Properties

Label 6040.2.a.s.1.12
Level $6040$
Weight $2$
Character 6040.1
Self dual yes
Analytic conductor $48.230$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 6040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.161432 q^{3} +1.00000 q^{5} -4.19061 q^{7} -2.97394 q^{9} +O(q^{10})\) \(q+0.161432 q^{3} +1.00000 q^{5} -4.19061 q^{7} -2.97394 q^{9} -5.12337 q^{11} +3.40106 q^{13} +0.161432 q^{15} +0.946395 q^{17} -7.49403 q^{19} -0.676498 q^{21} +0.0845329 q^{23} +1.00000 q^{25} -0.964384 q^{27} -7.04163 q^{29} +0.299430 q^{31} -0.827075 q^{33} -4.19061 q^{35} +6.50414 q^{37} +0.549040 q^{39} +3.06155 q^{41} -3.23185 q^{43} -2.97394 q^{45} +10.8083 q^{47} +10.5612 q^{49} +0.152778 q^{51} +1.20685 q^{53} -5.12337 q^{55} -1.20978 q^{57} -9.73216 q^{59} -1.97977 q^{61} +12.4626 q^{63} +3.40106 q^{65} -6.22539 q^{67} +0.0136463 q^{69} +5.16645 q^{71} +7.01109 q^{73} +0.161432 q^{75} +21.4700 q^{77} -0.877371 q^{79} +8.76614 q^{81} -8.30430 q^{83} +0.946395 q^{85} -1.13674 q^{87} -1.68009 q^{89} -14.2525 q^{91} +0.0483376 q^{93} -7.49403 q^{95} -0.0683939 q^{97} +15.2366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) 0.161432 0.0932027 0.0466014 0.998914i \(-0.485161\pi\)
0.0466014 + 0.998914i \(0.485161\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.19061 −1.58390 −0.791950 0.610585i \(-0.790934\pi\)
−0.791950 + 0.610585i \(0.790934\pi\)
\(8\) 0 0
\(9\) −2.97394 −0.991313
\(10\) 0 0
\(11\) −5.12337 −1.54475 −0.772377 0.635165i \(-0.780932\pi\)
−0.772377 + 0.635165i \(0.780932\pi\)
\(12\) 0 0
\(13\) 3.40106 0.943285 0.471643 0.881790i \(-0.343661\pi\)
0.471643 + 0.881790i \(0.343661\pi\)
\(14\) 0 0
\(15\) 0.161432 0.0416815
\(16\) 0 0
\(17\) 0.946395 0.229535 0.114767 0.993392i \(-0.463388\pi\)
0.114767 + 0.993392i \(0.463388\pi\)
\(18\) 0 0
\(19\) −7.49403 −1.71925 −0.859624 0.510927i \(-0.829302\pi\)
−0.859624 + 0.510927i \(0.829302\pi\)
\(20\) 0 0
\(21\) −0.676498 −0.147624
\(22\) 0 0
\(23\) 0.0845329 0.0176263 0.00881316 0.999961i \(-0.497195\pi\)
0.00881316 + 0.999961i \(0.497195\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −0.964384 −0.185596
\(28\) 0 0
\(29\) −7.04163 −1.30760 −0.653799 0.756669i \(-0.726826\pi\)
−0.653799 + 0.756669i \(0.726826\pi\)
\(30\) 0 0
\(31\) 0.299430 0.0537793 0.0268896 0.999638i \(-0.491440\pi\)
0.0268896 + 0.999638i \(0.491440\pi\)
\(32\) 0 0
\(33\) −0.827075 −0.143975
\(34\) 0 0
\(35\) −4.19061 −0.708342
\(36\) 0 0
\(37\) 6.50414 1.06927 0.534637 0.845082i \(-0.320448\pi\)
0.534637 + 0.845082i \(0.320448\pi\)
\(38\) 0 0
\(39\) 0.549040 0.0879168
\(40\) 0 0
\(41\) 3.06155 0.478133 0.239067 0.971003i \(-0.423159\pi\)
0.239067 + 0.971003i \(0.423159\pi\)
\(42\) 0 0
\(43\) −3.23185 −0.492852 −0.246426 0.969162i \(-0.579256\pi\)
−0.246426 + 0.969162i \(0.579256\pi\)
\(44\) 0 0
\(45\) −2.97394 −0.443329
\(46\) 0 0
\(47\) 10.8083 1.57656 0.788280 0.615317i \(-0.210972\pi\)
0.788280 + 0.615317i \(0.210972\pi\)
\(48\) 0 0
\(49\) 10.5612 1.50874
\(50\) 0 0
\(51\) 0.152778 0.0213932
\(52\) 0 0
\(53\) 1.20685 0.165773 0.0828867 0.996559i \(-0.473586\pi\)
0.0828867 + 0.996559i \(0.473586\pi\)
\(54\) 0 0
\(55\) −5.12337 −0.690835
\(56\) 0 0
\(57\) −1.20978 −0.160239
\(58\) 0 0
\(59\) −9.73216 −1.26702 −0.633510 0.773735i \(-0.718386\pi\)
−0.633510 + 0.773735i \(0.718386\pi\)
\(60\) 0 0
\(61\) −1.97977 −0.253483 −0.126742 0.991936i \(-0.540452\pi\)
−0.126742 + 0.991936i \(0.540452\pi\)
\(62\) 0 0
\(63\) 12.4626 1.57014
\(64\) 0 0
\(65\) 3.40106 0.421850
\(66\) 0 0
\(67\) −6.22539 −0.760552 −0.380276 0.924873i \(-0.624171\pi\)
−0.380276 + 0.924873i \(0.624171\pi\)
\(68\) 0 0
\(69\) 0.0136463 0.00164282
\(70\) 0 0
\(71\) 5.16645 0.613145 0.306572 0.951847i \(-0.400818\pi\)
0.306572 + 0.951847i \(0.400818\pi\)
\(72\) 0 0
\(73\) 7.01109 0.820586 0.410293 0.911954i \(-0.365427\pi\)
0.410293 + 0.911954i \(0.365427\pi\)
\(74\) 0 0
\(75\) 0.161432 0.0186405
\(76\) 0 0
\(77\) 21.4700 2.44674
\(78\) 0 0
\(79\) −0.877371 −0.0987119 −0.0493560 0.998781i \(-0.515717\pi\)
−0.0493560 + 0.998781i \(0.515717\pi\)
\(80\) 0 0
\(81\) 8.76614 0.974015
\(82\) 0 0
\(83\) −8.30430 −0.911515 −0.455757 0.890104i \(-0.650632\pi\)
−0.455757 + 0.890104i \(0.650632\pi\)
\(84\) 0 0
\(85\) 0.946395 0.102651
\(86\) 0 0
\(87\) −1.13674 −0.121872
\(88\) 0 0
\(89\) −1.68009 −0.178089 −0.0890446 0.996028i \(-0.528381\pi\)
−0.0890446 + 0.996028i \(0.528381\pi\)
\(90\) 0 0
\(91\) −14.2525 −1.49407
\(92\) 0 0
\(93\) 0.0483376 0.00501238
\(94\) 0 0
\(95\) −7.49403 −0.768871
\(96\) 0 0
\(97\) −0.0683939 −0.00694434 −0.00347217 0.999994i \(-0.501105\pi\)
−0.00347217 + 0.999994i \(0.501105\pi\)
\(98\) 0 0
\(99\) 15.2366 1.53133
\(100\) 0 0
\(101\) 14.5066 1.44346 0.721730 0.692175i \(-0.243347\pi\)
0.721730 + 0.692175i \(0.243347\pi\)
\(102\) 0 0
\(103\) 4.37358 0.430942 0.215471 0.976510i \(-0.430871\pi\)
0.215471 + 0.976510i \(0.430871\pi\)
\(104\) 0 0
\(105\) −0.676498 −0.0660194
\(106\) 0 0
\(107\) 8.34505 0.806746 0.403373 0.915036i \(-0.367838\pi\)
0.403373 + 0.915036i \(0.367838\pi\)
\(108\) 0 0
\(109\) −12.6827 −1.21478 −0.607390 0.794404i \(-0.707784\pi\)
−0.607390 + 0.794404i \(0.707784\pi\)
\(110\) 0 0
\(111\) 1.04998 0.0996593
\(112\) 0 0
\(113\) −8.45054 −0.794960 −0.397480 0.917611i \(-0.630115\pi\)
−0.397480 + 0.917611i \(0.630115\pi\)
\(114\) 0 0
\(115\) 0.0845329 0.00788273
\(116\) 0 0
\(117\) −10.1146 −0.935091
\(118\) 0 0
\(119\) −3.96597 −0.363560
\(120\) 0 0
\(121\) 15.2489 1.38626
\(122\) 0 0
\(123\) 0.494231 0.0445633
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.6730 1.30202 0.651009 0.759070i \(-0.274346\pi\)
0.651009 + 0.759070i \(0.274346\pi\)
\(128\) 0 0
\(129\) −0.521723 −0.0459351
\(130\) 0 0
\(131\) 19.5392 1.70715 0.853576 0.520969i \(-0.174429\pi\)
0.853576 + 0.520969i \(0.174429\pi\)
\(132\) 0 0
\(133\) 31.4045 2.72312
\(134\) 0 0
\(135\) −0.964384 −0.0830010
\(136\) 0 0
\(137\) −0.978415 −0.0835917 −0.0417958 0.999126i \(-0.513308\pi\)
−0.0417958 + 0.999126i \(0.513308\pi\)
\(138\) 0 0
\(139\) 5.63898 0.478292 0.239146 0.970984i \(-0.423133\pi\)
0.239146 + 0.970984i \(0.423133\pi\)
\(140\) 0 0
\(141\) 1.74481 0.146940
\(142\) 0 0
\(143\) −17.4249 −1.45714
\(144\) 0 0
\(145\) −7.04163 −0.584775
\(146\) 0 0
\(147\) 1.70491 0.140619
\(148\) 0 0
\(149\) −7.99834 −0.655250 −0.327625 0.944808i \(-0.606248\pi\)
−0.327625 + 0.944808i \(0.606248\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788
\(152\) 0 0
\(153\) −2.81452 −0.227541
\(154\) 0 0
\(155\) 0.299430 0.0240508
\(156\) 0 0
\(157\) 20.1455 1.60778 0.803892 0.594775i \(-0.202759\pi\)
0.803892 + 0.594775i \(0.202759\pi\)
\(158\) 0 0
\(159\) 0.194824 0.0154505
\(160\) 0 0
\(161\) −0.354244 −0.0279183
\(162\) 0 0
\(163\) 0.216522 0.0169593 0.00847964 0.999964i \(-0.497301\pi\)
0.00847964 + 0.999964i \(0.497301\pi\)
\(164\) 0 0
\(165\) −0.827075 −0.0643877
\(166\) 0 0
\(167\) 20.1317 1.55784 0.778920 0.627123i \(-0.215768\pi\)
0.778920 + 0.627123i \(0.215768\pi\)
\(168\) 0 0
\(169\) −1.43276 −0.110212
\(170\) 0 0
\(171\) 22.2868 1.70431
\(172\) 0 0
\(173\) −18.0911 −1.37544 −0.687720 0.725976i \(-0.741388\pi\)
−0.687720 + 0.725976i \(0.741388\pi\)
\(174\) 0 0
\(175\) −4.19061 −0.316780
\(176\) 0 0
\(177\) −1.57108 −0.118090
\(178\) 0 0
\(179\) 6.74050 0.503809 0.251904 0.967752i \(-0.418943\pi\)
0.251904 + 0.967752i \(0.418943\pi\)
\(180\) 0 0
\(181\) 2.26368 0.168258 0.0841289 0.996455i \(-0.473189\pi\)
0.0841289 + 0.996455i \(0.473189\pi\)
\(182\) 0 0
\(183\) −0.319597 −0.0236253
\(184\) 0 0
\(185\) 6.50414 0.478194
\(186\) 0 0
\(187\) −4.84873 −0.354574
\(188\) 0 0
\(189\) 4.04136 0.293965
\(190\) 0 0
\(191\) −6.81958 −0.493447 −0.246724 0.969086i \(-0.579354\pi\)
−0.246724 + 0.969086i \(0.579354\pi\)
\(192\) 0 0
\(193\) −16.2436 −1.16924 −0.584619 0.811308i \(-0.698756\pi\)
−0.584619 + 0.811308i \(0.698756\pi\)
\(194\) 0 0
\(195\) 0.549040 0.0393176
\(196\) 0 0
\(197\) 26.5182 1.88935 0.944673 0.328012i \(-0.106379\pi\)
0.944673 + 0.328012i \(0.106379\pi\)
\(198\) 0 0
\(199\) −11.7836 −0.835318 −0.417659 0.908604i \(-0.637149\pi\)
−0.417659 + 0.908604i \(0.637149\pi\)
\(200\) 0 0
\(201\) −1.00498 −0.0708856
\(202\) 0 0
\(203\) 29.5087 2.07110
\(204\) 0 0
\(205\) 3.06155 0.213828
\(206\) 0 0
\(207\) −0.251396 −0.0174732
\(208\) 0 0
\(209\) 38.3947 2.65582
\(210\) 0 0
\(211\) −1.30468 −0.0898176 −0.0449088 0.998991i \(-0.514300\pi\)
−0.0449088 + 0.998991i \(0.514300\pi\)
\(212\) 0 0
\(213\) 0.834029 0.0571468
\(214\) 0 0
\(215\) −3.23185 −0.220410
\(216\) 0 0
\(217\) −1.25480 −0.0851810
\(218\) 0 0
\(219\) 1.13181 0.0764808
\(220\) 0 0
\(221\) 3.21875 0.216517
\(222\) 0 0
\(223\) −3.63488 −0.243409 −0.121705 0.992566i \(-0.538836\pi\)
−0.121705 + 0.992566i \(0.538836\pi\)
\(224\) 0 0
\(225\) −2.97394 −0.198263
\(226\) 0 0
\(227\) 18.3226 1.21612 0.608058 0.793892i \(-0.291949\pi\)
0.608058 + 0.793892i \(0.291949\pi\)
\(228\) 0 0
\(229\) 14.5483 0.961376 0.480688 0.876892i \(-0.340387\pi\)
0.480688 + 0.876892i \(0.340387\pi\)
\(230\) 0 0
\(231\) 3.46595 0.228043
\(232\) 0 0
\(233\) −26.5748 −1.74097 −0.870485 0.492195i \(-0.836195\pi\)
−0.870485 + 0.492195i \(0.836195\pi\)
\(234\) 0 0
\(235\) 10.8083 0.705059
\(236\) 0 0
\(237\) −0.141636 −0.00920022
\(238\) 0 0
\(239\) 28.8526 1.86632 0.933160 0.359461i \(-0.117039\pi\)
0.933160 + 0.359461i \(0.117039\pi\)
\(240\) 0 0
\(241\) 14.9374 0.962205 0.481102 0.876664i \(-0.340237\pi\)
0.481102 + 0.876664i \(0.340237\pi\)
\(242\) 0 0
\(243\) 4.30829 0.276377
\(244\) 0 0
\(245\) 10.5612 0.674730
\(246\) 0 0
\(247\) −25.4877 −1.62174
\(248\) 0 0
\(249\) −1.34058 −0.0849557
\(250\) 0 0
\(251\) −24.1778 −1.52609 −0.763044 0.646346i \(-0.776296\pi\)
−0.763044 + 0.646346i \(0.776296\pi\)
\(252\) 0 0
\(253\) −0.433093 −0.0272283
\(254\) 0 0
\(255\) 0.152778 0.00956735
\(256\) 0 0
\(257\) 9.16663 0.571799 0.285899 0.958260i \(-0.407708\pi\)
0.285899 + 0.958260i \(0.407708\pi\)
\(258\) 0 0
\(259\) −27.2563 −1.69362
\(260\) 0 0
\(261\) 20.9414 1.29624
\(262\) 0 0
\(263\) 17.3915 1.07241 0.536203 0.844089i \(-0.319858\pi\)
0.536203 + 0.844089i \(0.319858\pi\)
\(264\) 0 0
\(265\) 1.20685 0.0741362
\(266\) 0 0
\(267\) −0.271220 −0.0165984
\(268\) 0 0
\(269\) −11.9765 −0.730218 −0.365109 0.930965i \(-0.618968\pi\)
−0.365109 + 0.930965i \(0.618968\pi\)
\(270\) 0 0
\(271\) 12.5814 0.764267 0.382133 0.924107i \(-0.375189\pi\)
0.382133 + 0.924107i \(0.375189\pi\)
\(272\) 0 0
\(273\) −2.30081 −0.139251
\(274\) 0 0
\(275\) −5.12337 −0.308951
\(276\) 0 0
\(277\) −4.98067 −0.299259 −0.149630 0.988742i \(-0.547808\pi\)
−0.149630 + 0.988742i \(0.547808\pi\)
\(278\) 0 0
\(279\) −0.890488 −0.0533121
\(280\) 0 0
\(281\) −12.8595 −0.767133 −0.383567 0.923513i \(-0.625304\pi\)
−0.383567 + 0.923513i \(0.625304\pi\)
\(282\) 0 0
\(283\) 9.19862 0.546801 0.273401 0.961900i \(-0.411852\pi\)
0.273401 + 0.961900i \(0.411852\pi\)
\(284\) 0 0
\(285\) −1.20978 −0.0716609
\(286\) 0 0
\(287\) −12.8297 −0.757316
\(288\) 0 0
\(289\) −16.1043 −0.947314
\(290\) 0 0
\(291\) −0.0110409 −0.000647232 0
\(292\) 0 0
\(293\) −17.1170 −0.999988 −0.499994 0.866029i \(-0.666665\pi\)
−0.499994 + 0.866029i \(0.666665\pi\)
\(294\) 0 0
\(295\) −9.73216 −0.566628
\(296\) 0 0
\(297\) 4.94090 0.286700
\(298\) 0 0
\(299\) 0.287502 0.0166267
\(300\) 0 0
\(301\) 13.5434 0.780628
\(302\) 0 0
\(303\) 2.34183 0.134534
\(304\) 0 0
\(305\) −1.97977 −0.113361
\(306\) 0 0
\(307\) −19.2058 −1.09614 −0.548068 0.836434i \(-0.684636\pi\)
−0.548068 + 0.836434i \(0.684636\pi\)
\(308\) 0 0
\(309\) 0.706035 0.0401649
\(310\) 0 0
\(311\) −23.2230 −1.31686 −0.658428 0.752644i \(-0.728778\pi\)
−0.658428 + 0.752644i \(0.728778\pi\)
\(312\) 0 0
\(313\) −15.5337 −0.878016 −0.439008 0.898483i \(-0.644670\pi\)
−0.439008 + 0.898483i \(0.644670\pi\)
\(314\) 0 0
\(315\) 12.4626 0.702189
\(316\) 0 0
\(317\) −12.3799 −0.695327 −0.347664 0.937619i \(-0.613025\pi\)
−0.347664 + 0.937619i \(0.613025\pi\)
\(318\) 0 0
\(319\) 36.0768 2.01992
\(320\) 0 0
\(321\) 1.34716 0.0751909
\(322\) 0 0
\(323\) −7.09232 −0.394627
\(324\) 0 0
\(325\) 3.40106 0.188657
\(326\) 0 0
\(327\) −2.04739 −0.113221
\(328\) 0 0
\(329\) −45.2935 −2.49711
\(330\) 0 0
\(331\) 13.5304 0.743698 0.371849 0.928293i \(-0.378724\pi\)
0.371849 + 0.928293i \(0.378724\pi\)
\(332\) 0 0
\(333\) −19.3429 −1.05999
\(334\) 0 0
\(335\) −6.22539 −0.340129
\(336\) 0 0
\(337\) 20.5353 1.11863 0.559315 0.828955i \(-0.311064\pi\)
0.559315 + 0.828955i \(0.311064\pi\)
\(338\) 0 0
\(339\) −1.36419 −0.0740924
\(340\) 0 0
\(341\) −1.53409 −0.0830757
\(342\) 0 0
\(343\) −14.9236 −0.805797
\(344\) 0 0
\(345\) 0.0136463 0.000734692 0
\(346\) 0 0
\(347\) −16.0348 −0.860796 −0.430398 0.902639i \(-0.641627\pi\)
−0.430398 + 0.902639i \(0.641627\pi\)
\(348\) 0 0
\(349\) 20.5976 1.10256 0.551282 0.834319i \(-0.314139\pi\)
0.551282 + 0.834319i \(0.314139\pi\)
\(350\) 0 0
\(351\) −3.27993 −0.175070
\(352\) 0 0
\(353\) 1.38401 0.0736634 0.0368317 0.999321i \(-0.488273\pi\)
0.0368317 + 0.999321i \(0.488273\pi\)
\(354\) 0 0
\(355\) 5.16645 0.274207
\(356\) 0 0
\(357\) −0.640234 −0.0338848
\(358\) 0 0
\(359\) 4.24361 0.223969 0.111985 0.993710i \(-0.464279\pi\)
0.111985 + 0.993710i \(0.464279\pi\)
\(360\) 0 0
\(361\) 37.1605 1.95582
\(362\) 0 0
\(363\) 2.46166 0.129204
\(364\) 0 0
\(365\) 7.01109 0.366977
\(366\) 0 0
\(367\) −14.7777 −0.771388 −0.385694 0.922627i \(-0.626038\pi\)
−0.385694 + 0.922627i \(0.626038\pi\)
\(368\) 0 0
\(369\) −9.10485 −0.473980
\(370\) 0 0
\(371\) −5.05743 −0.262569
\(372\) 0 0
\(373\) −11.5535 −0.598218 −0.299109 0.954219i \(-0.596689\pi\)
−0.299109 + 0.954219i \(0.596689\pi\)
\(374\) 0 0
\(375\) 0.161432 0.00833630
\(376\) 0 0
\(377\) −23.9490 −1.23344
\(378\) 0 0
\(379\) 20.1521 1.03514 0.517571 0.855640i \(-0.326836\pi\)
0.517571 + 0.855640i \(0.326836\pi\)
\(380\) 0 0
\(381\) 2.36869 0.121352
\(382\) 0 0
\(383\) −4.49393 −0.229629 −0.114814 0.993387i \(-0.536627\pi\)
−0.114814 + 0.993387i \(0.536627\pi\)
\(384\) 0 0
\(385\) 21.4700 1.09421
\(386\) 0 0
\(387\) 9.61131 0.488570
\(388\) 0 0
\(389\) −6.85928 −0.347779 −0.173890 0.984765i \(-0.555634\pi\)
−0.173890 + 0.984765i \(0.555634\pi\)
\(390\) 0 0
\(391\) 0.0800015 0.00404585
\(392\) 0 0
\(393\) 3.15426 0.159111
\(394\) 0 0
\(395\) −0.877371 −0.0441453
\(396\) 0 0
\(397\) −17.7721 −0.891954 −0.445977 0.895044i \(-0.647144\pi\)
−0.445977 + 0.895044i \(0.647144\pi\)
\(398\) 0 0
\(399\) 5.06969 0.253802
\(400\) 0 0
\(401\) 19.9637 0.996940 0.498470 0.866907i \(-0.333895\pi\)
0.498470 + 0.866907i \(0.333895\pi\)
\(402\) 0 0
\(403\) 1.01838 0.0507292
\(404\) 0 0
\(405\) 8.76614 0.435593
\(406\) 0 0
\(407\) −33.3231 −1.65177
\(408\) 0 0
\(409\) 15.6507 0.773875 0.386938 0.922106i \(-0.373533\pi\)
0.386938 + 0.922106i \(0.373533\pi\)
\(410\) 0 0
\(411\) −0.157947 −0.00779097
\(412\) 0 0
\(413\) 40.7837 2.00683
\(414\) 0 0
\(415\) −8.30430 −0.407642
\(416\) 0 0
\(417\) 0.910310 0.0445781
\(418\) 0 0
\(419\) −25.8757 −1.26411 −0.632056 0.774923i \(-0.717789\pi\)
−0.632056 + 0.774923i \(0.717789\pi\)
\(420\) 0 0
\(421\) 8.79000 0.428398 0.214199 0.976790i \(-0.431286\pi\)
0.214199 + 0.976790i \(0.431286\pi\)
\(422\) 0 0
\(423\) −32.1434 −1.56286
\(424\) 0 0
\(425\) 0.946395 0.0459069
\(426\) 0 0
\(427\) 8.29643 0.401492
\(428\) 0 0
\(429\) −2.81293 −0.135810
\(430\) 0 0
\(431\) 12.4202 0.598261 0.299131 0.954212i \(-0.403303\pi\)
0.299131 + 0.954212i \(0.403303\pi\)
\(432\) 0 0
\(433\) 9.44335 0.453819 0.226909 0.973916i \(-0.427138\pi\)
0.226909 + 0.973916i \(0.427138\pi\)
\(434\) 0 0
\(435\) −1.13674 −0.0545026
\(436\) 0 0
\(437\) −0.633492 −0.0303040
\(438\) 0 0
\(439\) 19.6942 0.939953 0.469977 0.882679i \(-0.344262\pi\)
0.469977 + 0.882679i \(0.344262\pi\)
\(440\) 0 0
\(441\) −31.4084 −1.49564
\(442\) 0 0
\(443\) −15.3873 −0.731072 −0.365536 0.930797i \(-0.619114\pi\)
−0.365536 + 0.930797i \(0.619114\pi\)
\(444\) 0 0
\(445\) −1.68009 −0.0796439
\(446\) 0 0
\(447\) −1.29119 −0.0610710
\(448\) 0 0
\(449\) 16.9276 0.798863 0.399432 0.916763i \(-0.369208\pi\)
0.399432 + 0.916763i \(0.369208\pi\)
\(450\) 0 0
\(451\) −15.6854 −0.738598
\(452\) 0 0
\(453\) 0.161432 0.00758473
\(454\) 0 0
\(455\) −14.2525 −0.668169
\(456\) 0 0
\(457\) −7.23431 −0.338407 −0.169203 0.985581i \(-0.554119\pi\)
−0.169203 + 0.985581i \(0.554119\pi\)
\(458\) 0 0
\(459\) −0.912689 −0.0426007
\(460\) 0 0
\(461\) 22.6230 1.05366 0.526829 0.849971i \(-0.323381\pi\)
0.526829 + 0.849971i \(0.323381\pi\)
\(462\) 0 0
\(463\) −36.7217 −1.70660 −0.853302 0.521418i \(-0.825403\pi\)
−0.853302 + 0.521418i \(0.825403\pi\)
\(464\) 0 0
\(465\) 0.0483376 0.00224160
\(466\) 0 0
\(467\) −13.6994 −0.633932 −0.316966 0.948437i \(-0.602664\pi\)
−0.316966 + 0.948437i \(0.602664\pi\)
\(468\) 0 0
\(469\) 26.0882 1.20464
\(470\) 0 0
\(471\) 3.25212 0.149850
\(472\) 0 0
\(473\) 16.5579 0.761335
\(474\) 0 0
\(475\) −7.49403 −0.343850
\(476\) 0 0
\(477\) −3.58910 −0.164333
\(478\) 0 0
\(479\) 30.7364 1.40438 0.702191 0.711989i \(-0.252205\pi\)
0.702191 + 0.711989i \(0.252205\pi\)
\(480\) 0 0
\(481\) 22.1210 1.00863
\(482\) 0 0
\(483\) −0.0571863 −0.00260207
\(484\) 0 0
\(485\) −0.0683939 −0.00310560
\(486\) 0 0
\(487\) −32.1333 −1.45610 −0.728050 0.685524i \(-0.759573\pi\)
−0.728050 + 0.685524i \(0.759573\pi\)
\(488\) 0 0
\(489\) 0.0349535 0.00158065
\(490\) 0 0
\(491\) −27.3982 −1.23646 −0.618232 0.785996i \(-0.712151\pi\)
−0.618232 + 0.785996i \(0.712151\pi\)
\(492\) 0 0
\(493\) −6.66416 −0.300139
\(494\) 0 0
\(495\) 15.2366 0.684834
\(496\) 0 0
\(497\) −21.6506 −0.971160
\(498\) 0 0
\(499\) 25.7573 1.15305 0.576527 0.817078i \(-0.304408\pi\)
0.576527 + 0.817078i \(0.304408\pi\)
\(500\) 0 0
\(501\) 3.24990 0.145195
\(502\) 0 0
\(503\) 40.6555 1.81274 0.906370 0.422486i \(-0.138842\pi\)
0.906370 + 0.422486i \(0.138842\pi\)
\(504\) 0 0
\(505\) 14.5066 0.645535
\(506\) 0 0
\(507\) −0.231293 −0.0102721
\(508\) 0 0
\(509\) −37.1664 −1.64737 −0.823687 0.567045i \(-0.808086\pi\)
−0.823687 + 0.567045i \(0.808086\pi\)
\(510\) 0 0
\(511\) −29.3807 −1.29973
\(512\) 0 0
\(513\) 7.22713 0.319085
\(514\) 0 0
\(515\) 4.37358 0.192723
\(516\) 0 0
\(517\) −55.3751 −2.43540
\(518\) 0 0
\(519\) −2.92048 −0.128195
\(520\) 0 0
\(521\) 17.8001 0.779835 0.389918 0.920850i \(-0.372503\pi\)
0.389918 + 0.920850i \(0.372503\pi\)
\(522\) 0 0
\(523\) 11.9113 0.520846 0.260423 0.965495i \(-0.416138\pi\)
0.260423 + 0.965495i \(0.416138\pi\)
\(524\) 0 0
\(525\) −0.676498 −0.0295248
\(526\) 0 0
\(527\) 0.283379 0.0123442
\(528\) 0 0
\(529\) −22.9929 −0.999689
\(530\) 0 0
\(531\) 28.9429 1.25601
\(532\) 0 0
\(533\) 10.4125 0.451016
\(534\) 0 0
\(535\) 8.34505 0.360788
\(536\) 0 0
\(537\) 1.08813 0.0469563
\(538\) 0 0
\(539\) −54.1089 −2.33063
\(540\) 0 0
\(541\) −36.0172 −1.54850 −0.774250 0.632880i \(-0.781873\pi\)
−0.774250 + 0.632880i \(0.781873\pi\)
\(542\) 0 0
\(543\) 0.365430 0.0156821
\(544\) 0 0
\(545\) −12.6827 −0.543266
\(546\) 0 0
\(547\) 23.1274 0.988857 0.494429 0.869218i \(-0.335377\pi\)
0.494429 + 0.869218i \(0.335377\pi\)
\(548\) 0 0
\(549\) 5.88771 0.251281
\(550\) 0 0
\(551\) 52.7702 2.24808
\(552\) 0 0
\(553\) 3.67672 0.156350
\(554\) 0 0
\(555\) 1.04998 0.0445690
\(556\) 0 0
\(557\) 13.7989 0.584679 0.292339 0.956315i \(-0.405566\pi\)
0.292339 + 0.956315i \(0.405566\pi\)
\(558\) 0 0
\(559\) −10.9917 −0.464900
\(560\) 0 0
\(561\) −0.782740 −0.0330473
\(562\) 0 0
\(563\) 18.3822 0.774717 0.387358 0.921929i \(-0.373388\pi\)
0.387358 + 0.921929i \(0.373388\pi\)
\(564\) 0 0
\(565\) −8.45054 −0.355517
\(566\) 0 0
\(567\) −36.7354 −1.54274
\(568\) 0 0
\(569\) −4.06642 −0.170473 −0.0852366 0.996361i \(-0.527165\pi\)
−0.0852366 + 0.996361i \(0.527165\pi\)
\(570\) 0 0
\(571\) −8.78702 −0.367725 −0.183863 0.982952i \(-0.558860\pi\)
−0.183863 + 0.982952i \(0.558860\pi\)
\(572\) 0 0
\(573\) −1.10090 −0.0459906
\(574\) 0 0
\(575\) 0.0845329 0.00352526
\(576\) 0 0
\(577\) −21.1665 −0.881173 −0.440587 0.897710i \(-0.645229\pi\)
−0.440587 + 0.897710i \(0.645229\pi\)
\(578\) 0 0
\(579\) −2.62223 −0.108976
\(580\) 0 0
\(581\) 34.8000 1.44375
\(582\) 0 0
\(583\) −6.18313 −0.256079
\(584\) 0 0
\(585\) −10.1146 −0.418186
\(586\) 0 0
\(587\) 42.9895 1.77437 0.887184 0.461416i \(-0.152658\pi\)
0.887184 + 0.461416i \(0.152658\pi\)
\(588\) 0 0
\(589\) −2.24394 −0.0924600
\(590\) 0 0
\(591\) 4.28089 0.176092
\(592\) 0 0
\(593\) 10.7723 0.442363 0.221182 0.975233i \(-0.429009\pi\)
0.221182 + 0.975233i \(0.429009\pi\)
\(594\) 0 0
\(595\) −3.96597 −0.162589
\(596\) 0 0
\(597\) −1.90225 −0.0778539
\(598\) 0 0
\(599\) −19.7694 −0.807756 −0.403878 0.914813i \(-0.632338\pi\)
−0.403878 + 0.914813i \(0.632338\pi\)
\(600\) 0 0
\(601\) 32.8725 1.34090 0.670449 0.741956i \(-0.266102\pi\)
0.670449 + 0.741956i \(0.266102\pi\)
\(602\) 0 0
\(603\) 18.5139 0.753946
\(604\) 0 0
\(605\) 15.2489 0.619956
\(606\) 0 0
\(607\) 15.7733 0.640217 0.320108 0.947381i \(-0.396281\pi\)
0.320108 + 0.947381i \(0.396281\pi\)
\(608\) 0 0
\(609\) 4.76364 0.193033
\(610\) 0 0
\(611\) 36.7599 1.48715
\(612\) 0 0
\(613\) 4.83510 0.195288 0.0976439 0.995221i \(-0.468869\pi\)
0.0976439 + 0.995221i \(0.468869\pi\)
\(614\) 0 0
\(615\) 0.494231 0.0199293
\(616\) 0 0
\(617\) 29.7533 1.19782 0.598912 0.800815i \(-0.295600\pi\)
0.598912 + 0.800815i \(0.295600\pi\)
\(618\) 0 0
\(619\) 29.4984 1.18564 0.592820 0.805335i \(-0.298014\pi\)
0.592820 + 0.805335i \(0.298014\pi\)
\(620\) 0 0
\(621\) −0.0815222 −0.00327137
\(622\) 0 0
\(623\) 7.04059 0.282075
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.19813 0.247529
\(628\) 0 0
\(629\) 6.15549 0.245435
\(630\) 0 0
\(631\) −6.40356 −0.254922 −0.127461 0.991844i \(-0.540683\pi\)
−0.127461 + 0.991844i \(0.540683\pi\)
\(632\) 0 0
\(633\) −0.210616 −0.00837124
\(634\) 0 0
\(635\) 14.6730 0.582280
\(636\) 0 0
\(637\) 35.9193 1.42317
\(638\) 0 0
\(639\) −15.3647 −0.607818
\(640\) 0 0
\(641\) 45.1471 1.78320 0.891601 0.452822i \(-0.149583\pi\)
0.891601 + 0.452822i \(0.149583\pi\)
\(642\) 0 0
\(643\) −2.76855 −0.109181 −0.0545905 0.998509i \(-0.517385\pi\)
−0.0545905 + 0.998509i \(0.517385\pi\)
\(644\) 0 0
\(645\) −0.521723 −0.0205428
\(646\) 0 0
\(647\) −38.3507 −1.50772 −0.753861 0.657034i \(-0.771811\pi\)
−0.753861 + 0.657034i \(0.771811\pi\)
\(648\) 0 0
\(649\) 49.8615 1.95723
\(650\) 0 0
\(651\) −0.202564 −0.00793911
\(652\) 0 0
\(653\) 21.4333 0.838749 0.419375 0.907813i \(-0.362249\pi\)
0.419375 + 0.907813i \(0.362249\pi\)
\(654\) 0 0
\(655\) 19.5392 0.763461
\(656\) 0 0
\(657\) −20.8505 −0.813457
\(658\) 0 0
\(659\) 5.68358 0.221401 0.110701 0.993854i \(-0.464691\pi\)
0.110701 + 0.993854i \(0.464691\pi\)
\(660\) 0 0
\(661\) 4.80659 0.186955 0.0934774 0.995621i \(-0.470202\pi\)
0.0934774 + 0.995621i \(0.470202\pi\)
\(662\) 0 0
\(663\) 0.519609 0.0201799
\(664\) 0 0
\(665\) 31.4045 1.21782
\(666\) 0 0
\(667\) −0.595249 −0.0230481
\(668\) 0 0
\(669\) −0.586785 −0.0226864
\(670\) 0 0
\(671\) 10.1431 0.391569
\(672\) 0 0
\(673\) 7.42751 0.286309 0.143155 0.989700i \(-0.454275\pi\)
0.143155 + 0.989700i \(0.454275\pi\)
\(674\) 0 0
\(675\) −0.964384 −0.0371192
\(676\) 0 0
\(677\) 5.88835 0.226308 0.113154 0.993577i \(-0.463905\pi\)
0.113154 + 0.993577i \(0.463905\pi\)
\(678\) 0 0
\(679\) 0.286612 0.0109992
\(680\) 0 0
\(681\) 2.95786 0.113345
\(682\) 0 0
\(683\) 41.9378 1.60471 0.802353 0.596849i \(-0.203581\pi\)
0.802353 + 0.596849i \(0.203581\pi\)
\(684\) 0 0
\(685\) −0.978415 −0.0373833
\(686\) 0 0
\(687\) 2.34855 0.0896029
\(688\) 0 0
\(689\) 4.10457 0.156372
\(690\) 0 0
\(691\) 37.3562 1.42110 0.710549 0.703648i \(-0.248447\pi\)
0.710549 + 0.703648i \(0.248447\pi\)
\(692\) 0 0
\(693\) −63.8506 −2.42548
\(694\) 0 0
\(695\) 5.63898 0.213899
\(696\) 0 0
\(697\) 2.89743 0.109748
\(698\) 0 0
\(699\) −4.29001 −0.162263
\(700\) 0 0
\(701\) −50.1798 −1.89526 −0.947632 0.319366i \(-0.896530\pi\)
−0.947632 + 0.319366i \(0.896530\pi\)
\(702\) 0 0
\(703\) −48.7422 −1.83835
\(704\) 0 0
\(705\) 1.74481 0.0657134
\(706\) 0 0
\(707\) −60.7914 −2.28630
\(708\) 0 0
\(709\) −16.4870 −0.619182 −0.309591 0.950870i \(-0.600192\pi\)
−0.309591 + 0.950870i \(0.600192\pi\)
\(710\) 0 0
\(711\) 2.60925 0.0978544
\(712\) 0 0
\(713\) 0.0253117 0.000947931 0
\(714\) 0 0
\(715\) −17.4249 −0.651655
\(716\) 0 0
\(717\) 4.65773 0.173946
\(718\) 0 0
\(719\) −29.3085 −1.09302 −0.546512 0.837451i \(-0.684045\pi\)
−0.546512 + 0.837451i \(0.684045\pi\)
\(720\) 0 0
\(721\) −18.3280 −0.682569
\(722\) 0 0
\(723\) 2.41138 0.0896801
\(724\) 0 0
\(725\) −7.04163 −0.261519
\(726\) 0 0
\(727\) 12.2952 0.456004 0.228002 0.973661i \(-0.426781\pi\)
0.228002 + 0.973661i \(0.426781\pi\)
\(728\) 0 0
\(729\) −25.6029 −0.948256
\(730\) 0 0
\(731\) −3.05860 −0.113127
\(732\) 0 0
\(733\) 35.6390 1.31636 0.658179 0.752862i \(-0.271327\pi\)
0.658179 + 0.752862i \(0.271327\pi\)
\(734\) 0 0
\(735\) 1.70491 0.0628867
\(736\) 0 0
\(737\) 31.8950 1.17487
\(738\) 0 0
\(739\) −44.0036 −1.61870 −0.809349 0.587328i \(-0.800180\pi\)
−0.809349 + 0.587328i \(0.800180\pi\)
\(740\) 0 0
\(741\) −4.11452 −0.151151
\(742\) 0 0
\(743\) −6.19041 −0.227104 −0.113552 0.993532i \(-0.536223\pi\)
−0.113552 + 0.993532i \(0.536223\pi\)
\(744\) 0 0
\(745\) −7.99834 −0.293037
\(746\) 0 0
\(747\) 24.6965 0.903597
\(748\) 0 0
\(749\) −34.9708 −1.27781
\(750\) 0 0
\(751\) 8.56892 0.312685 0.156342 0.987703i \(-0.450030\pi\)
0.156342 + 0.987703i \(0.450030\pi\)
\(752\) 0 0
\(753\) −3.90307 −0.142236
\(754\) 0 0
\(755\) 1.00000 0.0363937
\(756\) 0 0
\(757\) −8.56594 −0.311335 −0.155667 0.987810i \(-0.549753\pi\)
−0.155667 + 0.987810i \(0.549753\pi\)
\(758\) 0 0
\(759\) −0.0699150 −0.00253775
\(760\) 0 0
\(761\) 41.4229 1.50158 0.750789 0.660542i \(-0.229674\pi\)
0.750789 + 0.660542i \(0.229674\pi\)
\(762\) 0 0
\(763\) 53.1481 1.92409
\(764\) 0 0
\(765\) −2.81452 −0.101759
\(766\) 0 0
\(767\) −33.0997 −1.19516
\(768\) 0 0
\(769\) −16.9301 −0.610516 −0.305258 0.952270i \(-0.598743\pi\)
−0.305258 + 0.952270i \(0.598743\pi\)
\(770\) 0 0
\(771\) 1.47979 0.0532932
\(772\) 0 0
\(773\) −5.46937 −0.196719 −0.0983597 0.995151i \(-0.531360\pi\)
−0.0983597 + 0.995151i \(0.531360\pi\)
\(774\) 0 0
\(775\) 0.299430 0.0107559
\(776\) 0 0
\(777\) −4.40004 −0.157850
\(778\) 0 0
\(779\) −22.9433 −0.822030
\(780\) 0 0
\(781\) −26.4696 −0.947158
\(782\) 0 0
\(783\) 6.79083 0.242685
\(784\) 0 0
\(785\) 20.1455 0.719023
\(786\) 0 0
\(787\) 19.9943 0.712719 0.356359 0.934349i \(-0.384018\pi\)
0.356359 + 0.934349i \(0.384018\pi\)
\(788\) 0 0
\(789\) 2.80754 0.0999512
\(790\) 0 0
\(791\) 35.4129 1.25914
\(792\) 0 0
\(793\) −6.73332 −0.239107
\(794\) 0 0
\(795\) 0.194824 0.00690969
\(796\) 0 0
\(797\) 40.4212 1.43179 0.715896 0.698206i \(-0.246018\pi\)
0.715896 + 0.698206i \(0.246018\pi\)
\(798\) 0 0
\(799\) 10.2290 0.361875
\(800\) 0 0
\(801\) 4.99648 0.176542
\(802\) 0 0
\(803\) −35.9204 −1.26760
\(804\) 0 0
\(805\) −0.354244 −0.0124855
\(806\) 0 0
\(807\) −1.93338 −0.0680583
\(808\) 0 0
\(809\) −25.0223 −0.879736 −0.439868 0.898062i \(-0.644975\pi\)
−0.439868 + 0.898062i \(0.644975\pi\)
\(810\) 0 0
\(811\) 15.6427 0.549290 0.274645 0.961546i \(-0.411440\pi\)
0.274645 + 0.961546i \(0.411440\pi\)
\(812\) 0 0
\(813\) 2.03104 0.0712317
\(814\) 0 0
\(815\) 0.216522 0.00758442
\(816\) 0 0
\(817\) 24.2195 0.847335
\(818\) 0 0
\(819\) 42.3862 1.48109
\(820\) 0 0
\(821\) 42.8869 1.49676 0.748381 0.663269i \(-0.230832\pi\)
0.748381 + 0.663269i \(0.230832\pi\)
\(822\) 0 0
\(823\) −41.4506 −1.44488 −0.722439 0.691434i \(-0.756979\pi\)
−0.722439 + 0.691434i \(0.756979\pi\)
\(824\) 0 0
\(825\) −0.827075 −0.0287951
\(826\) 0 0
\(827\) −16.4717 −0.572777 −0.286389 0.958114i \(-0.592455\pi\)
−0.286389 + 0.958114i \(0.592455\pi\)
\(828\) 0 0
\(829\) 42.5502 1.47783 0.738915 0.673798i \(-0.235338\pi\)
0.738915 + 0.673798i \(0.235338\pi\)
\(830\) 0 0
\(831\) −0.804039 −0.0278918
\(832\) 0 0
\(833\) 9.99506 0.346308
\(834\) 0 0
\(835\) 20.1317 0.696687
\(836\) 0 0
\(837\) −0.288766 −0.00998121
\(838\) 0 0
\(839\) 5.51874 0.190528 0.0952640 0.995452i \(-0.469630\pi\)
0.0952640 + 0.995452i \(0.469630\pi\)
\(840\) 0 0
\(841\) 20.5845 0.709811
\(842\) 0 0
\(843\) −2.07593 −0.0714989
\(844\) 0 0
\(845\) −1.43276 −0.0492885
\(846\) 0 0
\(847\) −63.9022 −2.19570
\(848\) 0 0
\(849\) 1.48495 0.0509634
\(850\) 0 0
\(851\) 0.549814 0.0188474
\(852\) 0 0
\(853\) 48.6688 1.66639 0.833193 0.552982i \(-0.186510\pi\)
0.833193 + 0.552982i \(0.186510\pi\)
\(854\) 0 0
\(855\) 22.2868 0.762192
\(856\) 0 0
\(857\) 21.8795 0.747391 0.373695 0.927552i \(-0.378091\pi\)
0.373695 + 0.927552i \(0.378091\pi\)
\(858\) 0 0
\(859\) 3.56079 0.121493 0.0607463 0.998153i \(-0.480652\pi\)
0.0607463 + 0.998153i \(0.480652\pi\)
\(860\) 0 0
\(861\) −2.07113 −0.0705839
\(862\) 0 0
\(863\) 33.8993 1.15395 0.576973 0.816763i \(-0.304234\pi\)
0.576973 + 0.816763i \(0.304234\pi\)
\(864\) 0 0
\(865\) −18.0911 −0.615116
\(866\) 0 0
\(867\) −2.59975 −0.0882922
\(868\) 0 0
\(869\) 4.49509 0.152486
\(870\) 0 0
\(871\) −21.1730 −0.717418
\(872\) 0 0
\(873\) 0.203399 0.00688402
\(874\) 0 0
\(875\) −4.19061 −0.141668
\(876\) 0 0
\(877\) −4.83378 −0.163225 −0.0816126 0.996664i \(-0.526007\pi\)
−0.0816126 + 0.996664i \(0.526007\pi\)
\(878\) 0 0
\(879\) −2.76324 −0.0932016
\(880\) 0 0
\(881\) 0.347938 0.0117223 0.00586116 0.999983i \(-0.498134\pi\)
0.00586116 + 0.999983i \(0.498134\pi\)
\(882\) 0 0
\(883\) −4.71668 −0.158729 −0.0793644 0.996846i \(-0.525289\pi\)
−0.0793644 + 0.996846i \(0.525289\pi\)
\(884\) 0 0
\(885\) −1.57108 −0.0528113
\(886\) 0 0
\(887\) −22.3660 −0.750977 −0.375489 0.926827i \(-0.622525\pi\)
−0.375489 + 0.926827i \(0.622525\pi\)
\(888\) 0 0
\(889\) −61.4887 −2.06227
\(890\) 0 0
\(891\) −44.9122 −1.50461
\(892\) 0 0
\(893\) −80.9981 −2.71050
\(894\) 0 0
\(895\) 6.74050 0.225310
\(896\) 0 0
\(897\) 0.0464119 0.00154965
\(898\) 0 0
\(899\) −2.10848 −0.0703216
\(900\) 0 0
\(901\) 1.14216 0.0380507
\(902\) 0 0
\(903\) 2.18634 0.0727567
\(904\) 0 0
\(905\) 2.26368 0.0752472
\(906\) 0 0
\(907\) −45.4408 −1.50884 −0.754419 0.656393i \(-0.772081\pi\)
−0.754419 + 0.656393i \(0.772081\pi\)
\(908\) 0 0
\(909\) −43.1417 −1.43092
\(910\) 0 0
\(911\) −29.3315 −0.971798 −0.485899 0.874015i \(-0.661508\pi\)
−0.485899 + 0.874015i \(0.661508\pi\)
\(912\) 0 0
\(913\) 42.5460 1.40807
\(914\) 0 0
\(915\) −0.319597 −0.0105656
\(916\) 0 0
\(917\) −81.8813 −2.70396
\(918\) 0 0
\(919\) 46.7346 1.54163 0.770816 0.637057i \(-0.219849\pi\)
0.770816 + 0.637057i \(0.219849\pi\)
\(920\) 0 0
\(921\) −3.10044 −0.102163
\(922\) 0 0
\(923\) 17.5714 0.578370
\(924\) 0 0
\(925\) 6.50414 0.213855
\(926\) 0 0
\(927\) −13.0068 −0.427198
\(928\) 0 0
\(929\) 22.9530 0.753062 0.376531 0.926404i \(-0.377117\pi\)
0.376531 + 0.926404i \(0.377117\pi\)
\(930\) 0 0
\(931\) −79.1459 −2.59390
\(932\) 0 0
\(933\) −3.74893 −0.122735
\(934\) 0 0
\(935\) −4.84873 −0.158570
\(936\) 0 0
\(937\) −15.7330 −0.513976 −0.256988 0.966415i \(-0.582730\pi\)
−0.256988 + 0.966415i \(0.582730\pi\)
\(938\) 0 0
\(939\) −2.50763 −0.0818335
\(940\) 0 0
\(941\) −42.5723 −1.38782 −0.693908 0.720064i \(-0.744113\pi\)
−0.693908 + 0.720064i \(0.744113\pi\)
\(942\) 0 0
\(943\) 0.258801 0.00842773
\(944\) 0 0
\(945\) 4.04136 0.131465
\(946\) 0 0
\(947\) −52.1864 −1.69583 −0.847916 0.530131i \(-0.822143\pi\)
−0.847916 + 0.530131i \(0.822143\pi\)
\(948\) 0 0
\(949\) 23.8452 0.774046
\(950\) 0 0
\(951\) −1.99852 −0.0648064
\(952\) 0 0
\(953\) −6.38819 −0.206934 −0.103467 0.994633i \(-0.532994\pi\)
−0.103467 + 0.994633i \(0.532994\pi\)
\(954\) 0 0
\(955\) −6.81958 −0.220676
\(956\) 0 0
\(957\) 5.82395 0.188262
\(958\) 0 0
\(959\) 4.10015 0.132401
\(960\) 0 0
\(961\) −30.9103 −0.997108
\(962\) 0 0
\(963\) −24.8177 −0.799738
\(964\) 0 0
\(965\) −16.2436 −0.522900
\(966\) 0 0
\(967\) 2.07412 0.0666992 0.0333496 0.999444i \(-0.489383\pi\)
0.0333496 + 0.999444i \(0.489383\pi\)
\(968\) 0 0
\(969\) −1.14493 −0.0367803
\(970\) 0 0
\(971\) −36.8743 −1.18335 −0.591676 0.806176i \(-0.701534\pi\)
−0.591676 + 0.806176i \(0.701534\pi\)
\(972\) 0 0
\(973\) −23.6307 −0.757567
\(974\) 0 0
\(975\) 0.549040 0.0175834
\(976\) 0 0
\(977\) −16.8655 −0.539576 −0.269788 0.962920i \(-0.586954\pi\)
−0.269788 + 0.962920i \(0.586954\pi\)
\(978\) 0 0
\(979\) 8.60772 0.275104
\(980\) 0 0
\(981\) 37.7175 1.20423
\(982\) 0 0
\(983\) 11.8953 0.379402 0.189701 0.981842i \(-0.439248\pi\)
0.189701 + 0.981842i \(0.439248\pi\)
\(984\) 0 0
\(985\) 26.5182 0.844942
\(986\) 0 0
\(987\) −7.31182 −0.232738
\(988\) 0 0
\(989\) −0.273197 −0.00868716
\(990\) 0 0
\(991\) −47.0984 −1.49613 −0.748066 0.663625i \(-0.769017\pi\)
−0.748066 + 0.663625i \(0.769017\pi\)
\(992\) 0 0
\(993\) 2.18424 0.0693147
\(994\) 0 0
\(995\) −11.7836 −0.373566
\(996\) 0 0
\(997\) −37.6403 −1.19208 −0.596041 0.802954i \(-0.703260\pi\)
−0.596041 + 0.802954i \(0.703260\pi\)
\(998\) 0 0
\(999\) −6.27249 −0.198453
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 6040.2.a.s.1.12 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6040.2.a.s.1.12 24 1.1 even 1 trivial