Properties

Label 8020.2.a.e.1.9
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $35$
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] = [8020,2,Mod(1,8020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8020.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: \(64.0400224211\)
Analytic rank: \(0\)
Dimension: \(35\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 8020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.64682 q^{3} -1.00000 q^{5} +1.19631 q^{7} -0.287991 q^{9} +O(q^{10})\) \(q-1.64682 q^{3} -1.00000 q^{5} +1.19631 q^{7} -0.287991 q^{9} -4.54953 q^{11} +0.333289 q^{13} +1.64682 q^{15} +3.17504 q^{17} -5.49241 q^{19} -1.97010 q^{21} -6.05584 q^{23} +1.00000 q^{25} +5.41472 q^{27} -7.32750 q^{29} -9.21716 q^{31} +7.49225 q^{33} -1.19631 q^{35} -9.97265 q^{37} -0.548867 q^{39} +4.20054 q^{41} +7.57956 q^{43} +0.287991 q^{45} +4.25633 q^{47} -5.56885 q^{49} -5.22871 q^{51} +10.7744 q^{53} +4.54953 q^{55} +9.04500 q^{57} +0.501509 q^{59} +13.6677 q^{61} -0.344527 q^{63} -0.333289 q^{65} -12.7643 q^{67} +9.97286 q^{69} -13.3089 q^{71} -2.58878 q^{73} -1.64682 q^{75} -5.44264 q^{77} -7.02408 q^{79} -8.05309 q^{81} -11.8318 q^{83} -3.17504 q^{85} +12.0671 q^{87} +4.84136 q^{89} +0.398717 q^{91} +15.1790 q^{93} +5.49241 q^{95} +7.00650 q^{97} +1.31023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.64682 −0.950791 −0.475395 0.879772i \(-0.657695\pi\)
−0.475395 + 0.879772i \(0.657695\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.19631 0.452162 0.226081 0.974108i \(-0.427409\pi\)
0.226081 + 0.974108i \(0.427409\pi\)
\(8\) 0 0
\(9\) −0.287991 −0.0959971
\(10\) 0 0
\(11\) −4.54953 −1.37174 −0.685868 0.727726i \(-0.740577\pi\)
−0.685868 + 0.727726i \(0.740577\pi\)
\(12\) 0 0
\(13\) 0.333289 0.0924378 0.0462189 0.998931i \(-0.485283\pi\)
0.0462189 + 0.998931i \(0.485283\pi\)
\(14\) 0 0
\(15\) 1.64682 0.425207
\(16\) 0 0
\(17\) 3.17504 0.770060 0.385030 0.922904i \(-0.374191\pi\)
0.385030 + 0.922904i \(0.374191\pi\)
\(18\) 0 0
\(19\) −5.49241 −1.26005 −0.630023 0.776577i \(-0.716954\pi\)
−0.630023 + 0.776577i \(0.716954\pi\)
\(20\) 0 0
\(21\) −1.97010 −0.429911
\(22\) 0 0
\(23\) −6.05584 −1.26273 −0.631365 0.775486i \(-0.717505\pi\)
−0.631365 + 0.775486i \(0.717505\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.41472 1.04206
\(28\) 0 0
\(29\) −7.32750 −1.36068 −0.680341 0.732896i \(-0.738168\pi\)
−0.680341 + 0.732896i \(0.738168\pi\)
\(30\) 0 0
\(31\) −9.21716 −1.65545 −0.827726 0.561133i \(-0.810366\pi\)
−0.827726 + 0.561133i \(0.810366\pi\)
\(32\) 0 0
\(33\) 7.49225 1.30423
\(34\) 0 0
\(35\) −1.19631 −0.202213
\(36\) 0 0
\(37\) −9.97265 −1.63949 −0.819747 0.572726i \(-0.805886\pi\)
−0.819747 + 0.572726i \(0.805886\pi\)
\(38\) 0 0
\(39\) −0.548867 −0.0878890
\(40\) 0 0
\(41\) 4.20054 0.656015 0.328007 0.944675i \(-0.393623\pi\)
0.328007 + 0.944675i \(0.393623\pi\)
\(42\) 0 0
\(43\) 7.57956 1.15587 0.577936 0.816082i \(-0.303858\pi\)
0.577936 + 0.816082i \(0.303858\pi\)
\(44\) 0 0
\(45\) 0.287991 0.0429312
\(46\) 0 0
\(47\) 4.25633 0.620849 0.310425 0.950598i \(-0.399529\pi\)
0.310425 + 0.950598i \(0.399529\pi\)
\(48\) 0 0
\(49\) −5.56885 −0.795549
\(50\) 0 0
\(51\) −5.22871 −0.732166
\(52\) 0 0
\(53\) 10.7744 1.47997 0.739986 0.672622i \(-0.234832\pi\)
0.739986 + 0.672622i \(0.234832\pi\)
\(54\) 0 0
\(55\) 4.54953 0.613459
\(56\) 0 0
\(57\) 9.04500 1.19804
\(58\) 0 0
\(59\) 0.501509 0.0652909 0.0326455 0.999467i \(-0.489607\pi\)
0.0326455 + 0.999467i \(0.489607\pi\)
\(60\) 0 0
\(61\) 13.6677 1.74997 0.874984 0.484152i \(-0.160872\pi\)
0.874984 + 0.484152i \(0.160872\pi\)
\(62\) 0 0
\(63\) −0.344527 −0.0434063
\(64\) 0 0
\(65\) −0.333289 −0.0413394
\(66\) 0 0
\(67\) −12.7643 −1.55941 −0.779706 0.626145i \(-0.784632\pi\)
−0.779706 + 0.626145i \(0.784632\pi\)
\(68\) 0 0
\(69\) 9.97286 1.20059
\(70\) 0 0
\(71\) −13.3089 −1.57947 −0.789736 0.613447i \(-0.789783\pi\)
−0.789736 + 0.613447i \(0.789783\pi\)
\(72\) 0 0
\(73\) −2.58878 −0.302993 −0.151497 0.988458i \(-0.548409\pi\)
−0.151497 + 0.988458i \(0.548409\pi\)
\(74\) 0 0
\(75\) −1.64682 −0.190158
\(76\) 0 0
\(77\) −5.44264 −0.620247
\(78\) 0 0
\(79\) −7.02408 −0.790271 −0.395135 0.918623i \(-0.629302\pi\)
−0.395135 + 0.918623i \(0.629302\pi\)
\(80\) 0 0
\(81\) −8.05309 −0.894787
\(82\) 0 0
\(83\) −11.8318 −1.29870 −0.649352 0.760488i \(-0.724960\pi\)
−0.649352 + 0.760488i \(0.724960\pi\)
\(84\) 0 0
\(85\) −3.17504 −0.344381
\(86\) 0 0
\(87\) 12.0671 1.29372
\(88\) 0 0
\(89\) 4.84136 0.513183 0.256592 0.966520i \(-0.417400\pi\)
0.256592 + 0.966520i \(0.417400\pi\)
\(90\) 0 0
\(91\) 0.398717 0.0417969
\(92\) 0 0
\(93\) 15.1790 1.57399
\(94\) 0 0
\(95\) 5.49241 0.563509
\(96\) 0 0
\(97\) 7.00650 0.711402 0.355701 0.934600i \(-0.384242\pi\)
0.355701 + 0.934600i \(0.384242\pi\)
\(98\) 0 0
\(99\) 1.31023 0.131683
\(100\) 0 0
\(101\) 3.32323 0.330674 0.165337 0.986237i \(-0.447129\pi\)
0.165337 + 0.986237i \(0.447129\pi\)
\(102\) 0 0
\(103\) 19.3891 1.91046 0.955232 0.295858i \(-0.0956056\pi\)
0.955232 + 0.295858i \(0.0956056\pi\)
\(104\) 0 0
\(105\) 1.97010 0.192262
\(106\) 0 0
\(107\) 5.05521 0.488706 0.244353 0.969686i \(-0.421424\pi\)
0.244353 + 0.969686i \(0.421424\pi\)
\(108\) 0 0
\(109\) −12.1619 −1.16490 −0.582451 0.812866i \(-0.697906\pi\)
−0.582451 + 0.812866i \(0.697906\pi\)
\(110\) 0 0
\(111\) 16.4231 1.55881
\(112\) 0 0
\(113\) −9.92463 −0.933631 −0.466815 0.884355i \(-0.654599\pi\)
−0.466815 + 0.884355i \(0.654599\pi\)
\(114\) 0 0
\(115\) 6.05584 0.564710
\(116\) 0 0
\(117\) −0.0959845 −0.00887377
\(118\) 0 0
\(119\) 3.79833 0.348192
\(120\) 0 0
\(121\) 9.69824 0.881659
\(122\) 0 0
\(123\) −6.91753 −0.623733
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −3.37688 −0.299650 −0.149825 0.988713i \(-0.547871\pi\)
−0.149825 + 0.988713i \(0.547871\pi\)
\(128\) 0 0
\(129\) −12.4822 −1.09899
\(130\) 0 0
\(131\) 12.1614 1.06255 0.531275 0.847200i \(-0.321713\pi\)
0.531275 + 0.847200i \(0.321713\pi\)
\(132\) 0 0
\(133\) −6.57062 −0.569745
\(134\) 0 0
\(135\) −5.41472 −0.466025
\(136\) 0 0
\(137\) −18.5676 −1.58633 −0.793167 0.609004i \(-0.791569\pi\)
−0.793167 + 0.609004i \(0.791569\pi\)
\(138\) 0 0
\(139\) −21.0714 −1.78725 −0.893627 0.448810i \(-0.851848\pi\)
−0.893627 + 0.448810i \(0.851848\pi\)
\(140\) 0 0
\(141\) −7.00940 −0.590298
\(142\) 0 0
\(143\) −1.51631 −0.126800
\(144\) 0 0
\(145\) 7.32750 0.608515
\(146\) 0 0
\(147\) 9.17087 0.756401
\(148\) 0 0
\(149\) 0.790632 0.0647711 0.0323855 0.999475i \(-0.489690\pi\)
0.0323855 + 0.999475i \(0.489690\pi\)
\(150\) 0 0
\(151\) −11.1287 −0.905637 −0.452819 0.891603i \(-0.649581\pi\)
−0.452819 + 0.891603i \(0.649581\pi\)
\(152\) 0 0
\(153\) −0.914384 −0.0739236
\(154\) 0 0
\(155\) 9.21716 0.740340
\(156\) 0 0
\(157\) −12.4415 −0.992941 −0.496470 0.868054i \(-0.665371\pi\)
−0.496470 + 0.868054i \(0.665371\pi\)
\(158\) 0 0
\(159\) −17.7434 −1.40714
\(160\) 0 0
\(161\) −7.24465 −0.570958
\(162\) 0 0
\(163\) 12.1490 0.951582 0.475791 0.879559i \(-0.342162\pi\)
0.475791 + 0.879559i \(0.342162\pi\)
\(164\) 0 0
\(165\) −7.49225 −0.583271
\(166\) 0 0
\(167\) 11.1773 0.864922 0.432461 0.901653i \(-0.357645\pi\)
0.432461 + 0.901653i \(0.357645\pi\)
\(168\) 0 0
\(169\) −12.8889 −0.991455
\(170\) 0 0
\(171\) 1.58177 0.120961
\(172\) 0 0
\(173\) −5.82741 −0.443050 −0.221525 0.975155i \(-0.571103\pi\)
−0.221525 + 0.975155i \(0.571103\pi\)
\(174\) 0 0
\(175\) 1.19631 0.0904324
\(176\) 0 0
\(177\) −0.825894 −0.0620780
\(178\) 0 0
\(179\) −24.1873 −1.80784 −0.903922 0.427697i \(-0.859325\pi\)
−0.903922 + 0.427697i \(0.859325\pi\)
\(180\) 0 0
\(181\) 10.9710 0.815466 0.407733 0.913101i \(-0.366319\pi\)
0.407733 + 0.913101i \(0.366319\pi\)
\(182\) 0 0
\(183\) −22.5082 −1.66385
\(184\) 0 0
\(185\) 9.97265 0.733204
\(186\) 0 0
\(187\) −14.4449 −1.05632
\(188\) 0 0
\(189\) 6.47768 0.471182
\(190\) 0 0
\(191\) 21.6240 1.56466 0.782328 0.622867i \(-0.214032\pi\)
0.782328 + 0.622867i \(0.214032\pi\)
\(192\) 0 0
\(193\) 23.9764 1.72586 0.862930 0.505324i \(-0.168627\pi\)
0.862930 + 0.505324i \(0.168627\pi\)
\(194\) 0 0
\(195\) 0.548867 0.0393052
\(196\) 0 0
\(197\) 5.45320 0.388524 0.194262 0.980950i \(-0.437769\pi\)
0.194262 + 0.980950i \(0.437769\pi\)
\(198\) 0 0
\(199\) 19.2752 1.36638 0.683192 0.730238i \(-0.260591\pi\)
0.683192 + 0.730238i \(0.260591\pi\)
\(200\) 0 0
\(201\) 21.0205 1.48268
\(202\) 0 0
\(203\) −8.76595 −0.615249
\(204\) 0 0
\(205\) −4.20054 −0.293379
\(206\) 0 0
\(207\) 1.74403 0.121218
\(208\) 0 0
\(209\) 24.9879 1.72845
\(210\) 0 0
\(211\) −1.33340 −0.0917949 −0.0458975 0.998946i \(-0.514615\pi\)
−0.0458975 + 0.998946i \(0.514615\pi\)
\(212\) 0 0
\(213\) 21.9173 1.50175
\(214\) 0 0
\(215\) −7.57956 −0.516922
\(216\) 0 0
\(217\) −11.0266 −0.748532
\(218\) 0 0
\(219\) 4.26324 0.288083
\(220\) 0 0
\(221\) 1.05821 0.0711827
\(222\) 0 0
\(223\) −19.4274 −1.30095 −0.650476 0.759526i \(-0.725431\pi\)
−0.650476 + 0.759526i \(0.725431\pi\)
\(224\) 0 0
\(225\) −0.287991 −0.0191994
\(226\) 0 0
\(227\) 2.15735 0.143188 0.0715941 0.997434i \(-0.477191\pi\)
0.0715941 + 0.997434i \(0.477191\pi\)
\(228\) 0 0
\(229\) 17.8844 1.18184 0.590918 0.806731i \(-0.298766\pi\)
0.590918 + 0.806731i \(0.298766\pi\)
\(230\) 0 0
\(231\) 8.96304 0.589725
\(232\) 0 0
\(233\) 18.9828 1.24361 0.621803 0.783173i \(-0.286400\pi\)
0.621803 + 0.783173i \(0.286400\pi\)
\(234\) 0 0
\(235\) −4.25633 −0.277652
\(236\) 0 0
\(237\) 11.5674 0.751382
\(238\) 0 0
\(239\) 11.1282 0.719825 0.359913 0.932986i \(-0.382807\pi\)
0.359913 + 0.932986i \(0.382807\pi\)
\(240\) 0 0
\(241\) 26.7728 1.72459 0.862293 0.506410i \(-0.169028\pi\)
0.862293 + 0.506410i \(0.169028\pi\)
\(242\) 0 0
\(243\) −2.98220 −0.191308
\(244\) 0 0
\(245\) 5.56885 0.355781
\(246\) 0 0
\(247\) −1.83056 −0.116476
\(248\) 0 0
\(249\) 19.4848 1.23480
\(250\) 0 0
\(251\) −14.9135 −0.941330 −0.470665 0.882312i \(-0.655986\pi\)
−0.470665 + 0.882312i \(0.655986\pi\)
\(252\) 0 0
\(253\) 27.5512 1.73213
\(254\) 0 0
\(255\) 5.22871 0.327435
\(256\) 0 0
\(257\) 6.06625 0.378402 0.189201 0.981938i \(-0.439410\pi\)
0.189201 + 0.981938i \(0.439410\pi\)
\(258\) 0 0
\(259\) −11.9304 −0.741317
\(260\) 0 0
\(261\) 2.11026 0.130622
\(262\) 0 0
\(263\) 20.5058 1.26444 0.632220 0.774789i \(-0.282144\pi\)
0.632220 + 0.774789i \(0.282144\pi\)
\(264\) 0 0
\(265\) −10.7744 −0.661864
\(266\) 0 0
\(267\) −7.97284 −0.487930
\(268\) 0 0
\(269\) 7.19171 0.438486 0.219243 0.975670i \(-0.429641\pi\)
0.219243 + 0.975670i \(0.429641\pi\)
\(270\) 0 0
\(271\) 5.94253 0.360983 0.180491 0.983577i \(-0.442231\pi\)
0.180491 + 0.983577i \(0.442231\pi\)
\(272\) 0 0
\(273\) −0.656614 −0.0397401
\(274\) 0 0
\(275\) −4.54953 −0.274347
\(276\) 0 0
\(277\) −25.8200 −1.55137 −0.775686 0.631119i \(-0.782596\pi\)
−0.775686 + 0.631119i \(0.782596\pi\)
\(278\) 0 0
\(279\) 2.65446 0.158919
\(280\) 0 0
\(281\) 4.06523 0.242511 0.121256 0.992621i \(-0.461308\pi\)
0.121256 + 0.992621i \(0.461308\pi\)
\(282\) 0 0
\(283\) −6.75585 −0.401593 −0.200797 0.979633i \(-0.564353\pi\)
−0.200797 + 0.979633i \(0.564353\pi\)
\(284\) 0 0
\(285\) −9.04500 −0.535780
\(286\) 0 0
\(287\) 5.02515 0.296625
\(288\) 0 0
\(289\) −6.91913 −0.407008
\(290\) 0 0
\(291\) −11.5384 −0.676394
\(292\) 0 0
\(293\) 2.99399 0.174911 0.0874554 0.996168i \(-0.472126\pi\)
0.0874554 + 0.996168i \(0.472126\pi\)
\(294\) 0 0
\(295\) −0.501509 −0.0291990
\(296\) 0 0
\(297\) −24.6345 −1.42944
\(298\) 0 0
\(299\) −2.01835 −0.116724
\(300\) 0 0
\(301\) 9.06749 0.522642
\(302\) 0 0
\(303\) −5.47276 −0.314402
\(304\) 0 0
\(305\) −13.6677 −0.782610
\(306\) 0 0
\(307\) −4.86355 −0.277578 −0.138789 0.990322i \(-0.544321\pi\)
−0.138789 + 0.990322i \(0.544321\pi\)
\(308\) 0 0
\(309\) −31.9303 −1.81645
\(310\) 0 0
\(311\) 11.1236 0.630764 0.315382 0.948965i \(-0.397867\pi\)
0.315382 + 0.948965i \(0.397867\pi\)
\(312\) 0 0
\(313\) 10.3437 0.584659 0.292330 0.956318i \(-0.405570\pi\)
0.292330 + 0.956318i \(0.405570\pi\)
\(314\) 0 0
\(315\) 0.344527 0.0194119
\(316\) 0 0
\(317\) −22.0691 −1.23952 −0.619762 0.784790i \(-0.712771\pi\)
−0.619762 + 0.784790i \(0.712771\pi\)
\(318\) 0 0
\(319\) 33.3367 1.86650
\(320\) 0 0
\(321\) −8.32501 −0.464657
\(322\) 0 0
\(323\) −17.4386 −0.970311
\(324\) 0 0
\(325\) 0.333289 0.0184876
\(326\) 0 0
\(327\) 20.0285 1.10758
\(328\) 0 0
\(329\) 5.09188 0.280724
\(330\) 0 0
\(331\) −4.14828 −0.228010 −0.114005 0.993480i \(-0.536368\pi\)
−0.114005 + 0.993480i \(0.536368\pi\)
\(332\) 0 0
\(333\) 2.87204 0.157387
\(334\) 0 0
\(335\) 12.7643 0.697391
\(336\) 0 0
\(337\) 1.30726 0.0712110 0.0356055 0.999366i \(-0.488664\pi\)
0.0356055 + 0.999366i \(0.488664\pi\)
\(338\) 0 0
\(339\) 16.3441 0.887687
\(340\) 0 0
\(341\) 41.9338 2.27084
\(342\) 0 0
\(343\) −15.0362 −0.811879
\(344\) 0 0
\(345\) −9.97286 −0.536921
\(346\) 0 0
\(347\) 12.3661 0.663845 0.331922 0.943307i \(-0.392303\pi\)
0.331922 + 0.943307i \(0.392303\pi\)
\(348\) 0 0
\(349\) −23.1833 −1.24097 −0.620487 0.784217i \(-0.713065\pi\)
−0.620487 + 0.784217i \(0.713065\pi\)
\(350\) 0 0
\(351\) 1.80467 0.0963261
\(352\) 0 0
\(353\) −2.30510 −0.122688 −0.0613440 0.998117i \(-0.519539\pi\)
−0.0613440 + 0.998117i \(0.519539\pi\)
\(354\) 0 0
\(355\) 13.3089 0.706361
\(356\) 0 0
\(357\) −6.25515 −0.331058
\(358\) 0 0
\(359\) 29.1883 1.54050 0.770250 0.637742i \(-0.220131\pi\)
0.770250 + 0.637742i \(0.220131\pi\)
\(360\) 0 0
\(361\) 11.1666 0.587715
\(362\) 0 0
\(363\) −15.9712 −0.838273
\(364\) 0 0
\(365\) 2.58878 0.135503
\(366\) 0 0
\(367\) −22.0547 −1.15125 −0.575624 0.817715i \(-0.695241\pi\)
−0.575624 + 0.817715i \(0.695241\pi\)
\(368\) 0 0
\(369\) −1.20972 −0.0629756
\(370\) 0 0
\(371\) 12.8895 0.669187
\(372\) 0 0
\(373\) −30.8032 −1.59493 −0.797463 0.603367i \(-0.793825\pi\)
−0.797463 + 0.603367i \(0.793825\pi\)
\(374\) 0 0
\(375\) 1.64682 0.0850413
\(376\) 0 0
\(377\) −2.44218 −0.125778
\(378\) 0 0
\(379\) −10.1672 −0.522252 −0.261126 0.965305i \(-0.584094\pi\)
−0.261126 + 0.965305i \(0.584094\pi\)
\(380\) 0 0
\(381\) 5.56111 0.284904
\(382\) 0 0
\(383\) 16.2823 0.831987 0.415993 0.909368i \(-0.363434\pi\)
0.415993 + 0.909368i \(0.363434\pi\)
\(384\) 0 0
\(385\) 5.44264 0.277383
\(386\) 0 0
\(387\) −2.18285 −0.110960
\(388\) 0 0
\(389\) 27.5335 1.39600 0.698001 0.716096i \(-0.254073\pi\)
0.698001 + 0.716096i \(0.254073\pi\)
\(390\) 0 0
\(391\) −19.2275 −0.972377
\(392\) 0 0
\(393\) −20.0277 −1.01026
\(394\) 0 0
\(395\) 7.02408 0.353420
\(396\) 0 0
\(397\) 31.5081 1.58134 0.790672 0.612240i \(-0.209731\pi\)
0.790672 + 0.612240i \(0.209731\pi\)
\(398\) 0 0
\(399\) 10.8206 0.541708
\(400\) 0 0
\(401\) −1.00000 −0.0499376
\(402\) 0 0
\(403\) −3.07198 −0.153026
\(404\) 0 0
\(405\) 8.05309 0.400161
\(406\) 0 0
\(407\) 45.3709 2.24895
\(408\) 0 0
\(409\) 15.6818 0.775415 0.387707 0.921783i \(-0.373267\pi\)
0.387707 + 0.921783i \(0.373267\pi\)
\(410\) 0 0
\(411\) 30.5774 1.50827
\(412\) 0 0
\(413\) 0.599959 0.0295221
\(414\) 0 0
\(415\) 11.8318 0.580798
\(416\) 0 0
\(417\) 34.7008 1.69930
\(418\) 0 0
\(419\) −28.6004 −1.39722 −0.698610 0.715503i \(-0.746198\pi\)
−0.698610 + 0.715503i \(0.746198\pi\)
\(420\) 0 0
\(421\) 0.206044 0.0100419 0.00502097 0.999987i \(-0.498402\pi\)
0.00502097 + 0.999987i \(0.498402\pi\)
\(422\) 0 0
\(423\) −1.22579 −0.0595998
\(424\) 0 0
\(425\) 3.17504 0.154012
\(426\) 0 0
\(427\) 16.3508 0.791269
\(428\) 0 0
\(429\) 2.49709 0.120560
\(430\) 0 0
\(431\) −31.2222 −1.50392 −0.751960 0.659208i \(-0.770892\pi\)
−0.751960 + 0.659208i \(0.770892\pi\)
\(432\) 0 0
\(433\) 17.7707 0.854007 0.427004 0.904250i \(-0.359569\pi\)
0.427004 + 0.904250i \(0.359569\pi\)
\(434\) 0 0
\(435\) −12.0671 −0.578571
\(436\) 0 0
\(437\) 33.2611 1.59110
\(438\) 0 0
\(439\) 14.7496 0.703960 0.351980 0.936008i \(-0.385509\pi\)
0.351980 + 0.936008i \(0.385509\pi\)
\(440\) 0 0
\(441\) 1.60378 0.0763705
\(442\) 0 0
\(443\) 23.7852 1.13007 0.565034 0.825067i \(-0.308863\pi\)
0.565034 + 0.825067i \(0.308863\pi\)
\(444\) 0 0
\(445\) −4.84136 −0.229503
\(446\) 0 0
\(447\) −1.30203 −0.0615837
\(448\) 0 0
\(449\) 25.2922 1.19361 0.596807 0.802385i \(-0.296436\pi\)
0.596807 + 0.802385i \(0.296436\pi\)
\(450\) 0 0
\(451\) −19.1105 −0.899879
\(452\) 0 0
\(453\) 18.3269 0.861071
\(454\) 0 0
\(455\) −0.398717 −0.0186921
\(456\) 0 0
\(457\) −33.7849 −1.58039 −0.790196 0.612854i \(-0.790021\pi\)
−0.790196 + 0.612854i \(0.790021\pi\)
\(458\) 0 0
\(459\) 17.1920 0.802452
\(460\) 0 0
\(461\) −16.2241 −0.755630 −0.377815 0.925881i \(-0.623324\pi\)
−0.377815 + 0.925881i \(0.623324\pi\)
\(462\) 0 0
\(463\) −19.0458 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(464\) 0 0
\(465\) −15.1790 −0.703909
\(466\) 0 0
\(467\) −22.6299 −1.04719 −0.523593 0.851969i \(-0.675409\pi\)
−0.523593 + 0.851969i \(0.675409\pi\)
\(468\) 0 0
\(469\) −15.2701 −0.705107
\(470\) 0 0
\(471\) 20.4889 0.944079
\(472\) 0 0
\(473\) −34.4835 −1.58555
\(474\) 0 0
\(475\) −5.49241 −0.252009
\(476\) 0 0
\(477\) −3.10292 −0.142073
\(478\) 0 0
\(479\) 29.5568 1.35049 0.675243 0.737595i \(-0.264039\pi\)
0.675243 + 0.737595i \(0.264039\pi\)
\(480\) 0 0
\(481\) −3.32378 −0.151551
\(482\) 0 0
\(483\) 11.9306 0.542862
\(484\) 0 0
\(485\) −7.00650 −0.318149
\(486\) 0 0
\(487\) 15.4466 0.699950 0.349975 0.936759i \(-0.386190\pi\)
0.349975 + 0.936759i \(0.386190\pi\)
\(488\) 0 0
\(489\) −20.0072 −0.904755
\(490\) 0 0
\(491\) 34.3249 1.54906 0.774531 0.632536i \(-0.217986\pi\)
0.774531 + 0.632536i \(0.217986\pi\)
\(492\) 0 0
\(493\) −23.2651 −1.04781
\(494\) 0 0
\(495\) −1.31023 −0.0588903
\(496\) 0 0
\(497\) −15.9215 −0.714177
\(498\) 0 0
\(499\) −11.0715 −0.495629 −0.247815 0.968807i \(-0.579712\pi\)
−0.247815 + 0.968807i \(0.579712\pi\)
\(500\) 0 0
\(501\) −18.4069 −0.822360
\(502\) 0 0
\(503\) 5.37509 0.239663 0.119832 0.992794i \(-0.461764\pi\)
0.119832 + 0.992794i \(0.461764\pi\)
\(504\) 0 0
\(505\) −3.32323 −0.147882
\(506\) 0 0
\(507\) 21.2257 0.942666
\(508\) 0 0
\(509\) −31.5753 −1.39955 −0.699775 0.714363i \(-0.746716\pi\)
−0.699775 + 0.714363i \(0.746716\pi\)
\(510\) 0 0
\(511\) −3.09697 −0.137002
\(512\) 0 0
\(513\) −29.7399 −1.31305
\(514\) 0 0
\(515\) −19.3891 −0.854385
\(516\) 0 0
\(517\) −19.3643 −0.851641
\(518\) 0 0
\(519\) 9.59668 0.421247
\(520\) 0 0
\(521\) 32.9402 1.44314 0.721568 0.692344i \(-0.243422\pi\)
0.721568 + 0.692344i \(0.243422\pi\)
\(522\) 0 0
\(523\) 21.0653 0.921121 0.460560 0.887628i \(-0.347648\pi\)
0.460560 + 0.887628i \(0.347648\pi\)
\(524\) 0 0
\(525\) −1.97010 −0.0859823
\(526\) 0 0
\(527\) −29.2649 −1.27480
\(528\) 0 0
\(529\) 13.6732 0.594485
\(530\) 0 0
\(531\) −0.144430 −0.00626774
\(532\) 0 0
\(533\) 1.40000 0.0606406
\(534\) 0 0
\(535\) −5.05521 −0.218556
\(536\) 0 0
\(537\) 39.8321 1.71888
\(538\) 0 0
\(539\) 25.3356 1.09128
\(540\) 0 0
\(541\) 31.9163 1.37219 0.686095 0.727512i \(-0.259324\pi\)
0.686095 + 0.727512i \(0.259324\pi\)
\(542\) 0 0
\(543\) −18.0672 −0.775337
\(544\) 0 0
\(545\) 12.1619 0.520960
\(546\) 0 0
\(547\) −11.2350 −0.480373 −0.240186 0.970727i \(-0.577209\pi\)
−0.240186 + 0.970727i \(0.577209\pi\)
\(548\) 0 0
\(549\) −3.93618 −0.167992
\(550\) 0 0
\(551\) 40.2456 1.71452
\(552\) 0 0
\(553\) −8.40297 −0.357330
\(554\) 0 0
\(555\) −16.4231 −0.697123
\(556\) 0 0
\(557\) 33.8271 1.43330 0.716650 0.697433i \(-0.245674\pi\)
0.716650 + 0.697433i \(0.245674\pi\)
\(558\) 0 0
\(559\) 2.52619 0.106846
\(560\) 0 0
\(561\) 23.7882 1.00434
\(562\) 0 0
\(563\) −5.92063 −0.249525 −0.124762 0.992187i \(-0.539817\pi\)
−0.124762 + 0.992187i \(0.539817\pi\)
\(564\) 0 0
\(565\) 9.92463 0.417532
\(566\) 0 0
\(567\) −9.63397 −0.404589
\(568\) 0 0
\(569\) 33.1626 1.39025 0.695125 0.718889i \(-0.255349\pi\)
0.695125 + 0.718889i \(0.255349\pi\)
\(570\) 0 0
\(571\) 0.383667 0.0160559 0.00802797 0.999968i \(-0.497445\pi\)
0.00802797 + 0.999968i \(0.497445\pi\)
\(572\) 0 0
\(573\) −35.6107 −1.48766
\(574\) 0 0
\(575\) −6.05584 −0.252546
\(576\) 0 0
\(577\) 32.4167 1.34952 0.674762 0.738035i \(-0.264246\pi\)
0.674762 + 0.738035i \(0.264246\pi\)
\(578\) 0 0
\(579\) −39.4848 −1.64093
\(580\) 0 0
\(581\) −14.1544 −0.587225
\(582\) 0 0
\(583\) −49.0183 −2.03013
\(584\) 0 0
\(585\) 0.0959845 0.00396847
\(586\) 0 0
\(587\) 20.7747 0.857464 0.428732 0.903432i \(-0.358960\pi\)
0.428732 + 0.903432i \(0.358960\pi\)
\(588\) 0 0
\(589\) 50.6245 2.08594
\(590\) 0 0
\(591\) −8.98042 −0.369405
\(592\) 0 0
\(593\) 36.1912 1.48619 0.743097 0.669183i \(-0.233356\pi\)
0.743097 + 0.669183i \(0.233356\pi\)
\(594\) 0 0
\(595\) −3.79833 −0.155716
\(596\) 0 0
\(597\) −31.7428 −1.29915
\(598\) 0 0
\(599\) 13.6938 0.559514 0.279757 0.960071i \(-0.409746\pi\)
0.279757 + 0.960071i \(0.409746\pi\)
\(600\) 0 0
\(601\) 40.4133 1.64849 0.824246 0.566231i \(-0.191599\pi\)
0.824246 + 0.566231i \(0.191599\pi\)
\(602\) 0 0
\(603\) 3.67602 0.149699
\(604\) 0 0
\(605\) −9.69824 −0.394290
\(606\) 0 0
\(607\) −7.48952 −0.303990 −0.151995 0.988381i \(-0.548570\pi\)
−0.151995 + 0.988381i \(0.548570\pi\)
\(608\) 0 0
\(609\) 14.4359 0.584973
\(610\) 0 0
\(611\) 1.41859 0.0573899
\(612\) 0 0
\(613\) 21.9249 0.885536 0.442768 0.896636i \(-0.353996\pi\)
0.442768 + 0.896636i \(0.353996\pi\)
\(614\) 0 0
\(615\) 6.91753 0.278942
\(616\) 0 0
\(617\) 23.7412 0.955785 0.477892 0.878418i \(-0.341401\pi\)
0.477892 + 0.878418i \(0.341401\pi\)
\(618\) 0 0
\(619\) −17.6731 −0.710343 −0.355171 0.934801i \(-0.615577\pi\)
−0.355171 + 0.934801i \(0.615577\pi\)
\(620\) 0 0
\(621\) −32.7907 −1.31584
\(622\) 0 0
\(623\) 5.79176 0.232042
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −41.1505 −1.64339
\(628\) 0 0
\(629\) −31.6635 −1.26251
\(630\) 0 0
\(631\) 17.7660 0.707253 0.353627 0.935387i \(-0.384948\pi\)
0.353627 + 0.935387i \(0.384948\pi\)
\(632\) 0 0
\(633\) 2.19586 0.0872778
\(634\) 0 0
\(635\) 3.37688 0.134007
\(636\) 0 0
\(637\) −1.85604 −0.0735388
\(638\) 0 0
\(639\) 3.83284 0.151625
\(640\) 0 0
\(641\) 23.3977 0.924153 0.462077 0.886840i \(-0.347104\pi\)
0.462077 + 0.886840i \(0.347104\pi\)
\(642\) 0 0
\(643\) 10.8853 0.429274 0.214637 0.976694i \(-0.431143\pi\)
0.214637 + 0.976694i \(0.431143\pi\)
\(644\) 0 0
\(645\) 12.4822 0.491484
\(646\) 0 0
\(647\) 36.3724 1.42995 0.714973 0.699153i \(-0.246439\pi\)
0.714973 + 0.699153i \(0.246439\pi\)
\(648\) 0 0
\(649\) −2.28163 −0.0895619
\(650\) 0 0
\(651\) 18.1588 0.711698
\(652\) 0 0
\(653\) −0.0196962 −0.000770770 0 −0.000385385 1.00000i \(-0.500123\pi\)
−0.000385385 1.00000i \(0.500123\pi\)
\(654\) 0 0
\(655\) −12.1614 −0.475187
\(656\) 0 0
\(657\) 0.745545 0.0290865
\(658\) 0 0
\(659\) 17.1757 0.669071 0.334535 0.942383i \(-0.391421\pi\)
0.334535 + 0.942383i \(0.391421\pi\)
\(660\) 0 0
\(661\) 28.7296 1.11745 0.558726 0.829352i \(-0.311290\pi\)
0.558726 + 0.829352i \(0.311290\pi\)
\(662\) 0 0
\(663\) −1.74267 −0.0676798
\(664\) 0 0
\(665\) 6.57062 0.254798
\(666\) 0 0
\(667\) 44.3741 1.71817
\(668\) 0 0
\(669\) 31.9933 1.23693
\(670\) 0 0
\(671\) −62.1816 −2.40049
\(672\) 0 0
\(673\) −23.2239 −0.895215 −0.447608 0.894230i \(-0.647724\pi\)
−0.447608 + 0.894230i \(0.647724\pi\)
\(674\) 0 0
\(675\) 5.41472 0.208413
\(676\) 0 0
\(677\) −4.33551 −0.166627 −0.0833136 0.996523i \(-0.526550\pi\)
−0.0833136 + 0.996523i \(0.526550\pi\)
\(678\) 0 0
\(679\) 8.38193 0.321669
\(680\) 0 0
\(681\) −3.55276 −0.136142
\(682\) 0 0
\(683\) 4.30852 0.164861 0.0824305 0.996597i \(-0.473732\pi\)
0.0824305 + 0.996597i \(0.473732\pi\)
\(684\) 0 0
\(685\) 18.5676 0.709430
\(686\) 0 0
\(687\) −29.4524 −1.12368
\(688\) 0 0
\(689\) 3.59098 0.136805
\(690\) 0 0
\(691\) −8.30153 −0.315805 −0.157903 0.987455i \(-0.550473\pi\)
−0.157903 + 0.987455i \(0.550473\pi\)
\(692\) 0 0
\(693\) 1.56743 0.0595419
\(694\) 0 0
\(695\) 21.0714 0.799284
\(696\) 0 0
\(697\) 13.3369 0.505171
\(698\) 0 0
\(699\) −31.2613 −1.18241
\(700\) 0 0
\(701\) −14.5585 −0.549865 −0.274933 0.961463i \(-0.588656\pi\)
−0.274933 + 0.961463i \(0.588656\pi\)
\(702\) 0 0
\(703\) 54.7739 2.06584
\(704\) 0 0
\(705\) 7.00940 0.263989
\(706\) 0 0
\(707\) 3.97561 0.149518
\(708\) 0 0
\(709\) −19.1840 −0.720469 −0.360235 0.932862i \(-0.617303\pi\)
−0.360235 + 0.932862i \(0.617303\pi\)
\(710\) 0 0
\(711\) 2.02288 0.0758637
\(712\) 0 0
\(713\) 55.8176 2.09039
\(714\) 0 0
\(715\) 1.51631 0.0567068
\(716\) 0 0
\(717\) −18.3262 −0.684403
\(718\) 0 0
\(719\) −42.4592 −1.58346 −0.791730 0.610871i \(-0.790819\pi\)
−0.791730 + 0.610871i \(0.790819\pi\)
\(720\) 0 0
\(721\) 23.1953 0.863839
\(722\) 0 0
\(723\) −44.0899 −1.63972
\(724\) 0 0
\(725\) −7.32750 −0.272136
\(726\) 0 0
\(727\) 12.1127 0.449236 0.224618 0.974447i \(-0.427887\pi\)
0.224618 + 0.974447i \(0.427887\pi\)
\(728\) 0 0
\(729\) 29.0704 1.07668
\(730\) 0 0
\(731\) 24.0654 0.890091
\(732\) 0 0
\(733\) −0.928387 −0.0342908 −0.0171454 0.999853i \(-0.505458\pi\)
−0.0171454 + 0.999853i \(0.505458\pi\)
\(734\) 0 0
\(735\) −9.17087 −0.338273
\(736\) 0 0
\(737\) 58.0718 2.13910
\(738\) 0 0
\(739\) 28.7212 1.05653 0.528263 0.849081i \(-0.322843\pi\)
0.528263 + 0.849081i \(0.322843\pi\)
\(740\) 0 0
\(741\) 3.01460 0.110744
\(742\) 0 0
\(743\) −23.3516 −0.856686 −0.428343 0.903616i \(-0.640902\pi\)
−0.428343 + 0.903616i \(0.640902\pi\)
\(744\) 0 0
\(745\) −0.790632 −0.0289665
\(746\) 0 0
\(747\) 3.40745 0.124672
\(748\) 0 0
\(749\) 6.04759 0.220974
\(750\) 0 0
\(751\) −1.95874 −0.0714755 −0.0357377 0.999361i \(-0.511378\pi\)
−0.0357377 + 0.999361i \(0.511378\pi\)
\(752\) 0 0
\(753\) 24.5598 0.895007
\(754\) 0 0
\(755\) 11.1287 0.405013
\(756\) 0 0
\(757\) −42.9548 −1.56122 −0.780610 0.625018i \(-0.785092\pi\)
−0.780610 + 0.625018i \(0.785092\pi\)
\(758\) 0 0
\(759\) −45.3718 −1.64689
\(760\) 0 0
\(761\) 32.6348 1.18301 0.591504 0.806302i \(-0.298534\pi\)
0.591504 + 0.806302i \(0.298534\pi\)
\(762\) 0 0
\(763\) −14.5494 −0.526725
\(764\) 0 0
\(765\) 0.914384 0.0330596
\(766\) 0 0
\(767\) 0.167148 0.00603535
\(768\) 0 0
\(769\) −29.1553 −1.05137 −0.525683 0.850680i \(-0.676190\pi\)
−0.525683 + 0.850680i \(0.676190\pi\)
\(770\) 0 0
\(771\) −9.99001 −0.359781
\(772\) 0 0
\(773\) 24.8016 0.892051 0.446026 0.895020i \(-0.352839\pi\)
0.446026 + 0.895020i \(0.352839\pi\)
\(774\) 0 0
\(775\) −9.21716 −0.331090
\(776\) 0 0
\(777\) 19.6471 0.704837
\(778\) 0 0
\(779\) −23.0711 −0.826609
\(780\) 0 0
\(781\) 60.5491 2.16662
\(782\) 0 0
\(783\) −39.6764 −1.41792
\(784\) 0 0
\(785\) 12.4415 0.444057
\(786\) 0 0
\(787\) −35.1673 −1.25358 −0.626790 0.779189i \(-0.715631\pi\)
−0.626790 + 0.779189i \(0.715631\pi\)
\(788\) 0 0
\(789\) −33.7693 −1.20222
\(790\) 0 0
\(791\) −11.8729 −0.422152
\(792\) 0 0
\(793\) 4.55529 0.161763
\(794\) 0 0
\(795\) 17.7434 0.629294
\(796\) 0 0
\(797\) −3.45504 −0.122384 −0.0611919 0.998126i \(-0.519490\pi\)
−0.0611919 + 0.998126i \(0.519490\pi\)
\(798\) 0 0
\(799\) 13.5140 0.478091
\(800\) 0 0
\(801\) −1.39427 −0.0492641
\(802\) 0 0
\(803\) 11.7777 0.415627
\(804\) 0 0
\(805\) 7.24465 0.255340
\(806\) 0 0
\(807\) −11.8434 −0.416908
\(808\) 0 0
\(809\) 34.6555 1.21842 0.609211 0.793008i \(-0.291486\pi\)
0.609211 + 0.793008i \(0.291486\pi\)
\(810\) 0 0
\(811\) −27.6538 −0.971057 −0.485528 0.874221i \(-0.661373\pi\)
−0.485528 + 0.874221i \(0.661373\pi\)
\(812\) 0 0
\(813\) −9.78626 −0.343219
\(814\) 0 0
\(815\) −12.1490 −0.425560
\(816\) 0 0
\(817\) −41.6301 −1.45645
\(818\) 0 0
\(819\) −0.114827 −0.00401238
\(820\) 0 0
\(821\) −35.1004 −1.22501 −0.612505 0.790466i \(-0.709838\pi\)
−0.612505 + 0.790466i \(0.709838\pi\)
\(822\) 0 0
\(823\) −14.1490 −0.493204 −0.246602 0.969117i \(-0.579314\pi\)
−0.246602 + 0.969117i \(0.579314\pi\)
\(824\) 0 0
\(825\) 7.49225 0.260847
\(826\) 0 0
\(827\) −54.1327 −1.88238 −0.941189 0.337880i \(-0.890290\pi\)
−0.941189 + 0.337880i \(0.890290\pi\)
\(828\) 0 0
\(829\) −8.39897 −0.291708 −0.145854 0.989306i \(-0.546593\pi\)
−0.145854 + 0.989306i \(0.546593\pi\)
\(830\) 0 0
\(831\) 42.5208 1.47503
\(832\) 0 0
\(833\) −17.6813 −0.612621
\(834\) 0 0
\(835\) −11.1773 −0.386805
\(836\) 0 0
\(837\) −49.9084 −1.72509
\(838\) 0 0
\(839\) −20.3732 −0.703361 −0.351681 0.936120i \(-0.614390\pi\)
−0.351681 + 0.936120i \(0.614390\pi\)
\(840\) 0 0
\(841\) 24.6922 0.851456
\(842\) 0 0
\(843\) −6.69469 −0.230577
\(844\) 0 0
\(845\) 12.8889 0.443392
\(846\) 0 0
\(847\) 11.6021 0.398653
\(848\) 0 0
\(849\) 11.1256 0.381831
\(850\) 0 0
\(851\) 60.3927 2.07024
\(852\) 0 0
\(853\) −40.6038 −1.39025 −0.695123 0.718891i \(-0.744650\pi\)
−0.695123 + 0.718891i \(0.744650\pi\)
\(854\) 0 0
\(855\) −1.58177 −0.0540953
\(856\) 0 0
\(857\) 44.5582 1.52208 0.761039 0.648706i \(-0.224690\pi\)
0.761039 + 0.648706i \(0.224690\pi\)
\(858\) 0 0
\(859\) 19.1624 0.653812 0.326906 0.945057i \(-0.393994\pi\)
0.326906 + 0.945057i \(0.393994\pi\)
\(860\) 0 0
\(861\) −8.27550 −0.282028
\(862\) 0 0
\(863\) −9.56387 −0.325558 −0.162779 0.986663i \(-0.552046\pi\)
−0.162779 + 0.986663i \(0.552046\pi\)
\(864\) 0 0
\(865\) 5.82741 0.198138
\(866\) 0 0
\(867\) 11.3945 0.386979
\(868\) 0 0
\(869\) 31.9563 1.08404
\(870\) 0 0
\(871\) −4.25422 −0.144149
\(872\) 0 0
\(873\) −2.01781 −0.0682926
\(874\) 0 0
\(875\) −1.19631 −0.0404426
\(876\) 0 0
\(877\) 31.6092 1.06737 0.533684 0.845684i \(-0.320807\pi\)
0.533684 + 0.845684i \(0.320807\pi\)
\(878\) 0 0
\(879\) −4.93055 −0.166303
\(880\) 0 0
\(881\) −20.6076 −0.694286 −0.347143 0.937812i \(-0.612848\pi\)
−0.347143 + 0.937812i \(0.612848\pi\)
\(882\) 0 0
\(883\) 30.9465 1.04143 0.520716 0.853730i \(-0.325665\pi\)
0.520716 + 0.853730i \(0.325665\pi\)
\(884\) 0 0
\(885\) 0.825894 0.0277621
\(886\) 0 0
\(887\) −38.6004 −1.29608 −0.648038 0.761608i \(-0.724410\pi\)
−0.648038 + 0.761608i \(0.724410\pi\)
\(888\) 0 0
\(889\) −4.03979 −0.135490
\(890\) 0 0
\(891\) 36.6378 1.22741
\(892\) 0 0
\(893\) −23.3775 −0.782298
\(894\) 0 0
\(895\) 24.1873 0.808493
\(896\) 0 0
\(897\) 3.32385 0.110980
\(898\) 0 0
\(899\) 67.5387 2.25254
\(900\) 0 0
\(901\) 34.2090 1.13967
\(902\) 0 0
\(903\) −14.9325 −0.496923
\(904\) 0 0
\(905\) −10.9710 −0.364687
\(906\) 0 0
\(907\) −29.6058 −0.983044 −0.491522 0.870865i \(-0.663559\pi\)
−0.491522 + 0.870865i \(0.663559\pi\)
\(908\) 0 0
\(909\) −0.957062 −0.0317437
\(910\) 0 0
\(911\) −9.60413 −0.318199 −0.159100 0.987263i \(-0.550859\pi\)
−0.159100 + 0.987263i \(0.550859\pi\)
\(912\) 0 0
\(913\) 53.8290 1.78148
\(914\) 0 0
\(915\) 22.5082 0.744098
\(916\) 0 0
\(917\) 14.5488 0.480445
\(918\) 0 0
\(919\) 11.7499 0.387595 0.193797 0.981042i \(-0.437920\pi\)
0.193797 + 0.981042i \(0.437920\pi\)
\(920\) 0 0
\(921\) 8.00938 0.263918
\(922\) 0 0
\(923\) −4.43570 −0.146003
\(924\) 0 0
\(925\) −9.97265 −0.327899
\(926\) 0 0
\(927\) −5.58389 −0.183399
\(928\) 0 0
\(929\) −15.0897 −0.495078 −0.247539 0.968878i \(-0.579622\pi\)
−0.247539 + 0.968878i \(0.579622\pi\)
\(930\) 0 0
\(931\) 30.5864 1.00243
\(932\) 0 0
\(933\) −18.3186 −0.599724
\(934\) 0 0
\(935\) 14.4449 0.472400
\(936\) 0 0
\(937\) 27.4595 0.897064 0.448532 0.893767i \(-0.351947\pi\)
0.448532 + 0.893767i \(0.351947\pi\)
\(938\) 0 0
\(939\) −17.0341 −0.555888
\(940\) 0 0
\(941\) 19.0253 0.620208 0.310104 0.950703i \(-0.399636\pi\)
0.310104 + 0.950703i \(0.399636\pi\)
\(942\) 0 0
\(943\) −25.4378 −0.828369
\(944\) 0 0
\(945\) −6.47768 −0.210719
\(946\) 0 0
\(947\) 12.7366 0.413884 0.206942 0.978353i \(-0.433649\pi\)
0.206942 + 0.978353i \(0.433649\pi\)
\(948\) 0 0
\(949\) −0.862811 −0.0280080
\(950\) 0 0
\(951\) 36.3438 1.17853
\(952\) 0 0
\(953\) −60.2907 −1.95301 −0.976504 0.215499i \(-0.930862\pi\)
−0.976504 + 0.215499i \(0.930862\pi\)
\(954\) 0 0
\(955\) −21.6240 −0.699735
\(956\) 0 0
\(957\) −54.8994 −1.77465
\(958\) 0 0
\(959\) −22.2125 −0.717280
\(960\) 0 0
\(961\) 53.9561 1.74052
\(962\) 0 0
\(963\) −1.45586 −0.0469144
\(964\) 0 0
\(965\) −23.9764 −0.771828
\(966\) 0 0
\(967\) −2.26848 −0.0729496 −0.0364748 0.999335i \(-0.511613\pi\)
−0.0364748 + 0.999335i \(0.511613\pi\)
\(968\) 0 0
\(969\) 28.7182 0.922562
\(970\) 0 0
\(971\) 17.0183 0.546144 0.273072 0.961994i \(-0.411960\pi\)
0.273072 + 0.961994i \(0.411960\pi\)
\(972\) 0 0
\(973\) −25.2079 −0.808128
\(974\) 0 0
\(975\) −0.548867 −0.0175778
\(976\) 0 0
\(977\) 17.3760 0.555908 0.277954 0.960594i \(-0.410344\pi\)
0.277954 + 0.960594i \(0.410344\pi\)
\(978\) 0 0
\(979\) −22.0259 −0.703952
\(980\) 0 0
\(981\) 3.50253 0.111827
\(982\) 0 0
\(983\) −28.1829 −0.898896 −0.449448 0.893306i \(-0.648379\pi\)
−0.449448 + 0.893306i \(0.648379\pi\)
\(984\) 0 0
\(985\) −5.45320 −0.173753
\(986\) 0 0
\(987\) −8.38540 −0.266910
\(988\) 0 0
\(989\) −45.9006 −1.45955
\(990\) 0 0
\(991\) 22.7891 0.723919 0.361959 0.932194i \(-0.382108\pi\)
0.361959 + 0.932194i \(0.382108\pi\)
\(992\) 0 0
\(993\) 6.83146 0.216790
\(994\) 0 0
\(995\) −19.2752 −0.611066
\(996\) 0 0
\(997\) 13.3945 0.424208 0.212104 0.977247i \(-0.431968\pi\)
0.212104 + 0.977247i \(0.431968\pi\)
\(998\) 0 0
\(999\) −53.9991 −1.70846
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 8020.2.a.e.1.9 35
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.e.1.9 35 1.1 even 1 trivial