Properties

Label 6008.2.a.d.1.16
Level $6008$
Weight $2$
Character 6008.1
Self dual yes
Analytic conductor $47.974$
Analytic rank $0$
Dimension $49$
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] = [6008,2,Mod(1,6008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, 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("6008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6008 = 2^{3} \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6008.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: \(47.9741215344\)
Analytic rank: \(0\)
Dimension: \(49\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 6008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.08636 q^{3} -0.0864029 q^{5} -0.643099 q^{7} -1.81982 q^{9} +O(q^{10})\) \(q-1.08636 q^{3} -0.0864029 q^{5} -0.643099 q^{7} -1.81982 q^{9} -4.53359 q^{11} +4.55688 q^{13} +0.0938648 q^{15} +2.52956 q^{17} +4.06564 q^{19} +0.698638 q^{21} +0.0799297 q^{23} -4.99253 q^{25} +5.23607 q^{27} -4.92904 q^{29} +5.43721 q^{31} +4.92512 q^{33} +0.0555656 q^{35} -1.43701 q^{37} -4.95042 q^{39} -4.75696 q^{41} +1.56498 q^{43} +0.157238 q^{45} +5.54900 q^{47} -6.58642 q^{49} -2.74802 q^{51} -12.8813 q^{53} +0.391715 q^{55} -4.41676 q^{57} -6.90937 q^{59} -7.25651 q^{61} +1.17032 q^{63} -0.393728 q^{65} -1.62308 q^{67} -0.0868325 q^{69} +5.50057 q^{71} -11.0785 q^{73} +5.42370 q^{75} +2.91555 q^{77} +8.91385 q^{79} -0.228807 q^{81} +1.11686 q^{83} -0.218562 q^{85} +5.35472 q^{87} +2.69683 q^{89} -2.93053 q^{91} -5.90677 q^{93} -0.351283 q^{95} +11.3347 q^{97} +8.25031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 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
\(3\) −1.08636 −0.627211 −0.313606 0.949553i \(-0.601537\pi\)
−0.313606 + 0.949553i \(0.601537\pi\)
\(4\) 0 0
\(5\) −0.0864029 −0.0386406 −0.0193203 0.999813i \(-0.506150\pi\)
−0.0193203 + 0.999813i \(0.506150\pi\)
\(6\) 0 0
\(7\) −0.643099 −0.243069 −0.121534 0.992587i \(-0.538781\pi\)
−0.121534 + 0.992587i \(0.538781\pi\)
\(8\) 0 0
\(9\) −1.81982 −0.606606
\(10\) 0 0
\(11\) −4.53359 −1.36693 −0.683464 0.729984i \(-0.739528\pi\)
−0.683464 + 0.729984i \(0.739528\pi\)
\(12\) 0 0
\(13\) 4.55688 1.26385 0.631926 0.775029i \(-0.282265\pi\)
0.631926 + 0.775029i \(0.282265\pi\)
\(14\) 0 0
\(15\) 0.0938648 0.0242358
\(16\) 0 0
\(17\) 2.52956 0.613509 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(18\) 0 0
\(19\) 4.06564 0.932722 0.466361 0.884595i \(-0.345565\pi\)
0.466361 + 0.884595i \(0.345565\pi\)
\(20\) 0 0
\(21\) 0.698638 0.152455
\(22\) 0 0
\(23\) 0.0799297 0.0166665 0.00833324 0.999965i \(-0.497347\pi\)
0.00833324 + 0.999965i \(0.497347\pi\)
\(24\) 0 0
\(25\) −4.99253 −0.998507
\(26\) 0 0
\(27\) 5.23607 1.00768
\(28\) 0 0
\(29\) −4.92904 −0.915299 −0.457649 0.889133i \(-0.651309\pi\)
−0.457649 + 0.889133i \(0.651309\pi\)
\(30\) 0 0
\(31\) 5.43721 0.976551 0.488276 0.872689i \(-0.337626\pi\)
0.488276 + 0.872689i \(0.337626\pi\)
\(32\) 0 0
\(33\) 4.92512 0.857353
\(34\) 0 0
\(35\) 0.0555656 0.00939231
\(36\) 0 0
\(37\) −1.43701 −0.236243 −0.118122 0.992999i \(-0.537687\pi\)
−0.118122 + 0.992999i \(0.537687\pi\)
\(38\) 0 0
\(39\) −4.95042 −0.792702
\(40\) 0 0
\(41\) −4.75696 −0.742913 −0.371456 0.928450i \(-0.621141\pi\)
−0.371456 + 0.928450i \(0.621141\pi\)
\(42\) 0 0
\(43\) 1.56498 0.238657 0.119328 0.992855i \(-0.461926\pi\)
0.119328 + 0.992855i \(0.461926\pi\)
\(44\) 0 0
\(45\) 0.157238 0.0234396
\(46\) 0 0
\(47\) 5.54900 0.809404 0.404702 0.914449i \(-0.367375\pi\)
0.404702 + 0.914449i \(0.367375\pi\)
\(48\) 0 0
\(49\) −6.58642 −0.940918
\(50\) 0 0
\(51\) −2.74802 −0.384800
\(52\) 0 0
\(53\) −12.8813 −1.76939 −0.884693 0.466174i \(-0.845632\pi\)
−0.884693 + 0.466174i \(0.845632\pi\)
\(54\) 0 0
\(55\) 0.391715 0.0528189
\(56\) 0 0
\(57\) −4.41676 −0.585014
\(58\) 0 0
\(59\) −6.90937 −0.899524 −0.449762 0.893149i \(-0.648491\pi\)
−0.449762 + 0.893149i \(0.648491\pi\)
\(60\) 0 0
\(61\) −7.25651 −0.929101 −0.464550 0.885547i \(-0.653784\pi\)
−0.464550 + 0.885547i \(0.653784\pi\)
\(62\) 0 0
\(63\) 1.17032 0.147447
\(64\) 0 0
\(65\) −0.393728 −0.0488359
\(66\) 0 0
\(67\) −1.62308 −0.198290 −0.0991452 0.995073i \(-0.531611\pi\)
−0.0991452 + 0.995073i \(0.531611\pi\)
\(68\) 0 0
\(69\) −0.0868325 −0.0104534
\(70\) 0 0
\(71\) 5.50057 0.652797 0.326399 0.945232i \(-0.394165\pi\)
0.326399 + 0.945232i \(0.394165\pi\)
\(72\) 0 0
\(73\) −11.0785 −1.29664 −0.648318 0.761370i \(-0.724527\pi\)
−0.648318 + 0.761370i \(0.724527\pi\)
\(74\) 0 0
\(75\) 5.42370 0.626275
\(76\) 0 0
\(77\) 2.91555 0.332257
\(78\) 0 0
\(79\) 8.91385 1.00289 0.501443 0.865191i \(-0.332803\pi\)
0.501443 + 0.865191i \(0.332803\pi\)
\(80\) 0 0
\(81\) −0.228807 −0.0254231
\(82\) 0 0
\(83\) 1.11686 0.122591 0.0612955 0.998120i \(-0.480477\pi\)
0.0612955 + 0.998120i \(0.480477\pi\)
\(84\) 0 0
\(85\) −0.218562 −0.0237063
\(86\) 0 0
\(87\) 5.35472 0.574086
\(88\) 0 0
\(89\) 2.69683 0.285863 0.142932 0.989733i \(-0.454347\pi\)
0.142932 + 0.989733i \(0.454347\pi\)
\(90\) 0 0
\(91\) −2.93053 −0.307203
\(92\) 0 0
\(93\) −5.90677 −0.612504
\(94\) 0 0
\(95\) −0.351283 −0.0360409
\(96\) 0 0
\(97\) 11.3347 1.15087 0.575433 0.817849i \(-0.304834\pi\)
0.575433 + 0.817849i \(0.304834\pi\)
\(98\) 0 0
\(99\) 8.25031 0.829187
\(100\) 0 0
\(101\) 16.0514 1.59717 0.798586 0.601881i \(-0.205582\pi\)
0.798586 + 0.601881i \(0.205582\pi\)
\(102\) 0 0
\(103\) −1.46182 −0.144037 −0.0720187 0.997403i \(-0.522944\pi\)
−0.0720187 + 0.997403i \(0.522944\pi\)
\(104\) 0 0
\(105\) −0.0603644 −0.00589096
\(106\) 0 0
\(107\) 6.83803 0.661057 0.330529 0.943796i \(-0.392773\pi\)
0.330529 + 0.943796i \(0.392773\pi\)
\(108\) 0 0
\(109\) 19.2483 1.84365 0.921825 0.387607i \(-0.126698\pi\)
0.921825 + 0.387607i \(0.126698\pi\)
\(110\) 0 0
\(111\) 1.56111 0.148174
\(112\) 0 0
\(113\) −13.9421 −1.31156 −0.655781 0.754951i \(-0.727661\pi\)
−0.655781 + 0.754951i \(0.727661\pi\)
\(114\) 0 0
\(115\) −0.00690615 −0.000644002 0
\(116\) 0 0
\(117\) −8.29269 −0.766660
\(118\) 0 0
\(119\) −1.62676 −0.149125
\(120\) 0 0
\(121\) 9.55343 0.868494
\(122\) 0 0
\(123\) 5.16778 0.465963
\(124\) 0 0
\(125\) 0.863384 0.0772234
\(126\) 0 0
\(127\) 7.37287 0.654236 0.327118 0.944983i \(-0.393922\pi\)
0.327118 + 0.944983i \(0.393922\pi\)
\(128\) 0 0
\(129\) −1.70013 −0.149688
\(130\) 0 0
\(131\) −2.87682 −0.251349 −0.125675 0.992072i \(-0.540110\pi\)
−0.125675 + 0.992072i \(0.540110\pi\)
\(132\) 0 0
\(133\) −2.61461 −0.226715
\(134\) 0 0
\(135\) −0.452411 −0.0389374
\(136\) 0 0
\(137\) 5.58750 0.477373 0.238686 0.971097i \(-0.423283\pi\)
0.238686 + 0.971097i \(0.423283\pi\)
\(138\) 0 0
\(139\) 6.53750 0.554503 0.277252 0.960797i \(-0.410576\pi\)
0.277252 + 0.960797i \(0.410576\pi\)
\(140\) 0 0
\(141\) −6.02822 −0.507668
\(142\) 0 0
\(143\) −20.6590 −1.72759
\(144\) 0 0
\(145\) 0.425883 0.0353677
\(146\) 0 0
\(147\) 7.15524 0.590154
\(148\) 0 0
\(149\) −6.88365 −0.563930 −0.281965 0.959425i \(-0.590986\pi\)
−0.281965 + 0.959425i \(0.590986\pi\)
\(150\) 0 0
\(151\) 19.1390 1.55751 0.778754 0.627329i \(-0.215852\pi\)
0.778754 + 0.627329i \(0.215852\pi\)
\(152\) 0 0
\(153\) −4.60334 −0.372158
\(154\) 0 0
\(155\) −0.469790 −0.0377345
\(156\) 0 0
\(157\) −0.944598 −0.0753872 −0.0376936 0.999289i \(-0.512001\pi\)
−0.0376936 + 0.999289i \(0.512001\pi\)
\(158\) 0 0
\(159\) 13.9938 1.10978
\(160\) 0 0
\(161\) −0.0514027 −0.00405110
\(162\) 0 0
\(163\) 15.1266 1.18481 0.592404 0.805641i \(-0.298179\pi\)
0.592404 + 0.805641i \(0.298179\pi\)
\(164\) 0 0
\(165\) −0.425545 −0.0331286
\(166\) 0 0
\(167\) −23.0445 −1.78323 −0.891617 0.452791i \(-0.850428\pi\)
−0.891617 + 0.452791i \(0.850428\pi\)
\(168\) 0 0
\(169\) 7.76516 0.597320
\(170\) 0 0
\(171\) −7.39872 −0.565795
\(172\) 0 0
\(173\) 15.1542 1.15215 0.576077 0.817395i \(-0.304583\pi\)
0.576077 + 0.817395i \(0.304583\pi\)
\(174\) 0 0
\(175\) 3.21069 0.242706
\(176\) 0 0
\(177\) 7.50608 0.564191
\(178\) 0 0
\(179\) 12.6301 0.944018 0.472009 0.881594i \(-0.343529\pi\)
0.472009 + 0.881594i \(0.343529\pi\)
\(180\) 0 0
\(181\) 12.4885 0.928260 0.464130 0.885767i \(-0.346367\pi\)
0.464130 + 0.885767i \(0.346367\pi\)
\(182\) 0 0
\(183\) 7.88319 0.582742
\(184\) 0 0
\(185\) 0.124162 0.00912856
\(186\) 0 0
\(187\) −11.4680 −0.838623
\(188\) 0 0
\(189\) −3.36731 −0.244936
\(190\) 0 0
\(191\) 26.4480 1.91371 0.956857 0.290560i \(-0.0938416\pi\)
0.956857 + 0.290560i \(0.0938416\pi\)
\(192\) 0 0
\(193\) 5.15147 0.370811 0.185405 0.982662i \(-0.440640\pi\)
0.185405 + 0.982662i \(0.440640\pi\)
\(194\) 0 0
\(195\) 0.427731 0.0306304
\(196\) 0 0
\(197\) −2.71102 −0.193152 −0.0965760 0.995326i \(-0.530789\pi\)
−0.0965760 + 0.995326i \(0.530789\pi\)
\(198\) 0 0
\(199\) −21.4862 −1.52312 −0.761558 0.648097i \(-0.775565\pi\)
−0.761558 + 0.648097i \(0.775565\pi\)
\(200\) 0 0
\(201\) 1.76325 0.124370
\(202\) 0 0
\(203\) 3.16986 0.222480
\(204\) 0 0
\(205\) 0.411015 0.0287066
\(206\) 0 0
\(207\) −0.145457 −0.0101100
\(208\) 0 0
\(209\) −18.4319 −1.27496
\(210\) 0 0
\(211\) 24.2749 1.67116 0.835578 0.549372i \(-0.185133\pi\)
0.835578 + 0.549372i \(0.185133\pi\)
\(212\) 0 0
\(213\) −5.97560 −0.409442
\(214\) 0 0
\(215\) −0.135218 −0.00922182
\(216\) 0 0
\(217\) −3.49666 −0.237369
\(218\) 0 0
\(219\) 12.0352 0.813265
\(220\) 0 0
\(221\) 11.5269 0.775384
\(222\) 0 0
\(223\) 1.65697 0.110959 0.0554796 0.998460i \(-0.482331\pi\)
0.0554796 + 0.998460i \(0.482331\pi\)
\(224\) 0 0
\(225\) 9.08550 0.605700
\(226\) 0 0
\(227\) −19.8621 −1.31829 −0.659146 0.752015i \(-0.729082\pi\)
−0.659146 + 0.752015i \(0.729082\pi\)
\(228\) 0 0
\(229\) 22.2503 1.47034 0.735171 0.677881i \(-0.237102\pi\)
0.735171 + 0.677881i \(0.237102\pi\)
\(230\) 0 0
\(231\) −3.16734 −0.208396
\(232\) 0 0
\(233\) 12.9240 0.846682 0.423341 0.905970i \(-0.360857\pi\)
0.423341 + 0.905970i \(0.360857\pi\)
\(234\) 0 0
\(235\) −0.479450 −0.0312758
\(236\) 0 0
\(237\) −9.68366 −0.629021
\(238\) 0 0
\(239\) 20.5055 1.32639 0.663196 0.748446i \(-0.269200\pi\)
0.663196 + 0.748446i \(0.269200\pi\)
\(240\) 0 0
\(241\) 11.9383 0.769016 0.384508 0.923122i \(-0.374371\pi\)
0.384508 + 0.923122i \(0.374371\pi\)
\(242\) 0 0
\(243\) −15.4596 −0.991736
\(244\) 0 0
\(245\) 0.569086 0.0363576
\(246\) 0 0
\(247\) 18.5266 1.17882
\(248\) 0 0
\(249\) −1.21331 −0.0768905
\(250\) 0 0
\(251\) −23.6715 −1.49413 −0.747066 0.664750i \(-0.768538\pi\)
−0.747066 + 0.664750i \(0.768538\pi\)
\(252\) 0 0
\(253\) −0.362368 −0.0227819
\(254\) 0 0
\(255\) 0.237437 0.0148689
\(256\) 0 0
\(257\) 5.08886 0.317434 0.158717 0.987324i \(-0.449264\pi\)
0.158717 + 0.987324i \(0.449264\pi\)
\(258\) 0 0
\(259\) 0.924140 0.0574233
\(260\) 0 0
\(261\) 8.96995 0.555226
\(262\) 0 0
\(263\) −16.1800 −0.997703 −0.498852 0.866687i \(-0.666245\pi\)
−0.498852 + 0.866687i \(0.666245\pi\)
\(264\) 0 0
\(265\) 1.11298 0.0683701
\(266\) 0 0
\(267\) −2.92973 −0.179297
\(268\) 0 0
\(269\) 26.3752 1.60813 0.804063 0.594544i \(-0.202667\pi\)
0.804063 + 0.594544i \(0.202667\pi\)
\(270\) 0 0
\(271\) 17.7424 1.07777 0.538887 0.842378i \(-0.318845\pi\)
0.538887 + 0.842378i \(0.318845\pi\)
\(272\) 0 0
\(273\) 3.18361 0.192681
\(274\) 0 0
\(275\) 22.6341 1.36489
\(276\) 0 0
\(277\) 4.52925 0.272136 0.136068 0.990700i \(-0.456553\pi\)
0.136068 + 0.990700i \(0.456553\pi\)
\(278\) 0 0
\(279\) −9.89473 −0.592382
\(280\) 0 0
\(281\) −28.0818 −1.67522 −0.837610 0.546268i \(-0.816048\pi\)
−0.837610 + 0.546268i \(0.816048\pi\)
\(282\) 0 0
\(283\) 9.06552 0.538889 0.269445 0.963016i \(-0.413160\pi\)
0.269445 + 0.963016i \(0.413160\pi\)
\(284\) 0 0
\(285\) 0.381621 0.0226052
\(286\) 0 0
\(287\) 3.05920 0.180579
\(288\) 0 0
\(289\) −10.6013 −0.623607
\(290\) 0 0
\(291\) −12.3136 −0.721836
\(292\) 0 0
\(293\) 8.04959 0.470262 0.235131 0.971964i \(-0.424448\pi\)
0.235131 + 0.971964i \(0.424448\pi\)
\(294\) 0 0
\(295\) 0.596990 0.0347581
\(296\) 0 0
\(297\) −23.7382 −1.37743
\(298\) 0 0
\(299\) 0.364230 0.0210640
\(300\) 0 0
\(301\) −1.00643 −0.0580099
\(302\) 0 0
\(303\) −17.4376 −1.00176
\(304\) 0 0
\(305\) 0.626983 0.0359010
\(306\) 0 0
\(307\) −10.7127 −0.611409 −0.305704 0.952126i \(-0.598892\pi\)
−0.305704 + 0.952126i \(0.598892\pi\)
\(308\) 0 0
\(309\) 1.58807 0.0903419
\(310\) 0 0
\(311\) 13.9805 0.792760 0.396380 0.918087i \(-0.370266\pi\)
0.396380 + 0.918087i \(0.370266\pi\)
\(312\) 0 0
\(313\) 5.56354 0.314470 0.157235 0.987561i \(-0.449742\pi\)
0.157235 + 0.987561i \(0.449742\pi\)
\(314\) 0 0
\(315\) −0.101119 −0.00569743
\(316\) 0 0
\(317\) 6.26371 0.351805 0.175903 0.984408i \(-0.443716\pi\)
0.175903 + 0.984408i \(0.443716\pi\)
\(318\) 0 0
\(319\) 22.3462 1.25115
\(320\) 0 0
\(321\) −7.42858 −0.414623
\(322\) 0 0
\(323\) 10.2843 0.572233
\(324\) 0 0
\(325\) −22.7504 −1.26196
\(326\) 0 0
\(327\) −20.9106 −1.15636
\(328\) 0 0
\(329\) −3.56855 −0.196741
\(330\) 0 0
\(331\) −25.6294 −1.40872 −0.704361 0.709842i \(-0.748766\pi\)
−0.704361 + 0.709842i \(0.748766\pi\)
\(332\) 0 0
\(333\) 2.61510 0.143306
\(334\) 0 0
\(335\) 0.140239 0.00766205
\(336\) 0 0
\(337\) 25.0011 1.36189 0.680947 0.732332i \(-0.261568\pi\)
0.680947 + 0.732332i \(0.261568\pi\)
\(338\) 0 0
\(339\) 15.1462 0.822627
\(340\) 0 0
\(341\) −24.6501 −1.33488
\(342\) 0 0
\(343\) 8.73742 0.471776
\(344\) 0 0
\(345\) 0.00750258 0.000403925 0
\(346\) 0 0
\(347\) −13.8856 −0.745416 −0.372708 0.927949i \(-0.621571\pi\)
−0.372708 + 0.927949i \(0.621571\pi\)
\(348\) 0 0
\(349\) −26.7614 −1.43250 −0.716252 0.697842i \(-0.754144\pi\)
−0.716252 + 0.697842i \(0.754144\pi\)
\(350\) 0 0
\(351\) 23.8601 1.27356
\(352\) 0 0
\(353\) −8.94995 −0.476358 −0.238179 0.971221i \(-0.576550\pi\)
−0.238179 + 0.971221i \(0.576550\pi\)
\(354\) 0 0
\(355\) −0.475265 −0.0252244
\(356\) 0 0
\(357\) 1.76725 0.0935327
\(358\) 0 0
\(359\) 33.1713 1.75071 0.875356 0.483479i \(-0.160627\pi\)
0.875356 + 0.483479i \(0.160627\pi\)
\(360\) 0 0
\(361\) −2.47058 −0.130030
\(362\) 0 0
\(363\) −10.3785 −0.544729
\(364\) 0 0
\(365\) 0.957212 0.0501027
\(366\) 0 0
\(367\) 10.7634 0.561844 0.280922 0.959731i \(-0.409360\pi\)
0.280922 + 0.959731i \(0.409360\pi\)
\(368\) 0 0
\(369\) 8.65680 0.450655
\(370\) 0 0
\(371\) 8.28397 0.430082
\(372\) 0 0
\(373\) −25.4079 −1.31557 −0.657786 0.753205i \(-0.728507\pi\)
−0.657786 + 0.753205i \(0.728507\pi\)
\(374\) 0 0
\(375\) −0.937947 −0.0484354
\(376\) 0 0
\(377\) −22.4610 −1.15680
\(378\) 0 0
\(379\) −0.795661 −0.0408703 −0.0204352 0.999791i \(-0.506505\pi\)
−0.0204352 + 0.999791i \(0.506505\pi\)
\(380\) 0 0
\(381\) −8.00960 −0.410344
\(382\) 0 0
\(383\) 26.2933 1.34352 0.671762 0.740767i \(-0.265538\pi\)
0.671762 + 0.740767i \(0.265538\pi\)
\(384\) 0 0
\(385\) −0.251912 −0.0128386
\(386\) 0 0
\(387\) −2.84797 −0.144770
\(388\) 0 0
\(389\) 20.8254 1.05589 0.527946 0.849278i \(-0.322962\pi\)
0.527946 + 0.849278i \(0.322962\pi\)
\(390\) 0 0
\(391\) 0.202187 0.0102250
\(392\) 0 0
\(393\) 3.12527 0.157649
\(394\) 0 0
\(395\) −0.770182 −0.0387521
\(396\) 0 0
\(397\) −21.5081 −1.07946 −0.539730 0.841838i \(-0.681474\pi\)
−0.539730 + 0.841838i \(0.681474\pi\)
\(398\) 0 0
\(399\) 2.84041 0.142198
\(400\) 0 0
\(401\) 10.2060 0.509662 0.254831 0.966986i \(-0.417980\pi\)
0.254831 + 0.966986i \(0.417980\pi\)
\(402\) 0 0
\(403\) 24.7767 1.23422
\(404\) 0 0
\(405\) 0.0197696 0.000982361 0
\(406\) 0 0
\(407\) 6.51482 0.322927
\(408\) 0 0
\(409\) 19.0291 0.940929 0.470465 0.882419i \(-0.344086\pi\)
0.470465 + 0.882419i \(0.344086\pi\)
\(410\) 0 0
\(411\) −6.07005 −0.299413
\(412\) 0 0
\(413\) 4.44341 0.218646
\(414\) 0 0
\(415\) −0.0964998 −0.00473699
\(416\) 0 0
\(417\) −7.10209 −0.347791
\(418\) 0 0
\(419\) −16.7730 −0.819415 −0.409707 0.912217i \(-0.634369\pi\)
−0.409707 + 0.912217i \(0.634369\pi\)
\(420\) 0 0
\(421\) −13.9413 −0.679458 −0.339729 0.940523i \(-0.610335\pi\)
−0.339729 + 0.940523i \(0.610335\pi\)
\(422\) 0 0
\(423\) −10.0982 −0.490990
\(424\) 0 0
\(425\) −12.6289 −0.612593
\(426\) 0 0
\(427\) 4.66665 0.225835
\(428\) 0 0
\(429\) 22.4432 1.08357
\(430\) 0 0
\(431\) −1.42592 −0.0686842 −0.0343421 0.999410i \(-0.510934\pi\)
−0.0343421 + 0.999410i \(0.510934\pi\)
\(432\) 0 0
\(433\) −4.39205 −0.211069 −0.105534 0.994416i \(-0.533655\pi\)
−0.105534 + 0.994416i \(0.533655\pi\)
\(434\) 0 0
\(435\) −0.462663 −0.0221830
\(436\) 0 0
\(437\) 0.324965 0.0155452
\(438\) 0 0
\(439\) −0.163858 −0.00782052 −0.00391026 0.999992i \(-0.501245\pi\)
−0.00391026 + 0.999992i \(0.501245\pi\)
\(440\) 0 0
\(441\) 11.9861 0.570766
\(442\) 0 0
\(443\) 25.0145 1.18847 0.594236 0.804290i \(-0.297454\pi\)
0.594236 + 0.804290i \(0.297454\pi\)
\(444\) 0 0
\(445\) −0.233014 −0.0110459
\(446\) 0 0
\(447\) 7.47813 0.353703
\(448\) 0 0
\(449\) 28.0945 1.32586 0.662932 0.748680i \(-0.269312\pi\)
0.662932 + 0.748680i \(0.269312\pi\)
\(450\) 0 0
\(451\) 21.5661 1.01551
\(452\) 0 0
\(453\) −20.7919 −0.976887
\(454\) 0 0
\(455\) 0.253206 0.0118705
\(456\) 0 0
\(457\) 2.60049 0.121646 0.0608228 0.998149i \(-0.480628\pi\)
0.0608228 + 0.998149i \(0.480628\pi\)
\(458\) 0 0
\(459\) 13.2450 0.618222
\(460\) 0 0
\(461\) −26.4245 −1.23071 −0.615356 0.788249i \(-0.710988\pi\)
−0.615356 + 0.788249i \(0.710988\pi\)
\(462\) 0 0
\(463\) 18.2995 0.850450 0.425225 0.905088i \(-0.360195\pi\)
0.425225 + 0.905088i \(0.360195\pi\)
\(464\) 0 0
\(465\) 0.510362 0.0236675
\(466\) 0 0
\(467\) −6.59517 −0.305188 −0.152594 0.988289i \(-0.548763\pi\)
−0.152594 + 0.988289i \(0.548763\pi\)
\(468\) 0 0
\(469\) 1.04380 0.0481982
\(470\) 0 0
\(471\) 1.02618 0.0472837
\(472\) 0 0
\(473\) −7.09496 −0.326226
\(474\) 0 0
\(475\) −20.2978 −0.931329
\(476\) 0 0
\(477\) 23.4417 1.07332
\(478\) 0 0
\(479\) 11.8696 0.542336 0.271168 0.962532i \(-0.412590\pi\)
0.271168 + 0.962532i \(0.412590\pi\)
\(480\) 0 0
\(481\) −6.54828 −0.298576
\(482\) 0 0
\(483\) 0.0558419 0.00254089
\(484\) 0 0
\(485\) −0.979353 −0.0444701
\(486\) 0 0
\(487\) 29.2560 1.32571 0.662857 0.748746i \(-0.269344\pi\)
0.662857 + 0.748746i \(0.269344\pi\)
\(488\) 0 0
\(489\) −16.4330 −0.743125
\(490\) 0 0
\(491\) 13.9712 0.630513 0.315256 0.949007i \(-0.397910\pi\)
0.315256 + 0.949007i \(0.397910\pi\)
\(492\) 0 0
\(493\) −12.4683 −0.561544
\(494\) 0 0
\(495\) −0.712851 −0.0320403
\(496\) 0 0
\(497\) −3.53741 −0.158674
\(498\) 0 0
\(499\) −9.22901 −0.413147 −0.206574 0.978431i \(-0.566231\pi\)
−0.206574 + 0.978431i \(0.566231\pi\)
\(500\) 0 0
\(501\) 25.0346 1.11846
\(502\) 0 0
\(503\) 39.1056 1.74363 0.871817 0.489833i \(-0.162942\pi\)
0.871817 + 0.489833i \(0.162942\pi\)
\(504\) 0 0
\(505\) −1.38689 −0.0617156
\(506\) 0 0
\(507\) −8.43577 −0.374646
\(508\) 0 0
\(509\) −20.4622 −0.906970 −0.453485 0.891264i \(-0.649819\pi\)
−0.453485 + 0.891264i \(0.649819\pi\)
\(510\) 0 0
\(511\) 7.12455 0.315172
\(512\) 0 0
\(513\) 21.2880 0.939886
\(514\) 0 0
\(515\) 0.126306 0.00556569
\(516\) 0 0
\(517\) −25.1569 −1.10640
\(518\) 0 0
\(519\) −16.4630 −0.722644
\(520\) 0 0
\(521\) −10.2992 −0.451216 −0.225608 0.974218i \(-0.572437\pi\)
−0.225608 + 0.974218i \(0.572437\pi\)
\(522\) 0 0
\(523\) 36.3558 1.58973 0.794864 0.606788i \(-0.207542\pi\)
0.794864 + 0.606788i \(0.207542\pi\)
\(524\) 0 0
\(525\) −3.48798 −0.152228
\(526\) 0 0
\(527\) 13.7538 0.599123
\(528\) 0 0
\(529\) −22.9936 −0.999722
\(530\) 0 0
\(531\) 12.5738 0.545656
\(532\) 0 0
\(533\) −21.6769 −0.938931
\(534\) 0 0
\(535\) −0.590826 −0.0255436
\(536\) 0 0
\(537\) −13.7209 −0.592099
\(538\) 0 0
\(539\) 29.8601 1.28617
\(540\) 0 0
\(541\) 30.6247 1.31666 0.658330 0.752730i \(-0.271263\pi\)
0.658330 + 0.752730i \(0.271263\pi\)
\(542\) 0 0
\(543\) −13.5670 −0.582215
\(544\) 0 0
\(545\) −1.66311 −0.0712396
\(546\) 0 0
\(547\) 15.5797 0.666140 0.333070 0.942902i \(-0.391916\pi\)
0.333070 + 0.942902i \(0.391916\pi\)
\(548\) 0 0
\(549\) 13.2055 0.563598
\(550\) 0 0
\(551\) −20.0397 −0.853719
\(552\) 0 0
\(553\) −5.73249 −0.243770
\(554\) 0 0
\(555\) −0.134885 −0.00572554
\(556\) 0 0
\(557\) −24.1697 −1.02410 −0.512052 0.858955i \(-0.671114\pi\)
−0.512052 + 0.858955i \(0.671114\pi\)
\(558\) 0 0
\(559\) 7.13141 0.301626
\(560\) 0 0
\(561\) 12.4584 0.525994
\(562\) 0 0
\(563\) 22.0486 0.929239 0.464619 0.885510i \(-0.346191\pi\)
0.464619 + 0.885510i \(0.346191\pi\)
\(564\) 0 0
\(565\) 1.20464 0.0506795
\(566\) 0 0
\(567\) 0.147146 0.00617955
\(568\) 0 0
\(569\) 26.3055 1.10278 0.551392 0.834246i \(-0.314097\pi\)
0.551392 + 0.834246i \(0.314097\pi\)
\(570\) 0 0
\(571\) −47.5564 −1.99017 −0.995086 0.0990110i \(-0.968432\pi\)
−0.995086 + 0.0990110i \(0.968432\pi\)
\(572\) 0 0
\(573\) −28.7321 −1.20030
\(574\) 0 0
\(575\) −0.399052 −0.0166416
\(576\) 0 0
\(577\) 34.1394 1.42124 0.710620 0.703576i \(-0.248414\pi\)
0.710620 + 0.703576i \(0.248414\pi\)
\(578\) 0 0
\(579\) −5.59636 −0.232577
\(580\) 0 0
\(581\) −0.718250 −0.0297980
\(582\) 0 0
\(583\) 58.3986 2.41862
\(584\) 0 0
\(585\) 0.716513 0.0296242
\(586\) 0 0
\(587\) 1.89205 0.0780934 0.0390467 0.999237i \(-0.487568\pi\)
0.0390467 + 0.999237i \(0.487568\pi\)
\(588\) 0 0
\(589\) 22.1057 0.910850
\(590\) 0 0
\(591\) 2.94514 0.121147
\(592\) 0 0
\(593\) 19.1452 0.786197 0.393099 0.919496i \(-0.371403\pi\)
0.393099 + 0.919496i \(0.371403\pi\)
\(594\) 0 0
\(595\) 0.140557 0.00576226
\(596\) 0 0
\(597\) 23.3418 0.955315
\(598\) 0 0
\(599\) −41.6458 −1.70160 −0.850801 0.525488i \(-0.823883\pi\)
−0.850801 + 0.525488i \(0.823883\pi\)
\(600\) 0 0
\(601\) 39.2507 1.60107 0.800536 0.599285i \(-0.204548\pi\)
0.800536 + 0.599285i \(0.204548\pi\)
\(602\) 0 0
\(603\) 2.95371 0.120284
\(604\) 0 0
\(605\) −0.825444 −0.0335591
\(606\) 0 0
\(607\) 30.8704 1.25299 0.626496 0.779425i \(-0.284489\pi\)
0.626496 + 0.779425i \(0.284489\pi\)
\(608\) 0 0
\(609\) −3.44361 −0.139542
\(610\) 0 0
\(611\) 25.2861 1.02297
\(612\) 0 0
\(613\) 15.5488 0.628009 0.314004 0.949422i \(-0.398329\pi\)
0.314004 + 0.949422i \(0.398329\pi\)
\(614\) 0 0
\(615\) −0.446511 −0.0180051
\(616\) 0 0
\(617\) −9.93307 −0.399890 −0.199945 0.979807i \(-0.564076\pi\)
−0.199945 + 0.979807i \(0.564076\pi\)
\(618\) 0 0
\(619\) 4.08575 0.164220 0.0821100 0.996623i \(-0.473834\pi\)
0.0821100 + 0.996623i \(0.473834\pi\)
\(620\) 0 0
\(621\) 0.418517 0.0167945
\(622\) 0 0
\(623\) −1.73433 −0.0694844
\(624\) 0 0
\(625\) 24.8881 0.995523
\(626\) 0 0
\(627\) 20.0238 0.799672
\(628\) 0 0
\(629\) −3.63501 −0.144937
\(630\) 0 0
\(631\) 13.1626 0.523995 0.261997 0.965069i \(-0.415619\pi\)
0.261997 + 0.965069i \(0.415619\pi\)
\(632\) 0 0
\(633\) −26.3714 −1.04817
\(634\) 0 0
\(635\) −0.637037 −0.0252801
\(636\) 0 0
\(637\) −30.0135 −1.18918
\(638\) 0 0
\(639\) −10.0100 −0.395991
\(640\) 0 0
\(641\) −12.3850 −0.489176 −0.244588 0.969627i \(-0.578653\pi\)
−0.244588 + 0.969627i \(0.578653\pi\)
\(642\) 0 0
\(643\) −22.6135 −0.891788 −0.445894 0.895086i \(-0.647114\pi\)
−0.445894 + 0.895086i \(0.647114\pi\)
\(644\) 0 0
\(645\) 0.146896 0.00578403
\(646\) 0 0
\(647\) −9.93742 −0.390680 −0.195340 0.980736i \(-0.562581\pi\)
−0.195340 + 0.980736i \(0.562581\pi\)
\(648\) 0 0
\(649\) 31.3243 1.22958
\(650\) 0 0
\(651\) 3.79864 0.148880
\(652\) 0 0
\(653\) −14.1952 −0.555500 −0.277750 0.960653i \(-0.589589\pi\)
−0.277750 + 0.960653i \(0.589589\pi\)
\(654\) 0 0
\(655\) 0.248566 0.00971227
\(656\) 0 0
\(657\) 20.1608 0.786547
\(658\) 0 0
\(659\) −42.0553 −1.63824 −0.819120 0.573622i \(-0.805538\pi\)
−0.819120 + 0.573622i \(0.805538\pi\)
\(660\) 0 0
\(661\) −44.6652 −1.73728 −0.868638 0.495447i \(-0.835004\pi\)
−0.868638 + 0.495447i \(0.835004\pi\)
\(662\) 0 0
\(663\) −12.5224 −0.486330
\(664\) 0 0
\(665\) 0.225910 0.00876041
\(666\) 0 0
\(667\) −0.393976 −0.0152548
\(668\) 0 0
\(669\) −1.80007 −0.0695949
\(670\) 0 0
\(671\) 32.8980 1.27001
\(672\) 0 0
\(673\) 5.03801 0.194201 0.0971005 0.995275i \(-0.469043\pi\)
0.0971005 + 0.995275i \(0.469043\pi\)
\(674\) 0 0
\(675\) −26.1412 −1.00618
\(676\) 0 0
\(677\) 32.0307 1.23104 0.615521 0.788121i \(-0.288946\pi\)
0.615521 + 0.788121i \(0.288946\pi\)
\(678\) 0 0
\(679\) −7.28935 −0.279739
\(680\) 0 0
\(681\) 21.5774 0.826848
\(682\) 0 0
\(683\) 48.5034 1.85593 0.927965 0.372667i \(-0.121557\pi\)
0.927965 + 0.372667i \(0.121557\pi\)
\(684\) 0 0
\(685\) −0.482776 −0.0184459
\(686\) 0 0
\(687\) −24.1719 −0.922216
\(688\) 0 0
\(689\) −58.6986 −2.23624
\(690\) 0 0
\(691\) 21.6212 0.822507 0.411254 0.911521i \(-0.365091\pi\)
0.411254 + 0.911521i \(0.365091\pi\)
\(692\) 0 0
\(693\) −5.30577 −0.201549
\(694\) 0 0
\(695\) −0.564859 −0.0214263
\(696\) 0 0
\(697\) −12.0330 −0.455784
\(698\) 0 0
\(699\) −14.0402 −0.531049
\(700\) 0 0
\(701\) 16.6093 0.627323 0.313662 0.949535i \(-0.398444\pi\)
0.313662 + 0.949535i \(0.398444\pi\)
\(702\) 0 0
\(703\) −5.84237 −0.220349
\(704\) 0 0
\(705\) 0.520856 0.0196166
\(706\) 0 0
\(707\) −10.3226 −0.388222
\(708\) 0 0
\(709\) −25.0103 −0.939283 −0.469642 0.882857i \(-0.655617\pi\)
−0.469642 + 0.882857i \(0.655617\pi\)
\(710\) 0 0
\(711\) −16.2216 −0.608357
\(712\) 0 0
\(713\) 0.434594 0.0162757
\(714\) 0 0
\(715\) 1.78500 0.0667552
\(716\) 0 0
\(717\) −22.2764 −0.831928
\(718\) 0 0
\(719\) −33.2422 −1.23972 −0.619862 0.784711i \(-0.712811\pi\)
−0.619862 + 0.784711i \(0.712811\pi\)
\(720\) 0 0
\(721\) 0.940095 0.0350110
\(722\) 0 0
\(723\) −12.9693 −0.482335
\(724\) 0 0
\(725\) 24.6084 0.913932
\(726\) 0 0
\(727\) 3.42728 0.127111 0.0635554 0.997978i \(-0.479756\pi\)
0.0635554 + 0.997978i \(0.479756\pi\)
\(728\) 0 0
\(729\) 17.4812 0.647451
\(730\) 0 0
\(731\) 3.95870 0.146418
\(732\) 0 0
\(733\) 10.3373 0.381815 0.190908 0.981608i \(-0.438857\pi\)
0.190908 + 0.981608i \(0.438857\pi\)
\(734\) 0 0
\(735\) −0.618233 −0.0228039
\(736\) 0 0
\(737\) 7.35836 0.271049
\(738\) 0 0
\(739\) −43.7360 −1.60886 −0.804428 0.594050i \(-0.797528\pi\)
−0.804428 + 0.594050i \(0.797528\pi\)
\(740\) 0 0
\(741\) −20.1266 −0.739370
\(742\) 0 0
\(743\) −29.3659 −1.07733 −0.538665 0.842520i \(-0.681071\pi\)
−0.538665 + 0.842520i \(0.681071\pi\)
\(744\) 0 0
\(745\) 0.594767 0.0217906
\(746\) 0 0
\(747\) −2.03248 −0.0743645
\(748\) 0 0
\(749\) −4.39753 −0.160682
\(750\) 0 0
\(751\) 1.00000 0.0364905
\(752\) 0 0
\(753\) 25.7158 0.937136
\(754\) 0 0
\(755\) −1.65366 −0.0601830
\(756\) 0 0
\(757\) 49.3449 1.79347 0.896735 0.442568i \(-0.145933\pi\)
0.896735 + 0.442568i \(0.145933\pi\)
\(758\) 0 0
\(759\) 0.393663 0.0142891
\(760\) 0 0
\(761\) 12.6088 0.457068 0.228534 0.973536i \(-0.426607\pi\)
0.228534 + 0.973536i \(0.426607\pi\)
\(762\) 0 0
\(763\) −12.3785 −0.448133
\(764\) 0 0
\(765\) 0.397742 0.0143804
\(766\) 0 0
\(767\) −31.4852 −1.13686
\(768\) 0 0
\(769\) 8.02880 0.289526 0.144763 0.989466i \(-0.453758\pi\)
0.144763 + 0.989466i \(0.453758\pi\)
\(770\) 0 0
\(771\) −5.52834 −0.199098
\(772\) 0 0
\(773\) −45.3549 −1.63130 −0.815652 0.578543i \(-0.803621\pi\)
−0.815652 + 0.578543i \(0.803621\pi\)
\(774\) 0 0
\(775\) −27.1454 −0.975093
\(776\) 0 0
\(777\) −1.00395 −0.0360165
\(778\) 0 0
\(779\) −19.3401 −0.692931
\(780\) 0 0
\(781\) −24.9373 −0.892327
\(782\) 0 0
\(783\) −25.8088 −0.922330
\(784\) 0 0
\(785\) 0.0816160 0.00291300
\(786\) 0 0
\(787\) −11.3514 −0.404635 −0.202318 0.979320i \(-0.564847\pi\)
−0.202318 + 0.979320i \(0.564847\pi\)
\(788\) 0 0
\(789\) 17.5774 0.625771
\(790\) 0 0
\(791\) 8.96615 0.318800
\(792\) 0 0
\(793\) −33.0670 −1.17424
\(794\) 0 0
\(795\) −1.20910 −0.0428825
\(796\) 0 0
\(797\) −45.8609 −1.62448 −0.812238 0.583326i \(-0.801751\pi\)
−0.812238 + 0.583326i \(0.801751\pi\)
\(798\) 0 0
\(799\) 14.0365 0.496577
\(800\) 0 0
\(801\) −4.90774 −0.173406
\(802\) 0 0
\(803\) 50.2252 1.77241
\(804\) 0 0
\(805\) 0.00444134 0.000156537 0
\(806\) 0 0
\(807\) −28.6531 −1.00864
\(808\) 0 0
\(809\) 13.5190 0.475302 0.237651 0.971351i \(-0.423623\pi\)
0.237651 + 0.971351i \(0.423623\pi\)
\(810\) 0 0
\(811\) −37.7329 −1.32498 −0.662491 0.749070i \(-0.730501\pi\)
−0.662491 + 0.749070i \(0.730501\pi\)
\(812\) 0 0
\(813\) −19.2747 −0.675992
\(814\) 0 0
\(815\) −1.30698 −0.0457816
\(816\) 0 0
\(817\) 6.36263 0.222600
\(818\) 0 0
\(819\) 5.33302 0.186351
\(820\) 0 0
\(821\) 48.6436 1.69767 0.848836 0.528656i \(-0.177304\pi\)
0.848836 + 0.528656i \(0.177304\pi\)
\(822\) 0 0
\(823\) −39.9530 −1.39268 −0.696338 0.717714i \(-0.745188\pi\)
−0.696338 + 0.717714i \(0.745188\pi\)
\(824\) 0 0
\(825\) −24.5888 −0.856073
\(826\) 0 0
\(827\) −10.5132 −0.365581 −0.182790 0.983152i \(-0.558513\pi\)
−0.182790 + 0.983152i \(0.558513\pi\)
\(828\) 0 0
\(829\) 21.7750 0.756277 0.378139 0.925749i \(-0.376564\pi\)
0.378139 + 0.925749i \(0.376564\pi\)
\(830\) 0 0
\(831\) −4.92040 −0.170687
\(832\) 0 0
\(833\) −16.6608 −0.577261
\(834\) 0 0
\(835\) 1.99111 0.0689051
\(836\) 0 0
\(837\) 28.4696 0.984052
\(838\) 0 0
\(839\) 18.8305 0.650102 0.325051 0.945696i \(-0.394619\pi\)
0.325051 + 0.945696i \(0.394619\pi\)
\(840\) 0 0
\(841\) −4.70461 −0.162228
\(842\) 0 0
\(843\) 30.5070 1.05072
\(844\) 0 0
\(845\) −0.670932 −0.0230808
\(846\) 0 0
\(847\) −6.14380 −0.211104
\(848\) 0 0
\(849\) −9.84844 −0.337998
\(850\) 0 0
\(851\) −0.114860 −0.00393734
\(852\) 0 0
\(853\) −3.39978 −0.116406 −0.0582031 0.998305i \(-0.518537\pi\)
−0.0582031 + 0.998305i \(0.518537\pi\)
\(854\) 0 0
\(855\) 0.639271 0.0218626
\(856\) 0 0
\(857\) 49.7786 1.70040 0.850202 0.526457i \(-0.176480\pi\)
0.850202 + 0.526457i \(0.176480\pi\)
\(858\) 0 0
\(859\) −14.2516 −0.486257 −0.243128 0.969994i \(-0.578174\pi\)
−0.243128 + 0.969994i \(0.578174\pi\)
\(860\) 0 0
\(861\) −3.32340 −0.113261
\(862\) 0 0
\(863\) 11.3168 0.385228 0.192614 0.981275i \(-0.438303\pi\)
0.192614 + 0.981275i \(0.438303\pi\)
\(864\) 0 0
\(865\) −1.30937 −0.0445199
\(866\) 0 0
\(867\) 11.5169 0.391133
\(868\) 0 0
\(869\) −40.4117 −1.37087
\(870\) 0 0
\(871\) −7.39617 −0.250610
\(872\) 0 0
\(873\) −20.6271 −0.698122
\(874\) 0 0
\(875\) −0.555242 −0.0187706
\(876\) 0 0
\(877\) −39.5164 −1.33437 −0.667187 0.744890i \(-0.732502\pi\)
−0.667187 + 0.744890i \(0.732502\pi\)
\(878\) 0 0
\(879\) −8.74477 −0.294954
\(880\) 0 0
\(881\) 34.1496 1.15053 0.575265 0.817967i \(-0.304899\pi\)
0.575265 + 0.817967i \(0.304899\pi\)
\(882\) 0 0
\(883\) 22.6502 0.762241 0.381121 0.924525i \(-0.375538\pi\)
0.381121 + 0.924525i \(0.375538\pi\)
\(884\) 0 0
\(885\) −0.648547 −0.0218007
\(886\) 0 0
\(887\) 9.49135 0.318688 0.159344 0.987223i \(-0.449062\pi\)
0.159344 + 0.987223i \(0.449062\pi\)
\(888\) 0 0
\(889\) −4.74148 −0.159024
\(890\) 0 0
\(891\) 1.03732 0.0347515
\(892\) 0 0
\(893\) 22.5602 0.754949
\(894\) 0 0
\(895\) −1.09128 −0.0364774
\(896\) 0 0
\(897\) −0.395685 −0.0132115
\(898\) 0 0
\(899\) −26.8002 −0.893836
\(900\) 0 0
\(901\) −32.5841 −1.08553
\(902\) 0 0
\(903\) 1.09335 0.0363845
\(904\) 0 0
\(905\) −1.07904 −0.0358685
\(906\) 0 0
\(907\) 22.7746 0.756218 0.378109 0.925761i \(-0.376575\pi\)
0.378109 + 0.925761i \(0.376575\pi\)
\(908\) 0 0
\(909\) −29.2106 −0.968854
\(910\) 0 0
\(911\) 10.5037 0.348002 0.174001 0.984745i \(-0.444330\pi\)
0.174001 + 0.984745i \(0.444330\pi\)
\(912\) 0 0
\(913\) −5.06337 −0.167573
\(914\) 0 0
\(915\) −0.681131 −0.0225175
\(916\) 0 0
\(917\) 1.85008 0.0610951
\(918\) 0 0
\(919\) −5.97329 −0.197041 −0.0985204 0.995135i \(-0.531411\pi\)
−0.0985204 + 0.995135i \(0.531411\pi\)
\(920\) 0 0
\(921\) 11.6379 0.383482
\(922\) 0 0
\(923\) 25.0654 0.825038
\(924\) 0 0
\(925\) 7.17432 0.235890
\(926\) 0 0
\(927\) 2.66025 0.0873740
\(928\) 0 0
\(929\) −36.1852 −1.18720 −0.593600 0.804760i \(-0.702294\pi\)
−0.593600 + 0.804760i \(0.702294\pi\)
\(930\) 0 0
\(931\) −26.7780 −0.877614
\(932\) 0 0
\(933\) −15.1879 −0.497228
\(934\) 0 0
\(935\) 0.990868 0.0324049
\(936\) 0 0
\(937\) 56.8861 1.85839 0.929193 0.369594i \(-0.120503\pi\)
0.929193 + 0.369594i \(0.120503\pi\)
\(938\) 0 0
\(939\) −6.04402 −0.197239
\(940\) 0 0
\(941\) −29.0907 −0.948328 −0.474164 0.880436i \(-0.657250\pi\)
−0.474164 + 0.880436i \(0.657250\pi\)
\(942\) 0 0
\(943\) −0.380222 −0.0123817
\(944\) 0 0
\(945\) 0.290945 0.00946445
\(946\) 0 0
\(947\) −9.58176 −0.311366 −0.155683 0.987807i \(-0.549758\pi\)
−0.155683 + 0.987807i \(0.549758\pi\)
\(948\) 0 0
\(949\) −50.4832 −1.63876
\(950\) 0 0
\(951\) −6.80466 −0.220656
\(952\) 0 0
\(953\) −52.2933 −1.69395 −0.846973 0.531636i \(-0.821577\pi\)
−0.846973 + 0.531636i \(0.821577\pi\)
\(954\) 0 0
\(955\) −2.28519 −0.0739470
\(956\) 0 0
\(957\) −24.2761 −0.784734
\(958\) 0 0
\(959\) −3.59332 −0.116034
\(960\) 0 0
\(961\) −1.43678 −0.0463479
\(962\) 0 0
\(963\) −12.4440 −0.401001
\(964\) 0 0
\(965\) −0.445102 −0.0143283
\(966\) 0 0
\(967\) 10.9187 0.351122 0.175561 0.984469i \(-0.443826\pi\)
0.175561 + 0.984469i \(0.443826\pi\)
\(968\) 0 0
\(969\) −11.1725 −0.358911
\(970\) 0 0
\(971\) 18.2093 0.584363 0.292182 0.956363i \(-0.405619\pi\)
0.292182 + 0.956363i \(0.405619\pi\)
\(972\) 0 0
\(973\) −4.20426 −0.134782
\(974\) 0 0
\(975\) 24.7151 0.791518
\(976\) 0 0
\(977\) −42.4537 −1.35821 −0.679107 0.734039i \(-0.737633\pi\)
−0.679107 + 0.734039i \(0.737633\pi\)
\(978\) 0 0
\(979\) −12.2263 −0.390755
\(980\) 0 0
\(981\) −35.0283 −1.11837
\(982\) 0 0
\(983\) 56.3144 1.79615 0.898076 0.439841i \(-0.144965\pi\)
0.898076 + 0.439841i \(0.144965\pi\)
\(984\) 0 0
\(985\) 0.234240 0.00746350
\(986\) 0 0
\(987\) 3.87674 0.123398
\(988\) 0 0
\(989\) 0.125088 0.00397757
\(990\) 0 0
\(991\) 25.6246 0.813992 0.406996 0.913430i \(-0.366576\pi\)
0.406996 + 0.913430i \(0.366576\pi\)
\(992\) 0 0
\(993\) 27.8428 0.883566
\(994\) 0 0
\(995\) 1.85647 0.0588540
\(996\) 0 0
\(997\) 51.3571 1.62650 0.813248 0.581917i \(-0.197697\pi\)
0.813248 + 0.581917i \(0.197697\pi\)
\(998\) 0 0
\(999\) −7.52428 −0.238058
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 6008.2.a.d.1.16 49
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6008.2.a.d.1.16 49 1.1 even 1 trivial