Properties

Label 930.2.a.j.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +1.00000 q^{20} +4.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +2.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} -4.00000 q^{42} -4.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{56} -4.00000 q^{59} +1.00000 q^{60} -1.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -2.00000 q^{66} +4.00000 q^{67} -6.00000 q^{69} -4.00000 q^{70} -16.0000 q^{71} -1.00000 q^{72} +4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +8.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +8.00000 q^{83} +4.00000 q^{84} +4.00000 q^{86} -2.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +8.00000 q^{91} -6.00000 q^{92} +1.00000 q^{93} -4.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.00000 −1.06904
\(15\) 1.00000 0.258199
\(16\) 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 \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 4.00000 0.872872
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 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
\(55\) 2.00000 0.269680
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −1.00000 −0.127000
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −2.00000 −0.246183
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) −4.00000 −0.478091
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 8.00000 0.838628
\(92\) −6.00000 −0.625543
\(93\) 1.00000 0.103695
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −9.00000 −0.909137
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −2.00000 −0.196116
\(105\) 4.00000 0.390360
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −2.00000 −0.190693
\(111\) −2.00000 −0.189832
\(112\) 4.00000 0.377964
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −10.0000 −0.901670
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) −4.00000 −0.356348
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) 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
\(134\) −4.00000 −0.345547
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 6.00000 0.510754
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 4.00000 0.338062
\(141\) 4.00000 0.336861
\(142\) 16.0000 1.34269
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 9.00000 0.742307
\(148\) −2.00000 −0.164399
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 1.00000 0.0803219
\(156\) 2.00000 0.160128
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −4.00000 −0.318223
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) −24.0000 −1.89146
\(162\) −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
\(165\) 2.00000 0.155700
\(166\) −8.00000 −0.620920
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 1.00000 0.0745356
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) −2.00000 −0.147043
\(186\) −1.00000 −0.0733236
\(187\) 0 0
\(188\) 4.00000 0.291730
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −14.0000 −1.00514
\(195\) 2.00000 0.143223
\(196\) 9.00000 0.642857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −2.00000 −0.142134
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 12.0000 0.836080
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −16.0000 −1.09630
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) 2.00000 0.135457
\(219\) 4.00000 0.270295
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −4.00000 −0.267261
\(225\) 1.00000 0.0666667
\(226\) −2.00000 −0.133038
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 6.00000 0.395628
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) −2.00000 −0.130744
\(235\) 4.00000 0.260931
\(236\) −4.00000 −0.260378
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 1.00000 0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 9.00000 0.574989
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) −1.00000 −0.0635001
\(249\) 8.00000 0.506979
\(250\) −1.00000 −0.0632456
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 4.00000 0.251976
\(253\) −12.0000 −0.754434
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) −8.00000 −0.497096
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) 10.0000 0.616626 0.308313 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) −2.00000 −0.123091
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 12.0000 0.724947
\(275\) 2.00000 0.120605
\(276\) −6.00000 −0.361158
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 6.00000 0.359856
\(279\) 1.00000 0.0598684
\(280\) −4.00000 −0.239046
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −4.00000 −0.238197
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −40.0000 −2.36113
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 4.00000 0.234082
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) −9.00000 −0.524891
\(295\) −4.00000 −0.232889
\(296\) 2.00000 0.116248
\(297\) 2.00000 0.116052
\(298\) 2.00000 0.115857
\(299\) −12.0000 −0.693978
\(300\) 1.00000 0.0577350
\(301\) −16.0000 −0.922225
\(302\) −4.00000 −0.230174
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 8.00000 0.455842
\(309\) −12.0000 −0.682656
\(310\) −1.00000 −0.0567962
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −2.00000 −0.113228
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) −18.0000 −1.01580
\(315\) 4.00000 0.225374
\(316\) 4.00000 0.225018
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) 24.0000 1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 12.0000 0.664619
\(327\) −2.00000 −0.110600
\(328\) 10.0000 0.552158
\(329\) 16.0000 0.882109
\(330\) −2.00000 −0.110096
\(331\) 18.0000 0.989369 0.494685 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) 8.00000 0.439057
\(333\) −2.00000 −0.109599
\(334\) −14.0000 −0.766046
\(335\) 4.00000 0.218543
\(336\) 4.00000 0.218218
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 9.00000 0.489535
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 4.00000 0.215666
\(345\) −6.00000 −0.323029
\(346\) 14.0000 0.752645
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −4.00000 −0.213809
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) 36.0000 1.91609 0.958043 0.286623i \(-0.0925328\pi\)
0.958043 + 0.286623i \(0.0925328\pi\)
\(354\) 4.00000 0.212598
\(355\) −16.0000 −0.849192
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) 20.0000 1.05118
\(363\) −7.00000 −0.367405
\(364\) 8.00000 0.419314
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) −6.00000 −0.312772
\(369\) −10.0000 −0.520579
\(370\) 2.00000 0.103975
\(371\) 24.0000 1.24602
\(372\) 1.00000 0.0518476
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.00000 0.407718
\(386\) 10.0000 0.508987
\(387\) −4.00000 −0.203331
\(388\) 14.0000 0.710742
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) 4.00000 0.201773
\(394\) 6.00000 0.302276
\(395\) 4.00000 0.201262
\(396\) 2.00000 0.100504
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −4.00000 −0.199502
\(403\) 2.00000 0.0996271
\(404\) −10.0000 −0.497519
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 10.0000 0.493865
\(411\) −12.0000 −0.591916
\(412\) −12.0000 −0.591198
\(413\) −16.0000 −0.787309
\(414\) 6.00000 0.294884
\(415\) 8.00000 0.392705
\(416\) −2.00000 −0.0980581
\(417\) −6.00000 −0.293821
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 4.00000 0.195180
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 4.00000 0.194717
\(423\) 4.00000 0.194487
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 4.00000 0.193122
\(430\) 4.00000 0.192897
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 1.00000 0.0481125
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 6.00000 0.284427
\(446\) 10.0000 0.473514
\(447\) −2.00000 −0.0945968
\(448\) 4.00000 0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −20.0000 −0.941763
\(452\) 2.00000 0.0940721
\(453\) 4.00000 0.187936
\(454\) 20.0000 0.938647
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 20.0000 0.934539
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) −8.00000 −0.372194
\(463\) −18.0000 −0.836531 −0.418265 0.908325i \(-0.637362\pi\)
−0.418265 + 0.908325i \(0.637362\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 22.0000 1.01913
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 2.00000 0.0924500
\(469\) 16.0000 0.738811
\(470\) −4.00000 −0.184506
\(471\) 18.0000 0.829396
\(472\) 4.00000 0.184115
\(473\) −8.00000 −0.367840
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −4.00000 −0.182384
\(482\) 22.0000 1.00207
\(483\) −24.0000 −1.09204
\(484\) −7.00000 −0.318182
\(485\) 14.0000 0.635707
\(486\) −1.00000 −0.0453609
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) −9.00000 −0.406579
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 1.00000 0.0449013
\(497\) −64.0000 −2.87079
\(498\) −8.00000 −0.358489
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 1.00000 0.0447214
\(501\) 14.0000 0.625474
\(502\) −2.00000 −0.0892644
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −4.00000 −0.178174
\(505\) −10.0000 −0.444994
\(506\) 12.0000 0.533465
\(507\) −9.00000 −0.399704
\(508\) 18.0000 0.798621
\(509\) −4.00000 −0.177297 −0.0886484 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −12.0000 −0.528783
\(516\) −4.00000 −0.176090
\(517\) 8.00000 0.351840
\(518\) 8.00000 0.351500
\(519\) −14.0000 −0.614532
\(520\) −2.00000 −0.0877058
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 4.00000 0.174741
\(525\) 4.00000 0.174574
\(526\) −10.0000 −0.436021
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 13.0000 0.565217
\(530\) −6.00000 −0.260623
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) −6.00000 −0.259645
\(535\) 12.0000 0.518805
\(536\) −4.00000 −0.172774
\(537\) −2.00000 −0.0863064
\(538\) 24.0000 1.03471
\(539\) 18.0000 0.775315
\(540\) 1.00000 0.0430331
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) −8.00000 −0.342368
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) 16.0000 0.680389
\(554\) 2.00000 0.0849719
\(555\) −2.00000 −0.0848953
\(556\) −6.00000 −0.254457
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −8.00000 −0.338364
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 4.00000 0.168430
\(565\) 2.00000 0.0841406
\(566\) 4.00000 0.168133
\(567\) 4.00000 0.167984
\(568\) 16.0000 0.671345
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 40.0000 1.66957
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 17.0000 0.707107
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) −14.0000 −0.580319
\(583\) 12.0000 0.496989
\(584\) −4.00000 −0.165521
\(585\) 2.00000 0.0826898
\(586\) 2.00000 0.0826192
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 4.00000 0.164677
\(591\) −6.00000 −0.246807
\(592\) −2.00000 −0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 20.0000 0.818546
\(598\) 12.0000 0.490716
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 16.0000 0.652111
\(603\) 4.00000 0.162893
\(604\) 4.00000 0.162758
\(605\) −7.00000 −0.284590
\(606\) 10.0000 0.406222
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −42.0000 −1.69636 −0.848182 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) 28.0000 1.12999
\(615\) −10.0000 −0.403239
\(616\) −8.00000 −0.322329
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 12.0000 0.482711
\(619\) 34.0000 1.36658 0.683288 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 1.00000 0.0401610
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 0 0
\(630\) −4.00000 −0.159364
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −4.00000 −0.159111
\(633\) −4.00000 −0.158986
\(634\) 6.00000 0.238290
\(635\) 18.0000 0.714308
\(636\) 6.00000 0.237915
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) −1.00000 −0.0395285
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −12.0000 −0.473602
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) −24.0000 −0.945732
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.00000 −0.314027
\(650\) −2.00000 −0.0784465
\(651\) 4.00000 0.156772
\(652\) −12.0000 −0.469956
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 2.00000 0.0782062
\(655\) 4.00000 0.156293
\(656\) −10.0000 −0.390434
\(657\) 4.00000 0.156055
\(658\) −16.0000 −0.623745
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 2.00000 0.0778499
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −18.0000 −0.699590
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 14.0000 0.541676
\(669\) −10.0000 −0.386622
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 32.0000 1.23259
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) −2.00000 −0.0765840
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −8.00000 −0.305441
\(687\) −20.0000 −0.763048
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 6.00000 0.228416
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −14.0000 −0.532200
\(693\) 8.00000 0.303895
\(694\) −24.0000 −0.911028
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) 14.0000 0.529908
\(699\) −22.0000 −0.832116
\(700\) 4.00000 0.151186
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 4.00000 0.150649
\(706\) −36.0000 −1.35488
\(707\) −40.0000 −1.50435
\(708\) −4.00000 −0.150329
\(709\) −32.0000 −1.20179 −0.600893 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 16.0000 0.600469
\(711\) 4.00000 0.150012
\(712\) −6.00000 −0.224860
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −2.00000 −0.0747435
\(717\) −8.00000 −0.298765
\(718\) −24.0000 −0.895672
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 1.00000 0.0372678
\(721\) −48.0000 −1.78761
\(722\) 19.0000 0.707107
\(723\) −22.0000 −0.818189
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 0 0
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −26.0000 −0.959678
\(735\) 9.00000 0.331970
\(736\) 6.00000 0.221163
\(737\) 8.00000 0.294684
\(738\) 10.0000 0.368105
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 26.0000 0.953847 0.476924 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) −1.00000 −0.0366618
\(745\) −2.00000 −0.0732743
\(746\) −6.00000 −0.219676
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) −1.00000 −0.0365148
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 4.00000 0.145865
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) 4.00000 0.145575
\(756\) 4.00000 0.145479
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 16.0000 0.581146
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) −18.0000 −0.652071
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) −8.00000 −0.288863
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) −8.00000 −0.288300
\(771\) 18.0000 0.648254
\(772\) −10.0000 −0.359908
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) 4.00000 0.143777
\(775\) 1.00000 0.0359211
\(776\) −14.0000 −0.502571
\(777\) −8.00000 −0.286998
\(778\) −20.0000 −0.717035
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 18.0000 0.642448
\(786\) −4.00000 −0.142675
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −6.00000 −0.213741
\(789\) 10.0000 0.356009
\(790\) −4.00000 −0.142314
\(791\) 8.00000 0.284447
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) 6.00000 0.212798
\(796\) 20.0000 0.708881
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) −30.0000 −1.05934
\(803\) 8.00000 0.282314
\(804\) 4.00000 0.141069
\(805\) −24.0000 −0.845889
\(806\) −2.00000 −0.0704470
\(807\) −24.0000 −0.844840
\(808\) 10.0000 0.351799
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 0 0
\(818\) 18.0000 0.629355
\(819\) 8.00000 0.279543
\(820\) −10.0000 −0.349215
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 12.0000 0.418548
\(823\) 18.0000 0.627441 0.313720 0.949515i \(-0.398425\pi\)
0.313720 + 0.949515i \(0.398425\pi\)
\(824\) 12.0000 0.418040
\(825\) 2.00000 0.0696311
\(826\) 16.0000 0.556711
\(827\) −56.0000 −1.94731 −0.973655 0.228024i \(-0.926773\pi\)
−0.973655 + 0.228024i \(0.926773\pi\)
\(828\) −6.00000 −0.208514
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) −8.00000 −0.277684
\(831\) −2.00000 −0.0693792
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 6.00000 0.207763
\(835\) 14.0000 0.484490
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) −36.0000 −1.24360
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) −4.00000 −0.138013
\(841\) −29.0000 −1.00000
\(842\) 30.0000 1.03387
\(843\) 6.00000 0.206651
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) −4.00000 −0.137523
\(847\) −28.0000 −0.962091
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) −16.0000 −0.548151
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) −4.00000 −0.136558
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) −4.00000 −0.136399
\(861\) −40.0000 −1.36320
\(862\) −40.0000 −1.36241
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −14.0000 −0.476014
\(866\) 20.0000 0.679628
\(867\) −17.0000 −0.577350
\(868\) 4.00000 0.135769
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 2.00000 0.0677285
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 4.00000 0.135225
\(876\) 4.00000 0.135147
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −24.0000 −0.809961
\(879\) −2.00000 −0.0674583
\(880\) 2.00000 0.0674200
\(881\) 22.0000 0.741199 0.370599 0.928793i \(-0.379152\pi\)
0.370599 + 0.928793i \(0.379152\pi\)
\(882\) −9.00000 −0.303046
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) −12.0000 −0.403148
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 2.00000 0.0671156
\(889\) 72.0000 2.41480
\(890\) −6.00000 −0.201120
\(891\) 2.00000 0.0670025
\(892\) −10.0000 −0.334825
\(893\) 0 0
\(894\) 2.00000 0.0668900
\(895\) −2.00000 −0.0668526
\(896\) −4.00000 −0.133631
\(897\) −12.0000 −0.400668
\(898\) −6.00000 −0.200223
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 20.0000 0.665927
\(903\) −16.0000 −0.532447
\(904\) −2.00000 −0.0665190
\(905\) −20.0000 −0.664822
\(906\) −4.00000 −0.132891
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −20.0000 −0.663723
\(909\) −10.0000 −0.331679
\(910\) −8.00000 −0.265197
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 6.00000 0.197814
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) −32.0000 −1.05329
\(924\) 8.00000 0.263181
\(925\) −2.00000 −0.0657596
\(926\) 18.0000 0.591517
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) −1.00000 −0.0327913
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) −16.0000 −0.522419
\(939\) 28.0000 0.913745
\(940\) 4.00000 0.130466
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) −18.0000 −0.586472
\(943\) 60.0000 1.95387
\(944\) −4.00000 −0.130189
\(945\) 4.00000 0.130120
\(946\) 8.00000 0.260102
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 4.00000 0.129914
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −12.0000 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) −48.0000 −1.55000
\(960\) 1.00000 0.0322749
\(961\) 1.00000 0.0322581
\(962\) 4.00000 0.128965
\(963\) 12.0000 0.386695
\(964\) −22.0000 −0.708572
\(965\) −10.0000 −0.321911
\(966\) 24.0000 0.772187
\(967\) 26.0000 0.836104 0.418052 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 1.00000 0.0320750
\(973\) −24.0000 −0.769405
\(974\) −10.0000 −0.320421
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 12.0000 0.383718
\(979\) 12.0000 0.383522
\(980\) 9.00000 0.287494
\(981\) −2.00000 −0.0638551
\(982\) −6.00000 −0.191468
\(983\) 10.0000 0.318950 0.159475 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 10.0000 0.318788
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) −2.00000 −0.0635642
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 18.0000 0.571213
\(994\) 64.0000 2.02996
\(995\) 20.0000 0.634043
\(996\) 8.00000 0.253490
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) −22.0000 −0.696398
\(999\) −2.00000 −0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 930.2.a.j.1.1 1
3.2 odd 2 2790.2.a.v.1.1 1
4.3 odd 2 7440.2.a.g.1.1 1
5.2 odd 4 4650.2.d.h.3349.1 2
5.3 odd 4 4650.2.d.h.3349.2 2
5.4 even 2 4650.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.j.1.1 1 1.1 even 1 trivial
2790.2.a.v.1.1 1 3.2 odd 2
4650.2.a.x.1.1 1 5.4 even 2
4650.2.d.h.3349.1 2 5.2 odd 4
4650.2.d.h.3349.2 2 5.3 odd 4
7440.2.a.g.1.1 1 4.3 odd 2