Properties

Label 3150.2.a.bl.1.1
Level 3150
Weight 2
Character 3150.1
Self dual yes
Analytic conductor 25.153
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1050)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 3150.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +4.00000 q^{19} -2.00000 q^{22} +7.00000 q^{23} -1.00000 q^{26} +1.00000 q^{28} -1.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} +6.00000 q^{37} +4.00000 q^{38} +3.00000 q^{41} +1.00000 q^{43} -2.00000 q^{44} +7.00000 q^{46} -12.0000 q^{47} +1.00000 q^{49} -1.00000 q^{52} +11.0000 q^{53} +1.00000 q^{56} -1.00000 q^{58} +3.00000 q^{59} +5.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +12.0000 q^{67} -1.00000 q^{68} -4.00000 q^{71} +14.0000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -2.00000 q^{77} -2.00000 q^{79} +3.00000 q^{82} -3.00000 q^{83} +1.00000 q^{86} -2.00000 q^{88} -10.0000 q^{89} -1.00000 q^{91} +7.00000 q^{92} -12.0000 q^{94} +10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 7.00000 1.03209
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 7.00000 0.729800
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 17.0000 1.67506 0.837530 0.546392i \(-0.183999\pi\)
0.837530 + 0.546392i \(0.183999\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 3.00000 0.276172
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.00000 0.452679
\(123\) 0 0
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −5.00000 −0.409616 −0.204808 0.978802i \(-0.565657\pi\)
−0.204808 + 0.978802i \(0.565657\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) 0 0
\(161\) 7.00000 0.551677
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) 7.00000 0.516047
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) 0 0
\(206\) 17.0000 1.18445
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 11.0000 0.755483
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 0 0
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) 13.0000 0.870544 0.435272 0.900299i \(-0.356652\pi\)
0.435272 + 0.900299i \(0.356652\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 7.00000 0.464606 0.232303 0.972643i \(-0.425374\pi\)
0.232303 + 0.972643i \(0.425374\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) 0 0
\(238\) −1.00000 −0.0648204
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) −14.0000 −0.880172
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.00000 −0.311891 −0.155946 0.987766i \(-0.549842\pi\)
−0.155946 + 0.987766i \(0.549842\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) 11.0000 0.678289 0.339145 0.940734i \(-0.389862\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 3.00000 0.177084
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −5.00000 −0.289642
\(299\) −7.00000 −0.404820
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 22.0000 1.26596
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) −34.0000 −1.92796 −0.963982 0.265969i \(-0.914308\pi\)
−0.963982 + 0.265969i \(0.914308\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −17.0000 −0.954815 −0.477408 0.878682i \(-0.658423\pi\)
−0.477408 + 0.878682i \(0.658423\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 7.00000 0.390095
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 0 0
\(326\) 19.0000 1.05231
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) −3.00000 −0.164646
\(333\) 0 0
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) −27.0000 −1.47078 −0.735392 0.677642i \(-0.763002\pi\)
−0.735392 + 0.677642i \(0.763002\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 26.0000 1.39575 0.697877 0.716218i \(-0.254128\pi\)
0.697877 + 0.716218i \(0.254128\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −18.0000 −0.946059
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 7.00000 0.364900
\(369\) 0 0
\(370\) 0 0
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 13.0000 0.665138
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 27.0000 1.36024
\(395\) 0 0
\(396\) 0 0
\(397\) −3.00000 −0.150566 −0.0752828 0.997162i \(-0.523986\pi\)
−0.0752828 + 0.997162i \(0.523986\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 0 0
\(403\) −3.00000 −0.149441
\(404\) 0 0
\(405\) 0 0
\(406\) −1.00000 −0.0496292
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 17.0000 0.837530
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) −8.00000 −0.391293
\(419\) 25.0000 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 15.0000 0.730189
\(423\) 0 0
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 0 0
\(431\) −27.0000 −1.30054 −0.650272 0.759701i \(-0.725345\pi\)
−0.650272 + 0.759701i \(0.725345\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 28.0000 1.33942
\(438\) 0 0
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.00000 0.0475651
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 13.0000 0.615568
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 7.00000 0.328526
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) −39.0000 −1.80470 −0.902352 0.430999i \(-0.858161\pi\)
−0.902352 + 0.430999i \(0.858161\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 3.00000 0.138086
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) 0 0
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 5.00000 0.226339
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) 1.00000 0.0450377
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −27.0000 −1.20869 −0.604343 0.796724i \(-0.706564\pi\)
−0.604343 + 0.796724i \(0.706564\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 15.0000 0.669483
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −14.0000 −0.622376
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −5.00000 −0.220541
\(515\) 0 0
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) −3.00000 −0.130682
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) 15.0000 0.641354 0.320677 0.947189i \(-0.396090\pi\)
0.320677 + 0.947189i \(0.396090\pi\)
\(548\) 4.00000 0.170872
\(549\) 0 0
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 41.0000 1.72794 0.863972 0.503540i \(-0.167969\pi\)
0.863972 + 0.503540i \(0.167969\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 0 0
\(581\) −3.00000 −0.124461
\(582\) 0 0
\(583\) −22.0000 −0.911147
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 0 0
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5.00000 −0.204808
\(597\) 0 0
\(598\) −7.00000 −0.286251
\(599\) 45.0000 1.83865 0.919325 0.393499i \(-0.128735\pi\)
0.919325 + 0.393499i \(0.128735\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 1.00000 0.0407570
\(603\) 0 0
\(604\) 22.0000 0.895167
\(605\) 0 0
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −34.0000 −1.36328
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) −18.0000 −0.719425
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 26.0000 1.03504 0.517522 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 0 0
\(634\) −17.0000 −0.675156
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 7.00000 0.275839
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 0 0
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 19.0000 0.744097
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −3.00000 −0.116598
\(663\) 0 0
\(664\) −3.00000 −0.116423
\(665\) 0 0
\(666\) 0 0
\(667\) −7.00000 −0.271041
\(668\) −2.00000 −0.0773823
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) −27.0000 −1.04077 −0.520387 0.853931i \(-0.674212\pi\)
−0.520387 + 0.853931i \(0.674212\pi\)
\(674\) −27.0000 −1.04000
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −20.0000 −0.768662 −0.384331 0.923195i \(-0.625568\pi\)
−0.384331 + 0.923195i \(0.625568\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) −6.00000 −0.229752
\(683\) −6.00000 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 26.0000 0.986947
\(695\) 0 0
\(696\) 0 0
\(697\) −3.00000 −0.113633
\(698\) −1.00000 −0.0378506
\(699\) 0 0
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 21.0000 0.786456
\(714\) 0 0
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) 9.00000 0.335877
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) 17.0000 0.633113
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) 0 0
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) 0 0
\(731\) −1.00000 −0.0369863
\(732\) 0 0
\(733\) −19.0000 −0.701781 −0.350891 0.936416i \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) −13.0000 −0.479839
\(735\) 0 0
\(736\) 7.00000 0.258023
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 11.0000 0.404642 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.0000 0.403823
\(743\) −3.00000 −0.110059 −0.0550297 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 1.00000 0.0364179
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −5.00000 −0.181608
\(759\) 0 0
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 13.0000 0.470323
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) −3.00000 −0.108324
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −20.0000 −0.719350 −0.359675 0.933078i \(-0.617112\pi\)
−0.359675 + 0.933078i \(0.617112\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −22.0000 −0.788738
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −7.00000 −0.250319
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) 27.0000 0.961835
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −5.00000 −0.177555
\(794\) −3.00000 −0.106466
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) −16.0000 −0.564980
\(803\) −28.0000 −0.988099
\(804\) 0 0
\(805\) 0 0
\(806\) −3.00000 −0.105670
\(807\) 0 0
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) −1.00000 −0.0350931
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) 18.0000 0.627441 0.313720 0.949515i \(-0.398425\pi\)
0.313720 + 0.949515i \(0.398425\pi\)
\(824\) 17.0000 0.592223
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) −50.0000 −1.73867 −0.869335 0.494223i \(-0.835453\pi\)
−0.869335 + 0.494223i \(0.835453\pi\)
\(828\) 0 0
\(829\) 3.00000 0.104194 0.0520972 0.998642i \(-0.483409\pi\)
0.0520972 + 0.998642i \(0.483409\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 0 0
\(838\) 25.0000 0.863611
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) 15.0000 0.516321
\(845\) 0 0
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 11.0000 0.377742
\(849\) 0 0
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) 0 0
\(853\) −43.0000 −1.47229 −0.736146 0.676823i \(-0.763356\pi\)
−0.736146 + 0.676823i \(0.763356\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −27.0000 −0.919624
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 3.00000 0.101827
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) −4.00000 −0.135457
\(873\) 0 0
\(874\) 28.0000 0.947114
\(875\) 0 0
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 7.00000 0.236239
\(879\) 0 0
\(880\) 0 0
\(881\) −5.00000 −0.168454 −0.0842271 0.996447i \(-0.526842\pi\)
−0.0842271 + 0.996447i \(0.526842\pi\)
\(882\) 0 0
\(883\) −29.0000 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 50.0000 1.67884 0.839418 0.543487i \(-0.182896\pi\)
0.839418 + 0.543487i \(0.182896\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 13.0000 0.435272
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 16.0000 0.533927
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −11.0000 −0.366463
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 11.0000 0.365249 0.182625 0.983183i \(-0.441541\pi\)
0.182625 + 0.983183i \(0.441541\pi\)
\(908\) 7.00000 0.232303
\(909\) 0 0
\(910\) 0 0
\(911\) 19.0000 0.629498 0.314749 0.949175i \(-0.398080\pi\)
0.314749 + 0.949175i \(0.398080\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −32.0000 −1.05386
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) −1.00000 −0.0328266
\(929\) −25.0000 −0.820223 −0.410112 0.912035i \(-0.634510\pi\)
−0.410112 + 0.912035i \(0.634510\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 20.0000 0.655122
\(933\) 0 0
\(934\) −39.0000 −1.27612
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 21.0000 0.683854
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) −32.0000 −1.03986 −0.519930 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) −14.0000 −0.454459
\(950\) 0 0
\(951\) 0 0
\(952\) −1.00000 −0.0324102
\(953\) −40.0000 −1.29573 −0.647864 0.761756i \(-0.724337\pi\)
−0.647864 + 0.761756i \(0.724337\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 44.0000 1.41203 0.706014 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) −30.0000 −0.957338
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.00000 0.0318465
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 7.00000 0.222587
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 3.00000 0.0952501
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −27.0000 −0.854670
\(999\) 0 0
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 3150.2.a.bl.1.1 1
3.2 odd 2 1050.2.a.e.1.1 1
5.2 odd 4 3150.2.g.g.2899.2 2
5.3 odd 4 3150.2.g.g.2899.1 2
5.4 even 2 3150.2.a.c.1.1 1
12.11 even 2 8400.2.a.bt.1.1 1
15.2 even 4 1050.2.g.j.799.1 2
15.8 even 4 1050.2.g.j.799.2 2
15.14 odd 2 1050.2.a.o.1.1 yes 1
21.20 even 2 7350.2.a.bj.1.1 1
60.59 even 2 8400.2.a.t.1.1 1
105.104 even 2 7350.2.a.ca.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.e.1.1 1 3.2 odd 2
1050.2.a.o.1.1 yes 1 15.14 odd 2
1050.2.g.j.799.1 2 15.2 even 4
1050.2.g.j.799.2 2 15.8 even 4
3150.2.a.c.1.1 1 5.4 even 2
3150.2.a.bl.1.1 1 1.1 even 1 trivial
3150.2.g.g.2899.1 2 5.3 odd 4
3150.2.g.g.2899.2 2 5.2 odd 4
7350.2.a.bj.1.1 1 21.20 even 2
7350.2.a.ca.1.1 1 105.104 even 2
8400.2.a.t.1.1 1 60.59 even 2
8400.2.a.bt.1.1 1 12.11 even 2