Properties

Label 8007.2.a.a.1.1
Level 8007
Weight 2
Character 8007.1
Self dual yes
Analytic conductor 63.936
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 8007.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{21} +4.00000 q^{22} -6.00000 q^{23} +3.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{29} +5.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -2.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -6.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} -5.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -6.00000 q^{56} +4.00000 q^{57} -4.00000 q^{58} +4.00000 q^{59} +12.0000 q^{61} +2.00000 q^{63} +7.00000 q^{64} -4.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} +6.00000 q^{69} +8.00000 q^{71} -3.00000 q^{72} -4.00000 q^{73} -2.00000 q^{74} +5.00000 q^{75} +4.00000 q^{76} +8.00000 q^{77} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -16.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} +4.00000 q^{87} -12.0000 q^{88} -6.00000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -5.00000 q^{96} -12.0000 q^{97} -3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 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 \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 4.00000 0.852803
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 3.00000 0.612372
\(25\) −5.00000 −1.00000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) −4.00000 −0.696311
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −5.00000 −0.707107
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 4.00000 0.529813
\(58\) −4.00000 −0.525226
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −3.00000 −0.353553
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −2.00000 −0.232495
\(75\) 5.00000 0.577350
\(76\) 4.00000 0.458831
\(77\) 8.00000 0.911685
\(78\) −2.00000 −0.226455
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 4.00000 0.428845
\(88\) −12.0000 −1.27920
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −3.00000 −0.303046
\(99\) 4.00000 0.402015
\(100\) 5.00000 0.500000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 12.0000 1.08643
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −3.00000 −0.265165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 4.00000 0.348155
\(133\) −8.00000 −0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 6.00000 0.510754
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 8.00000 0.668994
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 3.00000 0.247436
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 5.00000 0.408248
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 12.0000 0.973329
\(153\) 1.00000 0.0808452
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 1.00000 0.0798087
\(158\) −10.0000 −0.795557
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 1.00000 0.0785674
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 6.00000 0.462910
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 4.00000 0.303239
\(175\) −10.0000 −0.755929
\(176\) −4.00000 −0.301511
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −24.0000 −1.78391 −0.891953 0.452128i \(-0.850665\pi\)
−0.891953 + 0.452128i \(0.850665\pi\)
\(182\) 4.00000 0.296500
\(183\) −12.0000 −0.887066
\(184\) 18.0000 1.32698
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −7.00000 −0.505181
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 4.00000 0.284268
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 15.0000 1.06066
\(201\) 4.00000 0.282138
\(202\) 2.00000 0.140720
\(203\) −8.00000 −0.561490
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 2.00000 0.137361
\(213\) −8.00000 −0.548151
\(214\) −14.0000 −0.957020
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 2.00000 0.134231
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 10.0000 0.668153
\(225\) −5.00000 −0.333333
\(226\) 6.00000 0.399114
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 12.0000 0.787839
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 10.0000 0.649570
\(238\) 2.00000 0.129641
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −12.0000 −0.768221
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) −2.00000 −0.125988
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −4.00000 −0.249029
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 14.0000 0.864923
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 12.0000 0.738549
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) −32.0000 −1.95107 −0.975537 0.219834i \(-0.929448\pi\)
−0.975537 + 0.219834i \(0.929448\pi\)
\(270\) 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
\(273\) −4.00000 −0.242091
\(274\) 10.0000 0.604122
\(275\) −20.0000 −1.20605
\(276\) −6.00000 −0.361158
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 2.00000 0.119952
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 4.00000 0.234082
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) −12.0000 −0.693978
\(300\) −5.00000 −0.288675
\(301\) 8.00000 0.461112
\(302\) 8.00000 0.460348
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −8.00000 −0.455842
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 6.00000 0.339683
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 1.00000 0.0564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 2.00000 0.112154
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 14.0000 0.781404
\(322\) −12.0000 −0.668734
\(323\) −4.00000 −0.222566
\(324\) −1.00000 −0.0555556
\(325\) −10.0000 −0.554700
\(326\) −14.0000 −0.775388
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 16.0000 0.878114
\(333\) −2.00000 −0.109599
\(334\) −20.0000 −1.09435
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −20.0000 −1.07990
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −4.00000 −0.214423
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −10.0000 −0.534522
\(351\) −2.00000 −0.106752
\(352\) 20.0000 1.06600
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −2.00000 −0.105851
\(358\) −4.00000 −0.211407
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −24.0000 −1.26141
\(363\) −5.00000 −0.262432
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) −12.0000 −0.627250
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) −2.00000 −0.102869
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 4.00000 0.203331
\(388\) 12.0000 0.609208
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 9.00000 0.454569
\(393\) −14.0000 −0.706207
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) −4.00000 −0.200502
\(399\) 8.00000 0.400501
\(400\) 5.00000 0.250000
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) −8.00000 −0.396545
\(408\) 3.00000 0.148522
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) −8.00000 −0.394132
\(413\) 8.00000 0.393654
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) −2.00000 −0.0979404
\(418\) −16.0000 −0.782586
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −2.00000 −0.0973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −5.00000 −0.242536
\(426\) −8.00000 −0.387601
\(427\) 24.0000 1.16144
\(428\) 14.0000 0.676716
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 24.0000 1.14808
\(438\) 4.00000 0.191127
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 2.00000 0.0951303
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 6.00000 0.283790
\(448\) 14.0000 0.661438
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) −5.00000 −0.235702
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) 10.0000 0.469323
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 6.00000 0.280362
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −8.00000 −0.372194
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −1.00000 −0.0460776
\(472\) −12.0000 −0.552345
\(473\) 16.0000 0.735681
\(474\) 10.0000 0.459315
\(475\) 20.0000 0.917663
\(476\) −2.00000 −0.0916698
\(477\) −2.00000 −0.0915737
\(478\) 24.0000 1.09773
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −16.0000 −0.728780
\(483\) 12.0000 0.546019
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −36.0000 −1.62964
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 16.0000 0.716977
\(499\) −30.0000 −1.34298 −0.671492 0.741012i \(-0.734346\pi\)
−0.671492 + 0.741012i \(0.734346\pi\)
\(500\) 0 0
\(501\) 20.0000 0.893534
\(502\) 4.00000 0.178529
\(503\) 26.0000 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) −24.0000 −1.06693
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −11.0000 −0.486136
\(513\) 4.00000 0.176604
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) −4.00000 −0.175075
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −14.0000 −0.611593
\(525\) 10.0000 0.436436
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 8.00000 0.346844
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 4.00000 0.172613
\(538\) −32.0000 −1.37962
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −20.0000 −0.859074
\(543\) 24.0000 1.02994
\(544\) 5.00000 0.214373
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −10.0000 −0.427179
\(549\) 12.0000 0.512148
\(550\) −20.0000 −0.852803
\(551\) 16.0000 0.681623
\(552\) −18.0000 −0.766131
\(553\) −20.0000 −0.850487
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 6.00000 0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −24.0000 −1.00880
\(567\) 2.00000 0.0839921
\(568\) −24.0000 −1.00702
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −8.00000 −0.334497
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) 7.00000 0.291667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) −32.0000 −1.32758
\(582\) 12.0000 0.497416
\(583\) −8.00000 −0.331326
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 2.00000 0.0821995
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 4.00000 0.163709
\(598\) −12.0000 −0.490716
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) −15.0000 −0.612372
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −20.0000 −0.811107
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −24.0000 −0.966988
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −8.00000 −0.321807
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) −8.00000 −0.320771
\(623\) −12.0000 −0.480770
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) 6.00000 0.239808
\(627\) 16.0000 0.638978
\(628\) −1.00000 −0.0399043
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 30.0000 1.19334
\(633\) 2.00000 0.0794929
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) −6.00000 −0.237729
\(638\) −16.0000 −0.633446
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 14.0000 0.552536
\(643\) −46.0000 −1.81406 −0.907031 0.421063i \(-0.861657\pi\)
−0.907031 + 0.421063i \(0.861657\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) −3.00000 −0.117851
\(649\) 16.0000 0.628055
\(650\) −10.0000 −0.392232
\(651\) 0 0
\(652\) 14.0000 0.548282
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −12.0000 −0.466393
\(663\) −2.00000 −0.0776736
\(664\) 48.0000 1.86276
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 24.0000 0.929284
\(668\) 20.0000 0.773823
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) −10.0000 −0.385758
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 16.0000 0.616297
\(675\) 5.00000 0.192450
\(676\) 9.00000 0.346154
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −6.00000 −0.230429
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 0 0
\(683\) 34.0000 1.30097 0.650487 0.759517i \(-0.274565\pi\)
0.650487 + 0.759517i \(0.274565\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −6.00000 −0.228914
\(688\) −4.00000 −0.152499
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) 6.00000 0.228086
\(693\) 8.00000 0.303895
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) 0 0
\(698\) −26.0000 −0.984115
\(699\) −10.0000 −0.378235
\(700\) 10.0000 0.377964
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 8.00000 0.301726
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 4.00000 0.150435
\(708\) 4.00000 0.150329
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −24.0000 −0.896296
\(718\) 4.00000 0.149279
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) 16.0000 0.595046
\(724\) 24.0000 0.891953
\(725\) 20.0000 0.742781
\(726\) −5.00000 −0.185567
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −12.0000 −0.444750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 12.0000 0.443533
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −30.0000 −1.10581
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) −4.00000 −0.146845
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −38.0000 −1.39128
\(747\) −16.0000 −0.585409
\(748\) −4.00000 −0.146254
\(749\) −28.0000 −1.02310
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 10.0000 0.363216
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 8.00000 0.288863
\(768\) 17.0000 0.613435
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −10.0000 −0.359908
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 36.0000 1.29232
\(777\) 4.00000 0.143499
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) −6.00000 −0.214560
\(783\) 4.00000 0.142948
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −14.0000 −0.499363
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 22.0000 0.783718
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) −12.0000 −0.426401
\(793\) 24.0000 0.852265
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) −25.0000 −0.883883
\(801\) −6.00000 −0.212000
\(802\) 4.00000 0.141245
\(803\) −16.0000 −0.564628
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 32.0000 1.12645
\(808\) −6.00000 −0.211079
\(809\) 40.0000 1.40633 0.703163 0.711029i \(-0.251771\pi\)
0.703163 + 0.711029i \(0.251771\pi\)
\(810\) 0 0
\(811\) 46.0000 1.61528 0.807639 0.589677i \(-0.200745\pi\)
0.807639 + 0.589677i \(0.200745\pi\)
\(812\) 8.00000 0.280745
\(813\) 20.0000 0.701431
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) −16.0000 −0.559769
\(818\) −14.0000 −0.489499
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −10.0000 −0.348790
\(823\) −54.0000 −1.88232 −0.941161 0.337959i \(-0.890263\pi\)
−0.941161 + 0.337959i \(0.890263\pi\)
\(824\) −24.0000 −0.836080
\(825\) 20.0000 0.696311
\(826\) 8.00000 0.278356
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 6.00000 0.208514
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 14.0000 0.485363
\(833\) −3.00000 −0.103944
\(834\) −2.00000 −0.0692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −16.0000 −0.552711
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 10.0000 0.344623
\(843\) −6.00000 −0.206651
\(844\) 2.00000 0.0688428
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 2.00000 0.0686803
\(849\) 24.0000 0.823678
\(850\) −5.00000 −0.171499
\(851\) 12.0000 0.411355
\(852\) 8.00000 0.274075
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 42.0000 1.43553
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) −8.00000 −0.273115
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 18.0000 0.611665
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −42.0000 −1.42230
\(873\) −12.0000 −0.406138
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 10.0000 0.337484
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −44.0000 −1.48240 −0.741199 0.671286i \(-0.765742\pi\)
−0.741199 + 0.671286i \(0.765742\pi\)
\(882\) −3.00000 −0.101015
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 38.0000 1.27592 0.637958 0.770072i \(-0.279780\pi\)
0.637958 + 0.770072i \(0.279780\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 4.00000 0.133930
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 12.0000 0.400668
\(898\) −16.0000 −0.533927
\(899\) 0 0
\(900\) 5.00000 0.166667
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) −10.0000 −0.331862
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −4.00000 −0.132453
\(913\) −64.0000 −2.11809
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 28.0000 0.924641
\(918\) −1.00000 −0.0330049
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 18.0000 0.592798
\(923\) 16.0000 0.526646
\(924\) 8.00000 0.263181
\(925\) 10.0000 0.328798
\(926\) 8.00000 0.262896
\(927\) 8.00000 0.262754
\(928\) −20.0000 −0.656532
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −10.0000 −0.327561
\(933\) 8.00000 0.261908
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −8.00000 −0.261209
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −1.00000 −0.0325818
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 14.0000 0.454939 0.227469 0.973785i \(-0.426955\pi\)
0.227469 + 0.973785i \(0.426955\pi\)
\(948\) −10.0000 −0.324785
\(949\) −8.00000 −0.259691
\(950\) 20.0000 0.648886
\(951\) 22.0000 0.713399
\(952\) −6.00000 −0.194461
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 16.0000 0.517207
\(958\) −30.0000 −0.969256
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −4.00000 −0.128965
\(963\) −14.0000 −0.451144
\(964\) 16.0000 0.515325
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −15.0000 −0.482118
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 1.00000 0.0320750
\(973\) 4.00000 0.128234
\(974\) 28.0000 0.897178
\(975\) 10.0000 0.320256
\(976\) −12.0000 −0.384111
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 14.0000 0.447671
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) −46.0000 −1.46717 −0.733586 0.679597i \(-0.762155\pi\)
−0.733586 + 0.679597i \(0.762155\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) 40.0000 1.26681 0.633406 0.773819i \(-0.281656\pi\)
0.633406 + 0.773819i \(0.281656\pi\)
\(998\) −30.0000 −0.949633
\(999\) 2.00000 0.0632772
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 8007.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8007.2.a.a.1.1 1 1.1 even 1 trivial