Properties

Label 294.2.a.g.1.1
Level $294$
Weight $2$
Character 294.1
Self dual yes
Analytic conductor $2.348$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.34760181943\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 294.1

$q$-expansion

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 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
\(21\) 0 0
\(22\) −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
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 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
\(35\) 0 0
\(36\) 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 \(0.344787\pi\)
\(42\) 0 0
\(43\) −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
\(49\) 0 0
\(50\) −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
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 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
\(70\) 0 0
\(71\) 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 \(0.429548\pi\)
\(84\) 0 0
\(85\) −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 \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 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 \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −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 \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 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
\(112\) 0 0
\(113\) −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
\(119\) 0 0
\(120\) 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
\(126\) 0 0
\(127\) 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 \(0.161711\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 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 \(0.554260\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) −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
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 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 \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 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
\(175\) 0 0
\(176\) −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 \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) −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
\(189\) 0 0
\(190\) 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
\(196\) 0 0
\(197\) −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 \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) −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
\(210\) 0 0
\(211\) 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 \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −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 \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) −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
\(238\) 0 0
\(239\) 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 \(0.520518\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 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 \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −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 \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) −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
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 2.00000 0.121716
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 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
\(280\) 0 0
\(281\) 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 \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 8.00000 0.474713
\(285\) 8.00000 0.473879
\(286\) 24.0000 1.41915
\(287\) 0 0
\(288\) 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 \(0.840002\pi\)
\(294\) 0 0
\(295\) −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
\(301\) 0 0
\(302\) −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 \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) −6.00000 −0.339683
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 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
\(322\) 0 0
\(323\) −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
\(336\) 0 0
\(337\) 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
\(343\) 0 0
\(344\) −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 \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) −4.00000 −0.213201
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −4.00000 −0.212598
\(355\) 16.0000 0.849192
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −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
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) −6.00000 −0.313625
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) 6.00000 0.312348
\(370\) −20.0000 −1.03975
\(371\) 0 0
\(372\) 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
\(378\) 0 0
\(379\) −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 \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 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
\(392\) 0 0
\(393\) 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 \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 40.0000 1.98273
\(408\) −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
\(413\) 0 0
\(414\) 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 \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 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
\(427\) 0 0
\(428\) 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 \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 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 \(0.305885\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) 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
\(448\) 0 0
\(449\) 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
\(455\) 0 0
\(456\) 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 \(-0.671214\pi\)
−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 \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) 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
\(476\) 0 0
\(477\) 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
\(483\) 0 0
\(484\) 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
\(490\) 0 0
\(491\) 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
\(497\) 0 0
\(498\) 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 \(0.679709\pi\)
\(504\) 0 0
\(505\) 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 \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −4.00000 −0.177123
\(511\) 0 0
\(512\) 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
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) −12.0000 −0.526235
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −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
\(525\) 0 0
\(526\) −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
\(532\) 0 0
\(533\) −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 \(-0.276899\pi\)
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
\(546\) 0 0
\(547\) −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
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 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
\(567\) 0 0
\(568\) 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
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −13.0000 −0.540729
\(579\) 2.00000 0.0831172
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) 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 \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 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 \(0.620500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 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 \(0.677914\pi\)
\(602\) 0 0
\(603\) 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 \(-0.927439\pi\)
−0.974130 + 0.225989i \(0.927439\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −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
\(616\) 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 \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) −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
\(630\) 0 0
\(631\) 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
\(637\) 0 0
\(638\) 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 \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −8.00000 −0.314756
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 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 \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) −4.00000 −0.155464
\(663\) 12.0000 0.466041
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) 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 \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −14.0000 −0.537667
\(679\) 0 0
\(680\) −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
\(686\) 0 0
\(687\) 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 \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −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
\(700\) 0 0
\(701\) −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
\(707\) 0 0
\(708\) −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
\(714\) 0 0
\(715\) 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
\(721\) 0 0
\(722\) −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 \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 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
\(735\) 0 0
\(736\) 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
\(742\) 0 0
\(743\) 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
\(749\) 0 0
\(750\) −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
\(756\) 0 0
\(757\) 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 \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 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 \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −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
\(784\) 0 0
\(785\) 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
\(791\) 0 0
\(792\) −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 \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 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
\(805\) 0 0
\(806\) 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 \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) 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
\(819\) 0 0
\(820\) 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
\(826\) 0 0
\(827\) −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 \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 8.00000 0.277684
\(831\) −10.0000 −0.346896
\(832\) −6.00000 −0.208013
\(833\) 0 0
\(834\) −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 \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 0 0
\(841\) −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 \(0.577041\pi\)
\(854\) 0 0
\(855\) 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 \(0.754643\pi\)
\(858\) 24.0000 0.819346
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 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
\(875\) 0 0
\(876\) −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 \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 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
\(891\) −4.00000 −0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) −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
\(903\) 0 0
\(904\) −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
\(910\) 0 0
\(911\) 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
\(917\) 0 0
\(918\) −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
\(924\) 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 \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −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 \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −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
\(945\) 0 0
\(946\) 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
\(952\) 0 0
\(953\) 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
\(959\) 0 0
\(960\) 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
\(966\) 0 0
\(967\) 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 \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 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
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 12.0000 0.382935
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 6.00000 0.191273
\(985\) −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
\(994\) 0 0
\(995\) −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 294.2.a.g.1.1 1
3.2 odd 2 882.2.a.b.1.1 1
4.3 odd 2 2352.2.a.l.1.1 1
5.4 even 2 7350.2.a.f.1.1 1
7.2 even 3 294.2.e.a.67.1 2
7.3 odd 6 294.2.e.c.79.1 2
7.4 even 3 294.2.e.a.79.1 2
7.5 odd 6 294.2.e.c.67.1 2
7.6 odd 2 42.2.a.a.1.1 1
8.3 odd 2 9408.2.a.bw.1.1 1
8.5 even 2 9408.2.a.n.1.1 1
12.11 even 2 7056.2.a.k.1.1 1
21.2 odd 6 882.2.g.j.361.1 2
21.5 even 6 882.2.g.h.361.1 2
21.11 odd 6 882.2.g.j.667.1 2
21.17 even 6 882.2.g.h.667.1 2
21.20 even 2 126.2.a.a.1.1 1
28.3 even 6 2352.2.q.i.961.1 2
28.11 odd 6 2352.2.q.n.961.1 2
28.19 even 6 2352.2.q.i.1537.1 2
28.23 odd 6 2352.2.q.n.1537.1 2
28.27 even 2 336.2.a.d.1.1 1
35.13 even 4 1050.2.g.a.799.1 2
35.27 even 4 1050.2.g.a.799.2 2
35.34 odd 2 1050.2.a.i.1.1 1
56.13 odd 2 1344.2.a.q.1.1 1
56.27 even 2 1344.2.a.i.1.1 1
63.13 odd 6 1134.2.f.g.379.1 2
63.20 even 6 1134.2.f.j.757.1 2
63.34 odd 6 1134.2.f.g.757.1 2
63.41 even 6 1134.2.f.j.379.1 2
77.76 even 2 5082.2.a.d.1.1 1
84.83 odd 2 1008.2.a.j.1.1 1
91.90 odd 2 7098.2.a.f.1.1 1
105.62 odd 4 3150.2.g.r.2899.1 2
105.83 odd 4 3150.2.g.r.2899.2 2
105.104 even 2 3150.2.a.bo.1.1 1
112.13 odd 4 5376.2.c.bc.2689.1 2
112.27 even 4 5376.2.c.e.2689.1 2
112.69 odd 4 5376.2.c.bc.2689.2 2
112.83 even 4 5376.2.c.e.2689.2 2
140.139 even 2 8400.2.a.k.1.1 1
168.83 odd 2 4032.2.a.m.1.1 1
168.125 even 2 4032.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 7.6 odd 2
126.2.a.a.1.1 1 21.20 even 2
294.2.a.g.1.1 1 1.1 even 1 trivial
294.2.e.a.67.1 2 7.2 even 3
294.2.e.a.79.1 2 7.4 even 3
294.2.e.c.67.1 2 7.5 odd 6
294.2.e.c.79.1 2 7.3 odd 6
336.2.a.d.1.1 1 28.27 even 2
882.2.a.b.1.1 1 3.2 odd 2
882.2.g.h.361.1 2 21.5 even 6
882.2.g.h.667.1 2 21.17 even 6
882.2.g.j.361.1 2 21.2 odd 6
882.2.g.j.667.1 2 21.11 odd 6
1008.2.a.j.1.1 1 84.83 odd 2
1050.2.a.i.1.1 1 35.34 odd 2
1050.2.g.a.799.1 2 35.13 even 4
1050.2.g.a.799.2 2 35.27 even 4
1134.2.f.g.379.1 2 63.13 odd 6
1134.2.f.g.757.1 2 63.34 odd 6
1134.2.f.j.379.1 2 63.41 even 6
1134.2.f.j.757.1 2 63.20 even 6
1344.2.a.i.1.1 1 56.27 even 2
1344.2.a.q.1.1 1 56.13 odd 2
2352.2.a.l.1.1 1 4.3 odd 2
2352.2.q.i.961.1 2 28.3 even 6
2352.2.q.i.1537.1 2 28.19 even 6
2352.2.q.n.961.1 2 28.11 odd 6
2352.2.q.n.1537.1 2 28.23 odd 6
3150.2.a.bo.1.1 1 105.104 even 2
3150.2.g.r.2899.1 2 105.62 odd 4
3150.2.g.r.2899.2 2 105.83 odd 4
4032.2.a.e.1.1 1 168.125 even 2
4032.2.a.m.1.1 1 168.83 odd 2
5082.2.a.d.1.1 1 77.76 even 2
5376.2.c.e.2689.1 2 112.27 even 4
5376.2.c.e.2689.2 2 112.83 even 4
5376.2.c.bc.2689.1 2 112.13 odd 4
5376.2.c.bc.2689.2 2 112.69 odd 4
7056.2.a.k.1.1 1 12.11 even 2
7098.2.a.f.1.1 1 91.90 odd 2
7350.2.a.f.1.1 1 5.4 even 2
8400.2.a.k.1.1 1 140.139 even 2
9408.2.a.n.1.1 1 8.5 even 2
9408.2.a.bw.1.1 1 8.3 odd 2