Properties

Label 8016.2.a.bd.1.9
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $10$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-2.53098\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.53098 q^{5} -1.95354 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.53098 q^{5} -1.95354 q^{7} +1.00000 q^{9} +2.15240 q^{11} -0.0320505 q^{13} +1.53098 q^{15} -2.27829 q^{17} +5.52833 q^{19} -1.95354 q^{21} -6.90191 q^{23} -2.65610 q^{25} +1.00000 q^{27} -2.80212 q^{29} -1.86516 q^{31} +2.15240 q^{33} -2.99083 q^{35} -7.30674 q^{37} -0.0320505 q^{39} -11.8322 q^{41} -1.12240 q^{43} +1.53098 q^{45} +0.929399 q^{47} -3.18367 q^{49} -2.27829 q^{51} -11.1295 q^{53} +3.29529 q^{55} +5.52833 q^{57} -8.11120 q^{59} -7.44769 q^{61} -1.95354 q^{63} -0.0490687 q^{65} -10.5639 q^{67} -6.90191 q^{69} +7.98837 q^{71} +10.4948 q^{73} -2.65610 q^{75} -4.20481 q^{77} +5.86119 q^{79} +1.00000 q^{81} +14.0793 q^{83} -3.48802 q^{85} -2.80212 q^{87} +7.24231 q^{89} +0.0626121 q^{91} -1.86516 q^{93} +8.46376 q^{95} +2.96086 q^{97} +2.15240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.53098 0.684675 0.342338 0.939577i \(-0.388781\pi\)
0.342338 + 0.939577i \(0.388781\pi\)
\(6\) 0 0
\(7\) −1.95354 −0.738369 −0.369185 0.929356i \(-0.620363\pi\)
−0.369185 + 0.929356i \(0.620363\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.15240 0.648974 0.324487 0.945890i \(-0.394808\pi\)
0.324487 + 0.945890i \(0.394808\pi\)
\(12\) 0 0
\(13\) −0.0320505 −0.00888922 −0.00444461 0.999990i \(-0.501415\pi\)
−0.00444461 + 0.999990i \(0.501415\pi\)
\(14\) 0 0
\(15\) 1.53098 0.395297
\(16\) 0 0
\(17\) −2.27829 −0.552567 −0.276283 0.961076i \(-0.589103\pi\)
−0.276283 + 0.961076i \(0.589103\pi\)
\(18\) 0 0
\(19\) 5.52833 1.26829 0.634143 0.773216i \(-0.281353\pi\)
0.634143 + 0.773216i \(0.281353\pi\)
\(20\) 0 0
\(21\) −1.95354 −0.426298
\(22\) 0 0
\(23\) −6.90191 −1.43915 −0.719574 0.694416i \(-0.755663\pi\)
−0.719574 + 0.694416i \(0.755663\pi\)
\(24\) 0 0
\(25\) −2.65610 −0.531220
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.80212 −0.520341 −0.260171 0.965563i \(-0.583779\pi\)
−0.260171 + 0.965563i \(0.583779\pi\)
\(30\) 0 0
\(31\) −1.86516 −0.334992 −0.167496 0.985873i \(-0.553568\pi\)
−0.167496 + 0.985873i \(0.553568\pi\)
\(32\) 0 0
\(33\) 2.15240 0.374686
\(34\) 0 0
\(35\) −2.99083 −0.505543
\(36\) 0 0
\(37\) −7.30674 −1.20122 −0.600611 0.799542i \(-0.705076\pi\)
−0.600611 + 0.799542i \(0.705076\pi\)
\(38\) 0 0
\(39\) −0.0320505 −0.00513219
\(40\) 0 0
\(41\) −11.8322 −1.84787 −0.923936 0.382548i \(-0.875047\pi\)
−0.923936 + 0.382548i \(0.875047\pi\)
\(42\) 0 0
\(43\) −1.12240 −0.171165 −0.0855825 0.996331i \(-0.527275\pi\)
−0.0855825 + 0.996331i \(0.527275\pi\)
\(44\) 0 0
\(45\) 1.53098 0.228225
\(46\) 0 0
\(47\) 0.929399 0.135567 0.0677834 0.997700i \(-0.478407\pi\)
0.0677834 + 0.997700i \(0.478407\pi\)
\(48\) 0 0
\(49\) −3.18367 −0.454811
\(50\) 0 0
\(51\) −2.27829 −0.319024
\(52\) 0 0
\(53\) −11.1295 −1.52876 −0.764379 0.644768i \(-0.776954\pi\)
−0.764379 + 0.644768i \(0.776954\pi\)
\(54\) 0 0
\(55\) 3.29529 0.444337
\(56\) 0 0
\(57\) 5.52833 0.732245
\(58\) 0 0
\(59\) −8.11120 −1.05599 −0.527994 0.849248i \(-0.677056\pi\)
−0.527994 + 0.849248i \(0.677056\pi\)
\(60\) 0 0
\(61\) −7.44769 −0.953580 −0.476790 0.879017i \(-0.658200\pi\)
−0.476790 + 0.879017i \(0.658200\pi\)
\(62\) 0 0
\(63\) −1.95354 −0.246123
\(64\) 0 0
\(65\) −0.0490687 −0.00608623
\(66\) 0 0
\(67\) −10.5639 −1.29058 −0.645290 0.763938i \(-0.723263\pi\)
−0.645290 + 0.763938i \(0.723263\pi\)
\(68\) 0 0
\(69\) −6.90191 −0.830892
\(70\) 0 0
\(71\) 7.98837 0.948045 0.474023 0.880513i \(-0.342801\pi\)
0.474023 + 0.880513i \(0.342801\pi\)
\(72\) 0 0
\(73\) 10.4948 1.22833 0.614164 0.789179i \(-0.289493\pi\)
0.614164 + 0.789179i \(0.289493\pi\)
\(74\) 0 0
\(75\) −2.65610 −0.306700
\(76\) 0 0
\(77\) −4.20481 −0.479183
\(78\) 0 0
\(79\) 5.86119 0.659435 0.329717 0.944080i \(-0.393047\pi\)
0.329717 + 0.944080i \(0.393047\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.0793 1.54541 0.772704 0.634767i \(-0.218904\pi\)
0.772704 + 0.634767i \(0.218904\pi\)
\(84\) 0 0
\(85\) −3.48802 −0.378329
\(86\) 0 0
\(87\) −2.80212 −0.300419
\(88\) 0 0
\(89\) 7.24231 0.767684 0.383842 0.923399i \(-0.374601\pi\)
0.383842 + 0.923399i \(0.374601\pi\)
\(90\) 0 0
\(91\) 0.0626121 0.00656353
\(92\) 0 0
\(93\) −1.86516 −0.193408
\(94\) 0 0
\(95\) 8.46376 0.868364
\(96\) 0 0
\(97\) 2.96086 0.300629 0.150315 0.988638i \(-0.451971\pi\)
0.150315 + 0.988638i \(0.451971\pi\)
\(98\) 0 0
\(99\) 2.15240 0.216325
\(100\) 0 0
\(101\) −10.7676 −1.07142 −0.535709 0.844403i \(-0.679955\pi\)
−0.535709 + 0.844403i \(0.679955\pi\)
\(102\) 0 0
\(103\) 2.59773 0.255962 0.127981 0.991777i \(-0.459150\pi\)
0.127981 + 0.991777i \(0.459150\pi\)
\(104\) 0 0
\(105\) −2.99083 −0.291876
\(106\) 0 0
\(107\) 2.49290 0.240998 0.120499 0.992713i \(-0.461551\pi\)
0.120499 + 0.992713i \(0.461551\pi\)
\(108\) 0 0
\(109\) 8.57800 0.821624 0.410812 0.911720i \(-0.365245\pi\)
0.410812 + 0.911720i \(0.365245\pi\)
\(110\) 0 0
\(111\) −7.30674 −0.693525
\(112\) 0 0
\(113\) 17.3592 1.63301 0.816507 0.577335i \(-0.195907\pi\)
0.816507 + 0.577335i \(0.195907\pi\)
\(114\) 0 0
\(115\) −10.5667 −0.985349
\(116\) 0 0
\(117\) −0.0320505 −0.00296307
\(118\) 0 0
\(119\) 4.45074 0.407998
\(120\) 0 0
\(121\) −6.36715 −0.578832
\(122\) 0 0
\(123\) −11.8322 −1.06687
\(124\) 0 0
\(125\) −11.7213 −1.04839
\(126\) 0 0
\(127\) −19.8473 −1.76116 −0.880580 0.473897i \(-0.842847\pi\)
−0.880580 + 0.473897i \(0.842847\pi\)
\(128\) 0 0
\(129\) −1.12240 −0.0988221
\(130\) 0 0
\(131\) −8.00587 −0.699476 −0.349738 0.936847i \(-0.613729\pi\)
−0.349738 + 0.936847i \(0.613729\pi\)
\(132\) 0 0
\(133\) −10.7998 −0.936463
\(134\) 0 0
\(135\) 1.53098 0.131766
\(136\) 0 0
\(137\) 12.5851 1.07522 0.537609 0.843194i \(-0.319328\pi\)
0.537609 + 0.843194i \(0.319328\pi\)
\(138\) 0 0
\(139\) 0.988330 0.0838291 0.0419145 0.999121i \(-0.486654\pi\)
0.0419145 + 0.999121i \(0.486654\pi\)
\(140\) 0 0
\(141\) 0.929399 0.0782695
\(142\) 0 0
\(143\) −0.0689857 −0.00576888
\(144\) 0 0
\(145\) −4.28999 −0.356265
\(146\) 0 0
\(147\) −3.18367 −0.262585
\(148\) 0 0
\(149\) −8.45810 −0.692915 −0.346457 0.938066i \(-0.612615\pi\)
−0.346457 + 0.938066i \(0.612615\pi\)
\(150\) 0 0
\(151\) 0.136644 0.0111199 0.00555997 0.999985i \(-0.498230\pi\)
0.00555997 + 0.999985i \(0.498230\pi\)
\(152\) 0 0
\(153\) −2.27829 −0.184189
\(154\) 0 0
\(155\) −2.85552 −0.229361
\(156\) 0 0
\(157\) 11.3727 0.907639 0.453820 0.891094i \(-0.350061\pi\)
0.453820 + 0.891094i \(0.350061\pi\)
\(158\) 0 0
\(159\) −11.1295 −0.882629
\(160\) 0 0
\(161\) 13.4832 1.06262
\(162\) 0 0
\(163\) 11.0067 0.862114 0.431057 0.902325i \(-0.358141\pi\)
0.431057 + 0.902325i \(0.358141\pi\)
\(164\) 0 0
\(165\) 3.29529 0.256538
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −12.9990 −0.999921
\(170\) 0 0
\(171\) 5.52833 0.422762
\(172\) 0 0
\(173\) −17.6800 −1.34418 −0.672091 0.740468i \(-0.734604\pi\)
−0.672091 + 0.740468i \(0.734604\pi\)
\(174\) 0 0
\(175\) 5.18880 0.392236
\(176\) 0 0
\(177\) −8.11120 −0.609675
\(178\) 0 0
\(179\) 23.2929 1.74100 0.870498 0.492172i \(-0.163797\pi\)
0.870498 + 0.492172i \(0.163797\pi\)
\(180\) 0 0
\(181\) 12.4704 0.926918 0.463459 0.886118i \(-0.346608\pi\)
0.463459 + 0.886118i \(0.346608\pi\)
\(182\) 0 0
\(183\) −7.44769 −0.550549
\(184\) 0 0
\(185\) −11.1865 −0.822446
\(186\) 0 0
\(187\) −4.90380 −0.358602
\(188\) 0 0
\(189\) −1.95354 −0.142099
\(190\) 0 0
\(191\) 10.6430 0.770103 0.385051 0.922895i \(-0.374184\pi\)
0.385051 + 0.922895i \(0.374184\pi\)
\(192\) 0 0
\(193\) −3.66898 −0.264099 −0.132050 0.991243i \(-0.542156\pi\)
−0.132050 + 0.991243i \(0.542156\pi\)
\(194\) 0 0
\(195\) −0.0490687 −0.00351389
\(196\) 0 0
\(197\) −5.11526 −0.364447 −0.182223 0.983257i \(-0.558329\pi\)
−0.182223 + 0.983257i \(0.558329\pi\)
\(198\) 0 0
\(199\) 0.690922 0.0489781 0.0244891 0.999700i \(-0.492204\pi\)
0.0244891 + 0.999700i \(0.492204\pi\)
\(200\) 0 0
\(201\) −10.5639 −0.745117
\(202\) 0 0
\(203\) 5.47406 0.384204
\(204\) 0 0
\(205\) −18.1148 −1.26519
\(206\) 0 0
\(207\) −6.90191 −0.479716
\(208\) 0 0
\(209\) 11.8992 0.823085
\(210\) 0 0
\(211\) −15.3286 −1.05526 −0.527631 0.849474i \(-0.676919\pi\)
−0.527631 + 0.849474i \(0.676919\pi\)
\(212\) 0 0
\(213\) 7.98837 0.547354
\(214\) 0 0
\(215\) −1.71838 −0.117192
\(216\) 0 0
\(217\) 3.64367 0.247348
\(218\) 0 0
\(219\) 10.4948 0.709175
\(220\) 0 0
\(221\) 0.0730204 0.00491188
\(222\) 0 0
\(223\) 23.7330 1.58928 0.794641 0.607079i \(-0.207659\pi\)
0.794641 + 0.607079i \(0.207659\pi\)
\(224\) 0 0
\(225\) −2.65610 −0.177073
\(226\) 0 0
\(227\) 20.9222 1.38866 0.694329 0.719658i \(-0.255701\pi\)
0.694329 + 0.719658i \(0.255701\pi\)
\(228\) 0 0
\(229\) −11.6369 −0.768988 −0.384494 0.923127i \(-0.625624\pi\)
−0.384494 + 0.923127i \(0.625624\pi\)
\(230\) 0 0
\(231\) −4.20481 −0.276656
\(232\) 0 0
\(233\) −7.90044 −0.517575 −0.258788 0.965934i \(-0.583323\pi\)
−0.258788 + 0.965934i \(0.583323\pi\)
\(234\) 0 0
\(235\) 1.42289 0.0928192
\(236\) 0 0
\(237\) 5.86119 0.380725
\(238\) 0 0
\(239\) −21.2822 −1.37663 −0.688316 0.725411i \(-0.741650\pi\)
−0.688316 + 0.725411i \(0.741650\pi\)
\(240\) 0 0
\(241\) −17.6365 −1.13607 −0.568033 0.823006i \(-0.692295\pi\)
−0.568033 + 0.823006i \(0.692295\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.87414 −0.311398
\(246\) 0 0
\(247\) −0.177186 −0.0112741
\(248\) 0 0
\(249\) 14.0793 0.892242
\(250\) 0 0
\(251\) −29.1497 −1.83992 −0.919958 0.392017i \(-0.871777\pi\)
−0.919958 + 0.392017i \(0.871777\pi\)
\(252\) 0 0
\(253\) −14.8557 −0.933970
\(254\) 0 0
\(255\) −3.48802 −0.218428
\(256\) 0 0
\(257\) −28.8771 −1.80131 −0.900653 0.434539i \(-0.856911\pi\)
−0.900653 + 0.434539i \(0.856911\pi\)
\(258\) 0 0
\(259\) 14.2740 0.886945
\(260\) 0 0
\(261\) −2.80212 −0.173447
\(262\) 0 0
\(263\) −3.69587 −0.227897 −0.113948 0.993487i \(-0.536350\pi\)
−0.113948 + 0.993487i \(0.536350\pi\)
\(264\) 0 0
\(265\) −17.0391 −1.04670
\(266\) 0 0
\(267\) 7.24231 0.443222
\(268\) 0 0
\(269\) −11.9065 −0.725952 −0.362976 0.931799i \(-0.618239\pi\)
−0.362976 + 0.931799i \(0.618239\pi\)
\(270\) 0 0
\(271\) −9.42035 −0.572245 −0.286123 0.958193i \(-0.592366\pi\)
−0.286123 + 0.958193i \(0.592366\pi\)
\(272\) 0 0
\(273\) 0.0626121 0.00378945
\(274\) 0 0
\(275\) −5.71700 −0.344748
\(276\) 0 0
\(277\) −9.72432 −0.584278 −0.292139 0.956376i \(-0.594367\pi\)
−0.292139 + 0.956376i \(0.594367\pi\)
\(278\) 0 0
\(279\) −1.86516 −0.111664
\(280\) 0 0
\(281\) −9.54649 −0.569496 −0.284748 0.958602i \(-0.591910\pi\)
−0.284748 + 0.958602i \(0.591910\pi\)
\(282\) 0 0
\(283\) −0.881411 −0.0523944 −0.0261972 0.999657i \(-0.508340\pi\)
−0.0261972 + 0.999657i \(0.508340\pi\)
\(284\) 0 0
\(285\) 8.46376 0.501350
\(286\) 0 0
\(287\) 23.1146 1.36441
\(288\) 0 0
\(289\) −11.8094 −0.694670
\(290\) 0 0
\(291\) 2.96086 0.173569
\(292\) 0 0
\(293\) 7.37269 0.430717 0.215358 0.976535i \(-0.430908\pi\)
0.215358 + 0.976535i \(0.430908\pi\)
\(294\) 0 0
\(295\) −12.4181 −0.723009
\(296\) 0 0
\(297\) 2.15240 0.124895
\(298\) 0 0
\(299\) 0.221210 0.0127929
\(300\) 0 0
\(301\) 2.19266 0.126383
\(302\) 0 0
\(303\) −10.7676 −0.618583
\(304\) 0 0
\(305\) −11.4023 −0.652892
\(306\) 0 0
\(307\) 11.3416 0.647299 0.323650 0.946177i \(-0.395090\pi\)
0.323650 + 0.946177i \(0.395090\pi\)
\(308\) 0 0
\(309\) 2.59773 0.147779
\(310\) 0 0
\(311\) 23.2572 1.31879 0.659397 0.751795i \(-0.270812\pi\)
0.659397 + 0.751795i \(0.270812\pi\)
\(312\) 0 0
\(313\) 24.6287 1.39210 0.696049 0.717994i \(-0.254940\pi\)
0.696049 + 0.717994i \(0.254940\pi\)
\(314\) 0 0
\(315\) −2.99083 −0.168514
\(316\) 0 0
\(317\) 14.6228 0.821299 0.410649 0.911793i \(-0.365302\pi\)
0.410649 + 0.911793i \(0.365302\pi\)
\(318\) 0 0
\(319\) −6.03130 −0.337688
\(320\) 0 0
\(321\) 2.49290 0.139140
\(322\) 0 0
\(323\) −12.5951 −0.700812
\(324\) 0 0
\(325\) 0.0851294 0.00472213
\(326\) 0 0
\(327\) 8.57800 0.474365
\(328\) 0 0
\(329\) −1.81562 −0.100098
\(330\) 0 0
\(331\) −14.4030 −0.791660 −0.395830 0.918324i \(-0.629543\pi\)
−0.395830 + 0.918324i \(0.629543\pi\)
\(332\) 0 0
\(333\) −7.30674 −0.400407
\(334\) 0 0
\(335\) −16.1731 −0.883628
\(336\) 0 0
\(337\) −18.4014 −1.00239 −0.501193 0.865335i \(-0.667105\pi\)
−0.501193 + 0.865335i \(0.667105\pi\)
\(338\) 0 0
\(339\) 17.3592 0.942821
\(340\) 0 0
\(341\) −4.01458 −0.217402
\(342\) 0 0
\(343\) 19.8942 1.07419
\(344\) 0 0
\(345\) −10.5667 −0.568891
\(346\) 0 0
\(347\) −4.12491 −0.221437 −0.110718 0.993852i \(-0.535315\pi\)
−0.110718 + 0.993852i \(0.535315\pi\)
\(348\) 0 0
\(349\) −21.8747 −1.17093 −0.585463 0.810699i \(-0.699087\pi\)
−0.585463 + 0.810699i \(0.699087\pi\)
\(350\) 0 0
\(351\) −0.0320505 −0.00171073
\(352\) 0 0
\(353\) −20.7177 −1.10269 −0.551347 0.834276i \(-0.685886\pi\)
−0.551347 + 0.834276i \(0.685886\pi\)
\(354\) 0 0
\(355\) 12.2300 0.649103
\(356\) 0 0
\(357\) 4.45074 0.235558
\(358\) 0 0
\(359\) 27.2856 1.44008 0.720040 0.693933i \(-0.244123\pi\)
0.720040 + 0.693933i \(0.244123\pi\)
\(360\) 0 0
\(361\) 11.5624 0.608548
\(362\) 0 0
\(363\) −6.36715 −0.334189
\(364\) 0 0
\(365\) 16.0674 0.841006
\(366\) 0 0
\(367\) −17.1725 −0.896398 −0.448199 0.893934i \(-0.647934\pi\)
−0.448199 + 0.893934i \(0.647934\pi\)
\(368\) 0 0
\(369\) −11.8322 −0.615957
\(370\) 0 0
\(371\) 21.7420 1.12879
\(372\) 0 0
\(373\) 16.5701 0.857967 0.428984 0.903312i \(-0.358872\pi\)
0.428984 + 0.903312i \(0.358872\pi\)
\(374\) 0 0
\(375\) −11.7213 −0.605287
\(376\) 0 0
\(377\) 0.0898095 0.00462543
\(378\) 0 0
\(379\) −32.1485 −1.65136 −0.825680 0.564139i \(-0.809208\pi\)
−0.825680 + 0.564139i \(0.809208\pi\)
\(380\) 0 0
\(381\) −19.8473 −1.01681
\(382\) 0 0
\(383\) −35.1609 −1.79664 −0.898319 0.439343i \(-0.855211\pi\)
−0.898319 + 0.439343i \(0.855211\pi\)
\(384\) 0 0
\(385\) −6.43749 −0.328085
\(386\) 0 0
\(387\) −1.12240 −0.0570550
\(388\) 0 0
\(389\) −22.1942 −1.12529 −0.562646 0.826698i \(-0.690217\pi\)
−0.562646 + 0.826698i \(0.690217\pi\)
\(390\) 0 0
\(391\) 15.7246 0.795225
\(392\) 0 0
\(393\) −8.00587 −0.403843
\(394\) 0 0
\(395\) 8.97336 0.451499
\(396\) 0 0
\(397\) 9.87019 0.495371 0.247685 0.968841i \(-0.420330\pi\)
0.247685 + 0.968841i \(0.420330\pi\)
\(398\) 0 0
\(399\) −10.7998 −0.540667
\(400\) 0 0
\(401\) 19.5475 0.976157 0.488079 0.872800i \(-0.337698\pi\)
0.488079 + 0.872800i \(0.337698\pi\)
\(402\) 0 0
\(403\) 0.0597793 0.00297782
\(404\) 0 0
\(405\) 1.53098 0.0760750
\(406\) 0 0
\(407\) −15.7271 −0.779562
\(408\) 0 0
\(409\) 6.76824 0.334668 0.167334 0.985900i \(-0.446484\pi\)
0.167334 + 0.985900i \(0.446484\pi\)
\(410\) 0 0
\(411\) 12.5851 0.620778
\(412\) 0 0
\(413\) 15.8456 0.779709
\(414\) 0 0
\(415\) 21.5552 1.05810
\(416\) 0 0
\(417\) 0.988330 0.0483987
\(418\) 0 0
\(419\) −1.13322 −0.0553615 −0.0276807 0.999617i \(-0.508812\pi\)
−0.0276807 + 0.999617i \(0.508812\pi\)
\(420\) 0 0
\(421\) 4.12164 0.200876 0.100438 0.994943i \(-0.467976\pi\)
0.100438 + 0.994943i \(0.467976\pi\)
\(422\) 0 0
\(423\) 0.929399 0.0451889
\(424\) 0 0
\(425\) 6.05136 0.293534
\(426\) 0 0
\(427\) 14.5494 0.704094
\(428\) 0 0
\(429\) −0.0689857 −0.00333066
\(430\) 0 0
\(431\) 6.83221 0.329096 0.164548 0.986369i \(-0.447383\pi\)
0.164548 + 0.986369i \(0.447383\pi\)
\(432\) 0 0
\(433\) 31.5894 1.51809 0.759046 0.651037i \(-0.225666\pi\)
0.759046 + 0.651037i \(0.225666\pi\)
\(434\) 0 0
\(435\) −4.28999 −0.205690
\(436\) 0 0
\(437\) −38.1560 −1.82525
\(438\) 0 0
\(439\) −4.58677 −0.218915 −0.109457 0.993991i \(-0.534911\pi\)
−0.109457 + 0.993991i \(0.534911\pi\)
\(440\) 0 0
\(441\) −3.18367 −0.151604
\(442\) 0 0
\(443\) 30.2286 1.43621 0.718103 0.695937i \(-0.245011\pi\)
0.718103 + 0.695937i \(0.245011\pi\)
\(444\) 0 0
\(445\) 11.0878 0.525614
\(446\) 0 0
\(447\) −8.45810 −0.400054
\(448\) 0 0
\(449\) −40.1741 −1.89593 −0.947966 0.318370i \(-0.896865\pi\)
−0.947966 + 0.318370i \(0.896865\pi\)
\(450\) 0 0
\(451\) −25.4676 −1.19922
\(452\) 0 0
\(453\) 0.136644 0.00642010
\(454\) 0 0
\(455\) 0.0958578 0.00449388
\(456\) 0 0
\(457\) 29.5905 1.38418 0.692092 0.721809i \(-0.256689\pi\)
0.692092 + 0.721809i \(0.256689\pi\)
\(458\) 0 0
\(459\) −2.27829 −0.106341
\(460\) 0 0
\(461\) −13.0716 −0.608807 −0.304404 0.952543i \(-0.598457\pi\)
−0.304404 + 0.952543i \(0.598457\pi\)
\(462\) 0 0
\(463\) 12.0242 0.558810 0.279405 0.960173i \(-0.409863\pi\)
0.279405 + 0.960173i \(0.409863\pi\)
\(464\) 0 0
\(465\) −2.85552 −0.132422
\(466\) 0 0
\(467\) −7.55381 −0.349549 −0.174774 0.984609i \(-0.555920\pi\)
−0.174774 + 0.984609i \(0.555920\pi\)
\(468\) 0 0
\(469\) 20.6369 0.952925
\(470\) 0 0
\(471\) 11.3727 0.524026
\(472\) 0 0
\(473\) −2.41587 −0.111082
\(474\) 0 0
\(475\) −14.6838 −0.673738
\(476\) 0 0
\(477\) −11.1295 −0.509586
\(478\) 0 0
\(479\) 26.2810 1.20081 0.600405 0.799696i \(-0.295006\pi\)
0.600405 + 0.799696i \(0.295006\pi\)
\(480\) 0 0
\(481\) 0.234185 0.0106779
\(482\) 0 0
\(483\) 13.4832 0.613506
\(484\) 0 0
\(485\) 4.53301 0.205834
\(486\) 0 0
\(487\) −6.63742 −0.300770 −0.150385 0.988628i \(-0.548051\pi\)
−0.150385 + 0.988628i \(0.548051\pi\)
\(488\) 0 0
\(489\) 11.0067 0.497742
\(490\) 0 0
\(491\) −13.0991 −0.591156 −0.295578 0.955319i \(-0.595512\pi\)
−0.295578 + 0.955319i \(0.595512\pi\)
\(492\) 0 0
\(493\) 6.38405 0.287523
\(494\) 0 0
\(495\) 3.29529 0.148112
\(496\) 0 0
\(497\) −15.6056 −0.700008
\(498\) 0 0
\(499\) −22.1353 −0.990912 −0.495456 0.868633i \(-0.664999\pi\)
−0.495456 + 0.868633i \(0.664999\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −22.7703 −1.01528 −0.507639 0.861570i \(-0.669482\pi\)
−0.507639 + 0.861570i \(0.669482\pi\)
\(504\) 0 0
\(505\) −16.4850 −0.733573
\(506\) 0 0
\(507\) −12.9990 −0.577305
\(508\) 0 0
\(509\) 1.95493 0.0866506 0.0433253 0.999061i \(-0.486205\pi\)
0.0433253 + 0.999061i \(0.486205\pi\)
\(510\) 0 0
\(511\) −20.5021 −0.906960
\(512\) 0 0
\(513\) 5.52833 0.244082
\(514\) 0 0
\(515\) 3.97707 0.175251
\(516\) 0 0
\(517\) 2.00044 0.0879794
\(518\) 0 0
\(519\) −17.6800 −0.776064
\(520\) 0 0
\(521\) −7.26095 −0.318108 −0.159054 0.987270i \(-0.550844\pi\)
−0.159054 + 0.987270i \(0.550844\pi\)
\(522\) 0 0
\(523\) 25.6991 1.12374 0.561871 0.827225i \(-0.310082\pi\)
0.561871 + 0.827225i \(0.310082\pi\)
\(524\) 0 0
\(525\) 5.18880 0.226458
\(526\) 0 0
\(527\) 4.24937 0.185106
\(528\) 0 0
\(529\) 24.6364 1.07115
\(530\) 0 0
\(531\) −8.11120 −0.351996
\(532\) 0 0
\(533\) 0.379227 0.0164261
\(534\) 0 0
\(535\) 3.81659 0.165006
\(536\) 0 0
\(537\) 23.2929 1.00516
\(538\) 0 0
\(539\) −6.85256 −0.295160
\(540\) 0 0
\(541\) 21.5546 0.926703 0.463352 0.886174i \(-0.346647\pi\)
0.463352 + 0.886174i \(0.346647\pi\)
\(542\) 0 0
\(543\) 12.4704 0.535156
\(544\) 0 0
\(545\) 13.1328 0.562545
\(546\) 0 0
\(547\) 11.8562 0.506936 0.253468 0.967344i \(-0.418429\pi\)
0.253468 + 0.967344i \(0.418429\pi\)
\(548\) 0 0
\(549\) −7.44769 −0.317860
\(550\) 0 0
\(551\) −15.4911 −0.659941
\(552\) 0 0
\(553\) −11.4501 −0.486907
\(554\) 0 0
\(555\) −11.1865 −0.474840
\(556\) 0 0
\(557\) −44.9125 −1.90300 −0.951502 0.307642i \(-0.900460\pi\)
−0.951502 + 0.307642i \(0.900460\pi\)
\(558\) 0 0
\(559\) 0.0359736 0.00152152
\(560\) 0 0
\(561\) −4.90380 −0.207039
\(562\) 0 0
\(563\) 8.14999 0.343481 0.171741 0.985142i \(-0.445061\pi\)
0.171741 + 0.985142i \(0.445061\pi\)
\(564\) 0 0
\(565\) 26.5766 1.11808
\(566\) 0 0
\(567\) −1.95354 −0.0820410
\(568\) 0 0
\(569\) 39.1874 1.64282 0.821410 0.570338i \(-0.193188\pi\)
0.821410 + 0.570338i \(0.193188\pi\)
\(570\) 0 0
\(571\) −6.40716 −0.268131 −0.134066 0.990972i \(-0.542803\pi\)
−0.134066 + 0.990972i \(0.542803\pi\)
\(572\) 0 0
\(573\) 10.6430 0.444619
\(574\) 0 0
\(575\) 18.3322 0.764504
\(576\) 0 0
\(577\) 45.5385 1.89579 0.947895 0.318582i \(-0.103207\pi\)
0.947895 + 0.318582i \(0.103207\pi\)
\(578\) 0 0
\(579\) −3.66898 −0.152478
\(580\) 0 0
\(581\) −27.5046 −1.14108
\(582\) 0 0
\(583\) −23.9552 −0.992125
\(584\) 0 0
\(585\) −0.0490687 −0.00202874
\(586\) 0 0
\(587\) 36.1537 1.49222 0.746111 0.665821i \(-0.231919\pi\)
0.746111 + 0.665821i \(0.231919\pi\)
\(588\) 0 0
\(589\) −10.3112 −0.424866
\(590\) 0 0
\(591\) −5.11526 −0.210414
\(592\) 0 0
\(593\) 20.9600 0.860724 0.430362 0.902656i \(-0.358386\pi\)
0.430362 + 0.902656i \(0.358386\pi\)
\(594\) 0 0
\(595\) 6.81399 0.279346
\(596\) 0 0
\(597\) 0.690922 0.0282775
\(598\) 0 0
\(599\) −22.3654 −0.913826 −0.456913 0.889512i \(-0.651045\pi\)
−0.456913 + 0.889512i \(0.651045\pi\)
\(600\) 0 0
\(601\) 8.29446 0.338338 0.169169 0.985587i \(-0.445892\pi\)
0.169169 + 0.985587i \(0.445892\pi\)
\(602\) 0 0
\(603\) −10.5639 −0.430193
\(604\) 0 0
\(605\) −9.74799 −0.396312
\(606\) 0 0
\(607\) 4.58092 0.185934 0.0929670 0.995669i \(-0.470365\pi\)
0.0929670 + 0.995669i \(0.470365\pi\)
\(608\) 0 0
\(609\) 5.47406 0.221820
\(610\) 0 0
\(611\) −0.0297877 −0.00120508
\(612\) 0 0
\(613\) 8.46801 0.342020 0.171010 0.985269i \(-0.445297\pi\)
0.171010 + 0.985269i \(0.445297\pi\)
\(614\) 0 0
\(615\) −18.1148 −0.730459
\(616\) 0 0
\(617\) 23.3651 0.940642 0.470321 0.882495i \(-0.344138\pi\)
0.470321 + 0.882495i \(0.344138\pi\)
\(618\) 0 0
\(619\) 11.4277 0.459318 0.229659 0.973271i \(-0.426239\pi\)
0.229659 + 0.973271i \(0.426239\pi\)
\(620\) 0 0
\(621\) −6.90191 −0.276964
\(622\) 0 0
\(623\) −14.1482 −0.566834
\(624\) 0 0
\(625\) −4.66464 −0.186586
\(626\) 0 0
\(627\) 11.8992 0.475208
\(628\) 0 0
\(629\) 16.6469 0.663755
\(630\) 0 0
\(631\) 3.40993 0.135747 0.0678736 0.997694i \(-0.478379\pi\)
0.0678736 + 0.997694i \(0.478379\pi\)
\(632\) 0 0
\(633\) −15.3286 −0.609255
\(634\) 0 0
\(635\) −30.3858 −1.20582
\(636\) 0 0
\(637\) 0.102038 0.00404291
\(638\) 0 0
\(639\) 7.98837 0.316015
\(640\) 0 0
\(641\) −19.8009 −0.782088 −0.391044 0.920372i \(-0.627886\pi\)
−0.391044 + 0.920372i \(0.627886\pi\)
\(642\) 0 0
\(643\) 39.9862 1.57690 0.788451 0.615097i \(-0.210883\pi\)
0.788451 + 0.615097i \(0.210883\pi\)
\(644\) 0 0
\(645\) −1.71838 −0.0676611
\(646\) 0 0
\(647\) 23.7997 0.935664 0.467832 0.883817i \(-0.345035\pi\)
0.467832 + 0.883817i \(0.345035\pi\)
\(648\) 0 0
\(649\) −17.4586 −0.685309
\(650\) 0 0
\(651\) 3.64367 0.142807
\(652\) 0 0
\(653\) −3.24958 −0.127166 −0.0635829 0.997977i \(-0.520253\pi\)
−0.0635829 + 0.997977i \(0.520253\pi\)
\(654\) 0 0
\(655\) −12.2568 −0.478914
\(656\) 0 0
\(657\) 10.4948 0.409443
\(658\) 0 0
\(659\) −47.1312 −1.83597 −0.917985 0.396616i \(-0.870184\pi\)
−0.917985 + 0.396616i \(0.870184\pi\)
\(660\) 0 0
\(661\) 49.8545 1.93912 0.969558 0.244861i \(-0.0787424\pi\)
0.969558 + 0.244861i \(0.0787424\pi\)
\(662\) 0 0
\(663\) 0.0730204 0.00283588
\(664\) 0 0
\(665\) −16.5343 −0.641173
\(666\) 0 0
\(667\) 19.3400 0.748848
\(668\) 0 0
\(669\) 23.7330 0.917573
\(670\) 0 0
\(671\) −16.0305 −0.618849
\(672\) 0 0
\(673\) 29.5519 1.13914 0.569571 0.821942i \(-0.307109\pi\)
0.569571 + 0.821942i \(0.307109\pi\)
\(674\) 0 0
\(675\) −2.65610 −0.102233
\(676\) 0 0
\(677\) −1.97155 −0.0757729 −0.0378864 0.999282i \(-0.512063\pi\)
−0.0378864 + 0.999282i \(0.512063\pi\)
\(678\) 0 0
\(679\) −5.78416 −0.221976
\(680\) 0 0
\(681\) 20.9222 0.801742
\(682\) 0 0
\(683\) −14.7594 −0.564751 −0.282375 0.959304i \(-0.591122\pi\)
−0.282375 + 0.959304i \(0.591122\pi\)
\(684\) 0 0
\(685\) 19.2676 0.736175
\(686\) 0 0
\(687\) −11.6369 −0.443976
\(688\) 0 0
\(689\) 0.356707 0.0135895
\(690\) 0 0
\(691\) 8.62826 0.328234 0.164117 0.986441i \(-0.447522\pi\)
0.164117 + 0.986441i \(0.447522\pi\)
\(692\) 0 0
\(693\) −4.20481 −0.159728
\(694\) 0 0
\(695\) 1.51311 0.0573957
\(696\) 0 0
\(697\) 26.9571 1.02107
\(698\) 0 0
\(699\) −7.90044 −0.298822
\(700\) 0 0
\(701\) −33.1406 −1.25170 −0.625851 0.779943i \(-0.715248\pi\)
−0.625851 + 0.779943i \(0.715248\pi\)
\(702\) 0 0
\(703\) −40.3941 −1.52349
\(704\) 0 0
\(705\) 1.42289 0.0535892
\(706\) 0 0
\(707\) 21.0350 0.791102
\(708\) 0 0
\(709\) 35.0277 1.31549 0.657746 0.753240i \(-0.271510\pi\)
0.657746 + 0.753240i \(0.271510\pi\)
\(710\) 0 0
\(711\) 5.86119 0.219812
\(712\) 0 0
\(713\) 12.8732 0.482104
\(714\) 0 0
\(715\) −0.105616 −0.00394981
\(716\) 0 0
\(717\) −21.2822 −0.794798
\(718\) 0 0
\(719\) −45.7903 −1.70769 −0.853846 0.520526i \(-0.825736\pi\)
−0.853846 + 0.520526i \(0.825736\pi\)
\(720\) 0 0
\(721\) −5.07477 −0.188994
\(722\) 0 0
\(723\) −17.6365 −0.655908
\(724\) 0 0
\(725\) 7.44271 0.276415
\(726\) 0 0
\(727\) −12.5668 −0.466076 −0.233038 0.972468i \(-0.574867\pi\)
−0.233038 + 0.972468i \(0.574867\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.55716 0.0945800
\(732\) 0 0
\(733\) 35.3051 1.30402 0.652012 0.758209i \(-0.273925\pi\)
0.652012 + 0.758209i \(0.273925\pi\)
\(734\) 0 0
\(735\) −4.87414 −0.179785
\(736\) 0 0
\(737\) −22.7377 −0.837553
\(738\) 0 0
\(739\) 30.2423 1.11248 0.556241 0.831021i \(-0.312243\pi\)
0.556241 + 0.831021i \(0.312243\pi\)
\(740\) 0 0
\(741\) −0.177186 −0.00650908
\(742\) 0 0
\(743\) −16.5002 −0.605332 −0.302666 0.953097i \(-0.597877\pi\)
−0.302666 + 0.953097i \(0.597877\pi\)
\(744\) 0 0
\(745\) −12.9492 −0.474422
\(746\) 0 0
\(747\) 14.0793 0.515136
\(748\) 0 0
\(749\) −4.86999 −0.177946
\(750\) 0 0
\(751\) −43.4211 −1.58446 −0.792229 0.610224i \(-0.791080\pi\)
−0.792229 + 0.610224i \(0.791080\pi\)
\(752\) 0 0
\(753\) −29.1497 −1.06228
\(754\) 0 0
\(755\) 0.209199 0.00761354
\(756\) 0 0
\(757\) −10.8584 −0.394655 −0.197327 0.980338i \(-0.563226\pi\)
−0.197327 + 0.980338i \(0.563226\pi\)
\(758\) 0 0
\(759\) −14.8557 −0.539228
\(760\) 0 0
\(761\) −42.4505 −1.53883 −0.769415 0.638750i \(-0.779452\pi\)
−0.769415 + 0.638750i \(0.779452\pi\)
\(762\) 0 0
\(763\) −16.7575 −0.606662
\(764\) 0 0
\(765\) −3.48802 −0.126110
\(766\) 0 0
\(767\) 0.259968 0.00938691
\(768\) 0 0
\(769\) 24.3994 0.879865 0.439933 0.898031i \(-0.355002\pi\)
0.439933 + 0.898031i \(0.355002\pi\)
\(770\) 0 0
\(771\) −28.8771 −1.03998
\(772\) 0 0
\(773\) −6.70612 −0.241203 −0.120601 0.992701i \(-0.538482\pi\)
−0.120601 + 0.992701i \(0.538482\pi\)
\(774\) 0 0
\(775\) 4.95405 0.177955
\(776\) 0 0
\(777\) 14.2740 0.512078
\(778\) 0 0
\(779\) −65.4120 −2.34363
\(780\) 0 0
\(781\) 17.1942 0.615257
\(782\) 0 0
\(783\) −2.80212 −0.100140
\(784\) 0 0
\(785\) 17.4114 0.621438
\(786\) 0 0
\(787\) 44.6064 1.59005 0.795023 0.606579i \(-0.207459\pi\)
0.795023 + 0.606579i \(0.207459\pi\)
\(788\) 0 0
\(789\) −3.69587 −0.131576
\(790\) 0 0
\(791\) −33.9119 −1.20577
\(792\) 0 0
\(793\) 0.238703 0.00847658
\(794\) 0 0
\(795\) −17.0391 −0.604314
\(796\) 0 0
\(797\) 23.7264 0.840433 0.420217 0.907424i \(-0.361954\pi\)
0.420217 + 0.907424i \(0.361954\pi\)
\(798\) 0 0
\(799\) −2.11744 −0.0749097
\(800\) 0 0
\(801\) 7.24231 0.255895
\(802\) 0 0
\(803\) 22.5891 0.797153
\(804\) 0 0
\(805\) 20.6425 0.727552
\(806\) 0 0
\(807\) −11.9065 −0.419128
\(808\) 0 0
\(809\) −39.3941 −1.38502 −0.692512 0.721407i \(-0.743496\pi\)
−0.692512 + 0.721407i \(0.743496\pi\)
\(810\) 0 0
\(811\) 40.3386 1.41648 0.708240 0.705971i \(-0.249489\pi\)
0.708240 + 0.705971i \(0.249489\pi\)
\(812\) 0 0
\(813\) −9.42035 −0.330386
\(814\) 0 0
\(815\) 16.8511 0.590268
\(816\) 0 0
\(817\) −6.20502 −0.217086
\(818\) 0 0
\(819\) 0.0626121 0.00218784
\(820\) 0 0
\(821\) −42.3862 −1.47929 −0.739645 0.672997i \(-0.765007\pi\)
−0.739645 + 0.672997i \(0.765007\pi\)
\(822\) 0 0
\(823\) 6.66255 0.232242 0.116121 0.993235i \(-0.462954\pi\)
0.116121 + 0.993235i \(0.462954\pi\)
\(824\) 0 0
\(825\) −5.71700 −0.199040
\(826\) 0 0
\(827\) 19.1082 0.664456 0.332228 0.943199i \(-0.392200\pi\)
0.332228 + 0.943199i \(0.392200\pi\)
\(828\) 0 0
\(829\) −50.4467 −1.75209 −0.876044 0.482231i \(-0.839827\pi\)
−0.876044 + 0.482231i \(0.839827\pi\)
\(830\) 0 0
\(831\) −9.72432 −0.337333
\(832\) 0 0
\(833\) 7.25333 0.251313
\(834\) 0 0
\(835\) 1.53098 0.0529818
\(836\) 0 0
\(837\) −1.86516 −0.0644693
\(838\) 0 0
\(839\) 27.7449 0.957862 0.478931 0.877853i \(-0.341024\pi\)
0.478931 + 0.877853i \(0.341024\pi\)
\(840\) 0 0
\(841\) −21.1481 −0.729245
\(842\) 0 0
\(843\) −9.54649 −0.328799
\(844\) 0 0
\(845\) −19.9012 −0.684621
\(846\) 0 0
\(847\) 12.4385 0.427392
\(848\) 0 0
\(849\) −0.881411 −0.0302499
\(850\) 0 0
\(851\) 50.4305 1.72873
\(852\) 0 0
\(853\) 49.0988 1.68111 0.840556 0.541725i \(-0.182229\pi\)
0.840556 + 0.541725i \(0.182229\pi\)
\(854\) 0 0
\(855\) 8.46376 0.289455
\(856\) 0 0
\(857\) 47.7973 1.63272 0.816362 0.577541i \(-0.195988\pi\)
0.816362 + 0.577541i \(0.195988\pi\)
\(858\) 0 0
\(859\) −34.6308 −1.18159 −0.590794 0.806822i \(-0.701186\pi\)
−0.590794 + 0.806822i \(0.701186\pi\)
\(860\) 0 0
\(861\) 23.1146 0.787743
\(862\) 0 0
\(863\) −53.2646 −1.81315 −0.906574 0.422047i \(-0.861312\pi\)
−0.906574 + 0.422047i \(0.861312\pi\)
\(864\) 0 0
\(865\) −27.0677 −0.920329
\(866\) 0 0
\(867\) −11.8094 −0.401068
\(868\) 0 0
\(869\) 12.6156 0.427956
\(870\) 0 0
\(871\) 0.338577 0.0114722
\(872\) 0 0
\(873\) 2.96086 0.100210
\(874\) 0 0
\(875\) 22.8981 0.774098
\(876\) 0 0
\(877\) −49.6892 −1.67789 −0.838943 0.544219i \(-0.816826\pi\)
−0.838943 + 0.544219i \(0.816826\pi\)
\(878\) 0 0
\(879\) 7.37269 0.248675
\(880\) 0 0
\(881\) 0.172379 0.00580759 0.00290379 0.999996i \(-0.499076\pi\)
0.00290379 + 0.999996i \(0.499076\pi\)
\(882\) 0 0
\(883\) −55.2229 −1.85840 −0.929200 0.369578i \(-0.879502\pi\)
−0.929200 + 0.369578i \(0.879502\pi\)
\(884\) 0 0
\(885\) −12.4181 −0.417429
\(886\) 0 0
\(887\) 39.2457 1.31774 0.658871 0.752256i \(-0.271034\pi\)
0.658871 + 0.752256i \(0.271034\pi\)
\(888\) 0 0
\(889\) 38.7725 1.30039
\(890\) 0 0
\(891\) 2.15240 0.0721083
\(892\) 0 0
\(893\) 5.13802 0.171937
\(894\) 0 0
\(895\) 35.6610 1.19202
\(896\) 0 0
\(897\) 0.221210 0.00738598
\(898\) 0 0
\(899\) 5.22640 0.174310
\(900\) 0 0
\(901\) 25.3563 0.844740
\(902\) 0 0
\(903\) 2.19266 0.0729673
\(904\) 0 0
\(905\) 19.0919 0.634638
\(906\) 0 0
\(907\) −17.0928 −0.567559 −0.283779 0.958890i \(-0.591588\pi\)
−0.283779 + 0.958890i \(0.591588\pi\)
\(908\) 0 0
\(909\) −10.7676 −0.357139
\(910\) 0 0
\(911\) 18.5880 0.615850 0.307925 0.951411i \(-0.400365\pi\)
0.307925 + 0.951411i \(0.400365\pi\)
\(912\) 0 0
\(913\) 30.3044 1.00293
\(914\) 0 0
\(915\) −11.4023 −0.376948
\(916\) 0 0
\(917\) 15.6398 0.516472
\(918\) 0 0
\(919\) −25.4303 −0.838868 −0.419434 0.907786i \(-0.637771\pi\)
−0.419434 + 0.907786i \(0.637771\pi\)
\(920\) 0 0
\(921\) 11.3416 0.373718
\(922\) 0 0
\(923\) −0.256032 −0.00842738
\(924\) 0 0
\(925\) 19.4074 0.638112
\(926\) 0 0
\(927\) 2.59773 0.0853205
\(928\) 0 0
\(929\) 6.00391 0.196982 0.0984909 0.995138i \(-0.468598\pi\)
0.0984909 + 0.995138i \(0.468598\pi\)
\(930\) 0 0
\(931\) −17.6004 −0.576830
\(932\) 0 0
\(933\) 23.2572 0.761406
\(934\) 0 0
\(935\) −7.50763 −0.245526
\(936\) 0 0
\(937\) −47.8575 −1.56344 −0.781719 0.623631i \(-0.785657\pi\)
−0.781719 + 0.623631i \(0.785657\pi\)
\(938\) 0 0
\(939\) 24.6287 0.803728
\(940\) 0 0
\(941\) −1.11153 −0.0362348 −0.0181174 0.999836i \(-0.505767\pi\)
−0.0181174 + 0.999836i \(0.505767\pi\)
\(942\) 0 0
\(943\) 81.6644 2.65936
\(944\) 0 0
\(945\) −2.99083 −0.0972919
\(946\) 0 0
\(947\) 15.8001 0.513434 0.256717 0.966487i \(-0.417359\pi\)
0.256717 + 0.966487i \(0.417359\pi\)
\(948\) 0 0
\(949\) −0.336365 −0.0109189
\(950\) 0 0
\(951\) 14.6228 0.474177
\(952\) 0 0
\(953\) 12.6651 0.410263 0.205131 0.978734i \(-0.434238\pi\)
0.205131 + 0.978734i \(0.434238\pi\)
\(954\) 0 0
\(955\) 16.2943 0.527270
\(956\) 0 0
\(957\) −6.03130 −0.194964
\(958\) 0 0
\(959\) −24.5855 −0.793908
\(960\) 0 0
\(961\) −27.5212 −0.887780
\(962\) 0 0
\(963\) 2.49290 0.0803327
\(964\) 0 0
\(965\) −5.61714 −0.180822
\(966\) 0 0
\(967\) 31.0282 0.997801 0.498901 0.866659i \(-0.333737\pi\)
0.498901 + 0.866659i \(0.333737\pi\)
\(968\) 0 0
\(969\) −12.5951 −0.404614
\(970\) 0 0
\(971\) −12.5965 −0.404242 −0.202121 0.979361i \(-0.564783\pi\)
−0.202121 + 0.979361i \(0.564783\pi\)
\(972\) 0 0
\(973\) −1.93074 −0.0618968
\(974\) 0 0
\(975\) 0.0851294 0.00272632
\(976\) 0 0
\(977\) −21.7953 −0.697295 −0.348647 0.937254i \(-0.613359\pi\)
−0.348647 + 0.937254i \(0.613359\pi\)
\(978\) 0 0
\(979\) 15.5884 0.498207
\(980\) 0 0
\(981\) 8.57800 0.273875
\(982\) 0 0
\(983\) −32.0842 −1.02333 −0.511664 0.859186i \(-0.670971\pi\)
−0.511664 + 0.859186i \(0.670971\pi\)
\(984\) 0 0
\(985\) −7.83136 −0.249528
\(986\) 0 0
\(987\) −1.81562 −0.0577918
\(988\) 0 0
\(989\) 7.74673 0.246332
\(990\) 0 0
\(991\) −33.9196 −1.07749 −0.538745 0.842469i \(-0.681101\pi\)
−0.538745 + 0.842469i \(0.681101\pi\)
\(992\) 0 0
\(993\) −14.4030 −0.457065
\(994\) 0 0
\(995\) 1.05779 0.0335341
\(996\) 0 0
\(997\) −55.7384 −1.76525 −0.882626 0.470076i \(-0.844226\pi\)
−0.882626 + 0.470076i \(0.844226\pi\)
\(998\) 0 0
\(999\) −7.30674 −0.231175
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 8016.2.a.bd.1.9 10
4.3 odd 2 4008.2.a.j.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.j.1.9 10 4.3 odd 2
8016.2.a.bd.1.9 10 1.1 even 1 trivial