Properties

Label 4004.2.a.k.1.10
Level $4004$
Weight $2$
Character 4004.1
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
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] = [4004,2,Mod(1,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, 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("4004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.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: \(31.9721009693\)
Analytic rank: \(0\)
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: \( x^{10} - x^{9} - 23x^{8} + 23x^{7} + 170x^{6} - 165x^{5} - 411x^{4} + 360x^{3} + 111x^{2} - 48x - 12 \) 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 1.10
Root \(3.12022\) of defining polynomial
Character \(\chi\) \(=\) 4004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.12022 q^{3} -0.246641 q^{5} -1.00000 q^{7} +6.73575 q^{9} +O(q^{10})\) \(q+3.12022 q^{3} -0.246641 q^{5} -1.00000 q^{7} +6.73575 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.769574 q^{15} -1.45111 q^{17} +3.30252 q^{19} -3.12022 q^{21} +8.80480 q^{23} -4.93917 q^{25} +11.6563 q^{27} -8.78276 q^{29} +8.87495 q^{31} +3.12022 q^{33} +0.246641 q^{35} -1.62416 q^{37} +3.12022 q^{39} -4.42584 q^{41} +8.05794 q^{43} -1.66131 q^{45} +8.80480 q^{47} +1.00000 q^{49} -4.52777 q^{51} -5.99343 q^{53} -0.246641 q^{55} +10.3046 q^{57} +1.90780 q^{59} +3.06693 q^{61} -6.73575 q^{63} -0.246641 q^{65} -0.264888 q^{67} +27.4729 q^{69} +2.08119 q^{71} +7.43101 q^{73} -15.4113 q^{75} -1.00000 q^{77} -11.2638 q^{79} +16.1630 q^{81} +14.9492 q^{83} +0.357903 q^{85} -27.4041 q^{87} -11.9320 q^{89} -1.00000 q^{91} +27.6918 q^{93} -0.814538 q^{95} +18.9656 q^{97} +6.73575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9} + 10 q^{11} + 10 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{19} - q^{21} + 4 q^{23} + 22 q^{25} - 5 q^{27} + 10 q^{29} - q^{31} + q^{33} - 4 q^{35} + 20 q^{37} + q^{39} + 8 q^{45} + 4 q^{47} + 10 q^{49} + 11 q^{51} - 5 q^{53} + 4 q^{55} + 16 q^{57} + 11 q^{59} + 12 q^{61} - 17 q^{63} + 4 q^{65} - 2 q^{67} + 10 q^{69} + 28 q^{71} + 11 q^{73} - 6 q^{75} - 10 q^{77} - 10 q^{79} + 46 q^{81} + 7 q^{83} + 33 q^{85} - 47 q^{87} + 30 q^{89} - 10 q^{91} + 41 q^{93} - 2 q^{95} + 55 q^{97} + 17 q^{99}+O(q^{100}) \) 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
\(3\) 3.12022 1.80146 0.900729 0.434382i \(-0.143033\pi\)
0.900729 + 0.434382i \(0.143033\pi\)
\(4\) 0 0
\(5\) −0.246641 −0.110301 −0.0551507 0.998478i \(-0.517564\pi\)
−0.0551507 + 0.998478i \(0.517564\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 6.73575 2.24525
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −0.769574 −0.198703
\(16\) 0 0
\(17\) −1.45111 −0.351945 −0.175973 0.984395i \(-0.556307\pi\)
−0.175973 + 0.984395i \(0.556307\pi\)
\(18\) 0 0
\(19\) 3.30252 0.757650 0.378825 0.925468i \(-0.376328\pi\)
0.378825 + 0.925468i \(0.376328\pi\)
\(20\) 0 0
\(21\) −3.12022 −0.680887
\(22\) 0 0
\(23\) 8.80480 1.83593 0.917963 0.396665i \(-0.129833\pi\)
0.917963 + 0.396665i \(0.129833\pi\)
\(24\) 0 0
\(25\) −4.93917 −0.987834
\(26\) 0 0
\(27\) 11.6563 2.24326
\(28\) 0 0
\(29\) −8.78276 −1.63092 −0.815458 0.578816i \(-0.803515\pi\)
−0.815458 + 0.578816i \(0.803515\pi\)
\(30\) 0 0
\(31\) 8.87495 1.59399 0.796994 0.603987i \(-0.206422\pi\)
0.796994 + 0.603987i \(0.206422\pi\)
\(32\) 0 0
\(33\) 3.12022 0.543160
\(34\) 0 0
\(35\) 0.246641 0.0416900
\(36\) 0 0
\(37\) −1.62416 −0.267011 −0.133506 0.991048i \(-0.542623\pi\)
−0.133506 + 0.991048i \(0.542623\pi\)
\(38\) 0 0
\(39\) 3.12022 0.499634
\(40\) 0 0
\(41\) −4.42584 −0.691199 −0.345600 0.938382i \(-0.612325\pi\)
−0.345600 + 0.938382i \(0.612325\pi\)
\(42\) 0 0
\(43\) 8.05794 1.22882 0.614412 0.788985i \(-0.289393\pi\)
0.614412 + 0.788985i \(0.289393\pi\)
\(44\) 0 0
\(45\) −1.66131 −0.247654
\(46\) 0 0
\(47\) 8.80480 1.28431 0.642156 0.766574i \(-0.278040\pi\)
0.642156 + 0.766574i \(0.278040\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.52777 −0.634014
\(52\) 0 0
\(53\) −5.99343 −0.823261 −0.411631 0.911351i \(-0.635041\pi\)
−0.411631 + 0.911351i \(0.635041\pi\)
\(54\) 0 0
\(55\) −0.246641 −0.0332571
\(56\) 0 0
\(57\) 10.3046 1.36487
\(58\) 0 0
\(59\) 1.90780 0.248375 0.124187 0.992259i \(-0.460368\pi\)
0.124187 + 0.992259i \(0.460368\pi\)
\(60\) 0 0
\(61\) 3.06693 0.392680 0.196340 0.980536i \(-0.437094\pi\)
0.196340 + 0.980536i \(0.437094\pi\)
\(62\) 0 0
\(63\) −6.73575 −0.848624
\(64\) 0 0
\(65\) −0.246641 −0.0305921
\(66\) 0 0
\(67\) −0.264888 −0.0323612 −0.0161806 0.999869i \(-0.505151\pi\)
−0.0161806 + 0.999869i \(0.505151\pi\)
\(68\) 0 0
\(69\) 27.4729 3.30734
\(70\) 0 0
\(71\) 2.08119 0.246992 0.123496 0.992345i \(-0.460589\pi\)
0.123496 + 0.992345i \(0.460589\pi\)
\(72\) 0 0
\(73\) 7.43101 0.869733 0.434867 0.900495i \(-0.356795\pi\)
0.434867 + 0.900495i \(0.356795\pi\)
\(74\) 0 0
\(75\) −15.4113 −1.77954
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −11.2638 −1.26728 −0.633638 0.773630i \(-0.718439\pi\)
−0.633638 + 0.773630i \(0.718439\pi\)
\(80\) 0 0
\(81\) 16.1630 1.79589
\(82\) 0 0
\(83\) 14.9492 1.64088 0.820442 0.571729i \(-0.193727\pi\)
0.820442 + 0.571729i \(0.193727\pi\)
\(84\) 0 0
\(85\) 0.357903 0.0388200
\(86\) 0 0
\(87\) −27.4041 −2.93803
\(88\) 0 0
\(89\) −11.9320 −1.26479 −0.632396 0.774645i \(-0.717929\pi\)
−0.632396 + 0.774645i \(0.717929\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 27.6918 2.87150
\(94\) 0 0
\(95\) −0.814538 −0.0835698
\(96\) 0 0
\(97\) 18.9656 1.92567 0.962834 0.270093i \(-0.0870545\pi\)
0.962834 + 0.270093i \(0.0870545\pi\)
\(98\) 0 0
\(99\) 6.73575 0.676968
\(100\) 0 0
\(101\) −13.5289 −1.34617 −0.673086 0.739564i \(-0.735032\pi\)
−0.673086 + 0.739564i \(0.735032\pi\)
\(102\) 0 0
\(103\) −3.20200 −0.315502 −0.157751 0.987479i \(-0.550424\pi\)
−0.157751 + 0.987479i \(0.550424\pi\)
\(104\) 0 0
\(105\) 0.769574 0.0751027
\(106\) 0 0
\(107\) 20.0200 1.93541 0.967704 0.252089i \(-0.0811176\pi\)
0.967704 + 0.252089i \(0.0811176\pi\)
\(108\) 0 0
\(109\) −6.98578 −0.669116 −0.334558 0.942375i \(-0.608587\pi\)
−0.334558 + 0.942375i \(0.608587\pi\)
\(110\) 0 0
\(111\) −5.06774 −0.481009
\(112\) 0 0
\(113\) −17.9225 −1.68601 −0.843004 0.537908i \(-0.819215\pi\)
−0.843004 + 0.537908i \(0.819215\pi\)
\(114\) 0 0
\(115\) −2.17163 −0.202505
\(116\) 0 0
\(117\) 6.73575 0.622720
\(118\) 0 0
\(119\) 1.45111 0.133023
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −13.8096 −1.24517
\(124\) 0 0
\(125\) 2.45141 0.219261
\(126\) 0 0
\(127\) −10.3375 −0.917307 −0.458654 0.888615i \(-0.651668\pi\)
−0.458654 + 0.888615i \(0.651668\pi\)
\(128\) 0 0
\(129\) 25.1425 2.21367
\(130\) 0 0
\(131\) −11.1582 −0.974894 −0.487447 0.873153i \(-0.662072\pi\)
−0.487447 + 0.873153i \(0.662072\pi\)
\(132\) 0 0
\(133\) −3.30252 −0.286365
\(134\) 0 0
\(135\) −2.87493 −0.247435
\(136\) 0 0
\(137\) 15.8367 1.35302 0.676512 0.736432i \(-0.263491\pi\)
0.676512 + 0.736432i \(0.263491\pi\)
\(138\) 0 0
\(139\) −11.2076 −0.950619 −0.475309 0.879819i \(-0.657664\pi\)
−0.475309 + 0.879819i \(0.657664\pi\)
\(140\) 0 0
\(141\) 27.4729 2.31363
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 2.16619 0.179892
\(146\) 0 0
\(147\) 3.12022 0.257351
\(148\) 0 0
\(149\) 16.9143 1.38567 0.692835 0.721096i \(-0.256361\pi\)
0.692835 + 0.721096i \(0.256361\pi\)
\(150\) 0 0
\(151\) −12.4047 −1.00948 −0.504739 0.863272i \(-0.668411\pi\)
−0.504739 + 0.863272i \(0.668411\pi\)
\(152\) 0 0
\(153\) −9.77429 −0.790204
\(154\) 0 0
\(155\) −2.18893 −0.175819
\(156\) 0 0
\(157\) −6.72448 −0.536672 −0.268336 0.963325i \(-0.586474\pi\)
−0.268336 + 0.963325i \(0.586474\pi\)
\(158\) 0 0
\(159\) −18.7008 −1.48307
\(160\) 0 0
\(161\) −8.80480 −0.693915
\(162\) 0 0
\(163\) 13.3375 1.04467 0.522337 0.852739i \(-0.325060\pi\)
0.522337 + 0.852739i \(0.325060\pi\)
\(164\) 0 0
\(165\) −0.769574 −0.0599112
\(166\) 0 0
\(167\) 0.484770 0.0375127 0.0187563 0.999824i \(-0.494029\pi\)
0.0187563 + 0.999824i \(0.494029\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 22.2449 1.70111
\(172\) 0 0
\(173\) 6.40825 0.487210 0.243605 0.969874i \(-0.421670\pi\)
0.243605 + 0.969874i \(0.421670\pi\)
\(174\) 0 0
\(175\) 4.93917 0.373366
\(176\) 0 0
\(177\) 5.95276 0.447437
\(178\) 0 0
\(179\) 12.9600 0.968676 0.484338 0.874881i \(-0.339060\pi\)
0.484338 + 0.874881i \(0.339060\pi\)
\(180\) 0 0
\(181\) 2.52546 0.187716 0.0938578 0.995586i \(-0.470080\pi\)
0.0938578 + 0.995586i \(0.470080\pi\)
\(182\) 0 0
\(183\) 9.56947 0.707396
\(184\) 0 0
\(185\) 0.400586 0.0294517
\(186\) 0 0
\(187\) −1.45111 −0.106115
\(188\) 0 0
\(189\) −11.6563 −0.847873
\(190\) 0 0
\(191\) −18.8302 −1.36251 −0.681253 0.732048i \(-0.738565\pi\)
−0.681253 + 0.732048i \(0.738565\pi\)
\(192\) 0 0
\(193\) −1.79522 −0.129223 −0.0646114 0.997910i \(-0.520581\pi\)
−0.0646114 + 0.997910i \(0.520581\pi\)
\(194\) 0 0
\(195\) −0.769574 −0.0551103
\(196\) 0 0
\(197\) −12.6981 −0.904702 −0.452351 0.891840i \(-0.649415\pi\)
−0.452351 + 0.891840i \(0.649415\pi\)
\(198\) 0 0
\(199\) 6.21430 0.440520 0.220260 0.975441i \(-0.429309\pi\)
0.220260 + 0.975441i \(0.429309\pi\)
\(200\) 0 0
\(201\) −0.826507 −0.0582973
\(202\) 0 0
\(203\) 8.78276 0.616429
\(204\) 0 0
\(205\) 1.09159 0.0762402
\(206\) 0 0
\(207\) 59.3069 4.12211
\(208\) 0 0
\(209\) 3.30252 0.228440
\(210\) 0 0
\(211\) 14.4466 0.994548 0.497274 0.867593i \(-0.334334\pi\)
0.497274 + 0.867593i \(0.334334\pi\)
\(212\) 0 0
\(213\) 6.49377 0.444946
\(214\) 0 0
\(215\) −1.98742 −0.135541
\(216\) 0 0
\(217\) −8.87495 −0.602471
\(218\) 0 0
\(219\) 23.1863 1.56679
\(220\) 0 0
\(221\) −1.45111 −0.0976120
\(222\) 0 0
\(223\) 8.34190 0.558615 0.279307 0.960202i \(-0.409895\pi\)
0.279307 + 0.960202i \(0.409895\pi\)
\(224\) 0 0
\(225\) −33.2690 −2.21793
\(226\) 0 0
\(227\) −20.8094 −1.38117 −0.690585 0.723251i \(-0.742647\pi\)
−0.690585 + 0.723251i \(0.742647\pi\)
\(228\) 0 0
\(229\) −9.94054 −0.656889 −0.328445 0.944523i \(-0.606524\pi\)
−0.328445 + 0.944523i \(0.606524\pi\)
\(230\) 0 0
\(231\) −3.12022 −0.205295
\(232\) 0 0
\(233\) −1.34684 −0.0882344 −0.0441172 0.999026i \(-0.514047\pi\)
−0.0441172 + 0.999026i \(0.514047\pi\)
\(234\) 0 0
\(235\) −2.17163 −0.141661
\(236\) 0 0
\(237\) −35.1455 −2.28294
\(238\) 0 0
\(239\) 15.0818 0.975558 0.487779 0.872967i \(-0.337807\pi\)
0.487779 + 0.872967i \(0.337807\pi\)
\(240\) 0 0
\(241\) −14.4514 −0.930895 −0.465448 0.885075i \(-0.654107\pi\)
−0.465448 + 0.885075i \(0.654107\pi\)
\(242\) 0 0
\(243\) 15.4631 0.991961
\(244\) 0 0
\(245\) −0.246641 −0.0157573
\(246\) 0 0
\(247\) 3.30252 0.210134
\(248\) 0 0
\(249\) 46.6446 2.95598
\(250\) 0 0
\(251\) −3.30327 −0.208500 −0.104250 0.994551i \(-0.533244\pi\)
−0.104250 + 0.994551i \(0.533244\pi\)
\(252\) 0 0
\(253\) 8.80480 0.553553
\(254\) 0 0
\(255\) 1.11673 0.0699326
\(256\) 0 0
\(257\) 30.8498 1.92436 0.962179 0.272417i \(-0.0878230\pi\)
0.962179 + 0.272417i \(0.0878230\pi\)
\(258\) 0 0
\(259\) 1.62416 0.100921
\(260\) 0 0
\(261\) −59.1584 −3.66181
\(262\) 0 0
\(263\) 4.90673 0.302562 0.151281 0.988491i \(-0.451660\pi\)
0.151281 + 0.988491i \(0.451660\pi\)
\(264\) 0 0
\(265\) 1.47823 0.0908068
\(266\) 0 0
\(267\) −37.2305 −2.27847
\(268\) 0 0
\(269\) −12.8566 −0.783881 −0.391940 0.919991i \(-0.628196\pi\)
−0.391940 + 0.919991i \(0.628196\pi\)
\(270\) 0 0
\(271\) −18.4601 −1.12137 −0.560686 0.828029i \(-0.689462\pi\)
−0.560686 + 0.828029i \(0.689462\pi\)
\(272\) 0 0
\(273\) −3.12022 −0.188844
\(274\) 0 0
\(275\) −4.93917 −0.297843
\(276\) 0 0
\(277\) −2.53780 −0.152482 −0.0762409 0.997089i \(-0.524292\pi\)
−0.0762409 + 0.997089i \(0.524292\pi\)
\(278\) 0 0
\(279\) 59.7794 3.57890
\(280\) 0 0
\(281\) −18.4565 −1.10102 −0.550511 0.834828i \(-0.685567\pi\)
−0.550511 + 0.834828i \(0.685567\pi\)
\(282\) 0 0
\(283\) 27.9169 1.65949 0.829745 0.558143i \(-0.188486\pi\)
0.829745 + 0.558143i \(0.188486\pi\)
\(284\) 0 0
\(285\) −2.54153 −0.150548
\(286\) 0 0
\(287\) 4.42584 0.261249
\(288\) 0 0
\(289\) −14.8943 −0.876135
\(290\) 0 0
\(291\) 59.1769 3.46901
\(292\) 0 0
\(293\) −24.2614 −1.41736 −0.708682 0.705528i \(-0.750710\pi\)
−0.708682 + 0.705528i \(0.750710\pi\)
\(294\) 0 0
\(295\) −0.470543 −0.0273961
\(296\) 0 0
\(297\) 11.6563 0.676369
\(298\) 0 0
\(299\) 8.80480 0.509194
\(300\) 0 0
\(301\) −8.05794 −0.464452
\(302\) 0 0
\(303\) −42.2130 −2.42507
\(304\) 0 0
\(305\) −0.756430 −0.0433131
\(306\) 0 0
\(307\) 4.88269 0.278670 0.139335 0.990245i \(-0.455504\pi\)
0.139335 + 0.990245i \(0.455504\pi\)
\(308\) 0 0
\(309\) −9.99093 −0.568364
\(310\) 0 0
\(311\) −27.9849 −1.58688 −0.793439 0.608650i \(-0.791712\pi\)
−0.793439 + 0.608650i \(0.791712\pi\)
\(312\) 0 0
\(313\) 19.1400 1.08186 0.540928 0.841069i \(-0.318073\pi\)
0.540928 + 0.841069i \(0.318073\pi\)
\(314\) 0 0
\(315\) 1.66131 0.0936044
\(316\) 0 0
\(317\) 3.21011 0.180298 0.0901488 0.995928i \(-0.471266\pi\)
0.0901488 + 0.995928i \(0.471266\pi\)
\(318\) 0 0
\(319\) −8.78276 −0.491740
\(320\) 0 0
\(321\) 62.4668 3.48655
\(322\) 0 0
\(323\) −4.79231 −0.266651
\(324\) 0 0
\(325\) −4.93917 −0.273976
\(326\) 0 0
\(327\) −21.7971 −1.20538
\(328\) 0 0
\(329\) −8.80480 −0.485424
\(330\) 0 0
\(331\) −30.0455 −1.65145 −0.825726 0.564072i \(-0.809234\pi\)
−0.825726 + 0.564072i \(0.809234\pi\)
\(332\) 0 0
\(333\) −10.9400 −0.599506
\(334\) 0 0
\(335\) 0.0653322 0.00356948
\(336\) 0 0
\(337\) −3.09963 −0.168848 −0.0844239 0.996430i \(-0.526905\pi\)
−0.0844239 + 0.996430i \(0.526905\pi\)
\(338\) 0 0
\(339\) −55.9221 −3.03727
\(340\) 0 0
\(341\) 8.87495 0.480606
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −6.77594 −0.364804
\(346\) 0 0
\(347\) 4.55586 0.244571 0.122286 0.992495i \(-0.460978\pi\)
0.122286 + 0.992495i \(0.460978\pi\)
\(348\) 0 0
\(349\) −7.72763 −0.413650 −0.206825 0.978378i \(-0.566313\pi\)
−0.206825 + 0.978378i \(0.566313\pi\)
\(350\) 0 0
\(351\) 11.6563 0.622169
\(352\) 0 0
\(353\) −12.2886 −0.654058 −0.327029 0.945014i \(-0.606047\pi\)
−0.327029 + 0.945014i \(0.606047\pi\)
\(354\) 0 0
\(355\) −0.513308 −0.0272436
\(356\) 0 0
\(357\) 4.52777 0.239635
\(358\) 0 0
\(359\) −26.6415 −1.40609 −0.703043 0.711148i \(-0.748176\pi\)
−0.703043 + 0.711148i \(0.748176\pi\)
\(360\) 0 0
\(361\) −8.09335 −0.425966
\(362\) 0 0
\(363\) 3.12022 0.163769
\(364\) 0 0
\(365\) −1.83279 −0.0959327
\(366\) 0 0
\(367\) −30.6145 −1.59807 −0.799033 0.601287i \(-0.794655\pi\)
−0.799033 + 0.601287i \(0.794655\pi\)
\(368\) 0 0
\(369\) −29.8113 −1.55191
\(370\) 0 0
\(371\) 5.99343 0.311163
\(372\) 0 0
\(373\) −22.7859 −1.17981 −0.589904 0.807473i \(-0.700834\pi\)
−0.589904 + 0.807473i \(0.700834\pi\)
\(374\) 0 0
\(375\) 7.64892 0.394989
\(376\) 0 0
\(377\) −8.78276 −0.452335
\(378\) 0 0
\(379\) 23.8546 1.22533 0.612665 0.790343i \(-0.290097\pi\)
0.612665 + 0.790343i \(0.290097\pi\)
\(380\) 0 0
\(381\) −32.2553 −1.65249
\(382\) 0 0
\(383\) −35.8150 −1.83006 −0.915032 0.403382i \(-0.867834\pi\)
−0.915032 + 0.403382i \(0.867834\pi\)
\(384\) 0 0
\(385\) 0.246641 0.0125700
\(386\) 0 0
\(387\) 54.2762 2.75902
\(388\) 0 0
\(389\) 13.1110 0.664753 0.332376 0.943147i \(-0.392150\pi\)
0.332376 + 0.943147i \(0.392150\pi\)
\(390\) 0 0
\(391\) −12.7767 −0.646146
\(392\) 0 0
\(393\) −34.8159 −1.75623
\(394\) 0 0
\(395\) 2.77812 0.139782
\(396\) 0 0
\(397\) −14.1320 −0.709262 −0.354631 0.935006i \(-0.615394\pi\)
−0.354631 + 0.935006i \(0.615394\pi\)
\(398\) 0 0
\(399\) −10.3046 −0.515874
\(400\) 0 0
\(401\) 17.1911 0.858482 0.429241 0.903190i \(-0.358781\pi\)
0.429241 + 0.903190i \(0.358781\pi\)
\(402\) 0 0
\(403\) 8.87495 0.442093
\(404\) 0 0
\(405\) −3.98647 −0.198089
\(406\) 0 0
\(407\) −1.62416 −0.0805069
\(408\) 0 0
\(409\) 37.6818 1.86324 0.931622 0.363429i \(-0.118394\pi\)
0.931622 + 0.363429i \(0.118394\pi\)
\(410\) 0 0
\(411\) 49.4140 2.43741
\(412\) 0 0
\(413\) −1.90780 −0.0938768
\(414\) 0 0
\(415\) −3.68708 −0.180992
\(416\) 0 0
\(417\) −34.9702 −1.71250
\(418\) 0 0
\(419\) −29.2049 −1.42675 −0.713376 0.700781i \(-0.752835\pi\)
−0.713376 + 0.700781i \(0.752835\pi\)
\(420\) 0 0
\(421\) −9.76379 −0.475858 −0.237929 0.971283i \(-0.576469\pi\)
−0.237929 + 0.971283i \(0.576469\pi\)
\(422\) 0 0
\(423\) 59.3069 2.88360
\(424\) 0 0
\(425\) 7.16726 0.347663
\(426\) 0 0
\(427\) −3.06693 −0.148419
\(428\) 0 0
\(429\) 3.12022 0.150645
\(430\) 0 0
\(431\) −8.80548 −0.424145 −0.212073 0.977254i \(-0.568021\pi\)
−0.212073 + 0.977254i \(0.568021\pi\)
\(432\) 0 0
\(433\) −7.63506 −0.366918 −0.183459 0.983027i \(-0.558729\pi\)
−0.183459 + 0.983027i \(0.558729\pi\)
\(434\) 0 0
\(435\) 6.75898 0.324068
\(436\) 0 0
\(437\) 29.0780 1.39099
\(438\) 0 0
\(439\) 26.1844 1.24971 0.624857 0.780739i \(-0.285157\pi\)
0.624857 + 0.780739i \(0.285157\pi\)
\(440\) 0 0
\(441\) 6.73575 0.320750
\(442\) 0 0
\(443\) −4.72069 −0.224287 −0.112143 0.993692i \(-0.535772\pi\)
−0.112143 + 0.993692i \(0.535772\pi\)
\(444\) 0 0
\(445\) 2.94293 0.139508
\(446\) 0 0
\(447\) 52.7761 2.49623
\(448\) 0 0
\(449\) −29.0257 −1.36981 −0.684903 0.728634i \(-0.740155\pi\)
−0.684903 + 0.728634i \(0.740155\pi\)
\(450\) 0 0
\(451\) −4.42584 −0.208404
\(452\) 0 0
\(453\) −38.7052 −1.81853
\(454\) 0 0
\(455\) 0.246641 0.0115627
\(456\) 0 0
\(457\) −29.7749 −1.39281 −0.696406 0.717648i \(-0.745219\pi\)
−0.696406 + 0.717648i \(0.745219\pi\)
\(458\) 0 0
\(459\) −16.9146 −0.789505
\(460\) 0 0
\(461\) 0.0121450 0.000565650 0 0.000282825 1.00000i \(-0.499910\pi\)
0.000282825 1.00000i \(0.499910\pi\)
\(462\) 0 0
\(463\) 8.37817 0.389366 0.194683 0.980866i \(-0.437632\pi\)
0.194683 + 0.980866i \(0.437632\pi\)
\(464\) 0 0
\(465\) −6.82993 −0.316730
\(466\) 0 0
\(467\) −30.6991 −1.42058 −0.710292 0.703907i \(-0.751437\pi\)
−0.710292 + 0.703907i \(0.751437\pi\)
\(468\) 0 0
\(469\) 0.264888 0.0122314
\(470\) 0 0
\(471\) −20.9818 −0.966792
\(472\) 0 0
\(473\) 8.05794 0.370504
\(474\) 0 0
\(475\) −16.3117 −0.748433
\(476\) 0 0
\(477\) −40.3702 −1.84843
\(478\) 0 0
\(479\) 24.3377 1.11202 0.556008 0.831177i \(-0.312332\pi\)
0.556008 + 0.831177i \(0.312332\pi\)
\(480\) 0 0
\(481\) −1.62416 −0.0740555
\(482\) 0 0
\(483\) −27.4729 −1.25006
\(484\) 0 0
\(485\) −4.67771 −0.212404
\(486\) 0 0
\(487\) 16.7875 0.760716 0.380358 0.924839i \(-0.375801\pi\)
0.380358 + 0.924839i \(0.375801\pi\)
\(488\) 0 0
\(489\) 41.6159 1.88194
\(490\) 0 0
\(491\) −31.4680 −1.42013 −0.710066 0.704135i \(-0.751335\pi\)
−0.710066 + 0.704135i \(0.751335\pi\)
\(492\) 0 0
\(493\) 12.7447 0.573993
\(494\) 0 0
\(495\) −1.66131 −0.0746704
\(496\) 0 0
\(497\) −2.08119 −0.0933543
\(498\) 0 0
\(499\) 24.5021 1.09686 0.548432 0.836195i \(-0.315225\pi\)
0.548432 + 0.836195i \(0.315225\pi\)
\(500\) 0 0
\(501\) 1.51259 0.0675774
\(502\) 0 0
\(503\) −26.2399 −1.16998 −0.584990 0.811041i \(-0.698901\pi\)
−0.584990 + 0.811041i \(0.698901\pi\)
\(504\) 0 0
\(505\) 3.33677 0.148484
\(506\) 0 0
\(507\) 3.12022 0.138574
\(508\) 0 0
\(509\) −5.58940 −0.247746 −0.123873 0.992298i \(-0.539532\pi\)
−0.123873 + 0.992298i \(0.539532\pi\)
\(510\) 0 0
\(511\) −7.43101 −0.328728
\(512\) 0 0
\(513\) 38.4953 1.69961
\(514\) 0 0
\(515\) 0.789745 0.0348003
\(516\) 0 0
\(517\) 8.80480 0.387234
\(518\) 0 0
\(519\) 19.9951 0.877689
\(520\) 0 0
\(521\) 15.0305 0.658497 0.329249 0.944243i \(-0.393205\pi\)
0.329249 + 0.944243i \(0.393205\pi\)
\(522\) 0 0
\(523\) −1.35702 −0.0593384 −0.0296692 0.999560i \(-0.509445\pi\)
−0.0296692 + 0.999560i \(0.509445\pi\)
\(524\) 0 0
\(525\) 15.4113 0.672603
\(526\) 0 0
\(527\) −12.8785 −0.560997
\(528\) 0 0
\(529\) 54.5244 2.37063
\(530\) 0 0
\(531\) 12.8505 0.557663
\(532\) 0 0
\(533\) −4.42584 −0.191704
\(534\) 0 0
\(535\) −4.93776 −0.213478
\(536\) 0 0
\(537\) 40.4380 1.74503
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −30.2691 −1.30137 −0.650685 0.759348i \(-0.725518\pi\)
−0.650685 + 0.759348i \(0.725518\pi\)
\(542\) 0 0
\(543\) 7.87996 0.338162
\(544\) 0 0
\(545\) 1.72298 0.0738044
\(546\) 0 0
\(547\) −17.6661 −0.755349 −0.377674 0.925938i \(-0.623276\pi\)
−0.377674 + 0.925938i \(0.623276\pi\)
\(548\) 0 0
\(549\) 20.6580 0.881663
\(550\) 0 0
\(551\) −29.0052 −1.23566
\(552\) 0 0
\(553\) 11.2638 0.478985
\(554\) 0 0
\(555\) 1.24991 0.0530559
\(556\) 0 0
\(557\) −9.11094 −0.386043 −0.193022 0.981195i \(-0.561829\pi\)
−0.193022 + 0.981195i \(0.561829\pi\)
\(558\) 0 0
\(559\) 8.05794 0.340815
\(560\) 0 0
\(561\) −4.52777 −0.191163
\(562\) 0 0
\(563\) 41.2123 1.73689 0.868446 0.495783i \(-0.165119\pi\)
0.868446 + 0.495783i \(0.165119\pi\)
\(564\) 0 0
\(565\) 4.42043 0.185969
\(566\) 0 0
\(567\) −16.1630 −0.678783
\(568\) 0 0
\(569\) 3.26404 0.136836 0.0684178 0.997657i \(-0.478205\pi\)
0.0684178 + 0.997657i \(0.478205\pi\)
\(570\) 0 0
\(571\) 1.17732 0.0492695 0.0246347 0.999697i \(-0.492158\pi\)
0.0246347 + 0.999697i \(0.492158\pi\)
\(572\) 0 0
\(573\) −58.7543 −2.45450
\(574\) 0 0
\(575\) −43.4884 −1.81359
\(576\) 0 0
\(577\) 1.81878 0.0757166 0.0378583 0.999283i \(-0.487946\pi\)
0.0378583 + 0.999283i \(0.487946\pi\)
\(578\) 0 0
\(579\) −5.60148 −0.232789
\(580\) 0 0
\(581\) −14.9492 −0.620196
\(582\) 0 0
\(583\) −5.99343 −0.248223
\(584\) 0 0
\(585\) −1.66131 −0.0686868
\(586\) 0 0
\(587\) 4.48078 0.184942 0.0924709 0.995715i \(-0.470524\pi\)
0.0924709 + 0.995715i \(0.470524\pi\)
\(588\) 0 0
\(589\) 29.3097 1.20769
\(590\) 0 0
\(591\) −39.6208 −1.62978
\(592\) 0 0
\(593\) 5.36331 0.220245 0.110122 0.993918i \(-0.464876\pi\)
0.110122 + 0.993918i \(0.464876\pi\)
\(594\) 0 0
\(595\) −0.357903 −0.0146726
\(596\) 0 0
\(597\) 19.3900 0.793579
\(598\) 0 0
\(599\) 23.7972 0.972326 0.486163 0.873868i \(-0.338396\pi\)
0.486163 + 0.873868i \(0.338396\pi\)
\(600\) 0 0
\(601\) 39.4756 1.61024 0.805122 0.593110i \(-0.202100\pi\)
0.805122 + 0.593110i \(0.202100\pi\)
\(602\) 0 0
\(603\) −1.78422 −0.0726589
\(604\) 0 0
\(605\) −0.246641 −0.0100274
\(606\) 0 0
\(607\) −19.1987 −0.779250 −0.389625 0.920974i \(-0.627395\pi\)
−0.389625 + 0.920974i \(0.627395\pi\)
\(608\) 0 0
\(609\) 27.4041 1.11047
\(610\) 0 0
\(611\) 8.80480 0.356204
\(612\) 0 0
\(613\) 47.4147 1.91506 0.957531 0.288332i \(-0.0931006\pi\)
0.957531 + 0.288332i \(0.0931006\pi\)
\(614\) 0 0
\(615\) 3.40601 0.137343
\(616\) 0 0
\(617\) −14.1718 −0.570536 −0.285268 0.958448i \(-0.592083\pi\)
−0.285268 + 0.958448i \(0.592083\pi\)
\(618\) 0 0
\(619\) 44.2115 1.77701 0.888504 0.458869i \(-0.151745\pi\)
0.888504 + 0.458869i \(0.151745\pi\)
\(620\) 0 0
\(621\) 102.632 4.11846
\(622\) 0 0
\(623\) 11.9320 0.478047
\(624\) 0 0
\(625\) 24.0912 0.963649
\(626\) 0 0
\(627\) 10.3046 0.411525
\(628\) 0 0
\(629\) 2.35684 0.0939733
\(630\) 0 0
\(631\) −22.9981 −0.915539 −0.457769 0.889071i \(-0.651351\pi\)
−0.457769 + 0.889071i \(0.651351\pi\)
\(632\) 0 0
\(633\) 45.0767 1.79164
\(634\) 0 0
\(635\) 2.54966 0.101180
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 14.0184 0.554559
\(640\) 0 0
\(641\) −43.7292 −1.72720 −0.863600 0.504178i \(-0.831796\pi\)
−0.863600 + 0.504178i \(0.831796\pi\)
\(642\) 0 0
\(643\) −6.02229 −0.237496 −0.118748 0.992924i \(-0.537888\pi\)
−0.118748 + 0.992924i \(0.537888\pi\)
\(644\) 0 0
\(645\) −6.20118 −0.244171
\(646\) 0 0
\(647\) −14.7985 −0.581789 −0.290895 0.956755i \(-0.593953\pi\)
−0.290895 + 0.956755i \(0.593953\pi\)
\(648\) 0 0
\(649\) 1.90780 0.0748878
\(650\) 0 0
\(651\) −27.6918 −1.08533
\(652\) 0 0
\(653\) −2.56561 −0.100400 −0.0502000 0.998739i \(-0.515986\pi\)
−0.0502000 + 0.998739i \(0.515986\pi\)
\(654\) 0 0
\(655\) 2.75207 0.107532
\(656\) 0 0
\(657\) 50.0534 1.95277
\(658\) 0 0
\(659\) 21.4914 0.837184 0.418592 0.908174i \(-0.362524\pi\)
0.418592 + 0.908174i \(0.362524\pi\)
\(660\) 0 0
\(661\) 45.3383 1.76346 0.881728 0.471759i \(-0.156381\pi\)
0.881728 + 0.471759i \(0.156381\pi\)
\(662\) 0 0
\(663\) −4.52777 −0.175844
\(664\) 0 0
\(665\) 0.814538 0.0315864
\(666\) 0 0
\(667\) −77.3304 −2.99424
\(668\) 0 0
\(669\) 26.0285 1.00632
\(670\) 0 0
\(671\) 3.06693 0.118397
\(672\) 0 0
\(673\) −42.1678 −1.62545 −0.812724 0.582649i \(-0.802016\pi\)
−0.812724 + 0.582649i \(0.802016\pi\)
\(674\) 0 0
\(675\) −57.5726 −2.21597
\(676\) 0 0
\(677\) −3.53077 −0.135698 −0.0678492 0.997696i \(-0.521614\pi\)
−0.0678492 + 0.997696i \(0.521614\pi\)
\(678\) 0 0
\(679\) −18.9656 −0.727834
\(680\) 0 0
\(681\) −64.9299 −2.48812
\(682\) 0 0
\(683\) 36.2139 1.38569 0.692844 0.721088i \(-0.256357\pi\)
0.692844 + 0.721088i \(0.256357\pi\)
\(684\) 0 0
\(685\) −3.90599 −0.149240
\(686\) 0 0
\(687\) −31.0166 −1.18336
\(688\) 0 0
\(689\) −5.99343 −0.228332
\(690\) 0 0
\(691\) 13.2732 0.504937 0.252469 0.967605i \(-0.418758\pi\)
0.252469 + 0.967605i \(0.418758\pi\)
\(692\) 0 0
\(693\) −6.73575 −0.255870
\(694\) 0 0
\(695\) 2.76426 0.104855
\(696\) 0 0
\(697\) 6.42236 0.243264
\(698\) 0 0
\(699\) −4.20243 −0.158950
\(700\) 0 0
\(701\) −17.4283 −0.658256 −0.329128 0.944285i \(-0.606755\pi\)
−0.329128 + 0.944285i \(0.606755\pi\)
\(702\) 0 0
\(703\) −5.36384 −0.202301
\(704\) 0 0
\(705\) −6.77594 −0.255197
\(706\) 0 0
\(707\) 13.5289 0.508805
\(708\) 0 0
\(709\) 30.8523 1.15868 0.579341 0.815085i \(-0.303310\pi\)
0.579341 + 0.815085i \(0.303310\pi\)
\(710\) 0 0
\(711\) −75.8700 −2.84535
\(712\) 0 0
\(713\) 78.1421 2.92645
\(714\) 0 0
\(715\) −0.246641 −0.00922386
\(716\) 0 0
\(717\) 47.0583 1.75743
\(718\) 0 0
\(719\) 45.0551 1.68027 0.840136 0.542375i \(-0.182475\pi\)
0.840136 + 0.542375i \(0.182475\pi\)
\(720\) 0 0
\(721\) 3.20200 0.119249
\(722\) 0 0
\(723\) −45.0914 −1.67697
\(724\) 0 0
\(725\) 43.3795 1.61107
\(726\) 0 0
\(727\) 15.8003 0.586001 0.293000 0.956112i \(-0.405346\pi\)
0.293000 + 0.956112i \(0.405346\pi\)
\(728\) 0 0
\(729\) −0.240761 −0.00891709
\(730\) 0 0
\(731\) −11.6929 −0.432479
\(732\) 0 0
\(733\) −20.1625 −0.744720 −0.372360 0.928088i \(-0.621451\pi\)
−0.372360 + 0.928088i \(0.621451\pi\)
\(734\) 0 0
\(735\) −0.769574 −0.0283862
\(736\) 0 0
\(737\) −0.264888 −0.00975727
\(738\) 0 0
\(739\) 19.2538 0.708264 0.354132 0.935195i \(-0.384776\pi\)
0.354132 + 0.935195i \(0.384776\pi\)
\(740\) 0 0
\(741\) 10.3046 0.378548
\(742\) 0 0
\(743\) 11.9952 0.440062 0.220031 0.975493i \(-0.429384\pi\)
0.220031 + 0.975493i \(0.429384\pi\)
\(744\) 0 0
\(745\) −4.17175 −0.152841
\(746\) 0 0
\(747\) 100.694 3.68419
\(748\) 0 0
\(749\) −20.0200 −0.731515
\(750\) 0 0
\(751\) 44.8063 1.63500 0.817502 0.575926i \(-0.195358\pi\)
0.817502 + 0.575926i \(0.195358\pi\)
\(752\) 0 0
\(753\) −10.3069 −0.375605
\(754\) 0 0
\(755\) 3.05950 0.111347
\(756\) 0 0
\(757\) −32.4954 −1.18107 −0.590533 0.807013i \(-0.701082\pi\)
−0.590533 + 0.807013i \(0.701082\pi\)
\(758\) 0 0
\(759\) 27.4729 0.997202
\(760\) 0 0
\(761\) −25.8480 −0.936989 −0.468494 0.883466i \(-0.655203\pi\)
−0.468494 + 0.883466i \(0.655203\pi\)
\(762\) 0 0
\(763\) 6.98578 0.252902
\(764\) 0 0
\(765\) 2.41074 0.0871606
\(766\) 0 0
\(767\) 1.90780 0.0688868
\(768\) 0 0
\(769\) −6.44814 −0.232526 −0.116263 0.993218i \(-0.537092\pi\)
−0.116263 + 0.993218i \(0.537092\pi\)
\(770\) 0 0
\(771\) 96.2581 3.46665
\(772\) 0 0
\(773\) −48.9792 −1.76166 −0.880830 0.473433i \(-0.843014\pi\)
−0.880830 + 0.473433i \(0.843014\pi\)
\(774\) 0 0
\(775\) −43.8349 −1.57460
\(776\) 0 0
\(777\) 5.06774 0.181804
\(778\) 0 0
\(779\) −14.6164 −0.523688
\(780\) 0 0
\(781\) 2.08119 0.0744710
\(782\) 0 0
\(783\) −102.375 −3.65857
\(784\) 0 0
\(785\) 1.65853 0.0591956
\(786\) 0 0
\(787\) −45.0923 −1.60737 −0.803683 0.595058i \(-0.797129\pi\)
−0.803683 + 0.595058i \(0.797129\pi\)
\(788\) 0 0
\(789\) 15.3100 0.545052
\(790\) 0 0
\(791\) 17.9225 0.637251
\(792\) 0 0
\(793\) 3.06693 0.108910
\(794\) 0 0
\(795\) 4.61239 0.163585
\(796\) 0 0
\(797\) −17.3417 −0.614276 −0.307138 0.951665i \(-0.599371\pi\)
−0.307138 + 0.951665i \(0.599371\pi\)
\(798\) 0 0
\(799\) −12.7767 −0.452007
\(800\) 0 0
\(801\) −80.3711 −2.83977
\(802\) 0 0
\(803\) 7.43101 0.262235
\(804\) 0 0
\(805\) 2.17163 0.0765397
\(806\) 0 0
\(807\) −40.1154 −1.41213
\(808\) 0 0
\(809\) −27.8915 −0.980612 −0.490306 0.871550i \(-0.663115\pi\)
−0.490306 + 0.871550i \(0.663115\pi\)
\(810\) 0 0
\(811\) −46.3954 −1.62916 −0.814581 0.580049i \(-0.803033\pi\)
−0.814581 + 0.580049i \(0.803033\pi\)
\(812\) 0 0
\(813\) −57.5995 −2.02010
\(814\) 0 0
\(815\) −3.28958 −0.115229
\(816\) 0 0
\(817\) 26.6115 0.931019
\(818\) 0 0
\(819\) −6.73575 −0.235366
\(820\) 0 0
\(821\) 23.7836 0.830053 0.415026 0.909809i \(-0.363772\pi\)
0.415026 + 0.909809i \(0.363772\pi\)
\(822\) 0 0
\(823\) −7.35161 −0.256261 −0.128131 0.991757i \(-0.540898\pi\)
−0.128131 + 0.991757i \(0.540898\pi\)
\(824\) 0 0
\(825\) −15.4113 −0.536552
\(826\) 0 0
\(827\) −1.38289 −0.0480879 −0.0240440 0.999711i \(-0.507654\pi\)
−0.0240440 + 0.999711i \(0.507654\pi\)
\(828\) 0 0
\(829\) 6.20434 0.215486 0.107743 0.994179i \(-0.465638\pi\)
0.107743 + 0.994179i \(0.465638\pi\)
\(830\) 0 0
\(831\) −7.91850 −0.274690
\(832\) 0 0
\(833\) −1.45111 −0.0502779
\(834\) 0 0
\(835\) −0.119564 −0.00413770
\(836\) 0 0
\(837\) 103.449 3.57573
\(838\) 0 0
\(839\) 20.3007 0.700857 0.350429 0.936589i \(-0.386036\pi\)
0.350429 + 0.936589i \(0.386036\pi\)
\(840\) 0 0
\(841\) 48.1368 1.65989
\(842\) 0 0
\(843\) −57.5883 −1.98345
\(844\) 0 0
\(845\) −0.246641 −0.00848472
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 87.1069 2.98950
\(850\) 0 0
\(851\) −14.3004 −0.490213
\(852\) 0 0
\(853\) −8.89492 −0.304556 −0.152278 0.988338i \(-0.548661\pi\)
−0.152278 + 0.988338i \(0.548661\pi\)
\(854\) 0 0
\(855\) −5.48652 −0.187635
\(856\) 0 0
\(857\) −52.5296 −1.79438 −0.897188 0.441648i \(-0.854394\pi\)
−0.897188 + 0.441648i \(0.854394\pi\)
\(858\) 0 0
\(859\) 6.46267 0.220503 0.110252 0.993904i \(-0.464834\pi\)
0.110252 + 0.993904i \(0.464834\pi\)
\(860\) 0 0
\(861\) 13.8096 0.470629
\(862\) 0 0
\(863\) −3.95039 −0.134473 −0.0672365 0.997737i \(-0.521418\pi\)
−0.0672365 + 0.997737i \(0.521418\pi\)
\(864\) 0 0
\(865\) −1.58054 −0.0537399
\(866\) 0 0
\(867\) −46.4734 −1.57832
\(868\) 0 0
\(869\) −11.2638 −0.382098
\(870\) 0 0
\(871\) −0.264888 −0.00897538
\(872\) 0 0
\(873\) 127.748 4.32360
\(874\) 0 0
\(875\) −2.45141 −0.0828727
\(876\) 0 0
\(877\) −0.409047 −0.0138125 −0.00690626 0.999976i \(-0.502198\pi\)
−0.00690626 + 0.999976i \(0.502198\pi\)
\(878\) 0 0
\(879\) −75.7007 −2.55332
\(880\) 0 0
\(881\) −1.29247 −0.0435444 −0.0217722 0.999763i \(-0.506931\pi\)
−0.0217722 + 0.999763i \(0.506931\pi\)
\(882\) 0 0
\(883\) −45.7346 −1.53909 −0.769547 0.638591i \(-0.779518\pi\)
−0.769547 + 0.638591i \(0.779518\pi\)
\(884\) 0 0
\(885\) −1.46820 −0.0493528
\(886\) 0 0
\(887\) −19.6200 −0.658774 −0.329387 0.944195i \(-0.606842\pi\)
−0.329387 + 0.944195i \(0.606842\pi\)
\(888\) 0 0
\(889\) 10.3375 0.346710
\(890\) 0 0
\(891\) 16.1630 0.541482
\(892\) 0 0
\(893\) 29.0780 0.973059
\(894\) 0 0
\(895\) −3.19647 −0.106846
\(896\) 0 0
\(897\) 27.4729 0.917292
\(898\) 0 0
\(899\) −77.9465 −2.59966
\(900\) 0 0
\(901\) 8.69711 0.289743
\(902\) 0 0
\(903\) −25.1425 −0.836690
\(904\) 0 0
\(905\) −0.622881 −0.0207053
\(906\) 0 0
\(907\) 12.8559 0.426873 0.213437 0.976957i \(-0.431534\pi\)
0.213437 + 0.976957i \(0.431534\pi\)
\(908\) 0 0
\(909\) −91.1269 −3.02249
\(910\) 0 0
\(911\) 38.6200 1.27954 0.639769 0.768567i \(-0.279030\pi\)
0.639769 + 0.768567i \(0.279030\pi\)
\(912\) 0 0
\(913\) 14.9492 0.494745
\(914\) 0 0
\(915\) −2.36023 −0.0780267
\(916\) 0 0
\(917\) 11.1582 0.368475
\(918\) 0 0
\(919\) −12.0157 −0.396360 −0.198180 0.980166i \(-0.563503\pi\)
−0.198180 + 0.980166i \(0.563503\pi\)
\(920\) 0 0
\(921\) 15.2350 0.502011
\(922\) 0 0
\(923\) 2.08119 0.0685033
\(924\) 0 0
\(925\) 8.02202 0.263763
\(926\) 0 0
\(927\) −21.5679 −0.708381
\(928\) 0 0
\(929\) −7.54856 −0.247660 −0.123830 0.992303i \(-0.539518\pi\)
−0.123830 + 0.992303i \(0.539518\pi\)
\(930\) 0 0
\(931\) 3.30252 0.108236
\(932\) 0 0
\(933\) −87.3189 −2.85869
\(934\) 0 0
\(935\) 0.357903 0.0117047
\(936\) 0 0
\(937\) −10.8822 −0.355504 −0.177752 0.984075i \(-0.556883\pi\)
−0.177752 + 0.984075i \(0.556883\pi\)
\(938\) 0 0
\(939\) 59.7209 1.94892
\(940\) 0 0
\(941\) 4.27321 0.139303 0.0696514 0.997571i \(-0.477811\pi\)
0.0696514 + 0.997571i \(0.477811\pi\)
\(942\) 0 0
\(943\) −38.9686 −1.26899
\(944\) 0 0
\(945\) 2.87493 0.0935215
\(946\) 0 0
\(947\) 1.27017 0.0412751 0.0206376 0.999787i \(-0.493430\pi\)
0.0206376 + 0.999787i \(0.493430\pi\)
\(948\) 0 0
\(949\) 7.43101 0.241221
\(950\) 0 0
\(951\) 10.0162 0.324799
\(952\) 0 0
\(953\) −13.2446 −0.429034 −0.214517 0.976720i \(-0.568818\pi\)
−0.214517 + 0.976720i \(0.568818\pi\)
\(954\) 0 0
\(955\) 4.64430 0.150286
\(956\) 0 0
\(957\) −27.4041 −0.885848
\(958\) 0 0
\(959\) −15.8367 −0.511395
\(960\) 0 0
\(961\) 47.7648 1.54080
\(962\) 0 0
\(963\) 134.850 4.34547
\(964\) 0 0
\(965\) 0.442776 0.0142535
\(966\) 0 0
\(967\) 55.8928 1.79739 0.898696 0.438572i \(-0.144516\pi\)
0.898696 + 0.438572i \(0.144516\pi\)
\(968\) 0 0
\(969\) −14.9531 −0.480361
\(970\) 0 0
\(971\) −47.5207 −1.52501 −0.762506 0.646982i \(-0.776031\pi\)
−0.762506 + 0.646982i \(0.776031\pi\)
\(972\) 0 0
\(973\) 11.2076 0.359300
\(974\) 0 0
\(975\) −15.4113 −0.493556
\(976\) 0 0
\(977\) −21.7370 −0.695428 −0.347714 0.937601i \(-0.613042\pi\)
−0.347714 + 0.937601i \(0.613042\pi\)
\(978\) 0 0
\(979\) −11.9320 −0.381349
\(980\) 0 0
\(981\) −47.0544 −1.50233
\(982\) 0 0
\(983\) −36.7735 −1.17289 −0.586446 0.809988i \(-0.699473\pi\)
−0.586446 + 0.809988i \(0.699473\pi\)
\(984\) 0 0
\(985\) 3.13187 0.0997898
\(986\) 0 0
\(987\) −27.4729 −0.874471
\(988\) 0 0
\(989\) 70.9485 2.25603
\(990\) 0 0
\(991\) −0.666569 −0.0211743 −0.0105871 0.999944i \(-0.503370\pi\)
−0.0105871 + 0.999944i \(0.503370\pi\)
\(992\) 0 0
\(993\) −93.7485 −2.97502
\(994\) 0 0
\(995\) −1.53270 −0.0485900
\(996\) 0 0
\(997\) −16.1041 −0.510021 −0.255011 0.966938i \(-0.582079\pi\)
−0.255011 + 0.966938i \(0.582079\pi\)
\(998\) 0 0
\(999\) −18.9318 −0.598976
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 4004.2.a.k.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.a.k.1.10 10 1.1 even 1 trivial