Properties

Label 5082.2.a.d.1.1
Level $5082$
Weight $2$
Character 5082.1
Self dual yes
Analytic conductor $40.580$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5082.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.5799743072\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5082.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +8.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} +6.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -2.00000 q^{45} -8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +12.0000 q^{65} +4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -6.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} +2.00000 q^{90} -6.00000 q^{91} +8.00000 q^{92} -8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 −0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 6.00000 0.960769
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −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
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) −8.00000 −0.963087
\(70\) 2.00000 0.239046
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 10.0000 1.16248
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) −6.00000 −0.628971
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 0.588348
\(105\) 2.00000 0.195180
\(106\) −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 \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 4.00000 0.374634
\(115\) −16.0000 −1.49201
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) −4.00000 −0.368230
\(119\) −2.00000 −0.183340
\(120\) −2.00000 −0.182574
\(121\) 0 0
\(122\) 6.00000 0.543214
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) −1.00000 −0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −12.0000 −1.05247
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) 2.00000 0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 8.00000 0.681005
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) 10.0000 0.827606
\(147\) −1.00000 −0.0824786
\(148\) −10.0000 −0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 2.00000 0.158114
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 1.00000 0.0771517
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 2.00000 0.151620
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 6.00000 0.444750
\(183\) 6.00000 0.443533
\(184\) −8.00000 −0.589768
\(185\) 20.0000 1.47043
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 8.00000 0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 14.0000 1.00514
\(195\) −12.0000 −0.859338
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −2.00000 −0.140720
\(203\) 2.00000 0.140372
\(204\) 2.00000 0.140028
\(205\) −12.0000 −0.838116
\(206\) −8.00000 −0.557386
\(207\) 8.00000 0.556038
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) 12.0000 0.820303
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −10.0000 −0.671156
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 14.0000 0.931266
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 2.00000 0.129099
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) −2.00000 −0.127775
\(246\) 6.00000 0.382546
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 4.00000 0.249029
\(259\) −10.0000 −0.621370
\(260\) 12.0000 0.744208
\(261\) 2.00000 0.123797
\(262\) −20.0000 −1.23560
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) −2.00000 −0.121716
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −2.00000 −0.121268
\(273\) 6.00000 0.363137
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 8.00000 0.474713
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) 14.0000 0.820695
\(292\) −10.0000 −0.585206
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 1.00000 0.0583212
\(295\) −8.00000 −0.465778
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −48.0000 −2.77591
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) 12.0000 0.687118
\(306\) 2.00000 0.114332
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −6.00000 −0.339683
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 10.0000 0.564333
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 12.0000 0.669775
\(322\) −8.00000 −0.445823
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −20.0000 −1.10770
\(327\) −2.00000 −0.110600
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) −10.0000 −0.547997
\(334\) −8.00000 −0.437741
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −23.0000 −1.25104
\(339\) 14.0000 0.760376
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 16.0000 0.861411
\(346\) 22.0000 1.18273
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −2.00000 −0.107211
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 1.00000 0.0534522
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 4.00000 0.212598
\(355\) −16.0000 −0.849192
\(356\) −6.00000 −0.317999
\(357\) 2.00000 0.105851
\(358\) 12.0000 0.634220
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 20.0000 1.04685
\(366\) −6.00000 −0.313625
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 8.00000 0.417029
\(369\) 6.00000 0.312348
\(370\) −20.0000 −1.03975
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 1.00000 0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000 0.203331
\(388\) −14.0000 −0.710742
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 12.0000 0.607644
\(391\) −16.0000 −0.809155
\(392\) −1.00000 −0.0505076
\(393\) −20.0000 −1.00887
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −8.00000 −0.401004
\(399\) −4.00000 −0.200250
\(400\) −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
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) −2.00000 −0.0993808
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 12.0000 0.592638
\(411\) −10.0000 −0.493264
\(412\) 8.00000 0.394132
\(413\) 4.00000 0.196827
\(414\) −8.00000 −0.393179
\(415\) −8.00000 −0.392705
\(416\) 6.00000 0.294174
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 2.00000 0.0975900
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 2.00000 0.0970143
\(426\) 8.00000 0.387601
\(427\) −6.00000 −0.290360
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −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
\(437\) 32.0000 1.53077
\(438\) −10.0000 −0.477818
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 10.0000 0.474579
\(445\) 12.0000 0.568855
\(446\) 16.0000 0.757622
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) 12.0000 0.562569
\(456\) 4.00000 0.187317
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 2.00000 0.0934539
\(459\) 2.00000 0.0933520
\(460\) −16.0000 −0.746004
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 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
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −6.00000 −0.277350
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −4.00000 −0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 60.0000 2.73576
\(482\) 2.00000 0.0910975
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) 28.0000 1.27141
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 6.00000 0.271607
\(489\) −20.0000 −0.904431
\(490\) 2.00000 0.0903508
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.00000 −0.180151
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 4.00000 0.179244
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 12.0000 0.536656
\(501\) −8.00000 −0.357414
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 4.00000 0.177123
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 30.0000 1.32324
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) 22.0000 0.965693
\(520\) −12.0000 −0.526235
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 20.0000 0.873704
\(525\) 1.00000 0.0436436
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 12.0000 0.521247
\(531\) 4.00000 0.173585
\(532\) 4.00000 0.173422
\(533\) −36.0000 −1.55933
\(534\) −6.00000 −0.259645
\(535\) 24.0000 1.03761
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) −22.0000 −0.948487
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 2.00000 0.0857493
\(545\) −4.00000 −0.171341
\(546\) −6.00000 −0.256776
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 10.0000 0.427179
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) −20.0000 −0.848953
\(556\) −4.00000 −0.169638
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 26.0000 1.09674
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 0 0
\(565\) 28.0000 1.17797
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −8.00000 −0.335673
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −8.00000 −0.335083
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 13.0000 0.540729
\(579\) 2.00000 0.0831172
\(580\) −4.00000 −0.166091
\(581\) 4.00000 0.165948
\(582\) −14.0000 −0.580319
\(583\) 0 0
\(584\) 10.0000 0.413803
\(585\) 12.0000 0.496139
\(586\) 30.0000 1.23929
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) −10.0000 −0.411345
\(592\) −10.0000 −0.410997
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) −6.00000 −0.245770
\(597\) −8.00000 −0.327418
\(598\) 48.0000 1.96287
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −4.00000 −0.163028
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) −48.0000 −1.94826 −0.974130 0.225989i \(-0.927439\pi\)
−0.974130 + 0.225989i \(0.927439\pi\)
\(608\) −4.00000 −0.162221
\(609\) −2.00000 −0.0810441
\(610\) −12.0000 −0.485866
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 28.0000 1.12999
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 8.00000 0.321807
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 8.00000 0.320771
\(623\) −6.00000 −0.240385
\(624\) 6.00000 0.240192
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 20.0000 0.797452
\(630\) 2.00000 0.0796819
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 2.00000 0.0790569
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −12.0000 −0.473602
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 8.00000 0.315244
\(645\) 8.00000 0.315000
\(646\) 8.00000 0.314756
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 2.00000 0.0782062
\(655\) −40.0000 −1.56293
\(656\) 6.00000 0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 4.00000 0.155464
\(663\) −12.0000 −0.466041
\(664\) −4.00000 −0.155230
\(665\) −8.00000 −0.310227
\(666\) 10.0000 0.387492
\(667\) 16.0000 0.619522
\(668\) 8.00000 0.309529
\(669\) 16.0000 0.618596
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −14.0000 −0.537667
\(679\) −14.0000 −0.537271
\(680\) −4.00000 −0.153393
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 4.00000 0.152944
\(685\) −20.0000 −0.764161
\(686\) −1.00000 −0.0381802
\(687\) 2.00000 0.0763048
\(688\) 4.00000 0.152499
\(689\) −36.0000 −1.37149
\(690\) −16.0000 −0.609110
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 8.00000 0.303457
\(696\) 2.00000 0.0758098
\(697\) −12.0000 −0.454532
\(698\) 22.0000 0.832712
\(699\) −22.0000 −0.832116
\(700\) −1.00000 −0.0377964
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −6.00000 −0.226455
\(703\) −40.0000 −1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 2.00000 0.0752177
\(708\) −4.00000 −0.150329
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) −18.0000 −0.668965
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) −8.00000 −0.295891
\(732\) 6.00000 0.221766
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −32.0000 −1.18114
\(735\) 2.00000 0.0737711
\(736\) −8.00000 −0.294884
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 20.0000 0.735215
\(741\) 24.0000 0.881662
\(742\) −6.00000 −0.220267
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 22.0000 0.805477
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 12.0000 0.438178
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 12.0000 0.437014
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) 16.0000 0.578103
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) −2.00000 −0.0719816
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 10.0000 0.358748
\(778\) 26.0000 0.932145
\(779\) 24.0000 0.859889
\(780\) −12.0000 −0.429669
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) 20.0000 0.713831
\(786\) 20.0000 0.713376
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 10.0000 0.356235
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) −6.00000 −0.212932
\(795\) 12.0000 0.425596
\(796\) 8.00000 0.283552
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) −22.0000 −0.774437
\(808\) −2.00000 −0.0703598
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 2.00000 0.0702728
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 0 0
\(815\) −40.0000 −1.40114
\(816\) 2.00000 0.0700140
\(817\) 16.0000 0.559769
\(818\) −22.0000 −0.769212
\(819\) −6.00000 −0.209657
\(820\) −12.0000 −0.419058
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 10.0000 0.348790
\(823\) −56.0000 −1.95204 −0.976019 0.217687i \(-0.930149\pi\)
−0.976019 + 0.217687i \(0.930149\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 8.00000 0.278019
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 8.00000 0.277684
\(831\) −10.0000 −0.346896
\(832\) −6.00000 −0.208013
\(833\) −2.00000 −0.0692959
\(834\) −4.00000 −0.138509
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) 56.0000 1.93333 0.966667 0.256036i \(-0.0824164\pi\)
0.966667 + 0.256036i \(0.0824164\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) −6.00000 −0.206774
\(843\) 26.0000 0.895488
\(844\) −20.0000 −0.688428
\(845\) −46.0000 −1.58245
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 4.00000 0.137280
\(850\) −2.00000 −0.0685994
\(851\) −80.0000 −2.74236
\(852\) −8.00000 −0.274075
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 6.00000 0.205316
\(855\) −8.00000 −0.273594
\(856\) 12.0000 0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −8.00000 −0.272798
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 1.00000 0.0340207
\(865\) 44.0000 1.49604
\(866\) −2.00000 −0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) −24.0000 −0.813209
\(872\) −2.00000 −0.0677285
\(873\) −14.0000 −0.473828
\(874\) −32.0000 −1.08242
\(875\) 12.0000 0.405674
\(876\) 10.0000 0.337869
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −24.0000 −0.809961
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 12.0000 0.403604
\(885\) 8.00000 0.268917
\(886\) 4.00000 0.134383
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −10.0000 −0.335578
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 24.0000 0.802232
\(896\) −1.00000 −0.0334077
\(897\) 48.0000 1.60267
\(898\) −34.0000 −1.13459
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 14.0000 0.465633
\(905\) 36.0000 1.19668
\(906\) 8.00000 0.265782
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −12.0000 −0.398234
\(909\) 2.00000 0.0663358
\(910\) −12.0000 −0.397796
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) −12.0000 −0.396708
\(916\) −2.00000 −0.0660819
\(917\) 20.0000 0.660458
\(918\) −2.00000 −0.0660098
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 16.0000 0.527504
\(921\) 28.0000 0.922631
\(922\) 22.0000 0.724531
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 32.0000 1.05159
\(927\) 8.00000 0.262754
\(928\) −2.00000 −0.0656532
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 22.0000 0.720634
\(933\) 8.00000 0.261908
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −4.00000 −0.130605
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) −10.0000 −0.325818
\(943\) 48.0000 1.56310
\(944\) 4.00000 0.130189
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) 2.00000 0.0648204
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) 10.0000 0.322917
\(960\) 2.00000 0.0645497
\(961\) −31.0000 −1.00000
\(962\) −60.0000 −1.93448
\(963\) −12.0000 −0.386695
\(964\) −2.00000 −0.0644157
\(965\) 4.00000 0.128765
\(966\) 8.00000 0.257396
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) −28.0000 −0.899026
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) −8.00000 −0.256337
\(975\) −6.00000 −0.192154
\(976\) −6.00000 −0.192055
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) −2.00000 −0.0638877
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 6.00000 0.191273
\(985\) −20.0000 −0.637253
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) −8.00000 −0.253745
\(995\) −16.0000 −0.507234
\(996\) −4.00000 −0.126745
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 44.0000 1.39280
\(999\) 10.0000 0.316386
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 5082.2.a.d.1.1 1
11.10 odd 2 42.2.a.a.1.1 1
33.32 even 2 126.2.a.a.1.1 1
44.43 even 2 336.2.a.d.1.1 1
55.32 even 4 1050.2.g.a.799.2 2
55.43 even 4 1050.2.g.a.799.1 2
55.54 odd 2 1050.2.a.i.1.1 1
77.10 even 6 294.2.e.a.79.1 2
77.32 odd 6 294.2.e.c.79.1 2
77.54 even 6 294.2.e.a.67.1 2
77.65 odd 6 294.2.e.c.67.1 2
77.76 even 2 294.2.a.g.1.1 1
88.21 odd 2 1344.2.a.q.1.1 1
88.43 even 2 1344.2.a.i.1.1 1
99.32 even 6 1134.2.f.j.379.1 2
99.43 odd 6 1134.2.f.g.757.1 2
99.65 even 6 1134.2.f.j.757.1 2
99.76 odd 6 1134.2.f.g.379.1 2
132.131 odd 2 1008.2.a.j.1.1 1
143.142 odd 2 7098.2.a.f.1.1 1
165.32 odd 4 3150.2.g.r.2899.1 2
165.98 odd 4 3150.2.g.r.2899.2 2
165.164 even 2 3150.2.a.bo.1.1 1
176.21 odd 4 5376.2.c.bc.2689.2 2
176.43 even 4 5376.2.c.e.2689.1 2
176.109 odd 4 5376.2.c.bc.2689.1 2
176.131 even 4 5376.2.c.e.2689.2 2
220.219 even 2 8400.2.a.k.1.1 1
231.32 even 6 882.2.g.h.667.1 2
231.65 even 6 882.2.g.h.361.1 2
231.131 odd 6 882.2.g.j.361.1 2
231.164 odd 6 882.2.g.j.667.1 2
231.230 odd 2 882.2.a.b.1.1 1
264.131 odd 2 4032.2.a.m.1.1 1
264.197 even 2 4032.2.a.e.1.1 1
308.87 odd 6 2352.2.q.n.961.1 2
308.131 odd 6 2352.2.q.n.1537.1 2
308.219 even 6 2352.2.q.i.1537.1 2
308.263 even 6 2352.2.q.i.961.1 2
308.307 odd 2 2352.2.a.l.1.1 1
385.384 even 2 7350.2.a.f.1.1 1
616.307 odd 2 9408.2.a.bw.1.1 1
616.461 even 2 9408.2.a.n.1.1 1
924.923 even 2 7056.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 11.10 odd 2
126.2.a.a.1.1 1 33.32 even 2
294.2.a.g.1.1 1 77.76 even 2
294.2.e.a.67.1 2 77.54 even 6
294.2.e.a.79.1 2 77.10 even 6
294.2.e.c.67.1 2 77.65 odd 6
294.2.e.c.79.1 2 77.32 odd 6
336.2.a.d.1.1 1 44.43 even 2
882.2.a.b.1.1 1 231.230 odd 2
882.2.g.h.361.1 2 231.65 even 6
882.2.g.h.667.1 2 231.32 even 6
882.2.g.j.361.1 2 231.131 odd 6
882.2.g.j.667.1 2 231.164 odd 6
1008.2.a.j.1.1 1 132.131 odd 2
1050.2.a.i.1.1 1 55.54 odd 2
1050.2.g.a.799.1 2 55.43 even 4
1050.2.g.a.799.2 2 55.32 even 4
1134.2.f.g.379.1 2 99.76 odd 6
1134.2.f.g.757.1 2 99.43 odd 6
1134.2.f.j.379.1 2 99.32 even 6
1134.2.f.j.757.1 2 99.65 even 6
1344.2.a.i.1.1 1 88.43 even 2
1344.2.a.q.1.1 1 88.21 odd 2
2352.2.a.l.1.1 1 308.307 odd 2
2352.2.q.i.961.1 2 308.263 even 6
2352.2.q.i.1537.1 2 308.219 even 6
2352.2.q.n.961.1 2 308.87 odd 6
2352.2.q.n.1537.1 2 308.131 odd 6
3150.2.a.bo.1.1 1 165.164 even 2
3150.2.g.r.2899.1 2 165.32 odd 4
3150.2.g.r.2899.2 2 165.98 odd 4
4032.2.a.e.1.1 1 264.197 even 2
4032.2.a.m.1.1 1 264.131 odd 2
5082.2.a.d.1.1 1 1.1 even 1 trivial
5376.2.c.e.2689.1 2 176.43 even 4
5376.2.c.e.2689.2 2 176.131 even 4
5376.2.c.bc.2689.1 2 176.109 odd 4
5376.2.c.bc.2689.2 2 176.21 odd 4
7056.2.a.k.1.1 1 924.923 even 2
7098.2.a.f.1.1 1 143.142 odd 2
7350.2.a.f.1.1 1 385.384 even 2
8400.2.a.k.1.1 1 220.219 even 2
9408.2.a.n.1.1 1 616.461 even 2
9408.2.a.bw.1.1 1 616.307 odd 2