Properties

Label 2541.2.a.j.1.1
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
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 \(\chi\) \(=\) 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -3.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} +5.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -2.00000 q^{45} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -3.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} +2.00000 q^{61} +1.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} +4.00000 q^{67} -6.00000 q^{68} -2.00000 q^{70} -3.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{78} +16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +12.0000 q^{83} -1.00000 q^{84} -12.0000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -14.0000 q^{89} -2.00000 q^{90} +2.00000 q^{91} +8.00000 q^{95} +5.00000 q^{96} +18.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 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
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) −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
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 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
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 −0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −2.00000 −0.338062
\(36\) −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 \(0.549916\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 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
\(51\) 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
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 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
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 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
\(70\) −2.00000 −0.239046
\(71\) 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 \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 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 \(0.271155\pi\)
\(84\) −1.00000 −0.109109
\(85\) −12.0000 −1.30158
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −2.00000 −0.210819
\(91\) 2.00000 0.209657
\(92\) 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 \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 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 \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) −2.00000 −0.195180
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) −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
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 6.00000 0.550019
\(120\) 6.00000 0.547723
\(121\) 0 0
\(122\) 2.00000 0.181071
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −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 \(0.555909\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 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 \(0.669953\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) −4.00000 −0.332182
\(146\) 6.00000 0.496564
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 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 \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 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
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −3.00000 −0.231455
\(169\) −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
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 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 \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 2.00000 0.148250
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 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 \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 18.0000 1.29232
\(195\) −4.00000 −0.286446
\(196\) −1.00000 −0.0714286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 3.00000 0.212132
\(201\) 4.00000 0.282138
\(202\) −14.0000 −0.985037
\(203\) 2.00000 0.140372
\(204\) −6.00000 −0.420084
\(205\) 4.00000 0.279372
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 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
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 6.00000 0.402694
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 5.00000 0.334077
\(225\) −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 \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 2.00000 0.129099
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −2.00000 −0.127775
\(246\) −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 \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) −17.0000 −1.06250
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 4.00000 0.249029
\(259\) 6.00000 0.372822
\(260\) 4.00000 0.248069
\(261\) 2.00000 0.123797
\(262\) −4.00000 −0.247121
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) −4.00000 −0.245256
\(267\) −14.0000 −0.856786
\(268\) −4.00000 −0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −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
\(273\) 2.00000 0.121046
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) 6.00000 0.358569
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 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
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 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 \(0.634103\pi\)
\(294\) 1.00000 0.0583212
\(295\) −24.0000 −1.39733
\(296\) −18.0000 −1.04623
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) −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 \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −6.00000 −0.339683
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −2.00000 −0.112867
\(315\) −2.00000 −0.112687
\(316\) −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
\(319\) 0 0
\(320\) −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
\(330\) 0 0
\(331\) −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
\(336\) −1.00000 −0.0545545
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) −14.0000 −0.760376
\(340\) 12.0000 0.650791
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −2.00000 −0.107211
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 6.00000 0.317554
\(358\) −4.00000 −0.211407
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 6.00000 0.316228
\(361\) −3.00000 −0.157895
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −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
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 1.00000 0.0514344
\(379\) 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
\(392\) −3.00000 −0.151523
\(393\) −4.00000 −0.201773
\(394\) −22.0000 −1.10834
\(395\) −32.0000 −1.61009
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 24.0000 1.20301
\(399\) −4.00000 −0.200250
\(400\) 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
\(406\) 2.00000 0.0992583
\(407\) 0 0
\(408\) −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
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 10.0000 0.490290
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) 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
\(427\) 2.00000 0.0967868
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 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 \(0.305885\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 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
\(448\) 7.00000 0.330719
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) −4.00000 −0.187523
\(456\) 12.0000 0.561951
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −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 \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) −36.0000 −1.65703
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 4.00000 0.183533
\(476\) −6.00000 −0.275010
\(477\) 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
\(484\) 0 0
\(485\) −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
\(490\) −2.00000 −0.0903508
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 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 \(0.679709\pi\)
\(504\) −3.00000 −0.133631
\(505\) 28.0000 1.24598
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) −12.0000 −0.531369
\(511\) 6.00000 0.265424
\(512\) −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
\(518\) 6.00000 0.263625
\(519\) 10.0000 0.438951
\(520\) 12.0000 0.526235
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 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
\(525\) −1.00000 −0.0436436
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −12.0000 −0.521247
\(531\) 12.0000 0.520756
\(532\) 4.00000 0.173422
\(533\) −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 \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −16.0000 −0.687259
\(543\) −26.0000 −1.11577
\(544\) 30.0000 1.28624
\(545\) −36.0000 −1.54207
\(546\) 2.00000 0.0855921
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −22.0000 −0.934690
\(555\) −12.0000 −0.509372
\(556\) 12.0000 0.508913
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 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
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 8.00000 0.335083
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 19.0000 0.790296
\(579\) −2.00000 −0.0831172
\(580\) 4.00000 0.166091
\(581\) 12.0000 0.497844
\(582\) 18.0000 0.746124
\(583\) 0 0
\(584\) −18.0000 −0.744845
\(585\) −4.00000 −0.165380
\(586\) −14.0000 −0.578335
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 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 \(0.460686\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 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 \(0.460950\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −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
\(609\) 2.00000 0.0810441
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −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 \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −14.0000 −0.560898
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 36.0000 1.43541
\(630\) −2.00000 −0.0796819
\(631\) −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
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 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 \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −24.0000 −0.944267
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) −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 \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −4.00000 −0.155464
\(663\) 12.0000 0.466041
\(664\) −36.0000 −1.39707
\(665\) 8.00000 0.310227
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 16.0000 0.618596
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 5.00000 0.192879
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 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
\(679\) 18.0000 0.690777
\(680\) 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
\(686\) 1.00000 0.0381802
\(687\) −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 \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −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
\(700\) 1.00000 0.0377964
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 2.00000 0.0754851
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) −14.0000 −0.526524
\(708\) −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
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −24.0000 −0.896296
\(718\) −32.0000 −1.19423
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 2.00000 0.0745356
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) 26.0000 0.966282
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −6.00000 −0.222375
\(729\) 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
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 0 0
\(738\) −2.00000 −0.0736210
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 12.0000 0.441129
\(741\) −8.00000 −0.293887
\(742\) 6.00000 0.220267
\(743\) 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
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 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
\(756\) −1.00000 −0.0363696
\(757\) −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
\(763\) 18.0000 0.651644
\(764\) 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 \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −54.0000 −1.93849
\(777\) 6.00000 0.215249
\(778\) 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
\(784\) −1.00000 −0.0357143
\(785\) 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
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 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 \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) −14.0000 −0.494666
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) −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 \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) −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
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) −6.00000 −0.210042
\(817\) −16.0000 −0.559769
\(818\) 22.0000 0.769212
\(819\) 2.00000 0.0698857
\(820\) −4.00000 −0.139686
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) −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
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −24.0000 −0.833052
\(831\) −22.0000 −0.763172
\(832\) 14.0000 0.485363
\(833\) 6.00000 0.207888
\(834\) −12.0000 −0.415526
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 6.00000 0.207020
\(841\) −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 \(0.445237\pi\)
\(854\) 2.00000 0.0684386
\(855\) 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 \(0.423143\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 8.00000 0.272798
\(861\) −2.00000 −0.0681598
\(862\) 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
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) 8.00000 0.271070
\(872\) −54.0000 −1.82867
\(873\) 18.0000 0.609208
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) −6.00000 −0.202721
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 24.0000 0.809961
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 1.00000 0.0336718
\(883\) −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
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 8.00000 0.267411
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 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
\(910\) −4.00000 −0.132599
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) −4.00000 −0.132236
\(916\) 10.0000 0.330409
\(917\) −4.00000 −0.132092
\(918\) 6.00000 0.198030
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 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 \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 4.00000 0.130605
\(939\) 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
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 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
\(952\) −18.0000 −0.583383
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 6.00000 0.194257
\(955\) 16.0000 0.517748
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −6.00000 −0.193750
\(960\) −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 \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) −36.0000 −1.15589
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −12.0000 −0.384702
\(974\) −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
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 18.0000 0.574696
\(982\) −20.0000 −0.638226
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 6.00000 0.191273
\(985\) 44.0000 1.40196
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 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 2541.2.a.j.1.1 1
3.2 odd 2 7623.2.a.g.1.1 1
11.10 odd 2 21.2.a.a.1.1 1
33.32 even 2 63.2.a.a.1.1 1
44.43 even 2 336.2.a.a.1.1 1
55.32 even 4 525.2.d.a.274.1 2
55.43 even 4 525.2.d.a.274.2 2
55.54 odd 2 525.2.a.d.1.1 1
77.10 even 6 147.2.e.c.79.1 2
77.32 odd 6 147.2.e.b.79.1 2
77.54 even 6 147.2.e.c.67.1 2
77.65 odd 6 147.2.e.b.67.1 2
77.76 even 2 147.2.a.a.1.1 1
88.21 odd 2 1344.2.a.g.1.1 1
88.43 even 2 1344.2.a.s.1.1 1
99.32 even 6 567.2.f.b.379.1 2
99.43 odd 6 567.2.f.g.190.1 2
99.65 even 6 567.2.f.b.190.1 2
99.76 odd 6 567.2.f.g.379.1 2
132.131 odd 2 1008.2.a.l.1.1 1
143.142 odd 2 3549.2.a.c.1.1 1
165.32 odd 4 1575.2.d.a.1324.2 2
165.98 odd 4 1575.2.d.a.1324.1 2
165.164 even 2 1575.2.a.c.1.1 1
176.21 odd 4 5376.2.c.r.2689.1 2
176.43 even 4 5376.2.c.l.2689.2 2
176.109 odd 4 5376.2.c.r.2689.2 2
176.131 even 4 5376.2.c.l.2689.1 2
187.186 odd 2 6069.2.a.b.1.1 1
209.208 even 2 7581.2.a.d.1.1 1
220.219 even 2 8400.2.a.bn.1.1 1
231.32 even 6 441.2.e.a.226.1 2
231.65 even 6 441.2.e.a.361.1 2
231.131 odd 6 441.2.e.b.361.1 2
231.164 odd 6 441.2.e.b.226.1 2
231.230 odd 2 441.2.a.f.1.1 1
264.131 odd 2 4032.2.a.k.1.1 1
264.197 even 2 4032.2.a.h.1.1 1
308.87 odd 6 2352.2.q.e.961.1 2
308.131 odd 6 2352.2.q.e.1537.1 2
308.219 even 6 2352.2.q.x.1537.1 2
308.263 even 6 2352.2.q.x.961.1 2
308.307 odd 2 2352.2.a.v.1.1 1
385.384 even 2 3675.2.a.n.1.1 1
616.307 odd 2 9408.2.a.m.1.1 1
616.461 even 2 9408.2.a.bv.1.1 1
924.923 even 2 7056.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 11.10 odd 2
63.2.a.a.1.1 1 33.32 even 2
147.2.a.a.1.1 1 77.76 even 2
147.2.e.b.67.1 2 77.65 odd 6
147.2.e.b.79.1 2 77.32 odd 6
147.2.e.c.67.1 2 77.54 even 6
147.2.e.c.79.1 2 77.10 even 6
336.2.a.a.1.1 1 44.43 even 2
441.2.a.f.1.1 1 231.230 odd 2
441.2.e.a.226.1 2 231.32 even 6
441.2.e.a.361.1 2 231.65 even 6
441.2.e.b.226.1 2 231.164 odd 6
441.2.e.b.361.1 2 231.131 odd 6
525.2.a.d.1.1 1 55.54 odd 2
525.2.d.a.274.1 2 55.32 even 4
525.2.d.a.274.2 2 55.43 even 4
567.2.f.b.190.1 2 99.65 even 6
567.2.f.b.379.1 2 99.32 even 6
567.2.f.g.190.1 2 99.43 odd 6
567.2.f.g.379.1 2 99.76 odd 6
1008.2.a.l.1.1 1 132.131 odd 2
1344.2.a.g.1.1 1 88.21 odd 2
1344.2.a.s.1.1 1 88.43 even 2
1575.2.a.c.1.1 1 165.164 even 2
1575.2.d.a.1324.1 2 165.98 odd 4
1575.2.d.a.1324.2 2 165.32 odd 4
2352.2.a.v.1.1 1 308.307 odd 2
2352.2.q.e.961.1 2 308.87 odd 6
2352.2.q.e.1537.1 2 308.131 odd 6
2352.2.q.x.961.1 2 308.263 even 6
2352.2.q.x.1537.1 2 308.219 even 6
2541.2.a.j.1.1 1 1.1 even 1 trivial
3549.2.a.c.1.1 1 143.142 odd 2
3675.2.a.n.1.1 1 385.384 even 2
4032.2.a.h.1.1 1 264.197 even 2
4032.2.a.k.1.1 1 264.131 odd 2
5376.2.c.l.2689.1 2 176.131 even 4
5376.2.c.l.2689.2 2 176.43 even 4
5376.2.c.r.2689.1 2 176.21 odd 4
5376.2.c.r.2689.2 2 176.109 odd 4
6069.2.a.b.1.1 1 187.186 odd 2
7056.2.a.p.1.1 1 924.923 even 2
7581.2.a.d.1.1 1 209.208 even 2
7623.2.a.g.1.1 1 3.2 odd 2
8400.2.a.bn.1.1 1 220.219 even 2
9408.2.a.m.1.1 1 616.307 odd 2
9408.2.a.bv.1.1 1 616.461 even 2