Properties

Label 3072.2.a.o.1.3
Level $3072$
Weight $2$
Character 3072.1
Self dual yes
Analytic conductor $24.530$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
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 1.3
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 \(-0.466226\pi\)
0.105906 + 0.994376i \(0.466226\pi\)
\(6\) 0 0
\(7\) −4.55765 −1.72263 −0.861314 0.508072i \(-0.830358\pi\)
−0.861314 + 0.508072i \(0.830358\pi\)
\(8\) 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 \(-0.496291\pi\)
0.0116517 + 0.999932i \(0.496291\pi\)
\(14\) 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 \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 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 \(-0.737144\pi\)
−0.677979 + 0.735082i \(0.737144\pi\)
\(30\) 0 0
\(31\) 0.557647 0.100156 0.0500782 0.998745i \(-0.484053\pi\)
0.0500782 + 0.998745i \(0.484053\pi\)
\(32\) 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 \(-0.670307\pi\)
−0.509871 + 0.860251i \(0.670307\pi\)
\(38\) 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 \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 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 \(-0.515264\pi\)
−0.0479352 + 0.998850i \(0.515264\pi\)
\(54\) 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 \(-0.420525\pi\)
0.247092 + 0.968992i \(0.420525\pi\)
\(62\) 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 \(-0.318086\pi\)
0.540893 + 0.841091i \(0.318086\pi\)
\(72\) 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 \(-0.712237\pi\)
−0.618445 + 0.785828i \(0.712237\pi\)
\(80\) 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 \(-0.489836\pi\)
0.0319242 + 0.999490i \(0.489836\pi\)
\(102\) 0 0
\(103\) 1.33686 0.131724 0.0658622 0.997829i \(-0.479020\pi\)
0.0658622 + 0.997829i \(0.479020\pi\)
\(104\) 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 \(-0.626542\pi\)
−0.387154 + 0.922015i \(0.626542\pi\)
\(110\) 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 \(-0.572644\pi\)
−0.226241 + 0.974071i \(0.572644\pi\)
\(128\) 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 \(-0.696482\pi\)
−0.578807 + 0.815465i \(0.696482\pi\)
\(150\) 0 0
\(151\) −9.97685 −0.811905 −0.405952 0.913894i \(-0.633060\pi\)
−0.405952 + 0.913894i \(0.633060\pi\)
\(152\) 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 \(-0.865509\pi\)
−0.912060 + 0.410056i \(0.865509\pi\)
\(158\) 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 \(-0.572528\pi\)
−0.225888 + 0.974153i \(0.572528\pi\)
\(168\) 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 \(-0.437563\pi\)
0.194897 + 0.980824i \(0.437563\pi\)
\(174\) 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 \(-0.692783\pi\)
−0.569294 + 0.822134i \(0.692783\pi\)
\(182\) 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 \(-0.398926\pi\)
0.312223 + 0.950009i \(0.398926\pi\)
\(192\) 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 \(-0.623161\pi\)
−0.377340 + 0.926075i \(0.623161\pi\)
\(198\) 0 0
\(199\) −3.68000 −0.260868 −0.130434 0.991457i \(-0.541637\pi\)
−0.130434 + 0.991457i \(0.541637\pi\)
\(200\) 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 \(-0.447955\pi\)
0.162777 + 0.986663i \(0.447955\pi\)
\(224\) 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 \(-0.108433\pi\)
0.942537 + 0.334101i \(0.108433\pi\)
\(230\) 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 \(-0.101424\pi\)
0.949665 + 0.313267i \(0.101424\pi\)
\(240\) 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 \(-0.365285\pi\)
0.410697 + 0.911772i \(0.365285\pi\)
\(264\) 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 \(-0.332160\pi\)
0.503188 + 0.864177i \(0.332160\pi\)
\(270\) 0 0
\(271\) −21.9769 −1.33500 −0.667499 0.744610i \(-0.732635\pi\)
−0.667499 + 0.744610i \(0.732635\pi\)
\(272\) 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 \(-0.654031\pi\)
−0.465237 + 0.885186i \(0.654031\pi\)
\(278\) 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 \(-0.851547\pi\)
−0.893203 + 0.449654i \(0.851547\pi\)
\(294\) 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 \(-0.224370\pi\)
0.761689 + 0.647943i \(0.224370\pi\)
\(312\) 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 \(-0.177983\pi\)
0.847706 + 0.530466i \(0.177983\pi\)
\(318\) 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 \(-0.225620\pi\)
0.759140 + 0.650927i \(0.225620\pi\)
\(350\) 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 \(-0.155214\pi\)
0.883452 + 0.468522i \(0.155214\pi\)
\(360\) 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 \(-0.505836\pi\)
−0.0183319 + 0.999832i \(0.505836\pi\)
\(368\) 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 \(-0.255773\pi\)
0.694168 + 0.719813i \(0.255773\pi\)
\(374\) 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 \(-0.725729\pi\)
−0.651188 + 0.758916i \(0.725729\pi\)
\(384\) 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 \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(390\) 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 \(-0.395965\pi\)
0.321047 + 0.947063i \(0.395965\pi\)
\(398\) 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 \(-0.526308\pi\)
−0.0825543 + 0.996587i \(0.526308\pi\)
\(422\) 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 \(-0.533984\pi\)
−0.106561 + 0.994306i \(0.533984\pi\)
\(432\) 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 \(-0.249761\pi\)
0.707637 + 0.706576i \(0.249761\pi\)
\(440\) 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 \(-0.659923\pi\)
−0.481542 + 0.876423i \(0.659923\pi\)
\(462\) 0 0
\(463\) −18.6435 −0.866437 −0.433219 0.901289i \(-0.642622\pi\)
−0.433219 + 0.901289i \(0.642622\pi\)
\(464\) 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 \(-0.492083\pi\)
0.0248706 + 0.999691i \(0.492083\pi\)
\(480\) 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 \(-0.795418\pi\)
−0.800473 + 0.599369i \(0.795418\pi\)
\(488\) 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 \(-0.263956\pi\)
0.675435 + 0.737420i \(0.263956\pi\)
\(504\) 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 \(-0.392385\pi\)
0.331677 + 0.943393i \(0.392385\pi\)
\(510\) 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 \(-0.312478\pi\)
0.555627 + 0.831432i \(0.312478\pi\)
\(542\) 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 \(-0.190891\pi\)
0.825504 + 0.564397i \(0.190891\pi\)
\(558\) 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 \(-0.458055\pi\)
0.131394 + 0.991330i \(0.458055\pi\)
\(600\) 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 \(-0.290808\pi\)
0.610900 + 0.791708i \(0.290808\pi\)
\(608\) 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 \(-0.522791\pi\)
−0.0715393 + 0.997438i \(0.522791\pi\)
\(614\) 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 \(-0.428534\pi\)
0.222635 + 0.974902i \(0.428534\pi\)
\(632\) 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 \(-0.542234\pi\)
−0.132292 + 0.991211i \(0.542234\pi\)
\(648\) 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 \(-0.757810\pi\)
−0.724243 + 0.689545i \(0.757810\pi\)
\(654\) 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 \(-0.393642\pi\)
0.327952 + 0.944694i \(0.393642\pi\)
\(662\) 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 \(-0.503871\pi\)
−0.0121603 + 0.999926i \(0.503871\pi\)
\(678\) 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 \(-0.590003\pi\)
−0.279000 + 0.960291i \(0.590003\pi\)
\(702\) 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 \(-0.640545\pi\)
−0.427327 + 0.904097i \(0.640545\pi\)
\(710\) 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 \(-0.696156\pi\)
−0.577972 + 0.816057i \(0.696156\pi\)
\(720\) 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 \(-0.775967\pi\)
−0.762375 + 0.647135i \(0.775967\pi\)
\(728\) 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 \(-0.498786\pi\)
0.00381477 + 0.999993i \(0.498786\pi\)
\(734\) 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 \(-0.233297\pi\)
0.743221 + 0.669046i \(0.233297\pi\)
\(744\) 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 \(-0.573743\pi\)
−0.229604 + 0.973284i \(0.573743\pi\)
\(752\) 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 \(-0.561733\pi\)
−0.192726 + 0.981253i \(0.561733\pi\)
\(758\) 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 \(-0.302343\pi\)
0.581814 + 0.813322i \(0.302343\pi\)
\(774\) 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 \(-0.359862\pi\)
0.426172 + 0.904642i \(0.359862\pi\)
\(798\) 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 \(-0.341498\pi\)
0.477625 + 0.878564i \(0.341498\pi\)
\(822\) 0 0
\(823\) 28.8560 1.00586 0.502929 0.864328i \(-0.332256\pi\)
0.502929 + 0.864328i \(0.332256\pi\)
\(824\) 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 \(-0.376715\pi\)
0.377700 + 0.925928i \(0.376715\pi\)
\(830\) 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 \(-0.221555\pi\)
0.767390 + 0.641181i \(0.221555\pi\)
\(840\) 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 \(-0.591524\pi\)
−0.283585 + 0.958947i \(0.591524\pi\)
\(854\) 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 \(-0.519762\pi\)
−0.0620443 + 0.998073i \(0.519762\pi\)
\(864\) 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 \(-0.905698\pi\)
−0.956435 + 0.291945i \(0.905698\pi\)
\(878\) 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 \(-0.355517\pi\)
0.438479 + 0.898742i \(0.355517\pi\)
\(888\) 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 \(-0.613399\pi\)
−0.348767 + 0.937210i \(0.613399\pi\)
\(912\) 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 \(-0.595309\pi\)
−0.294968 + 0.955507i \(0.595309\pi\)
\(920\) 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 \(-0.470911\pi\)
0.0912595 + 0.995827i \(0.470911\pi\)
\(942\) 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 \(-0.335342\pi\)
0.494524 + 0.869164i \(0.335342\pi\)
\(968\) 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 \(-0.235176\pi\)
0.739258 + 0.673422i \(0.235176\pi\)
\(984\) 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 \(-0.517384\pi\)
−0.0545873 + 0.998509i \(0.517384\pi\)
\(992\) 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 \(-0.336486\pi\)
0.491397 + 0.870936i \(0.336486\pi\)
\(998\) 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.o.1.3 4
3.2 odd 2 9216.2.a.bn.1.2 4
4.3 odd 2 3072.2.a.i.1.3 4
8.3 odd 2 3072.2.a.t.1.2 4
8.5 even 2 3072.2.a.n.1.2 4
12.11 even 2 9216.2.a.bo.1.2 4
16.3 odd 4 3072.2.d.f.1537.2 8
16.5 even 4 3072.2.d.i.1537.3 8
16.11 odd 4 3072.2.d.f.1537.7 8
16.13 even 4 3072.2.d.i.1537.6 8
24.5 odd 2 9216.2.a.x.1.3 4
24.11 even 2 9216.2.a.y.1.3 4
32.3 odd 8 48.2.j.a.13.4 8
32.5 even 8 384.2.j.a.97.2 8
32.11 odd 8 48.2.j.a.37.4 yes 8
32.13 even 8 384.2.j.a.289.2 8
32.19 odd 8 384.2.j.b.289.4 8
32.21 even 8 192.2.j.a.49.3 8
32.27 odd 8 384.2.j.b.97.4 8
32.29 even 8 192.2.j.a.145.3 8
96.5 odd 8 1152.2.k.f.865.2 8
96.11 even 8 144.2.k.b.37.1 8
96.29 odd 8 576.2.k.b.145.3 8
96.35 even 8 144.2.k.b.109.1 8
96.53 odd 8 576.2.k.b.433.3 8
96.59 even 8 1152.2.k.c.865.2 8
96.77 odd 8 1152.2.k.f.289.2 8
96.83 even 8 1152.2.k.c.289.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.4 8 32.3 odd 8
48.2.j.a.37.4 yes 8 32.11 odd 8
144.2.k.b.37.1 8 96.11 even 8
144.2.k.b.109.1 8 96.35 even 8
192.2.j.a.49.3 8 32.21 even 8
192.2.j.a.145.3 8 32.29 even 8
384.2.j.a.97.2 8 32.5 even 8
384.2.j.a.289.2 8 32.13 even 8
384.2.j.b.97.4 8 32.27 odd 8
384.2.j.b.289.4 8 32.19 odd 8
576.2.k.b.145.3 8 96.29 odd 8
576.2.k.b.433.3 8 96.53 odd 8
1152.2.k.c.289.2 8 96.83 even 8
1152.2.k.c.865.2 8 96.59 even 8
1152.2.k.f.289.2 8 96.77 odd 8
1152.2.k.f.865.2 8 96.5 odd 8
3072.2.a.i.1.3 4 4.3 odd 2
3072.2.a.n.1.2 4 8.5 even 2
3072.2.a.o.1.3 4 1.1 even 1 trivial
3072.2.a.t.1.2 4 8.3 odd 2
3072.2.d.f.1537.2 8 16.3 odd 4
3072.2.d.f.1537.7 8 16.11 odd 4
3072.2.d.i.1537.3 8 16.5 even 4
3072.2.d.i.1537.6 8 16.13 even 4
9216.2.a.x.1.3 4 24.5 odd 2
9216.2.a.y.1.3 4 24.11 even 2
9216.2.a.bn.1.2 4 3.2 odd 2
9216.2.a.bo.1.2 4 12.11 even 2