Properties

Label 4004.2.a.h.1.9
Level $4004$
Weight $2$
Character 4004.1
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14 \) 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 1.9
Root \(3.39317\) 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)
 
\(f(q)\) \(=\) \(q+3.39317 q^{3} -3.99177 q^{5} +1.00000 q^{7} +8.51361 q^{9} +O(q^{10})\) \(q+3.39317 q^{3} -3.99177 q^{5} +1.00000 q^{7} +8.51361 q^{9} +1.00000 q^{11} -1.00000 q^{13} -13.5448 q^{15} -0.471119 q^{17} +7.42544 q^{19} +3.39317 q^{21} -3.13086 q^{23} +10.9342 q^{25} +18.7086 q^{27} +0.836190 q^{29} +8.21252 q^{31} +3.39317 q^{33} -3.99177 q^{35} -10.4440 q^{37} -3.39317 q^{39} -4.29525 q^{41} -5.12434 q^{43} -33.9844 q^{45} -7.87553 q^{47} +1.00000 q^{49} -1.59859 q^{51} +10.7025 q^{53} -3.99177 q^{55} +25.1958 q^{57} +0.786985 q^{59} +12.4916 q^{61} +8.51361 q^{63} +3.99177 q^{65} +6.30389 q^{67} -10.6235 q^{69} +4.21357 q^{71} -2.43518 q^{73} +37.1017 q^{75} +1.00000 q^{77} +0.624254 q^{79} +37.9407 q^{81} -3.43620 q^{83} +1.88060 q^{85} +2.83734 q^{87} +5.92101 q^{89} -1.00000 q^{91} +27.8665 q^{93} -29.6407 q^{95} -2.95552 q^{97} +8.51361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) \) 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
\(3\) 3.39317 1.95905 0.979524 0.201327i \(-0.0645254\pi\)
0.979524 + 0.201327i \(0.0645254\pi\)
\(4\) 0 0
\(5\) −3.99177 −1.78517 −0.892587 0.450876i \(-0.851112\pi\)
−0.892587 + 0.450876i \(0.851112\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 8.51361 2.83787
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −13.5448 −3.49724
\(16\) 0 0
\(17\) −0.471119 −0.114263 −0.0571316 0.998367i \(-0.518195\pi\)
−0.0571316 + 0.998367i \(0.518195\pi\)
\(18\) 0 0
\(19\) 7.42544 1.70351 0.851757 0.523937i \(-0.175537\pi\)
0.851757 + 0.523937i \(0.175537\pi\)
\(20\) 0 0
\(21\) 3.39317 0.740451
\(22\) 0 0
\(23\) −3.13086 −0.652830 −0.326415 0.945227i \(-0.605841\pi\)
−0.326415 + 0.945227i \(0.605841\pi\)
\(24\) 0 0
\(25\) 10.9342 2.18684
\(26\) 0 0
\(27\) 18.7086 3.60048
\(28\) 0 0
\(29\) 0.836190 0.155277 0.0776383 0.996982i \(-0.475262\pi\)
0.0776383 + 0.996982i \(0.475262\pi\)
\(30\) 0 0
\(31\) 8.21252 1.47501 0.737506 0.675341i \(-0.236003\pi\)
0.737506 + 0.675341i \(0.236003\pi\)
\(32\) 0 0
\(33\) 3.39317 0.590675
\(34\) 0 0
\(35\) −3.99177 −0.674732
\(36\) 0 0
\(37\) −10.4440 −1.71699 −0.858495 0.512823i \(-0.828600\pi\)
−0.858495 + 0.512823i \(0.828600\pi\)
\(38\) 0 0
\(39\) −3.39317 −0.543342
\(40\) 0 0
\(41\) −4.29525 −0.670805 −0.335402 0.942075i \(-0.608872\pi\)
−0.335402 + 0.942075i \(0.608872\pi\)
\(42\) 0 0
\(43\) −5.12434 −0.781455 −0.390727 0.920506i \(-0.627776\pi\)
−0.390727 + 0.920506i \(0.627776\pi\)
\(44\) 0 0
\(45\) −33.9844 −5.06609
\(46\) 0 0
\(47\) −7.87553 −1.14876 −0.574382 0.818587i \(-0.694758\pi\)
−0.574382 + 0.818587i \(0.694758\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.59859 −0.223847
\(52\) 0 0
\(53\) 10.7025 1.47010 0.735050 0.678013i \(-0.237159\pi\)
0.735050 + 0.678013i \(0.237159\pi\)
\(54\) 0 0
\(55\) −3.99177 −0.538250
\(56\) 0 0
\(57\) 25.1958 3.33727
\(58\) 0 0
\(59\) 0.786985 0.102457 0.0512284 0.998687i \(-0.483686\pi\)
0.0512284 + 0.998687i \(0.483686\pi\)
\(60\) 0 0
\(61\) 12.4916 1.59939 0.799696 0.600406i \(-0.204994\pi\)
0.799696 + 0.600406i \(0.204994\pi\)
\(62\) 0 0
\(63\) 8.51361 1.07261
\(64\) 0 0
\(65\) 3.99177 0.495118
\(66\) 0 0
\(67\) 6.30389 0.770143 0.385072 0.922887i \(-0.374177\pi\)
0.385072 + 0.922887i \(0.374177\pi\)
\(68\) 0 0
\(69\) −10.6235 −1.27892
\(70\) 0 0
\(71\) 4.21357 0.500059 0.250030 0.968238i \(-0.419560\pi\)
0.250030 + 0.968238i \(0.419560\pi\)
\(72\) 0 0
\(73\) −2.43518 −0.285016 −0.142508 0.989794i \(-0.545517\pi\)
−0.142508 + 0.989794i \(0.545517\pi\)
\(74\) 0 0
\(75\) 37.1017 4.28413
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 0.624254 0.0702341 0.0351171 0.999383i \(-0.488820\pi\)
0.0351171 + 0.999383i \(0.488820\pi\)
\(80\) 0 0
\(81\) 37.9407 4.21563
\(82\) 0 0
\(83\) −3.43620 −0.377172 −0.188586 0.982057i \(-0.560390\pi\)
−0.188586 + 0.982057i \(0.560390\pi\)
\(84\) 0 0
\(85\) 1.88060 0.203980
\(86\) 0 0
\(87\) 2.83734 0.304194
\(88\) 0 0
\(89\) 5.92101 0.627626 0.313813 0.949485i \(-0.398394\pi\)
0.313813 + 0.949485i \(0.398394\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 27.8665 2.88962
\(94\) 0 0
\(95\) −29.6407 −3.04107
\(96\) 0 0
\(97\) −2.95552 −0.300087 −0.150044 0.988679i \(-0.547941\pi\)
−0.150044 + 0.988679i \(0.547941\pi\)
\(98\) 0 0
\(99\) 8.51361 0.855650
\(100\) 0 0
\(101\) −1.75909 −0.175036 −0.0875178 0.996163i \(-0.527893\pi\)
−0.0875178 + 0.996163i \(0.527893\pi\)
\(102\) 0 0
\(103\) 9.47902 0.933995 0.466998 0.884259i \(-0.345336\pi\)
0.466998 + 0.884259i \(0.345336\pi\)
\(104\) 0 0
\(105\) −13.5448 −1.32183
\(106\) 0 0
\(107\) 13.8890 1.34270 0.671352 0.741139i \(-0.265714\pi\)
0.671352 + 0.741139i \(0.265714\pi\)
\(108\) 0 0
\(109\) 13.6380 1.30628 0.653142 0.757236i \(-0.273451\pi\)
0.653142 + 0.757236i \(0.273451\pi\)
\(110\) 0 0
\(111\) −35.4384 −3.36366
\(112\) 0 0
\(113\) 5.13526 0.483085 0.241542 0.970390i \(-0.422347\pi\)
0.241542 + 0.970390i \(0.422347\pi\)
\(114\) 0 0
\(115\) 12.4977 1.16541
\(116\) 0 0
\(117\) −8.51361 −0.787083
\(118\) 0 0
\(119\) −0.471119 −0.0431874
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −14.5745 −1.31414
\(124\) 0 0
\(125\) −23.6880 −2.11872
\(126\) 0 0
\(127\) 16.4155 1.45664 0.728322 0.685235i \(-0.240300\pi\)
0.728322 + 0.685235i \(0.240300\pi\)
\(128\) 0 0
\(129\) −17.3878 −1.53091
\(130\) 0 0
\(131\) 21.7001 1.89594 0.947972 0.318354i \(-0.103130\pi\)
0.947972 + 0.318354i \(0.103130\pi\)
\(132\) 0 0
\(133\) 7.42544 0.643868
\(134\) 0 0
\(135\) −74.6805 −6.42747
\(136\) 0 0
\(137\) −14.4333 −1.23312 −0.616559 0.787308i \(-0.711474\pi\)
−0.616559 + 0.787308i \(0.711474\pi\)
\(138\) 0 0
\(139\) −6.55665 −0.556128 −0.278064 0.960563i \(-0.589693\pi\)
−0.278064 + 0.960563i \(0.589693\pi\)
\(140\) 0 0
\(141\) −26.7230 −2.25048
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −3.33788 −0.277196
\(146\) 0 0
\(147\) 3.39317 0.279864
\(148\) 0 0
\(149\) −12.9584 −1.06159 −0.530795 0.847500i \(-0.678107\pi\)
−0.530795 + 0.847500i \(0.678107\pi\)
\(150\) 0 0
\(151\) 20.4112 1.66104 0.830519 0.556990i \(-0.188044\pi\)
0.830519 + 0.556990i \(0.188044\pi\)
\(152\) 0 0
\(153\) −4.01092 −0.324264
\(154\) 0 0
\(155\) −32.7825 −2.63315
\(156\) 0 0
\(157\) −10.3074 −0.822621 −0.411310 0.911495i \(-0.634929\pi\)
−0.411310 + 0.911495i \(0.634929\pi\)
\(158\) 0 0
\(159\) 36.3154 2.88000
\(160\) 0 0
\(161\) −3.13086 −0.246746
\(162\) 0 0
\(163\) 5.63744 0.441559 0.220779 0.975324i \(-0.429140\pi\)
0.220779 + 0.975324i \(0.429140\pi\)
\(164\) 0 0
\(165\) −13.5448 −1.05446
\(166\) 0 0
\(167\) −6.07699 −0.470251 −0.235126 0.971965i \(-0.575550\pi\)
−0.235126 + 0.971965i \(0.575550\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 63.2173 4.83435
\(172\) 0 0
\(173\) −13.8355 −1.05189 −0.525946 0.850518i \(-0.676289\pi\)
−0.525946 + 0.850518i \(0.676289\pi\)
\(174\) 0 0
\(175\) 10.9342 0.826549
\(176\) 0 0
\(177\) 2.67038 0.200718
\(178\) 0 0
\(179\) 6.61263 0.494251 0.247126 0.968983i \(-0.420514\pi\)
0.247126 + 0.968983i \(0.420514\pi\)
\(180\) 0 0
\(181\) −2.57204 −0.191178 −0.0955889 0.995421i \(-0.530473\pi\)
−0.0955889 + 0.995421i \(0.530473\pi\)
\(182\) 0 0
\(183\) 42.3863 3.13328
\(184\) 0 0
\(185\) 41.6902 3.06512
\(186\) 0 0
\(187\) −0.471119 −0.0344516
\(188\) 0 0
\(189\) 18.7086 1.36085
\(190\) 0 0
\(191\) 13.8567 1.00263 0.501317 0.865264i \(-0.332849\pi\)
0.501317 + 0.865264i \(0.332849\pi\)
\(192\) 0 0
\(193\) −24.8940 −1.79191 −0.895956 0.444143i \(-0.853508\pi\)
−0.895956 + 0.444143i \(0.853508\pi\)
\(194\) 0 0
\(195\) 13.5448 0.969960
\(196\) 0 0
\(197\) −27.5198 −1.96071 −0.980354 0.197247i \(-0.936800\pi\)
−0.980354 + 0.197247i \(0.936800\pi\)
\(198\) 0 0
\(199\) 20.7468 1.47070 0.735352 0.677686i \(-0.237017\pi\)
0.735352 + 0.677686i \(0.237017\pi\)
\(200\) 0 0
\(201\) 21.3902 1.50875
\(202\) 0 0
\(203\) 0.836190 0.0586890
\(204\) 0 0
\(205\) 17.1456 1.19750
\(206\) 0 0
\(207\) −26.6549 −1.85265
\(208\) 0 0
\(209\) 7.42544 0.513629
\(210\) 0 0
\(211\) −15.7075 −1.08135 −0.540675 0.841231i \(-0.681831\pi\)
−0.540675 + 0.841231i \(0.681831\pi\)
\(212\) 0 0
\(213\) 14.2974 0.979640
\(214\) 0 0
\(215\) 20.4552 1.39503
\(216\) 0 0
\(217\) 8.21252 0.557502
\(218\) 0 0
\(219\) −8.26297 −0.558360
\(220\) 0 0
\(221\) 0.471119 0.0316909
\(222\) 0 0
\(223\) −12.1372 −0.812764 −0.406382 0.913703i \(-0.633210\pi\)
−0.406382 + 0.913703i \(0.633210\pi\)
\(224\) 0 0
\(225\) 93.0896 6.20598
\(226\) 0 0
\(227\) 14.8476 0.985468 0.492734 0.870180i \(-0.335998\pi\)
0.492734 + 0.870180i \(0.335998\pi\)
\(228\) 0 0
\(229\) 16.4440 1.08665 0.543325 0.839522i \(-0.317165\pi\)
0.543325 + 0.839522i \(0.317165\pi\)
\(230\) 0 0
\(231\) 3.39317 0.223254
\(232\) 0 0
\(233\) −2.39260 −0.156745 −0.0783724 0.996924i \(-0.524972\pi\)
−0.0783724 + 0.996924i \(0.524972\pi\)
\(234\) 0 0
\(235\) 31.4373 2.05074
\(236\) 0 0
\(237\) 2.11820 0.137592
\(238\) 0 0
\(239\) −1.98212 −0.128213 −0.0641063 0.997943i \(-0.520420\pi\)
−0.0641063 + 0.997943i \(0.520420\pi\)
\(240\) 0 0
\(241\) −21.8311 −1.40627 −0.703133 0.711058i \(-0.748216\pi\)
−0.703133 + 0.711058i \(0.748216\pi\)
\(242\) 0 0
\(243\) 72.6135 4.65816
\(244\) 0 0
\(245\) −3.99177 −0.255025
\(246\) 0 0
\(247\) −7.42544 −0.472470
\(248\) 0 0
\(249\) −11.6596 −0.738897
\(250\) 0 0
\(251\) −3.45012 −0.217770 −0.108885 0.994054i \(-0.534728\pi\)
−0.108885 + 0.994054i \(0.534728\pi\)
\(252\) 0 0
\(253\) −3.13086 −0.196836
\(254\) 0 0
\(255\) 6.38119 0.399606
\(256\) 0 0
\(257\) 1.74867 0.109079 0.0545394 0.998512i \(-0.482631\pi\)
0.0545394 + 0.998512i \(0.482631\pi\)
\(258\) 0 0
\(259\) −10.4440 −0.648961
\(260\) 0 0
\(261\) 7.11899 0.440655
\(262\) 0 0
\(263\) −11.1187 −0.685609 −0.342804 0.939407i \(-0.611377\pi\)
−0.342804 + 0.939407i \(0.611377\pi\)
\(264\) 0 0
\(265\) −42.7219 −2.62438
\(266\) 0 0
\(267\) 20.0910 1.22955
\(268\) 0 0
\(269\) −28.9283 −1.76379 −0.881895 0.471446i \(-0.843732\pi\)
−0.881895 + 0.471446i \(0.843732\pi\)
\(270\) 0 0
\(271\) −17.6428 −1.07172 −0.535862 0.844306i \(-0.680013\pi\)
−0.535862 + 0.844306i \(0.680013\pi\)
\(272\) 0 0
\(273\) −3.39317 −0.205364
\(274\) 0 0
\(275\) 10.9342 0.659358
\(276\) 0 0
\(277\) −25.8448 −1.55286 −0.776432 0.630202i \(-0.782972\pi\)
−0.776432 + 0.630202i \(0.782972\pi\)
\(278\) 0 0
\(279\) 69.9182 4.18589
\(280\) 0 0
\(281\) −15.8889 −0.947852 −0.473926 0.880565i \(-0.657164\pi\)
−0.473926 + 0.880565i \(0.657164\pi\)
\(282\) 0 0
\(283\) −12.3295 −0.732914 −0.366457 0.930435i \(-0.619429\pi\)
−0.366457 + 0.930435i \(0.619429\pi\)
\(284\) 0 0
\(285\) −100.576 −5.95760
\(286\) 0 0
\(287\) −4.29525 −0.253540
\(288\) 0 0
\(289\) −16.7780 −0.986944
\(290\) 0 0
\(291\) −10.0286 −0.587885
\(292\) 0 0
\(293\) 27.3645 1.59865 0.799325 0.600899i \(-0.205191\pi\)
0.799325 + 0.600899i \(0.205191\pi\)
\(294\) 0 0
\(295\) −3.14146 −0.182903
\(296\) 0 0
\(297\) 18.7086 1.08558
\(298\) 0 0
\(299\) 3.13086 0.181062
\(300\) 0 0
\(301\) −5.12434 −0.295362
\(302\) 0 0
\(303\) −5.96888 −0.342903
\(304\) 0 0
\(305\) −49.8637 −2.85519
\(306\) 0 0
\(307\) −3.83950 −0.219132 −0.109566 0.993980i \(-0.534946\pi\)
−0.109566 + 0.993980i \(0.534946\pi\)
\(308\) 0 0
\(309\) 32.1639 1.82974
\(310\) 0 0
\(311\) −5.89739 −0.334411 −0.167205 0.985922i \(-0.553474\pi\)
−0.167205 + 0.985922i \(0.553474\pi\)
\(312\) 0 0
\(313\) 15.9689 0.902618 0.451309 0.892368i \(-0.350957\pi\)
0.451309 + 0.892368i \(0.350957\pi\)
\(314\) 0 0
\(315\) −33.9844 −1.91480
\(316\) 0 0
\(317\) −33.0484 −1.85618 −0.928091 0.372353i \(-0.878551\pi\)
−0.928091 + 0.372353i \(0.878551\pi\)
\(318\) 0 0
\(319\) 0.836190 0.0468176
\(320\) 0 0
\(321\) 47.1279 2.63042
\(322\) 0 0
\(323\) −3.49827 −0.194649
\(324\) 0 0
\(325\) −10.9342 −0.606521
\(326\) 0 0
\(327\) 46.2761 2.55907
\(328\) 0 0
\(329\) −7.87553 −0.434192
\(330\) 0 0
\(331\) −20.3928 −1.12089 −0.560445 0.828191i \(-0.689370\pi\)
−0.560445 + 0.828191i \(0.689370\pi\)
\(332\) 0 0
\(333\) −88.9164 −4.87259
\(334\) 0 0
\(335\) −25.1637 −1.37484
\(336\) 0 0
\(337\) −11.0614 −0.602554 −0.301277 0.953537i \(-0.597413\pi\)
−0.301277 + 0.953537i \(0.597413\pi\)
\(338\) 0 0
\(339\) 17.4248 0.946386
\(340\) 0 0
\(341\) 8.21252 0.444733
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 42.4067 2.28310
\(346\) 0 0
\(347\) −28.4766 −1.52871 −0.764353 0.644798i \(-0.776942\pi\)
−0.764353 + 0.644798i \(0.776942\pi\)
\(348\) 0 0
\(349\) 31.6756 1.69556 0.847778 0.530351i \(-0.177940\pi\)
0.847778 + 0.530351i \(0.177940\pi\)
\(350\) 0 0
\(351\) −18.7086 −0.998592
\(352\) 0 0
\(353\) 1.01966 0.0542710 0.0271355 0.999632i \(-0.491361\pi\)
0.0271355 + 0.999632i \(0.491361\pi\)
\(354\) 0 0
\(355\) −16.8196 −0.892692
\(356\) 0 0
\(357\) −1.59859 −0.0846062
\(358\) 0 0
\(359\) 16.6690 0.879754 0.439877 0.898058i \(-0.355022\pi\)
0.439877 + 0.898058i \(0.355022\pi\)
\(360\) 0 0
\(361\) 36.1372 1.90196
\(362\) 0 0
\(363\) 3.39317 0.178095
\(364\) 0 0
\(365\) 9.72066 0.508802
\(366\) 0 0
\(367\) −21.5501 −1.12491 −0.562453 0.826829i \(-0.690142\pi\)
−0.562453 + 0.826829i \(0.690142\pi\)
\(368\) 0 0
\(369\) −36.5681 −1.90366
\(370\) 0 0
\(371\) 10.7025 0.555646
\(372\) 0 0
\(373\) −32.5768 −1.68677 −0.843383 0.537313i \(-0.819439\pi\)
−0.843383 + 0.537313i \(0.819439\pi\)
\(374\) 0 0
\(375\) −80.3775 −4.15067
\(376\) 0 0
\(377\) −0.836190 −0.0430660
\(378\) 0 0
\(379\) 23.5571 1.21005 0.605023 0.796208i \(-0.293164\pi\)
0.605023 + 0.796208i \(0.293164\pi\)
\(380\) 0 0
\(381\) 55.7008 2.85364
\(382\) 0 0
\(383\) 20.5312 1.04909 0.524547 0.851382i \(-0.324235\pi\)
0.524547 + 0.851382i \(0.324235\pi\)
\(384\) 0 0
\(385\) −3.99177 −0.203439
\(386\) 0 0
\(387\) −43.6266 −2.21767
\(388\) 0 0
\(389\) 2.72332 0.138078 0.0690390 0.997614i \(-0.478007\pi\)
0.0690390 + 0.997614i \(0.478007\pi\)
\(390\) 0 0
\(391\) 1.47501 0.0745944
\(392\) 0 0
\(393\) 73.6320 3.71425
\(394\) 0 0
\(395\) −2.49188 −0.125380
\(396\) 0 0
\(397\) −14.5807 −0.731785 −0.365892 0.930657i \(-0.619236\pi\)
−0.365892 + 0.930657i \(0.619236\pi\)
\(398\) 0 0
\(399\) 25.1958 1.26137
\(400\) 0 0
\(401\) −25.7872 −1.28775 −0.643877 0.765129i \(-0.722675\pi\)
−0.643877 + 0.765129i \(0.722675\pi\)
\(402\) 0 0
\(403\) −8.21252 −0.409095
\(404\) 0 0
\(405\) −151.451 −7.52564
\(406\) 0 0
\(407\) −10.4440 −0.517692
\(408\) 0 0
\(409\) −0.719307 −0.0355675 −0.0177837 0.999842i \(-0.505661\pi\)
−0.0177837 + 0.999842i \(0.505661\pi\)
\(410\) 0 0
\(411\) −48.9746 −2.41574
\(412\) 0 0
\(413\) 0.786985 0.0387250
\(414\) 0 0
\(415\) 13.7165 0.673317
\(416\) 0 0
\(417\) −22.2478 −1.08948
\(418\) 0 0
\(419\) 11.0098 0.537864 0.268932 0.963159i \(-0.413329\pi\)
0.268932 + 0.963159i \(0.413329\pi\)
\(420\) 0 0
\(421\) 28.0020 1.36473 0.682367 0.731009i \(-0.260950\pi\)
0.682367 + 0.731009i \(0.260950\pi\)
\(422\) 0 0
\(423\) −67.0492 −3.26004
\(424\) 0 0
\(425\) −5.15132 −0.249876
\(426\) 0 0
\(427\) 12.4916 0.604513
\(428\) 0 0
\(429\) −3.39317 −0.163824
\(430\) 0 0
\(431\) −12.7009 −0.611780 −0.305890 0.952067i \(-0.598954\pi\)
−0.305890 + 0.952067i \(0.598954\pi\)
\(432\) 0 0
\(433\) −12.7026 −0.610450 −0.305225 0.952280i \(-0.598732\pi\)
−0.305225 + 0.952280i \(0.598732\pi\)
\(434\) 0 0
\(435\) −11.3260 −0.543039
\(436\) 0 0
\(437\) −23.2480 −1.11210
\(438\) 0 0
\(439\) −33.5765 −1.60252 −0.801259 0.598317i \(-0.795836\pi\)
−0.801259 + 0.598317i \(0.795836\pi\)
\(440\) 0 0
\(441\) 8.51361 0.405410
\(442\) 0 0
\(443\) −10.8436 −0.515194 −0.257597 0.966252i \(-0.582931\pi\)
−0.257597 + 0.966252i \(0.582931\pi\)
\(444\) 0 0
\(445\) −23.6353 −1.12042
\(446\) 0 0
\(447\) −43.9699 −2.07971
\(448\) 0 0
\(449\) 7.65806 0.361406 0.180703 0.983538i \(-0.442163\pi\)
0.180703 + 0.983538i \(0.442163\pi\)
\(450\) 0 0
\(451\) −4.29525 −0.202255
\(452\) 0 0
\(453\) 69.2586 3.25405
\(454\) 0 0
\(455\) 3.99177 0.187137
\(456\) 0 0
\(457\) −29.8999 −1.39866 −0.699328 0.714801i \(-0.746517\pi\)
−0.699328 + 0.714801i \(0.746517\pi\)
\(458\) 0 0
\(459\) −8.81399 −0.411402
\(460\) 0 0
\(461\) −16.0757 −0.748719 −0.374360 0.927284i \(-0.622137\pi\)
−0.374360 + 0.927284i \(0.622137\pi\)
\(462\) 0 0
\(463\) −0.374768 −0.0174170 −0.00870848 0.999962i \(-0.502772\pi\)
−0.00870848 + 0.999962i \(0.502772\pi\)
\(464\) 0 0
\(465\) −111.237 −5.15847
\(466\) 0 0
\(467\) 7.50639 0.347355 0.173677 0.984803i \(-0.444435\pi\)
0.173677 + 0.984803i \(0.444435\pi\)
\(468\) 0 0
\(469\) 6.30389 0.291087
\(470\) 0 0
\(471\) −34.9748 −1.61155
\(472\) 0 0
\(473\) −5.12434 −0.235618
\(474\) 0 0
\(475\) 81.1914 3.72532
\(476\) 0 0
\(477\) 91.1168 4.17195
\(478\) 0 0
\(479\) −4.57393 −0.208988 −0.104494 0.994526i \(-0.533322\pi\)
−0.104494 + 0.994526i \(0.533322\pi\)
\(480\) 0 0
\(481\) 10.4440 0.476207
\(482\) 0 0
\(483\) −10.6235 −0.483388
\(484\) 0 0
\(485\) 11.7977 0.535707
\(486\) 0 0
\(487\) 26.8977 1.21885 0.609425 0.792843i \(-0.291400\pi\)
0.609425 + 0.792843i \(0.291400\pi\)
\(488\) 0 0
\(489\) 19.1288 0.865035
\(490\) 0 0
\(491\) 0.808321 0.0364790 0.0182395 0.999834i \(-0.494194\pi\)
0.0182395 + 0.999834i \(0.494194\pi\)
\(492\) 0 0
\(493\) −0.393945 −0.0177424
\(494\) 0 0
\(495\) −33.9844 −1.52748
\(496\) 0 0
\(497\) 4.21357 0.189005
\(498\) 0 0
\(499\) −7.93777 −0.355344 −0.177672 0.984090i \(-0.556857\pi\)
−0.177672 + 0.984090i \(0.556857\pi\)
\(500\) 0 0
\(501\) −20.6203 −0.921245
\(502\) 0 0
\(503\) −8.88207 −0.396032 −0.198016 0.980199i \(-0.563450\pi\)
−0.198016 + 0.980199i \(0.563450\pi\)
\(504\) 0 0
\(505\) 7.02186 0.312469
\(506\) 0 0
\(507\) 3.39317 0.150696
\(508\) 0 0
\(509\) 19.6020 0.868843 0.434421 0.900710i \(-0.356953\pi\)
0.434421 + 0.900710i \(0.356953\pi\)
\(510\) 0 0
\(511\) −2.43518 −0.107726
\(512\) 0 0
\(513\) 138.920 6.13346
\(514\) 0 0
\(515\) −37.8380 −1.66734
\(516\) 0 0
\(517\) −7.87553 −0.346365
\(518\) 0 0
\(519\) −46.9462 −2.06071
\(520\) 0 0
\(521\) 1.67605 0.0734292 0.0367146 0.999326i \(-0.488311\pi\)
0.0367146 + 0.999326i \(0.488311\pi\)
\(522\) 0 0
\(523\) 14.9196 0.652386 0.326193 0.945303i \(-0.394234\pi\)
0.326193 + 0.945303i \(0.394234\pi\)
\(524\) 0 0
\(525\) 37.1017 1.61925
\(526\) 0 0
\(527\) −3.86907 −0.168539
\(528\) 0 0
\(529\) −13.1977 −0.573813
\(530\) 0 0
\(531\) 6.70009 0.290759
\(532\) 0 0
\(533\) 4.29525 0.186048
\(534\) 0 0
\(535\) −55.4418 −2.39696
\(536\) 0 0
\(537\) 22.4378 0.968262
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 7.34604 0.315831 0.157915 0.987453i \(-0.449523\pi\)
0.157915 + 0.987453i \(0.449523\pi\)
\(542\) 0 0
\(543\) −8.72736 −0.374527
\(544\) 0 0
\(545\) −54.4397 −2.33194
\(546\) 0 0
\(547\) −29.3767 −1.25606 −0.628029 0.778190i \(-0.716138\pi\)
−0.628029 + 0.778190i \(0.716138\pi\)
\(548\) 0 0
\(549\) 106.349 4.53886
\(550\) 0 0
\(551\) 6.20908 0.264516
\(552\) 0 0
\(553\) 0.624254 0.0265460
\(554\) 0 0
\(555\) 141.462 6.00472
\(556\) 0 0
\(557\) −1.41854 −0.0601056 −0.0300528 0.999548i \(-0.509568\pi\)
−0.0300528 + 0.999548i \(0.509568\pi\)
\(558\) 0 0
\(559\) 5.12434 0.216737
\(560\) 0 0
\(561\) −1.59859 −0.0674924
\(562\) 0 0
\(563\) −5.98712 −0.252327 −0.126163 0.992009i \(-0.540266\pi\)
−0.126163 + 0.992009i \(0.540266\pi\)
\(564\) 0 0
\(565\) −20.4988 −0.862390
\(566\) 0 0
\(567\) 37.9407 1.59336
\(568\) 0 0
\(569\) −5.60728 −0.235069 −0.117535 0.993069i \(-0.537499\pi\)
−0.117535 + 0.993069i \(0.537499\pi\)
\(570\) 0 0
\(571\) 37.6024 1.57361 0.786805 0.617201i \(-0.211733\pi\)
0.786805 + 0.617201i \(0.211733\pi\)
\(572\) 0 0
\(573\) 47.0180 1.96421
\(574\) 0 0
\(575\) −34.2335 −1.42764
\(576\) 0 0
\(577\) −21.1990 −0.882527 −0.441263 0.897378i \(-0.645470\pi\)
−0.441263 + 0.897378i \(0.645470\pi\)
\(578\) 0 0
\(579\) −84.4697 −3.51044
\(580\) 0 0
\(581\) −3.43620 −0.142557
\(582\) 0 0
\(583\) 10.7025 0.443252
\(584\) 0 0
\(585\) 33.9844 1.40508
\(586\) 0 0
\(587\) 23.1120 0.953935 0.476967 0.878921i \(-0.341736\pi\)
0.476967 + 0.878921i \(0.341736\pi\)
\(588\) 0 0
\(589\) 60.9816 2.51270
\(590\) 0 0
\(591\) −93.3795 −3.84112
\(592\) 0 0
\(593\) −36.6448 −1.50482 −0.752412 0.658693i \(-0.771110\pi\)
−0.752412 + 0.658693i \(0.771110\pi\)
\(594\) 0 0
\(595\) 1.88060 0.0770970
\(596\) 0 0
\(597\) 70.3975 2.88118
\(598\) 0 0
\(599\) 24.7963 1.01315 0.506575 0.862196i \(-0.330911\pi\)
0.506575 + 0.862196i \(0.330911\pi\)
\(600\) 0 0
\(601\) 4.57196 0.186494 0.0932471 0.995643i \(-0.470275\pi\)
0.0932471 + 0.995643i \(0.470275\pi\)
\(602\) 0 0
\(603\) 53.6689 2.18557
\(604\) 0 0
\(605\) −3.99177 −0.162288
\(606\) 0 0
\(607\) 7.24592 0.294103 0.147051 0.989129i \(-0.453022\pi\)
0.147051 + 0.989129i \(0.453022\pi\)
\(608\) 0 0
\(609\) 2.83734 0.114975
\(610\) 0 0
\(611\) 7.87553 0.318610
\(612\) 0 0
\(613\) 9.24990 0.373600 0.186800 0.982398i \(-0.440188\pi\)
0.186800 + 0.982398i \(0.440188\pi\)
\(614\) 0 0
\(615\) 58.1781 2.34597
\(616\) 0 0
\(617\) −33.5364 −1.35013 −0.675063 0.737760i \(-0.735884\pi\)
−0.675063 + 0.737760i \(0.735884\pi\)
\(618\) 0 0
\(619\) −11.6828 −0.469573 −0.234787 0.972047i \(-0.575439\pi\)
−0.234787 + 0.972047i \(0.575439\pi\)
\(620\) 0 0
\(621\) −58.5741 −2.35050
\(622\) 0 0
\(623\) 5.92101 0.237220
\(624\) 0 0
\(625\) 39.8860 1.59544
\(626\) 0 0
\(627\) 25.1958 1.00622
\(628\) 0 0
\(629\) 4.92038 0.196189
\(630\) 0 0
\(631\) −35.5606 −1.41565 −0.707823 0.706389i \(-0.750323\pi\)
−0.707823 + 0.706389i \(0.750323\pi\)
\(632\) 0 0
\(633\) −53.2983 −2.11842
\(634\) 0 0
\(635\) −65.5271 −2.60036
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 35.8727 1.41910
\(640\) 0 0
\(641\) −30.5085 −1.20501 −0.602506 0.798114i \(-0.705831\pi\)
−0.602506 + 0.798114i \(0.705831\pi\)
\(642\) 0 0
\(643\) 47.9522 1.89105 0.945525 0.325549i \(-0.105549\pi\)
0.945525 + 0.325549i \(0.105549\pi\)
\(644\) 0 0
\(645\) 69.4079 2.73294
\(646\) 0 0
\(647\) 33.7878 1.32834 0.664168 0.747583i \(-0.268786\pi\)
0.664168 + 0.747583i \(0.268786\pi\)
\(648\) 0 0
\(649\) 0.786985 0.0308919
\(650\) 0 0
\(651\) 27.8665 1.09217
\(652\) 0 0
\(653\) −0.683141 −0.0267334 −0.0133667 0.999911i \(-0.504255\pi\)
−0.0133667 + 0.999911i \(0.504255\pi\)
\(654\) 0 0
\(655\) −86.6217 −3.38459
\(656\) 0 0
\(657\) −20.7321 −0.808837
\(658\) 0 0
\(659\) −15.3796 −0.599103 −0.299552 0.954080i \(-0.596837\pi\)
−0.299552 + 0.954080i \(0.596837\pi\)
\(660\) 0 0
\(661\) 14.2849 0.555617 0.277809 0.960636i \(-0.410392\pi\)
0.277809 + 0.960636i \(0.410392\pi\)
\(662\) 0 0
\(663\) 1.59859 0.0620840
\(664\) 0 0
\(665\) −29.6407 −1.14942
\(666\) 0 0
\(667\) −2.61799 −0.101369
\(668\) 0 0
\(669\) −41.1834 −1.59224
\(670\) 0 0
\(671\) 12.4916 0.482235
\(672\) 0 0
\(673\) −20.2419 −0.780269 −0.390135 0.920758i \(-0.627572\pi\)
−0.390135 + 0.920758i \(0.627572\pi\)
\(674\) 0 0
\(675\) 204.564 7.87367
\(676\) 0 0
\(677\) −9.15471 −0.351844 −0.175922 0.984404i \(-0.556291\pi\)
−0.175922 + 0.984404i \(0.556291\pi\)
\(678\) 0 0
\(679\) −2.95552 −0.113422
\(680\) 0 0
\(681\) 50.3804 1.93058
\(682\) 0 0
\(683\) 14.3870 0.550504 0.275252 0.961372i \(-0.411239\pi\)
0.275252 + 0.961372i \(0.411239\pi\)
\(684\) 0 0
\(685\) 57.6143 2.20133
\(686\) 0 0
\(687\) 55.7973 2.12880
\(688\) 0 0
\(689\) −10.7025 −0.407732
\(690\) 0 0
\(691\) 7.92175 0.301358 0.150679 0.988583i \(-0.451854\pi\)
0.150679 + 0.988583i \(0.451854\pi\)
\(692\) 0 0
\(693\) 8.51361 0.323405
\(694\) 0 0
\(695\) 26.1726 0.992785
\(696\) 0 0
\(697\) 2.02357 0.0766483
\(698\) 0 0
\(699\) −8.11852 −0.307071
\(700\) 0 0
\(701\) −18.3253 −0.692136 −0.346068 0.938209i \(-0.612483\pi\)
−0.346068 + 0.938209i \(0.612483\pi\)
\(702\) 0 0
\(703\) −77.5516 −2.92491
\(704\) 0 0
\(705\) 106.672 4.01750
\(706\) 0 0
\(707\) −1.75909 −0.0661572
\(708\) 0 0
\(709\) 36.6975 1.37820 0.689102 0.724664i \(-0.258005\pi\)
0.689102 + 0.724664i \(0.258005\pi\)
\(710\) 0 0
\(711\) 5.31466 0.199315
\(712\) 0 0
\(713\) −25.7123 −0.962931
\(714\) 0 0
\(715\) 3.99177 0.149284
\(716\) 0 0
\(717\) −6.72566 −0.251174
\(718\) 0 0
\(719\) 46.7262 1.74259 0.871296 0.490757i \(-0.163280\pi\)
0.871296 + 0.490757i \(0.163280\pi\)
\(720\) 0 0
\(721\) 9.47902 0.353017
\(722\) 0 0
\(723\) −74.0767 −2.75494
\(724\) 0 0
\(725\) 9.14308 0.339565
\(726\) 0 0
\(727\) −2.79457 −0.103645 −0.0518225 0.998656i \(-0.516503\pi\)
−0.0518225 + 0.998656i \(0.516503\pi\)
\(728\) 0 0
\(729\) 132.568 4.90992
\(730\) 0 0
\(731\) 2.41418 0.0892915
\(732\) 0 0
\(733\) −0.797036 −0.0294392 −0.0147196 0.999892i \(-0.504686\pi\)
−0.0147196 + 0.999892i \(0.504686\pi\)
\(734\) 0 0
\(735\) −13.5448 −0.499606
\(736\) 0 0
\(737\) 6.30389 0.232207
\(738\) 0 0
\(739\) 8.15617 0.300030 0.150015 0.988684i \(-0.452068\pi\)
0.150015 + 0.988684i \(0.452068\pi\)
\(740\) 0 0
\(741\) −25.1958 −0.925591
\(742\) 0 0
\(743\) −31.7867 −1.16614 −0.583071 0.812421i \(-0.698149\pi\)
−0.583071 + 0.812421i \(0.698149\pi\)
\(744\) 0 0
\(745\) 51.7268 1.89512
\(746\) 0 0
\(747\) −29.2544 −1.07036
\(748\) 0 0
\(749\) 13.8890 0.507494
\(750\) 0 0
\(751\) −8.84752 −0.322851 −0.161425 0.986885i \(-0.551609\pi\)
−0.161425 + 0.986885i \(0.551609\pi\)
\(752\) 0 0
\(753\) −11.7068 −0.426621
\(754\) 0 0
\(755\) −81.4767 −2.96524
\(756\) 0 0
\(757\) 17.8038 0.647091 0.323545 0.946213i \(-0.395125\pi\)
0.323545 + 0.946213i \(0.395125\pi\)
\(758\) 0 0
\(759\) −10.6235 −0.385610
\(760\) 0 0
\(761\) 18.9426 0.686669 0.343335 0.939213i \(-0.388444\pi\)
0.343335 + 0.939213i \(0.388444\pi\)
\(762\) 0 0
\(763\) 13.6380 0.493729
\(764\) 0 0
\(765\) 16.0107 0.578867
\(766\) 0 0
\(767\) −0.786985 −0.0284164
\(768\) 0 0
\(769\) −9.83217 −0.354557 −0.177279 0.984161i \(-0.556729\pi\)
−0.177279 + 0.984161i \(0.556729\pi\)
\(770\) 0 0
\(771\) 5.93352 0.213691
\(772\) 0 0
\(773\) −21.1236 −0.759762 −0.379881 0.925035i \(-0.624035\pi\)
−0.379881 + 0.925035i \(0.624035\pi\)
\(774\) 0 0
\(775\) 89.7974 3.22562
\(776\) 0 0
\(777\) −35.4384 −1.27135
\(778\) 0 0
\(779\) −31.8941 −1.14273
\(780\) 0 0
\(781\) 4.21357 0.150774
\(782\) 0 0
\(783\) 15.6440 0.559069
\(784\) 0 0
\(785\) 41.1448 1.46852
\(786\) 0 0
\(787\) 42.0638 1.49941 0.749707 0.661770i \(-0.230195\pi\)
0.749707 + 0.661770i \(0.230195\pi\)
\(788\) 0 0
\(789\) −37.7277 −1.34314
\(790\) 0 0
\(791\) 5.13526 0.182589
\(792\) 0 0
\(793\) −12.4916 −0.443591
\(794\) 0 0
\(795\) −144.963 −5.14129
\(796\) 0 0
\(797\) 17.5229 0.620695 0.310347 0.950623i \(-0.399555\pi\)
0.310347 + 0.950623i \(0.399555\pi\)
\(798\) 0 0
\(799\) 3.71031 0.131261
\(800\) 0 0
\(801\) 50.4092 1.78112
\(802\) 0 0
\(803\) −2.43518 −0.0859355
\(804\) 0 0
\(805\) 12.4977 0.440485
\(806\) 0 0
\(807\) −98.1587 −3.45535
\(808\) 0 0
\(809\) −33.8780 −1.19109 −0.595543 0.803323i \(-0.703063\pi\)
−0.595543 + 0.803323i \(0.703063\pi\)
\(810\) 0 0
\(811\) 0.516687 0.0181433 0.00907167 0.999959i \(-0.497112\pi\)
0.00907167 + 0.999959i \(0.497112\pi\)
\(812\) 0 0
\(813\) −59.8650 −2.09956
\(814\) 0 0
\(815\) −22.5034 −0.788259
\(816\) 0 0
\(817\) −38.0505 −1.33122
\(818\) 0 0
\(819\) −8.51361 −0.297490
\(820\) 0 0
\(821\) −12.0218 −0.419565 −0.209782 0.977748i \(-0.567276\pi\)
−0.209782 + 0.977748i \(0.567276\pi\)
\(822\) 0 0
\(823\) −26.6058 −0.927419 −0.463709 0.885987i \(-0.653482\pi\)
−0.463709 + 0.885987i \(0.653482\pi\)
\(824\) 0 0
\(825\) 37.1017 1.29171
\(826\) 0 0
\(827\) 2.84556 0.0989498 0.0494749 0.998775i \(-0.484245\pi\)
0.0494749 + 0.998775i \(0.484245\pi\)
\(828\) 0 0
\(829\) 30.3942 1.05563 0.527816 0.849358i \(-0.323011\pi\)
0.527816 + 0.849358i \(0.323011\pi\)
\(830\) 0 0
\(831\) −87.6958 −3.04213
\(832\) 0 0
\(833\) −0.471119 −0.0163233
\(834\) 0 0
\(835\) 24.2579 0.839480
\(836\) 0 0
\(837\) 153.645 5.31074
\(838\) 0 0
\(839\) −18.1903 −0.627998 −0.313999 0.949423i \(-0.601669\pi\)
−0.313999 + 0.949423i \(0.601669\pi\)
\(840\) 0 0
\(841\) −28.3008 −0.975889
\(842\) 0 0
\(843\) −53.9138 −1.85689
\(844\) 0 0
\(845\) −3.99177 −0.137321
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −41.8362 −1.43581
\(850\) 0 0
\(851\) 32.6988 1.12090
\(852\) 0 0
\(853\) 53.4823 1.83120 0.915600 0.402090i \(-0.131716\pi\)
0.915600 + 0.402090i \(0.131716\pi\)
\(854\) 0 0
\(855\) −252.349 −8.63015
\(856\) 0 0
\(857\) 16.7600 0.572509 0.286255 0.958154i \(-0.407590\pi\)
0.286255 + 0.958154i \(0.407590\pi\)
\(858\) 0 0
\(859\) 26.3561 0.899259 0.449630 0.893215i \(-0.351556\pi\)
0.449630 + 0.893215i \(0.351556\pi\)
\(860\) 0 0
\(861\) −14.5745 −0.496698
\(862\) 0 0
\(863\) 16.5357 0.562882 0.281441 0.959579i \(-0.409188\pi\)
0.281441 + 0.959579i \(0.409188\pi\)
\(864\) 0 0
\(865\) 55.2281 1.87781
\(866\) 0 0
\(867\) −56.9308 −1.93347
\(868\) 0 0
\(869\) 0.624254 0.0211764
\(870\) 0 0
\(871\) −6.30389 −0.213599
\(872\) 0 0
\(873\) −25.1621 −0.851608
\(874\) 0 0
\(875\) −23.6880 −0.800801
\(876\) 0 0
\(877\) 1.97842 0.0668066 0.0334033 0.999442i \(-0.489365\pi\)
0.0334033 + 0.999442i \(0.489365\pi\)
\(878\) 0 0
\(879\) 92.8523 3.13183
\(880\) 0 0
\(881\) 51.9104 1.74891 0.874453 0.485110i \(-0.161220\pi\)
0.874453 + 0.485110i \(0.161220\pi\)
\(882\) 0 0
\(883\) 15.0452 0.506311 0.253155 0.967426i \(-0.418532\pi\)
0.253155 + 0.967426i \(0.418532\pi\)
\(884\) 0 0
\(885\) −10.6595 −0.358316
\(886\) 0 0
\(887\) −14.5478 −0.488468 −0.244234 0.969716i \(-0.578537\pi\)
−0.244234 + 0.969716i \(0.578537\pi\)
\(888\) 0 0
\(889\) 16.4155 0.550560
\(890\) 0 0
\(891\) 37.9407 1.27106
\(892\) 0 0
\(893\) −58.4793 −1.95694
\(894\) 0 0
\(895\) −26.3961 −0.882324
\(896\) 0 0
\(897\) 10.6235 0.354710
\(898\) 0 0
\(899\) 6.86722 0.229035
\(900\) 0 0
\(901\) −5.04215 −0.167978
\(902\) 0 0
\(903\) −17.3878 −0.578629
\(904\) 0 0
\(905\) 10.2670 0.341286
\(906\) 0 0
\(907\) −24.1688 −0.802513 −0.401256 0.915966i \(-0.631426\pi\)
−0.401256 + 0.915966i \(0.631426\pi\)
\(908\) 0 0
\(909\) −14.9762 −0.496728
\(910\) 0 0
\(911\) −2.11985 −0.0702337 −0.0351168 0.999383i \(-0.511180\pi\)
−0.0351168 + 0.999383i \(0.511180\pi\)
\(912\) 0 0
\(913\) −3.43620 −0.113722
\(914\) 0 0
\(915\) −169.196 −5.59345
\(916\) 0 0
\(917\) 21.7001 0.716599
\(918\) 0 0
\(919\) 21.2016 0.699376 0.349688 0.936866i \(-0.386288\pi\)
0.349688 + 0.936866i \(0.386288\pi\)
\(920\) 0 0
\(921\) −13.0281 −0.429290
\(922\) 0 0
\(923\) −4.21357 −0.138691
\(924\) 0 0
\(925\) −114.197 −3.75479
\(926\) 0 0
\(927\) 80.7006 2.65056
\(928\) 0 0
\(929\) 13.3529 0.438095 0.219047 0.975714i \(-0.429705\pi\)
0.219047 + 0.975714i \(0.429705\pi\)
\(930\) 0 0
\(931\) 7.42544 0.243359
\(932\) 0 0
\(933\) −20.0109 −0.655126
\(934\) 0 0
\(935\) 1.88060 0.0615021
\(936\) 0 0
\(937\) −43.4451 −1.41929 −0.709645 0.704559i \(-0.751145\pi\)
−0.709645 + 0.704559i \(0.751145\pi\)
\(938\) 0 0
\(939\) 54.1854 1.76827
\(940\) 0 0
\(941\) −23.1935 −0.756088 −0.378044 0.925788i \(-0.623403\pi\)
−0.378044 + 0.925788i \(0.623403\pi\)
\(942\) 0 0
\(943\) 13.4478 0.437921
\(944\) 0 0
\(945\) −74.6805 −2.42936
\(946\) 0 0
\(947\) −8.55784 −0.278092 −0.139046 0.990286i \(-0.544404\pi\)
−0.139046 + 0.990286i \(0.544404\pi\)
\(948\) 0 0
\(949\) 2.43518 0.0790491
\(950\) 0 0
\(951\) −112.139 −3.63635
\(952\) 0 0
\(953\) −13.1413 −0.425687 −0.212844 0.977086i \(-0.568273\pi\)
−0.212844 + 0.977086i \(0.568273\pi\)
\(954\) 0 0
\(955\) −55.3126 −1.78987
\(956\) 0 0
\(957\) 2.83734 0.0917180
\(958\) 0 0
\(959\) −14.4333 −0.466075
\(960\) 0 0
\(961\) 36.4454 1.17566
\(962\) 0 0
\(963\) 118.246 3.81042
\(964\) 0 0
\(965\) 99.3712 3.19887
\(966\) 0 0
\(967\) −32.7140 −1.05201 −0.526005 0.850481i \(-0.676311\pi\)
−0.526005 + 0.850481i \(0.676311\pi\)
\(968\) 0 0
\(969\) −11.8702 −0.381326
\(970\) 0 0
\(971\) 36.3279 1.16582 0.582908 0.812538i \(-0.301915\pi\)
0.582908 + 0.812538i \(0.301915\pi\)
\(972\) 0 0
\(973\) −6.55665 −0.210197
\(974\) 0 0
\(975\) −37.1017 −1.18820
\(976\) 0 0
\(977\) 25.4953 0.815668 0.407834 0.913056i \(-0.366284\pi\)
0.407834 + 0.913056i \(0.366284\pi\)
\(978\) 0 0
\(979\) 5.92101 0.189236
\(980\) 0 0
\(981\) 116.109 3.70706
\(982\) 0 0
\(983\) 28.2147 0.899908 0.449954 0.893052i \(-0.351440\pi\)
0.449954 + 0.893052i \(0.351440\pi\)
\(984\) 0 0
\(985\) 109.853 3.50020
\(986\) 0 0
\(987\) −26.7230 −0.850603
\(988\) 0 0
\(989\) 16.0436 0.510157
\(990\) 0 0
\(991\) −5.36613 −0.170461 −0.0852303 0.996361i \(-0.527163\pi\)
−0.0852303 + 0.996361i \(0.527163\pi\)
\(992\) 0 0
\(993\) −69.1963 −2.19588
\(994\) 0 0
\(995\) −82.8165 −2.62546
\(996\) 0 0
\(997\) 12.8886 0.408186 0.204093 0.978952i \(-0.434576\pi\)
0.204093 + 0.978952i \(0.434576\pi\)
\(998\) 0 0
\(999\) −195.393 −6.18198
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.h.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.a.h.1.9 9 1.1 even 1 trivial