Properties

Label 6762.2.a.bc.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +12.0000 q^{43} +4.00000 q^{44} +2.00000 q^{45} +1.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} -2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} +10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} -1.00000 q^{69} -16.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +2.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +16.0000 q^{83} -12.0000 q^{85} +12.0000 q^{86} +2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +1.00000 q^{92} +4.00000 q^{93} +12.0000 q^{94} -1.00000 q^{96} +2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 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
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 4.00000 0.603023
\(45\) 2.00000 0.298142
\(46\) 1.00000 0.147442
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −2.00000 −0.258199
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 12.0000 1.29399
\(87\) 2.00000 0.214423
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 4.00000 0.414781
\(94\) 12.0000 1.23771
\(95\) 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
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 6.00000 0.594089
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 8.00000 0.762770
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) −2.00000 −0.172133
\(136\) −6.00000 −0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −16.0000 −1.34269
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 2.00000 0.160128
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) −8.00000 −0.622799
\(166\) 16.0000 1.24184
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 1.00000 0.0737210
\(185\) 12.0000 0.882258
\(186\) 4.00000 0.293294
\(187\) −24.0000 −1.75505
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 2.00000 0.143592
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 6.00000 0.412082
\(213\) 16.0000 1.09630
\(214\) −12.0000 −0.820303
\(215\) 24.0000 1.63679
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) 8.00000 0.539360
\(221\) 12.0000 0.807207
\(222\) −6.00000 −0.402694
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 24.0000 1.56559
\(236\) 4.00000 0.260378
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −2.00000 −0.129099
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) −16.0000 −1.01396
\(250\) −12.0000 −0.758947
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −8.00000 −0.501965
\(255\) 12.0000 0.751469
\(256\) 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
\(261\) −2.00000 −0.123797
\(262\) −4.00000 −0.247121
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −4.00000 −0.246183
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −2.00000 −0.121716
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −4.00000 −0.241209
\(276\) −1.00000 −0.0601929
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −12.0000 −0.714590
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −4.00000 −0.234888
\(291\) −2.00000 −0.117242
\(292\) −2.00000 −0.117041
\(293\) 34.0000 1.98630 0.993151 0.116841i \(-0.0372769\pi\)
0.993151 + 0.116841i \(0.0372769\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) 6.00000 0.347571
\(299\) −2.00000 −0.115663
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) 20.0000 1.14520
\(306\) −6.00000 −0.342997
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 2.00000 0.113228
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) −6.00000 −0.336463
\(319\) −8.00000 −0.447914
\(320\) 2.00000 0.111803
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 16.0000 0.878114
\(333\) 6.00000 0.328798
\(334\) −20.0000 −1.09435
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) −12.0000 −0.650791
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) −2.00000 −0.107676
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 2.00000 0.107211
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −4.00000 −0.212598
\(355\) −32.0000 −1.69838
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 2.00000 0.105409
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) −10.0000 −0.522708
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) 12.0000 0.623850
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −24.0000 −1.24101
\(375\) 12.0000 0.619677
\(376\) 12.0000 0.618853
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 8.00000 0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 12.0000 0.609994
\(388\) 2.00000 0.101535
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) 4.00000 0.202548
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) −18.0000 −0.906827
\(395\) 16.0000 0.805047
\(396\) 4.00000 0.201008
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −4.00000 −0.199502
\(403\) 8.00000 0.398508
\(404\) 14.0000 0.696526
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 6.00000 0.297044
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 12.0000 0.592638
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 32.0000 1.57082
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 28.0000 1.36302
\(423\) 12.0000 0.583460
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 8.00000 0.386244
\(430\) 24.0000 1.15738
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −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
\(435\) 4.00000 0.191785
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 2.00000 0.0955637
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −6.00000 −0.284747
\(445\) −12.0000 −0.568855
\(446\) 4.00000 0.189405
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 18.0000 0.841085
\(459\) 6.00000 0.280056
\(460\) 2.00000 0.0932505
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) 10.0000 0.463241
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 24.0000 1.10704
\(471\) −18.0000 −0.829396
\(472\) 4.00000 0.184115
\(473\) 48.0000 2.20704
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −12.0000 −0.547153
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 4.00000 0.181631
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 10.0000 0.452679
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −12.0000 −0.536656
\(501\) 20.0000 0.893534
\(502\) −24.0000 −1.07117
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) 4.00000 0.177822
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 48.0000 2.11104
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) −4.00000 −0.175412
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 24.0000 1.04546
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) 12.0000 0.521247
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) −24.0000 −1.03761
\(536\) 4.00000 0.172774
\(537\) −4.00000 −0.172613
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −20.0000 −0.859074
\(543\) −2.00000 −0.0858282
\(544\) −6.00000 −0.257248
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 10.0000 0.427179
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) −12.0000 −0.509372
\(556\) −4.00000 −0.169638
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) −4.00000 −0.169334
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) −22.0000 −0.928014
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −12.0000 −0.505291
\(565\) −12.0000 −0.504844
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) −16.0000 −0.671345
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −8.00000 −0.334497
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 19.0000 0.790296
\(579\) −18.0000 −0.748054
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) −2.00000 −0.0827606
\(585\) −4.00000 −0.165380
\(586\) 34.0000 1.40453
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) 18.0000 0.740421
\(592\) 6.00000 0.246598
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −2.00000 −0.0817861
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 1.00000 0.0408248
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) −14.0000 −0.568711
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) −24.0000 −0.970936
\(612\) −6.00000 −0.242536
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 20.0000 0.807134
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) −8.00000 −0.321288
\(621\) −1.00000 −0.0401286
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000 0.318223
\(633\) −28.0000 −1.11290
\(634\) −10.0000 −0.397151
\(635\) −16.0000 −0.634941
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) −16.0000 −0.632950
\(640\) 2.00000 0.0790569
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 12.0000 0.473602
\(643\) −24.0000 −0.946468 −0.473234 0.880937i \(-0.656913\pi\)
−0.473234 + 0.880937i \(0.656913\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 2.00000 0.0782062
\(655\) −8.00000 −0.312586
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −8.00000 −0.311400
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 28.0000 1.08825
\(663\) −12.0000 −0.466041
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −2.00000 −0.0774403
\(668\) −20.0000 −0.773823
\(669\) −4.00000 −0.154649
\(670\) 8.00000 0.309067
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) −12.0000 −0.460179
\(681\) −24.0000 −0.919682
\(682\) −16.0000 −0.612672
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 20.0000 0.764161
\(686\) 0 0
\(687\) −18.0000 −0.686743
\(688\) 12.0000 0.457496
\(689\) −12.0000 −0.457164
\(690\) −2.00000 −0.0761387
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −8.00000 −0.303457
\(696\) 2.00000 0.0758098
\(697\) −36.0000 −1.36360
\(698\) 6.00000 0.227103
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) −24.0000 −0.903892
\(706\) 14.0000 0.526897
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −32.0000 −1.20094
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 4.00000 0.149487
\(717\) −8.00000 −0.298765
\(718\) −16.0000 −0.597115
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) −10.0000 −0.371904
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) −5.00000 −0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) −72.0000 −2.66302
\(732\) −10.0000 −0.369611
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 16.0000 0.589368
\(738\) 6.00000 0.220863
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 4.00000 0.146647
\(745\) 12.0000 0.439646
\(746\) 22.0000 0.805477
\(747\) 16.0000 0.585409
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 12.0000 0.437595
\(753\) 24.0000 0.874609
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 20.0000 0.726433
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) −1.00000 −0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 18.0000 0.647834
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 12.0000 0.431331
\(775\) 4.00000 0.143684
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 38.0000 1.36237
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) −64.0000 −2.29010
\(782\) −6.00000 −0.214560
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 36.0000 1.28490
\(786\) 4.00000 0.142675
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) −18.0000 −0.641223
\(789\) 24.0000 0.854423
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −20.0000 −0.710221
\(794\) −34.0000 −1.20661
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) −8.00000 −0.282314
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −6.00000 −0.211210
\(808\) 14.0000 0.492518
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 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
\(813\) 20.0000 0.701431
\(814\) 24.0000 0.841200
\(815\) −8.00000 −0.280228
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −10.0000 −0.348790
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 1.00000 0.0347524
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 32.0000 1.11074
\(831\) 10.0000 0.346896
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) −40.0000 −1.38426
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −16.0000 −0.552711
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) 22.0000 0.757720
\(844\) 28.0000 0.963800
\(845\) −18.0000 −0.619219
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 24.0000 0.823678
\(850\) 6.00000 0.205798
\(851\) 6.00000 0.205677
\(852\) 16.0000 0.548151
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 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 \(0.377384\pi\)
\(858\) 8.00000 0.273115
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 12.0000 0.408012
\(866\) 2.00000 0.0679628
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 4.00000 0.135613
\(871\) −8.00000 −0.271070
\(872\) −2.00000 −0.0677285
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −26.0000 −0.877958 −0.438979 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) −20.0000 −0.674967
\(879\) −34.0000 −1.14679
\(880\) 8.00000 0.269680
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 12.0000 0.403604
\(885\) −8.00000 −0.268917
\(886\) 20.0000 0.671913
\(887\) 44.0000 1.47738 0.738688 0.674048i \(-0.235446\pi\)
0.738688 + 0.674048i \(0.235446\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) 4.00000 0.134005
\(892\) 4.00000 0.133930
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) −30.0000 −1.00111
\(899\) 8.00000 0.266815
\(900\) −1.00000 −0.0333333
\(901\) −36.0000 −1.19933
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 24.0000 0.796468
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 64.0000 2.11809
\(914\) −6.00000 −0.198462
\(915\) −20.0000 −0.661180
\(916\) 18.0000 0.594737
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 2.00000 0.0659380
\(921\) −20.0000 −0.659022
\(922\) 22.0000 0.724531
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −32.0000 −1.05159
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) 12.0000 0.392862
\(934\) −8.00000 −0.261768
\(935\) −48.0000 −1.56977
\(936\) −2.00000 −0.0653720
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 24.0000 0.782794
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) −18.0000 −0.586472
\(943\) 6.00000 0.195387
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −8.00000 −0.259828
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 6.00000 0.194257
\(955\) 16.0000 0.517748
\(956\) 8.00000 0.258738
\(957\) 8.00000 0.258603
\(958\) −40.0000 −1.29234
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) 10.0000 0.322078
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) −2.00000 −0.0640513
\(976\) 10.0000 0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 4.00000 0.127906
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) −12.0000 −0.382935
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −6.00000 −0.191273
\(985\) −36.0000 −1.14706
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 8.00000 0.254257
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) −4.00000 −0.127000
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) −36.0000 −1.13956
\(999\) −6.00000 −0.189832
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.bc.1.1 1
7.6 odd 2 966.2.a.j.1.1 1
21.20 even 2 2898.2.a.f.1.1 1
28.27 even 2 7728.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.j.1.1 1 7.6 odd 2
2898.2.a.f.1.1 1 21.20 even 2
6762.2.a.bc.1.1 1 1.1 even 1 trivial
7728.2.a.b.1.1 1 28.27 even 2