(545\) −3.82880 −0.164008
\(546\) 0 0
\(547\) −19.4249 −0.830549 −0.415275 0.909696i \(-0.636315\pi\)
−0.415275 + 0.909696i \(0.636315\pi\)
\(548\) 0 0
\(549\) −3.85970 −0.164728
\(550\) 0 0
\(551\) 26.4120 1.12519
\(552\) 0 0
\(553\) 50.1055 2.13070
\(554\) 0 0
\(555\) −2.93783 −0.124704
\(556\) 0 0
\(557\) −38.9652 −1.65101 −0.825504 0.564397i \(-0.809109\pi\)
−0.825504 + 0.564397i \(0.809109\pi\)
\(558\) 0 0
\(559\) −0.191354 −0.00809342
\(560\) 0 0
\(561\) −12.6533 −0.534224
\(562\) 0 0
\(563\) −28.1327 −1.18565 −0.592826 0.805330i \(-0.701988\pi\)
−0.592826 + 0.805330i \(0.701988\pi\)
\(564\) 0 0
\(565\) −4.52377 −0.190316
\(566\) 0 0
\(567\) 4.55765 0.191403
\(568\) 0 0
\(569\) −13.4849 −0.565317 −0.282658 0.959221i \(-0.591216\pi\)
−0.282658 + 0.959221i \(0.591216\pi\)
\(570\) 0 0
\(571\) −20.9706 −0.877591 −0.438795 0.898587i \(-0.644595\pi\)
−0.438795 + 0.898587i \(0.644595\pi\)
\(572\) 0 0
\(573\) −8.63001 −0.360524
\(574\) 0 0
\(575\) −13.5077 −0.563308
\(576\) 0 0
\(577\) −11.6176 −0.483648 −0.241824 0.970320i \(-0.577746\pi\)
−0.241824 + 0.970320i \(0.577746\pi\)
\(578\) 0 0
\(579\) 11.4514 0.475903
\(580\) 0 0
\(581\) −68.5907 −2.84562
\(582\) 0 0
\(583\) 2.44158 0.101120
\(584\) 0 0
\(585\) 0.0397948 0.00164531
\(586\) 0 0
\(587\) 24.0796 0.993871 0.496936 0.867787i \(-0.334459\pi\)
0.496936 + 0.867787i \(0.334459\pi\)
\(588\) 0 0
\(589\) −2.01704 −0.0831108
\(590\) 0 0
\(591\) 10.5925 0.435715
\(592\) 0 0
\(593\) −41.5372 −1.70573 −0.852865 0.522132i \(-0.825137\pi\)
−0.852865 + 0.522132i \(0.825137\pi\)
\(594\) 0 0
\(595\) 7.80785 0.320091
\(596\) 0 0
\(597\) 3.68000 0.150612
\(598\) 0 0
\(599\) −6.43160 −0.262788 −0.131394 0.991330i \(-0.541945\pi\)
−0.131394 + 0.991330i \(0.541945\pi\)
\(600\) 0 0
\(601\) −3.45844 −0.141073 −0.0705364 0.997509i \(-0.522471\pi\)
−0.0705364 + 0.997509i \(0.522471\pi\)
\(602\) 0 0
\(603\) −5.33962 −0.217446
\(604\) 0 0
\(605\) −0.586182 −0.0238317
\(606\) 0 0
\(607\) −30.1019 −1.22180 −0.610900 0.791708i \(-0.709192\pi\)
−0.610900 + 0.791708i \(0.709192\pi\)
\(608\) 0 0
\(609\) 33.2802 1.34858
\(610\) 0 0
\(611\) 0.237649 0.00961424
\(612\) 0 0
\(613\) 3.54246 0.143079 0.0715393 0.997438i \(-0.477209\pi\)
0.0715393 + 0.997438i \(0.477209\pi\)
\(614\) 0 0
\(615\) 4.39236 0.177117
\(616\) 0 0
\(617\) −22.9098 −0.922315 −0.461157 0.887318i \(-0.652566\pi\)
−0.461157 + 0.887318i \(0.652566\pi\)
\(618\) 0 0
\(619\) 40.4612 1.62627 0.813136 0.582074i \(-0.197758\pi\)
0.813136 + 0.582074i \(0.197758\pi\)
\(620\) 0 0
\(621\) 2.82843 0.113501
\(622\) 0 0
\(623\) −66.7847 −2.67567
\(624\) 0 0
\(625\) 21.6855 0.867420
\(626\) 0 0
\(627\) 12.6533 0.505325
\(628\) 0 0
\(629\) −22.4361 −0.894584
\(630\) 0 0
\(631\) −11.1851 −0.445270 −0.222635 0.974902i \(-0.571466\pi\)
−0.222635 + 0.974902i \(0.571466\pi\)
\(632\) 0 0
\(633\) 14.3102 0.568779
\(634\) 0 0
\(635\) −2.41512 −0.0958409
\(636\) 0 0
\(637\) −1.15716 −0.0458482
\(638\) 0 0
\(639\) −9.11529 −0.360595
\(640\) 0 0
\(641\) −6.69312 −0.264362 −0.132181 0.991226i \(-0.542198\pi\)
−0.132181 + 0.991226i \(0.542198\pi\)
\(642\) 0 0
\(643\) 25.3724 1.00059 0.500294 0.865856i \(-0.333225\pi\)
0.500294 + 0.865856i \(0.333225\pi\)
\(644\) 0 0
\(645\) −1.07866 −0.0424720
\(646\) 0 0
\(647\) 6.72999 0.264583 0.132292 0.991211i \(-0.457766\pi\)
0.132292 + 0.991211i \(0.457766\pi\)
\(648\) 0 0
\(649\) 19.7890 0.776786
\(650\) 0 0
\(651\) −2.54156 −0.0996116
\(652\) 0 0
\(653\) 37.0144 1.44849 0.724243 0.689545i \(-0.242190\pi\)
0.724243 + 0.689545i \(0.242190\pi\)
\(654\) 0 0
\(655\) 1.41921 0.0554529
\(656\) 0 0
\(657\) −0.541560 −0.0211283
\(658\) 0 0
\(659\) 19.7624 0.769832 0.384916 0.922952i \(-0.374230\pi\)
0.384916 + 0.922952i \(0.374230\pi\)
\(660\) 0 0
\(661\) −16.8632 −0.655904 −0.327952 0.944694i \(-0.606358\pi\)
−0.327952 + 0.944694i \(0.606358\pi\)
\(662\) 0 0
\(663\) 0.303911 0.0118029
\(664\) 0 0
\(665\) −7.80785 −0.302776
\(666\) 0 0
\(667\) 20.6533 0.799700
\(668\) 0 0
\(669\) −4.86156 −0.187959
\(670\) 0 0
\(671\) −13.5021 −0.521244
\(672\) 0 0
\(673\) −37.3066 −1.43807 −0.719033 0.694976i \(-0.755415\pi\)
−0.719033 + 0.694976i \(0.755415\pi\)
\(674\) 0 0
\(675\) −4.77568 −0.183816
\(676\) 0 0
\(677\) 0.632805 0.0243207 0.0121603 0.999926i \(-0.496129\pi\)
0.0121603 + 0.999926i \(0.496129\pi\)
\(678\) 0 0
\(679\) 19.6764 0.755113
\(680\) 0 0
\(681\) −15.0496 −0.576702
\(682\) 0 0
\(683\) 6.04606 0.231346 0.115673 0.993287i \(-0.463098\pi\)
0.115673 + 0.993287i \(0.463098\pi\)
\(684\) 0 0
\(685\) 1.60058 0.0611549
\(686\) 0 0
\(687\) −28.5264 −1.08835
\(688\) 0 0
\(689\) −0.0586426 −0.00223410
\(690\) 0 0
\(691\) 28.3955 1.08021 0.540107 0.841596i \(-0.318384\pi\)
0.540107 + 0.841596i \(0.318384\pi\)
\(692\) 0 0
\(693\) 15.9437 0.605652
\(694\) 0 0
\(695\) 3.93926 0.149425
\(696\) 0 0
\(697\) 33.5443 1.27058
\(698\) 0 0
\(699\) 13.5702 0.513272
\(700\) 0 0
\(701\) 14.7738 0.558000 0.279000 0.960291i \(-0.409997\pi\)
0.279000 + 0.960291i \(0.409997\pi\)
\(702\) 0 0
\(703\) 22.4361 0.846192
\(704\) 0 0
\(705\) 1.33962 0.0504529
\(706\) 0 0
\(707\) −2.92450 −0.109987
\(708\) 0 0
\(709\) 22.7569 0.854655 0.427327 0.904097i \(-0.359455\pi\)
0.427327 + 0.904097i \(0.359455\pi\)
\(710\) 0 0
\(711\) 10.9937 0.412296
\(712\) 0 0
\(713\) −1.57726 −0.0590690
\(714\) 0 0
\(715\) 0.139211 0.00520621
\(716\) 0 0
\(717\) −29.3629 −1.09658
\(718\) 0 0
\(719\) 30.9957 1.15594 0.577972 0.816057i \(-0.303844\pi\)
0.577972 + 0.816057i \(0.303844\pi\)
\(720\) 0 0
\(721\) −6.09292 −0.226912
\(722\) 0 0
\(723\) −24.0063 −0.892803
\(724\) 0 0
\(725\) −34.8723 −1.29512
\(726\) 0 0
\(727\) 41.1117 1.52475 0.762375 0.647135i \(-0.224033\pi\)
0.762375 + 0.647135i \(0.224033\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.23765 −0.304680
\(732\) 0 0
\(733\) −0.206562 −0.00762954 −0.00381477 0.999993i \(-0.501214\pi\)
−0.00381477 + 0.999993i \(0.501214\pi\)
\(734\) 0 0
\(735\) −6.52284 −0.240599
\(736\) 0 0
\(737\) −18.6792 −0.688058
\(738\) 0 0
\(739\) 2.13215 0.0784325 0.0392162 0.999231i \(-0.487514\pi\)
0.0392162 + 0.999231i \(0.487514\pi\)
\(740\) 0 0
\(741\) −0.303911 −0.0111644
\(742\) 0 0
\(743\) −40.5175 −1.48644 −0.743221 0.669046i \(-0.766703\pi\)
−0.743221 + 0.669046i \(0.766703\pi\)
\(744\) 0 0
\(745\) −6.69256 −0.245196
\(746\) 0 0
\(747\) −15.0496 −0.550636
\(748\) 0 0
\(749\) −39.0761 −1.42781
\(750\) 0 0
\(751\) 12.5843 0.459208 0.229604 0.973284i \(-0.426257\pi\)
0.229604 + 0.973284i \(0.426257\pi\)
\(752\) 0 0
\(753\) 22.2837 0.812064
\(754\) 0 0
\(755\) −4.72529 −0.171971
\(756\) 0 0
\(757\) 10.6052 0.385452 0.192726 0.981253i \(-0.438267\pi\)
0.192726 + 0.981253i \(0.438267\pi\)
\(758\) 0 0
\(759\) 9.89450 0.359148
\(760\) 0 0
\(761\) 42.8182 1.55216 0.776079 0.630635i \(-0.217206\pi\)
0.776079 + 0.630635i \(0.217206\pi\)
\(762\) 0 0
\(763\) 36.8441 1.33385
\(764\) 0 0
\(765\) 1.71313 0.0619384
\(766\) 0 0
\(767\) −0.475298 −0.0171620
\(768\) 0 0
\(769\) 12.7455 0.459614 0.229807 0.973236i \(-0.426190\pi\)
0.229807 + 0.973236i \(0.426190\pi\)
\(770\) 0 0
\(771\) 8.66038 0.311896
\(772\) 0 0
\(773\) −32.3522 −1.16363 −0.581814 0.813322i \(-0.697657\pi\)
−0.581814 + 0.813322i \(0.697657\pi\)
\(774\) 0 0
\(775\) 2.66314 0.0956630
\(776\) 0 0
\(777\) 28.2704 1.01419
\(778\) 0 0
\(779\) −33.5443 −1.20185
\(780\) 0 0
\(781\) −31.8874 −1.14102
\(782\) 0 0
\(783\) 7.30205 0.260954
\(784\) 0 0
\(785\) −10.8253 −0.386370
\(786\) 0 0
\(787\) 7.36056 0.262376 0.131188 0.991358i \(-0.458121\pi\)
0.131188 + 0.991358i \(0.458121\pi\)
\(788\) 0 0
\(789\) −13.3208 −0.474232
\(790\) 0 0
\(791\) 43.5317 1.54781
\(792\) 0 0
\(793\) 0.324298 0.0115162
\(794\) 0 0
\(795\) −0.330566 −0.0117240
\(796\) 0 0
\(797\) −24.0627 −0.852344 −0.426172 0.904642i \(-0.640138\pi\)
−0.426172 + 0.904642i \(0.640138\pi\)
\(798\) 0 0
\(799\) 10.2306 0.361932
\(800\) 0 0
\(801\) −14.6533 −0.517750
\(802\) 0 0
\(803\) −1.89450 −0.0668556
\(804\) 0 0
\(805\) −6.10550 −0.215190
\(806\) 0 0
\(807\) −16.5058 −0.581032
\(808\) 0 0
\(809\) −7.83586 −0.275494 −0.137747 0.990467i \(-0.543986\pi\)
−0.137747 + 0.990467i \(0.543986\pi\)
\(810\) 0 0
\(811\) −45.7351 −1.60598 −0.802988 0.595995i \(-0.796758\pi\)
−0.802988 + 0.595995i \(0.796758\pi\)
\(812\) 0 0
\(813\) 21.9769 0.770762
\(814\) 0 0
\(815\) −5.02684 −0.176083
\(816\) 0 0
\(817\) 8.23765 0.288199
\(818\) 0 0
\(819\) −0.382941 −0.0133810
\(820\) 0 0
\(821\) −27.3709 −0.955250 −0.477625 0.878564i \(-0.658502\pi\)
−0.477625 + 0.878564i \(0.658502\pi\)
\(822\) 0 0
\(823\) −28.8560 −1.00586 −0.502929 0.864328i \(-0.667744\pi\)
−0.502929 + 0.864328i \(0.667744\pi\)
\(824\) 0 0
\(825\) −16.7064 −0.581644
\(826\) 0 0
\(827\) −14.4227 −0.501528 −0.250764 0.968048i \(-0.580682\pi\)
−0.250764 + 0.968048i \(0.580682\pi\)
\(828\) 0 0
\(829\) −21.7497 −0.755400 −0.377700 0.925928i \(-0.623285\pi\)
−0.377700 + 0.925928i \(0.623285\pi\)
\(830\) 0 0
\(831\) 15.4862 0.537210
\(832\) 0 0
\(833\) −49.8147 −1.72598
\(834\) 0 0
\(835\) −2.76513 −0.0956914
\(836\) 0 0
\(837\) −0.557647 −0.0192751
\(838\) 0 0
\(839\) −44.4557 −1.53478 −0.767390 0.641181i \(-0.778445\pi\)
−0.767390 + 0.641181i \(0.778445\pi\)
\(840\) 0 0
\(841\) 24.3200 0.838620
\(842\) 0 0
\(843\) 22.8910 0.788407
\(844\) 0 0
\(845\) 6.15379 0.211697
\(846\) 0 0
\(847\) 5.64077 0.193819
\(848\) 0 0
\(849\) 6.34315 0.217696
\(850\) 0 0
\(851\) 17.5443 0.601411
\(852\) 0 0
\(853\) 16.5648 0.567169 0.283585 0.958947i \(-0.408476\pi\)
0.283585 + 0.958947i \(0.408476\pi\)
\(854\) 0 0
\(855\) −1.71313 −0.0585879
\(856\) 0 0
\(857\) −19.0888 −0.652062 −0.326031 0.945359i \(-0.605711\pi\)
−0.326031 + 0.945359i \(0.605711\pi\)
\(858\) 0 0
\(859\) −53.9272 −1.83997 −0.919987 0.391949i \(-0.871801\pi\)
−0.919987 + 0.391949i \(0.871801\pi\)
\(860\) 0 0
\(861\) −42.2672 −1.44046
\(862\) 0 0
\(863\) 3.64533 0.124089 0.0620443 0.998073i \(-0.480238\pi\)
0.0620443 + 0.998073i \(0.480238\pi\)
\(864\) 0 0
\(865\) 2.42826 0.0825632
\(866\) 0 0
\(867\) −3.91688 −0.133024
\(868\) 0 0
\(869\) 38.4586 1.30462
\(870\) 0 0
\(871\) 0.448643 0.0152017
\(872\) 0 0
\(873\) 4.31724 0.146116
\(874\) 0 0
\(875\) 21.1020 0.713377
\(876\) 0 0
\(877\) 56.6481 1.91287 0.956435 0.291945i \(-0.0943023\pi\)
0.956435 + 0.291945i \(0.0943023\pi\)
\(878\) 0 0
\(879\) 30.5783 1.03138
\(880\) 0 0
\(881\) 20.0118 0.674214 0.337107 0.941466i \(-0.390552\pi\)
0.337107 + 0.941466i \(0.390552\pi\)
\(882\) 0 0
\(883\) −15.0292 −0.505773 −0.252887 0.967496i \(-0.581380\pi\)
−0.252887 + 0.967496i \(0.581380\pi\)
\(884\) 0 0
\(885\) −2.67923 −0.0900614
\(886\) 0 0
\(887\) −26.1180 −0.876958 −0.438479 0.898742i \(-0.644483\pi\)
−0.438479 + 0.898742i \(0.644483\pi\)
\(888\) 0 0
\(889\) 23.2404 0.779458
\(890\) 0 0
\(891\) 3.49824 0.117195
\(892\) 0 0
\(893\) −10.2306 −0.342354
\(894\) 0 0
\(895\) −6.21805 −0.207847
\(896\) 0 0
\(897\) −0.237649 −0.00793486
\(898\) 0 0
\(899\) −4.07197 −0.135808
\(900\) 0 0
\(901\) −2.52452 −0.0841038
\(902\) 0 0
\(903\) 10.3798 0.345418
\(904\) 0 0
\(905\) −7.25507 −0.241167
\(906\) 0 0
\(907\) −51.2480 −1.70166 −0.850831 0.525439i \(-0.823901\pi\)
−0.850831 + 0.525439i \(0.823901\pi\)
\(908\) 0 0
\(909\) −0.641669 −0.0212828
\(910\) 0 0
\(911\) 21.0535 0.697533 0.348767 0.937210i \(-0.386601\pi\)
0.348767 + 0.937210i \(0.386601\pi\)
\(912\) 0 0
\(913\) −52.6470 −1.74236
\(914\) 0 0
\(915\) 1.82805 0.0604336
\(916\) 0 0
\(917\) −13.6569 −0.450989
\(918\) 0 0
\(919\) 17.8839 0.589937 0.294968 0.955507i \(-0.404691\pi\)
0.294968 + 0.955507i \(0.404691\pi\)
\(920\) 0 0
\(921\) −17.1286 −0.564407
\(922\) 0 0
\(923\) 0.765881 0.0252093
\(924\) 0 0
\(925\) −29.6228 −0.973992
\(926\) 0 0
\(927\) −1.33686 −0.0439081
\(928\) 0 0
\(929\) 10.2774 0.337192 0.168596 0.985685i \(-0.446077\pi\)
0.168596 + 0.985685i \(0.446077\pi\)
\(930\) 0 0
\(931\) 49.8147 1.63261
\(932\) 0 0
\(933\) −26.8651 −0.879523
\(934\) 0 0
\(935\) 5.99294 0.195990
\(936\) 0 0
\(937\) −13.5780 −0.443574 −0.221787 0.975095i \(-0.571189\pi\)
−0.221787 + 0.975095i \(0.571189\pi\)
\(938\) 0 0
\(939\) −19.6890 −0.642527
\(940\) 0 0
\(941\) −5.59890 −0.182519 −0.0912595 0.995827i \(-0.529089\pi\)
−0.0912595 + 0.995827i \(0.529089\pi\)
\(942\) 0 0
\(943\) −26.2306 −0.854186
\(944\) 0 0
\(945\) −2.15862 −0.0702199
\(946\) 0 0
\(947\) −46.9106 −1.52439 −0.762194 0.647348i \(-0.775878\pi\)
−0.762194 + 0.647348i \(0.775878\pi\)
\(948\) 0 0
\(949\) 0.0455027 0.00147708
\(950\) 0 0
\(951\) −30.1860 −0.978847
\(952\) 0 0
\(953\) 5.59115 0.181115 0.0905576 0.995891i \(-0.471135\pi\)
0.0905576 + 0.995891i \(0.471135\pi\)
\(954\) 0 0
\(955\) 4.08740 0.132265
\(956\) 0 0
\(957\) 25.5443 0.825730
\(958\) 0 0
\(959\) −15.4022 −0.497362
\(960\) 0 0
\(961\) −30.6890 −0.989969
\(962\) 0 0
\(963\) −8.57373 −0.276285
\(964\) 0 0
\(965\) −5.42367 −0.174594
\(966\) 0 0
\(967\) −30.7561 −0.989048 −0.494524 0.869164i \(-0.664658\pi\)
−0.494524 + 0.869164i \(0.664658\pi\)
\(968\) 0 0
\(969\) −13.0831 −0.420290
\(970\) 0 0
\(971\) −11.3668 −0.364779 −0.182389 0.983226i \(-0.558383\pi\)
−0.182389 + 0.983226i \(0.558383\pi\)
\(972\) 0 0
\(973\) −37.9070 −1.21524
\(974\) 0 0
\(975\) 0.401260 0.0128506
\(976\) 0 0
\(977\) −22.8323 −0.730471 −0.365235 0.930915i \(-0.619012\pi\)
−0.365235 + 0.930915i \(0.619012\pi\)
\(978\) 0 0
\(979\) −51.2608 −1.63830
\(980\) 0 0
\(981\) 8.08402 0.258103
\(982\) 0 0
\(983\) −46.3557 −1.47852 −0.739258 0.673422i \(-0.764824\pi\)
−0.739258 + 0.673422i \(0.764824\pi\)
\(984\) 0 0
\(985\) −5.01686 −0.159850
\(986\) 0 0
\(987\) −12.8910 −0.410324
\(988\) 0 0
\(989\) 6.44158 0.204830
\(990\) 0 0
\(991\) 3.43683 0.109175 0.0545873 0.998509i \(-0.482616\pi\)
0.0545873 + 0.998509i \(0.482616\pi\)
\(992\) 0 0
\(993\) −20.7784 −0.659383
\(994\) 0 0
\(995\) −1.74294 −0.0552550
\(996\) 0 0
\(997\) −31.0320 −0.982794 −0.491397 0.870936i \(-0.663514\pi\)
−0.491397 + 0.870936i \(0.663514\pi\)
\(998\) 0 0
\(999\) 6.20285 0.196249
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 3072.2.a.t.1.2 4
3.2 odd 2 9216.2.a.y.1.3 4
4.3 odd 2 3072.2.a.n.1.2 4
8.3 odd 2 3072.2.a.o.1.3 4
8.5 even 2 3072.2.a.i.1.3 4
12.11 even 2 9216.2.a.x.1.3 4
16.3 odd 4 3072.2.d.i.1537.3 8
16.5 even 4 3072.2.d.f.1537.2 8
16.11 odd 4 3072.2.d.i.1537.6 8
16.13 even 4 3072.2.d.f.1537.7 8
24.5 odd 2 9216.2.a.bo.1.2 4
24.11 even 2 9216.2.a.bn.1.2 4
32.3 odd 8 384.2.j.a.289.2 8
32.5 even 8 48.2.j.a.37.4 yes 8
32.11 odd 8 384.2.j.a.97.2 8
32.13 even 8 48.2.j.a.13.4 8
32.19 odd 8 192.2.j.a.145.3 8
32.21 even 8 384.2.j.b.97.4 8
32.27 odd 8 192.2.j.a.49.3 8
32.29 even 8 384.2.j.b.289.4 8
96.5 odd 8 144.2.k.b.37.1 8
96.11 even 8 1152.2.k.f.865.2 8
96.29 odd 8 1152.2.k.c.289.2 8
96.35 even 8 1152.2.k.f.289.2 8
96.53 odd 8 1152.2.k.c.865.2 8
96.59 even 8 576.2.k.b.433.3 8
96.77 odd 8 144.2.k.b.109.1 8
96.83 even 8 576.2.k.b.145.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.4 8 32.13 even 8
48.2.j.a.37.4 yes 8 32.5 even 8
144.2.k.b.37.1 8 96.5 odd 8
144.2.k.b.109.1 8 96.77 odd 8
192.2.j.a.49.3 8 32.27 odd 8
192.2.j.a.145.3 8 32.19 odd 8
384.2.j.a.97.2 8 32.11 odd 8
384.2.j.a.289.2 8 32.3 odd 8
384.2.j.b.97.4 8 32.21 even 8
384.2.j.b.289.4 8 32.29 even 8
576.2.k.b.145.3 8 96.83 even 8
576.2.k.b.433.3 8 96.59 even 8
1152.2.k.c.289.2 8 96.29 odd 8
1152.2.k.c.865.2 8 96.53 odd 8
1152.2.k.f.289.2 8 96.35 even 8
1152.2.k.f.865.2 8 96.11 even 8
3072.2.a.i.1.3 4 8.5 even 2
3072.2.a.n.1.2 4 4.3 odd 2
3072.2.a.o.1.3 4 8.3 odd 2
3072.2.a.t.1.2 4 1.1 even 1 trivial
3072.2.d.f.1537.2 8 16.5 even 4
3072.2.d.f.1537.7 8 16.13 even 4
3072.2.d.i.1537.3 8 16.3 odd 4
3072.2.d.i.1537.6 8 16.11 odd 4
9216.2.a.x.1.3 4 12.11 even 2
9216.2.a.y.1.3 4 3.2 odd 2
9216.2.a.bn.1.2 4 24.11 even 2
9216.2.a.bo.1.2 4 24.5 odd 2