Properties

Label 4730.2.a.k.1.1
Level $4730$
Weight $2$
Character 4730.1
Self dual yes
Analytic conductor $37.769$
Analytic rank $0$
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] = [4730,2,Mod(1,4730)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4730, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4730.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4730.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: \(37.7692401561\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 4730.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} +6.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -4.00000 q^{27} +2.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{38} +12.0000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -1.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -2.00000 q^{47} +2.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +6.00000 q^{52} -4.00000 q^{53} -4.00000 q^{54} +1.00000 q^{55} +4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +1.00000 q^{64} +6.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} +2.00000 q^{68} -12.0000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +2.00000 q^{75} +2.00000 q^{76} +12.0000 q^{78} +10.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{85} -1.00000 q^{86} +4.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} -6.00000 q^{92} -2.00000 q^{94} +2.00000 q^{95} +2.00000 q^{96} +6.00000 q^{97} -7.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 2.00000 0.324443
\(39\) 12.0000 1.92154
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) −6.00000 −0.884652
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 2.00000 0.288675
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 6.00000 0.832050
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −4.00000 −0.544331
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 2.00000 0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 6.00000 0.697486
\(75\) 2.00000 0.230940
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 12.0000 1.35873
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 2.00000 0.220863
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −1.00000 −0.107833
\(87\) 4.00000 0.428845
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 2.00000 0.205196
\(96\) 2.00000 0.204124
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −7.00000 −0.707107
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 4.00000 0.396059
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −4.00000 −0.384900
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 1.00000 0.0953463
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 4.00000 0.374634
\(115\) −6.00000 −0.559503
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 4.00000 0.360668
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.00000 −0.176090
\(130\) 6.00000 0.526235
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) −4.00000 −0.344265
\(136\) 2.00000 0.171499
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −12.0000 −1.02151
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) −8.00000 −0.671345
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) 14.0000 1.15865
\(147\) −14.0000 −1.15470
\(148\) 6.00000 0.493197
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 2.00000 0.163299
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 2.00000 0.162221
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 12.0000 0.960769
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 10.0000 0.795557
\(159\) −8.00000 −0.634441
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 2.00000 0.156174
\(165\) 2.00000 0.155700
\(166\) −12.0000 −0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 2.00000 0.152944
\(172\) −1.00000 −0.0762493
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −8.00000 −0.601317
\(178\) 6.00000 0.449719
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 1.00000 0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) −6.00000 −0.442326
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 2.00000 0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 6.00000 0.430775
\(195\) 12.0000 0.859338
\(196\) −7.00000 −0.500000
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 1.00000 0.0710669
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) 16.0000 1.12855
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 2.00000 0.139686
\(206\) −14.0000 −0.975426
\(207\) −6.00000 −0.417029
\(208\) 6.00000 0.416025
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) −4.00000 −0.274721
\(213\) −16.0000 −1.09630
\(214\) −4.00000 −0.273434
\(215\) −1.00000 −0.0681994
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) 28.0000 1.89206
\(220\) 1.00000 0.0674200
\(221\) 12.0000 0.807207
\(222\) 12.0000 0.805387
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −4.00000 −0.266076
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 4.00000 0.264906
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −30.0000 −1.96537 −0.982683 0.185296i \(-0.940675\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 6.00000 0.392232
\(235\) −2.00000 −0.130466
\(236\) −4.00000 −0.260378
\(237\) 20.0000 1.29914
\(238\) 0 0
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 2.00000 0.129099
\(241\) −24.0000 −1.54598 −0.772988 0.634421i \(-0.781239\pi\)
−0.772988 + 0.634421i \(0.781239\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) −2.00000 −0.128037
\(245\) −7.00000 −0.447214
\(246\) 4.00000 0.255031
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −4.00000 −0.250982
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) 2.00000 0.123797
\(262\) −10.0000 −0.617802
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 2.00000 0.123091
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 8.00000 0.488678
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) −4.00000 −0.243432
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 1.00000 0.0603023
\(276\) −12.0000 −0.722315
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) −8.00000 −0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) −4.00000 −0.238197
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −8.00000 −0.474713
\(285\) 4.00000 0.236940
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 12.0000 0.703452
\(292\) 14.0000 0.819288
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −14.0000 −0.816497
\(295\) −4.00000 −0.232889
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) −36.0000 −2.08193
\(300\) 2.00000 0.115470
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) 16.0000 0.919176
\(304\) 2.00000 0.114708
\(305\) −2.00000 −0.114520
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −28.0000 −1.59286
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 12.0000 0.679366
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −4.00000 −0.224662 −0.112331 0.993671i \(-0.535832\pi\)
−0.112331 + 0.993671i \(0.535832\pi\)
\(318\) −8.00000 −0.448618
\(319\) 2.00000 0.111979
\(320\) 1.00000 0.0559017
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) −11.0000 −0.611111
\(325\) 6.00000 0.332820
\(326\) −14.0000 −0.775388
\(327\) −8.00000 −0.442401
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 2.00000 0.110096
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) 12.0000 0.656611
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 23.0000 1.25104
\(339\) −8.00000 −0.434500
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 0 0
\(344\) −1.00000 −0.0539164
\(345\) −12.0000 −0.646058
\(346\) 2.00000 0.107521
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 4.00000 0.214423
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) −24.0000 −1.28103
\(352\) 1.00000 0.0533002
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −8.00000 −0.425195
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) 14.0000 0.735824
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) −4.00000 −0.209083
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −6.00000 −0.312772
\(369\) 2.00000 0.104116
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 2.00000 0.103418
\(375\) 2.00000 0.103280
\(376\) −2.00000 −0.103142
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 2.00000 0.102598
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −1.00000 −0.0508329
\(388\) 6.00000 0.304604
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 12.0000 0.607644
\(391\) −12.0000 −0.606866
\(392\) −7.00000 −0.353553
\(393\) −20.0000 −1.00887
\(394\) −6.00000 −0.302276
\(395\) 10.0000 0.503155
\(396\) 1.00000 0.0502519
\(397\) 16.0000 0.803017 0.401508 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 16.0000 0.798007
\(403\) 0 0
\(404\) 8.00000 0.398015
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 4.00000 0.198030
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 2.00000 0.0987730
\(411\) 24.0000 1.18383
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −12.0000 −0.589057
\(416\) 6.00000 0.294174
\(417\) −16.0000 −0.783523
\(418\) 2.00000 0.0978232
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 22.0000 1.07094
\(423\) −2.00000 −0.0972433
\(424\) −4.00000 −0.194257
\(425\) 2.00000 0.0970143
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 12.0000 0.579365
\(430\) −1.00000 −0.0482243
\(431\) 22.0000 1.05970 0.529851 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) −4.00000 −0.192450
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) −4.00000 −0.191565
\(437\) −12.0000 −0.574038
\(438\) 28.0000 1.33789
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 1.00000 0.0476731
\(441\) −7.00000 −0.333333
\(442\) 12.0000 0.570782
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 12.0000 0.569495
\(445\) 6.00000 0.284427
\(446\) −12.0000 −0.568216
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 2.00000 0.0941763
\(452\) −4.00000 −0.188144
\(453\) −8.00000 −0.375873
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −30.0000 −1.40334 −0.701670 0.712502i \(-0.747562\pi\)
−0.701670 + 0.712502i \(0.747562\pi\)
\(458\) 10.0000 0.467269
\(459\) −8.00000 −0.373408
\(460\) −6.00000 −0.279751
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −30.0000 −1.38972
\(467\) 14.0000 0.647843 0.323921 0.946084i \(-0.394999\pi\)
0.323921 + 0.946084i \(0.394999\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) −2.00000 −0.0922531
\(471\) 36.0000 1.65879
\(472\) −4.00000 −0.184115
\(473\) −1.00000 −0.0459800
\(474\) 20.0000 0.918630
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) −2.00000 −0.0914779
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 2.00000 0.0912871
\(481\) 36.0000 1.64146
\(482\) −24.0000 −1.09317
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 6.00000 0.272446
\(486\) −10.0000 −0.453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −28.0000 −1.26620
\(490\) −7.00000 −0.316228
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 4.00000 0.180334
\(493\) 4.00000 0.180151
\(494\) 12.0000 0.539906
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) −24.0000 −1.07547
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.0000 1.07224
\(502\) 20.0000 0.892644
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) −6.00000 −0.266733
\(507\) 46.0000 2.04293
\(508\) −4.00000 −0.177471
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −8.00000 −0.352865
\(515\) −14.0000 −0.616914
\(516\) −2.00000 −0.0880451
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 6.00000 0.263117
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 2.00000 0.0875376
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 13.0000 0.565217
\(530\) −4.00000 −0.173749
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 12.0000 0.519291
\(535\) −4.00000 −0.172935
\(536\) 8.00000 0.345547
\(537\) 48.0000 2.07135
\(538\) −26.0000 −1.12094
\(539\) −7.00000 −0.301511
\(540\) −4.00000 −0.172133
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −14.0000 −0.601351
\(543\) 28.0000 1.20160
\(544\) 2.00000 0.0857493
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 12.0000 0.512615
\(549\) −2.00000 −0.0853579
\(550\) 1.00000 0.0426401
\(551\) 4.00000 0.170406
\(552\) −12.0000 −0.510754
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) 12.0000 0.509372
\(556\) −8.00000 −0.339276
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −6.00000 −0.253095
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −4.00000 −0.168430
\(565\) −4.00000 −0.168281
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 4.00000 0.167542
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −13.0000 −0.540729
\(579\) −4.00000 −0.166234
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) 12.0000 0.497416
\(583\) −4.00000 −0.165663
\(584\) 14.0000 0.579324
\(585\) 6.00000 0.248069
\(586\) 6.00000 0.247858
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) −14.0000 −0.577350
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) −12.0000 −0.493614
\(592\) 6.00000 0.246598
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 16.0000 0.654836
\(598\) −36.0000 −1.47215
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 2.00000 0.0816497
\(601\) 12.0000 0.489490 0.244745 0.969587i \(-0.421296\pi\)
0.244745 + 0.969587i \(0.421296\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) −4.00000 −0.162758
\(605\) 1.00000 0.0406558
\(606\) 16.0000 0.649956
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) −12.0000 −0.485468
\(612\) 2.00000 0.0808452
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −20.0000 −0.807134
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) −28.0000 −1.12633
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 24.0000 0.963087
\(622\) 0 0
\(623\) 0 0
\(624\) 12.0000 0.480384
\(625\) 1.00000 0.0400000
\(626\) 24.0000 0.959233
\(627\) 4.00000 0.159745
\(628\) 18.0000 0.718278
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 10.0000 0.397779
\(633\) 44.0000 1.74884
\(634\) −4.00000 −0.158860
\(635\) −4.00000 −0.158735
\(636\) −8.00000 −0.317221
\(637\) −42.0000 −1.66410
\(638\) 2.00000 0.0791808
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) −8.00000 −0.315735
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) 4.00000 0.157378
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −11.0000 −0.432121
\(649\) −4.00000 −0.157014
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −14.0000 −0.548282
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) −8.00000 −0.312825
\(655\) −10.0000 −0.390732
\(656\) 2.00000 0.0780869
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 2.00000 0.0778499
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −28.0000 −1.08825
\(663\) 24.0000 0.932083
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −12.0000 −0.464642
\(668\) 12.0000 0.464294
\(669\) −24.0000 −0.927894
\(670\) 8.00000 0.309067
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −2.00000 −0.0770371
\(675\) −4.00000 −0.153960
\(676\) 23.0000 0.884615
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −8.00000 −0.307238
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) −40.0000 −1.53280
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 2.00000 0.0764719
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) −1.00000 −0.0381246
\(689\) −24.0000 −0.914327
\(690\) −12.0000 −0.456832
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −8.00000 −0.303457
\(696\) 4.00000 0.151620
\(697\) 4.00000 0.151511
\(698\) −6.00000 −0.227103
\(699\) −60.0000 −2.26941
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) −24.0000 −0.905822
\(703\) 12.0000 0.452589
\(704\) 1.00000 0.0376889
\(705\) −4.00000 −0.150649
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) −8.00000 −0.300235
\(711\) 10.0000 0.375029
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 24.0000 0.896922
\(717\) −4.00000 −0.149383
\(718\) −14.0000 −0.522475
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) −48.0000 −1.78514
\(724\) 14.0000 0.520306
\(725\) 2.00000 0.0742781
\(726\) 2.00000 0.0742270
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 14.0000 0.518163
\(731\) −2.00000 −0.0739727
\(732\) −4.00000 −0.147844
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 10.0000 0.369107
\(735\) −14.0000 −0.516398
\(736\) −6.00000 −0.221163
\(737\) 8.00000 0.294684
\(738\) 2.00000 0.0736210
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 6.00000 0.220564
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 6.00000 0.219676
\(747\) −12.0000 −0.439057
\(748\) 2.00000 0.0731272
\(749\) 0 0
\(750\) 2.00000 0.0730297
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 40.0000 1.45768
\(754\) 12.0000 0.437014
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 12.0000 0.435860
\(759\) −12.0000 −0.435572
\(760\) 2.00000 0.0725476
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) −32.0000 −1.15621
\(767\) −24.0000 −0.866590
\(768\) 2.00000 0.0721688
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −16.0000 −0.576226
\(772\) −2.00000 −0.0719816
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 4.00000 0.143315
\(780\) 12.0000 0.429669
\(781\) −8.00000 −0.286263
\(782\) −12.0000 −0.429119
\(783\) −8.00000 −0.285897
\(784\) −7.00000 −0.250000
\(785\) 18.0000 0.642448
\(786\) −20.0000 −0.713376
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −6.00000 −0.213741
\(789\) −48.0000 −1.70885
\(790\) 10.0000 0.355784
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −12.0000 −0.426132
\(794\) 16.0000 0.567819
\(795\) −8.00000 −0.283731
\(796\) 8.00000 0.283552
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) 2.00000 0.0706225
\(803\) 14.0000 0.494049
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) −52.0000 −1.83049
\(808\) 8.00000 0.281439
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −11.0000 −0.386501
\(811\) 22.0000 0.772524 0.386262 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 0 0
\(813\) −28.0000 −0.982003
\(814\) 6.00000 0.210300
\(815\) −14.0000 −0.490399
\(816\) 4.00000 0.140028
\(817\) −2.00000 −0.0699711
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 48.0000 1.67521 0.837606 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(822\) 24.0000 0.837096
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) −14.0000 −0.487713
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) −6.00000 −0.208514
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) −12.0000 −0.416526
\(831\) −12.0000 −0.416275
\(832\) 6.00000 0.208013
\(833\) −14.0000 −0.485071
\(834\) −16.0000 −0.554035
\(835\) 12.0000 0.415277
\(836\) 2.00000 0.0691714
\(837\) 0 0
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −22.0000 −0.758170
\(843\) −12.0000 −0.413302
\(844\) 22.0000 0.757271
\(845\) 23.0000 0.791224
\(846\) −2.00000 −0.0687614
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 24.0000 0.823678
\(850\) 2.00000 0.0685994
\(851\) −36.0000 −1.23406
\(852\) −16.0000 −0.548151
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) −4.00000 −0.136717
\(857\) 50.0000 1.70797 0.853984 0.520300i \(-0.174180\pi\)
0.853984 + 0.520300i \(0.174180\pi\)
\(858\) 12.0000 0.409673
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 0 0
\(862\) 22.0000 0.749323
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) −4.00000 −0.136083
\(865\) 2.00000 0.0680020
\(866\) −16.0000 −0.543702
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) 10.0000 0.339227
\(870\) 4.00000 0.135613
\(871\) 48.0000 1.62642
\(872\) −4.00000 −0.135457
\(873\) 6.00000 0.203069
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 28.0000 0.946032
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 14.0000 0.472477
\(879\) 12.0000 0.404750
\(880\) 1.00000 0.0337100
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) −7.00000 −0.235702
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 12.0000 0.403604
\(885\) −8.00000 −0.268917
\(886\) 28.0000 0.940678
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 12.0000 0.402694
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) −11.0000 −0.368514
\(892\) −12.0000 −0.401790
\(893\) −4.00000 −0.133855
\(894\) −12.0000 −0.401340
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −72.0000 −2.40401
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −8.00000 −0.266519
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −4.00000 −0.133038
\(905\) 14.0000 0.465376
\(906\) −8.00000 −0.265782
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) −20.0000 −0.663723
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 4.00000 0.132453
\(913\) −12.0000 −0.397142
\(914\) −30.0000 −0.992312
\(915\) −4.00000 −0.132236
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) −8.00000 −0.264039
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) −6.00000 −0.197814
\(921\) −40.0000 −1.31804
\(922\) 12.0000 0.395199
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −8.00000 −0.262896
\(927\) −14.0000 −0.459820
\(928\) 2.00000 0.0656532
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −14.0000 −0.458831
\(932\) −30.0000 −0.982683
\(933\) 0 0
\(934\) 14.0000 0.458094
\(935\) 2.00000 0.0654070
\(936\) 6.00000 0.196116
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 48.0000 1.56642
\(940\) −2.00000 −0.0652328
\(941\) 8.00000 0.260793 0.130396 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 36.0000 1.17294
\(943\) −12.0000 −0.390774
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −1.00000 −0.0325128
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 20.0000 0.649570
\(949\) 84.0000 2.72676
\(950\) 2.00000 0.0648886
\(951\) −8.00000 −0.259418
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −4.00000 −0.129505
\(955\) 0 0
\(956\) −2.00000 −0.0646846
\(957\) 4.00000 0.129302
\(958\) −6.00000 −0.193851
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −31.0000 −1.00000
\(962\) 36.0000 1.16069
\(963\) −4.00000 −0.128898
\(964\) −24.0000 −0.772988
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 1.00000 0.0321412
\(969\) 8.00000 0.256997
\(970\) 6.00000 0.192648
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) −18.0000 −0.576757
\(975\) 12.0000 0.384308
\(976\) −2.00000 −0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −28.0000 −0.895341
\(979\) 6.00000 0.191761
\(980\) −7.00000 −0.223607
\(981\) −4.00000 −0.127710
\(982\) −6.00000 −0.191468
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 4.00000 0.127515
\(985\) −6.00000 −0.191176
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 6.00000 0.190789
\(990\) 1.00000 0.0317821
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −56.0000 −1.77711
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) −24.0000 −0.760469
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −4.00000 −0.126618
\(999\) −24.0000 −0.759326
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 4730.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.k.1.1 1 1.1 even 1 trivial