Properties

Label 4023.2.a.e.1.2
Level $4023$
Weight $2$
Character 4023.1
Self dual yes
Analytic conductor $32.124$
Analytic rank $1$
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: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
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)
 
\(f(q)\) \(=\) \(q-2.73865 q^{2} +5.50022 q^{4} -2.48832 q^{5} +2.47965 q^{7} -9.58588 q^{8} +O(q^{10})\) \(q-2.73865 q^{2} +5.50022 q^{4} -2.48832 q^{5} +2.47965 q^{7} -9.58588 q^{8} +6.81463 q^{10} +3.17008 q^{11} +6.77948 q^{13} -6.79090 q^{14} +15.2520 q^{16} +3.00920 q^{17} -6.29624 q^{19} -13.6863 q^{20} -8.68174 q^{22} +2.42607 q^{23} +1.19172 q^{25} -18.5666 q^{26} +13.6386 q^{28} -0.314777 q^{29} -7.36807 q^{31} -22.5981 q^{32} -8.24116 q^{34} -6.17015 q^{35} -10.4814 q^{37} +17.2432 q^{38} +23.8527 q^{40} -6.72800 q^{41} -7.53813 q^{43} +17.4361 q^{44} -6.64416 q^{46} -8.67317 q^{47} -0.851331 q^{49} -3.26369 q^{50} +37.2886 q^{52} +4.75661 q^{53} -7.88816 q^{55} -23.7696 q^{56} +0.862065 q^{58} +6.54400 q^{59} -8.08687 q^{61} +20.1786 q^{62} +31.3843 q^{64} -16.8695 q^{65} -7.75384 q^{67} +16.5513 q^{68} +16.8979 q^{70} +5.97566 q^{71} +1.31360 q^{73} +28.7050 q^{74} -34.6307 q^{76} +7.86069 q^{77} +14.3076 q^{79} -37.9517 q^{80} +18.4257 q^{82} -4.52789 q^{83} -7.48785 q^{85} +20.6443 q^{86} -30.3880 q^{88} +4.07376 q^{89} +16.8107 q^{91} +13.3439 q^{92} +23.7528 q^{94} +15.6670 q^{95} +8.45212 q^{97} +2.33150 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73865 −1.93652 −0.968260 0.249946i \(-0.919587\pi\)
−0.968260 + 0.249946i \(0.919587\pi\)
\(3\) 0 0
\(4\) 5.50022 2.75011
\(5\) −2.48832 −1.11281 −0.556404 0.830912i \(-0.687819\pi\)
−0.556404 + 0.830912i \(0.687819\pi\)
\(6\) 0 0
\(7\) 2.47965 0.937220 0.468610 0.883405i \(-0.344755\pi\)
0.468610 + 0.883405i \(0.344755\pi\)
\(8\) −9.58588 −3.38912
\(9\) 0 0
\(10\) 6.81463 2.15498
\(11\) 3.17008 0.955815 0.477907 0.878410i \(-0.341395\pi\)
0.477907 + 0.878410i \(0.341395\pi\)
\(12\) 0 0
\(13\) 6.77948 1.88029 0.940145 0.340775i \(-0.110689\pi\)
0.940145 + 0.340775i \(0.110689\pi\)
\(14\) −6.79090 −1.81494
\(15\) 0 0
\(16\) 15.2520 3.81299
\(17\) 3.00920 0.729839 0.364920 0.931039i \(-0.381096\pi\)
0.364920 + 0.931039i \(0.381096\pi\)
\(18\) 0 0
\(19\) −6.29624 −1.44446 −0.722228 0.691655i \(-0.756882\pi\)
−0.722228 + 0.691655i \(0.756882\pi\)
\(20\) −13.6863 −3.06034
\(21\) 0 0
\(22\) −8.68174 −1.85095
\(23\) 2.42607 0.505871 0.252935 0.967483i \(-0.418604\pi\)
0.252935 + 0.967483i \(0.418604\pi\)
\(24\) 0 0
\(25\) 1.19172 0.238343
\(26\) −18.5666 −3.64122
\(27\) 0 0
\(28\) 13.6386 2.57746
\(29\) −0.314777 −0.0584526 −0.0292263 0.999573i \(-0.509304\pi\)
−0.0292263 + 0.999573i \(0.509304\pi\)
\(30\) 0 0
\(31\) −7.36807 −1.32334 −0.661672 0.749793i \(-0.730153\pi\)
−0.661672 + 0.749793i \(0.730153\pi\)
\(32\) −22.5981 −3.99481
\(33\) 0 0
\(34\) −8.24116 −1.41335
\(35\) −6.17015 −1.04295
\(36\) 0 0
\(37\) −10.4814 −1.72313 −0.861567 0.507643i \(-0.830517\pi\)
−0.861567 + 0.507643i \(0.830517\pi\)
\(38\) 17.2432 2.79722
\(39\) 0 0
\(40\) 23.8527 3.77144
\(41\) −6.72800 −1.05074 −0.525369 0.850875i \(-0.676073\pi\)
−0.525369 + 0.850875i \(0.676073\pi\)
\(42\) 0 0
\(43\) −7.53813 −1.14955 −0.574777 0.818310i \(-0.694911\pi\)
−0.574777 + 0.818310i \(0.694911\pi\)
\(44\) 17.4361 2.62859
\(45\) 0 0
\(46\) −6.64416 −0.979628
\(47\) −8.67317 −1.26511 −0.632556 0.774515i \(-0.717994\pi\)
−0.632556 + 0.774515i \(0.717994\pi\)
\(48\) 0 0
\(49\) −0.851331 −0.121619
\(50\) −3.26369 −0.461556
\(51\) 0 0
\(52\) 37.2886 5.17100
\(53\) 4.75661 0.653371 0.326685 0.945133i \(-0.394068\pi\)
0.326685 + 0.945133i \(0.394068\pi\)
\(54\) 0 0
\(55\) −7.88816 −1.06364
\(56\) −23.7696 −3.17635
\(57\) 0 0
\(58\) 0.862065 0.113195
\(59\) 6.54400 0.851956 0.425978 0.904733i \(-0.359930\pi\)
0.425978 + 0.904733i \(0.359930\pi\)
\(60\) 0 0
\(61\) −8.08687 −1.03542 −0.517709 0.855557i \(-0.673215\pi\)
−0.517709 + 0.855557i \(0.673215\pi\)
\(62\) 20.1786 2.56268
\(63\) 0 0
\(64\) 31.3843 3.92304
\(65\) −16.8695 −2.09240
\(66\) 0 0
\(67\) −7.75384 −0.947283 −0.473641 0.880718i \(-0.657061\pi\)
−0.473641 + 0.880718i \(0.657061\pi\)
\(68\) 16.5513 2.00714
\(69\) 0 0
\(70\) 16.8979 2.01969
\(71\) 5.97566 0.709181 0.354590 0.935022i \(-0.384620\pi\)
0.354590 + 0.935022i \(0.384620\pi\)
\(72\) 0 0
\(73\) 1.31360 0.153745 0.0768726 0.997041i \(-0.475507\pi\)
0.0768726 + 0.997041i \(0.475507\pi\)
\(74\) 28.7050 3.33688
\(75\) 0 0
\(76\) −34.6307 −3.97241
\(77\) 7.86069 0.895808
\(78\) 0 0
\(79\) 14.3076 1.60973 0.804867 0.593455i \(-0.202237\pi\)
0.804867 + 0.593455i \(0.202237\pi\)
\(80\) −37.9517 −4.24313
\(81\) 0 0
\(82\) 18.4257 2.03477
\(83\) −4.52789 −0.497001 −0.248500 0.968632i \(-0.579938\pi\)
−0.248500 + 0.968632i \(0.579938\pi\)
\(84\) 0 0
\(85\) −7.48785 −0.812171
\(86\) 20.6443 2.22613
\(87\) 0 0
\(88\) −30.3880 −3.23937
\(89\) 4.07376 0.431817 0.215909 0.976414i \(-0.430729\pi\)
0.215909 + 0.976414i \(0.430729\pi\)
\(90\) 0 0
\(91\) 16.8107 1.76225
\(92\) 13.3439 1.39120
\(93\) 0 0
\(94\) 23.7528 2.44991
\(95\) 15.6670 1.60740
\(96\) 0 0
\(97\) 8.45212 0.858183 0.429091 0.903261i \(-0.358834\pi\)
0.429091 + 0.903261i \(0.358834\pi\)
\(98\) 2.33150 0.235517
\(99\) 0 0
\(100\) 6.55469 0.655469
\(101\) −13.2721 −1.32062 −0.660311 0.750992i \(-0.729576\pi\)
−0.660311 + 0.750992i \(0.729576\pi\)
\(102\) 0 0
\(103\) −9.97594 −0.982959 −0.491479 0.870889i \(-0.663544\pi\)
−0.491479 + 0.870889i \(0.663544\pi\)
\(104\) −64.9873 −6.37253
\(105\) 0 0
\(106\) −13.0267 −1.26527
\(107\) −0.379534 −0.0366909 −0.0183455 0.999832i \(-0.505840\pi\)
−0.0183455 + 0.999832i \(0.505840\pi\)
\(108\) 0 0
\(109\) −2.55120 −0.244360 −0.122180 0.992508i \(-0.538989\pi\)
−0.122180 + 0.992508i \(0.538989\pi\)
\(110\) 21.6029 2.05976
\(111\) 0 0
\(112\) 37.8195 3.57361
\(113\) −19.1068 −1.79742 −0.898709 0.438545i \(-0.855494\pi\)
−0.898709 + 0.438545i \(0.855494\pi\)
\(114\) 0 0
\(115\) −6.03683 −0.562937
\(116\) −1.73134 −0.160751
\(117\) 0 0
\(118\) −17.9217 −1.64983
\(119\) 7.46178 0.684020
\(120\) 0 0
\(121\) −0.950605 −0.0864186
\(122\) 22.1471 2.00511
\(123\) 0 0
\(124\) −40.5260 −3.63934
\(125\) 9.47621 0.847578
\(126\) 0 0
\(127\) 9.06679 0.804547 0.402274 0.915520i \(-0.368220\pi\)
0.402274 + 0.915520i \(0.368220\pi\)
\(128\) −40.7546 −3.60223
\(129\) 0 0
\(130\) 46.1997 4.05198
\(131\) −3.91774 −0.342295 −0.171147 0.985245i \(-0.554747\pi\)
−0.171147 + 0.985245i \(0.554747\pi\)
\(132\) 0 0
\(133\) −15.6125 −1.35377
\(134\) 21.2351 1.83443
\(135\) 0 0
\(136\) −28.8459 −2.47351
\(137\) −22.7783 −1.94608 −0.973040 0.230637i \(-0.925919\pi\)
−0.973040 + 0.230637i \(0.925919\pi\)
\(138\) 0 0
\(139\) −3.47254 −0.294537 −0.147268 0.989097i \(-0.547048\pi\)
−0.147268 + 0.989097i \(0.547048\pi\)
\(140\) −33.9372 −2.86822
\(141\) 0 0
\(142\) −16.3653 −1.37334
\(143\) 21.4915 1.79721
\(144\) 0 0
\(145\) 0.783264 0.0650466
\(146\) −3.59749 −0.297731
\(147\) 0 0
\(148\) −57.6501 −4.73881
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) −15.0059 −1.22116 −0.610581 0.791953i \(-0.709064\pi\)
−0.610581 + 0.791953i \(0.709064\pi\)
\(152\) 60.3550 4.89543
\(153\) 0 0
\(154\) −21.5277 −1.73475
\(155\) 18.3341 1.47263
\(156\) 0 0
\(157\) 4.35510 0.347575 0.173788 0.984783i \(-0.444399\pi\)
0.173788 + 0.984783i \(0.444399\pi\)
\(158\) −39.1836 −3.11728
\(159\) 0 0
\(160\) 56.2311 4.44546
\(161\) 6.01581 0.474112
\(162\) 0 0
\(163\) 14.3238 1.12193 0.560965 0.827840i \(-0.310430\pi\)
0.560965 + 0.827840i \(0.310430\pi\)
\(164\) −37.0055 −2.88964
\(165\) 0 0
\(166\) 12.4003 0.962452
\(167\) −25.4508 −1.96944 −0.984722 0.174137i \(-0.944287\pi\)
−0.984722 + 0.174137i \(0.944287\pi\)
\(168\) 0 0
\(169\) 32.9614 2.53549
\(170\) 20.5066 1.57279
\(171\) 0 0
\(172\) −41.4614 −3.16140
\(173\) −0.602157 −0.0457812 −0.0228906 0.999738i \(-0.507287\pi\)
−0.0228906 + 0.999738i \(0.507287\pi\)
\(174\) 0 0
\(175\) 2.95504 0.223380
\(176\) 48.3499 3.64451
\(177\) 0 0
\(178\) −11.1566 −0.836223
\(179\) 10.6481 0.795880 0.397940 0.917411i \(-0.369725\pi\)
0.397940 + 0.917411i \(0.369725\pi\)
\(180\) 0 0
\(181\) 4.00089 0.297383 0.148692 0.988884i \(-0.452494\pi\)
0.148692 + 0.988884i \(0.452494\pi\)
\(182\) −46.0388 −3.41262
\(183\) 0 0
\(184\) −23.2560 −1.71446
\(185\) 26.0811 1.91752
\(186\) 0 0
\(187\) 9.53941 0.697591
\(188\) −47.7043 −3.47920
\(189\) 0 0
\(190\) −42.9065 −3.11277
\(191\) −0.911105 −0.0659252 −0.0329626 0.999457i \(-0.510494\pi\)
−0.0329626 + 0.999457i \(0.510494\pi\)
\(192\) 0 0
\(193\) −3.52960 −0.254066 −0.127033 0.991899i \(-0.540545\pi\)
−0.127033 + 0.991899i \(0.540545\pi\)
\(194\) −23.1474 −1.66189
\(195\) 0 0
\(196\) −4.68251 −0.334465
\(197\) −14.8739 −1.05972 −0.529862 0.848084i \(-0.677756\pi\)
−0.529862 + 0.848084i \(0.677756\pi\)
\(198\) 0 0
\(199\) 12.2759 0.870213 0.435107 0.900379i \(-0.356711\pi\)
0.435107 + 0.900379i \(0.356711\pi\)
\(200\) −11.4236 −0.807773
\(201\) 0 0
\(202\) 36.3476 2.55741
\(203\) −0.780537 −0.0547829
\(204\) 0 0
\(205\) 16.7414 1.16927
\(206\) 27.3206 1.90352
\(207\) 0 0
\(208\) 103.400 7.16953
\(209\) −19.9596 −1.38063
\(210\) 0 0
\(211\) 12.0371 0.828671 0.414335 0.910124i \(-0.364014\pi\)
0.414335 + 0.910124i \(0.364014\pi\)
\(212\) 26.1624 1.79684
\(213\) 0 0
\(214\) 1.03941 0.0710527
\(215\) 18.7572 1.27923
\(216\) 0 0
\(217\) −18.2702 −1.24026
\(218\) 6.98684 0.473208
\(219\) 0 0
\(220\) −43.3866 −2.92512
\(221\) 20.4008 1.37231
\(222\) 0 0
\(223\) −12.9425 −0.866697 −0.433348 0.901226i \(-0.642668\pi\)
−0.433348 + 0.901226i \(0.642668\pi\)
\(224\) −56.0353 −3.74402
\(225\) 0 0
\(226\) 52.3269 3.48074
\(227\) 23.4952 1.55943 0.779715 0.626135i \(-0.215364\pi\)
0.779715 + 0.626135i \(0.215364\pi\)
\(228\) 0 0
\(229\) 14.0115 0.925909 0.462954 0.886382i \(-0.346789\pi\)
0.462954 + 0.886382i \(0.346789\pi\)
\(230\) 16.5328 1.09014
\(231\) 0 0
\(232\) 3.01741 0.198103
\(233\) −0.283778 −0.0185909 −0.00929545 0.999957i \(-0.502959\pi\)
−0.00929545 + 0.999957i \(0.502959\pi\)
\(234\) 0 0
\(235\) 21.5816 1.40783
\(236\) 35.9934 2.34297
\(237\) 0 0
\(238\) −20.4352 −1.32462
\(239\) −27.1056 −1.75331 −0.876657 0.481115i \(-0.840232\pi\)
−0.876657 + 0.481115i \(0.840232\pi\)
\(240\) 0 0
\(241\) 21.6800 1.39653 0.698265 0.715839i \(-0.253956\pi\)
0.698265 + 0.715839i \(0.253956\pi\)
\(242\) 2.60338 0.167351
\(243\) 0 0
\(244\) −44.4795 −2.84751
\(245\) 2.11838 0.135338
\(246\) 0 0
\(247\) −42.6852 −2.71599
\(248\) 70.6294 4.48497
\(249\) 0 0
\(250\) −25.9521 −1.64135
\(251\) 19.3183 1.21936 0.609681 0.792647i \(-0.291298\pi\)
0.609681 + 0.792647i \(0.291298\pi\)
\(252\) 0 0
\(253\) 7.69083 0.483518
\(254\) −24.8308 −1.55802
\(255\) 0 0
\(256\) 48.8441 3.05276
\(257\) 1.23987 0.0773407 0.0386703 0.999252i \(-0.487688\pi\)
0.0386703 + 0.999252i \(0.487688\pi\)
\(258\) 0 0
\(259\) −25.9903 −1.61496
\(260\) −92.7859 −5.75434
\(261\) 0 0
\(262\) 10.7293 0.662861
\(263\) −27.1610 −1.67482 −0.837409 0.546577i \(-0.815930\pi\)
−0.837409 + 0.546577i \(0.815930\pi\)
\(264\) 0 0
\(265\) −11.8360 −0.727077
\(266\) 42.7571 2.62161
\(267\) 0 0
\(268\) −42.6478 −2.60513
\(269\) −4.77174 −0.290938 −0.145469 0.989363i \(-0.546469\pi\)
−0.145469 + 0.989363i \(0.546469\pi\)
\(270\) 0 0
\(271\) 18.7670 1.14002 0.570008 0.821639i \(-0.306940\pi\)
0.570008 + 0.821639i \(0.306940\pi\)
\(272\) 45.8963 2.78287
\(273\) 0 0
\(274\) 62.3818 3.76862
\(275\) 3.77783 0.227812
\(276\) 0 0
\(277\) −3.49008 −0.209699 −0.104849 0.994488i \(-0.533436\pi\)
−0.104849 + 0.994488i \(0.533436\pi\)
\(278\) 9.51007 0.570376
\(279\) 0 0
\(280\) 59.1464 3.53467
\(281\) −21.0452 −1.25545 −0.627727 0.778434i \(-0.716015\pi\)
−0.627727 + 0.778434i \(0.716015\pi\)
\(282\) 0 0
\(283\) 4.35846 0.259084 0.129542 0.991574i \(-0.458649\pi\)
0.129542 + 0.991574i \(0.458649\pi\)
\(284\) 32.8675 1.95032
\(285\) 0 0
\(286\) −58.8577 −3.48033
\(287\) −16.6831 −0.984772
\(288\) 0 0
\(289\) −7.94469 −0.467335
\(290\) −2.14509 −0.125964
\(291\) 0 0
\(292\) 7.22509 0.422816
\(293\) −25.5865 −1.49478 −0.747390 0.664385i \(-0.768693\pi\)
−0.747390 + 0.664385i \(0.768693\pi\)
\(294\) 0 0
\(295\) −16.2835 −0.948064
\(296\) 100.474 5.83991
\(297\) 0 0
\(298\) 2.73865 0.158646
\(299\) 16.4475 0.951183
\(300\) 0 0
\(301\) −18.6919 −1.07738
\(302\) 41.0959 2.36481
\(303\) 0 0
\(304\) −96.0299 −5.50769
\(305\) 20.1227 1.15222
\(306\) 0 0
\(307\) −16.4103 −0.936584 −0.468292 0.883574i \(-0.655131\pi\)
−0.468292 + 0.883574i \(0.655131\pi\)
\(308\) 43.2355 2.46357
\(309\) 0 0
\(310\) −50.2107 −2.85178
\(311\) 13.3945 0.759533 0.379767 0.925082i \(-0.376004\pi\)
0.379767 + 0.925082i \(0.376004\pi\)
\(312\) 0 0
\(313\) 2.20881 0.124850 0.0624248 0.998050i \(-0.480117\pi\)
0.0624248 + 0.998050i \(0.480117\pi\)
\(314\) −11.9271 −0.673086
\(315\) 0 0
\(316\) 78.6951 4.42694
\(317\) 15.5054 0.870872 0.435436 0.900220i \(-0.356594\pi\)
0.435436 + 0.900220i \(0.356594\pi\)
\(318\) 0 0
\(319\) −0.997867 −0.0558698
\(320\) −78.0941 −4.36559
\(321\) 0 0
\(322\) −16.4752 −0.918127
\(323\) −18.9467 −1.05422
\(324\) 0 0
\(325\) 8.07921 0.448154
\(326\) −39.2280 −2.17264
\(327\) 0 0
\(328\) 64.4938 3.56107
\(329\) −21.5064 −1.18569
\(330\) 0 0
\(331\) −23.7541 −1.30564 −0.652822 0.757511i \(-0.726415\pi\)
−0.652822 + 0.757511i \(0.726415\pi\)
\(332\) −24.9044 −1.36681
\(333\) 0 0
\(334\) 69.7009 3.81387
\(335\) 19.2940 1.05414
\(336\) 0 0
\(337\) 21.8512 1.19031 0.595156 0.803610i \(-0.297090\pi\)
0.595156 + 0.803610i \(0.297090\pi\)
\(338\) −90.2697 −4.91003
\(339\) 0 0
\(340\) −41.1848 −2.23356
\(341\) −23.3574 −1.26487
\(342\) 0 0
\(343\) −19.4686 −1.05120
\(344\) 72.2596 3.89598
\(345\) 0 0
\(346\) 1.64910 0.0886562
\(347\) 9.38008 0.503549 0.251775 0.967786i \(-0.418986\pi\)
0.251775 + 0.967786i \(0.418986\pi\)
\(348\) 0 0
\(349\) −6.67456 −0.357281 −0.178640 0.983914i \(-0.557170\pi\)
−0.178640 + 0.983914i \(0.557170\pi\)
\(350\) −8.09282 −0.432580
\(351\) 0 0
\(352\) −71.6376 −3.81830
\(353\) 8.47972 0.451330 0.225665 0.974205i \(-0.427545\pi\)
0.225665 + 0.974205i \(0.427545\pi\)
\(354\) 0 0
\(355\) −14.8693 −0.789183
\(356\) 22.4066 1.18754
\(357\) 0 0
\(358\) −29.1616 −1.54124
\(359\) −23.1074 −1.21956 −0.609782 0.792569i \(-0.708743\pi\)
−0.609782 + 0.792569i \(0.708743\pi\)
\(360\) 0 0
\(361\) 20.6426 1.08645
\(362\) −10.9570 −0.575889
\(363\) 0 0
\(364\) 92.4628 4.84637
\(365\) −3.26865 −0.171089
\(366\) 0 0
\(367\) −28.6452 −1.49527 −0.747633 0.664112i \(-0.768810\pi\)
−0.747633 + 0.664112i \(0.768810\pi\)
\(368\) 37.0023 1.92888
\(369\) 0 0
\(370\) −71.4270 −3.71331
\(371\) 11.7947 0.612352
\(372\) 0 0
\(373\) 19.5542 1.01248 0.506238 0.862394i \(-0.331036\pi\)
0.506238 + 0.862394i \(0.331036\pi\)
\(374\) −26.1251 −1.35090
\(375\) 0 0
\(376\) 83.1400 4.28762
\(377\) −2.13402 −0.109908
\(378\) 0 0
\(379\) 4.63031 0.237843 0.118922 0.992904i \(-0.462056\pi\)
0.118922 + 0.992904i \(0.462056\pi\)
\(380\) 86.1720 4.42053
\(381\) 0 0
\(382\) 2.49520 0.127666
\(383\) 2.09803 0.107204 0.0536021 0.998562i \(-0.482930\pi\)
0.0536021 + 0.998562i \(0.482930\pi\)
\(384\) 0 0
\(385\) −19.5599 −0.996863
\(386\) 9.66634 0.492004
\(387\) 0 0
\(388\) 46.4885 2.36010
\(389\) 17.4299 0.883729 0.441864 0.897082i \(-0.354317\pi\)
0.441864 + 0.897082i \(0.354317\pi\)
\(390\) 0 0
\(391\) 7.30054 0.369204
\(392\) 8.16076 0.412181
\(393\) 0 0
\(394\) 40.7345 2.05217
\(395\) −35.6019 −1.79133
\(396\) 0 0
\(397\) 24.4745 1.22834 0.614169 0.789175i \(-0.289491\pi\)
0.614169 + 0.789175i \(0.289491\pi\)
\(398\) −33.6193 −1.68519
\(399\) 0 0
\(400\) 18.1760 0.908800
\(401\) 36.3412 1.81479 0.907396 0.420276i \(-0.138067\pi\)
0.907396 + 0.420276i \(0.138067\pi\)
\(402\) 0 0
\(403\) −49.9517 −2.48827
\(404\) −72.9994 −3.63185
\(405\) 0 0
\(406\) 2.13762 0.106088
\(407\) −33.2269 −1.64700
\(408\) 0 0
\(409\) 0.706028 0.0349108 0.0174554 0.999848i \(-0.494443\pi\)
0.0174554 + 0.999848i \(0.494443\pi\)
\(410\) −45.8488 −2.26431
\(411\) 0 0
\(412\) −54.8698 −2.70324
\(413\) 16.2268 0.798471
\(414\) 0 0
\(415\) 11.2668 0.553067
\(416\) −153.203 −7.51140
\(417\) 0 0
\(418\) 54.6623 2.67362
\(419\) −18.2145 −0.889835 −0.444918 0.895571i \(-0.646767\pi\)
−0.444918 + 0.895571i \(0.646767\pi\)
\(420\) 0 0
\(421\) −29.6571 −1.44540 −0.722699 0.691163i \(-0.757099\pi\)
−0.722699 + 0.691163i \(0.757099\pi\)
\(422\) −32.9655 −1.60474
\(423\) 0 0
\(424\) −45.5963 −2.21435
\(425\) 3.58611 0.173952
\(426\) 0 0
\(427\) −20.0526 −0.970414
\(428\) −2.08752 −0.100904
\(429\) 0 0
\(430\) −51.3696 −2.47726
\(431\) −5.95610 −0.286895 −0.143448 0.989658i \(-0.545819\pi\)
−0.143448 + 0.989658i \(0.545819\pi\)
\(432\) 0 0
\(433\) −27.6458 −1.32857 −0.664285 0.747479i \(-0.731264\pi\)
−0.664285 + 0.747479i \(0.731264\pi\)
\(434\) 50.0358 2.40180
\(435\) 0 0
\(436\) −14.0321 −0.672017
\(437\) −15.2751 −0.730708
\(438\) 0 0
\(439\) 38.8197 1.85276 0.926381 0.376587i \(-0.122902\pi\)
0.926381 + 0.376587i \(0.122902\pi\)
\(440\) 75.6149 3.60480
\(441\) 0 0
\(442\) −55.8708 −2.65750
\(443\) −17.2197 −0.818133 −0.409067 0.912505i \(-0.634146\pi\)
−0.409067 + 0.912505i \(0.634146\pi\)
\(444\) 0 0
\(445\) −10.1368 −0.480530
\(446\) 35.4451 1.67838
\(447\) 0 0
\(448\) 77.8221 3.67675
\(449\) −14.6566 −0.691687 −0.345843 0.938292i \(-0.612407\pi\)
−0.345843 + 0.938292i \(0.612407\pi\)
\(450\) 0 0
\(451\) −21.3283 −1.00431
\(452\) −105.092 −4.94310
\(453\) 0 0
\(454\) −64.3451 −3.01987
\(455\) −41.8304 −1.96104
\(456\) 0 0
\(457\) −36.8089 −1.72185 −0.860925 0.508732i \(-0.830114\pi\)
−0.860925 + 0.508732i \(0.830114\pi\)
\(458\) −38.3727 −1.79304
\(459\) 0 0
\(460\) −33.2039 −1.54814
\(461\) −3.08406 −0.143639 −0.0718194 0.997418i \(-0.522881\pi\)
−0.0718194 + 0.997418i \(0.522881\pi\)
\(462\) 0 0
\(463\) −34.7752 −1.61614 −0.808071 0.589086i \(-0.799488\pi\)
−0.808071 + 0.589086i \(0.799488\pi\)
\(464\) −4.80096 −0.222879
\(465\) 0 0
\(466\) 0.777169 0.0360017
\(467\) −4.65470 −0.215394 −0.107697 0.994184i \(-0.534348\pi\)
−0.107697 + 0.994184i \(0.534348\pi\)
\(468\) 0 0
\(469\) −19.2268 −0.887812
\(470\) −59.1045 −2.72629
\(471\) 0 0
\(472\) −62.7300 −2.88738
\(473\) −23.8965 −1.09876
\(474\) 0 0
\(475\) −7.50332 −0.344276
\(476\) 41.0414 1.88113
\(477\) 0 0
\(478\) 74.2328 3.39533
\(479\) 32.7789 1.49770 0.748852 0.662737i \(-0.230605\pi\)
0.748852 + 0.662737i \(0.230605\pi\)
\(480\) 0 0
\(481\) −71.0586 −3.23999
\(482\) −59.3739 −2.70441
\(483\) 0 0
\(484\) −5.22853 −0.237661
\(485\) −21.0315 −0.954993
\(486\) 0 0
\(487\) 26.1514 1.18503 0.592517 0.805558i \(-0.298134\pi\)
0.592517 + 0.805558i \(0.298134\pi\)
\(488\) 77.5198 3.50915
\(489\) 0 0
\(490\) −5.80151 −0.262085
\(491\) 7.84533 0.354055 0.177027 0.984206i \(-0.443352\pi\)
0.177027 + 0.984206i \(0.443352\pi\)
\(492\) 0 0
\(493\) −0.947228 −0.0426610
\(494\) 116.900 5.25958
\(495\) 0 0
\(496\) −112.378 −5.04590
\(497\) 14.8176 0.664658
\(498\) 0 0
\(499\) −19.1048 −0.855247 −0.427623 0.903957i \(-0.640649\pi\)
−0.427623 + 0.903957i \(0.640649\pi\)
\(500\) 52.1212 2.33093
\(501\) 0 0
\(502\) −52.9062 −2.36132
\(503\) 4.05590 0.180844 0.0904218 0.995904i \(-0.471178\pi\)
0.0904218 + 0.995904i \(0.471178\pi\)
\(504\) 0 0
\(505\) 33.0251 1.46960
\(506\) −21.0625 −0.936343
\(507\) 0 0
\(508\) 49.8693 2.21259
\(509\) 8.75481 0.388050 0.194025 0.980997i \(-0.437846\pi\)
0.194025 + 0.980997i \(0.437846\pi\)
\(510\) 0 0
\(511\) 3.25727 0.144093
\(512\) −52.2578 −2.30949
\(513\) 0 0
\(514\) −3.39556 −0.149772
\(515\) 24.8233 1.09384
\(516\) 0 0
\(517\) −27.4946 −1.20921
\(518\) 71.1783 3.12739
\(519\) 0 0
\(520\) 161.709 7.09140
\(521\) −28.4166 −1.24496 −0.622478 0.782637i \(-0.713874\pi\)
−0.622478 + 0.782637i \(0.713874\pi\)
\(522\) 0 0
\(523\) −33.3726 −1.45928 −0.729642 0.683830i \(-0.760313\pi\)
−0.729642 + 0.683830i \(0.760313\pi\)
\(524\) −21.5484 −0.941348
\(525\) 0 0
\(526\) 74.3845 3.24332
\(527\) −22.1720 −0.965829
\(528\) 0 0
\(529\) −17.1142 −0.744095
\(530\) 32.4146 1.40800
\(531\) 0 0
\(532\) −85.8720 −3.72302
\(533\) −45.6123 −1.97569
\(534\) 0 0
\(535\) 0.944400 0.0408300
\(536\) 74.3274 3.21045
\(537\) 0 0
\(538\) 13.0681 0.563407
\(539\) −2.69879 −0.116245
\(540\) 0 0
\(541\) 13.8890 0.597134 0.298567 0.954389i \(-0.403491\pi\)
0.298567 + 0.954389i \(0.403491\pi\)
\(542\) −51.3963 −2.20766
\(543\) 0 0
\(544\) −68.0022 −2.91557
\(545\) 6.34818 0.271926
\(546\) 0 0
\(547\) 30.7903 1.31650 0.658248 0.752801i \(-0.271298\pi\)
0.658248 + 0.752801i \(0.271298\pi\)
\(548\) −125.285 −5.35193
\(549\) 0 0
\(550\) −10.3462 −0.441162
\(551\) 1.98191 0.0844322
\(552\) 0 0
\(553\) 35.4779 1.50867
\(554\) 9.55812 0.406086
\(555\) 0 0
\(556\) −19.0997 −0.810008
\(557\) 9.45861 0.400774 0.200387 0.979717i \(-0.435780\pi\)
0.200387 + 0.979717i \(0.435780\pi\)
\(558\) 0 0
\(559\) −51.1046 −2.16149
\(560\) −94.1070 −3.97674
\(561\) 0 0
\(562\) 57.6356 2.43121
\(563\) 34.3037 1.44573 0.722864 0.690990i \(-0.242825\pi\)
0.722864 + 0.690990i \(0.242825\pi\)
\(564\) 0 0
\(565\) 47.5438 2.00018
\(566\) −11.9363 −0.501721
\(567\) 0 0
\(568\) −57.2820 −2.40350
\(569\) 44.1953 1.85276 0.926381 0.376587i \(-0.122902\pi\)
0.926381 + 0.376587i \(0.122902\pi\)
\(570\) 0 0
\(571\) 25.1800 1.05375 0.526876 0.849942i \(-0.323363\pi\)
0.526876 + 0.849942i \(0.323363\pi\)
\(572\) 118.208 4.94252
\(573\) 0 0
\(574\) 45.6892 1.90703
\(575\) 2.89118 0.120571
\(576\) 0 0
\(577\) −4.75420 −0.197920 −0.0989600 0.995091i \(-0.531552\pi\)
−0.0989600 + 0.995091i \(0.531552\pi\)
\(578\) 21.7578 0.905003
\(579\) 0 0
\(580\) 4.30812 0.178885
\(581\) −11.2276 −0.465799
\(582\) 0 0
\(583\) 15.0788 0.624501
\(584\) −12.5920 −0.521061
\(585\) 0 0
\(586\) 70.0726 2.89467
\(587\) 17.7706 0.733469 0.366735 0.930326i \(-0.380476\pi\)
0.366735 + 0.930326i \(0.380476\pi\)
\(588\) 0 0
\(589\) 46.3911 1.91151
\(590\) 44.5950 1.83595
\(591\) 0 0
\(592\) −159.862 −6.57030
\(593\) 15.6395 0.642239 0.321120 0.947039i \(-0.395941\pi\)
0.321120 + 0.947039i \(0.395941\pi\)
\(594\) 0 0
\(595\) −18.5673 −0.761183
\(596\) −5.50022 −0.225298
\(597\) 0 0
\(598\) −45.0440 −1.84199
\(599\) −4.52110 −0.184727 −0.0923635 0.995725i \(-0.529442\pi\)
−0.0923635 + 0.995725i \(0.529442\pi\)
\(600\) 0 0
\(601\) 7.15979 0.292054 0.146027 0.989281i \(-0.453351\pi\)
0.146027 + 0.989281i \(0.453351\pi\)
\(602\) 51.1907 2.08638
\(603\) 0 0
\(604\) −82.5357 −3.35833
\(605\) 2.36540 0.0961674
\(606\) 0 0
\(607\) −26.2276 −1.06455 −0.532273 0.846573i \(-0.678662\pi\)
−0.532273 + 0.846573i \(0.678662\pi\)
\(608\) 142.283 5.77033
\(609\) 0 0
\(610\) −55.1090 −2.23130
\(611\) −58.7996 −2.37878
\(612\) 0 0
\(613\) 0.252061 0.0101807 0.00509033 0.999987i \(-0.498380\pi\)
0.00509033 + 0.999987i \(0.498380\pi\)
\(614\) 44.9421 1.81371
\(615\) 0 0
\(616\) −75.3516 −3.03600
\(617\) 8.17449 0.329093 0.164546 0.986369i \(-0.447384\pi\)
0.164546 + 0.986369i \(0.447384\pi\)
\(618\) 0 0
\(619\) 23.4564 0.942791 0.471396 0.881922i \(-0.343750\pi\)
0.471396 + 0.881922i \(0.343750\pi\)
\(620\) 100.841 4.04989
\(621\) 0 0
\(622\) −36.6829 −1.47085
\(623\) 10.1015 0.404708
\(624\) 0 0
\(625\) −29.5384 −1.18154
\(626\) −6.04917 −0.241774
\(627\) 0 0
\(628\) 23.9540 0.955869
\(629\) −31.5407 −1.25761
\(630\) 0 0
\(631\) 25.6859 1.02254 0.511270 0.859420i \(-0.329175\pi\)
0.511270 + 0.859420i \(0.329175\pi\)
\(632\) −137.151 −5.45558
\(633\) 0 0
\(634\) −42.4640 −1.68646
\(635\) −22.5610 −0.895307
\(636\) 0 0
\(637\) −5.77158 −0.228678
\(638\) 2.73281 0.108193
\(639\) 0 0
\(640\) 101.410 4.00860
\(641\) −20.7646 −0.820154 −0.410077 0.912051i \(-0.634498\pi\)
−0.410077 + 0.912051i \(0.634498\pi\)
\(642\) 0 0
\(643\) −46.6968 −1.84154 −0.920771 0.390104i \(-0.872439\pi\)
−0.920771 + 0.390104i \(0.872439\pi\)
\(644\) 33.0882 1.30386
\(645\) 0 0
\(646\) 51.8883 2.04152
\(647\) 25.8031 1.01443 0.507213 0.861821i \(-0.330676\pi\)
0.507213 + 0.861821i \(0.330676\pi\)
\(648\) 0 0
\(649\) 20.7450 0.814312
\(650\) −22.1262 −0.867859
\(651\) 0 0
\(652\) 78.7842 3.08543
\(653\) −34.7608 −1.36029 −0.680147 0.733076i \(-0.738084\pi\)
−0.680147 + 0.733076i \(0.738084\pi\)
\(654\) 0 0
\(655\) 9.74858 0.380909
\(656\) −102.615 −4.00645
\(657\) 0 0
\(658\) 58.8987 2.29611
\(659\) −28.7064 −1.11824 −0.559122 0.829085i \(-0.688862\pi\)
−0.559122 + 0.829085i \(0.688862\pi\)
\(660\) 0 0
\(661\) 22.5659 0.877711 0.438856 0.898558i \(-0.355384\pi\)
0.438856 + 0.898558i \(0.355384\pi\)
\(662\) 65.0543 2.52841
\(663\) 0 0
\(664\) 43.4038 1.68440
\(665\) 38.8487 1.50649
\(666\) 0 0
\(667\) −0.763671 −0.0295695
\(668\) −139.985 −5.41618
\(669\) 0 0
\(670\) −52.8396 −2.04137
\(671\) −25.6360 −0.989667
\(672\) 0 0
\(673\) −4.25388 −0.163975 −0.0819874 0.996633i \(-0.526127\pi\)
−0.0819874 + 0.996633i \(0.526127\pi\)
\(674\) −59.8430 −2.30506
\(675\) 0 0
\(676\) 181.295 6.97287
\(677\) 38.8249 1.49216 0.746081 0.665855i \(-0.231933\pi\)
0.746081 + 0.665855i \(0.231933\pi\)
\(678\) 0 0
\(679\) 20.9583 0.804306
\(680\) 71.7776 2.75255
\(681\) 0 0
\(682\) 63.9677 2.44945
\(683\) −8.06370 −0.308549 −0.154274 0.988028i \(-0.549304\pi\)
−0.154274 + 0.988028i \(0.549304\pi\)
\(684\) 0 0
\(685\) 56.6795 2.16561
\(686\) 53.3176 2.03568
\(687\) 0 0
\(688\) −114.971 −4.38324
\(689\) 32.2474 1.22853
\(690\) 0 0
\(691\) −8.24490 −0.313651 −0.156825 0.987626i \(-0.550126\pi\)
−0.156825 + 0.987626i \(0.550126\pi\)
\(692\) −3.31200 −0.125903
\(693\) 0 0
\(694\) −25.6888 −0.975133
\(695\) 8.64077 0.327763
\(696\) 0 0
\(697\) −20.2459 −0.766869
\(698\) 18.2793 0.691881
\(699\) 0 0
\(700\) 16.2534 0.614319
\(701\) −35.0902 −1.32534 −0.662669 0.748913i \(-0.730576\pi\)
−0.662669 + 0.748913i \(0.730576\pi\)
\(702\) 0 0
\(703\) 65.9935 2.48899
\(704\) 99.4907 3.74970
\(705\) 0 0
\(706\) −23.2230 −0.874010
\(707\) −32.9101 −1.23771
\(708\) 0 0
\(709\) 3.80885 0.143044 0.0715221 0.997439i \(-0.477214\pi\)
0.0715221 + 0.997439i \(0.477214\pi\)
\(710\) 40.7220 1.52827
\(711\) 0 0
\(712\) −39.0505 −1.46348
\(713\) −17.8755 −0.669441
\(714\) 0 0
\(715\) −53.4776 −1.99995
\(716\) 58.5671 2.18876
\(717\) 0 0
\(718\) 63.2832 2.36171
\(719\) 2.73421 0.101969 0.0509844 0.998699i \(-0.483764\pi\)
0.0509844 + 0.998699i \(0.483764\pi\)
\(720\) 0 0
\(721\) −24.7369 −0.921249
\(722\) −56.5329 −2.10394
\(723\) 0 0
\(724\) 22.0057 0.817837
\(725\) −0.375124 −0.0139318
\(726\) 0 0
\(727\) 3.03525 0.112571 0.0562856 0.998415i \(-0.482074\pi\)
0.0562856 + 0.998415i \(0.482074\pi\)
\(728\) −161.146 −5.97246
\(729\) 0 0
\(730\) 8.95170 0.331317
\(731\) −22.6838 −0.838989
\(732\) 0 0
\(733\) −5.33373 −0.197006 −0.0985029 0.995137i \(-0.531405\pi\)
−0.0985029 + 0.995137i \(0.531405\pi\)
\(734\) 78.4492 2.89561
\(735\) 0 0
\(736\) −54.8245 −2.02086
\(737\) −24.5803 −0.905426
\(738\) 0 0
\(739\) 25.4637 0.936697 0.468348 0.883544i \(-0.344849\pi\)
0.468348 + 0.883544i \(0.344849\pi\)
\(740\) 143.452 5.27339
\(741\) 0 0
\(742\) −32.3017 −1.18583
\(743\) −49.4663 −1.81474 −0.907372 0.420330i \(-0.861914\pi\)
−0.907372 + 0.420330i \(0.861914\pi\)
\(744\) 0 0
\(745\) 2.48832 0.0911648
\(746\) −53.5521 −1.96068
\(747\) 0 0
\(748\) 52.4688 1.91845
\(749\) −0.941111 −0.0343874
\(750\) 0 0
\(751\) 6.14112 0.224093 0.112046 0.993703i \(-0.464259\pi\)
0.112046 + 0.993703i \(0.464259\pi\)
\(752\) −132.283 −4.82386
\(753\) 0 0
\(754\) 5.84435 0.212839
\(755\) 37.3394 1.35892
\(756\) 0 0
\(757\) −3.19697 −0.116196 −0.0580979 0.998311i \(-0.518504\pi\)
−0.0580979 + 0.998311i \(0.518504\pi\)
\(758\) −12.6808 −0.460588
\(759\) 0 0
\(760\) −150.182 −5.44768
\(761\) 1.19046 0.0431540 0.0215770 0.999767i \(-0.493131\pi\)
0.0215770 + 0.999767i \(0.493131\pi\)
\(762\) 0 0
\(763\) −6.32607 −0.229019
\(764\) −5.01127 −0.181302
\(765\) 0 0
\(766\) −5.74577 −0.207603
\(767\) 44.3649 1.60192
\(768\) 0 0
\(769\) −22.1117 −0.797367 −0.398684 0.917088i \(-0.630533\pi\)
−0.398684 + 0.917088i \(0.630533\pi\)
\(770\) 53.5677 1.93045
\(771\) 0 0
\(772\) −19.4135 −0.698709
\(773\) 51.8878 1.86628 0.933138 0.359519i \(-0.117059\pi\)
0.933138 + 0.359519i \(0.117059\pi\)
\(774\) 0 0
\(775\) −8.78064 −0.315410
\(776\) −81.0210 −2.90848
\(777\) 0 0
\(778\) −47.7343 −1.71136
\(779\) 42.3611 1.51774
\(780\) 0 0
\(781\) 18.9433 0.677845
\(782\) −19.9936 −0.714971
\(783\) 0 0
\(784\) −12.9845 −0.463731
\(785\) −10.8369 −0.386785
\(786\) 0 0
\(787\) −28.3399 −1.01021 −0.505104 0.863058i \(-0.668546\pi\)
−0.505104 + 0.863058i \(0.668546\pi\)
\(788\) −81.8098 −2.91435
\(789\) 0 0
\(790\) 97.5012 3.46894
\(791\) −47.3782 −1.68458
\(792\) 0 0
\(793\) −54.8248 −1.94689
\(794\) −67.0270 −2.37870
\(795\) 0 0
\(796\) 67.5200 2.39318
\(797\) −49.6000 −1.75692 −0.878461 0.477814i \(-0.841429\pi\)
−0.878461 + 0.477814i \(0.841429\pi\)
\(798\) 0 0
\(799\) −26.0993 −0.923328
\(800\) −26.9305 −0.952135
\(801\) 0 0
\(802\) −99.5259 −3.51438
\(803\) 4.16422 0.146952
\(804\) 0 0
\(805\) −14.9692 −0.527596
\(806\) 136.800 4.81859
\(807\) 0 0
\(808\) 127.225 4.47575
\(809\) 38.3130 1.34701 0.673506 0.739182i \(-0.264788\pi\)
0.673506 + 0.739182i \(0.264788\pi\)
\(810\) 0 0
\(811\) 26.4768 0.929727 0.464863 0.885382i \(-0.346103\pi\)
0.464863 + 0.885382i \(0.346103\pi\)
\(812\) −4.29312 −0.150659
\(813\) 0 0
\(814\) 90.9970 3.18944
\(815\) −35.6422 −1.24849
\(816\) 0 0
\(817\) 47.4618 1.66048
\(818\) −1.93356 −0.0676055
\(819\) 0 0
\(820\) 92.0813 3.21562
\(821\) −7.34677 −0.256404 −0.128202 0.991748i \(-0.540921\pi\)
−0.128202 + 0.991748i \(0.540921\pi\)
\(822\) 0 0
\(823\) −7.73931 −0.269776 −0.134888 0.990861i \(-0.543067\pi\)
−0.134888 + 0.990861i \(0.543067\pi\)
\(824\) 95.6282 3.33137
\(825\) 0 0
\(826\) −44.4397 −1.54625
\(827\) 34.0565 1.18426 0.592130 0.805842i \(-0.298287\pi\)
0.592130 + 0.805842i \(0.298287\pi\)
\(828\) 0 0
\(829\) −38.2306 −1.32780 −0.663901 0.747820i \(-0.731101\pi\)
−0.663901 + 0.747820i \(0.731101\pi\)
\(830\) −30.8559 −1.07102
\(831\) 0 0
\(832\) 212.769 7.37645
\(833\) −2.56183 −0.0887621
\(834\) 0 0
\(835\) 63.3297 2.19161
\(836\) −109.782 −3.79689
\(837\) 0 0
\(838\) 49.8831 1.72318
\(839\) 44.7835 1.54610 0.773049 0.634346i \(-0.218731\pi\)
0.773049 + 0.634346i \(0.218731\pi\)
\(840\) 0 0
\(841\) −28.9009 −0.996583
\(842\) 81.2205 2.79904
\(843\) 0 0
\(844\) 66.2069 2.27894
\(845\) −82.0183 −2.82151
\(846\) 0 0
\(847\) −2.35717 −0.0809932
\(848\) 72.5477 2.49130
\(849\) 0 0
\(850\) −9.82112 −0.336862
\(851\) −25.4287 −0.871683
\(852\) 0 0
\(853\) −5.17950 −0.177343 −0.0886713 0.996061i \(-0.528262\pi\)
−0.0886713 + 0.996061i \(0.528262\pi\)
\(854\) 54.9171 1.87923
\(855\) 0 0
\(856\) 3.63816 0.124350
\(857\) 2.43044 0.0830221 0.0415110 0.999138i \(-0.486783\pi\)
0.0415110 + 0.999138i \(0.486783\pi\)
\(858\) 0 0
\(859\) −27.9715 −0.954376 −0.477188 0.878801i \(-0.658344\pi\)
−0.477188 + 0.878801i \(0.658344\pi\)
\(860\) 103.169 3.51803
\(861\) 0 0
\(862\) 16.3117 0.555578
\(863\) 33.4019 1.13701 0.568507 0.822678i \(-0.307521\pi\)
0.568507 + 0.822678i \(0.307521\pi\)
\(864\) 0 0
\(865\) 1.49836 0.0509457
\(866\) 75.7121 2.57280
\(867\) 0 0
\(868\) −100.490 −3.41086
\(869\) 45.3563 1.53861
\(870\) 0 0
\(871\) −52.5670 −1.78117
\(872\) 24.4555 0.828166
\(873\) 0 0
\(874\) 41.8332 1.41503
\(875\) 23.4977 0.794367
\(876\) 0 0
\(877\) 15.2112 0.513645 0.256822 0.966459i \(-0.417324\pi\)
0.256822 + 0.966459i \(0.417324\pi\)
\(878\) −106.314 −3.58791
\(879\) 0 0
\(880\) −120.310 −4.05564
\(881\) 35.0040 1.17932 0.589658 0.807653i \(-0.299263\pi\)
0.589658 + 0.807653i \(0.299263\pi\)
\(882\) 0 0
\(883\) −43.7983 −1.47393 −0.736966 0.675930i \(-0.763742\pi\)
−0.736966 + 0.675930i \(0.763742\pi\)
\(884\) 112.209 3.77400
\(885\) 0 0
\(886\) 47.1588 1.58433
\(887\) −13.5202 −0.453965 −0.226983 0.973899i \(-0.572886\pi\)
−0.226983 + 0.973899i \(0.572886\pi\)
\(888\) 0 0
\(889\) 22.4825 0.754038
\(890\) 27.7612 0.930556
\(891\) 0 0
\(892\) −71.1868 −2.38351
\(893\) 54.6083 1.82740
\(894\) 0 0
\(895\) −26.4959 −0.885662
\(896\) −101.057 −3.37608
\(897\) 0 0
\(898\) 40.1393 1.33946
\(899\) 2.31930 0.0773529
\(900\) 0 0
\(901\) 14.3136 0.476856
\(902\) 58.4108 1.94487
\(903\) 0 0
\(904\) 183.156 6.09167
\(905\) −9.95547 −0.330931
\(906\) 0 0
\(907\) −36.0600 −1.19735 −0.598676 0.800991i \(-0.704306\pi\)
−0.598676 + 0.800991i \(0.704306\pi\)
\(908\) 129.229 4.28860
\(909\) 0 0
\(910\) 114.559 3.79760
\(911\) 28.8375 0.955430 0.477715 0.878515i \(-0.341465\pi\)
0.477715 + 0.878515i \(0.341465\pi\)
\(912\) 0 0
\(913\) −14.3538 −0.475041
\(914\) 100.807 3.33440
\(915\) 0 0
\(916\) 77.0665 2.54635
\(917\) −9.71464 −0.320806
\(918\) 0 0
\(919\) 25.1788 0.830572 0.415286 0.909691i \(-0.363682\pi\)
0.415286 + 0.909691i \(0.363682\pi\)
\(920\) 57.8683 1.90786
\(921\) 0 0
\(922\) 8.44616 0.278159
\(923\) 40.5119 1.33347
\(924\) 0 0
\(925\) −12.4909 −0.410697
\(926\) 95.2372 3.12969
\(927\) 0 0
\(928\) 7.11335 0.233507
\(929\) −53.1220 −1.74288 −0.871439 0.490504i \(-0.836813\pi\)
−0.871439 + 0.490504i \(0.836813\pi\)
\(930\) 0 0
\(931\) 5.36018 0.175673
\(932\) −1.56084 −0.0511270
\(933\) 0 0
\(934\) 12.7476 0.417114
\(935\) −23.7371 −0.776285
\(936\) 0 0
\(937\) −48.0773 −1.57062 −0.785309 0.619104i \(-0.787496\pi\)
−0.785309 + 0.619104i \(0.787496\pi\)
\(938\) 52.6556 1.71927
\(939\) 0 0
\(940\) 118.703 3.87168
\(941\) 59.0197 1.92399 0.961994 0.273069i \(-0.0880388\pi\)
0.961994 + 0.273069i \(0.0880388\pi\)
\(942\) 0 0
\(943\) −16.3226 −0.531537
\(944\) 99.8089 3.24850
\(945\) 0 0
\(946\) 65.4441 2.12777
\(947\) −14.7766 −0.480176 −0.240088 0.970751i \(-0.577176\pi\)
−0.240088 + 0.970751i \(0.577176\pi\)
\(948\) 0 0
\(949\) 8.90553 0.289086
\(950\) 20.5490 0.666697
\(951\) 0 0
\(952\) −71.5277 −2.31823
\(953\) −10.8816 −0.352489 −0.176244 0.984346i \(-0.556395\pi\)
−0.176244 + 0.984346i \(0.556395\pi\)
\(954\) 0 0
\(955\) 2.26712 0.0733622
\(956\) −149.087 −4.82181
\(957\) 0 0
\(958\) −89.7699 −2.90033
\(959\) −56.4822 −1.82390
\(960\) 0 0
\(961\) 23.2885 0.751240
\(962\) 194.605 6.27431
\(963\) 0 0
\(964\) 119.245 3.84061
\(965\) 8.78275 0.282727
\(966\) 0 0
\(967\) −28.0491 −0.901998 −0.450999 0.892524i \(-0.648932\pi\)
−0.450999 + 0.892524i \(0.648932\pi\)
\(968\) 9.11238 0.292883
\(969\) 0 0
\(970\) 57.5981 1.84936
\(971\) −0.970323 −0.0311391 −0.0155696 0.999879i \(-0.504956\pi\)
−0.0155696 + 0.999879i \(0.504956\pi\)
\(972\) 0 0
\(973\) −8.61068 −0.276046
\(974\) −71.6197 −2.29484
\(975\) 0 0
\(976\) −123.341 −3.94804
\(977\) −11.2790 −0.360847 −0.180423 0.983589i \(-0.557747\pi\)
−0.180423 + 0.983589i \(0.557747\pi\)
\(978\) 0 0
\(979\) 12.9141 0.412737
\(980\) 11.6516 0.372195
\(981\) 0 0
\(982\) −21.4856 −0.685634
\(983\) −14.7779 −0.471341 −0.235671 0.971833i \(-0.575729\pi\)
−0.235671 + 0.971833i \(0.575729\pi\)
\(984\) 0 0
\(985\) 37.0110 1.17927
\(986\) 2.59413 0.0826139
\(987\) 0 0
\(988\) −234.778 −7.46928
\(989\) −18.2880 −0.581526
\(990\) 0 0
\(991\) 17.4544 0.554457 0.277228 0.960804i \(-0.410584\pi\)
0.277228 + 0.960804i \(0.410584\pi\)
\(992\) 166.504 5.28651
\(993\) 0 0
\(994\) −40.5801 −1.28712
\(995\) −30.5462 −0.968381
\(996\) 0 0
\(997\) 14.0910 0.446268 0.223134 0.974788i \(-0.428371\pi\)
0.223134 + 0.974788i \(0.428371\pi\)
\(998\) 52.3213 1.65620
\(999\) 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.e.1.2 25
3.2 odd 2 4023.2.a.f.1.24 yes 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4023.2.a.e.1.2 25 1.1 even 1 trivial
4023.2.a.f.1.24 yes 25 3.2 odd 2