Properties

Label 6762.2.a.q.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
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] = [6762,2,Mod(1,6762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, 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("6762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.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: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.00000 q^{20} +6.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +1.00000 q^{36} +2.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} -12.0000 q^{43} -6.00000 q^{44} +2.00000 q^{45} +1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -12.0000 q^{55} -6.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} -4.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +6.00000 q^{66} -12.0000 q^{67} -1.00000 q^{69} -1.00000 q^{72} +10.0000 q^{73} -1.00000 q^{75} -2.00000 q^{78} -6.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -14.0000 q^{83} +12.0000 q^{86} +6.00000 q^{87} +6.00000 q^{88} -2.00000 q^{90} -1.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} +6.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −6.00000 −0.904534
\(45\) 2.00000 0.298142
\(46\) 1.00000 0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 2.00000 0.258199
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 6.00000 0.738549
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 6.00000 0.643268
\(88\) 6.00000 0.639602
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −8.00000 −0.829561
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 12.0000 1.14416
\(111\) 0 0
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 25.0000 2.27273
\(122\) 4.00000 0.362143
\(123\) −10.0000 −0.901670
\(124\) −8.00000 −0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.0000 −1.05654
\(130\) −4.00000 −0.350823
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 1.00000 0.0851257
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 2.00000 0.160128
\(157\) −16.0000 −1.27694 −0.638470 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) 6.00000 0.477334
\(159\) 2.00000 0.158610
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −10.0000 −0.780869
\(165\) −12.0000 −0.934199
\(166\) 14.0000 1.08661
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 2.00000 0.149071
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −6.00000 −0.430775
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 6.00000 0.426401
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) −20.0000 −1.39686
\(206\) 14.0000 0.975426
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) −14.0000 −0.957020
\(215\) −24.0000 −1.63679
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 16.0000 1.08366
\(219\) 10.0000 0.675737
\(220\) −12.0000 −0.809040
\(221\) 0 0
\(222\) 0 0
\(223\) −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
\(226\) 8.00000 0.532152
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 0 0
\(229\) 24.0000 1.58596 0.792982 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 16.0000 1.04372
\(236\) 12.0000 0.781133
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 2.00000 0.129099
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −25.0000 −1.60706
\(243\) 1.00000 0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) −14.0000 −0.887214
\(250\) 12.0000 0.758947
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) −8.00000 −0.494242
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 6.00000 0.369274
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −2.00000 −0.121716
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 6.00000 0.361814
\(276\) −1.00000 −0.0601929
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 12.0000 0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) −8.00000 −0.476393
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −12.0000 −0.704664
\(291\) 6.00000 0.351726
\(292\) 10.0000 0.585206
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) −6.00000 −0.348155
\(298\) 18.0000 1.04271
\(299\) −2.00000 −0.115663
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 12.0000 0.690522
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 16.0000 0.908739
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −2.00000 −0.113228
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −2.00000 −0.112154
\(319\) −36.0000 −2.01561
\(320\) 2.00000 0.111803
\(321\) 14.0000 0.781404
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 12.0000 0.664619
\(327\) −16.0000 −0.884802
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 12.0000 0.660578
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 9.00000 0.489535
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) 48.0000 2.59935
\(342\) 0 0
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) −2.00000 −0.107676
\(346\) −18.0000 −0.967686
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 6.00000 0.321634
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 6.00000 0.319801
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −2.00000 −0.105409
\(361\) −19.0000 −1.00000
\(362\) 8.00000 0.420471
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 20.0000 1.04685
\(366\) 4.00000 0.209083
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) −8.00000 −0.412568
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −12.0000 −0.609994
\(388\) 6.00000 0.304604
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 6.00000 0.302276
\(395\) −12.0000 −0.603786
\(396\) −6.00000 −0.301511
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 12.0000 0.598506
\(403\) −16.0000 −0.797017
\(404\) 6.00000 0.298511
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 20.0000 0.987730
\(411\) −12.0000 −0.591916
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −28.0000 −1.37447
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) −20.0000 −0.973585
\(423\) 8.00000 0.388973
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 14.0000 0.676716
\(429\) −12.0000 −0.579365
\(430\) 24.0000 1.15738
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 12.0000 0.572078
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 12.0000 0.568216
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 1.00000 0.0471405
\(451\) 60.0000 2.82529
\(452\) −8.00000 −0.376288
\(453\) −12.0000 −0.563809
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −24.0000 −1.12145
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 6.00000 0.278543
\(465\) −16.0000 −0.741982
\(466\) −10.0000 −0.463241
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) −16.0000 −0.738025
\(471\) −16.0000 −0.737241
\(472\) −12.0000 −0.552345
\(473\) 72.0000 3.31056
\(474\) 6.00000 0.275589
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 16.0000 0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 12.0000 0.544892
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 4.00000 0.181071
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 14.0000 0.627355
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) −6.00000 −0.267793
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) −6.00000 −0.266733
\(507\) −9.00000 −0.399704
\(508\) 12.0000 0.532414
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −28.0000 −1.23383
\(516\) −12.0000 −0.528271
\(517\) −48.0000 −2.11104
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) −4.00000 −0.175412
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) −6.00000 −0.262613
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) −6.00000 −0.261116
\(529\) 1.00000 0.0434783
\(530\) −4.00000 −0.173749
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) 28.0000 1.21055
\(536\) 12.0000 0.518321
\(537\) −16.0000 −0.690451
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 28.0000 1.20270
\(543\) −8.00000 −0.343313
\(544\) 0 0
\(545\) −32.0000 −1.37073
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −12.0000 −0.512615
\(549\) −4.00000 −0.170716
\(550\) −6.00000 −0.255841
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −16.0000 −0.674919
\(563\) −14.0000 −0.590030 −0.295015 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) 8.00000 0.336861
\(565\) −16.0000 −0.673125
\(566\) 24.0000 1.00880
\(567\) 0 0
\(568\) 0 0
\(569\) 16.0000 0.670755 0.335377 0.942084i \(-0.391136\pi\)
0.335377 + 0.942084i \(0.391136\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 17.0000 0.707107
\(579\) 14.0000 0.581820
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) −12.0000 −0.496989
\(584\) −10.0000 −0.413803
\(585\) 4.00000 0.165380
\(586\) −6.00000 −0.247858
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −2.00000 −0.0818546
\(598\) 2.00000 0.0817861
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 1.00000 0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) −12.0000 −0.488273
\(605\) 50.0000 2.03279
\(606\) −6.00000 −0.243733
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) −20.0000 −0.807134
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 14.0000 0.563163
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −16.0000 −0.642575
\(621\) −1.00000 −0.0401286
\(622\) 16.0000 0.641542
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 6.00000 0.238667
\(633\) 20.0000 0.794929
\(634\) −22.0000 −0.873732
\(635\) 24.0000 0.952411
\(636\) 2.00000 0.0793052
\(637\) 0 0
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) −14.0000 −0.552536
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −72.0000 −2.82625
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 16.0000 0.625650
\(655\) 16.0000 0.625172
\(656\) −10.0000 −0.390434
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 42.0000 1.63609 0.818044 0.575156i \(-0.195059\pi\)
0.818044 + 0.575156i \(0.195059\pi\)
\(660\) −12.0000 −0.467099
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) −8.00000 −0.309529
\(669\) −12.0000 −0.463947
\(670\) 24.0000 0.927201
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 10.0000 0.385186
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 8.00000 0.307238
\(679\) 0 0
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) −48.0000 −1.83801
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −24.0000 −0.916993
\(686\) 0 0
\(687\) 24.0000 0.915657
\(688\) −12.0000 −0.457496
\(689\) 4.00000 0.152388
\(690\) 2.00000 0.0761387
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) −24.0000 −0.910372
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) 16.0000 0.602595
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) −16.0000 −0.597948
\(717\) −16.0000 −0.597531
\(718\) −24.0000 −0.895672
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 6.00000 0.223142
\(724\) −8.00000 −0.297318
\(725\) −6.00000 −0.222834
\(726\) −25.0000 −0.927837
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) −4.00000 −0.147844
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 72.0000 2.65215
\(738\) 10.0000 0.368105
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 8.00000 0.293294
\(745\) −36.0000 −1.31894
\(746\) 32.0000 1.17160
\(747\) −14.0000 −0.512233
\(748\) 0 0
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 8.00000 0.291730
\(753\) 6.00000 0.218652
\(754\) −12.0000 −0.437014
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) −32.0000 −1.16306 −0.581530 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −12.0000 −0.434714
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 12.0000 0.431331
\(775\) 8.00000 0.287368
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −32.0000 −1.14213
\(786\) −8.00000 −0.285351
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −6.00000 −0.213741
\(789\) 4.00000 0.142404
\(790\) 12.0000 0.426941
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) −8.00000 −0.284088
\(794\) 6.00000 0.212932
\(795\) 4.00000 0.141865
\(796\) −2.00000 −0.0708881
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 32.0000 1.12996
\(803\) −60.0000 −2.11735
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) −18.0000 −0.633630
\(808\) −6.00000 −0.211079
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −20.0000 −0.698430
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 12.0000 0.418548
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 14.0000 0.487713
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 18.0000 0.625921 0.312961 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 28.0000 0.971894
\(831\) 26.0000 0.901930
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) −26.0000 −0.898155
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −4.00000 −0.137849
\(843\) 16.0000 0.551069
\(844\) 20.0000 0.688428
\(845\) −18.0000 −0.619219
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 12.0000 0.409673
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 36.0000 1.22404
\(866\) −2.00000 −0.0679628
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) 36.0000 1.22122
\(870\) −12.0000 −0.406838
\(871\) −24.0000 −0.813209
\(872\) 16.0000 0.541828
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) −16.0000 −0.539974
\(879\) 6.00000 0.202375
\(880\) −12.0000 −0.404520
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) −36.0000 −1.20944
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) −12.0000 −0.401790
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) −32.0000 −1.06964
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) −2.00000 −0.0667409
\(899\) −48.0000 −1.60089
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) −60.0000 −1.99778
\(903\) 0 0
\(904\) 8.00000 0.266076
\(905\) −16.0000 −0.531858
\(906\) 12.0000 0.398673
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 10.0000 0.331862
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 84.0000 2.77999
\(914\) −26.0000 −0.860004
\(915\) −8.00000 −0.264472
\(916\) 24.0000 0.792982
\(917\) 0 0
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 2.00000 0.0659380
\(921\) 20.0000 0.659022
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) −14.0000 −0.459820
\(928\) −6.00000 −0.196960
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −16.0000 −0.523816
\(934\) 18.0000 0.588978
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 16.0000 0.521862
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 16.0000 0.521308
\(943\) 10.0000 0.325645
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −72.0000 −2.34092
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) −6.00000 −0.194871
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −36.0000 −1.16371
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 14.0000 0.451144
\(964\) 6.00000 0.193247
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) −12.0000 −0.385297
\(971\) 10.0000 0.320915 0.160458 0.987043i \(-0.448703\pi\)
0.160458 + 0.987043i \(0.448703\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) −2.00000 −0.0640513
\(976\) −4.00000 −0.128037
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 32.0000 1.02116
\(983\) −52.0000 −1.65854 −0.829271 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) 10.0000 0.318788
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 12.0000 0.381385
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000 0.254000
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) −14.0000 −0.443607
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 20.0000 0.633089
\(999\) 0 0
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 6762.2.a.q.1.1 1
7.6 odd 2 138.2.a.a.1.1 1
21.20 even 2 414.2.a.d.1.1 1
28.27 even 2 1104.2.a.e.1.1 1
35.13 even 4 3450.2.d.j.2899.2 2
35.27 even 4 3450.2.d.j.2899.1 2
35.34 odd 2 3450.2.a.y.1.1 1
56.13 odd 2 4416.2.a.z.1.1 1
56.27 even 2 4416.2.a.m.1.1 1
84.83 odd 2 3312.2.a.n.1.1 1
161.160 even 2 3174.2.a.b.1.1 1
483.482 odd 2 9522.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 1 7.6 odd 2
414.2.a.d.1.1 1 21.20 even 2
1104.2.a.e.1.1 1 28.27 even 2
3174.2.a.b.1.1 1 161.160 even 2
3312.2.a.n.1.1 1 84.83 odd 2
3450.2.a.y.1.1 1 35.34 odd 2
3450.2.d.j.2899.1 2 35.27 even 4
3450.2.d.j.2899.2 2 35.13 even 4
4416.2.a.m.1.1 1 56.27 even 2
4416.2.a.z.1.1 1 56.13 odd 2
6762.2.a.q.1.1 1 1.1 even 1 trivial
9522.2.a.i.1.1 1 483.482 odd 2