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Properties

Label 4620.2.a.n.1.1
Level $4620$
Weight $2$
Character 4620.1
Self dual yes
Analytic conductor $36.891$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4620,2,Mod(1,4620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4620.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4620.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: \(36.8908857338\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4620.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -4.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} -7.00000 q^{19} +1.00000 q^{21} -9.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.00000 q^{29} +2.00000 q^{31} -1.00000 q^{33} +1.00000 q^{35} -4.00000 q^{37} -4.00000 q^{39} -6.00000 q^{41} -1.00000 q^{43} +1.00000 q^{45} -6.00000 q^{47} +1.00000 q^{49} +3.00000 q^{51} +3.00000 q^{53} -1.00000 q^{55} -7.00000 q^{57} -9.00000 q^{59} -1.00000 q^{61} +1.00000 q^{63} -4.00000 q^{65} -10.0000 q^{67} -9.00000 q^{69} +6.00000 q^{71} +8.00000 q^{73} +1.00000 q^{75} -1.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +3.00000 q^{83} +3.00000 q^{85} -3.00000 q^{87} -15.0000 q^{89} -4.00000 q^{91} +2.00000 q^{93} -7.00000 q^{95} -1.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −9.00000 −1.87663 −0.938315 − 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) −7.00000 −0.718185
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 − 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −9.00000 −0.846649 −0.423324 − 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 − 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 − 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 − 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 − 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) −9.00000 −0.625543
\(208\) 0 0
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 − 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −13.0000 −0.870544 −0.435272 − 0.900299i \(-0.643348\pi\)
−0.435272 + 0.900299i \(0.643348\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 3.00000 0.199117 0.0995585 − 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 − 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 28.0000 1.78160
\(248\) 0 0
\(249\) 3.00000 0.190117
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 9.00000 0.565825
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) −15.0000 −0.917985
\(268\) 0 0
\(269\) 9.00000 0.548740 0.274370 − 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 − 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) −7.00000 −0.414644
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 0 0
\(293\) 15.0000 0.876309 0.438155 − 0.898900i \(-0.355632\pi\)
0.438155 + 0.898900i \(0.355632\pi\)
\(294\) 0 0
\(295\) −9.00000 −0.524000
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 36.0000 2.08193
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 − 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 − 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −21.0000 −1.16847
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 23.0000 1.26419 0.632097 − 0.774889i \(-0.282194\pi\)
0.632097 + 0.774889i \(0.282194\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 29.0000 1.57973 0.789865 − 0.613280i \(-0.210150\pi\)
0.789865 + 0.613280i \(0.210150\pi\)
\(338\) 0 0
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −9.00000 −0.484544
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 − 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 11.0000 0.588817 0.294408 − 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) 9.00000 0.475002 0.237501 − 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) −19.0000 −0.991792 −0.495896 − 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) 0 0
\(373\) 5.00000 0.258890 0.129445 − 0.991587i \(-0.458680\pi\)
0.129445 + 0.991587i \(0.458680\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 − 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 0 0
\(381\) 11.0000 0.563547
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −27.0000 −1.36545
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) −7.00000 −0.350438
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −34.0000 −1.68119 −0.840596 − 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) −9.00000 −0.442861
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 21.0000 1.02592 0.512959 − 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 − 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) 0 0
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 − 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 0 0
\(437\) 63.0000 3.01370
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 − 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −15.0000 −0.711068
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 0 0
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 17.0000 0.795226 0.397613 − 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) 2.00000 0.0927478
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) 11.0000 0.506853
\(472\) 0 0
\(473\) 1.00000 0.0459800
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) −9.00000 −0.409514
\(484\) 0 0
\(485\) −1.00000 −0.0454077
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 − 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −19.0000 −0.850557 −0.425278 − 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −15.0000 −0.668817 −0.334408 − 0.942428i \(-0.608537\pi\)
−0.334408 + 0.942428i \(0.608537\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −21.0000 −0.930809 −0.465404 − 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) −7.00000 −0.309058
\(514\) 0 0
\(515\) −7.00000 −0.308457
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 − 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 − 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) 0 0
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 21.0000 0.894630
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 − 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.00000 −0.375326
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) −3.00000 −0.124247
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 − 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 − 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 − 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) 0 0
\(623\) −15.0000 −0.600962
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.00000 0.279553
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 − 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 2.00000 0.0794929
\(634\) 0 0
\(635\) 11.0000 0.436522
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 − 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) 33.0000 1.29139 0.645695 − 0.763596i \(-0.276568\pi\)
0.645695 + 0.763596i \(0.276568\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 − 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −7.00000 −0.271448
\(666\) 0 0
\(667\) 27.0000 1.04544
\(668\) 0 0
\(669\) −13.0000 −0.502609
\(670\) 0 0
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −49.0000 −1.88881 −0.944406 − 0.328783i \(-0.893362\pi\)
−0.944406 + 0.328783i \(0.893362\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 27.0000 1.03769 0.518847 − 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 − 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 0 0
\(703\) 28.0000 1.05604
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 18.0000 0.676960
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 − 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) −15.0000 −0.560185
\(718\) 0 0
\(719\) −9.00000 −0.335643 −0.167822 − 0.985817i \(-0.553673\pi\)
−0.167822 + 0.985817i \(0.553673\pi\)
\(720\) 0 0
\(721\) −7.00000 −0.260694
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 35.0000 1.29808 0.649039 − 0.760755i \(-0.275171\pi\)
0.649039 + 0.760755i \(0.275171\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 28.0000 1.02861
\(742\) 0 0
\(743\) 18.0000 0.660356 0.330178 − 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 3.00000 0.109764
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 − 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) 36.0000 1.29988
\(768\) 0 0
\(769\) −37.0000 −1.33425 −0.667127 − 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.0000 0.647415 0.323708 − 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) 42.0000 1.50481
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 11.0000 0.392607
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 3.00000 0.106399
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) −15.0000 −0.529999
\(802\) 0 0
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) −9.00000 −0.317208
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) −25.0000 −0.876788
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 7.00000 0.244899
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 − 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 0 0
\(823\) −52.0000 −1.81261 −0.906303 − 0.422628i \(-0.861108\pi\)
−0.906303 + 0.422628i \(0.861108\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −24.0000 −0.834562 −0.417281 − 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 − 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) 39.0000 1.34643 0.673215 − 0.739447i \(-0.264913\pi\)
0.673215 + 0.739447i \(0.264913\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 − 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) −7.00000 −0.239395
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 − 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −9.00000 −0.306364 −0.153182 − 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 − 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 0 0
\(879\) 15.0000 0.505937
\(880\) 0 0
\(881\) 45.0000 1.51609 0.758044 − 0.652203i \(-0.226155\pi\)
0.758044 + 0.652203i \(0.226155\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) −9.00000 −0.302532
\(886\) 0 0
\(887\) 27.0000 0.906571 0.453286 − 0.891365i \(-0.350252\pi\)
0.453286 + 0.891365i \(0.350252\pi\)
\(888\) 0 0
\(889\) 11.0000 0.368928
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 42.0000 1.40548
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) 0 0
\(897\) 36.0000 1.20201
\(898\) 0 0
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) −1.00000 −0.0332779
\(904\) 0 0
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −3.00000 −0.0992855
\(914\) 0 0
\(915\) −1.00000 −0.0330590
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 14.0000 0.461817 0.230909 − 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −7.00000 −0.229910
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.00000 −0.0981105
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 − 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 54.0000 1.75848
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −33.0000 −1.07236 −0.536178 − 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.00000 0.0969762
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) −1.00000 −0.0321578 −0.0160789 − 0.999871i \(-0.505118\pi\)
−0.0160789 + 0.999871i \(0.505118\pi\)
\(968\) 0 0
\(969\) −21.0000 −0.674617
\(970\) 0 0
\(971\) −21.0000 −0.673922 −0.336961 − 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) 27.0000 0.863807 0.431903 − 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 0 0
\(979\) 15.0000 0.479402
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 30.0000 0.956851 0.478426 − 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 9.00000 0.286183
\(990\) 0 0
\(991\) 29.0000 0.921215 0.460608 − 0.887604i \(-0.347632\pi\)
0.460608 + 0.887604i \(0.347632\pi\)
\(992\) 0 0
\(993\) 23.0000 0.729883
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 4620.2.a.n.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4620.2.a.n.1.1 ✓ 1 1.1 even 1 trivial