Properties

Label 4023.2.a.f.1.9
Level $4023$
Weight $2$
Character 4023.1
Self dual yes
Analytic conductor $32.124$
Analytic rank $0$
Dimension $25$
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] = [4023,2,Mod(1,4023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4023, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4023.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4023 = 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4023.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: \(32.1238167332\)
Analytic rank: \(0\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 4023.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.599558 q^{2} -1.64053 q^{4} -0.261228 q^{5} -0.444015 q^{7} +2.18271 q^{8} +O(q^{10})\) \(q-0.599558 q^{2} -1.64053 q^{4} -0.261228 q^{5} -0.444015 q^{7} +2.18271 q^{8} +0.156622 q^{10} +3.27284 q^{11} +4.82387 q^{13} +0.266213 q^{14} +1.97240 q^{16} -0.628330 q^{17} +6.37907 q^{19} +0.428553 q^{20} -1.96226 q^{22} +6.91056 q^{23} -4.93176 q^{25} -2.89219 q^{26} +0.728419 q^{28} +3.92656 q^{29} +3.84367 q^{31} -5.54799 q^{32} +0.376720 q^{34} +0.115989 q^{35} -7.74015 q^{37} -3.82463 q^{38} -0.570186 q^{40} -12.1523 q^{41} +3.58602 q^{43} -5.36920 q^{44} -4.14329 q^{46} +6.18985 q^{47} -6.80285 q^{49} +2.95688 q^{50} -7.91370 q^{52} +2.50738 q^{53} -0.854960 q^{55} -0.969155 q^{56} -2.35420 q^{58} -8.32542 q^{59} +2.21416 q^{61} -2.30451 q^{62} -0.618452 q^{64} -1.26013 q^{65} +3.22654 q^{67} +1.03079 q^{68} -0.0695423 q^{70} +9.85495 q^{71} -12.8578 q^{73} +4.64067 q^{74} -10.4651 q^{76} -1.45319 q^{77} +9.04869 q^{79} -0.515246 q^{80} +7.28599 q^{82} +5.99942 q^{83} +0.164138 q^{85} -2.15003 q^{86} +7.14367 q^{88} +4.41461 q^{89} -2.14187 q^{91} -11.3370 q^{92} -3.71118 q^{94} -1.66639 q^{95} -5.55167 q^{97} +4.07871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8} - 4 q^{10} + 12 q^{11} + 10 q^{14} + 35 q^{16} + 26 q^{17} + 30 q^{20} + 8 q^{22} + 26 q^{23} + 27 q^{25} + 16 q^{26} + 4 q^{28} + 20 q^{29} - 6 q^{31} + 49 q^{32} - 14 q^{34} + 16 q^{35} - 2 q^{37} + 27 q^{38} + 2 q^{40} + 35 q^{41} + 4 q^{43} + 22 q^{44} + 6 q^{46} + 38 q^{47} + 19 q^{49} + 22 q^{50} + 4 q^{52} + 36 q^{53} + 10 q^{55} + 79 q^{56} - 22 q^{58} + 15 q^{59} + 10 q^{61} - 14 q^{62} + 41 q^{64} + 80 q^{65} - 6 q^{67} + 33 q^{68} + 8 q^{70} + 26 q^{71} - 6 q^{73} + 75 q^{74} - 10 q^{76} + 47 q^{77} - 6 q^{79} + 66 q^{80} + 12 q^{82} + 40 q^{83} - 12 q^{85} + 4 q^{86} + 12 q^{88} + 30 q^{89} + 158 q^{92} + 18 q^{94} + 22 q^{95} - 20 q^{97} + 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.599558 −0.423952 −0.211976 0.977275i \(-0.567990\pi\)
−0.211976 + 0.977275i \(0.567990\pi\)
\(3\) 0 0
\(4\) −1.64053 −0.820265
\(5\) −0.261228 −0.116825 −0.0584125 0.998293i \(-0.518604\pi\)
−0.0584125 + 0.998293i \(0.518604\pi\)
\(6\) 0 0
\(7\) −0.444015 −0.167822 −0.0839109 0.996473i \(-0.526741\pi\)
−0.0839109 + 0.996473i \(0.526741\pi\)
\(8\) 2.18271 0.771705
\(9\) 0 0
\(10\) 0.156622 0.0495281
\(11\) 3.27284 0.986800 0.493400 0.869803i \(-0.335754\pi\)
0.493400 + 0.869803i \(0.335754\pi\)
\(12\) 0 0
\(13\) 4.82387 1.33790 0.668951 0.743307i \(-0.266744\pi\)
0.668951 + 0.743307i \(0.266744\pi\)
\(14\) 0.266213 0.0711483
\(15\) 0 0
\(16\) 1.97240 0.493099
\(17\) −0.628330 −0.152392 −0.0761962 0.997093i \(-0.524278\pi\)
−0.0761962 + 0.997093i \(0.524278\pi\)
\(18\) 0 0
\(19\) 6.37907 1.46346 0.731730 0.681595i \(-0.238713\pi\)
0.731730 + 0.681595i \(0.238713\pi\)
\(20\) 0.428553 0.0958274
\(21\) 0 0
\(22\) −1.96226 −0.418356
\(23\) 6.91056 1.44095 0.720476 0.693480i \(-0.243923\pi\)
0.720476 + 0.693480i \(0.243923\pi\)
\(24\) 0 0
\(25\) −4.93176 −0.986352
\(26\) −2.89219 −0.567206
\(27\) 0 0
\(28\) 0.728419 0.137658
\(29\) 3.92656 0.729144 0.364572 0.931175i \(-0.381215\pi\)
0.364572 + 0.931175i \(0.381215\pi\)
\(30\) 0 0
\(31\) 3.84367 0.690344 0.345172 0.938539i \(-0.387821\pi\)
0.345172 + 0.938539i \(0.387821\pi\)
\(32\) −5.54799 −0.980755
\(33\) 0 0
\(34\) 0.376720 0.0646070
\(35\) 0.115989 0.0196058
\(36\) 0 0
\(37\) −7.74015 −1.27247 −0.636236 0.771494i \(-0.719510\pi\)
−0.636236 + 0.771494i \(0.719510\pi\)
\(38\) −3.82463 −0.620436
\(39\) 0 0
\(40\) −0.570186 −0.0901543
\(41\) −12.1523 −1.89786 −0.948932 0.315479i \(-0.897835\pi\)
−0.948932 + 0.315479i \(0.897835\pi\)
\(42\) 0 0
\(43\) 3.58602 0.546863 0.273432 0.961891i \(-0.411841\pi\)
0.273432 + 0.961891i \(0.411841\pi\)
\(44\) −5.36920 −0.809437
\(45\) 0 0
\(46\) −4.14329 −0.610894
\(47\) 6.18985 0.902882 0.451441 0.892301i \(-0.350910\pi\)
0.451441 + 0.892301i \(0.350910\pi\)
\(48\) 0 0
\(49\) −6.80285 −0.971836
\(50\) 2.95688 0.418166
\(51\) 0 0
\(52\) −7.91370 −1.09743
\(53\) 2.50738 0.344415 0.172207 0.985061i \(-0.444910\pi\)
0.172207 + 0.985061i \(0.444910\pi\)
\(54\) 0 0
\(55\) −0.854960 −0.115283
\(56\) −0.969155 −0.129509
\(57\) 0 0
\(58\) −2.35420 −0.309122
\(59\) −8.32542 −1.08388 −0.541939 0.840418i \(-0.682309\pi\)
−0.541939 + 0.840418i \(0.682309\pi\)
\(60\) 0 0
\(61\) 2.21416 0.283494 0.141747 0.989903i \(-0.454728\pi\)
0.141747 + 0.989903i \(0.454728\pi\)
\(62\) −2.30451 −0.292673
\(63\) 0 0
\(64\) −0.618452 −0.0773065
\(65\) −1.26013 −0.156300
\(66\) 0 0
\(67\) 3.22654 0.394184 0.197092 0.980385i \(-0.436850\pi\)
0.197092 + 0.980385i \(0.436850\pi\)
\(68\) 1.03079 0.125002
\(69\) 0 0
\(70\) −0.0695423 −0.00831190
\(71\) 9.85495 1.16957 0.584783 0.811189i \(-0.301179\pi\)
0.584783 + 0.811189i \(0.301179\pi\)
\(72\) 0 0
\(73\) −12.8578 −1.50489 −0.752446 0.658654i \(-0.771126\pi\)
−0.752446 + 0.658654i \(0.771126\pi\)
\(74\) 4.64067 0.539467
\(75\) 0 0
\(76\) −10.4651 −1.20042
\(77\) −1.45319 −0.165606
\(78\) 0 0
\(79\) 9.04869 1.01806 0.509028 0.860750i \(-0.330005\pi\)
0.509028 + 0.860750i \(0.330005\pi\)
\(80\) −0.515246 −0.0576063
\(81\) 0 0
\(82\) 7.28599 0.804603
\(83\) 5.99942 0.658522 0.329261 0.944239i \(-0.393200\pi\)
0.329261 + 0.944239i \(0.393200\pi\)
\(84\) 0 0
\(85\) 0.164138 0.0178032
\(86\) −2.15003 −0.231844
\(87\) 0 0
\(88\) 7.14367 0.761518
\(89\) 4.41461 0.467948 0.233974 0.972243i \(-0.424827\pi\)
0.233974 + 0.972243i \(0.424827\pi\)
\(90\) 0 0
\(91\) −2.14187 −0.224529
\(92\) −11.3370 −1.18196
\(93\) 0 0
\(94\) −3.71118 −0.382778
\(95\) −1.66639 −0.170968
\(96\) 0 0
\(97\) −5.55167 −0.563687 −0.281844 0.959460i \(-0.590946\pi\)
−0.281844 + 0.959460i \(0.590946\pi\)
\(98\) 4.07871 0.412012
\(99\) 0 0
\(100\) 8.09070 0.809070
\(101\) 1.04054 0.103537 0.0517687 0.998659i \(-0.483514\pi\)
0.0517687 + 0.998659i \(0.483514\pi\)
\(102\) 0 0
\(103\) −7.72271 −0.760942 −0.380471 0.924793i \(-0.624238\pi\)
−0.380471 + 0.924793i \(0.624238\pi\)
\(104\) 10.5291 1.03246
\(105\) 0 0
\(106\) −1.50332 −0.146015
\(107\) 20.4149 1.97358 0.986790 0.162003i \(-0.0517955\pi\)
0.986790 + 0.162003i \(0.0517955\pi\)
\(108\) 0 0
\(109\) 8.94231 0.856518 0.428259 0.903656i \(-0.359127\pi\)
0.428259 + 0.903656i \(0.359127\pi\)
\(110\) 0.512598 0.0488743
\(111\) 0 0
\(112\) −0.875773 −0.0827528
\(113\) 5.51469 0.518778 0.259389 0.965773i \(-0.416479\pi\)
0.259389 + 0.965773i \(0.416479\pi\)
\(114\) 0 0
\(115\) −1.80524 −0.168339
\(116\) −6.44164 −0.598091
\(117\) 0 0
\(118\) 4.99157 0.459512
\(119\) 0.278988 0.0255748
\(120\) 0 0
\(121\) −0.288490 −0.0262263
\(122\) −1.32752 −0.120188
\(123\) 0 0
\(124\) −6.30566 −0.566265
\(125\) 2.59446 0.232055
\(126\) 0 0
\(127\) −9.17707 −0.814333 −0.407166 0.913354i \(-0.633483\pi\)
−0.407166 + 0.913354i \(0.633483\pi\)
\(128\) 11.4668 1.01353
\(129\) 0 0
\(130\) 0.755523 0.0662637
\(131\) 12.1510 1.06164 0.530818 0.847486i \(-0.321885\pi\)
0.530818 + 0.847486i \(0.321885\pi\)
\(132\) 0 0
\(133\) −2.83240 −0.245600
\(134\) −1.93450 −0.167115
\(135\) 0 0
\(136\) −1.37146 −0.117602
\(137\) 2.81072 0.240136 0.120068 0.992766i \(-0.461689\pi\)
0.120068 + 0.992766i \(0.461689\pi\)
\(138\) 0 0
\(139\) −11.6351 −0.986879 −0.493440 0.869780i \(-0.664261\pi\)
−0.493440 + 0.869780i \(0.664261\pi\)
\(140\) −0.190284 −0.0160819
\(141\) 0 0
\(142\) −5.90862 −0.495840
\(143\) 15.7878 1.32024
\(144\) 0 0
\(145\) −1.02573 −0.0851821
\(146\) 7.70901 0.638002
\(147\) 0 0
\(148\) 12.6979 1.04376
\(149\) 1.00000 0.0819232
\(150\) 0 0
\(151\) 12.5969 1.02512 0.512561 0.858651i \(-0.328697\pi\)
0.512561 + 0.858651i \(0.328697\pi\)
\(152\) 13.9237 1.12936
\(153\) 0 0
\(154\) 0.871273 0.0702092
\(155\) −1.00408 −0.0806494
\(156\) 0 0
\(157\) −21.5713 −1.72157 −0.860787 0.508965i \(-0.830028\pi\)
−0.860787 + 0.508965i \(0.830028\pi\)
\(158\) −5.42522 −0.431607
\(159\) 0 0
\(160\) 1.44929 0.114577
\(161\) −3.06839 −0.241823
\(162\) 0 0
\(163\) −6.17715 −0.483832 −0.241916 0.970297i \(-0.577776\pi\)
−0.241916 + 0.970297i \(0.577776\pi\)
\(164\) 19.9362 1.55675
\(165\) 0 0
\(166\) −3.59700 −0.279181
\(167\) −9.65140 −0.746848 −0.373424 0.927661i \(-0.621816\pi\)
−0.373424 + 0.927661i \(0.621816\pi\)
\(168\) 0 0
\(169\) 10.2697 0.789980
\(170\) −0.0984101 −0.00754771
\(171\) 0 0
\(172\) −5.88297 −0.448573
\(173\) 19.0109 1.44537 0.722685 0.691178i \(-0.242908\pi\)
0.722685 + 0.691178i \(0.242908\pi\)
\(174\) 0 0
\(175\) 2.18977 0.165531
\(176\) 6.45535 0.486590
\(177\) 0 0
\(178\) −2.64682 −0.198387
\(179\) 16.8228 1.25739 0.628696 0.777651i \(-0.283589\pi\)
0.628696 + 0.777651i \(0.283589\pi\)
\(180\) 0 0
\(181\) −4.57424 −0.340000 −0.170000 0.985444i \(-0.554377\pi\)
−0.170000 + 0.985444i \(0.554377\pi\)
\(182\) 1.28418 0.0951894
\(183\) 0 0
\(184\) 15.0838 1.11199
\(185\) 2.02195 0.148656
\(186\) 0 0
\(187\) −2.05643 −0.150381
\(188\) −10.1546 −0.740602
\(189\) 0 0
\(190\) 0.999101 0.0724824
\(191\) −10.9470 −0.792099 −0.396049 0.918229i \(-0.629619\pi\)
−0.396049 + 0.918229i \(0.629619\pi\)
\(192\) 0 0
\(193\) 1.22630 0.0882713 0.0441357 0.999026i \(-0.485947\pi\)
0.0441357 + 0.999026i \(0.485947\pi\)
\(194\) 3.32855 0.238976
\(195\) 0 0
\(196\) 11.1603 0.797163
\(197\) 23.0036 1.63894 0.819471 0.573121i \(-0.194268\pi\)
0.819471 + 0.573121i \(0.194268\pi\)
\(198\) 0 0
\(199\) 4.64001 0.328922 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(200\) −10.7646 −0.761172
\(201\) 0 0
\(202\) −0.623863 −0.0438949
\(203\) −1.74345 −0.122366
\(204\) 0 0
\(205\) 3.17452 0.221718
\(206\) 4.63022 0.322603
\(207\) 0 0
\(208\) 9.51459 0.659718
\(209\) 20.8777 1.44414
\(210\) 0 0
\(211\) −0.989304 −0.0681065 −0.0340532 0.999420i \(-0.510842\pi\)
−0.0340532 + 0.999420i \(0.510842\pi\)
\(212\) −4.11343 −0.282511
\(213\) 0 0
\(214\) −12.2399 −0.836703
\(215\) −0.936770 −0.0638872
\(216\) 0 0
\(217\) −1.70665 −0.115855
\(218\) −5.36144 −0.363122
\(219\) 0 0
\(220\) 1.40259 0.0945624
\(221\) −3.03098 −0.203886
\(222\) 0 0
\(223\) 17.9115 1.19944 0.599721 0.800209i \(-0.295278\pi\)
0.599721 + 0.800209i \(0.295278\pi\)
\(224\) 2.46339 0.164592
\(225\) 0 0
\(226\) −3.30638 −0.219937
\(227\) −11.8155 −0.784220 −0.392110 0.919918i \(-0.628255\pi\)
−0.392110 + 0.919918i \(0.628255\pi\)
\(228\) 0 0
\(229\) 7.12495 0.470830 0.235415 0.971895i \(-0.424355\pi\)
0.235415 + 0.971895i \(0.424355\pi\)
\(230\) 1.08234 0.0713677
\(231\) 0 0
\(232\) 8.57054 0.562683
\(233\) 6.65663 0.436091 0.218045 0.975939i \(-0.430032\pi\)
0.218045 + 0.975939i \(0.430032\pi\)
\(234\) 0 0
\(235\) −1.61696 −0.105479
\(236\) 13.6581 0.889066
\(237\) 0 0
\(238\) −0.167269 −0.0108425
\(239\) −25.9173 −1.67645 −0.838226 0.545323i \(-0.816407\pi\)
−0.838226 + 0.545323i \(0.816407\pi\)
\(240\) 0 0
\(241\) 3.81308 0.245622 0.122811 0.992430i \(-0.460809\pi\)
0.122811 + 0.992430i \(0.460809\pi\)
\(242\) 0.172967 0.0111187
\(243\) 0 0
\(244\) −3.63240 −0.232540
\(245\) 1.77710 0.113535
\(246\) 0 0
\(247\) 30.7718 1.95796
\(248\) 8.38963 0.532742
\(249\) 0 0
\(250\) −1.55553 −0.0983803
\(251\) −9.02429 −0.569608 −0.284804 0.958586i \(-0.591929\pi\)
−0.284804 + 0.958586i \(0.591929\pi\)
\(252\) 0 0
\(253\) 22.6172 1.42193
\(254\) 5.50219 0.345238
\(255\) 0 0
\(256\) −5.63810 −0.352381
\(257\) −17.5684 −1.09589 −0.547944 0.836515i \(-0.684589\pi\)
−0.547944 + 0.836515i \(0.684589\pi\)
\(258\) 0 0
\(259\) 3.43674 0.213549
\(260\) 2.06728 0.128208
\(261\) 0 0
\(262\) −7.28522 −0.450082
\(263\) 9.80603 0.604665 0.302333 0.953202i \(-0.402235\pi\)
0.302333 + 0.953202i \(0.402235\pi\)
\(264\) 0 0
\(265\) −0.654998 −0.0402362
\(266\) 1.69819 0.104123
\(267\) 0 0
\(268\) −5.29323 −0.323336
\(269\) 3.16294 0.192848 0.0964238 0.995340i \(-0.469260\pi\)
0.0964238 + 0.995340i \(0.469260\pi\)
\(270\) 0 0
\(271\) −9.46191 −0.574770 −0.287385 0.957815i \(-0.592786\pi\)
−0.287385 + 0.957815i \(0.592786\pi\)
\(272\) −1.23932 −0.0751446
\(273\) 0 0
\(274\) −1.68519 −0.101806
\(275\) −16.1409 −0.973332
\(276\) 0 0
\(277\) 18.4411 1.10802 0.554009 0.832511i \(-0.313097\pi\)
0.554009 + 0.832511i \(0.313097\pi\)
\(278\) 6.97594 0.418389
\(279\) 0 0
\(280\) 0.253171 0.0151299
\(281\) 0.316081 0.0188558 0.00942791 0.999956i \(-0.496999\pi\)
0.00942791 + 0.999956i \(0.496999\pi\)
\(282\) 0 0
\(283\) 20.5665 1.22255 0.611276 0.791418i \(-0.290657\pi\)
0.611276 + 0.791418i \(0.290657\pi\)
\(284\) −16.1673 −0.959355
\(285\) 0 0
\(286\) −9.46570 −0.559718
\(287\) 5.39578 0.318503
\(288\) 0 0
\(289\) −16.6052 −0.976777
\(290\) 0.614984 0.0361131
\(291\) 0 0
\(292\) 21.0936 1.23441
\(293\) 24.2034 1.41398 0.706989 0.707224i \(-0.250053\pi\)
0.706989 + 0.707224i \(0.250053\pi\)
\(294\) 0 0
\(295\) 2.17484 0.126624
\(296\) −16.8945 −0.981973
\(297\) 0 0
\(298\) −0.599558 −0.0347315
\(299\) 33.3357 1.92785
\(300\) 0 0
\(301\) −1.59225 −0.0917755
\(302\) −7.55259 −0.434603
\(303\) 0 0
\(304\) 12.5821 0.721631
\(305\) −0.578402 −0.0331192
\(306\) 0 0
\(307\) −28.2176 −1.61046 −0.805232 0.592960i \(-0.797959\pi\)
−0.805232 + 0.592960i \(0.797959\pi\)
\(308\) 2.38400 0.135841
\(309\) 0 0
\(310\) 0.602003 0.0341915
\(311\) −23.0808 −1.30879 −0.654396 0.756152i \(-0.727077\pi\)
−0.654396 + 0.756152i \(0.727077\pi\)
\(312\) 0 0
\(313\) −29.4823 −1.66644 −0.833219 0.552943i \(-0.813505\pi\)
−0.833219 + 0.552943i \(0.813505\pi\)
\(314\) 12.9332 0.729865
\(315\) 0 0
\(316\) −14.8446 −0.835076
\(317\) 22.9025 1.28633 0.643165 0.765728i \(-0.277621\pi\)
0.643165 + 0.765728i \(0.277621\pi\)
\(318\) 0 0
\(319\) 12.8510 0.719519
\(320\) 0.161557 0.00903132
\(321\) 0 0
\(322\) 1.83968 0.102521
\(323\) −4.00816 −0.223020
\(324\) 0 0
\(325\) −23.7902 −1.31964
\(326\) 3.70356 0.205121
\(327\) 0 0
\(328\) −26.5249 −1.46459
\(329\) −2.74838 −0.151523
\(330\) 0 0
\(331\) 19.8813 1.09277 0.546387 0.837533i \(-0.316003\pi\)
0.546387 + 0.837533i \(0.316003\pi\)
\(332\) −9.84222 −0.540162
\(333\) 0 0
\(334\) 5.78658 0.316627
\(335\) −0.842863 −0.0460505
\(336\) 0 0
\(337\) −12.1256 −0.660526 −0.330263 0.943889i \(-0.607137\pi\)
−0.330263 + 0.943889i \(0.607137\pi\)
\(338\) −6.15731 −0.334913
\(339\) 0 0
\(340\) −0.269273 −0.0146034
\(341\) 12.5797 0.681231
\(342\) 0 0
\(343\) 6.12867 0.330917
\(344\) 7.82724 0.422017
\(345\) 0 0
\(346\) −11.3981 −0.612767
\(347\) −12.1614 −0.652858 −0.326429 0.945222i \(-0.605845\pi\)
−0.326429 + 0.945222i \(0.605845\pi\)
\(348\) 0 0
\(349\) −11.3827 −0.609304 −0.304652 0.952464i \(-0.598540\pi\)
−0.304652 + 0.952464i \(0.598540\pi\)
\(350\) −1.31290 −0.0701773
\(351\) 0 0
\(352\) −18.1577 −0.967809
\(353\) 30.1093 1.60255 0.801277 0.598293i \(-0.204154\pi\)
0.801277 + 0.598293i \(0.204154\pi\)
\(354\) 0 0
\(355\) −2.57439 −0.136635
\(356\) −7.24230 −0.383841
\(357\) 0 0
\(358\) −10.0862 −0.533074
\(359\) 14.2687 0.753074 0.376537 0.926402i \(-0.377115\pi\)
0.376537 + 0.926402i \(0.377115\pi\)
\(360\) 0 0
\(361\) 21.6925 1.14171
\(362\) 2.74252 0.144144
\(363\) 0 0
\(364\) 3.51380 0.184173
\(365\) 3.35883 0.175809
\(366\) 0 0
\(367\) −18.1200 −0.945858 −0.472929 0.881101i \(-0.656803\pi\)
−0.472929 + 0.881101i \(0.656803\pi\)
\(368\) 13.6304 0.710533
\(369\) 0 0
\(370\) −1.21228 −0.0630232
\(371\) −1.11331 −0.0578003
\(372\) 0 0
\(373\) 34.0027 1.76059 0.880296 0.474425i \(-0.157344\pi\)
0.880296 + 0.474425i \(0.157344\pi\)
\(374\) 1.23295 0.0637542
\(375\) 0 0
\(376\) 13.5106 0.696758
\(377\) 18.9412 0.975522
\(378\) 0 0
\(379\) 0.110504 0.00567621 0.00283810 0.999996i \(-0.499097\pi\)
0.00283810 + 0.999996i \(0.499097\pi\)
\(380\) 2.73377 0.140239
\(381\) 0 0
\(382\) 6.56338 0.335812
\(383\) 4.21011 0.215126 0.107563 0.994198i \(-0.465695\pi\)
0.107563 + 0.994198i \(0.465695\pi\)
\(384\) 0 0
\(385\) 0.379615 0.0193470
\(386\) −0.735241 −0.0374228
\(387\) 0 0
\(388\) 9.10769 0.462373
\(389\) −8.13526 −0.412474 −0.206237 0.978502i \(-0.566122\pi\)
−0.206237 + 0.978502i \(0.566122\pi\)
\(390\) 0 0
\(391\) −4.34211 −0.219590
\(392\) −14.8487 −0.749970
\(393\) 0 0
\(394\) −13.7920 −0.694832
\(395\) −2.36377 −0.118934
\(396\) 0 0
\(397\) 16.2733 0.816734 0.408367 0.912818i \(-0.366098\pi\)
0.408367 + 0.912818i \(0.366098\pi\)
\(398\) −2.78196 −0.139447
\(399\) 0 0
\(400\) −9.72739 −0.486369
\(401\) −15.3552 −0.766801 −0.383400 0.923582i \(-0.625247\pi\)
−0.383400 + 0.923582i \(0.625247\pi\)
\(402\) 0 0
\(403\) 18.5414 0.923612
\(404\) −1.70703 −0.0849281
\(405\) 0 0
\(406\) 1.04530 0.0518774
\(407\) −25.3323 −1.25568
\(408\) 0 0
\(409\) 14.8831 0.735920 0.367960 0.929842i \(-0.380056\pi\)
0.367960 + 0.929842i \(0.380056\pi\)
\(410\) −1.90331 −0.0939977
\(411\) 0 0
\(412\) 12.6693 0.624174
\(413\) 3.69661 0.181898
\(414\) 0 0
\(415\) −1.56722 −0.0769317
\(416\) −26.7628 −1.31215
\(417\) 0 0
\(418\) −12.5174 −0.612246
\(419\) 30.4137 1.48580 0.742902 0.669400i \(-0.233449\pi\)
0.742902 + 0.669400i \(0.233449\pi\)
\(420\) 0 0
\(421\) 22.4259 1.09297 0.546485 0.837469i \(-0.315966\pi\)
0.546485 + 0.837469i \(0.315966\pi\)
\(422\) 0.593145 0.0288739
\(423\) 0 0
\(424\) 5.47288 0.265786
\(425\) 3.09877 0.150313
\(426\) 0 0
\(427\) −0.983119 −0.0475765
\(428\) −33.4912 −1.61886
\(429\) 0 0
\(430\) 0.561649 0.0270851
\(431\) −21.9702 −1.05827 −0.529134 0.848539i \(-0.677483\pi\)
−0.529134 + 0.848539i \(0.677483\pi\)
\(432\) 0 0
\(433\) 17.6950 0.850367 0.425183 0.905107i \(-0.360210\pi\)
0.425183 + 0.905107i \(0.360210\pi\)
\(434\) 1.02323 0.0491168
\(435\) 0 0
\(436\) −14.6701 −0.702572
\(437\) 44.0830 2.10878
\(438\) 0 0
\(439\) −3.03826 −0.145008 −0.0725040 0.997368i \(-0.523099\pi\)
−0.0725040 + 0.997368i \(0.523099\pi\)
\(440\) −1.86613 −0.0889642
\(441\) 0 0
\(442\) 1.81725 0.0864378
\(443\) −8.37147 −0.397741 −0.198870 0.980026i \(-0.563727\pi\)
−0.198870 + 0.980026i \(0.563727\pi\)
\(444\) 0 0
\(445\) −1.15322 −0.0546680
\(446\) −10.7390 −0.508506
\(447\) 0 0
\(448\) 0.274602 0.0129737
\(449\) −40.5209 −1.91230 −0.956149 0.292881i \(-0.905386\pi\)
−0.956149 + 0.292881i \(0.905386\pi\)
\(450\) 0 0
\(451\) −39.7725 −1.87281
\(452\) −9.04701 −0.425536
\(453\) 0 0
\(454\) 7.08406 0.332472
\(455\) 0.559517 0.0262306
\(456\) 0 0
\(457\) −34.8252 −1.62905 −0.814527 0.580126i \(-0.803003\pi\)
−0.814527 + 0.580126i \(0.803003\pi\)
\(458\) −4.27183 −0.199609
\(459\) 0 0
\(460\) 2.96154 0.138083
\(461\) 38.7754 1.80595 0.902976 0.429691i \(-0.141377\pi\)
0.902976 + 0.429691i \(0.141377\pi\)
\(462\) 0 0
\(463\) 9.24148 0.429488 0.214744 0.976670i \(-0.431108\pi\)
0.214744 + 0.976670i \(0.431108\pi\)
\(464\) 7.74473 0.359540
\(465\) 0 0
\(466\) −3.99104 −0.184881
\(467\) 29.2041 1.35140 0.675701 0.737175i \(-0.263841\pi\)
0.675701 + 0.737175i \(0.263841\pi\)
\(468\) 0 0
\(469\) −1.43263 −0.0661527
\(470\) 0.969464 0.0447181
\(471\) 0 0
\(472\) −18.1720 −0.836433
\(473\) 11.7365 0.539644
\(474\) 0 0
\(475\) −31.4600 −1.44349
\(476\) −0.457688 −0.0209781
\(477\) 0 0
\(478\) 15.5389 0.710735
\(479\) −13.5808 −0.620521 −0.310261 0.950652i \(-0.600416\pi\)
−0.310261 + 0.950652i \(0.600416\pi\)
\(480\) 0 0
\(481\) −37.3375 −1.70244
\(482\) −2.28616 −0.104132
\(483\) 0 0
\(484\) 0.473276 0.0215126
\(485\) 1.45026 0.0658527
\(486\) 0 0
\(487\) 20.2699 0.918515 0.459258 0.888303i \(-0.348115\pi\)
0.459258 + 0.888303i \(0.348115\pi\)
\(488\) 4.83287 0.218774
\(489\) 0 0
\(490\) −1.06547 −0.0481332
\(491\) 0.396394 0.0178890 0.00894451 0.999960i \(-0.497153\pi\)
0.00894451 + 0.999960i \(0.497153\pi\)
\(492\) 0 0
\(493\) −2.46717 −0.111116
\(494\) −18.4495 −0.830082
\(495\) 0 0
\(496\) 7.58125 0.340408
\(497\) −4.37574 −0.196279
\(498\) 0 0
\(499\) 6.61548 0.296149 0.148075 0.988976i \(-0.452692\pi\)
0.148075 + 0.988976i \(0.452692\pi\)
\(500\) −4.25629 −0.190347
\(501\) 0 0
\(502\) 5.41059 0.241486
\(503\) −20.0664 −0.894715 −0.447358 0.894355i \(-0.647635\pi\)
−0.447358 + 0.894355i \(0.647635\pi\)
\(504\) 0 0
\(505\) −0.271818 −0.0120957
\(506\) −13.5603 −0.602830
\(507\) 0 0
\(508\) 15.0552 0.667969
\(509\) 32.9530 1.46061 0.730307 0.683119i \(-0.239377\pi\)
0.730307 + 0.683119i \(0.239377\pi\)
\(510\) 0 0
\(511\) 5.70905 0.252554
\(512\) −19.5532 −0.864137
\(513\) 0 0
\(514\) 10.5333 0.464604
\(515\) 2.01739 0.0888969
\(516\) 0 0
\(517\) 20.2584 0.890964
\(518\) −2.06053 −0.0905343
\(519\) 0 0
\(520\) −2.75050 −0.120618
\(521\) 10.7886 0.472659 0.236329 0.971673i \(-0.424056\pi\)
0.236329 + 0.971673i \(0.424056\pi\)
\(522\) 0 0
\(523\) −3.19598 −0.139750 −0.0698751 0.997556i \(-0.522260\pi\)
−0.0698751 + 0.997556i \(0.522260\pi\)
\(524\) −19.9340 −0.870822
\(525\) 0 0
\(526\) −5.87929 −0.256349
\(527\) −2.41510 −0.105203
\(528\) 0 0
\(529\) 24.7559 1.07634
\(530\) 0.392710 0.0170582
\(531\) 0 0
\(532\) 4.64664 0.201457
\(533\) −58.6210 −2.53916
\(534\) 0 0
\(535\) −5.33295 −0.230563
\(536\) 7.04260 0.304194
\(537\) 0 0
\(538\) −1.89637 −0.0817581
\(539\) −22.2647 −0.959007
\(540\) 0 0
\(541\) 0.540896 0.0232549 0.0116275 0.999932i \(-0.496299\pi\)
0.0116275 + 0.999932i \(0.496299\pi\)
\(542\) 5.67297 0.243675
\(543\) 0 0
\(544\) 3.48597 0.149460
\(545\) −2.33599 −0.100063
\(546\) 0 0
\(547\) −3.20203 −0.136909 −0.0684545 0.997654i \(-0.521807\pi\)
−0.0684545 + 0.997654i \(0.521807\pi\)
\(548\) −4.61107 −0.196975
\(549\) 0 0
\(550\) 9.67740 0.412646
\(551\) 25.0478 1.06707
\(552\) 0 0
\(553\) −4.01775 −0.170852
\(554\) −11.0565 −0.469746
\(555\) 0 0
\(556\) 19.0878 0.809502
\(557\) −27.5147 −1.16583 −0.582917 0.812532i \(-0.698089\pi\)
−0.582917 + 0.812532i \(0.698089\pi\)
\(558\) 0 0
\(559\) 17.2985 0.731649
\(560\) 0.228777 0.00966759
\(561\) 0 0
\(562\) −0.189509 −0.00799396
\(563\) −23.7576 −1.00126 −0.500631 0.865661i \(-0.666899\pi\)
−0.500631 + 0.865661i \(0.666899\pi\)
\(564\) 0 0
\(565\) −1.44059 −0.0606062
\(566\) −12.3308 −0.518303
\(567\) 0 0
\(568\) 21.5105 0.902560
\(569\) 27.3845 1.14802 0.574009 0.818849i \(-0.305387\pi\)
0.574009 + 0.818849i \(0.305387\pi\)
\(570\) 0 0
\(571\) −27.1732 −1.13716 −0.568582 0.822627i \(-0.692508\pi\)
−0.568582 + 0.822627i \(0.692508\pi\)
\(572\) −25.9003 −1.08295
\(573\) 0 0
\(574\) −3.23509 −0.135030
\(575\) −34.0812 −1.42129
\(576\) 0 0
\(577\) 37.6523 1.56748 0.783742 0.621086i \(-0.213308\pi\)
0.783742 + 0.621086i \(0.213308\pi\)
\(578\) 9.95579 0.414106
\(579\) 0 0
\(580\) 1.68274 0.0698719
\(581\) −2.66383 −0.110514
\(582\) 0 0
\(583\) 8.20625 0.339868
\(584\) −28.0649 −1.16133
\(585\) 0 0
\(586\) −14.5114 −0.599459
\(587\) −13.3155 −0.549591 −0.274795 0.961503i \(-0.588610\pi\)
−0.274795 + 0.961503i \(0.588610\pi\)
\(588\) 0 0
\(589\) 24.5191 1.01029
\(590\) −1.30394 −0.0536824
\(591\) 0 0
\(592\) −15.2666 −0.627455
\(593\) −14.2693 −0.585969 −0.292984 0.956117i \(-0.594648\pi\)
−0.292984 + 0.956117i \(0.594648\pi\)
\(594\) 0 0
\(595\) −0.0728795 −0.00298777
\(596\) −1.64053 −0.0671987
\(597\) 0 0
\(598\) −19.9867 −0.817316
\(599\) 17.4816 0.714278 0.357139 0.934051i \(-0.383752\pi\)
0.357139 + 0.934051i \(0.383752\pi\)
\(600\) 0 0
\(601\) 15.4975 0.632157 0.316079 0.948733i \(-0.397634\pi\)
0.316079 + 0.948733i \(0.397634\pi\)
\(602\) 0.954644 0.0389084
\(603\) 0 0
\(604\) −20.6656 −0.840872
\(605\) 0.0753617 0.00306389
\(606\) 0 0
\(607\) 4.01090 0.162797 0.0813987 0.996682i \(-0.474061\pi\)
0.0813987 + 0.996682i \(0.474061\pi\)
\(608\) −35.3910 −1.43529
\(609\) 0 0
\(610\) 0.346786 0.0140409
\(611\) 29.8590 1.20797
\(612\) 0 0
\(613\) −12.5165 −0.505535 −0.252767 0.967527i \(-0.581341\pi\)
−0.252767 + 0.967527i \(0.581341\pi\)
\(614\) 16.9181 0.682759
\(615\) 0 0
\(616\) −3.17189 −0.127799
\(617\) 5.54908 0.223398 0.111699 0.993742i \(-0.464371\pi\)
0.111699 + 0.993742i \(0.464371\pi\)
\(618\) 0 0
\(619\) 43.4692 1.74717 0.873587 0.486668i \(-0.161788\pi\)
0.873587 + 0.486668i \(0.161788\pi\)
\(620\) 1.64722 0.0661539
\(621\) 0 0
\(622\) 13.8383 0.554865
\(623\) −1.96015 −0.0785319
\(624\) 0 0
\(625\) 23.9811 0.959242
\(626\) 17.6764 0.706490
\(627\) 0 0
\(628\) 35.3883 1.41215
\(629\) 4.86337 0.193915
\(630\) 0 0
\(631\) −5.42040 −0.215783 −0.107891 0.994163i \(-0.534410\pi\)
−0.107891 + 0.994163i \(0.534410\pi\)
\(632\) 19.7507 0.785639
\(633\) 0 0
\(634\) −13.7314 −0.545342
\(635\) 2.39731 0.0951344
\(636\) 0 0
\(637\) −32.8161 −1.30022
\(638\) −7.70493 −0.305041
\(639\) 0 0
\(640\) −2.99545 −0.118405
\(641\) 25.5927 1.01085 0.505425 0.862871i \(-0.331336\pi\)
0.505425 + 0.862871i \(0.331336\pi\)
\(642\) 0 0
\(643\) 0.497333 0.0196129 0.00980646 0.999952i \(-0.496878\pi\)
0.00980646 + 0.999952i \(0.496878\pi\)
\(644\) 5.03379 0.198359
\(645\) 0 0
\(646\) 2.40313 0.0945498
\(647\) 29.8394 1.17311 0.586554 0.809910i \(-0.300484\pi\)
0.586554 + 0.809910i \(0.300484\pi\)
\(648\) 0 0
\(649\) −27.2478 −1.06957
\(650\) 14.2636 0.559464
\(651\) 0 0
\(652\) 10.1338 0.396870
\(653\) −34.4463 −1.34799 −0.673995 0.738736i \(-0.735423\pi\)
−0.673995 + 0.738736i \(0.735423\pi\)
\(654\) 0 0
\(655\) −3.17418 −0.124025
\(656\) −23.9691 −0.935836
\(657\) 0 0
\(658\) 1.64782 0.0642386
\(659\) 5.25578 0.204736 0.102368 0.994747i \(-0.467358\pi\)
0.102368 + 0.994747i \(0.467358\pi\)
\(660\) 0 0
\(661\) 2.37505 0.0923786 0.0461893 0.998933i \(-0.485292\pi\)
0.0461893 + 0.998933i \(0.485292\pi\)
\(662\) −11.9200 −0.463284
\(663\) 0 0
\(664\) 13.0950 0.508184
\(665\) 0.739904 0.0286922
\(666\) 0 0
\(667\) 27.1347 1.05066
\(668\) 15.8334 0.612613
\(669\) 0 0
\(670\) 0.505346 0.0195232
\(671\) 7.24660 0.279752
\(672\) 0 0
\(673\) −25.2096 −0.971757 −0.485879 0.874026i \(-0.661500\pi\)
−0.485879 + 0.874026i \(0.661500\pi\)
\(674\) 7.27003 0.280031
\(675\) 0 0
\(676\) −16.8478 −0.647992
\(677\) 11.2836 0.433662 0.216831 0.976209i \(-0.430428\pi\)
0.216831 + 0.976209i \(0.430428\pi\)
\(678\) 0 0
\(679\) 2.46502 0.0945989
\(680\) 0.358265 0.0137388
\(681\) 0 0
\(682\) −7.54229 −0.288809
\(683\) −25.0490 −0.958473 −0.479237 0.877686i \(-0.659086\pi\)
−0.479237 + 0.877686i \(0.659086\pi\)
\(684\) 0 0
\(685\) −0.734239 −0.0280538
\(686\) −3.67449 −0.140293
\(687\) 0 0
\(688\) 7.07306 0.269658
\(689\) 12.0953 0.460793
\(690\) 0 0
\(691\) −29.1571 −1.10919 −0.554594 0.832121i \(-0.687126\pi\)
−0.554594 + 0.832121i \(0.687126\pi\)
\(692\) −31.1879 −1.18559
\(693\) 0 0
\(694\) 7.29147 0.276780
\(695\) 3.03943 0.115292
\(696\) 0 0
\(697\) 7.63563 0.289220
\(698\) 6.82462 0.258316
\(699\) 0 0
\(700\) −3.59239 −0.135780
\(701\) 19.8243 0.748753 0.374377 0.927277i \(-0.377857\pi\)
0.374377 + 0.927277i \(0.377857\pi\)
\(702\) 0 0
\(703\) −49.3750 −1.86221
\(704\) −2.02410 −0.0762860
\(705\) 0 0
\(706\) −18.0523 −0.679406
\(707\) −0.462014 −0.0173758
\(708\) 0 0
\(709\) −48.3680 −1.81650 −0.908250 0.418428i \(-0.862581\pi\)
−0.908250 + 0.418428i \(0.862581\pi\)
\(710\) 1.54350 0.0579265
\(711\) 0 0
\(712\) 9.63582 0.361118
\(713\) 26.5620 0.994753
\(714\) 0 0
\(715\) −4.12422 −0.154237
\(716\) −27.5983 −1.03140
\(717\) 0 0
\(718\) −8.55493 −0.319267
\(719\) −19.4064 −0.723736 −0.361868 0.932229i \(-0.617861\pi\)
−0.361868 + 0.932229i \(0.617861\pi\)
\(720\) 0 0
\(721\) 3.42900 0.127703
\(722\) −13.0060 −0.484031
\(723\) 0 0
\(724\) 7.50417 0.278890
\(725\) −19.3648 −0.719192
\(726\) 0 0
\(727\) 17.8775 0.663041 0.331521 0.943448i \(-0.392438\pi\)
0.331521 + 0.943448i \(0.392438\pi\)
\(728\) −4.67508 −0.173270
\(729\) 0 0
\(730\) −2.01381 −0.0745345
\(731\) −2.25320 −0.0833378
\(732\) 0 0
\(733\) 37.9664 1.40232 0.701161 0.713003i \(-0.252666\pi\)
0.701161 + 0.713003i \(0.252666\pi\)
\(734\) 10.8640 0.400998
\(735\) 0 0
\(736\) −38.3397 −1.41322
\(737\) 10.5600 0.388981
\(738\) 0 0
\(739\) 45.1250 1.65995 0.829976 0.557800i \(-0.188354\pi\)
0.829976 + 0.557800i \(0.188354\pi\)
\(740\) −3.31706 −0.121938
\(741\) 0 0
\(742\) 0.667495 0.0245045
\(743\) 3.12372 0.114598 0.0572990 0.998357i \(-0.481751\pi\)
0.0572990 + 0.998357i \(0.481751\pi\)
\(744\) 0 0
\(745\) −0.261228 −0.00957067
\(746\) −20.3866 −0.746406
\(747\) 0 0
\(748\) 3.37363 0.123352
\(749\) −9.06450 −0.331210
\(750\) 0 0
\(751\) −0.918626 −0.0335211 −0.0167606 0.999860i \(-0.505335\pi\)
−0.0167606 + 0.999860i \(0.505335\pi\)
\(752\) 12.2088 0.445210
\(753\) 0 0
\(754\) −11.3564 −0.413574
\(755\) −3.29067 −0.119760
\(756\) 0 0
\(757\) −49.7475 −1.80810 −0.904051 0.427424i \(-0.859421\pi\)
−0.904051 + 0.427424i \(0.859421\pi\)
\(758\) −0.0662536 −0.00240644
\(759\) 0 0
\(760\) −3.63726 −0.131937
\(761\) −29.8519 −1.08213 −0.541065 0.840981i \(-0.681979\pi\)
−0.541065 + 0.840981i \(0.681979\pi\)
\(762\) 0 0
\(763\) −3.97052 −0.143742
\(764\) 17.9589 0.649731
\(765\) 0 0
\(766\) −2.52420 −0.0912032
\(767\) −40.1607 −1.45012
\(768\) 0 0
\(769\) −40.8013 −1.47133 −0.735666 0.677345i \(-0.763131\pi\)
−0.735666 + 0.677345i \(0.763131\pi\)
\(770\) −0.227601 −0.00820218
\(771\) 0 0
\(772\) −2.01179 −0.0724059
\(773\) −23.0022 −0.827330 −0.413665 0.910429i \(-0.635752\pi\)
−0.413665 + 0.910429i \(0.635752\pi\)
\(774\) 0 0
\(775\) −18.9561 −0.680922
\(776\) −12.1177 −0.435000
\(777\) 0 0
\(778\) 4.87756 0.174869
\(779\) −77.5202 −2.77745
\(780\) 0 0
\(781\) 32.2537 1.15413
\(782\) 2.60335 0.0930956
\(783\) 0 0
\(784\) −13.4179 −0.479212
\(785\) 5.63503 0.201123
\(786\) 0 0
\(787\) 44.6790 1.59263 0.796317 0.604880i \(-0.206779\pi\)
0.796317 + 0.604880i \(0.206779\pi\)
\(788\) −37.7382 −1.34437
\(789\) 0 0
\(790\) 1.41722 0.0504224
\(791\) −2.44860 −0.0870623
\(792\) 0 0
\(793\) 10.6808 0.379287
\(794\) −9.75679 −0.346256
\(795\) 0 0
\(796\) −7.61207 −0.269803
\(797\) −29.5661 −1.04728 −0.523642 0.851939i \(-0.675427\pi\)
−0.523642 + 0.851939i \(0.675427\pi\)
\(798\) 0 0
\(799\) −3.88927 −0.137592
\(800\) 27.3613 0.967370
\(801\) 0 0
\(802\) 9.20632 0.325087
\(803\) −42.0816 −1.48503
\(804\) 0 0
\(805\) 0.801551 0.0282510
\(806\) −11.1166 −0.391567
\(807\) 0 0
\(808\) 2.27119 0.0799003
\(809\) 39.7260 1.39669 0.698346 0.715760i \(-0.253920\pi\)
0.698346 + 0.715760i \(0.253920\pi\)
\(810\) 0 0
\(811\) 17.2842 0.606930 0.303465 0.952843i \(-0.401857\pi\)
0.303465 + 0.952843i \(0.401857\pi\)
\(812\) 2.86018 0.100373
\(813\) 0 0
\(814\) 15.1882 0.532346
\(815\) 1.61365 0.0565236
\(816\) 0 0
\(817\) 22.8755 0.800312
\(818\) −8.92327 −0.311995
\(819\) 0 0
\(820\) −5.20789 −0.181867
\(821\) 44.8857 1.56652 0.783261 0.621694i \(-0.213555\pi\)
0.783261 + 0.621694i \(0.213555\pi\)
\(822\) 0 0
\(823\) −30.2619 −1.05486 −0.527431 0.849598i \(-0.676845\pi\)
−0.527431 + 0.849598i \(0.676845\pi\)
\(824\) −16.8564 −0.587222
\(825\) 0 0
\(826\) −2.21633 −0.0771160
\(827\) 38.7434 1.34724 0.673620 0.739077i \(-0.264738\pi\)
0.673620 + 0.739077i \(0.264738\pi\)
\(828\) 0 0
\(829\) 27.5153 0.955645 0.477823 0.878456i \(-0.341426\pi\)
0.477823 + 0.878456i \(0.341426\pi\)
\(830\) 0.939639 0.0326153
\(831\) 0 0
\(832\) −2.98333 −0.103428
\(833\) 4.27443 0.148100
\(834\) 0 0
\(835\) 2.52122 0.0872504
\(836\) −34.2505 −1.18458
\(837\) 0 0
\(838\) −18.2348 −0.629910
\(839\) 4.74016 0.163648 0.0818242 0.996647i \(-0.473925\pi\)
0.0818242 + 0.996647i \(0.473925\pi\)
\(840\) 0 0
\(841\) −13.5821 −0.468350
\(842\) −13.4456 −0.463367
\(843\) 0 0
\(844\) 1.62298 0.0558654
\(845\) −2.68275 −0.0922893
\(846\) 0 0
\(847\) 0.128094 0.00440135
\(848\) 4.94554 0.169831
\(849\) 0 0
\(850\) −1.85789 −0.0637253
\(851\) −53.4888 −1.83357
\(852\) 0 0
\(853\) 52.6087 1.80129 0.900643 0.434559i \(-0.143096\pi\)
0.900643 + 0.434559i \(0.143096\pi\)
\(854\) 0.589438 0.0201701
\(855\) 0 0
\(856\) 44.5598 1.52302
\(857\) −2.38074 −0.0813244 −0.0406622 0.999173i \(-0.512947\pi\)
−0.0406622 + 0.999173i \(0.512947\pi\)
\(858\) 0 0
\(859\) 4.31177 0.147116 0.0735578 0.997291i \(-0.476565\pi\)
0.0735578 + 0.997291i \(0.476565\pi\)
\(860\) 1.53680 0.0524044
\(861\) 0 0
\(862\) 13.1724 0.448654
\(863\) −33.2810 −1.13290 −0.566449 0.824097i \(-0.691683\pi\)
−0.566449 + 0.824097i \(0.691683\pi\)
\(864\) 0 0
\(865\) −4.96618 −0.168855
\(866\) −10.6092 −0.360514
\(867\) 0 0
\(868\) 2.79981 0.0950316
\(869\) 29.6149 1.00462
\(870\) 0 0
\(871\) 15.5644 0.527380
\(872\) 19.5185 0.660979
\(873\) 0 0
\(874\) −26.4303 −0.894019
\(875\) −1.15198 −0.0389439
\(876\) 0 0
\(877\) 3.71046 0.125293 0.0626467 0.998036i \(-0.480046\pi\)
0.0626467 + 0.998036i \(0.480046\pi\)
\(878\) 1.82161 0.0614764
\(879\) 0 0
\(880\) −1.68632 −0.0568459
\(881\) −31.9505 −1.07644 −0.538219 0.842805i \(-0.680903\pi\)
−0.538219 + 0.842805i \(0.680903\pi\)
\(882\) 0 0
\(883\) −1.05315 −0.0354412 −0.0177206 0.999843i \(-0.505641\pi\)
−0.0177206 + 0.999843i \(0.505641\pi\)
\(884\) 4.97242 0.167240
\(885\) 0 0
\(886\) 5.01919 0.168623
\(887\) 45.8752 1.54034 0.770169 0.637840i \(-0.220172\pi\)
0.770169 + 0.637840i \(0.220172\pi\)
\(888\) 0 0
\(889\) 4.07475 0.136663
\(890\) 0.691424 0.0231766
\(891\) 0 0
\(892\) −29.3843 −0.983860
\(893\) 39.4855 1.32133
\(894\) 0 0
\(895\) −4.39459 −0.146895
\(896\) −5.09141 −0.170092
\(897\) 0 0
\(898\) 24.2946 0.810722
\(899\) 15.0924 0.503360
\(900\) 0 0
\(901\) −1.57546 −0.0524862
\(902\) 23.8459 0.793982
\(903\) 0 0
\(904\) 12.0370 0.400344
\(905\) 1.19492 0.0397205
\(906\) 0 0
\(907\) 32.5597 1.08113 0.540563 0.841304i \(-0.318211\pi\)
0.540563 + 0.841304i \(0.318211\pi\)
\(908\) 19.3836 0.643268
\(909\) 0 0
\(910\) −0.335463 −0.0111205
\(911\) −43.9638 −1.45658 −0.728292 0.685267i \(-0.759686\pi\)
−0.728292 + 0.685267i \(0.759686\pi\)
\(912\) 0 0
\(913\) 19.6352 0.649829
\(914\) 20.8797 0.690640
\(915\) 0 0
\(916\) −11.6887 −0.386205
\(917\) −5.39521 −0.178165
\(918\) 0 0
\(919\) 4.69180 0.154768 0.0773840 0.997001i \(-0.475343\pi\)
0.0773840 + 0.997001i \(0.475343\pi\)
\(920\) −3.94031 −0.129908
\(921\) 0 0
\(922\) −23.2481 −0.765637
\(923\) 47.5390 1.56476
\(924\) 0 0
\(925\) 38.1726 1.25511
\(926\) −5.54081 −0.182082
\(927\) 0 0
\(928\) −21.7845 −0.715111
\(929\) −34.3079 −1.12561 −0.562803 0.826591i \(-0.690277\pi\)
−0.562803 + 0.826591i \(0.690277\pi\)
\(930\) 0 0
\(931\) −43.3959 −1.42224
\(932\) −10.9204 −0.357710
\(933\) 0 0
\(934\) −17.5095 −0.572930
\(935\) 0.537197 0.0175682
\(936\) 0 0
\(937\) −23.3932 −0.764223 −0.382111 0.924116i \(-0.624803\pi\)
−0.382111 + 0.924116i \(0.624803\pi\)
\(938\) 0.858945 0.0280456
\(939\) 0 0
\(940\) 2.65268 0.0865208
\(941\) −52.0248 −1.69596 −0.847979 0.530029i \(-0.822181\pi\)
−0.847979 + 0.530029i \(0.822181\pi\)
\(942\) 0 0
\(943\) −83.9790 −2.73473
\(944\) −16.4210 −0.534459
\(945\) 0 0
\(946\) −7.03671 −0.228783
\(947\) 49.5994 1.61176 0.805882 0.592076i \(-0.201692\pi\)
0.805882 + 0.592076i \(0.201692\pi\)
\(948\) 0 0
\(949\) −62.0244 −2.01340
\(950\) 18.8621 0.611968
\(951\) 0 0
\(952\) 0.608949 0.0197362
\(953\) 22.6710 0.734384 0.367192 0.930145i \(-0.380319\pi\)
0.367192 + 0.930145i \(0.380319\pi\)
\(954\) 0 0
\(955\) 2.85967 0.0925369
\(956\) 42.5181 1.37513
\(957\) 0 0
\(958\) 8.14246 0.263071
\(959\) −1.24800 −0.0403000
\(960\) 0 0
\(961\) −16.2262 −0.523425
\(962\) 22.3860 0.721754
\(963\) 0 0
\(964\) −6.25547 −0.201475
\(965\) −0.320346 −0.0103123
\(966\) 0 0
\(967\) 6.82815 0.219579 0.109789 0.993955i \(-0.464982\pi\)
0.109789 + 0.993955i \(0.464982\pi\)
\(968\) −0.629690 −0.0202390
\(969\) 0 0
\(970\) −0.869513 −0.0279184
\(971\) 21.5753 0.692383 0.346191 0.938164i \(-0.387475\pi\)
0.346191 + 0.938164i \(0.387475\pi\)
\(972\) 0 0
\(973\) 5.16617 0.165620
\(974\) −12.1530 −0.389406
\(975\) 0 0
\(976\) 4.36720 0.139791
\(977\) −42.6020 −1.36296 −0.681480 0.731837i \(-0.738663\pi\)
−0.681480 + 0.731837i \(0.738663\pi\)
\(978\) 0 0
\(979\) 14.4483 0.461771
\(980\) −2.91538 −0.0931285
\(981\) 0 0
\(982\) −0.237661 −0.00758408
\(983\) −41.6211 −1.32751 −0.663753 0.747952i \(-0.731038\pi\)
−0.663753 + 0.747952i \(0.731038\pi\)
\(984\) 0 0
\(985\) −6.00920 −0.191469
\(986\) 1.47921 0.0471078
\(987\) 0 0
\(988\) −50.4821 −1.60605
\(989\) 24.7814 0.788004
\(990\) 0 0
\(991\) −13.1346 −0.417233 −0.208616 0.977998i \(-0.566896\pi\)
−0.208616 + 0.977998i \(0.566896\pi\)
\(992\) −21.3247 −0.677058
\(993\) 0 0
\(994\) 2.62351 0.0832127
\(995\) −1.21210 −0.0384262
\(996\) 0 0
\(997\) 30.3719 0.961887 0.480943 0.876752i \(-0.340294\pi\)
0.480943 + 0.876752i \(0.340294\pi\)
\(998\) −3.96636 −0.125553
\(999\) 0 0
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 4023.2.a.f.1.9 yes 25
3.2 odd 2 4023.2.a.e.1.17 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4023.2.a.e.1.17 25 3.2 odd 2
4023.2.a.f.1.9 yes 25 1.1 even 1 trivial