Properties

Label 7942.2.a.k.1.1
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7942.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{20} +4.00000 q^{21} +1.00000 q^{22} -2.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{27} +2.00000 q^{28} -2.00000 q^{29} -4.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{39} -2.00000 q^{40} -2.00000 q^{41} -4.00000 q^{42} -1.00000 q^{44} +2.00000 q^{45} -8.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -12.0000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +4.00000 q^{54} -2.00000 q^{55} -2.00000 q^{56} +2.00000 q^{58} +2.00000 q^{59} +4.00000 q^{60} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +2.00000 q^{66} -10.0000 q^{67} -6.00000 q^{68} -4.00000 q^{70} -1.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} -2.00000 q^{75} -2.00000 q^{77} +4.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} -11.0000 q^{81} +2.00000 q^{82} +12.0000 q^{83} +4.00000 q^{84} -12.0000 q^{85} -4.00000 q^{87} +1.00000 q^{88} -16.0000 q^{89} -2.00000 q^{90} -4.00000 q^{91} -8.00000 q^{93} +8.00000 q^{94} -2.00000 q^{96} +8.00000 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −2.00000 −0.816497
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) 4.00000 0.872872
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −2.00000 −0.408248
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −4.00000 −0.769800
\(28\) 2.00000 0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −4.00000 −0.730297
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 6.00000 1.02899
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) −2.00000 −0.316228
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −4.00000 −0.617213
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.00000 −0.150756
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 2.00000 0.288675
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −12.0000 −1.68034
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 4.00000 0.544331
\(55\) −2.00000 −0.269680
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 4.00000 0.516398
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 2.00000 0.246183
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −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
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) −11.0000 −1.22222
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 4.00000 0.436436
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 1.00000 0.106600
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) −2.00000 −0.210819
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 3.00000 0.303046
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) 20.0000 1.99007 0.995037 0.0995037i \(-0.0317255\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) 12.0000 1.18818
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 2.00000 0.196116
\(105\) 8.00000 0.780720
\(106\) 2.00000 0.194257
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −4.00000 −0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 2.00000 0.190693
\(111\) −12.0000 −1.13899
\(112\) 2.00000 0.188982
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −2.00000 −0.184115
\(119\) −12.0000 −1.10004
\(120\) −4.00000 −0.365148
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) −2.00000 −0.178174
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) −8.00000 −0.688530
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 4.00000 0.338062
\(141\) −16.0000 −1.34744
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −14.0000 −1.15865
\(147\) −6.00000 −0.494872
\(148\) −6.00000 −0.493197
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 2.00000 0.163299
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 2.00000 0.161165
\(155\) −8.00000 −0.642575
\(156\) −4.00000 −0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000 0.636446
\(159\) −4.00000 −0.317221
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) 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
\(166\) −12.0000 −0.931381
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 4.00000 0.303239
\(175\) −2.00000 −0.151186
\(176\) −1.00000 −0.0753778
\(177\) 4.00000 0.300658
\(178\) 16.0000 1.19925
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 8.00000 0.586588
\(187\) 6.00000 0.438763
\(188\) −8.00000 −0.583460
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 2.00000 0.144338
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −8.00000 −0.574367
\(195\) −8.00000 −0.572892
\(196\) −3.00000 −0.214286
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 1.00000 0.0710669
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 1.00000 0.0707107
\(201\) −20.0000 −1.41069
\(202\) −20.0000 −1.40720
\(203\) −4.00000 −0.280745
\(204\) −12.0000 −0.840168
\(205\) −4.00000 −0.279372
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −8.00000 −0.552052
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) −8.00000 −0.543075
\(218\) −2.00000 −0.135457
\(219\) 28.0000 1.89206
\(220\) −2.00000 −0.134840
\(221\) 12.0000 0.807207
\(222\) 12.0000 0.805387
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −2.00000 −0.133631
\(225\) −1.00000 −0.0666667
\(226\) 12.0000 0.798228
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 2.00000 0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) −16.0000 −1.04372
\(236\) 2.00000 0.130189
\(237\) −16.0000 −1.03931
\(238\) 12.0000 0.777844
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 4.00000 0.258199
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 4.00000 0.255031
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 24.0000 1.52094
\(250\) 12.0000 0.758947
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) −24.0000 −1.50294
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) −4.00000 −0.248069
\(261\) −2.00000 −0.123797
\(262\) −4.00000 −0.247121
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 2.00000 0.123091
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) −32.0000 −1.95837
\(268\) −10.0000 −0.610847
\(269\) 26.0000 1.58525 0.792624 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 8.00000 0.486864
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) −6.00000 −0.363803
\(273\) −8.00000 −0.484182
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 4.00000 0.239904
\(279\) −4.00000 −0.239474
\(280\) −4.00000 −0.239046
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 16.0000 0.952786
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −4.00000 −0.236113
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) 16.0000 0.937937
\(292\) 14.0000 0.819288
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 6.00000 0.349927
\(295\) 4.00000 0.232889
\(296\) 6.00000 0.348743
\(297\) 4.00000 0.232104
\(298\) −12.0000 −0.695141
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 40.0000 2.29794
\(304\) 0 0
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −2.00000 −0.113961
\(309\) 8.00000 0.455104
\(310\) 8.00000 0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 4.00000 0.226455
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 10.0000 0.564333
\(315\) 4.00000 0.225374
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 4.00000 0.224309
\(319\) 2.00000 0.111979
\(320\) 2.00000 0.111803
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 4.00000 0.221201
\(328\) 2.00000 0.110432
\(329\) −16.0000 −0.882109
\(330\) 4.00000 0.220193
\(331\) −14.0000 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) 12.0000 0.658586
\(333\) −6.00000 −0.328798
\(334\) 8.00000 0.437741
\(335\) −20.0000 −1.09272
\(336\) 4.00000 0.218218
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 9.00000 0.489535
\(339\) −24.0000 −1.30350
\(340\) −12.0000 −0.650791
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −4.00000 −0.214423
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 2.00000 0.106904
\(351\) 8.00000 0.427008
\(352\) 1.00000 0.0533002
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −16.0000 −0.847998
\(357\) −24.0000 −1.27021
\(358\) 18.0000 0.951330
\(359\) 14.0000 0.738892 0.369446 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) −2.00000 −0.105409
\(361\) 0 0
\(362\) −2.00000 −0.105118
\(363\) 2.00000 0.104973
\(364\) −4.00000 −0.209657
\(365\) 28.0000 1.46559
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 12.0000 0.623850
\(371\) −4.00000 −0.207670
\(372\) −8.00000 −0.414781
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) −6.00000 −0.310253
\(375\) −24.0000 −1.23935
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) 8.00000 0.411476
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) −4.00000 −0.204658
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) −2.00000 −0.102062
\(385\) −4.00000 −0.203859
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 8.00000 0.405096
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 8.00000 0.403547
\(394\) −4.00000 −0.201517
\(395\) −16.0000 −0.805047
\(396\) −1.00000 −0.0502519
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 20.0000 0.997509
\(403\) 8.00000 0.398508
\(404\) 20.0000 0.995037
\(405\) −22.0000 −1.09319
\(406\) 4.00000 0.198517
\(407\) 6.00000 0.297409
\(408\) 12.0000 0.594089
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 4.00000 0.197546
\(411\) −12.0000 −0.591916
\(412\) 4.00000 0.197066
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 2.00000 0.0980581
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 8.00000 0.390360
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −12.0000 −0.584151
\(423\) −8.00000 −0.388973
\(424\) 2.00000 0.0971286
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −4.00000 −0.192450
\(433\) 40.0000 1.92228 0.961139 0.276066i \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 8.00000 0.384012
\(435\) −8.00000 −0.383571
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −28.0000 −1.33789
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 2.00000 0.0953463
\(441\) −3.00000 −0.142857
\(442\) −12.0000 −0.570782
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −12.0000 −0.569495
\(445\) −32.0000 −1.51695
\(446\) 8.00000 0.378811
\(447\) 24.0000 1.13516
\(448\) 2.00000 0.0944911
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 1.00000 0.0471405
\(451\) 2.00000 0.0941763
\(452\) −12.0000 −0.564433
\(453\) 16.0000 0.751746
\(454\) 28.0000 1.31411
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −18.0000 −0.841085
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 4.00000 0.186097
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −16.0000 −0.741982
\(466\) 6.00000 0.277945
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −20.0000 −0.923514
\(470\) 16.0000 0.738025
\(471\) −20.0000 −0.921551
\(472\) −2.00000 −0.0920575
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) −2.00000 −0.0915737
\(478\) 14.0000 0.640345
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) −4.00000 −0.182574
\(481\) 12.0000 0.547153
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 16.0000 0.726523
\(486\) 10.0000 0.453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 6.00000 0.271052
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) −4.00000 −0.180334
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −24.0000 −1.07547
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −12.0000 −0.536656
\(501\) −16.0000 −0.714827
\(502\) −20.0000 −0.892644
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 40.0000 1.77998
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) 8.00000 0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 24.0000 1.06274
\(511\) 28.0000 1.23865
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −8.00000 −0.352865
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 12.0000 0.527250
\(519\) 36.0000 1.58022
\(520\) 4.00000 0.175412
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 2.00000 0.0875376
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 4.00000 0.174741
\(525\) −4.00000 −0.174574
\(526\) 18.0000 0.784837
\(527\) 24.0000 1.04546
\(528\) −2.00000 −0.0870388
\(529\) −23.0000 −1.00000
\(530\) 4.00000 0.173749
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 32.0000 1.38478
\(535\) −24.0000 −1.03761
\(536\) 10.0000 0.431934
\(537\) −36.0000 −1.55351
\(538\) −26.0000 −1.12094
\(539\) 3.00000 0.129219
\(540\) −8.00000 −0.344265
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −14.0000 −0.601351
\(543\) 4.00000 0.171656
\(544\) 6.00000 0.257248
\(545\) 4.00000 0.171341
\(546\) 8.00000 0.342368
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 20.0000 0.849719
\(555\) −24.0000 −1.01874
\(556\) −4.00000 −0.169638
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 4.00000 0.169031
\(561\) 12.0000 0.506640
\(562\) 22.0000 0.928014
\(563\) −16.0000 −0.674320 −0.337160 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) −16.0000 −0.673722
\(565\) −24.0000 −1.00969
\(566\) −16.0000 −0.672530
\(567\) −22.0000 −0.923913
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 2.00000 0.0836242
\(573\) 8.00000 0.334205
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −19.0000 −0.790296
\(579\) 28.0000 1.16364
\(580\) −4.00000 −0.166091
\(581\) 24.0000 0.995688
\(582\) −16.0000 −0.663221
\(583\) 2.00000 0.0828315
\(584\) −14.0000 −0.579324
\(585\) −4.00000 −0.165380
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −6.00000 −0.247436
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 8.00000 0.329076
\(592\) −6.00000 −0.246598
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) −4.00000 −0.164122
\(595\) −24.0000 −0.983904
\(596\) 12.0000 0.491539
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 2.00000 0.0816497
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) 8.00000 0.325515
\(605\) 2.00000 0.0813116
\(606\) −40.0000 −1.62489
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) −6.00000 −0.242536
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −8.00000 −0.322854
\(615\) −8.00000 −0.322591
\(616\) 2.00000 0.0805823
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) −8.00000 −0.321807
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −32.0000 −1.28205
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 36.0000 1.43541
\(630\) −4.00000 −0.159364
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 8.00000 0.318223
\(633\) 24.0000 0.953914
\(634\) −18.0000 −0.714871
\(635\) 16.0000 0.634941
\(636\) −4.00000 −0.158610
\(637\) 6.00000 0.237729
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 24.0000 0.947204
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 11.0000 0.432121
\(649\) −2.00000 −0.0785069
\(650\) −2.00000 −0.0784465
\(651\) −16.0000 −0.627089
\(652\) 4.00000 0.156652
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) −4.00000 −0.156412
\(655\) 8.00000 0.312586
\(656\) −2.00000 −0.0780869
\(657\) 14.0000 0.546192
\(658\) 16.0000 0.623745
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) −4.00000 −0.155700
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) 14.0000 0.544125
\(663\) 24.0000 0.932083
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) −16.0000 −0.618596
\(670\) 20.0000 0.772667
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 18.0000 0.693334
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 24.0000 0.921714
\(679\) 16.0000 0.614024
\(680\) 12.0000 0.460179
\(681\) −56.0000 −2.14592
\(682\) −4.00000 −0.153168
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 20.0000 0.763604
\(687\) 36.0000 1.37349
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000 0.684257
\(693\) −2.00000 −0.0759737
\(694\) 4.00000 0.151838
\(695\) −8.00000 −0.303457
\(696\) 4.00000 0.151620
\(697\) 12.0000 0.454532
\(698\) 12.0000 0.454207
\(699\) −12.0000 −0.453882
\(700\) −2.00000 −0.0755929
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −8.00000 −0.301941
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) −32.0000 −1.20519
\(706\) −14.0000 −0.526897
\(707\) 40.0000 1.50435
\(708\) 4.00000 0.150329
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 16.0000 0.599625
\(713\) 0 0
\(714\) 24.0000 0.898177
\(715\) 4.00000 0.149592
\(716\) −18.0000 −0.672692
\(717\) −28.0000 −1.04568
\(718\) −14.0000 −0.522475
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 2.00000 0.0745356
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 52.0000 1.93390
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) −2.00000 −0.0742270
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 4.00000 0.148250
\(729\) 13.0000 0.481481
\(730\) −28.0000 −1.03633
\(731\) 0 0
\(732\) 0 0
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) −28.0000 −1.03350
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 2.00000 0.0736210
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 8.00000 0.293294
\(745\) 24.0000 0.879292
\(746\) 34.0000 1.24483
\(747\) 12.0000 0.439057
\(748\) 6.00000 0.219382
\(749\) −24.0000 −0.876941
\(750\) 24.0000 0.876356
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −8.00000 −0.291730
\(753\) 40.0000 1.45768
\(754\) −4.00000 −0.145671
\(755\) 16.0000 0.582300
\(756\) −8.00000 −0.290957
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) −16.0000 −0.579619
\(763\) 4.00000 0.144810
\(764\) 4.00000 0.144715
\(765\) −12.0000 −0.433861
\(766\) 28.0000 1.01168
\(767\) −4.00000 −0.144432
\(768\) 2.00000 0.0721688
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 4.00000 0.144150
\(771\) 16.0000 0.576226
\(772\) 14.0000 0.503871
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) −8.00000 −0.287183
\(777\) −24.0000 −0.860995
\(778\) 14.0000 0.501924
\(779\) 0 0
\(780\) −8.00000 −0.286446
\(781\) 0 0
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) −3.00000 −0.107143
\(785\) −20.0000 −0.713831
\(786\) −8.00000 −0.285351
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 4.00000 0.142494
\(789\) −36.0000 −1.28163
\(790\) 16.0000 0.569254
\(791\) −24.0000 −0.853342
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) −8.00000 −0.283731
\(796\) −12.0000 −0.425329
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 1.00000 0.0353553
\(801\) −16.0000 −0.565332
\(802\) 16.0000 0.564980
\(803\) −14.0000 −0.494049
\(804\) −20.0000 −0.705346
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 52.0000 1.83049
\(808\) −20.0000 −0.703598
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 22.0000 0.773001
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) −4.00000 −0.140372
\(813\) 28.0000 0.982003
\(814\) −6.00000 −0.210300
\(815\) 8.00000 0.280228
\(816\) −12.0000 −0.420084
\(817\) 0 0
\(818\) 6.00000 0.209785
\(819\) −4.00000 −0.139771
\(820\) −4.00000 −0.139686
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 12.0000 0.418548
\(823\) −12.0000 −0.418294 −0.209147 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) −4.00000 −0.139347
\(825\) 2.00000 0.0696311
\(826\) −4.00000 −0.139178
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −24.0000 −0.833052
\(831\) −40.0000 −1.38758
\(832\) −2.00000 −0.0693375
\(833\) 18.0000 0.623663
\(834\) 8.00000 0.277017
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) −36.0000 −1.24360
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −8.00000 −0.276026
\(841\) −25.0000 −0.862069
\(842\) −22.0000 −0.758170
\(843\) −44.0000 −1.51544
\(844\) 12.0000 0.413057
\(845\) −18.0000 −0.619219
\(846\) 8.00000 0.275046
\(847\) 2.00000 0.0687208
\(848\) −2.00000 −0.0686803
\(849\) 32.0000 1.09824
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) −4.00000 −0.136558
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 24.0000 0.817443
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 4.00000 0.136083
\(865\) 36.0000 1.22404
\(866\) −40.0000 −1.35926
\(867\) 38.0000 1.29055
\(868\) −8.00000 −0.271538
\(869\) 8.00000 0.271381
\(870\) 8.00000 0.271225
\(871\) 20.0000 0.677674
\(872\) −2.00000 −0.0677285
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 28.0000 0.946032
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −40.0000 −1.34993
\(879\) −12.0000 −0.404750
\(880\) −2.00000 −0.0674200
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 3.00000 0.101015
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 12.0000 0.403604
\(885\) 8.00000 0.268917
\(886\) 12.0000 0.403148
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 12.0000 0.402694
\(889\) 16.0000 0.536623
\(890\) 32.0000 1.07264
\(891\) 11.0000 0.368514
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) −24.0000 −0.802680
\(895\) −36.0000 −1.20335
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 36.0000 1.20134
\(899\) 8.00000 0.266815
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) 4.00000 0.132964
\(906\) −16.0000 −0.531564
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −28.0000 −0.929213
\(909\) 20.0000 0.663358
\(910\) 8.00000 0.265197
\(911\) 60.0000 1.98789 0.993944 0.109885i \(-0.0350482\pi\)
0.993944 + 0.109885i \(0.0350482\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) 18.0000 0.594737
\(917\) 8.00000 0.264183
\(918\) −24.0000 −0.792118
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) −4.00000 −0.131733
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 6.00000 0.197279
\(926\) 28.0000 0.920137
\(927\) 4.00000 0.131377
\(928\) 2.00000 0.0656532
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 48.0000 1.57145
\(934\) −28.0000 −0.916188
\(935\) 12.0000 0.392442
\(936\) 2.00000 0.0653720
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 20.0000 0.653023
\(939\) 52.0000 1.69696
\(940\) −16.0000 −0.521862
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) 2.00000 0.0650945
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −16.0000 −0.519656
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) 12.0000 0.388922
\(953\) −10.0000 −0.323932 −0.161966 0.986796i \(-0.551783\pi\)
−0.161966 + 0.986796i \(0.551783\pi\)
\(954\) 2.00000 0.0647524
\(955\) 8.00000 0.258874
\(956\) −14.0000 −0.452792
\(957\) 4.00000 0.129302
\(958\) −30.0000 −0.969256
\(959\) −12.0000 −0.387500
\(960\) 4.00000 0.129099
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) 26.0000 0.837404
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) 6.00000 0.192947 0.0964735 0.995336i \(-0.469244\pi\)
0.0964735 + 0.995336i \(0.469244\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −16.0000 −0.513729
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) −10.0000 −0.320750
\(973\) −8.00000 −0.256468
\(974\) 32.0000 1.02535
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) −8.00000 −0.255812
\(979\) 16.0000 0.511362
\(980\) −6.00000 −0.191663
\(981\) 2.00000 0.0638551
\(982\) 32.0000 1.02116
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 4.00000 0.127515
\(985\) 8.00000 0.254901
\(986\) −12.0000 −0.382158
\(987\) −32.0000 −1.01857
\(988\) 0 0
\(989\) 0 0
\(990\) 2.00000 0.0635642
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 4.00000 0.127000
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 24.0000 0.760469
\(997\) −44.0000 −1.39349 −0.696747 0.717317i \(-0.745370\pi\)
−0.696747 + 0.717317i \(0.745370\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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 7942.2.a.k.1.1 1
19.18 odd 2 7942.2.a.m.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7942.2.a.k.1.1 1 1.1 even 1 trivial
7942.2.a.m.1.1 yes 1 19.18 odd 2