Properties

Label 42.2.a.a.1.1
Level $42$
Weight $2$
Character 42.1
Self dual yes
Analytic conductor $0.335$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 42.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.335371688489\)
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 \(\chi\) \(=\) 42.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{77} -6.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -6.00000 q^{91} +8.00000 q^{92} +8.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 \(-0.647584\pi\)
−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
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 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 \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 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 \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) −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
\(55\) 8.00000 1.07872
\(56\) −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 \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 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 \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −10.0000 −1.16248
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) −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 \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 1.00000 0.109109
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 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
\(91\) −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
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −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 \(-0.303036\pi\)
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 \(0.530535\pi\)
\(110\) 8.00000 0.762770
\(111\) 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
\(121\) 5.00000 0.454545
\(122\) 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 \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 4.00000 0.348155
\(133\) 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 \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −24.0000 −2.00698
\(144\) 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 \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 4.00000 0.322329
\(155\) 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
\(165\) −8.00000 −0.622799
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 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 \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 2.00000 0.151620
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) −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
\(187\) −8.00000 −0.585018
\(188\) 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 \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −14.0000 −1.00514
\(195\) 12.0000 0.859338
\(196\) 1.00000 0.0714286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −4.00000 −0.284268
\(199\) 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
\(209\) 16.0000 1.10674
\(210\) −2.00000 −0.138013
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 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
\(220\) 8.00000 0.539360
\(221\) 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 \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 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
\(231\) −4.00000 −0.263181
\(232\) −2.00000 −0.131306
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 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 \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 5.00000 0.321412
\(243\) −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
\(253\) −32.0000 −2.01182
\(254\) 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 \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 4.00000 0.246183
\(265\) −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
\(275\) 4.00000 0.241209
\(276\) −8.00000 −0.481543
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) −8.00000 −0.473879
\(286\) −24.0000 −1.41915
\(287\) 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 \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −8.00000 −0.465778
\(296\) −10.0000 −0.581238
\(297\) 4.00000 0.232104
\(298\) 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 \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 4.00000 0.227921
\(309\) −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
\(319\) 8.00000 0.447914
\(320\) −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
\(330\) −8.00000 −0.440386
\(331\) −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 \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 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 \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 2.00000 0.107211
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 1.00000 0.0534522
\(351\) −6.00000 −0.320256
\(352\) −4.00000 −0.213201
\(353\) −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 \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −18.0000 −0.946059
\(363\) −5.00000 −0.262432
\(364\) −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 \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −8.00000 −0.413670
\(375\) −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
\(385\) −8.00000 −0.407718
\(386\) 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
\(396\) −4.00000 −0.201008
\(397\) 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
\(407\) 40.0000 1.98273
\(408\) −2.00000 −0.0990148
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 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
\(418\) 16.0000 0.782586
\(419\) −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
\(429\) 24.0000 1.15873
\(430\) 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 \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 8.00000 0.381385
\(441\) 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
\(451\) 24.0000 1.13012
\(452\) −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 \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −2.00000 −0.0934539
\(459\) −2.00000 −0.0933520
\(460\) −16.0000 −0.746004
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) −4.00000 −0.186097
\(463\) −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
\(473\) 16.0000 0.735681
\(474\) 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 \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 2.00000 0.0912871
\(481\) −60.0000 −2.73576
\(482\) 2.00000 0.0910975
\(483\) 8.00000 0.364013
\(484\) 5.00000 0.227273
\(485\) 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 \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 6.00000 0.270501
\(493\) −4.00000 −0.180151
\(494\) −24.0000 −1.07981
\(495\) 8.00000 0.359573
\(496\) 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 \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 4.00000 0.177998
\(506\) −32.0000 −1.42257
\(507\) −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 \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −20.0000 −0.873704
\(525\) −1.00000 −0.0436436
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 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
\(539\) −4.00000 −0.172292
\(540\) 2.00000 0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 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 \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 10.0000 0.427179
\(549\) 6.00000 0.256074
\(550\) 4.00000 0.170561
\(551\) 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 \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 2.00000 0.0845154
\(561\) 8.00000 0.337760
\(562\) 26.0000 1.09674
\(563\) 44.0000 1.85438 0.927189 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 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 \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −8.00000 −0.335083
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −24.0000 −1.00349
\(573\) 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
\(583\) −24.0000 −0.993978
\(584\) 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 \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 4.00000 0.164122
\(595\) 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 \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) −10.0000 −0.406558
\(606\) 2.00000 0.0812444
\(607\) 48.0000 1.94826 0.974130 0.225989i \(-0.0725612\pi\)
0.974130 + 0.225989i \(0.0725612\pi\)
\(608\) −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 \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) 28.0000 1.12999
\(615\) −12.0000 −0.483887
\(616\) 4.00000 0.161165
\(617\) −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
\(627\) −16.0000 −0.638978
\(628\) −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
\(638\) 8.00000 0.316723
\(639\) 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
\(649\) −16.0000 −0.628055
\(650\) −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 \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) −8.00000 −0.311400
\(661\) −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
\(671\) −24.0000 −0.926510
\(672\) 1.00000 0.0385758
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 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
\(693\) 4.00000 0.151947
\(694\) 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 \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) −6.00000 −0.226455
\(703\) 40.0000 1.50863
\(704\) −4.00000 −0.150756
\(705\) 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
\(715\) 48.0000 1.79510
\(716\) −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
\(726\) −5.00000 −0.185567
\(727\) −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 \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 32.0000 1.18114
\(735\) 2.00000 0.0737711
\(736\) 8.00000 0.294884
\(737\) −16.0000 −0.589368
\(738\) −6.00000 −0.220863
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 20.0000 0.735215
\(741\) 24.0000 0.881662
\(742\) −6.00000 −0.220267
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 22.0000 0.805477
\(747\) −4.00000 −0.146352
\(748\) −8.00000 −0.292509
\(749\) −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
\(759\) 32.0000 1.16153
\(760\) 8.00000 0.290191
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 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 \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −8.00000 −0.288300
\(771\) 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
\(781\) −32.0000 −1.14505
\(782\) 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 \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) −10.0000 −0.356235
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) −4.00000 −0.142134
\(793\) 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
\(803\) −40.0000 −1.41157
\(804\) −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 \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) −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 \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) −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
\(825\) −4.00000 −0.139262
\(826\) −4.00000 −0.139178
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 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
\(836\) 16.0000 0.553372
\(837\) 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
\(847\) −5.00000 −0.171802
\(848\) 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 \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −6.00000 −0.205316
\(855\) 8.00000 0.273594
\(856\) 12.0000 0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 24.0000 0.819346
\(859\) 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 \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −24.0000 −0.809961
\(879\) −30.0000 −1.01187
\(880\) 8.00000 0.269680
\(881\) 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 \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) −4.00000 −0.134005
\(892\) −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
\(902\) 24.0000 0.799113
\(903\) −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
\(913\) 16.0000 0.529523
\(914\) 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 \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −16.0000 −0.527504
\(921\) −28.0000 −0.922631
\(922\) 22.0000 0.724531
\(923\) 48.0000 1.57994
\(924\) −4.00000 −0.131590
\(925\) 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
\(935\) 16.0000 0.523256
\(936\) 6.00000 0.196116
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) −4.00000 −0.130605
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −26.0000 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 10.0000 0.325818
\(943\) −48.0000 −1.56310
\(944\) 4.00000 0.130189
\(945\) −2.00000 −0.0650600
\(946\) 16.0000 0.520205
\(947\) 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 \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) −8.00000 −0.258603
\(958\) −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 \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) 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
\(979\) 24.0000 0.767043
\(980\) −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
\(990\) 8.00000 0.254257
\(991\) −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 \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −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 42.2.a.a.1.1 1
3.2 odd 2 126.2.a.a.1.1 1
4.3 odd 2 336.2.a.d.1.1 1
5.2 odd 4 1050.2.g.a.799.2 2
5.3 odd 4 1050.2.g.a.799.1 2
5.4 even 2 1050.2.a.i.1.1 1
7.2 even 3 294.2.e.c.67.1 2
7.3 odd 6 294.2.e.a.79.1 2
7.4 even 3 294.2.e.c.79.1 2
7.5 odd 6 294.2.e.a.67.1 2
7.6 odd 2 294.2.a.g.1.1 1
8.3 odd 2 1344.2.a.i.1.1 1
8.5 even 2 1344.2.a.q.1.1 1
9.2 odd 6 1134.2.f.j.757.1 2
9.4 even 3 1134.2.f.g.379.1 2
9.5 odd 6 1134.2.f.j.379.1 2
9.7 even 3 1134.2.f.g.757.1 2
11.10 odd 2 5082.2.a.d.1.1 1
12.11 even 2 1008.2.a.j.1.1 1
13.12 even 2 7098.2.a.f.1.1 1
15.2 even 4 3150.2.g.r.2899.1 2
15.8 even 4 3150.2.g.r.2899.2 2
15.14 odd 2 3150.2.a.bo.1.1 1
16.3 odd 4 5376.2.c.e.2689.2 2
16.5 even 4 5376.2.c.bc.2689.2 2
16.11 odd 4 5376.2.c.e.2689.1 2
16.13 even 4 5376.2.c.bc.2689.1 2
20.19 odd 2 8400.2.a.k.1.1 1
21.2 odd 6 882.2.g.h.361.1 2
21.5 even 6 882.2.g.j.361.1 2
21.11 odd 6 882.2.g.h.667.1 2
21.17 even 6 882.2.g.j.667.1 2
21.20 even 2 882.2.a.b.1.1 1
24.5 odd 2 4032.2.a.e.1.1 1
24.11 even 2 4032.2.a.m.1.1 1
28.3 even 6 2352.2.q.n.961.1 2
28.11 odd 6 2352.2.q.i.961.1 2
28.19 even 6 2352.2.q.n.1537.1 2
28.23 odd 6 2352.2.q.i.1537.1 2
28.27 even 2 2352.2.a.l.1.1 1
35.34 odd 2 7350.2.a.f.1.1 1
56.13 odd 2 9408.2.a.n.1.1 1
56.27 even 2 9408.2.a.bw.1.1 1
84.83 odd 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 1.1 even 1 trivial
126.2.a.a.1.1 1 3.2 odd 2
294.2.a.g.1.1 1 7.6 odd 2
294.2.e.a.67.1 2 7.5 odd 6
294.2.e.a.79.1 2 7.3 odd 6
294.2.e.c.67.1 2 7.2 even 3
294.2.e.c.79.1 2 7.4 even 3
336.2.a.d.1.1 1 4.3 odd 2
882.2.a.b.1.1 1 21.20 even 2
882.2.g.h.361.1 2 21.2 odd 6
882.2.g.h.667.1 2 21.11 odd 6
882.2.g.j.361.1 2 21.5 even 6
882.2.g.j.667.1 2 21.17 even 6
1008.2.a.j.1.1 1 12.11 even 2
1050.2.a.i.1.1 1 5.4 even 2
1050.2.g.a.799.1 2 5.3 odd 4
1050.2.g.a.799.2 2 5.2 odd 4
1134.2.f.g.379.1 2 9.4 even 3
1134.2.f.g.757.1 2 9.7 even 3
1134.2.f.j.379.1 2 9.5 odd 6
1134.2.f.j.757.1 2 9.2 odd 6
1344.2.a.i.1.1 1 8.3 odd 2
1344.2.a.q.1.1 1 8.5 even 2
2352.2.a.l.1.1 1 28.27 even 2
2352.2.q.i.961.1 2 28.11 odd 6
2352.2.q.i.1537.1 2 28.23 odd 6
2352.2.q.n.961.1 2 28.3 even 6
2352.2.q.n.1537.1 2 28.19 even 6
3150.2.a.bo.1.1 1 15.14 odd 2
3150.2.g.r.2899.1 2 15.2 even 4
3150.2.g.r.2899.2 2 15.8 even 4
4032.2.a.e.1.1 1 24.5 odd 2
4032.2.a.m.1.1 1 24.11 even 2
5082.2.a.d.1.1 1 11.10 odd 2
5376.2.c.e.2689.1 2 16.11 odd 4
5376.2.c.e.2689.2 2 16.3 odd 4
5376.2.c.bc.2689.1 2 16.13 even 4
5376.2.c.bc.2689.2 2 16.5 even 4
7056.2.a.k.1.1 1 84.83 odd 2
7098.2.a.f.1.1 1 13.12 even 2
7350.2.a.f.1.1 1 35.34 odd 2
8400.2.a.k.1.1 1 20.19 odd 2
9408.2.a.n.1.1 1 56.13 odd 2
9408.2.a.bw.1.1 1 56.27 even 2