Properties

Label 147.2.a.a.1.1
Level $147$
Weight $2$
Character 147.1
Self dual yes
Analytic conductor $1.174$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -5.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +1.00000 q^{48} +1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} +4.00000 q^{57} +2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +2.00000 q^{61} +7.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +3.00000 q^{72} +6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{78} -16.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} +12.0000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +12.0000 q^{88} +14.0000 q^{89} -2.00000 q^{90} -8.00000 q^{95} +5.00000 q^{96} -18.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −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
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −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
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) −4.00000 −0.696311
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) 6.00000 0.948683
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 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
\(46\) 0 0
\(47\) 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
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 2.00000 0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 12.0000 1.27920
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 5.00000 0.510310
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 6.00000 0.594089
\(103\) −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
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) −8.00000 −0.762770
\(111\) −6.00000 −0.569495
\(112\) 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
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −6.00000 −0.547723
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 3.00000 0.265165
\(129\) 4.00000 0.352180
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −2.00000 −0.172133
\(136\) 18.0000 1.54349
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) −1.00000 −0.0833333
\(145\) −4.00000 −0.332182
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −12.0000 −0.973329
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 16.0000 1.27289
\(159\) −6.00000 −0.475831
\(160\) −10.0000 −0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 2.00000 0.156174
\(165\) −8.00000 −0.622799
\(166\) −12.0000 −0.931381
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 12.0000 0.901975
\(178\) −14.0000 −1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −2.00000 −0.149071
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −7.00000 −0.505181
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 18.0000 1.29232
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) −4.00000 −0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −3.00000 −0.212132
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −4.00000 −0.279372
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −8.00000 −0.545595
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) 18.0000 1.21911
\(219\) −6.00000 −0.405442
\(220\) −8.00000 −0.539360
\(221\) 12.0000 0.807207
\(222\) 6.00000 0.402694
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −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
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 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
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 12.0000 0.758947
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) −17.0000 −1.06250
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) −2.00000 −0.123797
\(262\) 4.00000 0.247121
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −12.0000 −0.738549
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) −4.00000 −0.244339
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 2.00000 0.121716
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) 18.0000 1.05518
\(292\) −6.00000 −0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 18.0000 1.04623
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 14.0000 0.804279
\(304\) 4.00000 0.229416
\(305\) 4.00000 0.229039
\(306\) −6.00000 −0.342997
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −6.00000 −0.339683
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −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
\(320\) 14.0000 0.782624
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) −1.00000 −0.0555556
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 18.0000 0.995402
\(328\) −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 \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) −8.00000 −0.437741
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 9.00000 0.489535
\(339\) 14.0000 0.760376
\(340\) −12.0000 −0.650791
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) −2.00000 −0.107211
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −20.0000 −1.06600
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 6.00000 0.316228
\(361\) −3.00000 −0.157895
\(362\) −26.0000 −1.36653
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 2.00000 0.104542
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) −12.0000 −0.623850
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −24.0000 −1.24101
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) 18.0000 0.913812
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) −22.0000 −1.10834
\(395\) −32.0000 −1.61009
\(396\) −4.00000 −0.201008
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) −18.0000 −0.891133
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 4.00000 0.197546
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −10.0000 −0.490290
\(417\) 12.0000 0.587643
\(418\) 16.0000 0.782586
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −8.00000 −0.386244
\(430\) 8.00000 0.385794
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 24.0000 1.14416
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 6.00000 0.284747
\(445\) 28.0000 1.32733
\(446\) 16.0000 0.757622
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) −8.00000 −0.376705
\(452\) 14.0000 0.658505
\(453\) −8.00000 −0.375873
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −10.0000 −0.467269
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) −36.0000 −1.65703
\(473\) −16.0000 −0.735681
\(474\) −16.0000 −0.734904
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 10.0000 0.456435
\(481\) 12.0000 0.547153
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −36.0000 −1.63468
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 6.00000 0.271607
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 12.0000 0.536656
\(501\) −8.00000 −0.357414
\(502\) −20.0000 −0.892644
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −28.0000 −1.24598
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) 26.0000 1.14681
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 12.0000 0.526235
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 2.00000 0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) −12.0000 −0.521247
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 14.0000 0.605839
\(535\) 8.00000 0.345870
\(536\) 12.0000 0.518321
\(537\) 4.00000 0.172613
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 16.0000 0.687259
\(543\) −26.0000 −1.11577
\(544\) −30.0000 −1.28624
\(545\) −36.0000 −1.54207
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 4.00000 0.170561
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) −12.0000 −0.509372
\(556\) 12.0000 0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 22.0000 0.928014
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −28.0000 −1.17797
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −8.00000 −0.335083
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −19.0000 −0.790296
\(579\) −2.00000 −0.0831172
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −18.0000 −0.746124
\(583\) 24.0000 0.993978
\(584\) 18.0000 0.744845
\(585\) 4.00000 0.165380
\(586\) 14.0000 0.578335
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) −22.0000 −0.904959
\(592\) −6.00000 −0.246598
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 3.00000 0.122474
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) 10.0000 0.406558
\(606\) −14.0000 −0.568711
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 4.00000 0.161427
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −8.00000 −0.321807
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) 16.0000 0.638978
\(628\) −2.00000 −0.0798087
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −48.0000 −1.90934
\(633\) −4.00000 −0.158986
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 6.00000 0.237171
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 24.0000 0.944267
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) 3.00000 0.117851
\(649\) −48.0000 −1.88416
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −18.0000 −0.703856
\(655\) −8.00000 −0.312586
\(656\) 2.00000 0.0780869
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 8.00000 0.311400
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 4.00000 0.155464
\(663\) −12.0000 −0.466041
\(664\) 36.0000 1.39707
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 16.0000 0.618596
\(670\) −8.00000 −0.309067
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) 9.00000 0.346154
\(677\) 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
\(680\) 36.0000 1.38054
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) −24.0000 −0.910372
\(696\) 6.00000 0.227429
\(697\) −12.0000 −0.454532
\(698\) −2.00000 −0.0757011
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 2.00000 0.0754851
\(703\) −24.0000 −0.905177
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 42.0000 1.57402
\(713\) 0 0
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) 4.00000 0.149487
\(717\) −24.0000 −0.896296
\(718\) −32.0000 −1.19423
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) −26.0000 −0.966282
\(725\) 2.00000 0.0742781
\(726\) 5.00000 0.185567
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) −24.0000 −0.887672
\(732\) 2.00000 0.0739221
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 2.00000 0.0736210
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −12.0000 −0.441129
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 10.0000 0.366126
\(747\) 12.0000 0.439057
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) 4.00000 0.145671
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) 12.0000 0.433861
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 17.0000 0.613435
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) −2.00000 −0.0719816
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −54.0000 −1.93849
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 8.00000 0.286630
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) −4.00000 −0.142675
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) −22.0000 −0.783718
\(789\) −16.0000 −0.569615
\(790\) 32.0000 1.13851
\(791\) 0 0
\(792\) 12.0000 0.426401
\(793\) 4.00000 0.142044
\(794\) −18.0000 −0.638796
\(795\) −12.0000 −0.425596
\(796\) 24.0000 0.850657
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 14.0000 0.494666
\(802\) 30.0000 1.05934
\(803\) 24.0000 0.846942
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) −42.0000 −1.47755
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) −24.0000 −0.841200
\(815\) 8.00000 0.280228
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) −6.00000 −0.209274
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −24.0000 −0.836080
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −24.0000 −0.833052
\(831\) −22.0000 −0.763172
\(832\) 14.0000 0.485363
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) 16.0000 0.553703
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −38.0000 −1.30957
\(843\) 22.0000 0.757720
\(844\) −4.00000 −0.137686
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −20.0000 −0.686398
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 12.0000 0.410152
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 8.00000 0.273115
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 5.00000 0.170103
\(865\) 20.0000 0.680020
\(866\) −14.0000 −0.475739
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) −4.00000 −0.135613
\(871\) 8.00000 0.271070
\(872\) −54.0000 −1.82867
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) −24.0000 −0.809961
\(879\) 14.0000 0.472208
\(880\) −8.00000 −0.269680
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −12.0000 −0.403604
\(885\) 24.0000 0.806751
\(886\) −36.0000 −1.20944
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −18.0000 −0.604040
\(889\) 0 0
\(890\) −28.0000 −0.938562
\(891\) 4.00000 0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) 8.00000 0.266371
\(903\) 0 0
\(904\) −42.0000 −1.39690
\(905\) 52.0000 1.72854
\(906\) 8.00000 0.265782
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −12.0000 −0.398234
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −4.00000 −0.132453
\(913\) 48.0000 1.58857
\(914\) −10.0000 −0.330771
\(915\) −4.00000 −0.132236
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −10.0000 −0.329332
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −16.0000 −0.525793
\(927\) −8.00000 −0.262754
\(928\) 10.0000 0.328266
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) −24.0000 −0.785725
\(934\) 36.0000 1.17796
\(935\) 48.0000 1.56977
\(936\) 6.00000 0.196116
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) −16.0000 −0.519656
\(949\) 12.0000 0.389536
\(950\) −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
\(955\) −16.0000 −0.517748
\(956\) −24.0000 −0.776215
\(957\) 8.00000 0.258603
\(958\) −16.0000 −0.516937
\(959\) 0 0
\(960\) −14.0000 −0.451848
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) 4.00000 0.128898
\(964\) 2.00000 0.0644157
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 15.0000 0.482118
\(969\) 24.0000 0.770991
\(970\) 36.0000 1.15589
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 4.00000 0.127906
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) −20.0000 −0.638226
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 6.00000 0.191273
\(985\) 44.0000 1.40196
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) −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
\(995\) −48.0000 −1.52170
\(996\) 12.0000 0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −4.00000 −0.126618
\(999\) −6.00000 −0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 147.2.a.a.1.1 1
3.2 odd 2 441.2.a.f.1.1 1
4.3 odd 2 2352.2.a.v.1.1 1
5.4 even 2 3675.2.a.n.1.1 1
7.2 even 3 147.2.e.c.67.1 2
7.3 odd 6 147.2.e.b.79.1 2
7.4 even 3 147.2.e.c.79.1 2
7.5 odd 6 147.2.e.b.67.1 2
7.6 odd 2 21.2.a.a.1.1 1
8.3 odd 2 9408.2.a.m.1.1 1
8.5 even 2 9408.2.a.bv.1.1 1
12.11 even 2 7056.2.a.p.1.1 1
21.2 odd 6 441.2.e.b.361.1 2
21.5 even 6 441.2.e.a.361.1 2
21.11 odd 6 441.2.e.b.226.1 2
21.17 even 6 441.2.e.a.226.1 2
21.20 even 2 63.2.a.a.1.1 1
28.3 even 6 2352.2.q.x.961.1 2
28.11 odd 6 2352.2.q.e.961.1 2
28.19 even 6 2352.2.q.x.1537.1 2
28.23 odd 6 2352.2.q.e.1537.1 2
28.27 even 2 336.2.a.a.1.1 1
35.13 even 4 525.2.d.a.274.2 2
35.27 even 4 525.2.d.a.274.1 2
35.34 odd 2 525.2.a.d.1.1 1
56.13 odd 2 1344.2.a.g.1.1 1
56.27 even 2 1344.2.a.s.1.1 1
63.13 odd 6 567.2.f.g.379.1 2
63.20 even 6 567.2.f.b.190.1 2
63.34 odd 6 567.2.f.g.190.1 2
63.41 even 6 567.2.f.b.379.1 2
77.76 even 2 2541.2.a.j.1.1 1
84.83 odd 2 1008.2.a.l.1.1 1
91.90 odd 2 3549.2.a.c.1.1 1
105.62 odd 4 1575.2.d.a.1324.2 2
105.83 odd 4 1575.2.d.a.1324.1 2
105.104 even 2 1575.2.a.c.1.1 1
112.13 odd 4 5376.2.c.r.2689.2 2
112.27 even 4 5376.2.c.l.2689.2 2
112.69 odd 4 5376.2.c.r.2689.1 2
112.83 even 4 5376.2.c.l.2689.1 2
119.118 odd 2 6069.2.a.b.1.1 1
133.132 even 2 7581.2.a.d.1.1 1
140.139 even 2 8400.2.a.bn.1.1 1
168.83 odd 2 4032.2.a.k.1.1 1
168.125 even 2 4032.2.a.h.1.1 1
231.230 odd 2 7623.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 7.6 odd 2
63.2.a.a.1.1 1 21.20 even 2
147.2.a.a.1.1 1 1.1 even 1 trivial
147.2.e.b.67.1 2 7.5 odd 6
147.2.e.b.79.1 2 7.3 odd 6
147.2.e.c.67.1 2 7.2 even 3
147.2.e.c.79.1 2 7.4 even 3
336.2.a.a.1.1 1 28.27 even 2
441.2.a.f.1.1 1 3.2 odd 2
441.2.e.a.226.1 2 21.17 even 6
441.2.e.a.361.1 2 21.5 even 6
441.2.e.b.226.1 2 21.11 odd 6
441.2.e.b.361.1 2 21.2 odd 6
525.2.a.d.1.1 1 35.34 odd 2
525.2.d.a.274.1 2 35.27 even 4
525.2.d.a.274.2 2 35.13 even 4
567.2.f.b.190.1 2 63.20 even 6
567.2.f.b.379.1 2 63.41 even 6
567.2.f.g.190.1 2 63.34 odd 6
567.2.f.g.379.1 2 63.13 odd 6
1008.2.a.l.1.1 1 84.83 odd 2
1344.2.a.g.1.1 1 56.13 odd 2
1344.2.a.s.1.1 1 56.27 even 2
1575.2.a.c.1.1 1 105.104 even 2
1575.2.d.a.1324.1 2 105.83 odd 4
1575.2.d.a.1324.2 2 105.62 odd 4
2352.2.a.v.1.1 1 4.3 odd 2
2352.2.q.e.961.1 2 28.11 odd 6
2352.2.q.e.1537.1 2 28.23 odd 6
2352.2.q.x.961.1 2 28.3 even 6
2352.2.q.x.1537.1 2 28.19 even 6
2541.2.a.j.1.1 1 77.76 even 2
3549.2.a.c.1.1 1 91.90 odd 2
3675.2.a.n.1.1 1 5.4 even 2
4032.2.a.h.1.1 1 168.125 even 2
4032.2.a.k.1.1 1 168.83 odd 2
5376.2.c.l.2689.1 2 112.83 even 4
5376.2.c.l.2689.2 2 112.27 even 4
5376.2.c.r.2689.1 2 112.69 odd 4
5376.2.c.r.2689.2 2 112.13 odd 4
6069.2.a.b.1.1 1 119.118 odd 2
7056.2.a.p.1.1 1 12.11 even 2
7581.2.a.d.1.1 1 133.132 even 2
7623.2.a.g.1.1 1 231.230 odd 2
8400.2.a.bn.1.1 1 140.139 even 2
9408.2.a.m.1.1 1 8.3 odd 2
9408.2.a.bv.1.1 1 8.5 even 2