Properties

Label 3234.2.a.s.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
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] = [3234,2,Mod(1,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, 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("3234.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.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: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -4.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +10.0000 q^{29} -4.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{39} +4.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.00000 q^{45} -6.00000 q^{46} +2.00000 q^{47} -1.00000 q^{48} +11.0000 q^{50} -2.00000 q^{51} -4.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +10.0000 q^{58} -4.00000 q^{60} +8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -16.0000 q^{65} -1.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +6.00000 q^{69} +2.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -11.0000 q^{75} +4.00000 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +8.00000 q^{85} +4.00000 q^{86} -10.0000 q^{87} +1.00000 q^{88} -10.0000 q^{89} +4.00000 q^{90} -6.00000 q^{92} -8.00000 q^{93} +2.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) −4.00000 −0.730297
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 4.00000 0.632456
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 4.00000 0.596285
\(46\) −6.00000 −0.884652
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 11.0000 1.55563
\(51\) −2.00000 −0.280056
\(52\) −4.00000 −0.554700
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −4.00000 −0.516398
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.0000 −1.98456
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) −11.0000 −1.27017
\(76\) 0 0
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 4.00000 0.431331
\(87\) −10.0000 −1.07211
\(88\) 1.00000 0.106600
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 4.00000 0.421637
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −8.00000 −0.829561
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 11.0000 1.10000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −2.00000 −0.198030
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 4.00000 0.381385
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) 10.0000 0.928477
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) −4.00000 −0.365148
\(121\) 1.00000 0.0909091
\(122\) 8.00000 0.724286
\(123\) 2.00000 0.180334
\(124\) 8.00000 0.718421
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −22.0000 −1.95218 −0.976092 0.217357i \(-0.930256\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) −16.0000 −1.40329
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) −4.00000 −0.344265
\(136\) 2.00000 0.171499
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 6.00000 0.510754
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 2.00000 0.167836
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 40.0000 3.32182
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −11.0000 −0.898146
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 32.0000 2.57030
\(156\) 4.00000 0.320256
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 10.0000 0.795557
\(159\) −4.00000 −0.317221
\(160\) 4.00000 0.316228
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.00000 −0.156174
\(165\) −4.00000 −0.311400
\(166\) −4.00000 −0.310460
\(167\) 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
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 4.00000 0.298142
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) −6.00000 −0.442326
\(185\) −8.00000 −0.588172
\(186\) −8.00000 −0.586588
\(187\) 2.00000 0.146254
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 0 0
\(191\) 22.0000 1.59186 0.795932 0.605386i \(-0.206981\pi\)
0.795932 + 0.605386i \(0.206981\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 16.0000 1.14578
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 1.00000 0.0710669
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 11.0000 0.777817
\(201\) 12.0000 0.846415
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −8.00000 −0.558744
\(206\) −4.00000 −0.278693
\(207\) −6.00000 −0.417029
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 4.00000 0.274721
\(213\) −2.00000 −0.137038
\(214\) −12.0000 −0.820303
\(215\) 16.0000 1.09119
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 20.0000 1.35457
\(219\) −6.00000 −0.405442
\(220\) 4.00000 0.269680
\(221\) −8.00000 −0.538138
\(222\) 2.00000 0.134231
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −24.0000 −1.58251
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.00000 −0.261488
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) −4.00000 −0.258199
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 24.0000 1.51789
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −22.0000 −1.38040
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) −16.0000 −0.992278
\(261\) 10.0000 0.618984
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −12.0000 −0.733017
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) −4.00000 −0.243432
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 11.0000 0.663325
\(276\) 6.00000 0.361158
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) −2.00000 −0.119098
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 40.0000 2.34888
\(291\) −2.00000 −0.117242
\(292\) 6.00000 0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −1.00000 −0.0580259
\(298\) −10.0000 −0.579284
\(299\) 24.0000 1.38796
\(300\) −11.0000 −0.635085
\(301\) 0 0
\(302\) 2.00000 0.115087
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 32.0000 1.83231
\(306\) 2.00000 0.114332
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 32.0000 1.81748
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 4.00000 0.226455
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −32.0000 −1.79730 −0.898650 0.438667i \(-0.855451\pi\)
−0.898650 + 0.438667i \(0.855451\pi\)
\(318\) −4.00000 −0.224309
\(319\) 10.0000 0.559893
\(320\) 4.00000 0.223607
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −44.0000 −2.44068
\(326\) 4.00000 0.221540
\(327\) −20.0000 −1.10600
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) 12.0000 0.656611
\(335\) −48.0000 −2.62252
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 3.00000 0.163178
\(339\) 6.00000 0.325875
\(340\) 8.00000 0.433861
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 24.0000 1.29212
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −10.0000 −0.536056
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 4.00000 0.210819
\(361\) −19.0000 −1.00000
\(362\) −2.00000 −0.105118
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) −8.00000 −0.418167
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −6.00000 −0.312772
\(369\) −2.00000 −0.104116
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 2.00000 0.103418
\(375\) −24.0000 −1.23935
\(376\) 2.00000 0.103142
\(377\) −40.0000 −2.06010
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 22.0000 1.12709
\(382\) 22.0000 1.12562
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 16.0000 0.810191
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 18.0000 0.906827
\(395\) 40.0000 2.01262
\(396\) 1.00000 0.0502519
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 12.0000 0.598506
\(403\) −32.0000 −1.59403
\(404\) −2.00000 −0.0995037
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −8.00000 −0.395092
\(411\) 2.00000 0.0986527
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −16.0000 −0.785409
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 12.0000 0.584151
\(423\) 2.00000 0.0972433
\(424\) 4.00000 0.194257
\(425\) 22.0000 1.06716
\(426\) −2.00000 −0.0969003
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 16.0000 0.771589
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) −40.0000 −1.91785
\(436\) 20.0000 0.957826
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 2.00000 0.0949158
\(445\) −40.0000 −1.89618
\(446\) 16.0000 0.757622
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 11.0000 0.518545
\(451\) −2.00000 −0.0941763
\(452\) −6.00000 −0.282216
\(453\) −2.00000 −0.0939682
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) −24.0000 −1.11901
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 10.0000 0.464238
\(465\) −32.0000 −1.48396
\(466\) −6.00000 −0.277945
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) −20.0000 −0.914779
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) −4.00000 −0.182574
\(481\) 8.00000 0.364769
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 8.00000 0.363261
\(486\) −1.00000 −0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 8.00000 0.362143
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 2.00000 0.0901670
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 24.0000 1.07331
\(501\) −12.0000 −0.536120
\(502\) 8.00000 0.357057
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) −6.00000 −0.266733
\(507\) −3.00000 −0.133235
\(508\) −22.0000 −0.976092
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) −8.00000 −0.354246
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) −16.0000 −0.701646
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 10.0000 0.437688
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 16.0000 0.696971
\(528\) −1.00000 −0.0435194
\(529\) 13.0000 0.565217
\(530\) 16.0000 0.694996
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 10.0000 0.432742
\(535\) −48.0000 −2.07522
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 20.0000 0.862261
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) 12.0000 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) −22.0000 −0.944981
\(543\) 2.00000 0.0858282
\(544\) 2.00000 0.0857493
\(545\) 80.0000 3.42682
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 8.00000 0.341432
\(550\) 11.0000 0.469042
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 8.00000 0.339887
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) 38.0000 1.61011 0.805056 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 8.00000 0.338667
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) 22.0000 0.928014
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −2.00000 −0.0842152
\(565\) −24.0000 −1.00969
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −4.00000 −0.167248
\(573\) −22.0000 −0.919063
\(574\) 0 0
\(575\) −66.0000 −2.75239
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) 40.0000 1.66091
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 4.00000 0.165663
\(584\) 6.00000 0.248282
\(585\) −16.0000 −0.661519
\(586\) −14.0000 −0.578335
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) −2.00000 −0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 20.0000 0.818546
\(598\) 24.0000 0.981433
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) −11.0000 −0.449073
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 2.00000 0.0813788
\(605\) 4.00000 0.162623
\(606\) 2.00000 0.0812444
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 32.0000 1.29564
\(611\) −8.00000 −0.323645
\(612\) 2.00000 0.0808452
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −8.00000 −0.322854
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 4.00000 0.160904
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 32.0000 1.28515
\(621\) 6.00000 0.240772
\(622\) −2.00000 −0.0801927
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 41.0000 1.64000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −48.0000 −1.91085 −0.955425 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) 10.0000 0.397779
\(633\) −12.0000 −0.476957
\(634\) −32.0000 −1.27088
\(635\) −88.0000 −3.49217
\(636\) −4.00000 −0.158610
\(637\) 0 0
\(638\) 10.0000 0.395904
\(639\) 2.00000 0.0791188
\(640\) 4.00000 0.158114
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 12.0000 0.473602
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) 22.0000 0.864909 0.432455 0.901656i \(-0.357648\pi\)
0.432455 + 0.901656i \(0.357648\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −44.0000 −1.72582
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −20.0000 −0.782062
\(655\) −48.0000 −1.87552
\(656\) −2.00000 −0.0780869
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −4.00000 −0.155700
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) −28.0000 −1.08825
\(663\) 8.00000 0.310694
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −60.0000 −2.32321
\(668\) 12.0000 0.464294
\(669\) −16.0000 −0.618596
\(670\) −48.0000 −1.85440
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −22.0000 −0.847408
\(675\) −11.0000 −0.423390
\(676\) 3.00000 0.115385
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) −12.0000 −0.459841
\(682\) 8.00000 0.306336
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 0 0
\(685\) −8.00000 −0.305664
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) −16.0000 −0.609551
\(690\) 24.0000 0.913664
\(691\) −52.0000 −1.97817 −0.989087 0.147335i \(-0.952930\pi\)
−0.989087 + 0.147335i \(0.952930\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) −4.00000 −0.151511
\(698\) 20.0000 0.757011
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 4.00000 0.150970
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) −8.00000 −0.301297
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 8.00000 0.300235
\(711\) 10.0000 0.375029
\(712\) −10.0000 −0.374766
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 20.0000 0.746914
\(718\) −20.0000 −0.746393
\(719\) −10.0000 −0.372937 −0.186469 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) −18.0000 −0.669427
\(724\) −2.00000 −0.0743294
\(725\) 110.000 4.08530
\(726\) −1.00000 −0.0371135
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 24.0000 0.888280
\(731\) 8.00000 0.295891
\(732\) −8.00000 −0.295689
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −12.0000 −0.442026
\(738\) −2.00000 −0.0736210
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) 44.0000 1.61420 0.807102 0.590412i \(-0.201035\pi\)
0.807102 + 0.590412i \(0.201035\pi\)
\(744\) −8.00000 −0.293294
\(745\) −40.0000 −1.46549
\(746\) 4.00000 0.146450
\(747\) −4.00000 −0.146352
\(748\) 2.00000 0.0731272
\(749\) 0 0
\(750\) −24.0000 −0.876356
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 2.00000 0.0729325
\(753\) −8.00000 −0.291536
\(754\) −40.0000 −1.45671
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −20.0000 −0.726433
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 22.0000 0.796976
\(763\) 0 0
\(764\) 22.0000 0.795932
\(765\) 8.00000 0.289241
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 14.0000 0.503871
\(773\) −4.00000 −0.143870 −0.0719350 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) 4.00000 0.143777
\(775\) 88.0000 3.16105
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) 0 0
\(780\) 16.0000 0.572892
\(781\) 2.00000 0.0715656
\(782\) −12.0000 −0.429119
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) −72.0000 −2.56979
\(786\) 12.0000 0.428026
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 18.0000 0.641223
\(789\) 16.0000 0.569615
\(790\) 40.0000 1.42314
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −32.0000 −1.13635
\(794\) −18.0000 −0.638796
\(795\) −16.0000 −0.567462
\(796\) −20.0000 −0.708881
\(797\) −28.0000 −0.991811 −0.495905 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 11.0000 0.388909
\(801\) −10.0000 −0.353333
\(802\) −18.0000 −0.635602
\(803\) 6.00000 0.211735
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) −20.0000 −0.704033
\(808\) −2.00000 −0.0703598
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 4.00000 0.140546
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 22.0000 0.771574
\(814\) −2.00000 −0.0701000
\(815\) 16.0000 0.560456
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 2.00000 0.0697580
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −4.00000 −0.139347
\(825\) −11.0000 −0.382971
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) −16.0000 −0.555368
\(831\) −8.00000 −0.277517
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 0 0
\(835\) 48.0000 1.66111
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −18.0000 −0.620321
\(843\) −22.0000 −0.757720
\(844\) 12.0000 0.413057
\(845\) 12.0000 0.412813
\(846\) 2.00000 0.0687614
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) 4.00000 0.137280
\(850\) 22.0000 0.754594
\(851\) 12.0000 0.411355
\(852\) −2.00000 −0.0685189
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 4.00000 0.136558
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 24.0000 0.816024
\(866\) 6.00000 0.203888
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 10.0000 0.339227
\(870\) −40.0000 −1.35613
\(871\) 48.0000 1.62642
\(872\) 20.0000 0.677285
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) −10.0000 −0.337484
\(879\) 14.0000 0.472208
\(880\) 4.00000 0.134840
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) −40.0000 −1.34080
\(891\) 1.00000 0.0335013
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 30.0000 1.00111
\(899\) 80.0000 2.66815
\(900\) 11.0000 0.366667
\(901\) 8.00000 0.266519
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −8.00000 −0.265929
\(906\) −2.00000 −0.0664455
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 12.0000 0.398234
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 2.00000 0.0662630 0.0331315 0.999451i \(-0.489452\pi\)
0.0331315 + 0.999451i \(0.489452\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) −2.00000 −0.0661541
\(915\) −32.0000 −1.05789
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) −24.0000 −0.791257
\(921\) 8.00000 0.263609
\(922\) 18.0000 0.592798
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) 4.00000 0.131448
\(927\) −4.00000 −0.131377
\(928\) 10.0000 0.328266
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) −32.0000 −1.04932
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 2.00000 0.0654771
\(934\) −28.0000 −0.916188
\(935\) 8.00000 0.261628
\(936\) −4.00000 −0.130744
\(937\) −58.0000 −1.89478 −0.947389 0.320085i \(-0.896288\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 8.00000 0.260931
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 18.0000 0.586472
\(943\) 12.0000 0.390774
\(944\) 0 0
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) −10.0000 −0.324785
\(949\) −24.0000 −0.779073
\(950\) 0 0
\(951\) 32.0000 1.03767
\(952\) 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
\(955\) 88.0000 2.84761
\(956\) −20.0000 −0.646846
\(957\) −10.0000 −0.323254
\(958\) −40.0000 −1.29234
\(959\) 0 0
\(960\) −4.00000 −0.129099
\(961\) 33.0000 1.06452
\(962\) 8.00000 0.257930
\(963\) −12.0000 −0.386695
\(964\) 18.0000 0.579741
\(965\) 56.0000 1.80270
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −32.0000 −1.02693 −0.513464 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) 44.0000 1.40913
\(976\) 8.00000 0.256074
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −4.00000 −0.127906
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 12.0000 0.382935
\(983\) 46.0000 1.46717 0.733586 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 2.00000 0.0637577
\(985\) 72.0000 2.29411
\(986\) 20.0000 0.636930
\(987\) 0 0
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 4.00000 0.127128
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 8.00000 0.254000
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) −80.0000 −2.53617
\(996\) 4.00000 0.126745
\(997\) 52.0000 1.64686 0.823428 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) −20.0000 −0.633089
\(999\) 2.00000 0.0632772
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 3234.2.a.s.1.1 1
3.2 odd 2 9702.2.a.a.1.1 1
7.6 odd 2 66.2.a.c.1.1 1
21.20 even 2 198.2.a.c.1.1 1
28.27 even 2 528.2.a.a.1.1 1
35.13 even 4 1650.2.c.m.199.1 2
35.27 even 4 1650.2.c.m.199.2 2
35.34 odd 2 1650.2.a.c.1.1 1
56.13 odd 2 2112.2.a.n.1.1 1
56.27 even 2 2112.2.a.bd.1.1 1
63.13 odd 6 1782.2.e.l.1189.1 2
63.20 even 6 1782.2.e.n.595.1 2
63.34 odd 6 1782.2.e.l.595.1 2
63.41 even 6 1782.2.e.n.1189.1 2
77.6 even 10 726.2.e.m.487.1 4
77.13 even 10 726.2.e.m.565.1 4
77.20 odd 10 726.2.e.e.565.1 4
77.27 odd 10 726.2.e.e.487.1 4
77.41 even 10 726.2.e.m.493.1 4
77.48 odd 10 726.2.e.e.511.1 4
77.62 even 10 726.2.e.m.511.1 4
77.69 odd 10 726.2.e.e.493.1 4
77.76 even 2 726.2.a.d.1.1 1
84.83 odd 2 1584.2.a.s.1.1 1
105.62 odd 4 4950.2.c.d.199.1 2
105.83 odd 4 4950.2.c.d.199.2 2
105.104 even 2 4950.2.a.bo.1.1 1
168.83 odd 2 6336.2.a.d.1.1 1
168.125 even 2 6336.2.a.c.1.1 1
231.230 odd 2 2178.2.a.m.1.1 1
308.307 odd 2 5808.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.c.1.1 1 7.6 odd 2
198.2.a.c.1.1 1 21.20 even 2
528.2.a.a.1.1 1 28.27 even 2
726.2.a.d.1.1 1 77.76 even 2
726.2.e.e.487.1 4 77.27 odd 10
726.2.e.e.493.1 4 77.69 odd 10
726.2.e.e.511.1 4 77.48 odd 10
726.2.e.e.565.1 4 77.20 odd 10
726.2.e.m.487.1 4 77.6 even 10
726.2.e.m.493.1 4 77.41 even 10
726.2.e.m.511.1 4 77.62 even 10
726.2.e.m.565.1 4 77.13 even 10
1584.2.a.s.1.1 1 84.83 odd 2
1650.2.a.c.1.1 1 35.34 odd 2
1650.2.c.m.199.1 2 35.13 even 4
1650.2.c.m.199.2 2 35.27 even 4
1782.2.e.l.595.1 2 63.34 odd 6
1782.2.e.l.1189.1 2 63.13 odd 6
1782.2.e.n.595.1 2 63.20 even 6
1782.2.e.n.1189.1 2 63.41 even 6
2112.2.a.n.1.1 1 56.13 odd 2
2112.2.a.bd.1.1 1 56.27 even 2
2178.2.a.m.1.1 1 231.230 odd 2
3234.2.a.s.1.1 1 1.1 even 1 trivial
4950.2.a.bo.1.1 1 105.104 even 2
4950.2.c.d.199.1 2 105.62 odd 4
4950.2.c.d.199.2 2 105.83 odd 4
5808.2.a.b.1.1 1 308.307 odd 2
6336.2.a.c.1.1 1 168.125 even 2
6336.2.a.d.1.1 1 168.83 odd 2
9702.2.a.a.1.1 1 3.2 odd 2