Properties

Label 3120.2.a.bg.1.2
Level $3120$
Weight $2$
Character 3120.1
Self dual yes
Analytic conductor $24.913$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3120.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +1.43845 q^{11} +1.00000 q^{13} +1.00000 q^{15} +5.68466 q^{17} +5.12311 q^{19} +2.56155 q^{21} +1.43845 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -1.12311 q^{31} +1.43845 q^{33} +2.56155 q^{35} -10.8078 q^{37} +1.00000 q^{39} -9.68466 q^{41} -6.24621 q^{43} +1.00000 q^{45} +1.12311 q^{47} -0.438447 q^{49} +5.68466 q^{51} -0.561553 q^{53} +1.43845 q^{55} +5.12311 q^{57} +8.00000 q^{59} +1.68466 q^{61} +2.56155 q^{63} +1.00000 q^{65} -2.24621 q^{67} +1.43845 q^{69} +7.68466 q^{71} -0.246211 q^{73} +1.00000 q^{75} +3.68466 q^{77} +8.80776 q^{79} +1.00000 q^{81} +8.00000 q^{83} +5.68466 q^{85} -2.00000 q^{87} -2.31534 q^{89} +2.56155 q^{91} -1.12311 q^{93} +5.12311 q^{95} +2.80776 q^{97} +1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} + 2 q^{15} - q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 7 q^{33} + q^{35} - q^{37} + 2 q^{39} - 7 q^{41} + 4 q^{43} + 2 q^{45} - 6 q^{47} - 5 q^{49} - q^{51} + 3 q^{53} + 7 q^{55} + 2 q^{57} + 16 q^{59} - 9 q^{61} + q^{63} + 2 q^{65} + 12 q^{67} + 7 q^{69} + 3 q^{71} + 16 q^{73} + 2 q^{75} - 5 q^{77} - 3 q^{79} + 2 q^{81} + 16 q^{83} - q^{85} - 4 q^{87} - 17 q^{89} + q^{91} + 6 q^{93} + 2 q^{95} - 15 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.56155 0.968176 0.484088 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.43845 0.433708 0.216854 0.976204i \(-0.430420\pi\)
0.216854 + 0.976204i \(0.430420\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 5.68466 1.37873 0.689366 0.724413i \(-0.257889\pi\)
0.689366 + 0.724413i \(0.257889\pi\)
\(18\) 0 0
\(19\) 5.12311 1.17532 0.587661 0.809108i \(-0.300049\pi\)
0.587661 + 0.809108i \(0.300049\pi\)
\(20\) 0 0
\(21\) 2.56155 0.558977
\(22\) 0 0
\(23\) 1.43845 0.299937 0.149968 0.988691i \(-0.452083\pi\)
0.149968 + 0.988691i \(0.452083\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −1.12311 −0.201716 −0.100858 0.994901i \(-0.532159\pi\)
−0.100858 + 0.994901i \(0.532159\pi\)
\(32\) 0 0
\(33\) 1.43845 0.250402
\(34\) 0 0
\(35\) 2.56155 0.432981
\(36\) 0 0
\(37\) −10.8078 −1.77679 −0.888393 0.459084i \(-0.848178\pi\)
−0.888393 + 0.459084i \(0.848178\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.68466 −1.51249 −0.756245 0.654289i \(-0.772968\pi\)
−0.756245 + 0.654289i \(0.772968\pi\)
\(42\) 0 0
\(43\) −6.24621 −0.952538 −0.476269 0.879300i \(-0.658011\pi\)
−0.476269 + 0.879300i \(0.658011\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.12311 0.163822 0.0819109 0.996640i \(-0.473898\pi\)
0.0819109 + 0.996640i \(0.473898\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 5.68466 0.796011
\(52\) 0 0
\(53\) −0.561553 −0.0771352 −0.0385676 0.999256i \(-0.512279\pi\)
−0.0385676 + 0.999256i \(0.512279\pi\)
\(54\) 0 0
\(55\) 1.43845 0.193960
\(56\) 0 0
\(57\) 5.12311 0.678572
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 1.68466 0.215698 0.107849 0.994167i \(-0.465604\pi\)
0.107849 + 0.994167i \(0.465604\pi\)
\(62\) 0 0
\(63\) 2.56155 0.322725
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −2.24621 −0.274418 −0.137209 0.990542i \(-0.543813\pi\)
−0.137209 + 0.990542i \(0.543813\pi\)
\(68\) 0 0
\(69\) 1.43845 0.173169
\(70\) 0 0
\(71\) 7.68466 0.912001 0.456001 0.889979i \(-0.349281\pi\)
0.456001 + 0.889979i \(0.349281\pi\)
\(72\) 0 0
\(73\) −0.246211 −0.0288168 −0.0144084 0.999896i \(-0.504587\pi\)
−0.0144084 + 0.999896i \(0.504587\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 3.68466 0.419906
\(78\) 0 0
\(79\) 8.80776 0.990951 0.495475 0.868622i \(-0.334994\pi\)
0.495475 + 0.868622i \(0.334994\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 5.68466 0.616588
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −2.31534 −0.245426 −0.122713 0.992442i \(-0.539159\pi\)
−0.122713 + 0.992442i \(0.539159\pi\)
\(90\) 0 0
\(91\) 2.56155 0.268524
\(92\) 0 0
\(93\) −1.12311 −0.116461
\(94\) 0 0
\(95\) 5.12311 0.525620
\(96\) 0 0
\(97\) 2.80776 0.285085 0.142543 0.989789i \(-0.454472\pi\)
0.142543 + 0.989789i \(0.454472\pi\)
\(98\) 0 0
\(99\) 1.43845 0.144569
\(100\) 0 0
\(101\) −7.12311 −0.708776 −0.354388 0.935099i \(-0.615311\pi\)
−0.354388 + 0.935099i \(0.615311\pi\)
\(102\) 0 0
\(103\) 2.24621 0.221326 0.110663 0.993858i \(-0.464703\pi\)
0.110663 + 0.993858i \(0.464703\pi\)
\(104\) 0 0
\(105\) 2.56155 0.249982
\(106\) 0 0
\(107\) −4.80776 −0.464784 −0.232392 0.972622i \(-0.574655\pi\)
−0.232392 + 0.972622i \(0.574655\pi\)
\(108\) 0 0
\(109\) −4.87689 −0.467122 −0.233561 0.972342i \(-0.575038\pi\)
−0.233561 + 0.972342i \(0.575038\pi\)
\(110\) 0 0
\(111\) −10.8078 −1.02583
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 1.43845 0.134136
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 14.5616 1.33486
\(120\) 0 0
\(121\) −8.93087 −0.811897
\(122\) 0 0
\(123\) −9.68466 −0.873236
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.36932 −0.653921 −0.326961 0.945038i \(-0.606024\pi\)
−0.326961 + 0.945038i \(0.606024\pi\)
\(128\) 0 0
\(129\) −6.24621 −0.549948
\(130\) 0 0
\(131\) 6.87689 0.600837 0.300419 0.953807i \(-0.402874\pi\)
0.300419 + 0.953807i \(0.402874\pi\)
\(132\) 0 0
\(133\) 13.1231 1.13792
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −0.246211 −0.0210352 −0.0105176 0.999945i \(-0.503348\pi\)
−0.0105176 + 0.999945i \(0.503348\pi\)
\(138\) 0 0
\(139\) 5.43845 0.461283 0.230642 0.973039i \(-0.425918\pi\)
0.230642 + 0.973039i \(0.425918\pi\)
\(140\) 0 0
\(141\) 1.12311 0.0945826
\(142\) 0 0
\(143\) 1.43845 0.120289
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) −0.438447 −0.0361625
\(148\) 0 0
\(149\) −16.5616 −1.35678 −0.678388 0.734704i \(-0.737321\pi\)
−0.678388 + 0.734704i \(0.737321\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 5.68466 0.459577
\(154\) 0 0
\(155\) −1.12311 −0.0902100
\(156\) 0 0
\(157\) −22.4924 −1.79509 −0.897545 0.440922i \(-0.854651\pi\)
−0.897545 + 0.440922i \(0.854651\pi\)
\(158\) 0 0
\(159\) −0.561553 −0.0445340
\(160\) 0 0
\(161\) 3.68466 0.290392
\(162\) 0 0
\(163\) −9.43845 −0.739276 −0.369638 0.929176i \(-0.620518\pi\)
−0.369638 + 0.929176i \(0.620518\pi\)
\(164\) 0 0
\(165\) 1.43845 0.111983
\(166\) 0 0
\(167\) 22.2462 1.72146 0.860732 0.509059i \(-0.170006\pi\)
0.860732 + 0.509059i \(0.170006\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.12311 0.391774
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 2.56155 0.193635
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) 14.8769 1.11195 0.555976 0.831199i \(-0.312345\pi\)
0.555976 + 0.831199i \(0.312345\pi\)
\(180\) 0 0
\(181\) 19.9309 1.48145 0.740725 0.671808i \(-0.234482\pi\)
0.740725 + 0.671808i \(0.234482\pi\)
\(182\) 0 0
\(183\) 1.68466 0.124534
\(184\) 0 0
\(185\) −10.8078 −0.794603
\(186\) 0 0
\(187\) 8.17708 0.597967
\(188\) 0 0
\(189\) 2.56155 0.186326
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 21.6847 1.56090 0.780448 0.625221i \(-0.214991\pi\)
0.780448 + 0.625221i \(0.214991\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −17.3693 −1.23751 −0.618756 0.785583i \(-0.712363\pi\)
−0.618756 + 0.785583i \(0.712363\pi\)
\(198\) 0 0
\(199\) −18.2462 −1.29344 −0.646720 0.762728i \(-0.723860\pi\)
−0.646720 + 0.762728i \(0.723860\pi\)
\(200\) 0 0
\(201\) −2.24621 −0.158436
\(202\) 0 0
\(203\) −5.12311 −0.359572
\(204\) 0 0
\(205\) −9.68466 −0.676406
\(206\) 0 0
\(207\) 1.43845 0.0999790
\(208\) 0 0
\(209\) 7.36932 0.509746
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 7.68466 0.526544
\(214\) 0 0
\(215\) −6.24621 −0.425988
\(216\) 0 0
\(217\) −2.87689 −0.195296
\(218\) 0 0
\(219\) −0.246211 −0.0166374
\(220\) 0 0
\(221\) 5.68466 0.382392
\(222\) 0 0
\(223\) −14.2462 −0.953997 −0.476998 0.878904i \(-0.658275\pi\)
−0.476998 + 0.878904i \(0.658275\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −0.630683 −0.0418599 −0.0209300 0.999781i \(-0.506663\pi\)
−0.0209300 + 0.999781i \(0.506663\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 3.68466 0.242433
\(232\) 0 0
\(233\) 6.31534 0.413732 0.206866 0.978369i \(-0.433674\pi\)
0.206866 + 0.978369i \(0.433674\pi\)
\(234\) 0 0
\(235\) 1.12311 0.0732633
\(236\) 0 0
\(237\) 8.80776 0.572126
\(238\) 0 0
\(239\) 15.6847 1.01456 0.507278 0.861782i \(-0.330652\pi\)
0.507278 + 0.861782i \(0.330652\pi\)
\(240\) 0 0
\(241\) 28.2462 1.81950 0.909749 0.415158i \(-0.136274\pi\)
0.909749 + 0.415158i \(0.136274\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.438447 −0.0280114
\(246\) 0 0
\(247\) 5.12311 0.325975
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 24.4924 1.54595 0.772974 0.634438i \(-0.218768\pi\)
0.772974 + 0.634438i \(0.218768\pi\)
\(252\) 0 0
\(253\) 2.06913 0.130085
\(254\) 0 0
\(255\) 5.68466 0.355987
\(256\) 0 0
\(257\) −18.4924 −1.15353 −0.576763 0.816912i \(-0.695684\pi\)
−0.576763 + 0.816912i \(0.695684\pi\)
\(258\) 0 0
\(259\) −27.6847 −1.72024
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 4.49242 0.277015 0.138507 0.990361i \(-0.455770\pi\)
0.138507 + 0.990361i \(0.455770\pi\)
\(264\) 0 0
\(265\) −0.561553 −0.0344959
\(266\) 0 0
\(267\) −2.31534 −0.141697
\(268\) 0 0
\(269\) −4.87689 −0.297349 −0.148675 0.988886i \(-0.547501\pi\)
−0.148675 + 0.988886i \(0.547501\pi\)
\(270\) 0 0
\(271\) −19.3693 −1.17660 −0.588301 0.808642i \(-0.700203\pi\)
−0.588301 + 0.808642i \(0.700203\pi\)
\(272\) 0 0
\(273\) 2.56155 0.155032
\(274\) 0 0
\(275\) 1.43845 0.0867416
\(276\) 0 0
\(277\) 10.4924 0.630429 0.315214 0.949021i \(-0.397924\pi\)
0.315214 + 0.949021i \(0.397924\pi\)
\(278\) 0 0
\(279\) −1.12311 −0.0672386
\(280\) 0 0
\(281\) 20.2462 1.20779 0.603894 0.797065i \(-0.293615\pi\)
0.603894 + 0.797065i \(0.293615\pi\)
\(282\) 0 0
\(283\) 18.7386 1.11390 0.556948 0.830547i \(-0.311972\pi\)
0.556948 + 0.830547i \(0.311972\pi\)
\(284\) 0 0
\(285\) 5.12311 0.303467
\(286\) 0 0
\(287\) −24.8078 −1.46436
\(288\) 0 0
\(289\) 15.3153 0.900902
\(290\) 0 0
\(291\) 2.80776 0.164594
\(292\) 0 0
\(293\) 23.6155 1.37963 0.689817 0.723984i \(-0.257691\pi\)
0.689817 + 0.723984i \(0.257691\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 1.43845 0.0834672
\(298\) 0 0
\(299\) 1.43845 0.0831875
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) −7.12311 −0.409212
\(304\) 0 0
\(305\) 1.68466 0.0964633
\(306\) 0 0
\(307\) −9.43845 −0.538681 −0.269340 0.963045i \(-0.586806\pi\)
−0.269340 + 0.963045i \(0.586806\pi\)
\(308\) 0 0
\(309\) 2.24621 0.127782
\(310\) 0 0
\(311\) −22.7386 −1.28939 −0.644695 0.764440i \(-0.723016\pi\)
−0.644695 + 0.764440i \(0.723016\pi\)
\(312\) 0 0
\(313\) 24.7386 1.39831 0.699155 0.714970i \(-0.253560\pi\)
0.699155 + 0.714970i \(0.253560\pi\)
\(314\) 0 0
\(315\) 2.56155 0.144327
\(316\) 0 0
\(317\) 11.1231 0.624736 0.312368 0.949961i \(-0.398878\pi\)
0.312368 + 0.949961i \(0.398878\pi\)
\(318\) 0 0
\(319\) −2.87689 −0.161075
\(320\) 0 0
\(321\) −4.80776 −0.268343
\(322\) 0 0
\(323\) 29.1231 1.62045
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −4.87689 −0.269693
\(328\) 0 0
\(329\) 2.87689 0.158608
\(330\) 0 0
\(331\) −26.2462 −1.44262 −0.721311 0.692611i \(-0.756460\pi\)
−0.721311 + 0.692611i \(0.756460\pi\)
\(332\) 0 0
\(333\) −10.8078 −0.592262
\(334\) 0 0
\(335\) −2.24621 −0.122724
\(336\) 0 0
\(337\) −11.1231 −0.605914 −0.302957 0.953004i \(-0.597974\pi\)
−0.302957 + 0.953004i \(0.597974\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −1.61553 −0.0874858
\(342\) 0 0
\(343\) −19.0540 −1.02882
\(344\) 0 0
\(345\) 1.43845 0.0774434
\(346\) 0 0
\(347\) −31.0540 −1.66706 −0.833532 0.552471i \(-0.813685\pi\)
−0.833532 + 0.552471i \(0.813685\pi\)
\(348\) 0 0
\(349\) −25.3693 −1.35799 −0.678994 0.734144i \(-0.737584\pi\)
−0.678994 + 0.734144i \(0.737584\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 12.2462 0.651800 0.325900 0.945404i \(-0.394333\pi\)
0.325900 + 0.945404i \(0.394333\pi\)
\(354\) 0 0
\(355\) 7.68466 0.407859
\(356\) 0 0
\(357\) 14.5616 0.770679
\(358\) 0 0
\(359\) −14.2462 −0.751886 −0.375943 0.926643i \(-0.622681\pi\)
−0.375943 + 0.926643i \(0.622681\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) 0 0
\(363\) −8.93087 −0.468749
\(364\) 0 0
\(365\) −0.246211 −0.0128873
\(366\) 0 0
\(367\) 2.24621 0.117251 0.0586256 0.998280i \(-0.481328\pi\)
0.0586256 + 0.998280i \(0.481328\pi\)
\(368\) 0 0
\(369\) −9.68466 −0.504163
\(370\) 0 0
\(371\) −1.43845 −0.0746805
\(372\) 0 0
\(373\) −23.1231 −1.19727 −0.598635 0.801022i \(-0.704290\pi\)
−0.598635 + 0.801022i \(0.704290\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −2.87689 −0.147776 −0.0738881 0.997267i \(-0.523541\pi\)
−0.0738881 + 0.997267i \(0.523541\pi\)
\(380\) 0 0
\(381\) −7.36932 −0.377542
\(382\) 0 0
\(383\) −27.3693 −1.39851 −0.699253 0.714874i \(-0.746484\pi\)
−0.699253 + 0.714874i \(0.746484\pi\)
\(384\) 0 0
\(385\) 3.68466 0.187788
\(386\) 0 0
\(387\) −6.24621 −0.317513
\(388\) 0 0
\(389\) −21.8617 −1.10843 −0.554217 0.832372i \(-0.686982\pi\)
−0.554217 + 0.832372i \(0.686982\pi\)
\(390\) 0 0
\(391\) 8.17708 0.413533
\(392\) 0 0
\(393\) 6.87689 0.346893
\(394\) 0 0
\(395\) 8.80776 0.443167
\(396\) 0 0
\(397\) 13.1922 0.662099 0.331050 0.943613i \(-0.392597\pi\)
0.331050 + 0.943613i \(0.392597\pi\)
\(398\) 0 0
\(399\) 13.1231 0.656977
\(400\) 0 0
\(401\) 12.2462 0.611547 0.305773 0.952104i \(-0.401085\pi\)
0.305773 + 0.952104i \(0.401085\pi\)
\(402\) 0 0
\(403\) −1.12311 −0.0559459
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −15.5464 −0.770606
\(408\) 0 0
\(409\) 14.4924 0.716604 0.358302 0.933606i \(-0.383356\pi\)
0.358302 + 0.933606i \(0.383356\pi\)
\(410\) 0 0
\(411\) −0.246211 −0.0121447
\(412\) 0 0
\(413\) 20.4924 1.00837
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 5.43845 0.266322
\(418\) 0 0
\(419\) −9.12311 −0.445693 −0.222846 0.974854i \(-0.571535\pi\)
−0.222846 + 0.974854i \(0.571535\pi\)
\(420\) 0 0
\(421\) −37.8617 −1.84527 −0.922634 0.385676i \(-0.873968\pi\)
−0.922634 + 0.385676i \(0.873968\pi\)
\(422\) 0 0
\(423\) 1.12311 0.0546073
\(424\) 0 0
\(425\) 5.68466 0.275746
\(426\) 0 0
\(427\) 4.31534 0.208834
\(428\) 0 0
\(429\) 1.43845 0.0694489
\(430\) 0 0
\(431\) −30.2462 −1.45691 −0.728454 0.685094i \(-0.759761\pi\)
−0.728454 + 0.685094i \(0.759761\pi\)
\(432\) 0 0
\(433\) 10.6307 0.510878 0.255439 0.966825i \(-0.417780\pi\)
0.255439 + 0.966825i \(0.417780\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) 7.36932 0.352522
\(438\) 0 0
\(439\) 21.9309 1.04670 0.523352 0.852117i \(-0.324681\pi\)
0.523352 + 0.852117i \(0.324681\pi\)
\(440\) 0 0
\(441\) −0.438447 −0.0208784
\(442\) 0 0
\(443\) 15.6847 0.745201 0.372600 0.927992i \(-0.378466\pi\)
0.372600 + 0.927992i \(0.378466\pi\)
\(444\) 0 0
\(445\) −2.31534 −0.109758
\(446\) 0 0
\(447\) −16.5616 −0.783335
\(448\) 0 0
\(449\) −1.68466 −0.0795039 −0.0397520 0.999210i \(-0.512657\pi\)
−0.0397520 + 0.999210i \(0.512657\pi\)
\(450\) 0 0
\(451\) −13.9309 −0.655979
\(452\) 0 0
\(453\) −4.00000 −0.187936
\(454\) 0 0
\(455\) 2.56155 0.120087
\(456\) 0 0
\(457\) −31.4384 −1.47063 −0.735314 0.677726i \(-0.762965\pi\)
−0.735314 + 0.677726i \(0.762965\pi\)
\(458\) 0 0
\(459\) 5.68466 0.265337
\(460\) 0 0
\(461\) −11.4384 −0.532742 −0.266371 0.963871i \(-0.585825\pi\)
−0.266371 + 0.963871i \(0.585825\pi\)
\(462\) 0 0
\(463\) 10.5616 0.490837 0.245418 0.969417i \(-0.421075\pi\)
0.245418 + 0.969417i \(0.421075\pi\)
\(464\) 0 0
\(465\) −1.12311 −0.0520828
\(466\) 0 0
\(467\) 20.1771 0.933684 0.466842 0.884341i \(-0.345392\pi\)
0.466842 + 0.884341i \(0.345392\pi\)
\(468\) 0 0
\(469\) −5.75379 −0.265685
\(470\) 0 0
\(471\) −22.4924 −1.03640
\(472\) 0 0
\(473\) −8.98485 −0.413124
\(474\) 0 0
\(475\) 5.12311 0.235064
\(476\) 0 0
\(477\) −0.561553 −0.0257117
\(478\) 0 0
\(479\) 35.5464 1.62416 0.812078 0.583549i \(-0.198336\pi\)
0.812078 + 0.583549i \(0.198336\pi\)
\(480\) 0 0
\(481\) −10.8078 −0.492792
\(482\) 0 0
\(483\) 3.68466 0.167658
\(484\) 0 0
\(485\) 2.80776 0.127494
\(486\) 0 0
\(487\) 7.68466 0.348225 0.174113 0.984726i \(-0.444294\pi\)
0.174113 + 0.984726i \(0.444294\pi\)
\(488\) 0 0
\(489\) −9.43845 −0.426821
\(490\) 0 0
\(491\) 40.4924 1.82740 0.913699 0.406392i \(-0.133213\pi\)
0.913699 + 0.406392i \(0.133213\pi\)
\(492\) 0 0
\(493\) −11.3693 −0.512048
\(494\) 0 0
\(495\) 1.43845 0.0646534
\(496\) 0 0
\(497\) 19.6847 0.882978
\(498\) 0 0
\(499\) −19.8617 −0.889134 −0.444567 0.895746i \(-0.646642\pi\)
−0.444567 + 0.895746i \(0.646642\pi\)
\(500\) 0 0
\(501\) 22.2462 0.993887
\(502\) 0 0
\(503\) 30.7386 1.37057 0.685284 0.728276i \(-0.259678\pi\)
0.685284 + 0.728276i \(0.259678\pi\)
\(504\) 0 0
\(505\) −7.12311 −0.316974
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −30.3153 −1.34370 −0.671852 0.740685i \(-0.734501\pi\)
−0.671852 + 0.740685i \(0.734501\pi\)
\(510\) 0 0
\(511\) −0.630683 −0.0278998
\(512\) 0 0
\(513\) 5.12311 0.226191
\(514\) 0 0
\(515\) 2.24621 0.0989799
\(516\) 0 0
\(517\) 1.61553 0.0710508
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −30.2462 −1.32257 −0.661287 0.750133i \(-0.729990\pi\)
−0.661287 + 0.750133i \(0.729990\pi\)
\(524\) 0 0
\(525\) 2.56155 0.111795
\(526\) 0 0
\(527\) −6.38447 −0.278112
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −9.68466 −0.419489
\(534\) 0 0
\(535\) −4.80776 −0.207858
\(536\) 0 0
\(537\) 14.8769 0.641985
\(538\) 0 0
\(539\) −0.630683 −0.0271654
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 19.9309 0.855315
\(544\) 0 0
\(545\) −4.87689 −0.208903
\(546\) 0 0
\(547\) −42.7386 −1.82737 −0.913686 0.406421i \(-0.866777\pi\)
−0.913686 + 0.406421i \(0.866777\pi\)
\(548\) 0 0
\(549\) 1.68466 0.0718995
\(550\) 0 0
\(551\) −10.2462 −0.436503
\(552\) 0 0
\(553\) 22.5616 0.959415
\(554\) 0 0
\(555\) −10.8078 −0.458764
\(556\) 0 0
\(557\) −34.0000 −1.44063 −0.720313 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 0 0
\(559\) −6.24621 −0.264187
\(560\) 0 0
\(561\) 8.17708 0.345237
\(562\) 0 0
\(563\) 21.4384 0.903523 0.451761 0.892139i \(-0.350796\pi\)
0.451761 + 0.892139i \(0.350796\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 2.56155 0.107575
\(568\) 0 0
\(569\) 7.12311 0.298616 0.149308 0.988791i \(-0.452295\pi\)
0.149308 + 0.988791i \(0.452295\pi\)
\(570\) 0 0
\(571\) −21.4384 −0.897171 −0.448586 0.893740i \(-0.648072\pi\)
−0.448586 + 0.893740i \(0.648072\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 1.43845 0.0599874
\(576\) 0 0
\(577\) 33.5464 1.39655 0.698277 0.715827i \(-0.253950\pi\)
0.698277 + 0.715827i \(0.253950\pi\)
\(578\) 0 0
\(579\) 21.6847 0.901184
\(580\) 0 0
\(581\) 20.4924 0.850169
\(582\) 0 0
\(583\) −0.807764 −0.0334542
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) −7.36932 −0.304164 −0.152082 0.988368i \(-0.548598\pi\)
−0.152082 + 0.988368i \(0.548598\pi\)
\(588\) 0 0
\(589\) −5.75379 −0.237081
\(590\) 0 0
\(591\) −17.3693 −0.714478
\(592\) 0 0
\(593\) 35.6155 1.46255 0.731277 0.682080i \(-0.238925\pi\)
0.731277 + 0.682080i \(0.238925\pi\)
\(594\) 0 0
\(595\) 14.5616 0.596965
\(596\) 0 0
\(597\) −18.2462 −0.746768
\(598\) 0 0
\(599\) −30.7386 −1.25595 −0.627973 0.778235i \(-0.716115\pi\)
−0.627973 + 0.778235i \(0.716115\pi\)
\(600\) 0 0
\(601\) 0.561553 0.0229062 0.0114531 0.999934i \(-0.496354\pi\)
0.0114531 + 0.999934i \(0.496354\pi\)
\(602\) 0 0
\(603\) −2.24621 −0.0914728
\(604\) 0 0
\(605\) −8.93087 −0.363091
\(606\) 0 0
\(607\) 22.7386 0.922933 0.461466 0.887158i \(-0.347324\pi\)
0.461466 + 0.887158i \(0.347324\pi\)
\(608\) 0 0
\(609\) −5.12311 −0.207599
\(610\) 0 0
\(611\) 1.12311 0.0454360
\(612\) 0 0
\(613\) −23.9309 −0.966559 −0.483279 0.875466i \(-0.660554\pi\)
−0.483279 + 0.875466i \(0.660554\pi\)
\(614\) 0 0
\(615\) −9.68466 −0.390523
\(616\) 0 0
\(617\) −16.2462 −0.654048 −0.327024 0.945016i \(-0.606046\pi\)
−0.327024 + 0.945016i \(0.606046\pi\)
\(618\) 0 0
\(619\) −31.3693 −1.26084 −0.630420 0.776255i \(-0.717117\pi\)
−0.630420 + 0.776255i \(0.717117\pi\)
\(620\) 0 0
\(621\) 1.43845 0.0577229
\(622\) 0 0
\(623\) −5.93087 −0.237615
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.36932 0.294302
\(628\) 0 0
\(629\) −61.4384 −2.44971
\(630\) 0 0
\(631\) −9.75379 −0.388292 −0.194146 0.980973i \(-0.562194\pi\)
−0.194146 + 0.980973i \(0.562194\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) −7.36932 −0.292442
\(636\) 0 0
\(637\) −0.438447 −0.0173719
\(638\) 0 0
\(639\) 7.68466 0.304000
\(640\) 0 0
\(641\) 25.3693 1.00203 0.501014 0.865439i \(-0.332961\pi\)
0.501014 + 0.865439i \(0.332961\pi\)
\(642\) 0 0
\(643\) 8.80776 0.347344 0.173672 0.984804i \(-0.444437\pi\)
0.173672 + 0.984804i \(0.444437\pi\)
\(644\) 0 0
\(645\) −6.24621 −0.245944
\(646\) 0 0
\(647\) −11.6847 −0.459371 −0.229686 0.973265i \(-0.573770\pi\)
−0.229686 + 0.973265i \(0.573770\pi\)
\(648\) 0 0
\(649\) 11.5076 0.451712
\(650\) 0 0
\(651\) −2.87689 −0.112754
\(652\) 0 0
\(653\) 40.2462 1.57496 0.787478 0.616343i \(-0.211386\pi\)
0.787478 + 0.616343i \(0.211386\pi\)
\(654\) 0 0
\(655\) 6.87689 0.268702
\(656\) 0 0
\(657\) −0.246211 −0.00960562
\(658\) 0 0
\(659\) 24.4924 0.954089 0.477045 0.878879i \(-0.341708\pi\)
0.477045 + 0.878879i \(0.341708\pi\)
\(660\) 0 0
\(661\) −39.1231 −1.52171 −0.760856 0.648920i \(-0.775221\pi\)
−0.760856 + 0.648920i \(0.775221\pi\)
\(662\) 0 0
\(663\) 5.68466 0.220774
\(664\) 0 0
\(665\) 13.1231 0.508892
\(666\) 0 0
\(667\) −2.87689 −0.111394
\(668\) 0 0
\(669\) −14.2462 −0.550790
\(670\) 0 0
\(671\) 2.42329 0.0935502
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 35.9309 1.38094 0.690468 0.723363i \(-0.257405\pi\)
0.690468 + 0.723363i \(0.257405\pi\)
\(678\) 0 0
\(679\) 7.19224 0.276013
\(680\) 0 0
\(681\) −0.630683 −0.0241678
\(682\) 0 0
\(683\) 28.4924 1.09023 0.545116 0.838361i \(-0.316486\pi\)
0.545116 + 0.838361i \(0.316486\pi\)
\(684\) 0 0
\(685\) −0.246211 −0.00940725
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) −0.561553 −0.0213935
\(690\) 0 0
\(691\) −37.1231 −1.41223 −0.706115 0.708097i \(-0.749554\pi\)
−0.706115 + 0.708097i \(0.749554\pi\)
\(692\) 0 0
\(693\) 3.68466 0.139969
\(694\) 0 0
\(695\) 5.43845 0.206292
\(696\) 0 0
\(697\) −55.0540 −2.08532
\(698\) 0 0
\(699\) 6.31534 0.238868
\(700\) 0 0
\(701\) 5.36932 0.202796 0.101398 0.994846i \(-0.467668\pi\)
0.101398 + 0.994846i \(0.467668\pi\)
\(702\) 0 0
\(703\) −55.3693 −2.08829
\(704\) 0 0
\(705\) 1.12311 0.0422986
\(706\) 0 0
\(707\) −18.2462 −0.686219
\(708\) 0 0
\(709\) 8.87689 0.333379 0.166689 0.986009i \(-0.446692\pi\)
0.166689 + 0.986009i \(0.446692\pi\)
\(710\) 0 0
\(711\) 8.80776 0.330317
\(712\) 0 0
\(713\) −1.61553 −0.0605020
\(714\) 0 0
\(715\) 1.43845 0.0537949
\(716\) 0 0
\(717\) 15.6847 0.585754
\(718\) 0 0
\(719\) −26.2462 −0.978819 −0.489409 0.872054i \(-0.662788\pi\)
−0.489409 + 0.872054i \(0.662788\pi\)
\(720\) 0 0
\(721\) 5.75379 0.214282
\(722\) 0 0
\(723\) 28.2462 1.05049
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 47.3693 1.75683 0.878415 0.477898i \(-0.158601\pi\)
0.878415 + 0.477898i \(0.158601\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −35.5076 −1.31330
\(732\) 0 0
\(733\) 32.4233 1.19758 0.598791 0.800905i \(-0.295648\pi\)
0.598791 + 0.800905i \(0.295648\pi\)
\(734\) 0 0
\(735\) −0.438447 −0.0161724
\(736\) 0 0
\(737\) −3.23106 −0.119017
\(738\) 0 0
\(739\) 27.8617 1.02491 0.512455 0.858714i \(-0.328736\pi\)
0.512455 + 0.858714i \(0.328736\pi\)
\(740\) 0 0
\(741\) 5.12311 0.188202
\(742\) 0 0
\(743\) 17.1231 0.628186 0.314093 0.949392i \(-0.398300\pi\)
0.314093 + 0.949392i \(0.398300\pi\)
\(744\) 0 0
\(745\) −16.5616 −0.606768
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) −12.3153 −0.449993
\(750\) 0 0
\(751\) 19.6847 0.718303 0.359152 0.933279i \(-0.383066\pi\)
0.359152 + 0.933279i \(0.383066\pi\)
\(752\) 0 0
\(753\) 24.4924 0.892553
\(754\) 0 0
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) −23.1231 −0.840424 −0.420212 0.907426i \(-0.638044\pi\)
−0.420212 + 0.907426i \(0.638044\pi\)
\(758\) 0 0
\(759\) 2.06913 0.0751047
\(760\) 0 0
\(761\) −10.4924 −0.380350 −0.190175 0.981750i \(-0.560906\pi\)
−0.190175 + 0.981750i \(0.560906\pi\)
\(762\) 0 0
\(763\) −12.4924 −0.452256
\(764\) 0 0
\(765\) 5.68466 0.205529
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −15.6155 −0.563110 −0.281555 0.959545i \(-0.590850\pi\)
−0.281555 + 0.959545i \(0.590850\pi\)
\(770\) 0 0
\(771\) −18.4924 −0.665988
\(772\) 0 0
\(773\) −12.8769 −0.463150 −0.231575 0.972817i \(-0.574388\pi\)
−0.231575 + 0.972817i \(0.574388\pi\)
\(774\) 0 0
\(775\) −1.12311 −0.0403431
\(776\) 0 0
\(777\) −27.6847 −0.993182
\(778\) 0 0
\(779\) −49.6155 −1.77766
\(780\) 0 0
\(781\) 11.0540 0.395542
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) −22.4924 −0.802789
\(786\) 0 0
\(787\) −30.7386 −1.09571 −0.547857 0.836572i \(-0.684556\pi\)
−0.547857 + 0.836572i \(0.684556\pi\)
\(788\) 0 0
\(789\) 4.49242 0.159934
\(790\) 0 0
\(791\) 5.12311 0.182157
\(792\) 0 0
\(793\) 1.68466 0.0598240
\(794\) 0 0
\(795\) −0.561553 −0.0199162
\(796\) 0 0
\(797\) −20.4233 −0.723430 −0.361715 0.932289i \(-0.617809\pi\)
−0.361715 + 0.932289i \(0.617809\pi\)
\(798\) 0 0
\(799\) 6.38447 0.225866
\(800\) 0 0
\(801\) −2.31534 −0.0818086
\(802\) 0 0
\(803\) −0.354162 −0.0124981
\(804\) 0 0
\(805\) 3.68466 0.129867
\(806\) 0 0
\(807\) −4.87689 −0.171675
\(808\) 0 0
\(809\) 14.4924 0.509526 0.254763 0.967003i \(-0.418002\pi\)
0.254763 + 0.967003i \(0.418002\pi\)
\(810\) 0 0
\(811\) 24.9848 0.877337 0.438668 0.898649i \(-0.355450\pi\)
0.438668 + 0.898649i \(0.355450\pi\)
\(812\) 0 0
\(813\) −19.3693 −0.679312
\(814\) 0 0
\(815\) −9.43845 −0.330614
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 2.56155 0.0895079
\(820\) 0 0
\(821\) −45.6847 −1.59441 −0.797203 0.603712i \(-0.793688\pi\)
−0.797203 + 0.603712i \(0.793688\pi\)
\(822\) 0 0
\(823\) 44.4924 1.55091 0.775454 0.631404i \(-0.217521\pi\)
0.775454 + 0.631404i \(0.217521\pi\)
\(824\) 0 0
\(825\) 1.43845 0.0500803
\(826\) 0 0
\(827\) −47.3693 −1.64719 −0.823596 0.567177i \(-0.808036\pi\)
−0.823596 + 0.567177i \(0.808036\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 10.4924 0.363978
\(832\) 0 0
\(833\) −2.49242 −0.0863573
\(834\) 0 0
\(835\) 22.2462 0.769862
\(836\) 0 0
\(837\) −1.12311 −0.0388202
\(838\) 0 0
\(839\) 11.1922 0.386399 0.193199 0.981160i \(-0.438114\pi\)
0.193199 + 0.981160i \(0.438114\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 20.2462 0.697317
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −22.8769 −0.786059
\(848\) 0 0
\(849\) 18.7386 0.643108
\(850\) 0 0
\(851\) −15.5464 −0.532924
\(852\) 0 0
\(853\) −35.4384 −1.21339 −0.606695 0.794935i \(-0.707505\pi\)
−0.606695 + 0.794935i \(0.707505\pi\)
\(854\) 0 0
\(855\) 5.12311 0.175207
\(856\) 0 0
\(857\) 51.7926 1.76920 0.884601 0.466349i \(-0.154431\pi\)
0.884601 + 0.466349i \(0.154431\pi\)
\(858\) 0 0
\(859\) −52.8078 −1.80178 −0.900889 0.434050i \(-0.857084\pi\)
−0.900889 + 0.434050i \(0.857084\pi\)
\(860\) 0 0
\(861\) −24.8078 −0.845446
\(862\) 0 0
\(863\) 55.8617 1.90156 0.950778 0.309873i \(-0.100287\pi\)
0.950778 + 0.309873i \(0.100287\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) 15.3153 0.520136
\(868\) 0 0
\(869\) 12.6695 0.429783
\(870\) 0 0
\(871\) −2.24621 −0.0761100
\(872\) 0 0
\(873\) 2.80776 0.0950284
\(874\) 0 0
\(875\) 2.56155 0.0865963
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 23.6155 0.796532
\(880\) 0 0
\(881\) −8.24621 −0.277822 −0.138911 0.990305i \(-0.544360\pi\)
−0.138911 + 0.990305i \(0.544360\pi\)
\(882\) 0 0
\(883\) 0.492423 0.0165713 0.00828567 0.999966i \(-0.497363\pi\)
0.00828567 + 0.999966i \(0.497363\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 11.0540 0.371156 0.185578 0.982630i \(-0.440584\pi\)
0.185578 + 0.982630i \(0.440584\pi\)
\(888\) 0 0
\(889\) −18.8769 −0.633111
\(890\) 0 0
\(891\) 1.43845 0.0481898
\(892\) 0 0
\(893\) 5.75379 0.192543
\(894\) 0 0
\(895\) 14.8769 0.497280
\(896\) 0 0
\(897\) 1.43845 0.0480284
\(898\) 0 0
\(899\) 2.24621 0.0749153
\(900\) 0 0
\(901\) −3.19224 −0.106349
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 0 0
\(905\) 19.9309 0.662525
\(906\) 0 0
\(907\) 38.8769 1.29089 0.645443 0.763808i \(-0.276673\pi\)
0.645443 + 0.763808i \(0.276673\pi\)
\(908\) 0 0
\(909\) −7.12311 −0.236259
\(910\) 0 0
\(911\) −34.2462 −1.13463 −0.567314 0.823502i \(-0.692017\pi\)
−0.567314 + 0.823502i \(0.692017\pi\)
\(912\) 0 0
\(913\) 11.5076 0.380845
\(914\) 0 0
\(915\) 1.68466 0.0556931
\(916\) 0 0
\(917\) 17.6155 0.581716
\(918\) 0 0
\(919\) 32.8078 1.08223 0.541114 0.840949i \(-0.318003\pi\)
0.541114 + 0.840949i \(0.318003\pi\)
\(920\) 0 0
\(921\) −9.43845 −0.311007
\(922\) 0 0
\(923\) 7.68466 0.252944
\(924\) 0 0
\(925\) −10.8078 −0.355357
\(926\) 0 0
\(927\) 2.24621 0.0737753
\(928\) 0 0
\(929\) −35.3002 −1.15816 −0.579081 0.815270i \(-0.696588\pi\)
−0.579081 + 0.815270i \(0.696588\pi\)
\(930\) 0 0
\(931\) −2.24621 −0.0736166
\(932\) 0 0
\(933\) −22.7386 −0.744429
\(934\) 0 0
\(935\) 8.17708 0.267419
\(936\) 0 0
\(937\) −48.2462 −1.57614 −0.788068 0.615589i \(-0.788918\pi\)
−0.788068 + 0.615589i \(0.788918\pi\)
\(938\) 0 0
\(939\) 24.7386 0.807315
\(940\) 0 0
\(941\) −49.5464 −1.61517 −0.807583 0.589754i \(-0.799225\pi\)
−0.807583 + 0.589754i \(0.799225\pi\)
\(942\) 0 0
\(943\) −13.9309 −0.453652
\(944\) 0 0
\(945\) 2.56155 0.0833273
\(946\) 0 0
\(947\) 1.26137 0.0409889 0.0204944 0.999790i \(-0.493476\pi\)
0.0204944 + 0.999790i \(0.493476\pi\)
\(948\) 0 0
\(949\) −0.246211 −0.00799236
\(950\) 0 0
\(951\) 11.1231 0.360691
\(952\) 0 0
\(953\) 9.19224 0.297766 0.148883 0.988855i \(-0.452432\pi\)
0.148883 + 0.988855i \(0.452432\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) −2.87689 −0.0929968
\(958\) 0 0
\(959\) −0.630683 −0.0203658
\(960\) 0 0
\(961\) −29.7386 −0.959311
\(962\) 0 0
\(963\) −4.80776 −0.154928
\(964\) 0 0
\(965\) 21.6847 0.698054
\(966\) 0 0
\(967\) 25.7538 0.828186 0.414093 0.910235i \(-0.364099\pi\)
0.414093 + 0.910235i \(0.364099\pi\)
\(968\) 0 0
\(969\) 29.1231 0.935569
\(970\) 0 0
\(971\) −24.4924 −0.785999 −0.393000 0.919539i \(-0.628563\pi\)
−0.393000 + 0.919539i \(0.628563\pi\)
\(972\) 0 0
\(973\) 13.9309 0.446603
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) 13.8617 0.443476 0.221738 0.975106i \(-0.428827\pi\)
0.221738 + 0.975106i \(0.428827\pi\)
\(978\) 0 0
\(979\) −3.33050 −0.106443
\(980\) 0 0
\(981\) −4.87689 −0.155707
\(982\) 0 0
\(983\) −12.9848 −0.414152 −0.207076 0.978325i \(-0.566395\pi\)
−0.207076 + 0.978325i \(0.566395\pi\)
\(984\) 0 0
\(985\) −17.3693 −0.553432
\(986\) 0 0
\(987\) 2.87689 0.0915726
\(988\) 0 0
\(989\) −8.98485 −0.285701
\(990\) 0 0
\(991\) −3.05398 −0.0970127 −0.0485064 0.998823i \(-0.515446\pi\)
−0.0485064 + 0.998823i \(0.515446\pi\)
\(992\) 0 0
\(993\) −26.2462 −0.832898
\(994\) 0 0
\(995\) −18.2462 −0.578444
\(996\) 0 0
\(997\) 19.1231 0.605635 0.302817 0.953049i \(-0.402073\pi\)
0.302817 + 0.953049i \(0.402073\pi\)
\(998\) 0 0
\(999\) −10.8078 −0.341943
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 3120.2.a.bg.1.2 2
3.2 odd 2 9360.2.a.ci.1.2 2
4.3 odd 2 1560.2.a.o.1.1 2
12.11 even 2 4680.2.a.y.1.1 2
20.19 odd 2 7800.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.o.1.1 2 4.3 odd 2
3120.2.a.bg.1.2 2 1.1 even 1 trivial
4680.2.a.y.1.1 2 12.11 even 2
7800.2.a.bd.1.2 2 20.19 odd 2
9360.2.a.ci.1.2 2 3.2 odd 2