Properties

Label 4840.2.a.k.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $2$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +1.00000 q^{5} -0.267949 q^{7} +O(q^{10})\) \(q-1.73205 q^{3} +1.00000 q^{5} -0.267949 q^{7} -3.46410 q^{13} -1.73205 q^{15} -2.00000 q^{19} +0.464102 q^{21} +7.46410 q^{23} +1.00000 q^{25} +5.19615 q^{27} +4.92820 q^{29} -1.46410 q^{31} -0.267949 q^{35} -4.00000 q^{37} +6.00000 q^{39} +1.92820 q^{41} -1.73205 q^{43} +6.66025 q^{47} -6.92820 q^{49} -7.46410 q^{53} +3.46410 q^{57} -1.46410 q^{59} -1.53590 q^{61} -3.46410 q^{65} +5.73205 q^{67} -12.9282 q^{69} +2.53590 q^{71} -1.07180 q^{73} -1.73205 q^{75} -14.3923 q^{79} -9.00000 q^{81} +7.46410 q^{83} -8.53590 q^{87} +15.9282 q^{89} +0.928203 q^{91} +2.53590 q^{93} -2.00000 q^{95} -14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 4 q^{7} - 4 q^{19} - 6 q^{21} + 8 q^{23} + 2 q^{25} - 4 q^{29} + 4 q^{31} - 4 q^{35} - 8 q^{37} + 12 q^{39} - 10 q^{41} - 4 q^{47} - 8 q^{53} + 4 q^{59} - 10 q^{61} + 8 q^{67} - 12 q^{69} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 18 q^{81} + 8 q^{83} - 24 q^{87} + 18 q^{89} - 12 q^{91} + 12 q^{93} - 4 q^{95} - 8 q^{97}+O(q^{100}) \) 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
\(3\) −1.73205 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.267949 −0.101275 −0.0506376 0.998717i \(-0.516125\pi\)
−0.0506376 + 0.998717i \(0.516125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 0 0
\(15\) −1.73205 −0.447214
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0.464102 0.101275
\(22\) 0 0
\(23\) 7.46410 1.55637 0.778186 0.628033i \(-0.216140\pi\)
0.778186 + 0.628033i \(0.216140\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 4.92820 0.915144 0.457572 0.889172i \(-0.348719\pi\)
0.457572 + 0.889172i \(0.348719\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.267949 −0.0452917
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 1.92820 0.301135 0.150567 0.988600i \(-0.451890\pi\)
0.150567 + 0.988600i \(0.451890\pi\)
\(42\) 0 0
\(43\) −1.73205 −0.264135 −0.132068 0.991241i \(-0.542162\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.66025 0.971498 0.485749 0.874098i \(-0.338547\pi\)
0.485749 + 0.874098i \(0.338547\pi\)
\(48\) 0 0
\(49\) −6.92820 −0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.46410 −1.02527 −0.512637 0.858606i \(-0.671331\pi\)
−0.512637 + 0.858606i \(0.671331\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) −1.46410 −0.190610 −0.0953049 0.995448i \(-0.530383\pi\)
−0.0953049 + 0.995448i \(0.530383\pi\)
\(60\) 0 0
\(61\) −1.53590 −0.196652 −0.0983258 0.995154i \(-0.531349\pi\)
−0.0983258 + 0.995154i \(0.531349\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) 5.73205 0.700281 0.350141 0.936697i \(-0.386134\pi\)
0.350141 + 0.936697i \(0.386134\pi\)
\(68\) 0 0
\(69\) −12.9282 −1.55637
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) −1.07180 −0.125444 −0.0627222 0.998031i \(-0.519978\pi\)
−0.0627222 + 0.998031i \(0.519978\pi\)
\(74\) 0 0
\(75\) −1.73205 −0.200000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.3923 −1.61926 −0.809630 0.586940i \(-0.800332\pi\)
−0.809630 + 0.586940i \(0.800332\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 7.46410 0.819292 0.409646 0.912245i \(-0.365652\pi\)
0.409646 + 0.912245i \(0.365652\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.53590 −0.915144
\(88\) 0 0
\(89\) 15.9282 1.68839 0.844193 0.536039i \(-0.180080\pi\)
0.844193 + 0.536039i \(0.180080\pi\)
\(90\) 0 0
\(91\) 0.928203 0.0973021
\(92\) 0 0
\(93\) 2.53590 0.262960
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −14.3923 −1.46132 −0.730659 0.682743i \(-0.760787\pi\)
−0.730659 + 0.682743i \(0.760787\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.3923 −1.13358 −0.566788 0.823863i \(-0.691814\pi\)
−0.566788 + 0.823863i \(0.691814\pi\)
\(102\) 0 0
\(103\) −2.39230 −0.235721 −0.117860 0.993030i \(-0.537604\pi\)
−0.117860 + 0.993030i \(0.537604\pi\)
\(104\) 0 0
\(105\) 0.464102 0.0452917
\(106\) 0 0
\(107\) −2.26795 −0.219251 −0.109625 0.993973i \(-0.534965\pi\)
−0.109625 + 0.993973i \(0.534965\pi\)
\(108\) 0 0
\(109\) −7.53590 −0.721808 −0.360904 0.932603i \(-0.617532\pi\)
−0.360904 + 0.932603i \(0.617532\pi\)
\(110\) 0 0
\(111\) 6.92820 0.657596
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 7.46410 0.696031
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −3.33975 −0.301135
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.6603 1.65583 0.827915 0.560854i \(-0.189527\pi\)
0.827915 + 0.560854i \(0.189527\pi\)
\(128\) 0 0
\(129\) 3.00000 0.264135
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 0 0
\(133\) 0.535898 0.0464683
\(134\) 0 0
\(135\) 5.19615 0.447214
\(136\) 0 0
\(137\) 10.3923 0.887875 0.443937 0.896058i \(-0.353581\pi\)
0.443937 + 0.896058i \(0.353581\pi\)
\(138\) 0 0
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) 0 0
\(141\) −11.5359 −0.971498
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.92820 0.409265
\(146\) 0 0
\(147\) 12.0000 0.989743
\(148\) 0 0
\(149\) −6.46410 −0.529560 −0.264780 0.964309i \(-0.585299\pi\)
−0.264780 + 0.964309i \(0.585299\pi\)
\(150\) 0 0
\(151\) −4.39230 −0.357441 −0.178720 0.983900i \(-0.557196\pi\)
−0.178720 + 0.983900i \(0.557196\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.46410 −0.117599
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) 12.9282 1.02527
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 8.12436 0.636349 0.318174 0.948032i \(-0.396930\pi\)
0.318174 + 0.948032i \(0.396930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.5167 1.58763 0.793813 0.608161i \(-0.208093\pi\)
0.793813 + 0.608161i \(0.208093\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.46410 −0.263371 −0.131685 0.991292i \(-0.542039\pi\)
−0.131685 + 0.991292i \(0.542039\pi\)
\(174\) 0 0
\(175\) −0.267949 −0.0202551
\(176\) 0 0
\(177\) 2.53590 0.190610
\(178\) 0 0
\(179\) 2.39230 0.178809 0.0894046 0.995995i \(-0.471504\pi\)
0.0894046 + 0.995995i \(0.471504\pi\)
\(180\) 0 0
\(181\) 17.3923 1.29276 0.646380 0.763016i \(-0.276282\pi\)
0.646380 + 0.763016i \(0.276282\pi\)
\(182\) 0 0
\(183\) 2.66025 0.196652
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.39230 −0.101275
\(190\) 0 0
\(191\) −11.0718 −0.801127 −0.400564 0.916269i \(-0.631186\pi\)
−0.400564 + 0.916269i \(0.631186\pi\)
\(192\) 0 0
\(193\) −15.8564 −1.14137 −0.570685 0.821169i \(-0.693322\pi\)
−0.570685 + 0.821169i \(0.693322\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −0.928203 −0.0661317 −0.0330659 0.999453i \(-0.510527\pi\)
−0.0330659 + 0.999453i \(0.510527\pi\)
\(198\) 0 0
\(199\) 6.39230 0.453138 0.226569 0.973995i \(-0.427249\pi\)
0.226569 + 0.973995i \(0.427249\pi\)
\(200\) 0 0
\(201\) −9.92820 −0.700281
\(202\) 0 0
\(203\) −1.32051 −0.0926815
\(204\) 0 0
\(205\) 1.92820 0.134672
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −6.53590 −0.449950 −0.224975 0.974365i \(-0.572230\pi\)
−0.224975 + 0.974365i \(0.572230\pi\)
\(212\) 0 0
\(213\) −4.39230 −0.300956
\(214\) 0 0
\(215\) −1.73205 −0.118125
\(216\) 0 0
\(217\) 0.392305 0.0266314
\(218\) 0 0
\(219\) 1.85641 0.125444
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.267949 0.0179432 0.00897160 0.999960i \(-0.497144\pi\)
0.00897160 + 0.999960i \(0.497144\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.80385 0.451587 0.225794 0.974175i \(-0.427503\pi\)
0.225794 + 0.974175i \(0.427503\pi\)
\(228\) 0 0
\(229\) −4.60770 −0.304485 −0.152243 0.988343i \(-0.548649\pi\)
−0.152243 + 0.988343i \(0.548649\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.3923 −0.942871 −0.471436 0.881900i \(-0.656264\pi\)
−0.471436 + 0.881900i \(0.656264\pi\)
\(234\) 0 0
\(235\) 6.66025 0.434467
\(236\) 0 0
\(237\) 24.9282 1.61926
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) −21.0000 −1.35273 −0.676364 0.736567i \(-0.736446\pi\)
−0.676364 + 0.736567i \(0.736446\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.92820 −0.442627
\(246\) 0 0
\(247\) 6.92820 0.440831
\(248\) 0 0
\(249\) −12.9282 −0.819292
\(250\) 0 0
\(251\) 10.5359 0.665020 0.332510 0.943100i \(-0.392104\pi\)
0.332510 + 0.943100i \(0.392104\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.2487 1.63735 0.818675 0.574257i \(-0.194709\pi\)
0.818675 + 0.574257i \(0.194709\pi\)
\(258\) 0 0
\(259\) 1.07180 0.0665982
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −26.3923 −1.62742 −0.813710 0.581272i \(-0.802555\pi\)
−0.813710 + 0.581272i \(0.802555\pi\)
\(264\) 0 0
\(265\) −7.46410 −0.458516
\(266\) 0 0
\(267\) −27.5885 −1.68839
\(268\) 0 0
\(269\) −19.2487 −1.17361 −0.586807 0.809727i \(-0.699615\pi\)
−0.586807 + 0.809727i \(0.699615\pi\)
\(270\) 0 0
\(271\) −18.9282 −1.14981 −0.574903 0.818221i \(-0.694960\pi\)
−0.574903 + 0.818221i \(0.694960\pi\)
\(272\) 0 0
\(273\) −1.60770 −0.0973021
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −31.8564 −1.90039 −0.950197 0.311650i \(-0.899118\pi\)
−0.950197 + 0.311650i \(0.899118\pi\)
\(282\) 0 0
\(283\) −18.5167 −1.10070 −0.550351 0.834934i \(-0.685506\pi\)
−0.550351 + 0.834934i \(0.685506\pi\)
\(284\) 0 0
\(285\) 3.46410 0.205196
\(286\) 0 0
\(287\) −0.516660 −0.0304975
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 24.9282 1.46132
\(292\) 0 0
\(293\) −30.2487 −1.76715 −0.883574 0.468291i \(-0.844870\pi\)
−0.883574 + 0.468291i \(0.844870\pi\)
\(294\) 0 0
\(295\) −1.46410 −0.0852433
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −25.8564 −1.49531
\(300\) 0 0
\(301\) 0.464102 0.0267504
\(302\) 0 0
\(303\) 19.7321 1.13358
\(304\) 0 0
\(305\) −1.53590 −0.0879453
\(306\) 0 0
\(307\) −26.3923 −1.50629 −0.753144 0.657855i \(-0.771464\pi\)
−0.753144 + 0.657855i \(0.771464\pi\)
\(308\) 0 0
\(309\) 4.14359 0.235721
\(310\) 0 0
\(311\) 11.0718 0.627824 0.313912 0.949452i \(-0.398360\pi\)
0.313912 + 0.949452i \(0.398360\pi\)
\(312\) 0 0
\(313\) 3.46410 0.195803 0.0979013 0.995196i \(-0.468787\pi\)
0.0979013 + 0.995196i \(0.468787\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −19.8564 −1.11525 −0.557623 0.830094i \(-0.688287\pi\)
−0.557623 + 0.830094i \(0.688287\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.92820 0.219251
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3.46410 −0.192154
\(326\) 0 0
\(327\) 13.0526 0.721808
\(328\) 0 0
\(329\) −1.78461 −0.0983887
\(330\) 0 0
\(331\) −20.9282 −1.15032 −0.575159 0.818042i \(-0.695060\pi\)
−0.575159 + 0.818042i \(0.695060\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.73205 0.313175
\(336\) 0 0
\(337\) 1.32051 0.0719327 0.0359663 0.999353i \(-0.488549\pi\)
0.0359663 + 0.999353i \(0.488549\pi\)
\(338\) 0 0
\(339\) 17.3205 0.940721
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 3.73205 0.201512
\(344\) 0 0
\(345\) −12.9282 −0.696031
\(346\) 0 0
\(347\) 5.73205 0.307713 0.153856 0.988093i \(-0.450831\pi\)
0.153856 + 0.988093i \(0.450831\pi\)
\(348\) 0 0
\(349\) −29.7128 −1.59049 −0.795245 0.606288i \(-0.792658\pi\)
−0.795245 + 0.606288i \(0.792658\pi\)
\(350\) 0 0
\(351\) −18.0000 −0.960769
\(352\) 0 0
\(353\) 33.4641 1.78111 0.890557 0.454871i \(-0.150315\pi\)
0.890557 + 0.454871i \(0.150315\pi\)
\(354\) 0 0
\(355\) 2.53590 0.134592
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.7846 −1.41364 −0.706819 0.707395i \(-0.749870\pi\)
−0.706819 + 0.707395i \(0.749870\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.07180 −0.0561004
\(366\) 0 0
\(367\) 33.5885 1.75330 0.876652 0.481126i \(-0.159772\pi\)
0.876652 + 0.481126i \(0.159772\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 12.3923 0.641649 0.320825 0.947139i \(-0.396040\pi\)
0.320825 + 0.947139i \(0.396040\pi\)
\(374\) 0 0
\(375\) −1.73205 −0.0894427
\(376\) 0 0
\(377\) −17.0718 −0.879242
\(378\) 0 0
\(379\) −6.39230 −0.328351 −0.164175 0.986431i \(-0.552496\pi\)
−0.164175 + 0.986431i \(0.552496\pi\)
\(380\) 0 0
\(381\) −32.3205 −1.65583
\(382\) 0 0
\(383\) 12.5359 0.640554 0.320277 0.947324i \(-0.396224\pi\)
0.320277 + 0.947324i \(0.396224\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −23.2487 −1.17876 −0.589378 0.807857i \(-0.700627\pi\)
−0.589378 + 0.807857i \(0.700627\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 24.2487 1.22319
\(394\) 0 0
\(395\) −14.3923 −0.724155
\(396\) 0 0
\(397\) −30.6410 −1.53783 −0.768914 0.639352i \(-0.779203\pi\)
−0.768914 + 0.639352i \(0.779203\pi\)
\(398\) 0 0
\(399\) −0.928203 −0.0464683
\(400\) 0 0
\(401\) −10.8564 −0.542143 −0.271072 0.962559i \(-0.587378\pi\)
−0.271072 + 0.962559i \(0.587378\pi\)
\(402\) 0 0
\(403\) 5.07180 0.252644
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 26.8564 1.32796 0.663982 0.747749i \(-0.268865\pi\)
0.663982 + 0.747749i \(0.268865\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 0.392305 0.0193041
\(414\) 0 0
\(415\) 7.46410 0.366398
\(416\) 0 0
\(417\) 14.5359 0.711826
\(418\) 0 0
\(419\) 15.8564 0.774636 0.387318 0.921946i \(-0.373402\pi\)
0.387318 + 0.921946i \(0.373402\pi\)
\(420\) 0 0
\(421\) 18.4641 0.899885 0.449943 0.893057i \(-0.351444\pi\)
0.449943 + 0.893057i \(0.351444\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.411543 0.0199159
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.3923 −1.46395 −0.731973 0.681334i \(-0.761400\pi\)
−0.731973 + 0.681334i \(0.761400\pi\)
\(432\) 0 0
\(433\) 29.1769 1.40215 0.701077 0.713086i \(-0.252703\pi\)
0.701077 + 0.713086i \(0.252703\pi\)
\(434\) 0 0
\(435\) −8.53590 −0.409265
\(436\) 0 0
\(437\) −14.9282 −0.714113
\(438\) 0 0
\(439\) −1.07180 −0.0511541 −0.0255770 0.999673i \(-0.508142\pi\)
−0.0255770 + 0.999673i \(0.508142\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.6603 −0.981598 −0.490799 0.871273i \(-0.663295\pi\)
−0.490799 + 0.871273i \(0.663295\pi\)
\(444\) 0 0
\(445\) 15.9282 0.755069
\(446\) 0 0
\(447\) 11.1962 0.529560
\(448\) 0 0
\(449\) 22.8564 1.07866 0.539330 0.842094i \(-0.318677\pi\)
0.539330 + 0.842094i \(0.318677\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 7.60770 0.357441
\(454\) 0 0
\(455\) 0.928203 0.0435148
\(456\) 0 0
\(457\) −6.24871 −0.292302 −0.146151 0.989262i \(-0.546689\pi\)
−0.146151 + 0.989262i \(0.546689\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 35.2487 1.64170 0.820848 0.571147i \(-0.193501\pi\)
0.820848 + 0.571147i \(0.193501\pi\)
\(462\) 0 0
\(463\) −19.7321 −0.917026 −0.458513 0.888688i \(-0.651618\pi\)
−0.458513 + 0.888688i \(0.651618\pi\)
\(464\) 0 0
\(465\) 2.53590 0.117599
\(466\) 0 0
\(467\) −15.3397 −0.709839 −0.354919 0.934897i \(-0.615492\pi\)
−0.354919 + 0.934897i \(0.615492\pi\)
\(468\) 0 0
\(469\) −1.53590 −0.0709212
\(470\) 0 0
\(471\) 34.6410 1.59617
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 28.5359 1.30384 0.651919 0.758288i \(-0.273964\pi\)
0.651919 + 0.758288i \(0.273964\pi\)
\(480\) 0 0
\(481\) 13.8564 0.631798
\(482\) 0 0
\(483\) 3.46410 0.157622
\(484\) 0 0
\(485\) −14.3923 −0.653521
\(486\) 0 0
\(487\) 6.67949 0.302677 0.151338 0.988482i \(-0.451642\pi\)
0.151338 + 0.988482i \(0.451642\pi\)
\(488\) 0 0
\(489\) −14.0718 −0.636349
\(490\) 0 0
\(491\) 32.2487 1.45536 0.727682 0.685915i \(-0.240598\pi\)
0.727682 + 0.685915i \(0.240598\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.679492 −0.0304794
\(498\) 0 0
\(499\) 33.3205 1.49163 0.745815 0.666153i \(-0.232060\pi\)
0.745815 + 0.666153i \(0.232060\pi\)
\(500\) 0 0
\(501\) −35.5359 −1.58763
\(502\) 0 0
\(503\) −38.6603 −1.72378 −0.861888 0.507099i \(-0.830718\pi\)
−0.861888 + 0.507099i \(0.830718\pi\)
\(504\) 0 0
\(505\) −11.3923 −0.506951
\(506\) 0 0
\(507\) 1.73205 0.0769231
\(508\) 0 0
\(509\) −10.4641 −0.463813 −0.231907 0.972738i \(-0.574496\pi\)
−0.231907 + 0.972738i \(0.574496\pi\)
\(510\) 0 0
\(511\) 0.287187 0.0127044
\(512\) 0 0
\(513\) −10.3923 −0.458831
\(514\) 0 0
\(515\) −2.39230 −0.105418
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) 0 0
\(523\) −9.32051 −0.407557 −0.203779 0.979017i \(-0.565322\pi\)
−0.203779 + 0.979017i \(0.565322\pi\)
\(524\) 0 0
\(525\) 0.464102 0.0202551
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 32.7128 1.42230
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.67949 −0.289321
\(534\) 0 0
\(535\) −2.26795 −0.0980520
\(536\) 0 0
\(537\) −4.14359 −0.178809
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −12.4641 −0.535874 −0.267937 0.963436i \(-0.586342\pi\)
−0.267937 + 0.963436i \(0.586342\pi\)
\(542\) 0 0
\(543\) −30.1244 −1.29276
\(544\) 0 0
\(545\) −7.53590 −0.322802
\(546\) 0 0
\(547\) 22.3923 0.957426 0.478713 0.877971i \(-0.341103\pi\)
0.478713 + 0.877971i \(0.341103\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.85641 −0.419897
\(552\) 0 0
\(553\) 3.85641 0.163991
\(554\) 0 0
\(555\) 6.92820 0.294086
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −32.6603 −1.37647 −0.688233 0.725490i \(-0.741613\pi\)
−0.688233 + 0.725490i \(0.741613\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) 0 0
\(567\) 2.41154 0.101275
\(568\) 0 0
\(569\) −18.7128 −0.784482 −0.392241 0.919863i \(-0.628300\pi\)
−0.392241 + 0.919863i \(0.628300\pi\)
\(570\) 0 0
\(571\) −16.2487 −0.679987 −0.339994 0.940428i \(-0.610425\pi\)
−0.339994 + 0.940428i \(0.610425\pi\)
\(572\) 0 0
\(573\) 19.1769 0.801127
\(574\) 0 0
\(575\) 7.46410 0.311275
\(576\) 0 0
\(577\) 8.39230 0.349376 0.174688 0.984624i \(-0.444108\pi\)
0.174688 + 0.984624i \(0.444108\pi\)
\(578\) 0 0
\(579\) 27.4641 1.14137
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.94744 0.204203 0.102101 0.994774i \(-0.467443\pi\)
0.102101 + 0.994774i \(0.467443\pi\)
\(588\) 0 0
\(589\) 2.92820 0.120655
\(590\) 0 0
\(591\) 1.60770 0.0661317
\(592\) 0 0
\(593\) 6.39230 0.262500 0.131250 0.991349i \(-0.458101\pi\)
0.131250 + 0.991349i \(0.458101\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11.0718 −0.453138
\(598\) 0 0
\(599\) −24.9282 −1.01854 −0.509269 0.860607i \(-0.670084\pi\)
−0.509269 + 0.860607i \(0.670084\pi\)
\(600\) 0 0
\(601\) −23.8564 −0.973123 −0.486562 0.873646i \(-0.661749\pi\)
−0.486562 + 0.873646i \(0.661749\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 27.4641 1.11473 0.557367 0.830266i \(-0.311812\pi\)
0.557367 + 0.830266i \(0.311812\pi\)
\(608\) 0 0
\(609\) 2.28719 0.0926815
\(610\) 0 0
\(611\) −23.0718 −0.933385
\(612\) 0 0
\(613\) −13.7128 −0.553855 −0.276928 0.960891i \(-0.589316\pi\)
−0.276928 + 0.960891i \(0.589316\pi\)
\(614\) 0 0
\(615\) −3.33975 −0.134672
\(616\) 0 0
\(617\) 11.3205 0.455746 0.227873 0.973691i \(-0.426823\pi\)
0.227873 + 0.973691i \(0.426823\pi\)
\(618\) 0 0
\(619\) −17.0718 −0.686173 −0.343087 0.939304i \(-0.611472\pi\)
−0.343087 + 0.939304i \(0.611472\pi\)
\(620\) 0 0
\(621\) 38.7846 1.55637
\(622\) 0 0
\(623\) −4.26795 −0.170992
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −27.7128 −1.10323 −0.551615 0.834099i \(-0.685988\pi\)
−0.551615 + 0.834099i \(0.685988\pi\)
\(632\) 0 0
\(633\) 11.3205 0.449950
\(634\) 0 0
\(635\) 18.6603 0.740510
\(636\) 0 0
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.7846 1.53190 0.765950 0.642900i \(-0.222269\pi\)
0.765950 + 0.642900i \(0.222269\pi\)
\(642\) 0 0
\(643\) −14.8038 −0.583807 −0.291903 0.956448i \(-0.594289\pi\)
−0.291903 + 0.956448i \(0.594289\pi\)
\(644\) 0 0
\(645\) 3.00000 0.118125
\(646\) 0 0
\(647\) 5.87564 0.230995 0.115498 0.993308i \(-0.463154\pi\)
0.115498 + 0.993308i \(0.463154\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.679492 −0.0266314
\(652\) 0 0
\(653\) 3.85641 0.150913 0.0754564 0.997149i \(-0.475959\pi\)
0.0754564 + 0.997149i \(0.475959\pi\)
\(654\) 0 0
\(655\) −14.0000 −0.547025
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.1769 1.83775 0.918876 0.394547i \(-0.129098\pi\)
0.918876 + 0.394547i \(0.129098\pi\)
\(660\) 0 0
\(661\) 4.32051 0.168048 0.0840241 0.996464i \(-0.473223\pi\)
0.0840241 + 0.996464i \(0.473223\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.535898 0.0207812
\(666\) 0 0
\(667\) 36.7846 1.42431
\(668\) 0 0
\(669\) −0.464102 −0.0179432
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 29.4641 1.13576 0.567879 0.823112i \(-0.307764\pi\)
0.567879 + 0.823112i \(0.307764\pi\)
\(674\) 0 0
\(675\) 5.19615 0.200000
\(676\) 0 0
\(677\) 39.7128 1.52629 0.763144 0.646229i \(-0.223655\pi\)
0.763144 + 0.646229i \(0.223655\pi\)
\(678\) 0 0
\(679\) 3.85641 0.147995
\(680\) 0 0
\(681\) −11.7846 −0.451587
\(682\) 0 0
\(683\) −19.5885 −0.749531 −0.374766 0.927120i \(-0.622277\pi\)
−0.374766 + 0.927120i \(0.622277\pi\)
\(684\) 0 0
\(685\) 10.3923 0.397070
\(686\) 0 0
\(687\) 7.98076 0.304485
\(688\) 0 0
\(689\) 25.8564 0.985051
\(690\) 0 0
\(691\) −43.5692 −1.65745 −0.828726 0.559655i \(-0.810934\pi\)
−0.828726 + 0.559655i \(0.810934\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.39230 −0.318338
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 24.9282 0.942871
\(700\) 0 0
\(701\) −13.7128 −0.517926 −0.258963 0.965887i \(-0.583381\pi\)
−0.258963 + 0.965887i \(0.583381\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −11.5359 −0.434467
\(706\) 0 0
\(707\) 3.05256 0.114803
\(708\) 0 0
\(709\) −21.3923 −0.803405 −0.401702 0.915770i \(-0.631581\pi\)
−0.401702 + 0.915770i \(0.631581\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.9282 −0.409264
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.2487 0.905585
\(718\) 0 0
\(719\) 32.6410 1.21730 0.608652 0.793437i \(-0.291710\pi\)
0.608652 + 0.793437i \(0.291710\pi\)
\(720\) 0 0
\(721\) 0.641016 0.0238727
\(722\) 0 0
\(723\) 36.3731 1.35273
\(724\) 0 0
\(725\) 4.92820 0.183029
\(726\) 0 0
\(727\) 16.8038 0.623220 0.311610 0.950210i \(-0.399132\pi\)
0.311610 + 0.950210i \(0.399132\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 28.0000 1.03420 0.517102 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(734\) 0 0
\(735\) 12.0000 0.442627
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −51.1769 −1.88257 −0.941287 0.337609i \(-0.890382\pi\)
−0.941287 + 0.337609i \(0.890382\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 4.26795 0.156576 0.0782879 0.996931i \(-0.475055\pi\)
0.0782879 + 0.996931i \(0.475055\pi\)
\(744\) 0 0
\(745\) −6.46410 −0.236826
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.607695 0.0222047
\(750\) 0 0
\(751\) 47.4641 1.73199 0.865995 0.500053i \(-0.166686\pi\)
0.865995 + 0.500053i \(0.166686\pi\)
\(752\) 0 0
\(753\) −18.2487 −0.665020
\(754\) 0 0
\(755\) −4.39230 −0.159852
\(756\) 0 0
\(757\) −25.3205 −0.920290 −0.460145 0.887844i \(-0.652202\pi\)
−0.460145 + 0.887844i \(0.652202\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.14359 −0.150205 −0.0751026 0.997176i \(-0.523928\pi\)
−0.0751026 + 0.997176i \(0.523928\pi\)
\(762\) 0 0
\(763\) 2.01924 0.0731013
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.07180 0.183132
\(768\) 0 0
\(769\) −15.8564 −0.571797 −0.285898 0.958260i \(-0.592292\pi\)
−0.285898 + 0.958260i \(0.592292\pi\)
\(770\) 0 0
\(771\) −45.4641 −1.63735
\(772\) 0 0
\(773\) 9.46410 0.340400 0.170200 0.985410i \(-0.445559\pi\)
0.170200 + 0.985410i \(0.445559\pi\)
\(774\) 0 0
\(775\) −1.46410 −0.0525921
\(776\) 0 0
\(777\) −1.85641 −0.0665982
\(778\) 0 0
\(779\) −3.85641 −0.138170
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 25.6077 0.915144
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 27.5885 0.983422 0.491711 0.870758i \(-0.336372\pi\)
0.491711 + 0.870758i \(0.336372\pi\)
\(788\) 0 0
\(789\) 45.7128 1.62742
\(790\) 0 0
\(791\) 2.67949 0.0952718
\(792\) 0 0
\(793\) 5.32051 0.188937
\(794\) 0 0
\(795\) 12.9282 0.458516
\(796\) 0 0
\(797\) −34.5359 −1.22332 −0.611662 0.791119i \(-0.709499\pi\)
−0.611662 + 0.791119i \(0.709499\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) 33.3397 1.17361
\(808\) 0 0
\(809\) −38.7846 −1.36359 −0.681797 0.731541i \(-0.738801\pi\)
−0.681797 + 0.731541i \(0.738801\pi\)
\(810\) 0 0
\(811\) 44.3923 1.55882 0.779412 0.626511i \(-0.215518\pi\)
0.779412 + 0.626511i \(0.215518\pi\)
\(812\) 0 0
\(813\) 32.7846 1.14981
\(814\) 0 0
\(815\) 8.12436 0.284584
\(816\) 0 0
\(817\) 3.46410 0.121194
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.1051 0.876175 0.438087 0.898932i \(-0.355656\pi\)
0.438087 + 0.898932i \(0.355656\pi\)
\(822\) 0 0
\(823\) 26.1244 0.910638 0.455319 0.890328i \(-0.349525\pi\)
0.455319 + 0.890328i \(0.349525\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.80385 0.0974993 0.0487497 0.998811i \(-0.484476\pi\)
0.0487497 + 0.998811i \(0.484476\pi\)
\(828\) 0 0
\(829\) −23.2487 −0.807461 −0.403731 0.914878i \(-0.632287\pi\)
−0.403731 + 0.914878i \(0.632287\pi\)
\(830\) 0 0
\(831\) −38.1051 −1.32185
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 20.5167 0.710008
\(836\) 0 0
\(837\) −7.60770 −0.262960
\(838\) 0 0
\(839\) 27.4641 0.948166 0.474083 0.880480i \(-0.342780\pi\)
0.474083 + 0.880480i \(0.342780\pi\)
\(840\) 0 0
\(841\) −4.71281 −0.162511
\(842\) 0 0
\(843\) 55.1769 1.90039
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 32.0718 1.10070
\(850\) 0 0
\(851\) −29.8564 −1.02346
\(852\) 0 0
\(853\) −13.0718 −0.447570 −0.223785 0.974639i \(-0.571841\pi\)
−0.223785 + 0.974639i \(0.571841\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 48.9282 1.67136 0.835678 0.549220i \(-0.185075\pi\)
0.835678 + 0.549220i \(0.185075\pi\)
\(858\) 0 0
\(859\) −39.0718 −1.33311 −0.666556 0.745455i \(-0.732232\pi\)
−0.666556 + 0.745455i \(0.732232\pi\)
\(860\) 0 0
\(861\) 0.894882 0.0304975
\(862\) 0 0
\(863\) −32.5167 −1.10688 −0.553440 0.832889i \(-0.686685\pi\)
−0.553440 + 0.832889i \(0.686685\pi\)
\(864\) 0 0
\(865\) −3.46410 −0.117783
\(866\) 0 0
\(867\) 29.4449 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −19.8564 −0.672809
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.267949 −0.00905834
\(876\) 0 0
\(877\) −35.6077 −1.20239 −0.601193 0.799104i \(-0.705308\pi\)
−0.601193 + 0.799104i \(0.705308\pi\)
\(878\) 0 0
\(879\) 52.3923 1.76715
\(880\) 0 0
\(881\) 0.856406 0.0288531 0.0144265 0.999896i \(-0.495408\pi\)
0.0144265 + 0.999896i \(0.495408\pi\)
\(882\) 0 0
\(883\) 11.4641 0.385798 0.192899 0.981219i \(-0.438211\pi\)
0.192899 + 0.981219i \(0.438211\pi\)
\(884\) 0 0
\(885\) 2.53590 0.0852433
\(886\) 0 0
\(887\) 12.5167 0.420268 0.210134 0.977673i \(-0.432610\pi\)
0.210134 + 0.977673i \(0.432610\pi\)
\(888\) 0 0
\(889\) −5.00000 −0.167695
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13.3205 −0.445754
\(894\) 0 0
\(895\) 2.39230 0.0799659
\(896\) 0 0
\(897\) 44.7846 1.49531
\(898\) 0 0
\(899\) −7.21539 −0.240647
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −0.803848 −0.0267504
\(904\) 0 0
\(905\) 17.3923 0.578140
\(906\) 0 0
\(907\) −52.9090 −1.75681 −0.878407 0.477914i \(-0.841393\pi\)
−0.878407 + 0.477914i \(0.841393\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.2487 −0.405818 −0.202909 0.979198i \(-0.565040\pi\)
−0.202909 + 0.979198i \(0.565040\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 2.66025 0.0879453
\(916\) 0 0
\(917\) 3.75129 0.123878
\(918\) 0 0
\(919\) 42.7846 1.41133 0.705667 0.708544i \(-0.250647\pi\)
0.705667 + 0.708544i \(0.250647\pi\)
\(920\) 0 0
\(921\) 45.7128 1.50629
\(922\) 0 0
\(923\) −8.78461 −0.289149
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −13.2154 −0.433583 −0.216791 0.976218i \(-0.569559\pi\)
−0.216791 + 0.976218i \(0.569559\pi\)
\(930\) 0 0
\(931\) 13.8564 0.454125
\(932\) 0 0
\(933\) −19.1769 −0.627824
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 39.0718 1.27642 0.638210 0.769862i \(-0.279675\pi\)
0.638210 + 0.769862i \(0.279675\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −20.4641 −0.667111 −0.333555 0.942731i \(-0.608248\pi\)
−0.333555 + 0.942731i \(0.608248\pi\)
\(942\) 0 0
\(943\) 14.3923 0.468678
\(944\) 0 0
\(945\) −1.39230 −0.0452917
\(946\) 0 0
\(947\) 49.3205 1.60270 0.801351 0.598195i \(-0.204115\pi\)
0.801351 + 0.598195i \(0.204115\pi\)
\(948\) 0 0
\(949\) 3.71281 0.120523
\(950\) 0 0
\(951\) 34.3923 1.11525
\(952\) 0 0
\(953\) −4.00000 −0.129573 −0.0647864 0.997899i \(-0.520637\pi\)
−0.0647864 + 0.997899i \(0.520637\pi\)
\(954\) 0 0
\(955\) −11.0718 −0.358275
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.78461 −0.0899197
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −15.8564 −0.510436
\(966\) 0 0
\(967\) −3.17691 −0.102163 −0.0510813 0.998694i \(-0.516267\pi\)
−0.0510813 + 0.998694i \(0.516267\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −0.784610 −0.0251793 −0.0125897 0.999921i \(-0.504008\pi\)
−0.0125897 + 0.999921i \(0.504008\pi\)
\(972\) 0 0
\(973\) 2.24871 0.0720904
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) 22.9282 0.733538 0.366769 0.930312i \(-0.380464\pi\)
0.366769 + 0.930312i \(0.380464\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.7321 −1.26726 −0.633628 0.773638i \(-0.718435\pi\)
−0.633628 + 0.773638i \(0.718435\pi\)
\(984\) 0 0
\(985\) −0.928203 −0.0295750
\(986\) 0 0
\(987\) 3.09103 0.0983887
\(988\) 0 0
\(989\) −12.9282 −0.411093
\(990\) 0 0
\(991\) 6.78461 0.215520 0.107760 0.994177i \(-0.465632\pi\)
0.107760 + 0.994177i \(0.465632\pi\)
\(992\) 0 0
\(993\) 36.2487 1.15032
\(994\) 0 0
\(995\) 6.39230 0.202650
\(996\) 0 0
\(997\) 2.67949 0.0848604 0.0424302 0.999099i \(-0.486490\pi\)
0.0424302 + 0.999099i \(0.486490\pi\)
\(998\) 0 0
\(999\) −20.7846 −0.657596
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 4840.2.a.k.1.1 2
4.3 odd 2 9680.2.a.br.1.2 2
11.10 odd 2 4840.2.a.l.1.1 yes 2
44.43 even 2 9680.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.k.1.1 2 1.1 even 1 trivial
4840.2.a.l.1.1 yes 2 11.10 odd 2
9680.2.a.bq.1.2 2 44.43 even 2
9680.2.a.br.1.2 2 4.3 odd 2