Properties

Label 6160.2.a.bf.1.1
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.10278 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.62721 q^{9} +O(q^{10})\) \(q-3.10278 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.62721 q^{9} -1.00000 q^{11} -3.62721 q^{13} -3.10278 q^{15} -4.20555 q^{17} +8.15165 q^{19} -3.10278 q^{21} -0.897225 q^{23} +1.00000 q^{25} -11.2544 q^{27} -7.30833 q^{29} +3.42166 q^{31} +3.10278 q^{33} +1.00000 q^{35} -1.10278 q^{37} +11.2544 q^{39} +12.3572 q^{41} -10.6761 q^{43} +6.62721 q^{45} +1.15667 q^{47} +1.00000 q^{49} +13.0489 q^{51} +11.3083 q^{53} -1.00000 q^{55} -25.2927 q^{57} -7.25443 q^{59} -3.15667 q^{61} +6.62721 q^{63} -3.62721 q^{65} +8.41110 q^{67} +2.78389 q^{69} +13.0489 q^{71} +0.205550 q^{73} -3.10278 q^{75} -1.00000 q^{77} -12.1517 q^{79} +15.0383 q^{81} -2.20555 q^{83} -4.20555 q^{85} +22.6761 q^{87} -7.45998 q^{89} -3.62721 q^{91} -10.6167 q^{93} +8.15165 q^{95} +2.25945 q^{97} -6.62721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} + O(q^{10}) \) \( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99} + O(q^{100}) \)

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
\(3\) −3.10278 −1.79139 −0.895694 0.444671i \(-0.853321\pi\)
−0.895694 + 0.444671i \(0.853321\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.62721 2.20907
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.62721 −1.00601 −0.503004 0.864284i \(-0.667772\pi\)
−0.503004 + 0.864284i \(0.667772\pi\)
\(14\) 0 0
\(15\) −3.10278 −0.801133
\(16\) 0 0
\(17\) −4.20555 −1.02000 −0.509998 0.860176i \(-0.670354\pi\)
−0.509998 + 0.860176i \(0.670354\pi\)
\(18\) 0 0
\(19\) 8.15165 1.87012 0.935058 0.354493i \(-0.115347\pi\)
0.935058 + 0.354493i \(0.115347\pi\)
\(20\) 0 0
\(21\) −3.10278 −0.677081
\(22\) 0 0
\(23\) −0.897225 −0.187084 −0.0935422 0.995615i \(-0.529819\pi\)
−0.0935422 + 0.995615i \(0.529819\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −11.2544 −2.16592
\(28\) 0 0
\(29\) −7.30833 −1.35712 −0.678561 0.734544i \(-0.737396\pi\)
−0.678561 + 0.734544i \(0.737396\pi\)
\(30\) 0 0
\(31\) 3.42166 0.614549 0.307274 0.951621i \(-0.400583\pi\)
0.307274 + 0.951621i \(0.400583\pi\)
\(32\) 0 0
\(33\) 3.10278 0.540124
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −1.10278 −0.181295 −0.0906476 0.995883i \(-0.528894\pi\)
−0.0906476 + 0.995883i \(0.528894\pi\)
\(38\) 0 0
\(39\) 11.2544 1.80215
\(40\) 0 0
\(41\) 12.3572 1.92987 0.964935 0.262488i \(-0.0845430\pi\)
0.964935 + 0.262488i \(0.0845430\pi\)
\(42\) 0 0
\(43\) −10.6761 −1.62809 −0.814044 0.580803i \(-0.802739\pi\)
−0.814044 + 0.580803i \(0.802739\pi\)
\(44\) 0 0
\(45\) 6.62721 0.987927
\(46\) 0 0
\(47\) 1.15667 0.168718 0.0843591 0.996435i \(-0.473116\pi\)
0.0843591 + 0.996435i \(0.473116\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 13.0489 1.82721
\(52\) 0 0
\(53\) 11.3083 1.55332 0.776659 0.629921i \(-0.216913\pi\)
0.776659 + 0.629921i \(0.216913\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −25.2927 −3.35011
\(58\) 0 0
\(59\) −7.25443 −0.944446 −0.472223 0.881479i \(-0.656548\pi\)
−0.472223 + 0.881479i \(0.656548\pi\)
\(60\) 0 0
\(61\) −3.15667 −0.404171 −0.202085 0.979368i \(-0.564772\pi\)
−0.202085 + 0.979368i \(0.564772\pi\)
\(62\) 0 0
\(63\) 6.62721 0.834950
\(64\) 0 0
\(65\) −3.62721 −0.449900
\(66\) 0 0
\(67\) 8.41110 1.02758 0.513790 0.857916i \(-0.328241\pi\)
0.513790 + 0.857916i \(0.328241\pi\)
\(68\) 0 0
\(69\) 2.78389 0.335141
\(70\) 0 0
\(71\) 13.0489 1.54862 0.774308 0.632809i \(-0.218098\pi\)
0.774308 + 0.632809i \(0.218098\pi\)
\(72\) 0 0
\(73\) 0.205550 0.0240578 0.0120289 0.999928i \(-0.496171\pi\)
0.0120289 + 0.999928i \(0.496171\pi\)
\(74\) 0 0
\(75\) −3.10278 −0.358278
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −12.1517 −1.36717 −0.683584 0.729872i \(-0.739580\pi\)
−0.683584 + 0.729872i \(0.739580\pi\)
\(80\) 0 0
\(81\) 15.0383 1.67092
\(82\) 0 0
\(83\) −2.20555 −0.242091 −0.121045 0.992647i \(-0.538625\pi\)
−0.121045 + 0.992647i \(0.538625\pi\)
\(84\) 0 0
\(85\) −4.20555 −0.456156
\(86\) 0 0
\(87\) 22.6761 2.43113
\(88\) 0 0
\(89\) −7.45998 −0.790756 −0.395378 0.918519i \(-0.629386\pi\)
−0.395378 + 0.918519i \(0.629386\pi\)
\(90\) 0 0
\(91\) −3.62721 −0.380235
\(92\) 0 0
\(93\) −10.6167 −1.10090
\(94\) 0 0
\(95\) 8.15165 0.836342
\(96\) 0 0
\(97\) 2.25945 0.229412 0.114706 0.993399i \(-0.463407\pi\)
0.114706 + 0.993399i \(0.463407\pi\)
\(98\) 0 0
\(99\) −6.62721 −0.666060
\(100\) 0 0
\(101\) 13.6655 1.35977 0.679885 0.733318i \(-0.262030\pi\)
0.679885 + 0.733318i \(0.262030\pi\)
\(102\) 0 0
\(103\) 5.04888 0.497481 0.248740 0.968570i \(-0.419983\pi\)
0.248740 + 0.968570i \(0.419983\pi\)
\(104\) 0 0
\(105\) −3.10278 −0.302800
\(106\) 0 0
\(107\) −12.9894 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(108\) 0 0
\(109\) 5.10278 0.488757 0.244379 0.969680i \(-0.421416\pi\)
0.244379 + 0.969680i \(0.421416\pi\)
\(110\) 0 0
\(111\) 3.42166 0.324770
\(112\) 0 0
\(113\) −10.4111 −0.979394 −0.489697 0.871893i \(-0.662893\pi\)
−0.489697 + 0.871893i \(0.662893\pi\)
\(114\) 0 0
\(115\) −0.897225 −0.0836667
\(116\) 0 0
\(117\) −24.0383 −2.22234
\(118\) 0 0
\(119\) −4.20555 −0.385522
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −38.3416 −3.45715
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −3.25443 −0.288784 −0.144392 0.989521i \(-0.546123\pi\)
−0.144392 + 0.989521i \(0.546123\pi\)
\(128\) 0 0
\(129\) 33.1255 2.91654
\(130\) 0 0
\(131\) 17.3083 1.51224 0.756118 0.654436i \(-0.227094\pi\)
0.756118 + 0.654436i \(0.227094\pi\)
\(132\) 0 0
\(133\) 8.15165 0.706838
\(134\) 0 0
\(135\) −11.2544 −0.968627
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −13.2005 −1.11965 −0.559827 0.828609i \(-0.689132\pi\)
−0.559827 + 0.828609i \(0.689132\pi\)
\(140\) 0 0
\(141\) −3.58890 −0.302240
\(142\) 0 0
\(143\) 3.62721 0.303323
\(144\) 0 0
\(145\) −7.30833 −0.606923
\(146\) 0 0
\(147\) −3.10278 −0.255913
\(148\) 0 0
\(149\) −23.6116 −1.93434 −0.967170 0.254131i \(-0.918211\pi\)
−0.967170 + 0.254131i \(0.918211\pi\)
\(150\) 0 0
\(151\) −11.5139 −0.936986 −0.468493 0.883467i \(-0.655203\pi\)
−0.468493 + 0.883467i \(0.655203\pi\)
\(152\) 0 0
\(153\) −27.8711 −2.25324
\(154\) 0 0
\(155\) 3.42166 0.274835
\(156\) 0 0
\(157\) −16.2056 −1.29334 −0.646672 0.762768i \(-0.723840\pi\)
−0.646672 + 0.762768i \(0.723840\pi\)
\(158\) 0 0
\(159\) −35.0872 −2.78260
\(160\) 0 0
\(161\) −0.897225 −0.0707112
\(162\) 0 0
\(163\) −6.31335 −0.494500 −0.247250 0.968952i \(-0.579527\pi\)
−0.247250 + 0.968952i \(0.579527\pi\)
\(164\) 0 0
\(165\) 3.10278 0.241551
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 0.156674 0.0120519
\(170\) 0 0
\(171\) 54.0227 4.13122
\(172\) 0 0
\(173\) 9.25443 0.703601 0.351800 0.936075i \(-0.385570\pi\)
0.351800 + 0.936075i \(0.385570\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 22.5089 1.69187
\(178\) 0 0
\(179\) −4.98944 −0.372928 −0.186464 0.982462i \(-0.559703\pi\)
−0.186464 + 0.982462i \(0.559703\pi\)
\(180\) 0 0
\(181\) −9.66553 −0.718433 −0.359216 0.933254i \(-0.616956\pi\)
−0.359216 + 0.933254i \(0.616956\pi\)
\(182\) 0 0
\(183\) 9.79445 0.724027
\(184\) 0 0
\(185\) −1.10278 −0.0810776
\(186\) 0 0
\(187\) 4.20555 0.307540
\(188\) 0 0
\(189\) −11.2544 −0.818639
\(190\) 0 0
\(191\) 9.15667 0.662554 0.331277 0.943534i \(-0.392521\pi\)
0.331277 + 0.943534i \(0.392521\pi\)
\(192\) 0 0
\(193\) 1.52946 0.110093 0.0550465 0.998484i \(-0.482469\pi\)
0.0550465 + 0.998484i \(0.482469\pi\)
\(194\) 0 0
\(195\) 11.2544 0.805946
\(196\) 0 0
\(197\) 14.3033 1.01907 0.509534 0.860451i \(-0.329818\pi\)
0.509534 + 0.860451i \(0.329818\pi\)
\(198\) 0 0
\(199\) 10.0978 0.715811 0.357905 0.933758i \(-0.383491\pi\)
0.357905 + 0.933758i \(0.383491\pi\)
\(200\) 0 0
\(201\) −26.0978 −1.84079
\(202\) 0 0
\(203\) −7.30833 −0.512944
\(204\) 0 0
\(205\) 12.3572 0.863064
\(206\) 0 0
\(207\) −5.94610 −0.413283
\(208\) 0 0
\(209\) −8.15165 −0.563861
\(210\) 0 0
\(211\) −10.8433 −0.746485 −0.373243 0.927734i \(-0.621754\pi\)
−0.373243 + 0.927734i \(0.621754\pi\)
\(212\) 0 0
\(213\) −40.4877 −2.77417
\(214\) 0 0
\(215\) −10.6761 −0.728103
\(216\) 0 0
\(217\) 3.42166 0.232278
\(218\) 0 0
\(219\) −0.637776 −0.0430969
\(220\) 0 0
\(221\) 15.2544 1.02612
\(222\) 0 0
\(223\) 9.45998 0.633487 0.316743 0.948511i \(-0.397411\pi\)
0.316743 + 0.948511i \(0.397411\pi\)
\(224\) 0 0
\(225\) 6.62721 0.441814
\(226\) 0 0
\(227\) 17.5678 1.16601 0.583007 0.812467i \(-0.301876\pi\)
0.583007 + 0.812467i \(0.301876\pi\)
\(228\) 0 0
\(229\) 25.7250 1.69995 0.849977 0.526820i \(-0.176616\pi\)
0.849977 + 0.526820i \(0.176616\pi\)
\(230\) 0 0
\(231\) 3.10278 0.204148
\(232\) 0 0
\(233\) 19.1567 1.25500 0.627498 0.778618i \(-0.284079\pi\)
0.627498 + 0.778618i \(0.284079\pi\)
\(234\) 0 0
\(235\) 1.15667 0.0754531
\(236\) 0 0
\(237\) 37.7038 2.44913
\(238\) 0 0
\(239\) 18.0539 1.16781 0.583905 0.811822i \(-0.301524\pi\)
0.583905 + 0.811822i \(0.301524\pi\)
\(240\) 0 0
\(241\) 6.15165 0.396263 0.198131 0.980175i \(-0.436513\pi\)
0.198131 + 0.980175i \(0.436513\pi\)
\(242\) 0 0
\(243\) −12.8972 −0.827357
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −29.5678 −1.88135
\(248\) 0 0
\(249\) 6.84333 0.433678
\(250\) 0 0
\(251\) 29.9789 1.89225 0.946125 0.323802i \(-0.104961\pi\)
0.946125 + 0.323802i \(0.104961\pi\)
\(252\) 0 0
\(253\) 0.897225 0.0564080
\(254\) 0 0
\(255\) 13.0489 0.817152
\(256\) 0 0
\(257\) 4.05390 0.252875 0.126438 0.991975i \(-0.459646\pi\)
0.126438 + 0.991975i \(0.459646\pi\)
\(258\) 0 0
\(259\) −1.10278 −0.0685231
\(260\) 0 0
\(261\) −48.4338 −2.99798
\(262\) 0 0
\(263\) 14.5089 0.894654 0.447327 0.894370i \(-0.352376\pi\)
0.447327 + 0.894370i \(0.352376\pi\)
\(264\) 0 0
\(265\) 11.3083 0.694665
\(266\) 0 0
\(267\) 23.1466 1.41655
\(268\) 0 0
\(269\) 4.37279 0.266614 0.133307 0.991075i \(-0.457440\pi\)
0.133307 + 0.991075i \(0.457440\pi\)
\(270\) 0 0
\(271\) 10.6167 0.644916 0.322458 0.946584i \(-0.395491\pi\)
0.322458 + 0.946584i \(0.395491\pi\)
\(272\) 0 0
\(273\) 11.2544 0.681149
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −19.7633 −1.18746 −0.593730 0.804664i \(-0.702345\pi\)
−0.593730 + 0.804664i \(0.702345\pi\)
\(278\) 0 0
\(279\) 22.6761 1.35758
\(280\) 0 0
\(281\) −27.0489 −1.61360 −0.806800 0.590824i \(-0.798803\pi\)
−0.806800 + 0.590824i \(0.798803\pi\)
\(282\) 0 0
\(283\) −27.6655 −1.64454 −0.822272 0.569094i \(-0.807294\pi\)
−0.822272 + 0.569094i \(0.807294\pi\)
\(284\) 0 0
\(285\) −25.2927 −1.49821
\(286\) 0 0
\(287\) 12.3572 0.729423
\(288\) 0 0
\(289\) 0.686652 0.0403913
\(290\) 0 0
\(291\) −7.01056 −0.410966
\(292\) 0 0
\(293\) −8.37279 −0.489143 −0.244572 0.969631i \(-0.578647\pi\)
−0.244572 + 0.969631i \(0.578647\pi\)
\(294\) 0 0
\(295\) −7.25443 −0.422369
\(296\) 0 0
\(297\) 11.2544 0.653048
\(298\) 0 0
\(299\) 3.25443 0.188208
\(300\) 0 0
\(301\) −10.6761 −0.615360
\(302\) 0 0
\(303\) −42.4011 −2.43588
\(304\) 0 0
\(305\) −3.15667 −0.180751
\(306\) 0 0
\(307\) 10.2056 0.582462 0.291231 0.956653i \(-0.405935\pi\)
0.291231 + 0.956653i \(0.405935\pi\)
\(308\) 0 0
\(309\) −15.6655 −0.891181
\(310\) 0 0
\(311\) 8.16724 0.463122 0.231561 0.972820i \(-0.425617\pi\)
0.231561 + 0.972820i \(0.425617\pi\)
\(312\) 0 0
\(313\) 34.5628 1.95360 0.976801 0.214149i \(-0.0686977\pi\)
0.976801 + 0.214149i \(0.0686977\pi\)
\(314\) 0 0
\(315\) 6.62721 0.373401
\(316\) 0 0
\(317\) 9.21057 0.517317 0.258659 0.965969i \(-0.416720\pi\)
0.258659 + 0.965969i \(0.416720\pi\)
\(318\) 0 0
\(319\) 7.30833 0.409188
\(320\) 0 0
\(321\) 40.3033 2.24951
\(322\) 0 0
\(323\) −34.2822 −1.90751
\(324\) 0 0
\(325\) −3.62721 −0.201202
\(326\) 0 0
\(327\) −15.8328 −0.875554
\(328\) 0 0
\(329\) 1.15667 0.0637695
\(330\) 0 0
\(331\) −16.2439 −0.892843 −0.446422 0.894823i \(-0.647302\pi\)
−0.446422 + 0.894823i \(0.647302\pi\)
\(332\) 0 0
\(333\) −7.30833 −0.400494
\(334\) 0 0
\(335\) 8.41110 0.459547
\(336\) 0 0
\(337\) −1.25443 −0.0683329 −0.0341665 0.999416i \(-0.510878\pi\)
−0.0341665 + 0.999416i \(0.510878\pi\)
\(338\) 0 0
\(339\) 32.3033 1.75447
\(340\) 0 0
\(341\) −3.42166 −0.185293
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 2.78389 0.149879
\(346\) 0 0
\(347\) −7.42166 −0.398416 −0.199208 0.979957i \(-0.563837\pi\)
−0.199208 + 0.979957i \(0.563837\pi\)
\(348\) 0 0
\(349\) −18.3033 −0.979753 −0.489877 0.871792i \(-0.662958\pi\)
−0.489877 + 0.871792i \(0.662958\pi\)
\(350\) 0 0
\(351\) 40.8222 2.17893
\(352\) 0 0
\(353\) 28.8761 1.53692 0.768460 0.639898i \(-0.221023\pi\)
0.768460 + 0.639898i \(0.221023\pi\)
\(354\) 0 0
\(355\) 13.0489 0.692562
\(356\) 0 0
\(357\) 13.0489 0.690620
\(358\) 0 0
\(359\) −4.15165 −0.219116 −0.109558 0.993980i \(-0.534943\pi\)
−0.109558 + 0.993980i \(0.534943\pi\)
\(360\) 0 0
\(361\) 47.4494 2.49734
\(362\) 0 0
\(363\) −3.10278 −0.162853
\(364\) 0 0
\(365\) 0.205550 0.0107590
\(366\) 0 0
\(367\) 5.26447 0.274803 0.137402 0.990515i \(-0.456125\pi\)
0.137402 + 0.990515i \(0.456125\pi\)
\(368\) 0 0
\(369\) 81.8938 4.26322
\(370\) 0 0
\(371\) 11.3083 0.587099
\(372\) 0 0
\(373\) 35.8711 1.85733 0.928667 0.370915i \(-0.120956\pi\)
0.928667 + 0.370915i \(0.120956\pi\)
\(374\) 0 0
\(375\) −3.10278 −0.160227
\(376\) 0 0
\(377\) 26.5089 1.36528
\(378\) 0 0
\(379\) 33.0177 1.69601 0.848003 0.529992i \(-0.177805\pi\)
0.848003 + 0.529992i \(0.177805\pi\)
\(380\) 0 0
\(381\) 10.0978 0.517323
\(382\) 0 0
\(383\) −2.95112 −0.150795 −0.0753977 0.997154i \(-0.524023\pi\)
−0.0753977 + 0.997154i \(0.524023\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −70.7527 −3.59656
\(388\) 0 0
\(389\) 1.89220 0.0959384 0.0479692 0.998849i \(-0.484725\pi\)
0.0479692 + 0.998849i \(0.484725\pi\)
\(390\) 0 0
\(391\) 3.77332 0.190825
\(392\) 0 0
\(393\) −53.7038 −2.70900
\(394\) 0 0
\(395\) −12.1517 −0.611416
\(396\) 0 0
\(397\) 5.36222 0.269122 0.134561 0.990905i \(-0.457038\pi\)
0.134561 + 0.990905i \(0.457038\pi\)
\(398\) 0 0
\(399\) −25.2927 −1.26622
\(400\) 0 0
\(401\) 23.7038 1.18371 0.591857 0.806043i \(-0.298395\pi\)
0.591857 + 0.806043i \(0.298395\pi\)
\(402\) 0 0
\(403\) −12.4111 −0.618241
\(404\) 0 0
\(405\) 15.0383 0.747260
\(406\) 0 0
\(407\) 1.10278 0.0546625
\(408\) 0 0
\(409\) 27.0816 1.33910 0.669551 0.742766i \(-0.266487\pi\)
0.669551 + 0.742766i \(0.266487\pi\)
\(410\) 0 0
\(411\) −31.0278 −1.53049
\(412\) 0 0
\(413\) −7.25443 −0.356967
\(414\) 0 0
\(415\) −2.20555 −0.108266
\(416\) 0 0
\(417\) 40.9583 2.00573
\(418\) 0 0
\(419\) 3.05892 0.149438 0.0747191 0.997205i \(-0.476194\pi\)
0.0747191 + 0.997205i \(0.476194\pi\)
\(420\) 0 0
\(421\) 21.6655 1.05591 0.527957 0.849271i \(-0.322958\pi\)
0.527957 + 0.849271i \(0.322958\pi\)
\(422\) 0 0
\(423\) 7.66553 0.372711
\(424\) 0 0
\(425\) −4.20555 −0.203999
\(426\) 0 0
\(427\) −3.15667 −0.152762
\(428\) 0 0
\(429\) −11.2544 −0.543369
\(430\) 0 0
\(431\) 34.3260 1.65343 0.826713 0.562623i \(-0.190208\pi\)
0.826713 + 0.562623i \(0.190208\pi\)
\(432\) 0 0
\(433\) −0.0538991 −0.00259023 −0.00129511 0.999999i \(-0.500412\pi\)
−0.00129511 + 0.999999i \(0.500412\pi\)
\(434\) 0 0
\(435\) 22.6761 1.08724
\(436\) 0 0
\(437\) −7.31386 −0.349870
\(438\) 0 0
\(439\) 7.14663 0.341090 0.170545 0.985350i \(-0.445447\pi\)
0.170545 + 0.985350i \(0.445447\pi\)
\(440\) 0 0
\(441\) 6.62721 0.315582
\(442\) 0 0
\(443\) −8.74557 −0.415515 −0.207757 0.978180i \(-0.566616\pi\)
−0.207757 + 0.978180i \(0.566616\pi\)
\(444\) 0 0
\(445\) −7.45998 −0.353637
\(446\) 0 0
\(447\) 73.2616 3.46515
\(448\) 0 0
\(449\) 39.7038 1.87374 0.936870 0.349678i \(-0.113709\pi\)
0.936870 + 0.349678i \(0.113709\pi\)
\(450\) 0 0
\(451\) −12.3572 −0.581878
\(452\) 0 0
\(453\) 35.7250 1.67851
\(454\) 0 0
\(455\) −3.62721 −0.170046
\(456\) 0 0
\(457\) 12.3133 0.575994 0.287997 0.957631i \(-0.407011\pi\)
0.287997 + 0.957631i \(0.407011\pi\)
\(458\) 0 0
\(459\) 47.3311 2.20922
\(460\) 0 0
\(461\) −15.0489 −0.700896 −0.350448 0.936582i \(-0.613971\pi\)
−0.350448 + 0.936582i \(0.613971\pi\)
\(462\) 0 0
\(463\) −31.4061 −1.45956 −0.729782 0.683680i \(-0.760378\pi\)
−0.729782 + 0.683680i \(0.760378\pi\)
\(464\) 0 0
\(465\) −10.6167 −0.492335
\(466\) 0 0
\(467\) 19.0716 0.882529 0.441264 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(468\) 0 0
\(469\) 8.41110 0.388389
\(470\) 0 0
\(471\) 50.2822 2.31688
\(472\) 0 0
\(473\) 10.6761 0.490887
\(474\) 0 0
\(475\) 8.15165 0.374023
\(476\) 0 0
\(477\) 74.9427 3.43139
\(478\) 0 0
\(479\) 9.49115 0.433662 0.216831 0.976209i \(-0.430428\pi\)
0.216831 + 0.976209i \(0.430428\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 2.78389 0.126671
\(484\) 0 0
\(485\) 2.25945 0.102596
\(486\) 0 0
\(487\) −0.0438527 −0.00198715 −0.000993577 1.00000i \(-0.500316\pi\)
−0.000993577 1.00000i \(0.500316\pi\)
\(488\) 0 0
\(489\) 19.5889 0.885841
\(490\) 0 0
\(491\) −1.56777 −0.0707527 −0.0353763 0.999374i \(-0.511263\pi\)
−0.0353763 + 0.999374i \(0.511263\pi\)
\(492\) 0 0
\(493\) 30.7355 1.38426
\(494\) 0 0
\(495\) −6.62721 −0.297871
\(496\) 0 0
\(497\) 13.0489 0.585322
\(498\) 0 0
\(499\) 42.3416 1.89547 0.947736 0.319057i \(-0.103366\pi\)
0.947736 + 0.319057i \(0.103366\pi\)
\(500\) 0 0
\(501\) −24.8222 −1.10897
\(502\) 0 0
\(503\) 36.4111 1.62349 0.811745 0.584012i \(-0.198518\pi\)
0.811745 + 0.584012i \(0.198518\pi\)
\(504\) 0 0
\(505\) 13.6655 0.608108
\(506\) 0 0
\(507\) −0.486125 −0.0215896
\(508\) 0 0
\(509\) −30.0766 −1.33312 −0.666562 0.745450i \(-0.732235\pi\)
−0.666562 + 0.745450i \(0.732235\pi\)
\(510\) 0 0
\(511\) 0.205550 0.00909300
\(512\) 0 0
\(513\) −91.7422 −4.05051
\(514\) 0 0
\(515\) 5.04888 0.222480
\(516\) 0 0
\(517\) −1.15667 −0.0508705
\(518\) 0 0
\(519\) −28.7144 −1.26042
\(520\) 0 0
\(521\) −33.2233 −1.45554 −0.727769 0.685823i \(-0.759443\pi\)
−0.727769 + 0.685823i \(0.759443\pi\)
\(522\) 0 0
\(523\) 15.5577 0.680292 0.340146 0.940373i \(-0.389524\pi\)
0.340146 + 0.940373i \(0.389524\pi\)
\(524\) 0 0
\(525\) −3.10278 −0.135416
\(526\) 0 0
\(527\) −14.3900 −0.626837
\(528\) 0 0
\(529\) −22.1950 −0.964999
\(530\) 0 0
\(531\) −48.0766 −2.08635
\(532\) 0 0
\(533\) −44.8222 −1.94147
\(534\) 0 0
\(535\) −12.9894 −0.561582
\(536\) 0 0
\(537\) 15.4811 0.668059
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 23.4161 1.00674 0.503369 0.864072i \(-0.332094\pi\)
0.503369 + 0.864072i \(0.332094\pi\)
\(542\) 0 0
\(543\) 29.9900 1.28699
\(544\) 0 0
\(545\) 5.10278 0.218579
\(546\) 0 0
\(547\) 8.41110 0.359633 0.179816 0.983700i \(-0.442450\pi\)
0.179816 + 0.983700i \(0.442450\pi\)
\(548\) 0 0
\(549\) −20.9200 −0.892842
\(550\) 0 0
\(551\) −59.5749 −2.53798
\(552\) 0 0
\(553\) −12.1517 −0.516741
\(554\) 0 0
\(555\) 3.42166 0.145242
\(556\) 0 0
\(557\) 37.0278 1.56892 0.784458 0.620182i \(-0.212941\pi\)
0.784458 + 0.620182i \(0.212941\pi\)
\(558\) 0 0
\(559\) 38.7244 1.63787
\(560\) 0 0
\(561\) −13.0489 −0.550924
\(562\) 0 0
\(563\) −22.9200 −0.965961 −0.482980 0.875631i \(-0.660446\pi\)
−0.482980 + 0.875631i \(0.660446\pi\)
\(564\) 0 0
\(565\) −10.4111 −0.437998
\(566\) 0 0
\(567\) 15.0383 0.631550
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −20.3033 −0.849667 −0.424833 0.905272i \(-0.639667\pi\)
−0.424833 + 0.905272i \(0.639667\pi\)
\(572\) 0 0
\(573\) −28.4111 −1.18689
\(574\) 0 0
\(575\) −0.897225 −0.0374169
\(576\) 0 0
\(577\) 16.4650 0.685447 0.342723 0.939436i \(-0.388651\pi\)
0.342723 + 0.939436i \(0.388651\pi\)
\(578\) 0 0
\(579\) −4.74557 −0.197219
\(580\) 0 0
\(581\) −2.20555 −0.0915016
\(582\) 0 0
\(583\) −11.3083 −0.468343
\(584\) 0 0
\(585\) −24.0383 −0.993862
\(586\) 0 0
\(587\) 30.6605 1.26549 0.632747 0.774358i \(-0.281927\pi\)
0.632747 + 0.774358i \(0.281927\pi\)
\(588\) 0 0
\(589\) 27.8922 1.14928
\(590\) 0 0
\(591\) −44.3799 −1.82555
\(592\) 0 0
\(593\) 18.3033 0.751627 0.375813 0.926695i \(-0.377363\pi\)
0.375813 + 0.926695i \(0.377363\pi\)
\(594\) 0 0
\(595\) −4.20555 −0.172411
\(596\) 0 0
\(597\) −31.3311 −1.28229
\(598\) 0 0
\(599\) −29.6555 −1.21169 −0.605845 0.795583i \(-0.707165\pi\)
−0.605845 + 0.795583i \(0.707165\pi\)
\(600\) 0 0
\(601\) 18.7783 0.765985 0.382992 0.923751i \(-0.374894\pi\)
0.382992 + 0.923751i \(0.374894\pi\)
\(602\) 0 0
\(603\) 55.7422 2.27000
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −14.6761 −0.595684 −0.297842 0.954615i \(-0.596267\pi\)
−0.297842 + 0.954615i \(0.596267\pi\)
\(608\) 0 0
\(609\) 22.6761 0.918881
\(610\) 0 0
\(611\) −4.19550 −0.169732
\(612\) 0 0
\(613\) −5.28560 −0.213483 −0.106742 0.994287i \(-0.534042\pi\)
−0.106742 + 0.994287i \(0.534042\pi\)
\(614\) 0 0
\(615\) −38.3416 −1.54608
\(616\) 0 0
\(617\) −44.6933 −1.79928 −0.899642 0.436629i \(-0.856172\pi\)
−0.899642 + 0.436629i \(0.856172\pi\)
\(618\) 0 0
\(619\) 17.5678 0.706108 0.353054 0.935603i \(-0.385143\pi\)
0.353054 + 0.935603i \(0.385143\pi\)
\(620\) 0 0
\(621\) 10.0978 0.405209
\(622\) 0 0
\(623\) −7.45998 −0.298878
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 25.2927 1.01009
\(628\) 0 0
\(629\) 4.63778 0.184920
\(630\) 0 0
\(631\) −4.74557 −0.188918 −0.0944592 0.995529i \(-0.530112\pi\)
−0.0944592 + 0.995529i \(0.530112\pi\)
\(632\) 0 0
\(633\) 33.6444 1.33724
\(634\) 0 0
\(635\) −3.25443 −0.129148
\(636\) 0 0
\(637\) −3.62721 −0.143715
\(638\) 0 0
\(639\) 86.4777 3.42100
\(640\) 0 0
\(641\) −2.52998 −0.0999281 −0.0499641 0.998751i \(-0.515911\pi\)
−0.0499641 + 0.998751i \(0.515911\pi\)
\(642\) 0 0
\(643\) 0.670549 0.0264439 0.0132219 0.999913i \(-0.495791\pi\)
0.0132219 + 0.999913i \(0.495791\pi\)
\(644\) 0 0
\(645\) 33.1255 1.30432
\(646\) 0 0
\(647\) −11.8922 −0.467531 −0.233765 0.972293i \(-0.575105\pi\)
−0.233765 + 0.972293i \(0.575105\pi\)
\(648\) 0 0
\(649\) 7.25443 0.284761
\(650\) 0 0
\(651\) −10.6167 −0.416099
\(652\) 0 0
\(653\) −11.4161 −0.446747 −0.223374 0.974733i \(-0.571707\pi\)
−0.223374 + 0.974733i \(0.571707\pi\)
\(654\) 0 0
\(655\) 17.3083 0.676292
\(656\) 0 0
\(657\) 1.36222 0.0531454
\(658\) 0 0
\(659\) −1.87108 −0.0728868 −0.0364434 0.999336i \(-0.511603\pi\)
−0.0364434 + 0.999336i \(0.511603\pi\)
\(660\) 0 0
\(661\) 33.1355 1.28882 0.644412 0.764679i \(-0.277102\pi\)
0.644412 + 0.764679i \(0.277102\pi\)
\(662\) 0 0
\(663\) −47.3311 −1.83819
\(664\) 0 0
\(665\) 8.15165 0.316107
\(666\) 0 0
\(667\) 6.55721 0.253896
\(668\) 0 0
\(669\) −29.3522 −1.13482
\(670\) 0 0
\(671\) 3.15667 0.121862
\(672\) 0 0
\(673\) −18.1955 −0.701385 −0.350693 0.936491i \(-0.614054\pi\)
−0.350693 + 0.936491i \(0.614054\pi\)
\(674\) 0 0
\(675\) −11.2544 −0.433183
\(676\) 0 0
\(677\) 15.9789 0.614118 0.307059 0.951690i \(-0.400655\pi\)
0.307059 + 0.951690i \(0.400655\pi\)
\(678\) 0 0
\(679\) 2.25945 0.0867097
\(680\) 0 0
\(681\) −54.5089 −2.08878
\(682\) 0 0
\(683\) 17.8711 0.683818 0.341909 0.939733i \(-0.388927\pi\)
0.341909 + 0.939733i \(0.388927\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) −79.8188 −3.04528
\(688\) 0 0
\(689\) −41.0177 −1.56265
\(690\) 0 0
\(691\) 2.20555 0.0839031 0.0419515 0.999120i \(-0.486642\pi\)
0.0419515 + 0.999120i \(0.486642\pi\)
\(692\) 0 0
\(693\) −6.62721 −0.251747
\(694\) 0 0
\(695\) −13.2005 −0.500725
\(696\) 0 0
\(697\) −51.9688 −1.96846
\(698\) 0 0
\(699\) −59.4389 −2.24818
\(700\) 0 0
\(701\) −25.6217 −0.967717 −0.483859 0.875146i \(-0.660765\pi\)
−0.483859 + 0.875146i \(0.660765\pi\)
\(702\) 0 0
\(703\) −8.98944 −0.339043
\(704\) 0 0
\(705\) −3.58890 −0.135166
\(706\) 0 0
\(707\) 13.6655 0.513945
\(708\) 0 0
\(709\) 5.48110 0.205847 0.102924 0.994689i \(-0.467180\pi\)
0.102924 + 0.994689i \(0.467180\pi\)
\(710\) 0 0
\(711\) −80.5316 −3.02017
\(712\) 0 0
\(713\) −3.07000 −0.114972
\(714\) 0 0
\(715\) 3.62721 0.135650
\(716\) 0 0
\(717\) −56.0172 −2.09200
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 5.04888 0.188030
\(722\) 0 0
\(723\) −19.0872 −0.709860
\(724\) 0 0
\(725\) −7.30833 −0.271424
\(726\) 0 0
\(727\) −18.5855 −0.689297 −0.344649 0.938732i \(-0.612002\pi\)
−0.344649 + 0.938732i \(0.612002\pi\)
\(728\) 0 0
\(729\) −5.09775 −0.188806
\(730\) 0 0
\(731\) 44.8988 1.66064
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 0 0
\(735\) −3.10278 −0.114448
\(736\) 0 0
\(737\) −8.41110 −0.309827
\(738\) 0 0
\(739\) −5.97887 −0.219936 −0.109968 0.993935i \(-0.535075\pi\)
−0.109968 + 0.993935i \(0.535075\pi\)
\(740\) 0 0
\(741\) 91.7422 3.37023
\(742\) 0 0
\(743\) 50.0978 1.83791 0.918954 0.394364i \(-0.129035\pi\)
0.918954 + 0.394364i \(0.129035\pi\)
\(744\) 0 0
\(745\) −23.6116 −0.865063
\(746\) 0 0
\(747\) −14.6167 −0.534795
\(748\) 0 0
\(749\) −12.9894 −0.474624
\(750\) 0 0
\(751\) 8.63778 0.315197 0.157598 0.987503i \(-0.449625\pi\)
0.157598 + 0.987503i \(0.449625\pi\)
\(752\) 0 0
\(753\) −93.0177 −3.38975
\(754\) 0 0
\(755\) −11.5139 −0.419033
\(756\) 0 0
\(757\) 25.5139 0.927318 0.463659 0.886014i \(-0.346536\pi\)
0.463659 + 0.886014i \(0.346536\pi\)
\(758\) 0 0
\(759\) −2.78389 −0.101049
\(760\) 0 0
\(761\) −32.6605 −1.18394 −0.591971 0.805959i \(-0.701650\pi\)
−0.591971 + 0.805959i \(0.701650\pi\)
\(762\) 0 0
\(763\) 5.10278 0.184733
\(764\) 0 0
\(765\) −27.8711 −1.00768
\(766\) 0 0
\(767\) 26.3133 0.950120
\(768\) 0 0
\(769\) 16.7683 0.604680 0.302340 0.953200i \(-0.402232\pi\)
0.302340 + 0.953200i \(0.402232\pi\)
\(770\) 0 0
\(771\) −12.5783 −0.452998
\(772\) 0 0
\(773\) −2.82220 −0.101507 −0.0507537 0.998711i \(-0.516162\pi\)
−0.0507537 + 0.998711i \(0.516162\pi\)
\(774\) 0 0
\(775\) 3.42166 0.122910
\(776\) 0 0
\(777\) 3.42166 0.122751
\(778\) 0 0
\(779\) 100.732 3.60908
\(780\) 0 0
\(781\) −13.0489 −0.466925
\(782\) 0 0
\(783\) 82.2510 2.93941
\(784\) 0 0
\(785\) −16.2056 −0.578401
\(786\) 0 0
\(787\) 4.82220 0.171893 0.0859464 0.996300i \(-0.472609\pi\)
0.0859464 + 0.996300i \(0.472609\pi\)
\(788\) 0 0
\(789\) −45.0177 −1.60267
\(790\) 0 0
\(791\) −10.4111 −0.370176
\(792\) 0 0
\(793\) 11.4499 0.406599
\(794\) 0 0
\(795\) −35.0872 −1.24441
\(796\) 0 0
\(797\) −0.540024 −0.0191286 −0.00956431 0.999954i \(-0.503044\pi\)
−0.00956431 + 0.999954i \(0.503044\pi\)
\(798\) 0 0
\(799\) −4.86445 −0.172092
\(800\) 0 0
\(801\) −49.4389 −1.74684
\(802\) 0 0
\(803\) −0.205550 −0.00725371
\(804\) 0 0
\(805\) −0.897225 −0.0316230
\(806\) 0 0
\(807\) −13.5678 −0.477608
\(808\) 0 0
\(809\) −17.8922 −0.629056 −0.314528 0.949248i \(-0.601846\pi\)
−0.314528 + 0.949248i \(0.601846\pi\)
\(810\) 0 0
\(811\) 12.2283 0.429393 0.214696 0.976681i \(-0.431124\pi\)
0.214696 + 0.976681i \(0.431124\pi\)
\(812\) 0 0
\(813\) −32.9411 −1.15529
\(814\) 0 0
\(815\) −6.31335 −0.221147
\(816\) 0 0
\(817\) −87.0278 −3.04472
\(818\) 0 0
\(819\) −24.0383 −0.839967
\(820\) 0 0
\(821\) 38.5316 1.34476 0.672381 0.740206i \(-0.265272\pi\)
0.672381 + 0.740206i \(0.265272\pi\)
\(822\) 0 0
\(823\) −21.9461 −0.764993 −0.382496 0.923957i \(-0.624936\pi\)
−0.382496 + 0.923957i \(0.624936\pi\)
\(824\) 0 0
\(825\) 3.10278 0.108025
\(826\) 0 0
\(827\) −9.68665 −0.336838 −0.168419 0.985716i \(-0.553866\pi\)
−0.168419 + 0.985716i \(0.553866\pi\)
\(828\) 0 0
\(829\) 31.0771 1.07935 0.539677 0.841872i \(-0.318546\pi\)
0.539677 + 0.841872i \(0.318546\pi\)
\(830\) 0 0
\(831\) 61.3210 2.12720
\(832\) 0 0
\(833\) −4.20555 −0.145714
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −38.5089 −1.33106
\(838\) 0 0
\(839\) 24.7738 0.855288 0.427644 0.903947i \(-0.359344\pi\)
0.427644 + 0.903947i \(0.359344\pi\)
\(840\) 0 0
\(841\) 24.4116 0.841780
\(842\) 0 0
\(843\) 83.9266 2.89058
\(844\) 0 0
\(845\) 0.156674 0.00538976
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 85.8399 2.94602
\(850\) 0 0
\(851\) 0.989437 0.0339175
\(852\) 0 0
\(853\) 1.58890 0.0544029 0.0272014 0.999630i \(-0.491340\pi\)
0.0272014 + 0.999630i \(0.491340\pi\)
\(854\) 0 0
\(855\) 54.0227 1.84754
\(856\) 0 0
\(857\) 19.1255 0.653315 0.326657 0.945143i \(-0.394078\pi\)
0.326657 + 0.945143i \(0.394078\pi\)
\(858\) 0 0
\(859\) −4.94108 −0.168587 −0.0842937 0.996441i \(-0.526863\pi\)
−0.0842937 + 0.996441i \(0.526863\pi\)
\(860\) 0 0
\(861\) −38.3416 −1.30668
\(862\) 0 0
\(863\) 19.3295 0.657982 0.328991 0.944333i \(-0.393291\pi\)
0.328991 + 0.944333i \(0.393291\pi\)
\(864\) 0 0
\(865\) 9.25443 0.314660
\(866\) 0 0
\(867\) −2.13053 −0.0723564
\(868\) 0 0
\(869\) 12.1517 0.412217
\(870\) 0 0
\(871\) −30.5089 −1.03375
\(872\) 0 0
\(873\) 14.9739 0.506788
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 16.4011 0.553824 0.276912 0.960895i \(-0.410689\pi\)
0.276912 + 0.960895i \(0.410689\pi\)
\(878\) 0 0
\(879\) 25.9789 0.876246
\(880\) 0 0
\(881\) 39.5678 1.33307 0.666536 0.745473i \(-0.267776\pi\)
0.666536 + 0.745473i \(0.267776\pi\)
\(882\) 0 0
\(883\) −55.4389 −1.86567 −0.932833 0.360309i \(-0.882671\pi\)
−0.932833 + 0.360309i \(0.882671\pi\)
\(884\) 0 0
\(885\) 22.5089 0.756627
\(886\) 0 0
\(887\) −44.1955 −1.48394 −0.741970 0.670433i \(-0.766108\pi\)
−0.741970 + 0.670433i \(0.766108\pi\)
\(888\) 0 0
\(889\) −3.25443 −0.109150
\(890\) 0 0
\(891\) −15.0383 −0.503802
\(892\) 0 0
\(893\) 9.42880 0.315523
\(894\) 0 0
\(895\) −4.98944 −0.166778
\(896\) 0 0
\(897\) −10.0978 −0.337154
\(898\) 0 0
\(899\) −25.0066 −0.834018
\(900\) 0 0
\(901\) −47.5577 −1.58438
\(902\) 0 0
\(903\) 33.1255 1.10235
\(904\) 0 0
\(905\) −9.66553 −0.321293
\(906\) 0 0
\(907\) −9.26447 −0.307622 −0.153811 0.988100i \(-0.549155\pi\)
−0.153811 + 0.988100i \(0.549155\pi\)
\(908\) 0 0
\(909\) 90.5644 3.00383
\(910\) 0 0
\(911\) 37.0489 1.22748 0.613742 0.789507i \(-0.289664\pi\)
0.613742 + 0.789507i \(0.289664\pi\)
\(912\) 0 0
\(913\) 2.20555 0.0729931
\(914\) 0 0
\(915\) 9.79445 0.323795
\(916\) 0 0
\(917\) 17.3083 0.571571
\(918\) 0 0
\(919\) 13.9149 0.459011 0.229506 0.973307i \(-0.426289\pi\)
0.229506 + 0.973307i \(0.426289\pi\)
\(920\) 0 0
\(921\) −31.6655 −1.04341
\(922\) 0 0
\(923\) −47.3311 −1.55792
\(924\) 0 0
\(925\) −1.10278 −0.0362590
\(926\) 0 0
\(927\) 33.4600 1.09897
\(928\) 0 0
\(929\) −32.0666 −1.05207 −0.526035 0.850463i \(-0.676322\pi\)
−0.526035 + 0.850463i \(0.676322\pi\)
\(930\) 0 0
\(931\) 8.15165 0.267160
\(932\) 0 0
\(933\) −25.3411 −0.829630
\(934\) 0 0
\(935\) 4.20555 0.137536
\(936\) 0 0
\(937\) 25.4499 0.831413 0.415706 0.909499i \(-0.363534\pi\)
0.415706 + 0.909499i \(0.363534\pi\)
\(938\) 0 0
\(939\) −107.240 −3.49966
\(940\) 0 0
\(941\) −3.49115 −0.113808 −0.0569041 0.998380i \(-0.518123\pi\)
−0.0569041 + 0.998380i \(0.518123\pi\)
\(942\) 0 0
\(943\) −11.0872 −0.361049
\(944\) 0 0
\(945\) −11.2544 −0.366107
\(946\) 0 0
\(947\) −30.6167 −0.994907 −0.497454 0.867491i \(-0.665732\pi\)
−0.497454 + 0.867491i \(0.665732\pi\)
\(948\) 0 0
\(949\) −0.745574 −0.0242024
\(950\) 0 0
\(951\) −28.5783 −0.926716
\(952\) 0 0
\(953\) −33.9406 −1.09944 −0.549721 0.835348i \(-0.685266\pi\)
−0.549721 + 0.835348i \(0.685266\pi\)
\(954\) 0 0
\(955\) 9.15667 0.296303
\(956\) 0 0
\(957\) −22.6761 −0.733014
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −19.2922 −0.622330
\(962\) 0 0
\(963\) −86.0838 −2.77401
\(964\) 0 0
\(965\) 1.52946 0.0492351
\(966\) 0 0
\(967\) 60.8021 1.95526 0.977632 0.210323i \(-0.0674515\pi\)
0.977632 + 0.210323i \(0.0674515\pi\)
\(968\) 0 0
\(969\) 106.370 3.41709
\(970\) 0 0
\(971\) −50.2933 −1.61399 −0.806994 0.590560i \(-0.798907\pi\)
−0.806994 + 0.590560i \(0.798907\pi\)
\(972\) 0 0
\(973\) −13.2005 −0.423189
\(974\) 0 0
\(975\) 11.2544 0.360430
\(976\) 0 0
\(977\) 28.0978 0.898927 0.449463 0.893299i \(-0.351615\pi\)
0.449463 + 0.893299i \(0.351615\pi\)
\(978\) 0 0
\(979\) 7.45998 0.238422
\(980\) 0 0
\(981\) 33.8172 1.07970
\(982\) 0 0
\(983\) 21.0489 0.671355 0.335677 0.941977i \(-0.391035\pi\)
0.335677 + 0.941977i \(0.391035\pi\)
\(984\) 0 0
\(985\) 14.3033 0.455741
\(986\) 0 0
\(987\) −3.58890 −0.114236
\(988\) 0 0
\(989\) 9.57885 0.304590
\(990\) 0 0
\(991\) 44.6832 1.41941 0.709705 0.704499i \(-0.248828\pi\)
0.709705 + 0.704499i \(0.248828\pi\)
\(992\) 0 0
\(993\) 50.4011 1.59943
\(994\) 0 0
\(995\) 10.0978 0.320120
\(996\) 0 0
\(997\) 9.47002 0.299919 0.149959 0.988692i \(-0.452086\pi\)
0.149959 + 0.988692i \(0.452086\pi\)
\(998\) 0 0
\(999\) 12.4111 0.392670
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 6160.2.a.bf.1.1 3
4.3 odd 2 770.2.a.m.1.3 3
12.11 even 2 6930.2.a.ce.1.1 3
20.3 even 4 3850.2.c.ba.1849.3 6
20.7 even 4 3850.2.c.ba.1849.4 6
20.19 odd 2 3850.2.a.bt.1.1 3
28.27 even 2 5390.2.a.ca.1.1 3
44.43 even 2 8470.2.a.ci.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.3 3 4.3 odd 2
3850.2.a.bt.1.1 3 20.19 odd 2
3850.2.c.ba.1849.3 6 20.3 even 4
3850.2.c.ba.1849.4 6 20.7 even 4
5390.2.a.ca.1.1 3 28.27 even 2
6160.2.a.bf.1.1 3 1.1 even 1 trivial
6930.2.a.ce.1.1 3 12.11 even 2
8470.2.a.ci.1.3 3 44.43 even 2