Properties

Label 1170.2.a.q.1.1
Level $1170$
Weight $2$
Character 1170.1
Self dual yes
Analytic conductor $9.342$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 1170.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.60555 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.60555 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{13} -2.60555 q^{14} +1.00000 q^{16} +4.60555 q^{17} +6.60555 q^{19} +1.00000 q^{20} +4.60555 q^{23} +1.00000 q^{25} +1.00000 q^{26} -2.60555 q^{28} -4.60555 q^{29} +2.00000 q^{31} +1.00000 q^{32} +4.60555 q^{34} -2.60555 q^{35} +11.2111 q^{37} +6.60555 q^{38} +1.00000 q^{40} +3.21110 q^{41} +5.21110 q^{43} +4.60555 q^{46} -9.21110 q^{47} -0.211103 q^{49} +1.00000 q^{50} +1.00000 q^{52} -2.60555 q^{56} -4.60555 q^{58} -9.21110 q^{59} -7.21110 q^{61} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -7.21110 q^{67} +4.60555 q^{68} -2.60555 q^{70} -12.0000 q^{71} +6.60555 q^{73} +11.2111 q^{74} +6.60555 q^{76} -1.21110 q^{79} +1.00000 q^{80} +3.21110 q^{82} +4.60555 q^{85} +5.21110 q^{86} +3.21110 q^{89} -2.60555 q^{91} +4.60555 q^{92} -9.21110 q^{94} +6.60555 q^{95} +6.60555 q^{97} -0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 2 q^{10} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 6 q^{19} + 2 q^{20} + 2 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} - 2 q^{29} + 4 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} + 8 q^{37} + 6 q^{38} + 2 q^{40} - 8 q^{41} - 4 q^{43} + 2 q^{46} - 4 q^{47} + 14 q^{49} + 2 q^{50} + 2 q^{52} + 2 q^{56} - 2 q^{58} - 4 q^{59} + 4 q^{62} + 2 q^{64} + 2 q^{65} + 2 q^{68} + 2 q^{70} - 24 q^{71} + 6 q^{73} + 8 q^{74} + 6 q^{76} + 12 q^{79} + 2 q^{80} - 8 q^{82} + 2 q^{85} - 4 q^{86} - 8 q^{89} + 2 q^{91} + 2 q^{92} - 4 q^{94} + 6 q^{95} + 6 q^{97} + 14 q^{98} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −2.60555 −0.696363
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.60555 1.11701 0.558505 0.829501i \(-0.311375\pi\)
0.558505 + 0.829501i \(0.311375\pi\)
\(18\) 0 0
\(19\) 6.60555 1.51542 0.757709 0.652593i \(-0.226319\pi\)
0.757709 + 0.652593i \(0.226319\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 4.60555 0.960324 0.480162 0.877180i \(-0.340578\pi\)
0.480162 + 0.877180i \(0.340578\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −2.60555 −0.492403
\(29\) −4.60555 −0.855229 −0.427615 0.903961i \(-0.640646\pi\)
−0.427615 + 0.903961i \(0.640646\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.60555 0.789846
\(35\) −2.60555 −0.440419
\(36\) 0 0
\(37\) 11.2111 1.84309 0.921547 0.388267i \(-0.126926\pi\)
0.921547 + 0.388267i \(0.126926\pi\)
\(38\) 6.60555 1.07156
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.21110 0.501490 0.250745 0.968053i \(-0.419324\pi\)
0.250745 + 0.968053i \(0.419324\pi\)
\(42\) 0 0
\(43\) 5.21110 0.794686 0.397343 0.917670i \(-0.369932\pi\)
0.397343 + 0.917670i \(0.369932\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.60555 0.679051
\(47\) −9.21110 −1.34358 −0.671789 0.740743i \(-0.734474\pi\)
−0.671789 + 0.740743i \(0.734474\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.60555 −0.348181
\(57\) 0 0
\(58\) −4.60555 −0.604739
\(59\) −9.21110 −1.19918 −0.599592 0.800306i \(-0.704670\pi\)
−0.599592 + 0.800306i \(0.704670\pi\)
\(60\) 0 0
\(61\) −7.21110 −0.923287 −0.461644 0.887066i \(-0.652740\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −7.21110 −0.880976 −0.440488 0.897758i \(-0.645195\pi\)
−0.440488 + 0.897758i \(0.645195\pi\)
\(68\) 4.60555 0.558505
\(69\) 0 0
\(70\) −2.60555 −0.311423
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 6.60555 0.773121 0.386561 0.922264i \(-0.373663\pi\)
0.386561 + 0.922264i \(0.373663\pi\)
\(74\) 11.2111 1.30326
\(75\) 0 0
\(76\) 6.60555 0.757709
\(77\) 0 0
\(78\) 0 0
\(79\) −1.21110 −0.136260 −0.0681298 0.997676i \(-0.521703\pi\)
−0.0681298 + 0.997676i \(0.521703\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 3.21110 0.354607
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.60555 0.499542
\(86\) 5.21110 0.561928
\(87\) 0 0
\(88\) 0 0
\(89\) 3.21110 0.340376 0.170188 0.985412i \(-0.445562\pi\)
0.170188 + 0.985412i \(0.445562\pi\)
\(90\) 0 0
\(91\) −2.60555 −0.273136
\(92\) 4.60555 0.480162
\(93\) 0 0
\(94\) −9.21110 −0.950053
\(95\) 6.60555 0.677715
\(96\) 0 0
\(97\) 6.60555 0.670692 0.335346 0.942095i \(-0.391147\pi\)
0.335346 + 0.942095i \(0.391147\pi\)
\(98\) −0.211103 −0.0213246
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.8167 −1.37481 −0.687404 0.726275i \(-0.741250\pi\)
−0.687404 + 0.726275i \(0.741250\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 18.6056 1.78209 0.891044 0.453916i \(-0.149974\pi\)
0.891044 + 0.453916i \(0.149974\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.60555 −0.246201
\(113\) 7.39445 0.695611 0.347806 0.937567i \(-0.386927\pi\)
0.347806 + 0.937567i \(0.386927\pi\)
\(114\) 0 0
\(115\) 4.60555 0.429470
\(116\) −4.60555 −0.427615
\(117\) 0 0
\(118\) −9.21110 −0.847951
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −7.21110 −0.652863
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.21110 −0.107468 −0.0537340 0.998555i \(-0.517112\pi\)
−0.0537340 + 0.998555i \(0.517112\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) −16.6056 −1.45083 −0.725417 0.688310i \(-0.758353\pi\)
−0.725417 + 0.688310i \(0.758353\pi\)
\(132\) 0 0
\(133\) −17.2111 −1.49239
\(134\) −7.21110 −0.622944
\(135\) 0 0
\(136\) 4.60555 0.394923
\(137\) 21.6333 1.84826 0.924129 0.382080i \(-0.124792\pi\)
0.924129 + 0.382080i \(0.124792\pi\)
\(138\) 0 0
\(139\) 14.4222 1.22328 0.611638 0.791138i \(-0.290511\pi\)
0.611638 + 0.791138i \(0.290511\pi\)
\(140\) −2.60555 −0.220209
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) −4.60555 −0.382470
\(146\) 6.60555 0.546679
\(147\) 0 0
\(148\) 11.2111 0.921547
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 8.42221 0.685389 0.342695 0.939447i \(-0.388660\pi\)
0.342695 + 0.939447i \(0.388660\pi\)
\(152\) 6.60555 0.535781
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 11.2111 0.894743 0.447372 0.894348i \(-0.352360\pi\)
0.447372 + 0.894348i \(0.352360\pi\)
\(158\) −1.21110 −0.0963501
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 3.21110 0.250745
\(165\) 0 0
\(166\) 0 0
\(167\) −18.4222 −1.42555 −0.712777 0.701391i \(-0.752563\pi\)
−0.712777 + 0.701391i \(0.752563\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 4.60555 0.353230
\(171\) 0 0
\(172\) 5.21110 0.397343
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 0 0
\(175\) −2.60555 −0.196961
\(176\) 0 0
\(177\) 0 0
\(178\) 3.21110 0.240682
\(179\) 11.0278 0.824253 0.412127 0.911127i \(-0.364786\pi\)
0.412127 + 0.911127i \(0.364786\pi\)
\(180\) 0 0
\(181\) −19.2111 −1.42795 −0.713975 0.700171i \(-0.753107\pi\)
−0.713975 + 0.700171i \(0.753107\pi\)
\(182\) −2.60555 −0.193136
\(183\) 0 0
\(184\) 4.60555 0.339526
\(185\) 11.2111 0.824257
\(186\) 0 0
\(187\) 0 0
\(188\) −9.21110 −0.671789
\(189\) 0 0
\(190\) 6.60555 0.479217
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −14.6056 −1.05133 −0.525665 0.850691i \(-0.676184\pi\)
−0.525665 + 0.850691i \(0.676184\pi\)
\(194\) 6.60555 0.474251
\(195\) 0 0
\(196\) −0.211103 −0.0150788
\(197\) −24.4222 −1.74001 −0.870005 0.493043i \(-0.835885\pi\)
−0.870005 + 0.493043i \(0.835885\pi\)
\(198\) 0 0
\(199\) 5.21110 0.369405 0.184703 0.982794i \(-0.440868\pi\)
0.184703 + 0.982794i \(0.440868\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −13.8167 −0.972136
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 3.21110 0.224273
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −25.2111 −1.73560 −0.867802 0.496910i \(-0.834468\pi\)
−0.867802 + 0.496910i \(0.834468\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 5.21110 0.355394
\(216\) 0 0
\(217\) −5.21110 −0.353753
\(218\) 18.6056 1.26013
\(219\) 0 0
\(220\) 0 0
\(221\) 4.60555 0.309803
\(222\) 0 0
\(223\) −17.3944 −1.16482 −0.582409 0.812896i \(-0.697890\pi\)
−0.582409 + 0.812896i \(0.697890\pi\)
\(224\) −2.60555 −0.174091
\(225\) 0 0
\(226\) 7.39445 0.491871
\(227\) 6.42221 0.426257 0.213128 0.977024i \(-0.431635\pi\)
0.213128 + 0.977024i \(0.431635\pi\)
\(228\) 0 0
\(229\) 0.183346 0.0121159 0.00605793 0.999982i \(-0.498072\pi\)
0.00605793 + 0.999982i \(0.498072\pi\)
\(230\) 4.60555 0.303681
\(231\) 0 0
\(232\) −4.60555 −0.302369
\(233\) −1.81665 −0.119013 −0.0595065 0.998228i \(-0.518953\pi\)
−0.0595065 + 0.998228i \(0.518953\pi\)
\(234\) 0 0
\(235\) −9.21110 −0.600866
\(236\) −9.21110 −0.599592
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) 18.4222 1.19163 0.595817 0.803120i \(-0.296828\pi\)
0.595817 + 0.803120i \(0.296828\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) −7.21110 −0.461644
\(245\) −0.211103 −0.0134868
\(246\) 0 0
\(247\) 6.60555 0.420301
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 4.60555 0.290700 0.145350 0.989380i \(-0.453569\pi\)
0.145350 + 0.989380i \(0.453569\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.21110 −0.0759913
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.0278 −0.687893 −0.343946 0.938989i \(-0.611764\pi\)
−0.343946 + 0.938989i \(0.611764\pi\)
\(258\) 0 0
\(259\) −29.2111 −1.81509
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) −16.6056 −1.02589
\(263\) −11.0278 −0.680001 −0.340000 0.940425i \(-0.610427\pi\)
−0.340000 + 0.940425i \(0.610427\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −17.2111 −1.05528
\(267\) 0 0
\(268\) −7.21110 −0.440488
\(269\) 10.1833 0.620890 0.310445 0.950591i \(-0.399522\pi\)
0.310445 + 0.950591i \(0.399522\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 4.60555 0.279253
\(273\) 0 0
\(274\) 21.6333 1.30692
\(275\) 0 0
\(276\) 0 0
\(277\) −31.2111 −1.87529 −0.937647 0.347590i \(-0.887000\pi\)
−0.937647 + 0.347590i \(0.887000\pi\)
\(278\) 14.4222 0.864986
\(279\) 0 0
\(280\) −2.60555 −0.155711
\(281\) −3.21110 −0.191558 −0.0957792 0.995403i \(-0.530534\pi\)
−0.0957792 + 0.995403i \(0.530534\pi\)
\(282\) 0 0
\(283\) −1.21110 −0.0719926 −0.0359963 0.999352i \(-0.511460\pi\)
−0.0359963 + 0.999352i \(0.511460\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −8.36669 −0.493870
\(288\) 0 0
\(289\) 4.21110 0.247712
\(290\) −4.60555 −0.270447
\(291\) 0 0
\(292\) 6.60555 0.386561
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) −9.21110 −0.536291
\(296\) 11.2111 0.651632
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 4.60555 0.266346
\(300\) 0 0
\(301\) −13.5778 −0.782611
\(302\) 8.42221 0.484643
\(303\) 0 0
\(304\) 6.60555 0.378854
\(305\) −7.21110 −0.412907
\(306\) 0 0
\(307\) 4.78890 0.273317 0.136658 0.990618i \(-0.456364\pi\)
0.136658 + 0.990618i \(0.456364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.00000 0.113592
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 11.2111 0.633689 0.316844 0.948478i \(-0.397377\pi\)
0.316844 + 0.948478i \(0.397377\pi\)
\(314\) 11.2111 0.632679
\(315\) 0 0
\(316\) −1.21110 −0.0681298
\(317\) −12.4222 −0.697701 −0.348850 0.937178i \(-0.613428\pi\)
−0.348850 + 0.937178i \(0.613428\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) 30.4222 1.69274
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 2.00000 0.110770
\(327\) 0 0
\(328\) 3.21110 0.177303
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −5.39445 −0.296506 −0.148253 0.988949i \(-0.547365\pi\)
−0.148253 + 0.988949i \(0.547365\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −18.4222 −1.00802
\(335\) −7.21110 −0.393985
\(336\) 0 0
\(337\) 26.8444 1.46231 0.731154 0.682212i \(-0.238982\pi\)
0.731154 + 0.682212i \(0.238982\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 4.60555 0.249771
\(341\) 0 0
\(342\) 0 0
\(343\) 18.7889 1.01451
\(344\) 5.21110 0.280964
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 14.7889 0.793910 0.396955 0.917838i \(-0.370067\pi\)
0.396955 + 0.917838i \(0.370067\pi\)
\(348\) 0 0
\(349\) −11.8167 −0.632531 −0.316265 0.948671i \(-0.602429\pi\)
−0.316265 + 0.948671i \(0.602429\pi\)
\(350\) −2.60555 −0.139273
\(351\) 0 0
\(352\) 0 0
\(353\) 21.6333 1.15142 0.575712 0.817652i \(-0.304725\pi\)
0.575712 + 0.817652i \(0.304725\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 3.21110 0.170188
\(357\) 0 0
\(358\) 11.0278 0.582835
\(359\) −30.4222 −1.60562 −0.802811 0.596233i \(-0.796663\pi\)
−0.802811 + 0.596233i \(0.796663\pi\)
\(360\) 0 0
\(361\) 24.6333 1.29649
\(362\) −19.2111 −1.00971
\(363\) 0 0
\(364\) −2.60555 −0.136568
\(365\) 6.60555 0.345750
\(366\) 0 0
\(367\) −6.78890 −0.354378 −0.177189 0.984177i \(-0.556700\pi\)
−0.177189 + 0.984177i \(0.556700\pi\)
\(368\) 4.60555 0.240081
\(369\) 0 0
\(370\) 11.2111 0.582837
\(371\) 0 0
\(372\) 0 0
\(373\) −0.788897 −0.0408476 −0.0204238 0.999791i \(-0.506502\pi\)
−0.0204238 + 0.999791i \(0.506502\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.21110 −0.475026
\(377\) −4.60555 −0.237198
\(378\) 0 0
\(379\) −26.6056 −1.36664 −0.683318 0.730121i \(-0.739464\pi\)
−0.683318 + 0.730121i \(0.739464\pi\)
\(380\) 6.60555 0.338858
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 18.4222 0.941331 0.470665 0.882312i \(-0.344014\pi\)
0.470665 + 0.882312i \(0.344014\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.6056 −0.743403
\(387\) 0 0
\(388\) 6.60555 0.335346
\(389\) −1.81665 −0.0921080 −0.0460540 0.998939i \(-0.514665\pi\)
−0.0460540 + 0.998939i \(0.514665\pi\)
\(390\) 0 0
\(391\) 21.2111 1.07269
\(392\) −0.211103 −0.0106623
\(393\) 0 0
\(394\) −24.4222 −1.23037
\(395\) −1.21110 −0.0609372
\(396\) 0 0
\(397\) 35.2111 1.76719 0.883597 0.468248i \(-0.155114\pi\)
0.883597 + 0.468248i \(0.155114\pi\)
\(398\) 5.21110 0.261209
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 33.6333 1.67957 0.839784 0.542921i \(-0.182682\pi\)
0.839784 + 0.542921i \(0.182682\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) −13.8167 −0.687404
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −13.6333 −0.674124 −0.337062 0.941483i \(-0.609433\pi\)
−0.337062 + 0.941483i \(0.609433\pi\)
\(410\) 3.21110 0.158585
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) −29.4500 −1.43872 −0.719362 0.694635i \(-0.755566\pi\)
−0.719362 + 0.694635i \(0.755566\pi\)
\(420\) 0 0
\(421\) −9.02776 −0.439986 −0.219993 0.975501i \(-0.570603\pi\)
−0.219993 + 0.975501i \(0.570603\pi\)
\(422\) −25.2111 −1.22726
\(423\) 0 0
\(424\) 0 0
\(425\) 4.60555 0.223402
\(426\) 0 0
\(427\) 18.7889 0.909258
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 5.21110 0.251302
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −22.8444 −1.09783 −0.548916 0.835877i \(-0.684959\pi\)
−0.548916 + 0.835877i \(0.684959\pi\)
\(434\) −5.21110 −0.250141
\(435\) 0 0
\(436\) 18.6056 0.891044
\(437\) 30.4222 1.45529
\(438\) 0 0
\(439\) 32.8444 1.56758 0.783789 0.621027i \(-0.213284\pi\)
0.783789 + 0.621027i \(0.213284\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.60555 0.219064
\(443\) −39.6333 −1.88304 −0.941518 0.336964i \(-0.890600\pi\)
−0.941518 + 0.336964i \(0.890600\pi\)
\(444\) 0 0
\(445\) 3.21110 0.152221
\(446\) −17.3944 −0.823651
\(447\) 0 0
\(448\) −2.60555 −0.123101
\(449\) −9.63331 −0.454624 −0.227312 0.973822i \(-0.572994\pi\)
−0.227312 + 0.973822i \(0.572994\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.39445 0.347806
\(453\) 0 0
\(454\) 6.42221 0.301409
\(455\) −2.60555 −0.122150
\(456\) 0 0
\(457\) −17.3944 −0.813678 −0.406839 0.913500i \(-0.633369\pi\)
−0.406839 + 0.913500i \(0.633369\pi\)
\(458\) 0.183346 0.00856720
\(459\) 0 0
\(460\) 4.60555 0.214735
\(461\) −3.21110 −0.149556 −0.0747780 0.997200i \(-0.523825\pi\)
−0.0747780 + 0.997200i \(0.523825\pi\)
\(462\) 0 0
\(463\) 12.1833 0.566208 0.283104 0.959089i \(-0.408636\pi\)
0.283104 + 0.959089i \(0.408636\pi\)
\(464\) −4.60555 −0.213807
\(465\) 0 0
\(466\) −1.81665 −0.0841549
\(467\) −5.57779 −0.258110 −0.129055 0.991637i \(-0.541194\pi\)
−0.129055 + 0.991637i \(0.541194\pi\)
\(468\) 0 0
\(469\) 18.7889 0.867591
\(470\) −9.21110 −0.424876
\(471\) 0 0
\(472\) −9.21110 −0.423975
\(473\) 0 0
\(474\) 0 0
\(475\) 6.60555 0.303083
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) 18.4222 0.842612
\(479\) 18.4222 0.841732 0.420866 0.907123i \(-0.361726\pi\)
0.420866 + 0.907123i \(0.361726\pi\)
\(480\) 0 0
\(481\) 11.2111 0.511182
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 6.60555 0.299943
\(486\) 0 0
\(487\) 2.97224 0.134685 0.0673426 0.997730i \(-0.478548\pi\)
0.0673426 + 0.997730i \(0.478548\pi\)
\(488\) −7.21110 −0.326431
\(489\) 0 0
\(490\) −0.211103 −0.00953664
\(491\) 32.2389 1.45492 0.727460 0.686150i \(-0.240701\pi\)
0.727460 + 0.686150i \(0.240701\pi\)
\(492\) 0 0
\(493\) −21.2111 −0.955300
\(494\) 6.60555 0.297198
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 31.2666 1.40250
\(498\) 0 0
\(499\) 27.8167 1.24524 0.622622 0.782523i \(-0.286067\pi\)
0.622622 + 0.782523i \(0.286067\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 4.60555 0.205556
\(503\) 20.2389 0.902406 0.451203 0.892421i \(-0.350995\pi\)
0.451203 + 0.892421i \(0.350995\pi\)
\(504\) 0 0
\(505\) −13.8167 −0.614833
\(506\) 0 0
\(507\) 0 0
\(508\) −1.21110 −0.0537340
\(509\) 9.63331 0.426989 0.213494 0.976944i \(-0.431515\pi\)
0.213494 + 0.976944i \(0.431515\pi\)
\(510\) 0 0
\(511\) −17.2111 −0.761374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −11.0278 −0.486413
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) −29.2111 −1.28346
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) −21.2111 −0.929275 −0.464638 0.885501i \(-0.653815\pi\)
−0.464638 + 0.885501i \(0.653815\pi\)
\(522\) 0 0
\(523\) 38.4222 1.68009 0.840043 0.542520i \(-0.182530\pi\)
0.840043 + 0.542520i \(0.182530\pi\)
\(524\) −16.6056 −0.725417
\(525\) 0 0
\(526\) −11.0278 −0.480833
\(527\) 9.21110 0.401242
\(528\) 0 0
\(529\) −1.78890 −0.0777781
\(530\) 0 0
\(531\) 0 0
\(532\) −17.2111 −0.746196
\(533\) 3.21110 0.139088
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −7.21110 −0.311472
\(537\) 0 0
\(538\) 10.1833 0.439035
\(539\) 0 0
\(540\) 0 0
\(541\) −9.02776 −0.388134 −0.194067 0.980988i \(-0.562168\pi\)
−0.194067 + 0.980988i \(0.562168\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) 4.60555 0.197461
\(545\) 18.6056 0.796974
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 21.6333 0.924129
\(549\) 0 0
\(550\) 0 0
\(551\) −30.4222 −1.29603
\(552\) 0 0
\(553\) 3.15559 0.134189
\(554\) −31.2111 −1.32603
\(555\) 0 0
\(556\) 14.4222 0.611638
\(557\) −18.8444 −0.798463 −0.399232 0.916850i \(-0.630723\pi\)
−0.399232 + 0.916850i \(0.630723\pi\)
\(558\) 0 0
\(559\) 5.21110 0.220406
\(560\) −2.60555 −0.110105
\(561\) 0 0
\(562\) −3.21110 −0.135452
\(563\) −15.6333 −0.658865 −0.329433 0.944179i \(-0.606857\pi\)
−0.329433 + 0.944179i \(0.606857\pi\)
\(564\) 0 0
\(565\) 7.39445 0.311087
\(566\) −1.21110 −0.0509064
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −26.7889 −1.12305 −0.561525 0.827460i \(-0.689785\pi\)
−0.561525 + 0.827460i \(0.689785\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −8.36669 −0.349219
\(575\) 4.60555 0.192065
\(576\) 0 0
\(577\) 27.8167 1.15802 0.579011 0.815320i \(-0.303439\pi\)
0.579011 + 0.815320i \(0.303439\pi\)
\(578\) 4.21110 0.175159
\(579\) 0 0
\(580\) −4.60555 −0.191235
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 6.60555 0.273340
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −6.42221 −0.265073 −0.132536 0.991178i \(-0.542312\pi\)
−0.132536 + 0.991178i \(0.542312\pi\)
\(588\) 0 0
\(589\) 13.2111 0.544354
\(590\) −9.21110 −0.379215
\(591\) 0 0
\(592\) 11.2111 0.460773
\(593\) 20.7889 0.853698 0.426849 0.904323i \(-0.359624\pi\)
0.426849 + 0.904323i \(0.359624\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 4.60555 0.188335
\(599\) −36.8444 −1.50542 −0.752711 0.658351i \(-0.771254\pi\)
−0.752711 + 0.658351i \(0.771254\pi\)
\(600\) 0 0
\(601\) 17.6333 0.719278 0.359639 0.933092i \(-0.382900\pi\)
0.359639 + 0.933092i \(0.382900\pi\)
\(602\) −13.5778 −0.553390
\(603\) 0 0
\(604\) 8.42221 0.342695
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) 41.2111 1.67271 0.836354 0.548190i \(-0.184683\pi\)
0.836354 + 0.548190i \(0.184683\pi\)
\(608\) 6.60555 0.267890
\(609\) 0 0
\(610\) −7.21110 −0.291969
\(611\) −9.21110 −0.372641
\(612\) 0 0
\(613\) 5.63331 0.227527 0.113764 0.993508i \(-0.463709\pi\)
0.113764 + 0.993508i \(0.463709\pi\)
\(614\) 4.78890 0.193264
\(615\) 0 0
\(616\) 0 0
\(617\) −18.8444 −0.758647 −0.379324 0.925264i \(-0.623843\pi\)
−0.379324 + 0.925264i \(0.623843\pi\)
\(618\) 0 0
\(619\) −2.60555 −0.104726 −0.0523630 0.998628i \(-0.516675\pi\)
−0.0523630 + 0.998628i \(0.516675\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −8.36669 −0.335204
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 11.2111 0.448086
\(627\) 0 0
\(628\) 11.2111 0.447372
\(629\) 51.6333 2.05875
\(630\) 0 0
\(631\) 23.2111 0.924019 0.462010 0.886875i \(-0.347129\pi\)
0.462010 + 0.886875i \(0.347129\pi\)
\(632\) −1.21110 −0.0481751
\(633\) 0 0
\(634\) −12.4222 −0.493349
\(635\) −1.21110 −0.0480611
\(636\) 0 0
\(637\) −0.211103 −0.00836419
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 48.0555 1.89512 0.947562 0.319571i \(-0.103539\pi\)
0.947562 + 0.319571i \(0.103539\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 30.4222 1.19695
\(647\) −35.0278 −1.37708 −0.688542 0.725197i \(-0.741749\pi\)
−0.688542 + 0.725197i \(0.741749\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) −2.78890 −0.109138 −0.0545690 0.998510i \(-0.517378\pi\)
−0.0545690 + 0.998510i \(0.517378\pi\)
\(654\) 0 0
\(655\) −16.6056 −0.648833
\(656\) 3.21110 0.125372
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) 13.8167 0.538220 0.269110 0.963109i \(-0.413270\pi\)
0.269110 + 0.963109i \(0.413270\pi\)
\(660\) 0 0
\(661\) −21.0278 −0.817885 −0.408942 0.912560i \(-0.634102\pi\)
−0.408942 + 0.912560i \(0.634102\pi\)
\(662\) −5.39445 −0.209661
\(663\) 0 0
\(664\) 0 0
\(665\) −17.2111 −0.667418
\(666\) 0 0
\(667\) −21.2111 −0.821297
\(668\) −18.4222 −0.712777
\(669\) 0 0
\(670\) −7.21110 −0.278589
\(671\) 0 0
\(672\) 0 0
\(673\) −19.2111 −0.740534 −0.370267 0.928925i \(-0.620734\pi\)
−0.370267 + 0.928925i \(0.620734\pi\)
\(674\) 26.8444 1.03401
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 9.21110 0.354011 0.177006 0.984210i \(-0.443359\pi\)
0.177006 + 0.984210i \(0.443359\pi\)
\(678\) 0 0
\(679\) −17.2111 −0.660501
\(680\) 4.60555 0.176615
\(681\) 0 0
\(682\) 0 0
\(683\) 5.57779 0.213428 0.106714 0.994290i \(-0.465967\pi\)
0.106714 + 0.994290i \(0.465967\pi\)
\(684\) 0 0
\(685\) 21.6333 0.826566
\(686\) 18.7889 0.717363
\(687\) 0 0
\(688\) 5.21110 0.198671
\(689\) 0 0
\(690\) 0 0
\(691\) −23.8167 −0.906028 −0.453014 0.891503i \(-0.649651\pi\)
−0.453014 + 0.891503i \(0.649651\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 14.7889 0.561379
\(695\) 14.4222 0.547065
\(696\) 0 0
\(697\) 14.7889 0.560169
\(698\) −11.8167 −0.447267
\(699\) 0 0
\(700\) −2.60555 −0.0984806
\(701\) 16.6056 0.627183 0.313592 0.949558i \(-0.398468\pi\)
0.313592 + 0.949558i \(0.398468\pi\)
\(702\) 0 0
\(703\) 74.0555 2.79306
\(704\) 0 0
\(705\) 0 0
\(706\) 21.6333 0.814180
\(707\) 36.0000 1.35392
\(708\) 0 0
\(709\) 31.4500 1.18113 0.590564 0.806991i \(-0.298905\pi\)
0.590564 + 0.806991i \(0.298905\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) 3.21110 0.120341
\(713\) 9.21110 0.344959
\(714\) 0 0
\(715\) 0 0
\(716\) 11.0278 0.412127
\(717\) 0 0
\(718\) −30.4222 −1.13535
\(719\) 21.2111 0.791041 0.395520 0.918457i \(-0.370564\pi\)
0.395520 + 0.918457i \(0.370564\pi\)
\(720\) 0 0
\(721\) 10.4222 0.388143
\(722\) 24.6333 0.916757
\(723\) 0 0
\(724\) −19.2111 −0.713975
\(725\) −4.60555 −0.171046
\(726\) 0 0
\(727\) −43.6333 −1.61827 −0.809135 0.587623i \(-0.800064\pi\)
−0.809135 + 0.587623i \(0.800064\pi\)
\(728\) −2.60555 −0.0965682
\(729\) 0 0
\(730\) 6.60555 0.244482
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 41.6333 1.53776 0.768881 0.639392i \(-0.220814\pi\)
0.768881 + 0.639392i \(0.220814\pi\)
\(734\) −6.78890 −0.250583
\(735\) 0 0
\(736\) 4.60555 0.169763
\(737\) 0 0
\(738\) 0 0
\(739\) −26.6056 −0.978701 −0.489351 0.872087i \(-0.662766\pi\)
−0.489351 + 0.872087i \(0.662766\pi\)
\(740\) 11.2111 0.412128
\(741\) 0 0
\(742\) 0 0
\(743\) −46.0555 −1.68961 −0.844806 0.535072i \(-0.820284\pi\)
−0.844806 + 0.535072i \(0.820284\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −0.788897 −0.0288836
\(747\) 0 0
\(748\) 0 0
\(749\) 31.2666 1.14246
\(750\) 0 0
\(751\) −35.2666 −1.28690 −0.643449 0.765489i \(-0.722497\pi\)
−0.643449 + 0.765489i \(0.722497\pi\)
\(752\) −9.21110 −0.335894
\(753\) 0 0
\(754\) −4.60555 −0.167724
\(755\) 8.42221 0.306515
\(756\) 0 0
\(757\) −34.8444 −1.26644 −0.633221 0.773971i \(-0.718268\pi\)
−0.633221 + 0.773971i \(0.718268\pi\)
\(758\) −26.6056 −0.966357
\(759\) 0 0
\(760\) 6.60555 0.239609
\(761\) −9.63331 −0.349207 −0.174604 0.984639i \(-0.555864\pi\)
−0.174604 + 0.984639i \(0.555864\pi\)
\(762\) 0 0
\(763\) −48.4777 −1.75501
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 18.4222 0.665621
\(767\) −9.21110 −0.332594
\(768\) 0 0
\(769\) −34.8444 −1.25652 −0.628261 0.778003i \(-0.716233\pi\)
−0.628261 + 0.778003i \(0.716233\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.6056 −0.525665
\(773\) −12.4222 −0.446796 −0.223398 0.974727i \(-0.571715\pi\)
−0.223398 + 0.974727i \(0.571715\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 6.60555 0.237125
\(777\) 0 0
\(778\) −1.81665 −0.0651302
\(779\) 21.2111 0.759967
\(780\) 0 0
\(781\) 0 0
\(782\) 21.2111 0.758507
\(783\) 0 0
\(784\) −0.211103 −0.00753938
\(785\) 11.2111 0.400141
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) −24.4222 −0.870005
\(789\) 0 0
\(790\) −1.21110 −0.0430891
\(791\) −19.2666 −0.685042
\(792\) 0 0
\(793\) −7.21110 −0.256074
\(794\) 35.2111 1.24960
\(795\) 0 0
\(796\) 5.21110 0.184703
\(797\) 42.4222 1.50267 0.751336 0.659920i \(-0.229410\pi\)
0.751336 + 0.659920i \(0.229410\pi\)
\(798\) 0 0
\(799\) −42.4222 −1.50079
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 33.6333 1.18763
\(803\) 0 0
\(804\) 0 0
\(805\) −12.0000 −0.422944
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) −13.8167 −0.486068
\(809\) 6.42221 0.225793 0.112896 0.993607i \(-0.463987\pi\)
0.112896 + 0.993607i \(0.463987\pi\)
\(810\) 0 0
\(811\) 49.0278 1.72160 0.860799 0.508946i \(-0.169965\pi\)
0.860799 + 0.508946i \(0.169965\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 34.4222 1.20428
\(818\) −13.6333 −0.476677
\(819\) 0 0
\(820\) 3.21110 0.112137
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) −0.972244 −0.0336862
\(834\) 0 0
\(835\) −18.4222 −0.637527
\(836\) 0 0
\(837\) 0 0
\(838\) −29.4500 −1.01733
\(839\) −42.4222 −1.46458 −0.732289 0.680994i \(-0.761548\pi\)
−0.732289 + 0.680994i \(0.761548\pi\)
\(840\) 0 0
\(841\) −7.78890 −0.268583
\(842\) −9.02776 −0.311117
\(843\) 0 0
\(844\) −25.2111 −0.867802
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 28.6611 0.984806
\(848\) 0 0
\(849\) 0 0
\(850\) 4.60555 0.157969
\(851\) 51.6333 1.76997
\(852\) 0 0
\(853\) 24.0555 0.823645 0.411823 0.911264i \(-0.364892\pi\)
0.411823 + 0.911264i \(0.364892\pi\)
\(854\) 18.7889 0.642943
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −29.4500 −1.00599 −0.502996 0.864289i \(-0.667769\pi\)
−0.502996 + 0.864289i \(0.667769\pi\)
\(858\) 0 0
\(859\) 4.36669 0.148990 0.0744948 0.997221i \(-0.476266\pi\)
0.0744948 + 0.997221i \(0.476266\pi\)
\(860\) 5.21110 0.177697
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −42.4222 −1.44407 −0.722034 0.691857i \(-0.756793\pi\)
−0.722034 + 0.691857i \(0.756793\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) −22.8444 −0.776285
\(867\) 0 0
\(868\) −5.21110 −0.176876
\(869\) 0 0
\(870\) 0 0
\(871\) −7.21110 −0.244339
\(872\) 18.6056 0.630063
\(873\) 0 0
\(874\) 30.4222 1.02905
\(875\) −2.60555 −0.0880837
\(876\) 0 0
\(877\) 26.0000 0.877958 0.438979 0.898497i \(-0.355340\pi\)
0.438979 + 0.898497i \(0.355340\pi\)
\(878\) 32.8444 1.10845
\(879\) 0 0
\(880\) 0 0
\(881\) 26.7889 0.902541 0.451270 0.892387i \(-0.350971\pi\)
0.451270 + 0.892387i \(0.350971\pi\)
\(882\) 0 0
\(883\) 14.4222 0.485346 0.242673 0.970108i \(-0.421976\pi\)
0.242673 + 0.970108i \(0.421976\pi\)
\(884\) 4.60555 0.154901
\(885\) 0 0
\(886\) −39.6333 −1.33151
\(887\) 4.60555 0.154639 0.0773196 0.997006i \(-0.475364\pi\)
0.0773196 + 0.997006i \(0.475364\pi\)
\(888\) 0 0
\(889\) 3.15559 0.105835
\(890\) 3.21110 0.107636
\(891\) 0 0
\(892\) −17.3944 −0.582409
\(893\) −60.8444 −2.03608
\(894\) 0 0
\(895\) 11.0278 0.368617
\(896\) −2.60555 −0.0870454
\(897\) 0 0
\(898\) −9.63331 −0.321468
\(899\) −9.21110 −0.307207
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 7.39445 0.245936
\(905\) −19.2111 −0.638599
\(906\) 0 0
\(907\) −12.3667 −0.410629 −0.205315 0.978696i \(-0.565822\pi\)
−0.205315 + 0.978696i \(0.565822\pi\)
\(908\) 6.42221 0.213128
\(909\) 0 0
\(910\) −2.60555 −0.0863732
\(911\) 15.6333 0.517955 0.258977 0.965883i \(-0.416615\pi\)
0.258977 + 0.965883i \(0.416615\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −17.3944 −0.575357
\(915\) 0 0
\(916\) 0.183346 0.00605793
\(917\) 43.2666 1.42879
\(918\) 0 0
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 4.60555 0.151841
\(921\) 0 0
\(922\) −3.21110 −0.105752
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 11.2111 0.368619
\(926\) 12.1833 0.400370
\(927\) 0 0
\(928\) −4.60555 −0.151185
\(929\) −40.0555 −1.31418 −0.657089 0.753813i \(-0.728213\pi\)
−0.657089 + 0.753813i \(0.728213\pi\)
\(930\) 0 0
\(931\) −1.39445 −0.0457012
\(932\) −1.81665 −0.0595065
\(933\) 0 0
\(934\) −5.57779 −0.182511
\(935\) 0 0
\(936\) 0 0
\(937\) −13.6333 −0.445381 −0.222690 0.974889i \(-0.571484\pi\)
−0.222690 + 0.974889i \(0.571484\pi\)
\(938\) 18.7889 0.613479
\(939\) 0 0
\(940\) −9.21110 −0.300433
\(941\) 21.6333 0.705226 0.352613 0.935769i \(-0.385293\pi\)
0.352613 + 0.935769i \(0.385293\pi\)
\(942\) 0 0
\(943\) 14.7889 0.481593
\(944\) −9.21110 −0.299796
\(945\) 0 0
\(946\) 0 0
\(947\) 42.4222 1.37854 0.689268 0.724506i \(-0.257932\pi\)
0.689268 + 0.724506i \(0.257932\pi\)
\(948\) 0 0
\(949\) 6.60555 0.214425
\(950\) 6.60555 0.214312
\(951\) 0 0
\(952\) −12.0000 −0.388922
\(953\) 16.6056 0.537907 0.268953 0.963153i \(-0.413322\pi\)
0.268953 + 0.963153i \(0.413322\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 18.4222 0.595817
\(957\) 0 0
\(958\) 18.4222 0.595194
\(959\) −56.3667 −1.82018
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 11.2111 0.361460
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −14.6056 −0.470169
\(966\) 0 0
\(967\) −26.6056 −0.855577 −0.427788 0.903879i \(-0.640707\pi\)
−0.427788 + 0.903879i \(0.640707\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 6.60555 0.212091
\(971\) −19.3944 −0.622397 −0.311199 0.950345i \(-0.600730\pi\)
−0.311199 + 0.950345i \(0.600730\pi\)
\(972\) 0 0
\(973\) −37.5778 −1.20469
\(974\) 2.97224 0.0952368
\(975\) 0 0
\(976\) −7.21110 −0.230822
\(977\) 57.6333 1.84385 0.921926 0.387365i \(-0.126615\pi\)
0.921926 + 0.387365i \(0.126615\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.211103 −0.00674342
\(981\) 0 0
\(982\) 32.2389 1.02878
\(983\) 27.6333 0.881366 0.440683 0.897663i \(-0.354736\pi\)
0.440683 + 0.897663i \(0.354736\pi\)
\(984\) 0 0
\(985\) −24.4222 −0.778156
\(986\) −21.2111 −0.675499
\(987\) 0 0
\(988\) 6.60555 0.210151
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −31.6333 −1.00487 −0.502433 0.864616i \(-0.667561\pi\)
−0.502433 + 0.864616i \(0.667561\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 31.2666 0.991717
\(995\) 5.21110 0.165203
\(996\) 0 0
\(997\) 4.78890 0.151666 0.0758330 0.997121i \(-0.475838\pi\)
0.0758330 + 0.997121i \(0.475838\pi\)
\(998\) 27.8167 0.880521
\(999\) 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 1170.2.a.q.1.1 yes 2
3.2 odd 2 1170.2.a.p.1.1 2
4.3 odd 2 9360.2.a.cn.1.2 2
5.2 odd 4 5850.2.e.bj.5149.3 4
5.3 odd 4 5850.2.e.bj.5149.2 4
5.4 even 2 5850.2.a.ce.1.2 2
12.11 even 2 9360.2.a.cf.1.2 2
15.2 even 4 5850.2.e.bl.5149.1 4
15.8 even 4 5850.2.e.bl.5149.4 4
15.14 odd 2 5850.2.a.ck.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1170.2.a.p.1.1 2 3.2 odd 2
1170.2.a.q.1.1 yes 2 1.1 even 1 trivial
5850.2.a.ce.1.2 2 5.4 even 2
5850.2.a.ck.1.2 2 15.14 odd 2
5850.2.e.bj.5149.2 4 5.3 odd 4
5850.2.e.bj.5149.3 4 5.2 odd 4
5850.2.e.bl.5149.1 4 15.2 even 4
5850.2.e.bl.5149.4 4 15.8 even 4
9360.2.a.cf.1.2 2 12.11 even 2
9360.2.a.cn.1.2 2 4.3 odd 2