Properties

Label 4008.2.a.f.1.1
Level $4008$
Weight $2$
Character 4008.1
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4008,2,Mod(1,4008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4008, 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("4008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.284897.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80442\) of defining polynomial
Character \(\chi\) \(=\) 4008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.40664 q^{5} +0.548479 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.40664 q^{5} +0.548479 q^{7} +1.00000 q^{9} -0.977163 q^{11} -1.94627 q^{13} -3.40664 q^{15} +1.38380 q^{17} -1.16100 q^{19} +0.548479 q^{21} +4.44417 q^{23} +6.60517 q^{25} +1.00000 q^{27} +3.71999 q^{29} +5.46700 q^{31} -0.977163 q^{33} -1.86847 q^{35} -2.93026 q^{37} -1.94627 q^{39} +6.35002 q^{41} -11.3309 q^{43} -3.40664 q^{45} +10.2154 q^{47} -6.69917 q^{49} +1.38380 q^{51} +3.23512 q^{53} +3.32884 q^{55} -1.16100 q^{57} -12.8823 q^{59} -14.9043 q^{61} +0.548479 q^{63} +6.63022 q^{65} -15.2508 q^{67} +4.44417 q^{69} -11.4987 q^{71} -0.389926 q^{73} +6.60517 q^{75} -0.535953 q^{77} -10.1425 q^{79} +1.00000 q^{81} -5.41931 q^{83} -4.71410 q^{85} +3.71999 q^{87} -17.1536 q^{89} -1.06749 q^{91} +5.46700 q^{93} +3.95512 q^{95} -2.62873 q^{97} -0.977163 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9} - 3 q^{11} - 14 q^{13} + q^{15} - 13 q^{17} - 2 q^{19} - 4 q^{21} - 5 q^{23} + 2 q^{25} + 5 q^{27} + 13 q^{29} + 2 q^{31} - 3 q^{33} - 12 q^{35} - 5 q^{37} - 14 q^{39} - 20 q^{41} - 20 q^{43} + q^{45} + q^{47} - 9 q^{49} - 13 q^{51} - 3 q^{53} - 3 q^{55} - 2 q^{57} - q^{59} - 34 q^{61} - 4 q^{63} - 22 q^{65} - 16 q^{67} - 5 q^{69} - 5 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{77} - 20 q^{79} + 5 q^{81} - 15 q^{83} - 27 q^{85} + 13 q^{87} - 48 q^{89} + 7 q^{91} + 2 q^{93} - 5 q^{95} - 21 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(5\) −3.40664 −1.52349 −0.761747 0.647875i \(-0.775658\pi\)
−0.761747 + 0.647875i \(0.775658\pi\)
\(6\) 0 0
\(7\) 0.548479 0.207306 0.103653 0.994614i \(-0.466947\pi\)
0.103653 + 0.994614i \(0.466947\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.977163 −0.294626 −0.147313 0.989090i \(-0.547062\pi\)
−0.147313 + 0.989090i \(0.547062\pi\)
\(12\) 0 0
\(13\) −1.94627 −0.539797 −0.269899 0.962889i \(-0.586990\pi\)
−0.269899 + 0.962889i \(0.586990\pi\)
\(14\) 0 0
\(15\) −3.40664 −0.879590
\(16\) 0 0
\(17\) 1.38380 0.335621 0.167810 0.985819i \(-0.446330\pi\)
0.167810 + 0.985819i \(0.446330\pi\)
\(18\) 0 0
\(19\) −1.16100 −0.266352 −0.133176 0.991092i \(-0.542518\pi\)
−0.133176 + 0.991092i \(0.542518\pi\)
\(20\) 0 0
\(21\) 0.548479 0.119688
\(22\) 0 0
\(23\) 4.44417 0.926673 0.463336 0.886182i \(-0.346652\pi\)
0.463336 + 0.886182i \(0.346652\pi\)
\(24\) 0 0
\(25\) 6.60517 1.32103
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.71999 0.690785 0.345393 0.938458i \(-0.387746\pi\)
0.345393 + 0.938458i \(0.387746\pi\)
\(30\) 0 0
\(31\) 5.46700 0.981903 0.490951 0.871187i \(-0.336649\pi\)
0.490951 + 0.871187i \(0.336649\pi\)
\(32\) 0 0
\(33\) −0.977163 −0.170102
\(34\) 0 0
\(35\) −1.86847 −0.315829
\(36\) 0 0
\(37\) −2.93026 −0.481732 −0.240866 0.970558i \(-0.577431\pi\)
−0.240866 + 0.970558i \(0.577431\pi\)
\(38\) 0 0
\(39\) −1.94627 −0.311652
\(40\) 0 0
\(41\) 6.35002 0.991706 0.495853 0.868407i \(-0.334856\pi\)
0.495853 + 0.868407i \(0.334856\pi\)
\(42\) 0 0
\(43\) −11.3309 −1.72794 −0.863970 0.503544i \(-0.832029\pi\)
−0.863970 + 0.503544i \(0.832029\pi\)
\(44\) 0 0
\(45\) −3.40664 −0.507831
\(46\) 0 0
\(47\) 10.2154 1.49008 0.745038 0.667022i \(-0.232431\pi\)
0.745038 + 0.667022i \(0.232431\pi\)
\(48\) 0 0
\(49\) −6.69917 −0.957024
\(50\) 0 0
\(51\) 1.38380 0.193771
\(52\) 0 0
\(53\) 3.23512 0.444378 0.222189 0.975004i \(-0.428680\pi\)
0.222189 + 0.975004i \(0.428680\pi\)
\(54\) 0 0
\(55\) 3.32884 0.448860
\(56\) 0 0
\(57\) −1.16100 −0.153779
\(58\) 0 0
\(59\) −12.8823 −1.67713 −0.838566 0.544800i \(-0.816605\pi\)
−0.838566 + 0.544800i \(0.816605\pi\)
\(60\) 0 0
\(61\) −14.9043 −1.90831 −0.954153 0.299321i \(-0.903240\pi\)
−0.954153 + 0.299321i \(0.903240\pi\)
\(62\) 0 0
\(63\) 0.548479 0.0691019
\(64\) 0 0
\(65\) 6.63022 0.822378
\(66\) 0 0
\(67\) −15.2508 −1.86318 −0.931591 0.363508i \(-0.881579\pi\)
−0.931591 + 0.363508i \(0.881579\pi\)
\(68\) 0 0
\(69\) 4.44417 0.535015
\(70\) 0 0
\(71\) −11.4987 −1.36464 −0.682322 0.731052i \(-0.739030\pi\)
−0.682322 + 0.731052i \(0.739030\pi\)
\(72\) 0 0
\(73\) −0.389926 −0.0456373 −0.0228187 0.999740i \(-0.507264\pi\)
−0.0228187 + 0.999740i \(0.507264\pi\)
\(74\) 0 0
\(75\) 6.60517 0.762699
\(76\) 0 0
\(77\) −0.535953 −0.0610776
\(78\) 0 0
\(79\) −10.1425 −1.14113 −0.570563 0.821254i \(-0.693275\pi\)
−0.570563 + 0.821254i \(0.693275\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.41931 −0.594847 −0.297423 0.954746i \(-0.596127\pi\)
−0.297423 + 0.954746i \(0.596127\pi\)
\(84\) 0 0
\(85\) −4.71410 −0.511316
\(86\) 0 0
\(87\) 3.71999 0.398825
\(88\) 0 0
\(89\) −17.1536 −1.81828 −0.909142 0.416487i \(-0.863261\pi\)
−0.909142 + 0.416487i \(0.863261\pi\)
\(90\) 0 0
\(91\) −1.06749 −0.111903
\(92\) 0 0
\(93\) 5.46700 0.566902
\(94\) 0 0
\(95\) 3.95512 0.405786
\(96\) 0 0
\(97\) −2.62873 −0.266907 −0.133453 0.991055i \(-0.542607\pi\)
−0.133453 + 0.991055i \(0.542607\pi\)
\(98\) 0 0
\(99\) −0.977163 −0.0982086
\(100\) 0 0
\(101\) 10.7613 1.07079 0.535393 0.844603i \(-0.320163\pi\)
0.535393 + 0.844603i \(0.320163\pi\)
\(102\) 0 0
\(103\) 0.545324 0.0537323 0.0268662 0.999639i \(-0.491447\pi\)
0.0268662 + 0.999639i \(0.491447\pi\)
\(104\) 0 0
\(105\) −1.86847 −0.182344
\(106\) 0 0
\(107\) −12.2316 −1.18248 −0.591239 0.806496i \(-0.701361\pi\)
−0.591239 + 0.806496i \(0.701361\pi\)
\(108\) 0 0
\(109\) 3.42284 0.327848 0.163924 0.986473i \(-0.447585\pi\)
0.163924 + 0.986473i \(0.447585\pi\)
\(110\) 0 0
\(111\) −2.93026 −0.278128
\(112\) 0 0
\(113\) 11.9535 1.12449 0.562243 0.826972i \(-0.309938\pi\)
0.562243 + 0.826972i \(0.309938\pi\)
\(114\) 0 0
\(115\) −15.1397 −1.41178
\(116\) 0 0
\(117\) −1.94627 −0.179932
\(118\) 0 0
\(119\) 0.758985 0.0695760
\(120\) 0 0
\(121\) −10.0452 −0.913196
\(122\) 0 0
\(123\) 6.35002 0.572562
\(124\) 0 0
\(125\) −5.46823 −0.489093
\(126\) 0 0
\(127\) −3.62801 −0.321934 −0.160967 0.986960i \(-0.551461\pi\)
−0.160967 + 0.986960i \(0.551461\pi\)
\(128\) 0 0
\(129\) −11.3309 −0.997626
\(130\) 0 0
\(131\) 6.01253 0.525317 0.262658 0.964889i \(-0.415401\pi\)
0.262658 + 0.964889i \(0.415401\pi\)
\(132\) 0 0
\(133\) −0.636786 −0.0552163
\(134\) 0 0
\(135\) −3.40664 −0.293197
\(136\) 0 0
\(137\) −3.87068 −0.330695 −0.165347 0.986235i \(-0.552875\pi\)
−0.165347 + 0.986235i \(0.552875\pi\)
\(138\) 0 0
\(139\) 5.82501 0.494071 0.247035 0.969006i \(-0.420544\pi\)
0.247035 + 0.969006i \(0.420544\pi\)
\(140\) 0 0
\(141\) 10.2154 0.860295
\(142\) 0 0
\(143\) 1.90182 0.159038
\(144\) 0 0
\(145\) −12.6727 −1.05241
\(146\) 0 0
\(147\) −6.69917 −0.552538
\(148\) 0 0
\(149\) 14.1095 1.15589 0.577947 0.816074i \(-0.303854\pi\)
0.577947 + 0.816074i \(0.303854\pi\)
\(150\) 0 0
\(151\) 20.7002 1.68456 0.842278 0.539043i \(-0.181214\pi\)
0.842278 + 0.539043i \(0.181214\pi\)
\(152\) 0 0
\(153\) 1.38380 0.111874
\(154\) 0 0
\(155\) −18.6241 −1.49592
\(156\) 0 0
\(157\) 6.25155 0.498928 0.249464 0.968384i \(-0.419746\pi\)
0.249464 + 0.968384i \(0.419746\pi\)
\(158\) 0 0
\(159\) 3.23512 0.256562
\(160\) 0 0
\(161\) 2.43753 0.192104
\(162\) 0 0
\(163\) 10.3332 0.809360 0.404680 0.914458i \(-0.367383\pi\)
0.404680 + 0.914458i \(0.367383\pi\)
\(164\) 0 0
\(165\) 3.32884 0.259150
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −9.21205 −0.708619
\(170\) 0 0
\(171\) −1.16100 −0.0887841
\(172\) 0 0
\(173\) −4.48522 −0.341005 −0.170503 0.985357i \(-0.554539\pi\)
−0.170503 + 0.985357i \(0.554539\pi\)
\(174\) 0 0
\(175\) 3.62280 0.273858
\(176\) 0 0
\(177\) −12.8823 −0.968292
\(178\) 0 0
\(179\) 20.1276 1.50441 0.752205 0.658929i \(-0.228990\pi\)
0.752205 + 0.658929i \(0.228990\pi\)
\(180\) 0 0
\(181\) −22.5001 −1.67242 −0.836210 0.548409i \(-0.815234\pi\)
−0.836210 + 0.548409i \(0.815234\pi\)
\(182\) 0 0
\(183\) −14.9043 −1.10176
\(184\) 0 0
\(185\) 9.98233 0.733916
\(186\) 0 0
\(187\) −1.35220 −0.0988824
\(188\) 0 0
\(189\) 0.548479 0.0398960
\(190\) 0 0
\(191\) −10.9408 −0.791651 −0.395826 0.918326i \(-0.629542\pi\)
−0.395826 + 0.918326i \(0.629542\pi\)
\(192\) 0 0
\(193\) 4.36096 0.313909 0.156954 0.987606i \(-0.449832\pi\)
0.156954 + 0.987606i \(0.449832\pi\)
\(194\) 0 0
\(195\) 6.63022 0.474800
\(196\) 0 0
\(197\) −7.29648 −0.519853 −0.259926 0.965628i \(-0.583698\pi\)
−0.259926 + 0.965628i \(0.583698\pi\)
\(198\) 0 0
\(199\) −22.7900 −1.61554 −0.807770 0.589498i \(-0.799325\pi\)
−0.807770 + 0.589498i \(0.799325\pi\)
\(200\) 0 0
\(201\) −15.2508 −1.07571
\(202\) 0 0
\(203\) 2.04034 0.143204
\(204\) 0 0
\(205\) −21.6322 −1.51086
\(206\) 0 0
\(207\) 4.44417 0.308891
\(208\) 0 0
\(209\) 1.13449 0.0784743
\(210\) 0 0
\(211\) −23.3625 −1.60834 −0.804169 0.594401i \(-0.797389\pi\)
−0.804169 + 0.594401i \(0.797389\pi\)
\(212\) 0 0
\(213\) −11.4987 −0.787877
\(214\) 0 0
\(215\) 38.6001 2.63251
\(216\) 0 0
\(217\) 2.99854 0.203554
\(218\) 0 0
\(219\) −0.389926 −0.0263487
\(220\) 0 0
\(221\) −2.69324 −0.181167
\(222\) 0 0
\(223\) 2.93666 0.196653 0.0983267 0.995154i \(-0.468651\pi\)
0.0983267 + 0.995154i \(0.468651\pi\)
\(224\) 0 0
\(225\) 6.60517 0.440345
\(226\) 0 0
\(227\) −20.3110 −1.34809 −0.674045 0.738690i \(-0.735445\pi\)
−0.674045 + 0.738690i \(0.735445\pi\)
\(228\) 0 0
\(229\) 3.90723 0.258197 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(230\) 0 0
\(231\) −0.535953 −0.0352631
\(232\) 0 0
\(233\) −26.0582 −1.70713 −0.853564 0.520988i \(-0.825564\pi\)
−0.853564 + 0.520988i \(0.825564\pi\)
\(234\) 0 0
\(235\) −34.8003 −2.27012
\(236\) 0 0
\(237\) −10.1425 −0.658829
\(238\) 0 0
\(239\) 22.1406 1.43216 0.716080 0.698018i \(-0.245935\pi\)
0.716080 + 0.698018i \(0.245935\pi\)
\(240\) 0 0
\(241\) −19.5932 −1.26211 −0.631053 0.775740i \(-0.717377\pi\)
−0.631053 + 0.775740i \(0.717377\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 22.8216 1.45802
\(246\) 0 0
\(247\) 2.25962 0.143776
\(248\) 0 0
\(249\) −5.41931 −0.343435
\(250\) 0 0
\(251\) 13.5771 0.856978 0.428489 0.903547i \(-0.359046\pi\)
0.428489 + 0.903547i \(0.359046\pi\)
\(252\) 0 0
\(253\) −4.34268 −0.273022
\(254\) 0 0
\(255\) −4.71410 −0.295208
\(256\) 0 0
\(257\) −19.8735 −1.23968 −0.619838 0.784730i \(-0.712802\pi\)
−0.619838 + 0.784730i \(0.712802\pi\)
\(258\) 0 0
\(259\) −1.60719 −0.0998657
\(260\) 0 0
\(261\) 3.71999 0.230262
\(262\) 0 0
\(263\) −19.8907 −1.22651 −0.613257 0.789884i \(-0.710141\pi\)
−0.613257 + 0.789884i \(0.710141\pi\)
\(264\) 0 0
\(265\) −11.0209 −0.677008
\(266\) 0 0
\(267\) −17.1536 −1.04979
\(268\) 0 0
\(269\) 13.4649 0.820972 0.410486 0.911867i \(-0.365359\pi\)
0.410486 + 0.911867i \(0.365359\pi\)
\(270\) 0 0
\(271\) 21.3536 1.29714 0.648570 0.761155i \(-0.275368\pi\)
0.648570 + 0.761155i \(0.275368\pi\)
\(272\) 0 0
\(273\) −1.06749 −0.0646072
\(274\) 0 0
\(275\) −6.45433 −0.389211
\(276\) 0 0
\(277\) 5.62547 0.338002 0.169001 0.985616i \(-0.445946\pi\)
0.169001 + 0.985616i \(0.445946\pi\)
\(278\) 0 0
\(279\) 5.46700 0.327301
\(280\) 0 0
\(281\) −12.1233 −0.723216 −0.361608 0.932330i \(-0.617772\pi\)
−0.361608 + 0.932330i \(0.617772\pi\)
\(282\) 0 0
\(283\) −2.83994 −0.168817 −0.0844084 0.996431i \(-0.526900\pi\)
−0.0844084 + 0.996431i \(0.526900\pi\)
\(284\) 0 0
\(285\) 3.95512 0.234281
\(286\) 0 0
\(287\) 3.48285 0.205586
\(288\) 0 0
\(289\) −15.0851 −0.887359
\(290\) 0 0
\(291\) −2.62873 −0.154099
\(292\) 0 0
\(293\) −17.8412 −1.04229 −0.521147 0.853467i \(-0.674496\pi\)
−0.521147 + 0.853467i \(0.674496\pi\)
\(294\) 0 0
\(295\) 43.8853 2.55510
\(296\) 0 0
\(297\) −0.977163 −0.0567007
\(298\) 0 0
\(299\) −8.64953 −0.500215
\(300\) 0 0
\(301\) −6.21474 −0.358211
\(302\) 0 0
\(303\) 10.7613 0.618219
\(304\) 0 0
\(305\) 50.7737 2.90729
\(306\) 0 0
\(307\) 17.9879 1.02662 0.513311 0.858203i \(-0.328419\pi\)
0.513311 + 0.858203i \(0.328419\pi\)
\(308\) 0 0
\(309\) 0.545324 0.0310224
\(310\) 0 0
\(311\) −8.91959 −0.505783 −0.252892 0.967495i \(-0.581382\pi\)
−0.252892 + 0.967495i \(0.581382\pi\)
\(312\) 0 0
\(313\) 3.72295 0.210434 0.105217 0.994449i \(-0.466446\pi\)
0.105217 + 0.994449i \(0.466446\pi\)
\(314\) 0 0
\(315\) −1.86847 −0.105276
\(316\) 0 0
\(317\) −17.1341 −0.962348 −0.481174 0.876625i \(-0.659790\pi\)
−0.481174 + 0.876625i \(0.659790\pi\)
\(318\) 0 0
\(319\) −3.63504 −0.203523
\(320\) 0 0
\(321\) −12.2316 −0.682704
\(322\) 0 0
\(323\) −1.60660 −0.0893933
\(324\) 0 0
\(325\) −12.8554 −0.713090
\(326\) 0 0
\(327\) 3.42284 0.189283
\(328\) 0 0
\(329\) 5.60296 0.308901
\(330\) 0 0
\(331\) 36.3810 1.99968 0.999839 0.0179188i \(-0.00570404\pi\)
0.999839 + 0.0179188i \(0.00570404\pi\)
\(332\) 0 0
\(333\) −2.93026 −0.160577
\(334\) 0 0
\(335\) 51.9539 2.83855
\(336\) 0 0
\(337\) 15.6702 0.853612 0.426806 0.904343i \(-0.359639\pi\)
0.426806 + 0.904343i \(0.359639\pi\)
\(338\) 0 0
\(339\) 11.9535 0.649223
\(340\) 0 0
\(341\) −5.34215 −0.289294
\(342\) 0 0
\(343\) −7.51371 −0.405702
\(344\) 0 0
\(345\) −15.1397 −0.815092
\(346\) 0 0
\(347\) 3.96072 0.212623 0.106311 0.994333i \(-0.466096\pi\)
0.106311 + 0.994333i \(0.466096\pi\)
\(348\) 0 0
\(349\) −10.7431 −0.575066 −0.287533 0.957771i \(-0.592835\pi\)
−0.287533 + 0.957771i \(0.592835\pi\)
\(350\) 0 0
\(351\) −1.94627 −0.103884
\(352\) 0 0
\(353\) 14.7105 0.782961 0.391480 0.920186i \(-0.371963\pi\)
0.391480 + 0.920186i \(0.371963\pi\)
\(354\) 0 0
\(355\) 39.1719 2.07903
\(356\) 0 0
\(357\) 0.758985 0.0401697
\(358\) 0 0
\(359\) −12.3440 −0.651490 −0.325745 0.945458i \(-0.605615\pi\)
−0.325745 + 0.945458i \(0.605615\pi\)
\(360\) 0 0
\(361\) −17.6521 −0.929056
\(362\) 0 0
\(363\) −10.0452 −0.527234
\(364\) 0 0
\(365\) 1.32833 0.0695282
\(366\) 0 0
\(367\) 21.3455 1.11422 0.557112 0.830437i \(-0.311909\pi\)
0.557112 + 0.830437i \(0.311909\pi\)
\(368\) 0 0
\(369\) 6.35002 0.330569
\(370\) 0 0
\(371\) 1.77440 0.0921221
\(372\) 0 0
\(373\) −0.0396748 −0.00205428 −0.00102714 0.999999i \(-0.500327\pi\)
−0.00102714 + 0.999999i \(0.500327\pi\)
\(374\) 0 0
\(375\) −5.46823 −0.282378
\(376\) 0 0
\(377\) −7.24009 −0.372884
\(378\) 0 0
\(379\) 31.2798 1.60673 0.803367 0.595485i \(-0.203040\pi\)
0.803367 + 0.595485i \(0.203040\pi\)
\(380\) 0 0
\(381\) −3.62801 −0.185868
\(382\) 0 0
\(383\) −18.7930 −0.960277 −0.480139 0.877193i \(-0.659414\pi\)
−0.480139 + 0.877193i \(0.659414\pi\)
\(384\) 0 0
\(385\) 1.82580 0.0930513
\(386\) 0 0
\(387\) −11.3309 −0.575980
\(388\) 0 0
\(389\) −34.3125 −1.73971 −0.869856 0.493306i \(-0.835788\pi\)
−0.869856 + 0.493306i \(0.835788\pi\)
\(390\) 0 0
\(391\) 6.14983 0.311011
\(392\) 0 0
\(393\) 6.01253 0.303292
\(394\) 0 0
\(395\) 34.5520 1.73850
\(396\) 0 0
\(397\) −16.8147 −0.843903 −0.421952 0.906618i \(-0.638655\pi\)
−0.421952 + 0.906618i \(0.638655\pi\)
\(398\) 0 0
\(399\) −0.636786 −0.0318792
\(400\) 0 0
\(401\) 37.8464 1.88996 0.944980 0.327129i \(-0.106081\pi\)
0.944980 + 0.327129i \(0.106081\pi\)
\(402\) 0 0
\(403\) −10.6402 −0.530028
\(404\) 0 0
\(405\) −3.40664 −0.169277
\(406\) 0 0
\(407\) 2.86334 0.141931
\(408\) 0 0
\(409\) −23.8406 −1.17884 −0.589422 0.807825i \(-0.700644\pi\)
−0.589422 + 0.807825i \(0.700644\pi\)
\(410\) 0 0
\(411\) −3.87068 −0.190927
\(412\) 0 0
\(413\) −7.06567 −0.347679
\(414\) 0 0
\(415\) 18.4616 0.906246
\(416\) 0 0
\(417\) 5.82501 0.285252
\(418\) 0 0
\(419\) −26.1953 −1.27972 −0.639862 0.768490i \(-0.721008\pi\)
−0.639862 + 0.768490i \(0.721008\pi\)
\(420\) 0 0
\(421\) −0.289799 −0.0141239 −0.00706196 0.999975i \(-0.502248\pi\)
−0.00706196 + 0.999975i \(0.502248\pi\)
\(422\) 0 0
\(423\) 10.2154 0.496692
\(424\) 0 0
\(425\) 9.14023 0.443366
\(426\) 0 0
\(427\) −8.17472 −0.395602
\(428\) 0 0
\(429\) 1.90182 0.0918207
\(430\) 0 0
\(431\) 16.1155 0.776258 0.388129 0.921605i \(-0.373122\pi\)
0.388129 + 0.921605i \(0.373122\pi\)
\(432\) 0 0
\(433\) 12.0680 0.579949 0.289974 0.957034i \(-0.406353\pi\)
0.289974 + 0.957034i \(0.406353\pi\)
\(434\) 0 0
\(435\) −12.6727 −0.607607
\(436\) 0 0
\(437\) −5.15969 −0.246822
\(438\) 0 0
\(439\) −10.5088 −0.501560 −0.250780 0.968044i \(-0.580687\pi\)
−0.250780 + 0.968044i \(0.580687\pi\)
\(440\) 0 0
\(441\) −6.69917 −0.319008
\(442\) 0 0
\(443\) −19.8417 −0.942708 −0.471354 0.881944i \(-0.656235\pi\)
−0.471354 + 0.881944i \(0.656235\pi\)
\(444\) 0 0
\(445\) 58.4362 2.77014
\(446\) 0 0
\(447\) 14.1095 0.667356
\(448\) 0 0
\(449\) −16.0031 −0.755234 −0.377617 0.925962i \(-0.623256\pi\)
−0.377617 + 0.925962i \(0.623256\pi\)
\(450\) 0 0
\(451\) −6.20500 −0.292182
\(452\) 0 0
\(453\) 20.7002 0.972579
\(454\) 0 0
\(455\) 3.63654 0.170484
\(456\) 0 0
\(457\) 23.4461 1.09676 0.548381 0.836228i \(-0.315244\pi\)
0.548381 + 0.836228i \(0.315244\pi\)
\(458\) 0 0
\(459\) 1.38380 0.0645902
\(460\) 0 0
\(461\) 24.6649 1.14876 0.574379 0.818590i \(-0.305244\pi\)
0.574379 + 0.818590i \(0.305244\pi\)
\(462\) 0 0
\(463\) −2.63528 −0.122472 −0.0612359 0.998123i \(-0.519504\pi\)
−0.0612359 + 0.998123i \(0.519504\pi\)
\(464\) 0 0
\(465\) −18.6241 −0.863672
\(466\) 0 0
\(467\) −11.7325 −0.542917 −0.271459 0.962450i \(-0.587506\pi\)
−0.271459 + 0.962450i \(0.587506\pi\)
\(468\) 0 0
\(469\) −8.36475 −0.386248
\(470\) 0 0
\(471\) 6.25155 0.288056
\(472\) 0 0
\(473\) 11.0721 0.509095
\(474\) 0 0
\(475\) −7.66862 −0.351861
\(476\) 0 0
\(477\) 3.23512 0.148126
\(478\) 0 0
\(479\) 26.1101 1.19300 0.596501 0.802612i \(-0.296557\pi\)
0.596501 + 0.802612i \(0.296557\pi\)
\(480\) 0 0
\(481\) 5.70307 0.260038
\(482\) 0 0
\(483\) 2.43753 0.110912
\(484\) 0 0
\(485\) 8.95512 0.406631
\(486\) 0 0
\(487\) 29.1754 1.32206 0.661032 0.750358i \(-0.270119\pi\)
0.661032 + 0.750358i \(0.270119\pi\)
\(488\) 0 0
\(489\) 10.3332 0.467284
\(490\) 0 0
\(491\) 8.31324 0.375171 0.187586 0.982248i \(-0.439934\pi\)
0.187586 + 0.982248i \(0.439934\pi\)
\(492\) 0 0
\(493\) 5.14772 0.231842
\(494\) 0 0
\(495\) 3.32884 0.149620
\(496\) 0 0
\(497\) −6.30679 −0.282898
\(498\) 0 0
\(499\) 33.0939 1.48148 0.740742 0.671789i \(-0.234474\pi\)
0.740742 + 0.671789i \(0.234474\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −22.4327 −1.00022 −0.500112 0.865961i \(-0.666708\pi\)
−0.500112 + 0.865961i \(0.666708\pi\)
\(504\) 0 0
\(505\) −36.6597 −1.63134
\(506\) 0 0
\(507\) −9.21205 −0.409121
\(508\) 0 0
\(509\) 7.47758 0.331438 0.165719 0.986173i \(-0.447006\pi\)
0.165719 + 0.986173i \(0.447006\pi\)
\(510\) 0 0
\(511\) −0.213866 −0.00946088
\(512\) 0 0
\(513\) −1.16100 −0.0512595
\(514\) 0 0
\(515\) −1.85772 −0.0818609
\(516\) 0 0
\(517\) −9.98215 −0.439014
\(518\) 0 0
\(519\) −4.48522 −0.196879
\(520\) 0 0
\(521\) −4.51154 −0.197654 −0.0988271 0.995105i \(-0.531509\pi\)
−0.0988271 + 0.995105i \(0.531509\pi\)
\(522\) 0 0
\(523\) 27.9098 1.22041 0.610206 0.792243i \(-0.291087\pi\)
0.610206 + 0.792243i \(0.291087\pi\)
\(524\) 0 0
\(525\) 3.62280 0.158112
\(526\) 0 0
\(527\) 7.56524 0.329547
\(528\) 0 0
\(529\) −3.24938 −0.141277
\(530\) 0 0
\(531\) −12.8823 −0.559044
\(532\) 0 0
\(533\) −12.3588 −0.535320
\(534\) 0 0
\(535\) 41.6688 1.80150
\(536\) 0 0
\(537\) 20.1276 0.868572
\(538\) 0 0
\(539\) 6.54618 0.281964
\(540\) 0 0
\(541\) −28.7584 −1.23642 −0.618211 0.786012i \(-0.712142\pi\)
−0.618211 + 0.786012i \(0.712142\pi\)
\(542\) 0 0
\(543\) −22.5001 −0.965572
\(544\) 0 0
\(545\) −11.6604 −0.499475
\(546\) 0 0
\(547\) −22.8889 −0.978659 −0.489330 0.872099i \(-0.662758\pi\)
−0.489330 + 0.872099i \(0.662758\pi\)
\(548\) 0 0
\(549\) −14.9043 −0.636102
\(550\) 0 0
\(551\) −4.31892 −0.183992
\(552\) 0 0
\(553\) −5.56298 −0.236562
\(554\) 0 0
\(555\) 9.98233 0.423727
\(556\) 0 0
\(557\) 41.8344 1.77258 0.886291 0.463129i \(-0.153273\pi\)
0.886291 + 0.463129i \(0.153273\pi\)
\(558\) 0 0
\(559\) 22.0529 0.932737
\(560\) 0 0
\(561\) −1.35220 −0.0570898
\(562\) 0 0
\(563\) 33.9234 1.42970 0.714851 0.699277i \(-0.246495\pi\)
0.714851 + 0.699277i \(0.246495\pi\)
\(564\) 0 0
\(565\) −40.7211 −1.71315
\(566\) 0 0
\(567\) 0.548479 0.0230340
\(568\) 0 0
\(569\) −26.0510 −1.09212 −0.546058 0.837747i \(-0.683872\pi\)
−0.546058 + 0.837747i \(0.683872\pi\)
\(570\) 0 0
\(571\) −3.91990 −0.164043 −0.0820214 0.996631i \(-0.526138\pi\)
−0.0820214 + 0.996631i \(0.526138\pi\)
\(572\) 0 0
\(573\) −10.9408 −0.457060
\(574\) 0 0
\(575\) 29.3545 1.22417
\(576\) 0 0
\(577\) −30.8384 −1.28382 −0.641910 0.766780i \(-0.721858\pi\)
−0.641910 + 0.766780i \(0.721858\pi\)
\(578\) 0 0
\(579\) 4.36096 0.181235
\(580\) 0 0
\(581\) −2.97238 −0.123315
\(582\) 0 0
\(583\) −3.16124 −0.130925
\(584\) 0 0
\(585\) 6.63022 0.274126
\(586\) 0 0
\(587\) 33.0392 1.36367 0.681837 0.731504i \(-0.261181\pi\)
0.681837 + 0.731504i \(0.261181\pi\)
\(588\) 0 0
\(589\) −6.34721 −0.261532
\(590\) 0 0
\(591\) −7.29648 −0.300137
\(592\) 0 0
\(593\) −39.8895 −1.63807 −0.819033 0.573747i \(-0.805489\pi\)
−0.819033 + 0.573747i \(0.805489\pi\)
\(594\) 0 0
\(595\) −2.58559 −0.105999
\(596\) 0 0
\(597\) −22.7900 −0.932732
\(598\) 0 0
\(599\) 7.18551 0.293592 0.146796 0.989167i \(-0.453104\pi\)
0.146796 + 0.989167i \(0.453104\pi\)
\(600\) 0 0
\(601\) −12.6866 −0.517499 −0.258749 0.965944i \(-0.583310\pi\)
−0.258749 + 0.965944i \(0.583310\pi\)
\(602\) 0 0
\(603\) −15.2508 −0.621061
\(604\) 0 0
\(605\) 34.2202 1.39125
\(606\) 0 0
\(607\) −25.5231 −1.03595 −0.517975 0.855396i \(-0.673314\pi\)
−0.517975 + 0.855396i \(0.673314\pi\)
\(608\) 0 0
\(609\) 2.04034 0.0826786
\(610\) 0 0
\(611\) −19.8820 −0.804339
\(612\) 0 0
\(613\) 8.66709 0.350060 0.175030 0.984563i \(-0.443998\pi\)
0.175030 + 0.984563i \(0.443998\pi\)
\(614\) 0 0
\(615\) −21.6322 −0.872294
\(616\) 0 0
\(617\) 41.3383 1.66422 0.832109 0.554613i \(-0.187134\pi\)
0.832109 + 0.554613i \(0.187134\pi\)
\(618\) 0 0
\(619\) −5.59969 −0.225071 −0.112535 0.993648i \(-0.535897\pi\)
−0.112535 + 0.993648i \(0.535897\pi\)
\(620\) 0 0
\(621\) 4.44417 0.178338
\(622\) 0 0
\(623\) −9.40842 −0.376940
\(624\) 0 0
\(625\) −14.3976 −0.575903
\(626\) 0 0
\(627\) 1.13449 0.0453071
\(628\) 0 0
\(629\) −4.05489 −0.161679
\(630\) 0 0
\(631\) 0.900479 0.0358475 0.0179237 0.999839i \(-0.494294\pi\)
0.0179237 + 0.999839i \(0.494294\pi\)
\(632\) 0 0
\(633\) −23.3625 −0.928574
\(634\) 0 0
\(635\) 12.3593 0.490464
\(636\) 0 0
\(637\) 13.0384 0.516599
\(638\) 0 0
\(639\) −11.4987 −0.454881
\(640\) 0 0
\(641\) 17.5091 0.691567 0.345783 0.938314i \(-0.387613\pi\)
0.345783 + 0.938314i \(0.387613\pi\)
\(642\) 0 0
\(643\) −3.47894 −0.137196 −0.0685980 0.997644i \(-0.521853\pi\)
−0.0685980 + 0.997644i \(0.521853\pi\)
\(644\) 0 0
\(645\) 38.6001 1.51988
\(646\) 0 0
\(647\) 9.60160 0.377478 0.188739 0.982027i \(-0.439560\pi\)
0.188739 + 0.982027i \(0.439560\pi\)
\(648\) 0 0
\(649\) 12.5881 0.494126
\(650\) 0 0
\(651\) 2.99854 0.117522
\(652\) 0 0
\(653\) 15.8659 0.620879 0.310440 0.950593i \(-0.399524\pi\)
0.310440 + 0.950593i \(0.399524\pi\)
\(654\) 0 0
\(655\) −20.4825 −0.800317
\(656\) 0 0
\(657\) −0.389926 −0.0152124
\(658\) 0 0
\(659\) −24.2593 −0.945008 −0.472504 0.881328i \(-0.656650\pi\)
−0.472504 + 0.881328i \(0.656650\pi\)
\(660\) 0 0
\(661\) 23.6372 0.919381 0.459690 0.888079i \(-0.347960\pi\)
0.459690 + 0.888079i \(0.347960\pi\)
\(662\) 0 0
\(663\) −2.69324 −0.104597
\(664\) 0 0
\(665\) 2.16930 0.0841218
\(666\) 0 0
\(667\) 16.5323 0.640132
\(668\) 0 0
\(669\) 2.93666 0.113538
\(670\) 0 0
\(671\) 14.5640 0.562236
\(672\) 0 0
\(673\) 37.6912 1.45289 0.726445 0.687224i \(-0.241171\pi\)
0.726445 + 0.687224i \(0.241171\pi\)
\(674\) 0 0
\(675\) 6.60517 0.254233
\(676\) 0 0
\(677\) −37.7588 −1.45119 −0.725594 0.688123i \(-0.758435\pi\)
−0.725594 + 0.688123i \(0.758435\pi\)
\(678\) 0 0
\(679\) −1.44180 −0.0553313
\(680\) 0 0
\(681\) −20.3110 −0.778320
\(682\) 0 0
\(683\) 24.5037 0.937608 0.468804 0.883302i \(-0.344685\pi\)
0.468804 + 0.883302i \(0.344685\pi\)
\(684\) 0 0
\(685\) 13.1860 0.503811
\(686\) 0 0
\(687\) 3.90723 0.149070
\(688\) 0 0
\(689\) −6.29641 −0.239874
\(690\) 0 0
\(691\) −34.7858 −1.32331 −0.661657 0.749807i \(-0.730146\pi\)
−0.661657 + 0.749807i \(0.730146\pi\)
\(692\) 0 0
\(693\) −0.535953 −0.0203592
\(694\) 0 0
\(695\) −19.8437 −0.752714
\(696\) 0 0
\(697\) 8.78715 0.332837
\(698\) 0 0
\(699\) −26.0582 −0.985611
\(700\) 0 0
\(701\) −14.1262 −0.533540 −0.266770 0.963760i \(-0.585956\pi\)
−0.266770 + 0.963760i \(0.585956\pi\)
\(702\) 0 0
\(703\) 3.40204 0.128310
\(704\) 0 0
\(705\) −34.8003 −1.31065
\(706\) 0 0
\(707\) 5.90233 0.221980
\(708\) 0 0
\(709\) −31.3280 −1.17655 −0.588275 0.808661i \(-0.700193\pi\)
−0.588275 + 0.808661i \(0.700193\pi\)
\(710\) 0 0
\(711\) −10.1425 −0.380375
\(712\) 0 0
\(713\) 24.2963 0.909903
\(714\) 0 0
\(715\) −6.47881 −0.242294
\(716\) 0 0
\(717\) 22.1406 0.826858
\(718\) 0 0
\(719\) 13.1522 0.490494 0.245247 0.969461i \(-0.421131\pi\)
0.245247 + 0.969461i \(0.421131\pi\)
\(720\) 0 0
\(721\) 0.299099 0.0111390
\(722\) 0 0
\(723\) −19.5932 −0.728677
\(724\) 0 0
\(725\) 24.5712 0.912551
\(726\) 0 0
\(727\) −43.9706 −1.63078 −0.815390 0.578913i \(-0.803477\pi\)
−0.815390 + 0.578913i \(0.803477\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −15.6796 −0.579932
\(732\) 0 0
\(733\) −29.8792 −1.10361 −0.551806 0.833972i \(-0.686061\pi\)
−0.551806 + 0.833972i \(0.686061\pi\)
\(734\) 0 0
\(735\) 22.8216 0.841789
\(736\) 0 0
\(737\) 14.9025 0.548941
\(738\) 0 0
\(739\) 42.1071 1.54894 0.774468 0.632613i \(-0.218018\pi\)
0.774468 + 0.632613i \(0.218018\pi\)
\(740\) 0 0
\(741\) 2.25962 0.0830093
\(742\) 0 0
\(743\) 5.40968 0.198462 0.0992310 0.995064i \(-0.468362\pi\)
0.0992310 + 0.995064i \(0.468362\pi\)
\(744\) 0 0
\(745\) −48.0659 −1.76100
\(746\) 0 0
\(747\) −5.41931 −0.198282
\(748\) 0 0
\(749\) −6.70880 −0.245134
\(750\) 0 0
\(751\) 24.2028 0.883173 0.441587 0.897219i \(-0.354416\pi\)
0.441587 + 0.897219i \(0.354416\pi\)
\(752\) 0 0
\(753\) 13.5771 0.494776
\(754\) 0 0
\(755\) −70.5180 −2.56641
\(756\) 0 0
\(757\) 11.1230 0.404273 0.202137 0.979357i \(-0.435212\pi\)
0.202137 + 0.979357i \(0.435212\pi\)
\(758\) 0 0
\(759\) −4.34268 −0.157629
\(760\) 0 0
\(761\) 46.2131 1.67522 0.837612 0.546266i \(-0.183951\pi\)
0.837612 + 0.546266i \(0.183951\pi\)
\(762\) 0 0
\(763\) 1.87736 0.0679648
\(764\) 0 0
\(765\) −4.71410 −0.170439
\(766\) 0 0
\(767\) 25.0724 0.905311
\(768\) 0 0
\(769\) −2.85940 −0.103113 −0.0515563 0.998670i \(-0.516418\pi\)
−0.0515563 + 0.998670i \(0.516418\pi\)
\(770\) 0 0
\(771\) −19.8735 −0.715727
\(772\) 0 0
\(773\) −15.3191 −0.550991 −0.275496 0.961302i \(-0.588842\pi\)
−0.275496 + 0.961302i \(0.588842\pi\)
\(774\) 0 0
\(775\) 36.1105 1.29713
\(776\) 0 0
\(777\) −1.60719 −0.0576575
\(778\) 0 0
\(779\) −7.37239 −0.264143
\(780\) 0 0
\(781\) 11.2361 0.402059
\(782\) 0 0
\(783\) 3.71999 0.132942
\(784\) 0 0
\(785\) −21.2968 −0.760114
\(786\) 0 0
\(787\) −22.1474 −0.789471 −0.394735 0.918795i \(-0.629164\pi\)
−0.394735 + 0.918795i \(0.629164\pi\)
\(788\) 0 0
\(789\) −19.8907 −0.708128
\(790\) 0 0
\(791\) 6.55622 0.233112
\(792\) 0 0
\(793\) 29.0078 1.03010
\(794\) 0 0
\(795\) −11.0209 −0.390871
\(796\) 0 0
\(797\) −3.24008 −0.114770 −0.0573848 0.998352i \(-0.518276\pi\)
−0.0573848 + 0.998352i \(0.518276\pi\)
\(798\) 0 0
\(799\) 14.1361 0.500100
\(800\) 0 0
\(801\) −17.1536 −0.606094
\(802\) 0 0
\(803\) 0.381021 0.0134459
\(804\) 0 0
\(805\) −8.30379 −0.292670
\(806\) 0 0
\(807\) 13.4649 0.473989
\(808\) 0 0
\(809\) 10.7561 0.378163 0.189081 0.981961i \(-0.439449\pi\)
0.189081 + 0.981961i \(0.439449\pi\)
\(810\) 0 0
\(811\) −56.3522 −1.97879 −0.989396 0.145241i \(-0.953604\pi\)
−0.989396 + 0.145241i \(0.953604\pi\)
\(812\) 0 0
\(813\) 21.3536 0.748904
\(814\) 0 0
\(815\) −35.2015 −1.23306
\(816\) 0 0
\(817\) 13.1552 0.460241
\(818\) 0 0
\(819\) −1.06749 −0.0373010
\(820\) 0 0
\(821\) −6.76637 −0.236148 −0.118074 0.993005i \(-0.537672\pi\)
−0.118074 + 0.993005i \(0.537672\pi\)
\(822\) 0 0
\(823\) 28.4265 0.990886 0.495443 0.868640i \(-0.335006\pi\)
0.495443 + 0.868640i \(0.335006\pi\)
\(824\) 0 0
\(825\) −6.45433 −0.224711
\(826\) 0 0
\(827\) 13.3921 0.465689 0.232844 0.972514i \(-0.425197\pi\)
0.232844 + 0.972514i \(0.425197\pi\)
\(828\) 0 0
\(829\) 23.5846 0.819126 0.409563 0.912282i \(-0.365681\pi\)
0.409563 + 0.912282i \(0.365681\pi\)
\(830\) 0 0
\(831\) 5.62547 0.195145
\(832\) 0 0
\(833\) −9.27031 −0.321197
\(834\) 0 0
\(835\) −3.40664 −0.117892
\(836\) 0 0
\(837\) 5.46700 0.188967
\(838\) 0 0
\(839\) −24.8349 −0.857394 −0.428697 0.903448i \(-0.641027\pi\)
−0.428697 + 0.903448i \(0.641027\pi\)
\(840\) 0 0
\(841\) −15.1617 −0.522816
\(842\) 0 0
\(843\) −12.1233 −0.417549
\(844\) 0 0
\(845\) 31.3821 1.07958
\(846\) 0 0
\(847\) −5.50956 −0.189311
\(848\) 0 0
\(849\) −2.83994 −0.0974664
\(850\) 0 0
\(851\) −13.0226 −0.446408
\(852\) 0 0
\(853\) −15.2493 −0.522127 −0.261063 0.965322i \(-0.584073\pi\)
−0.261063 + 0.965322i \(0.584073\pi\)
\(854\) 0 0
\(855\) 3.95512 0.135262
\(856\) 0 0
\(857\) 3.00641 0.102697 0.0513485 0.998681i \(-0.483648\pi\)
0.0513485 + 0.998681i \(0.483648\pi\)
\(858\) 0 0
\(859\) 26.7316 0.912070 0.456035 0.889962i \(-0.349269\pi\)
0.456035 + 0.889962i \(0.349269\pi\)
\(860\) 0 0
\(861\) 3.48285 0.118695
\(862\) 0 0
\(863\) 16.1526 0.549839 0.274920 0.961467i \(-0.411349\pi\)
0.274920 + 0.961467i \(0.411349\pi\)
\(864\) 0 0
\(865\) 15.2795 0.519519
\(866\) 0 0
\(867\) −15.0851 −0.512317
\(868\) 0 0
\(869\) 9.91092 0.336205
\(870\) 0 0
\(871\) 29.6821 1.00574
\(872\) 0 0
\(873\) −2.62873 −0.0889689
\(874\) 0 0
\(875\) −2.99921 −0.101392
\(876\) 0 0
\(877\) 52.9473 1.78790 0.893952 0.448162i \(-0.147921\pi\)
0.893952 + 0.448162i \(0.147921\pi\)
\(878\) 0 0
\(879\) −17.8412 −0.601769
\(880\) 0 0
\(881\) −42.3111 −1.42550 −0.712749 0.701420i \(-0.752550\pi\)
−0.712749 + 0.701420i \(0.752550\pi\)
\(882\) 0 0
\(883\) 21.9912 0.740063 0.370032 0.929019i \(-0.379347\pi\)
0.370032 + 0.929019i \(0.379347\pi\)
\(884\) 0 0
\(885\) 43.8853 1.47519
\(886\) 0 0
\(887\) 32.9683 1.10697 0.553483 0.832860i \(-0.313298\pi\)
0.553483 + 0.832860i \(0.313298\pi\)
\(888\) 0 0
\(889\) −1.98989 −0.0667386
\(890\) 0 0
\(891\) −0.977163 −0.0327362
\(892\) 0 0
\(893\) −11.8602 −0.396885
\(894\) 0 0
\(895\) −68.5676 −2.29196
\(896\) 0 0
\(897\) −8.64953 −0.288800
\(898\) 0 0
\(899\) 20.3372 0.678284
\(900\) 0 0
\(901\) 4.47676 0.149143
\(902\) 0 0
\(903\) −6.21474 −0.206813
\(904\) 0 0
\(905\) 76.6497 2.54792
\(906\) 0 0
\(907\) −20.7799 −0.689985 −0.344992 0.938606i \(-0.612118\pi\)
−0.344992 + 0.938606i \(0.612118\pi\)
\(908\) 0 0
\(909\) 10.7613 0.356929
\(910\) 0 0
\(911\) −15.6556 −0.518694 −0.259347 0.965784i \(-0.583507\pi\)
−0.259347 + 0.965784i \(0.583507\pi\)
\(912\) 0 0
\(913\) 5.29555 0.175257
\(914\) 0 0
\(915\) 50.7737 1.67853
\(916\) 0 0
\(917\) 3.29774 0.108901
\(918\) 0 0
\(919\) 12.9838 0.428296 0.214148 0.976801i \(-0.431303\pi\)
0.214148 + 0.976801i \(0.431303\pi\)
\(920\) 0 0
\(921\) 17.9879 0.592720
\(922\) 0 0
\(923\) 22.3795 0.736631
\(924\) 0 0
\(925\) −19.3549 −0.636384
\(926\) 0 0
\(927\) 0.545324 0.0179108
\(928\) 0 0
\(929\) 1.96973 0.0646246 0.0323123 0.999478i \(-0.489713\pi\)
0.0323123 + 0.999478i \(0.489713\pi\)
\(930\) 0 0
\(931\) 7.77776 0.254906
\(932\) 0 0
\(933\) −8.91959 −0.292014
\(934\) 0 0
\(935\) 4.60644 0.150647
\(936\) 0 0
\(937\) −43.8351 −1.43203 −0.716016 0.698084i \(-0.754036\pi\)
−0.716016 + 0.698084i \(0.754036\pi\)
\(938\) 0 0
\(939\) 3.72295 0.121494
\(940\) 0 0
\(941\) −5.35649 −0.174617 −0.0873083 0.996181i \(-0.527827\pi\)
−0.0873083 + 0.996181i \(0.527827\pi\)
\(942\) 0 0
\(943\) 28.2205 0.918987
\(944\) 0 0
\(945\) −1.86847 −0.0607813
\(946\) 0 0
\(947\) −36.7424 −1.19397 −0.596985 0.802253i \(-0.703635\pi\)
−0.596985 + 0.802253i \(0.703635\pi\)
\(948\) 0 0
\(949\) 0.758899 0.0246349
\(950\) 0 0
\(951\) −17.1341 −0.555612
\(952\) 0 0
\(953\) 21.0794 0.682828 0.341414 0.939913i \(-0.389094\pi\)
0.341414 + 0.939913i \(0.389094\pi\)
\(954\) 0 0
\(955\) 37.2715 1.20608
\(956\) 0 0
\(957\) −3.63504 −0.117504
\(958\) 0 0
\(959\) −2.12299 −0.0685549
\(960\) 0 0
\(961\) −1.11186 −0.0358666
\(962\) 0 0
\(963\) −12.2316 −0.394159
\(964\) 0 0
\(965\) −14.8562 −0.478238
\(966\) 0 0
\(967\) −2.90443 −0.0934001 −0.0467001 0.998909i \(-0.514871\pi\)
−0.0467001 + 0.998909i \(0.514871\pi\)
\(968\) 0 0
\(969\) −1.60660 −0.0516113
\(970\) 0 0
\(971\) 34.3856 1.10349 0.551743 0.834014i \(-0.313963\pi\)
0.551743 + 0.834014i \(0.313963\pi\)
\(972\) 0 0
\(973\) 3.19490 0.102424
\(974\) 0 0
\(975\) −12.8554 −0.411703
\(976\) 0 0
\(977\) 18.2981 0.585407 0.292704 0.956203i \(-0.405445\pi\)
0.292704 + 0.956203i \(0.405445\pi\)
\(978\) 0 0
\(979\) 16.7619 0.535713
\(980\) 0 0
\(981\) 3.42284 0.109283
\(982\) 0 0
\(983\) −28.5885 −0.911831 −0.455915 0.890023i \(-0.650688\pi\)
−0.455915 + 0.890023i \(0.650688\pi\)
\(984\) 0 0
\(985\) 24.8565 0.791992
\(986\) 0 0
\(987\) 5.60296 0.178344
\(988\) 0 0
\(989\) −50.3562 −1.60123
\(990\) 0 0
\(991\) 3.21688 0.102188 0.0510938 0.998694i \(-0.483729\pi\)
0.0510938 + 0.998694i \(0.483729\pi\)
\(992\) 0 0
\(993\) 36.3810 1.15452
\(994\) 0 0
\(995\) 77.6372 2.46126
\(996\) 0 0
\(997\) −37.7255 −1.19478 −0.597390 0.801951i \(-0.703796\pi\)
−0.597390 + 0.801951i \(0.703796\pi\)
\(998\) 0 0
\(999\) −2.93026 −0.0927094
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 4008.2.a.f.1.1 5
4.3 odd 2 8016.2.a.s.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.f.1.1 5 1.1 even 1 trivial
8016.2.a.s.1.1 5 4.3 odd 2