Properties

Label 4004.2.a.c.1.1
Level 4004
Weight 2
Character 4004.1
Self dual yes
Analytic conductor 31.972
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
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
Character \(\chi\) \(=\) 4004.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} +4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +8.00000 q^{15} +2.00000 q^{17} +4.00000 q^{19} +2.00000 q^{21} +8.00000 q^{23} +11.0000 q^{25} -4.00000 q^{27} -10.0000 q^{29} +8.00000 q^{31} +2.00000 q^{33} +4.00000 q^{35} +2.00000 q^{37} -2.00000 q^{39} -10.0000 q^{41} -10.0000 q^{43} +4.00000 q^{45} -12.0000 q^{47} +1.00000 q^{49} +4.00000 q^{51} -10.0000 q^{53} +4.00000 q^{55} +8.00000 q^{57} +12.0000 q^{59} +2.00000 q^{61} +1.00000 q^{63} -4.00000 q^{65} -4.00000 q^{67} +16.0000 q^{69} +2.00000 q^{73} +22.0000 q^{75} +1.00000 q^{77} +10.0000 q^{79} -11.0000 q^{81} -4.00000 q^{83} +8.00000 q^{85} -20.0000 q^{87} +12.0000 q^{89} -1.00000 q^{91} +16.0000 q^{93} +16.0000 q^{95} -8.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 8.00000 2.06559
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) −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
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 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
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 16.0000 1.92617
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 22.0000 2.54034
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) −20.0000 −2.14423
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 16.0000 1.65912
\(94\) 0 0
\(95\) 16.0000 1.64157
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 32.0000 2.98402
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −20.0000 −1.80334
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 0 0
\(129\) −20.0000 −1.76090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) −16.0000 −1.37706
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −40.0000 −3.32182
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 32.0000 2.57030
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) −20.0000 −1.58610
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) 24.0000 1.80395
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) −8.00000 −0.572892
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −26.0000 −1.84309 −0.921546 0.388270i \(-0.873073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) −10.0000 −0.701862
\(204\) 0 0
\(205\) −40.0000 −2.79372
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −40.0000 −2.72798
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −48.0000 −3.13117
\(236\) 0 0
\(237\) 20.0000 1.29914
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 16.0000 1.00196
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) −40.0000 −2.45718
\(266\) 0 0
\(267\) 24.0000 1.46878
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 11.0000 0.663325
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) 32.0000 1.89552
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 0 0
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 48.0000 2.79467
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) 20.0000 1.14897
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −28.0000 −1.59286
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −11.0000 −0.610170
\(326\) 0 0
\(327\) 32.0000 1.76960
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 64.0000 3.44564
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 48.0000 2.47871
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −28.0000 −1.43448
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) 0 0
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 40.0000 2.01262
\(396\) 0 0
\(397\) 32.0000 1.60603 0.803017 0.595956i \(-0.203227\pi\)
0.803017 + 0.595956i \(0.203227\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) −44.0000 −2.18638
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) 0 0
\(417\) −24.0000 −1.17529
\(418\) 0 0
\(419\) 2.00000 0.0977064 0.0488532 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) 0 0
\(421\) −38.0000 −1.85201 −0.926003 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) −80.0000 −3.83571
\(436\) 0 0
\(437\) 32.0000 1.53077
\(438\) 0 0
\(439\) 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
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 48.0000 2.27542
\(446\) 0 0
\(447\) −8.00000 −0.378387
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 0 0
\(453\) −32.0000 −1.50349
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 64.0000 2.96793
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 44.0000 2.02741
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 44.0000 2.01886
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) −32.0000 −1.45305
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 40.0000 1.77998
\(506\) 0 0
\(507\) 2.00000 0.0888231
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) −16.0000 −0.706417
\(514\) 0 0
\(515\) −56.0000 −2.46765
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) −36.0000 −1.58022
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 22.0000 0.960159
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 4.00000 0.171656
\(544\) 0 0
\(545\) 64.0000 2.74146
\(546\) 0 0
\(547\) 10.0000 0.427569 0.213785 0.976881i \(-0.431421\pi\)
0.213785 + 0.976881i \(0.431421\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 0 0
\(557\) −20.0000 −0.847427 −0.423714 0.905796i \(-0.639274\pi\)
−0.423714 + 0.905796i \(0.639274\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 0 0
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 0 0
\(565\) −24.0000 −1.00969
\(566\) 0 0
\(567\) −11.0000 −0.461957
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 46.0000 1.92504 0.962520 0.271211i \(-0.0874240\pi\)
0.962520 + 0.271211i \(0.0874240\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 88.0000 3.66985
\(576\) 0 0
\(577\) −16.0000 −0.666089 −0.333044 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) 0 0
\(597\) −52.0000 −2.12822
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 0 0
\(609\) −20.0000 −0.810441
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) −80.0000 −3.22591
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −32.0000 −1.28412
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 8.00000 0.319489
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) −56.0000 −2.22229
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) 0 0
\(645\) −80.0000 −3.15000
\(646\) 0 0
\(647\) −26.0000 −1.02217 −0.511083 0.859532i \(-0.670755\pi\)
−0.511083 + 0.859532i \(0.670755\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 0 0
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) −80.0000 −3.09761
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) −44.0000 −1.69356
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) 32.0000 1.22088
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) −48.0000 −1.82074
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −96.0000 −3.61557
\(706\) 0 0
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 44.0000 1.63638
\(724\) 0 0
\(725\) −110.000 −4.08530
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) −10.0000 −0.369358 −0.184679 0.982799i \(-0.559125\pi\)
−0.184679 + 0.982799i \(0.559125\pi\)
\(734\) 0 0
\(735\) 8.00000 0.295084
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) −16.0000 −0.586195
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) 0 0
\(755\) −64.0000 −2.32920
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 8.00000 0.289241
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) 0 0
\(773\) −40.0000 −1.43870 −0.719350 0.694648i \(-0.755560\pi\)
−0.719350 + 0.694648i \(0.755560\pi\)
\(774\) 0 0
\(775\) 88.0000 3.16105
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 40.0000 1.42948
\(784\) 0 0
\(785\) 88.0000 3.14085
\(786\) 0 0
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 0 0
\(789\) 36.0000 1.28163
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) −80.0000 −2.83731
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) −20.0000 −0.704033
\(808\) 0 0
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) −32.0000 −1.12229
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 22.0000 0.765942
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 54.0000 1.87550 0.937749 0.347314i \(-0.112906\pi\)
0.937749 + 0.347314i \(0.112906\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −32.0000 −1.10608
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 24.0000 0.826604
\(844\) 0 0
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −56.0000 −1.92192
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 16.0000 0.547188
\(856\) 0 0
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −20.0000 −0.681598
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −72.0000 −2.44807
\(866\) 0 0
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −24.0000 −0.810422 −0.405211 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) 0 0
\(879\) −36.0000 −1.21425
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 96.0000 3.22700
\(886\) 0 0
\(887\) 20.0000 0.671534 0.335767 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 0 0
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) −80.0000 −2.66815
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) −20.0000 −0.665558
\(904\) 0 0
\(905\) 8.00000 0.265929
\(906\) 0 0
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 0 0
\(915\) 16.0000 0.528944
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −46.0000 −1.51740 −0.758700 0.651440i \(-0.774165\pi\)
−0.758700 + 0.651440i \(0.774165\pi\)
\(920\) 0 0
\(921\) −40.0000 −1.31804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 22.0000 0.723356
\(926\) 0 0
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) 20.0000 0.654771
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 44.0000 1.43589
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 0 0
\(943\) −80.0000 −2.60516
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) 20.0000 0.648544
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) 48.0000 1.55324
\(956\) 0 0
\(957\) −20.0000 −0.646508
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 0 0
\(965\) 48.0000 1.54517
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) −22.0000 −0.704564
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 16.0000 0.510841
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) −80.0000 −2.54385
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 0 0
\(993\) 56.0000 1.77711
\(994\) 0 0
\(995\) −104.000 −3.29702
\(996\) 0 0
\(997\) 58.0000 1.83688 0.918439 0.395562i \(-0.129450\pi\)
0.918439 + 0.395562i \(0.129450\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 4004.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.a.c.1.1 1 1.1 even 1 trivial