Properties

Label 6039.2.a.i.1.10
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.33092\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.33092 q^{2} -0.228660 q^{4} +2.36819 q^{5} -2.92169 q^{7} -2.96616 q^{8} +O(q^{10})\) \(q+1.33092 q^{2} -0.228660 q^{4} +2.36819 q^{5} -2.92169 q^{7} -2.96616 q^{8} +3.15187 q^{10} -1.00000 q^{11} +4.67063 q^{13} -3.88853 q^{14} -3.49039 q^{16} +3.08651 q^{17} -5.01988 q^{19} -0.541511 q^{20} -1.33092 q^{22} +4.73072 q^{23} +0.608331 q^{25} +6.21622 q^{26} +0.668075 q^{28} +2.14373 q^{29} +1.22908 q^{31} +1.28690 q^{32} +4.10789 q^{34} -6.91913 q^{35} -5.71888 q^{37} -6.68104 q^{38} -7.02444 q^{40} +9.30697 q^{41} -3.00386 q^{43} +0.228660 q^{44} +6.29619 q^{46} +1.19144 q^{47} +1.53629 q^{49} +0.809639 q^{50} -1.06799 q^{52} +4.66301 q^{53} -2.36819 q^{55} +8.66621 q^{56} +2.85313 q^{58} +8.94746 q^{59} -1.00000 q^{61} +1.63581 q^{62} +8.69354 q^{64} +11.0609 q^{65} +8.99807 q^{67} -0.705762 q^{68} -9.20879 q^{70} -14.8610 q^{71} +9.79980 q^{73} -7.61136 q^{74} +1.14785 q^{76} +2.92169 q^{77} +11.9696 q^{79} -8.26592 q^{80} +12.3868 q^{82} -12.9510 q^{83} +7.30945 q^{85} -3.99788 q^{86} +2.96616 q^{88} +0.425469 q^{89} -13.6462 q^{91} -1.08173 q^{92} +1.58570 q^{94} -11.8880 q^{95} +5.28118 q^{97} +2.04467 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.33092 0.941100 0.470550 0.882373i \(-0.344055\pi\)
0.470550 + 0.882373i \(0.344055\pi\)
\(3\) 0 0
\(4\) −0.228660 −0.114330
\(5\) 2.36819 1.05909 0.529544 0.848283i \(-0.322363\pi\)
0.529544 + 0.848283i \(0.322363\pi\)
\(6\) 0 0
\(7\) −2.92169 −1.10430 −0.552148 0.833746i \(-0.686192\pi\)
−0.552148 + 0.833746i \(0.686192\pi\)
\(8\) −2.96616 −1.04870
\(9\) 0 0
\(10\) 3.15187 0.996708
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.67063 1.29540 0.647700 0.761895i \(-0.275731\pi\)
0.647700 + 0.761895i \(0.275731\pi\)
\(14\) −3.88853 −1.03925
\(15\) 0 0
\(16\) −3.49039 −0.872599
\(17\) 3.08651 0.748589 0.374294 0.927310i \(-0.377885\pi\)
0.374294 + 0.927310i \(0.377885\pi\)
\(18\) 0 0
\(19\) −5.01988 −1.15164 −0.575820 0.817577i \(-0.695317\pi\)
−0.575820 + 0.817577i \(0.695317\pi\)
\(20\) −0.541511 −0.121086
\(21\) 0 0
\(22\) −1.33092 −0.283752
\(23\) 4.73072 0.986423 0.493212 0.869909i \(-0.335823\pi\)
0.493212 + 0.869909i \(0.335823\pi\)
\(24\) 0 0
\(25\) 0.608331 0.121666
\(26\) 6.21622 1.21910
\(27\) 0 0
\(28\) 0.668075 0.126254
\(29\) 2.14373 0.398081 0.199040 0.979991i \(-0.436218\pi\)
0.199040 + 0.979991i \(0.436218\pi\)
\(30\) 0 0
\(31\) 1.22908 0.220750 0.110375 0.993890i \(-0.464795\pi\)
0.110375 + 0.993890i \(0.464795\pi\)
\(32\) 1.28690 0.227494
\(33\) 0 0
\(34\) 4.10789 0.704497
\(35\) −6.91913 −1.16955
\(36\) 0 0
\(37\) −5.71888 −0.940178 −0.470089 0.882619i \(-0.655778\pi\)
−0.470089 + 0.882619i \(0.655778\pi\)
\(38\) −6.68104 −1.08381
\(39\) 0 0
\(40\) −7.02444 −1.11066
\(41\) 9.30697 1.45350 0.726752 0.686900i \(-0.241029\pi\)
0.726752 + 0.686900i \(0.241029\pi\)
\(42\) 0 0
\(43\) −3.00386 −0.458084 −0.229042 0.973417i \(-0.573559\pi\)
−0.229042 + 0.973417i \(0.573559\pi\)
\(44\) 0.228660 0.0344718
\(45\) 0 0
\(46\) 6.29619 0.928323
\(47\) 1.19144 0.173789 0.0868944 0.996218i \(-0.472306\pi\)
0.0868944 + 0.996218i \(0.472306\pi\)
\(48\) 0 0
\(49\) 1.53629 0.219470
\(50\) 0.809639 0.114500
\(51\) 0 0
\(52\) −1.06799 −0.148103
\(53\) 4.66301 0.640514 0.320257 0.947331i \(-0.396231\pi\)
0.320257 + 0.947331i \(0.396231\pi\)
\(54\) 0 0
\(55\) −2.36819 −0.319327
\(56\) 8.66621 1.15807
\(57\) 0 0
\(58\) 2.85313 0.374634
\(59\) 8.94746 1.16486 0.582430 0.812881i \(-0.302102\pi\)
0.582430 + 0.812881i \(0.302102\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 1.63581 0.207747
\(63\) 0 0
\(64\) 8.69354 1.08669
\(65\) 11.0609 1.37194
\(66\) 0 0
\(67\) 8.99807 1.09929 0.549645 0.835398i \(-0.314763\pi\)
0.549645 + 0.835398i \(0.314763\pi\)
\(68\) −0.705762 −0.0855862
\(69\) 0 0
\(70\) −9.20879 −1.10066
\(71\) −14.8610 −1.76368 −0.881838 0.471553i \(-0.843694\pi\)
−0.881838 + 0.471553i \(0.843694\pi\)
\(72\) 0 0
\(73\) 9.79980 1.14698 0.573490 0.819213i \(-0.305589\pi\)
0.573490 + 0.819213i \(0.305589\pi\)
\(74\) −7.61136 −0.884802
\(75\) 0 0
\(76\) 1.14785 0.131667
\(77\) 2.92169 0.332958
\(78\) 0 0
\(79\) 11.9696 1.34669 0.673343 0.739331i \(-0.264858\pi\)
0.673343 + 0.739331i \(0.264858\pi\)
\(80\) −8.26592 −0.924158
\(81\) 0 0
\(82\) 12.3868 1.36789
\(83\) −12.9510 −1.42156 −0.710781 0.703414i \(-0.751658\pi\)
−0.710781 + 0.703414i \(0.751658\pi\)
\(84\) 0 0
\(85\) 7.30945 0.792821
\(86\) −3.99788 −0.431103
\(87\) 0 0
\(88\) 2.96616 0.316194
\(89\) 0.425469 0.0450997 0.0225498 0.999746i \(-0.492822\pi\)
0.0225498 + 0.999746i \(0.492822\pi\)
\(90\) 0 0
\(91\) −13.6462 −1.43051
\(92\) −1.08173 −0.112778
\(93\) 0 0
\(94\) 1.58570 0.163553
\(95\) −11.8880 −1.21969
\(96\) 0 0
\(97\) 5.28118 0.536222 0.268111 0.963388i \(-0.413601\pi\)
0.268111 + 0.963388i \(0.413601\pi\)
\(98\) 2.04467 0.206543
\(99\) 0 0
\(100\) −0.139101 −0.0139101
\(101\) 18.0972 1.80074 0.900368 0.435130i \(-0.143297\pi\)
0.900368 + 0.435130i \(0.143297\pi\)
\(102\) 0 0
\(103\) 5.40800 0.532867 0.266433 0.963853i \(-0.414155\pi\)
0.266433 + 0.963853i \(0.414155\pi\)
\(104\) −13.8538 −1.35848
\(105\) 0 0
\(106\) 6.20608 0.602788
\(107\) 5.18077 0.500844 0.250422 0.968137i \(-0.419431\pi\)
0.250422 + 0.968137i \(0.419431\pi\)
\(108\) 0 0
\(109\) −11.1822 −1.07106 −0.535530 0.844516i \(-0.679888\pi\)
−0.535530 + 0.844516i \(0.679888\pi\)
\(110\) −3.15187 −0.300519
\(111\) 0 0
\(112\) 10.1979 0.963607
\(113\) 8.56972 0.806172 0.403086 0.915162i \(-0.367938\pi\)
0.403086 + 0.915162i \(0.367938\pi\)
\(114\) 0 0
\(115\) 11.2033 1.04471
\(116\) −0.490186 −0.0455126
\(117\) 0 0
\(118\) 11.9083 1.09625
\(119\) −9.01784 −0.826664
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.33092 −0.120496
\(123\) 0 0
\(124\) −0.281042 −0.0252383
\(125\) −10.4003 −0.930232
\(126\) 0 0
\(127\) 0.489906 0.0434721 0.0217361 0.999764i \(-0.493081\pi\)
0.0217361 + 0.999764i \(0.493081\pi\)
\(128\) 8.99659 0.795193
\(129\) 0 0
\(130\) 14.7212 1.29114
\(131\) 4.25776 0.372002 0.186001 0.982550i \(-0.440447\pi\)
0.186001 + 0.982550i \(0.440447\pi\)
\(132\) 0 0
\(133\) 14.6665 1.27175
\(134\) 11.9757 1.03454
\(135\) 0 0
\(136\) −9.15509 −0.785042
\(137\) 20.9037 1.78592 0.892961 0.450134i \(-0.148624\pi\)
0.892961 + 0.450134i \(0.148624\pi\)
\(138\) 0 0
\(139\) 10.2136 0.866307 0.433153 0.901320i \(-0.357401\pi\)
0.433153 + 0.901320i \(0.357401\pi\)
\(140\) 1.58213 0.133714
\(141\) 0 0
\(142\) −19.7788 −1.65980
\(143\) −4.67063 −0.390578
\(144\) 0 0
\(145\) 5.07676 0.421602
\(146\) 13.0427 1.07942
\(147\) 0 0
\(148\) 1.30768 0.107491
\(149\) 13.3117 1.09054 0.545268 0.838262i \(-0.316428\pi\)
0.545268 + 0.838262i \(0.316428\pi\)
\(150\) 0 0
\(151\) −14.1469 −1.15126 −0.575628 0.817711i \(-0.695243\pi\)
−0.575628 + 0.817711i \(0.695243\pi\)
\(152\) 14.8898 1.20772
\(153\) 0 0
\(154\) 3.88853 0.313347
\(155\) 2.91070 0.233793
\(156\) 0 0
\(157\) 15.2651 1.21829 0.609144 0.793059i \(-0.291513\pi\)
0.609144 + 0.793059i \(0.291513\pi\)
\(158\) 15.9305 1.26737
\(159\) 0 0
\(160\) 3.04762 0.240936
\(161\) −13.8217 −1.08930
\(162\) 0 0
\(163\) 18.0609 1.41464 0.707321 0.706892i \(-0.249903\pi\)
0.707321 + 0.706892i \(0.249903\pi\)
\(164\) −2.12813 −0.166179
\(165\) 0 0
\(166\) −17.2368 −1.33783
\(167\) 18.8759 1.46066 0.730329 0.683096i \(-0.239367\pi\)
0.730329 + 0.683096i \(0.239367\pi\)
\(168\) 0 0
\(169\) 8.81480 0.678061
\(170\) 9.72827 0.746124
\(171\) 0 0
\(172\) 0.686863 0.0523728
\(173\) −20.7395 −1.57680 −0.788398 0.615165i \(-0.789089\pi\)
−0.788398 + 0.615165i \(0.789089\pi\)
\(174\) 0 0
\(175\) −1.77736 −0.134356
\(176\) 3.49039 0.263098
\(177\) 0 0
\(178\) 0.566264 0.0424433
\(179\) −10.9017 −0.814834 −0.407417 0.913242i \(-0.633570\pi\)
−0.407417 + 0.913242i \(0.633570\pi\)
\(180\) 0 0
\(181\) 9.68426 0.719825 0.359913 0.932986i \(-0.382806\pi\)
0.359913 + 0.932986i \(0.382806\pi\)
\(182\) −18.1619 −1.34625
\(183\) 0 0
\(184\) −14.0321 −1.03446
\(185\) −13.5434 −0.995731
\(186\) 0 0
\(187\) −3.08651 −0.225708
\(188\) −0.272434 −0.0198693
\(189\) 0 0
\(190\) −15.8220 −1.14785
\(191\) 10.8157 0.782597 0.391298 0.920264i \(-0.372026\pi\)
0.391298 + 0.920264i \(0.372026\pi\)
\(192\) 0 0
\(193\) −6.19562 −0.445970 −0.222985 0.974822i \(-0.571580\pi\)
−0.222985 + 0.974822i \(0.571580\pi\)
\(194\) 7.02881 0.504639
\(195\) 0 0
\(196\) −0.351288 −0.0250920
\(197\) 1.14995 0.0819308 0.0409654 0.999161i \(-0.486957\pi\)
0.0409654 + 0.999161i \(0.486957\pi\)
\(198\) 0 0
\(199\) 13.8061 0.978687 0.489344 0.872091i \(-0.337236\pi\)
0.489344 + 0.872091i \(0.337236\pi\)
\(200\) −1.80441 −0.127591
\(201\) 0 0
\(202\) 24.0858 1.69467
\(203\) −6.26332 −0.439599
\(204\) 0 0
\(205\) 22.0407 1.53939
\(206\) 7.19760 0.501481
\(207\) 0 0
\(208\) −16.3023 −1.13036
\(209\) 5.01988 0.347232
\(210\) 0 0
\(211\) 8.67180 0.596991 0.298496 0.954411i \(-0.403515\pi\)
0.298496 + 0.954411i \(0.403515\pi\)
\(212\) −1.06624 −0.0732300
\(213\) 0 0
\(214\) 6.89518 0.471345
\(215\) −7.11371 −0.485151
\(216\) 0 0
\(217\) −3.59100 −0.243773
\(218\) −14.8826 −1.00797
\(219\) 0 0
\(220\) 0.541511 0.0365087
\(221\) 14.4160 0.969722
\(222\) 0 0
\(223\) 0.540497 0.0361944 0.0180972 0.999836i \(-0.494239\pi\)
0.0180972 + 0.999836i \(0.494239\pi\)
\(224\) −3.75992 −0.251220
\(225\) 0 0
\(226\) 11.4056 0.758688
\(227\) −5.60176 −0.371802 −0.185901 0.982568i \(-0.559520\pi\)
−0.185901 + 0.982568i \(0.559520\pi\)
\(228\) 0 0
\(229\) −12.8413 −0.848578 −0.424289 0.905527i \(-0.639476\pi\)
−0.424289 + 0.905527i \(0.639476\pi\)
\(230\) 14.9106 0.983176
\(231\) 0 0
\(232\) −6.35865 −0.417466
\(233\) −26.1667 −1.71423 −0.857117 0.515121i \(-0.827747\pi\)
−0.857117 + 0.515121i \(0.827747\pi\)
\(234\) 0 0
\(235\) 2.82155 0.184058
\(236\) −2.04593 −0.133179
\(237\) 0 0
\(238\) −12.0020 −0.777974
\(239\) −17.1822 −1.11142 −0.555711 0.831376i \(-0.687554\pi\)
−0.555711 + 0.831376i \(0.687554\pi\)
\(240\) 0 0
\(241\) 20.5527 1.32392 0.661959 0.749540i \(-0.269725\pi\)
0.661959 + 0.749540i \(0.269725\pi\)
\(242\) 1.33092 0.0855546
\(243\) 0 0
\(244\) 0.228660 0.0146385
\(245\) 3.63823 0.232438
\(246\) 0 0
\(247\) −23.4460 −1.49183
\(248\) −3.64565 −0.231499
\(249\) 0 0
\(250\) −13.8420 −0.875442
\(251\) −22.6512 −1.42973 −0.714865 0.699262i \(-0.753512\pi\)
−0.714865 + 0.699262i \(0.753512\pi\)
\(252\) 0 0
\(253\) −4.73072 −0.297418
\(254\) 0.652024 0.0409116
\(255\) 0 0
\(256\) −5.41338 −0.338336
\(257\) 12.2924 0.766778 0.383389 0.923587i \(-0.374757\pi\)
0.383389 + 0.923587i \(0.374757\pi\)
\(258\) 0 0
\(259\) 16.7088 1.03824
\(260\) −2.52920 −0.156854
\(261\) 0 0
\(262\) 5.66672 0.350091
\(263\) −11.9232 −0.735219 −0.367609 0.929980i \(-0.619824\pi\)
−0.367609 + 0.929980i \(0.619824\pi\)
\(264\) 0 0
\(265\) 11.0429 0.678360
\(266\) 19.5200 1.19684
\(267\) 0 0
\(268\) −2.05750 −0.125682
\(269\) 8.16149 0.497615 0.248807 0.968553i \(-0.419961\pi\)
0.248807 + 0.968553i \(0.419961\pi\)
\(270\) 0 0
\(271\) 27.6984 1.68256 0.841278 0.540602i \(-0.181804\pi\)
0.841278 + 0.540602i \(0.181804\pi\)
\(272\) −10.7731 −0.653218
\(273\) 0 0
\(274\) 27.8211 1.68073
\(275\) −0.608331 −0.0366838
\(276\) 0 0
\(277\) −1.78976 −0.107536 −0.0537681 0.998553i \(-0.517123\pi\)
−0.0537681 + 0.998553i \(0.517123\pi\)
\(278\) 13.5935 0.815281
\(279\) 0 0
\(280\) 20.5233 1.22650
\(281\) 11.9883 0.715164 0.357582 0.933882i \(-0.383601\pi\)
0.357582 + 0.933882i \(0.383601\pi\)
\(282\) 0 0
\(283\) −18.0687 −1.07407 −0.537035 0.843560i \(-0.680456\pi\)
−0.537035 + 0.843560i \(0.680456\pi\)
\(284\) 3.39812 0.201641
\(285\) 0 0
\(286\) −6.21622 −0.367573
\(287\) −27.1921 −1.60510
\(288\) 0 0
\(289\) −7.47345 −0.439615
\(290\) 6.75675 0.396770
\(291\) 0 0
\(292\) −2.24082 −0.131134
\(293\) −23.4090 −1.36757 −0.683784 0.729684i \(-0.739667\pi\)
−0.683784 + 0.729684i \(0.739667\pi\)
\(294\) 0 0
\(295\) 21.1893 1.23369
\(296\) 16.9631 0.985962
\(297\) 0 0
\(298\) 17.7168 1.02630
\(299\) 22.0954 1.27781
\(300\) 0 0
\(301\) 8.77635 0.505860
\(302\) −18.8283 −1.08345
\(303\) 0 0
\(304\) 17.5214 1.00492
\(305\) −2.36819 −0.135602
\(306\) 0 0
\(307\) 8.48031 0.483997 0.241998 0.970277i \(-0.422197\pi\)
0.241998 + 0.970277i \(0.422197\pi\)
\(308\) −0.668075 −0.0380671
\(309\) 0 0
\(310\) 3.87390 0.220023
\(311\) 16.1743 0.917160 0.458580 0.888653i \(-0.348358\pi\)
0.458580 + 0.888653i \(0.348358\pi\)
\(312\) 0 0
\(313\) 0.716595 0.0405044 0.0202522 0.999795i \(-0.493553\pi\)
0.0202522 + 0.999795i \(0.493553\pi\)
\(314\) 20.3166 1.14653
\(315\) 0 0
\(316\) −2.73697 −0.153967
\(317\) 2.26108 0.126995 0.0634974 0.997982i \(-0.479775\pi\)
0.0634974 + 0.997982i \(0.479775\pi\)
\(318\) 0 0
\(319\) −2.14373 −0.120026
\(320\) 20.5880 1.15090
\(321\) 0 0
\(322\) −18.3955 −1.02514
\(323\) −15.4939 −0.862104
\(324\) 0 0
\(325\) 2.84129 0.157607
\(326\) 24.0376 1.33132
\(327\) 0 0
\(328\) −27.6060 −1.52429
\(329\) −3.48101 −0.191914
\(330\) 0 0
\(331\) −11.8409 −0.650836 −0.325418 0.945570i \(-0.605505\pi\)
−0.325418 + 0.945570i \(0.605505\pi\)
\(332\) 2.96139 0.162527
\(333\) 0 0
\(334\) 25.1222 1.37463
\(335\) 21.3092 1.16424
\(336\) 0 0
\(337\) 30.4451 1.65845 0.829225 0.558915i \(-0.188782\pi\)
0.829225 + 0.558915i \(0.188782\pi\)
\(338\) 11.7318 0.638124
\(339\) 0 0
\(340\) −1.67138 −0.0906433
\(341\) −1.22908 −0.0665585
\(342\) 0 0
\(343\) 15.9633 0.861936
\(344\) 8.90993 0.480391
\(345\) 0 0
\(346\) −27.6026 −1.48392
\(347\) −12.9041 −0.692728 −0.346364 0.938100i \(-0.612584\pi\)
−0.346364 + 0.938100i \(0.612584\pi\)
\(348\) 0 0
\(349\) −18.0493 −0.966159 −0.483080 0.875576i \(-0.660482\pi\)
−0.483080 + 0.875576i \(0.660482\pi\)
\(350\) −2.36552 −0.126442
\(351\) 0 0
\(352\) −1.28690 −0.0685919
\(353\) −11.3839 −0.605903 −0.302951 0.953006i \(-0.597972\pi\)
−0.302951 + 0.953006i \(0.597972\pi\)
\(354\) 0 0
\(355\) −35.1937 −1.86789
\(356\) −0.0972879 −0.00515625
\(357\) 0 0
\(358\) −14.5093 −0.766841
\(359\) −3.51661 −0.185599 −0.0927997 0.995685i \(-0.529582\pi\)
−0.0927997 + 0.995685i \(0.529582\pi\)
\(360\) 0 0
\(361\) 6.19918 0.326273
\(362\) 12.8889 0.677428
\(363\) 0 0
\(364\) 3.12033 0.163550
\(365\) 23.2078 1.21475
\(366\) 0 0
\(367\) −6.46554 −0.337498 −0.168749 0.985659i \(-0.553973\pi\)
−0.168749 + 0.985659i \(0.553973\pi\)
\(368\) −16.5121 −0.860751
\(369\) 0 0
\(370\) −18.0252 −0.937083
\(371\) −13.6239 −0.707317
\(372\) 0 0
\(373\) 4.51638 0.233849 0.116925 0.993141i \(-0.462696\pi\)
0.116925 + 0.993141i \(0.462696\pi\)
\(374\) −4.10789 −0.212414
\(375\) 0 0
\(376\) −3.53399 −0.182252
\(377\) 10.0126 0.515674
\(378\) 0 0
\(379\) −26.1477 −1.34312 −0.671558 0.740952i \(-0.734375\pi\)
−0.671558 + 0.740952i \(0.734375\pi\)
\(380\) 2.71832 0.139447
\(381\) 0 0
\(382\) 14.3948 0.736502
\(383\) 23.3062 1.19089 0.595445 0.803396i \(-0.296976\pi\)
0.595445 + 0.803396i \(0.296976\pi\)
\(384\) 0 0
\(385\) 6.91913 0.352631
\(386\) −8.24585 −0.419703
\(387\) 0 0
\(388\) −1.20760 −0.0613064
\(389\) −19.3335 −0.980249 −0.490125 0.871652i \(-0.663049\pi\)
−0.490125 + 0.871652i \(0.663049\pi\)
\(390\) 0 0
\(391\) 14.6014 0.738425
\(392\) −4.55688 −0.230157
\(393\) 0 0
\(394\) 1.53049 0.0771051
\(395\) 28.3463 1.42626
\(396\) 0 0
\(397\) 8.98364 0.450876 0.225438 0.974258i \(-0.427619\pi\)
0.225438 + 0.974258i \(0.427619\pi\)
\(398\) 18.3748 0.921043
\(399\) 0 0
\(400\) −2.12332 −0.106166
\(401\) 6.20996 0.310110 0.155055 0.987906i \(-0.450444\pi\)
0.155055 + 0.987906i \(0.450444\pi\)
\(402\) 0 0
\(403\) 5.74059 0.285959
\(404\) −4.13810 −0.205878
\(405\) 0 0
\(406\) −8.33596 −0.413707
\(407\) 5.71888 0.283474
\(408\) 0 0
\(409\) 27.1966 1.34478 0.672392 0.740196i \(-0.265267\pi\)
0.672392 + 0.740196i \(0.265267\pi\)
\(410\) 29.3343 1.44872
\(411\) 0 0
\(412\) −1.23660 −0.0609227
\(413\) −26.1417 −1.28635
\(414\) 0 0
\(415\) −30.6705 −1.50556
\(416\) 6.01063 0.294695
\(417\) 0 0
\(418\) 6.68104 0.326780
\(419\) −17.1176 −0.836250 −0.418125 0.908389i \(-0.637313\pi\)
−0.418125 + 0.908389i \(0.637313\pi\)
\(420\) 0 0
\(421\) −16.4220 −0.800357 −0.400179 0.916437i \(-0.631052\pi\)
−0.400179 + 0.916437i \(0.631052\pi\)
\(422\) 11.5414 0.561829
\(423\) 0 0
\(424\) −13.8312 −0.671705
\(425\) 1.87762 0.0910780
\(426\) 0 0
\(427\) 2.92169 0.141391
\(428\) −1.18464 −0.0572615
\(429\) 0 0
\(430\) −9.46776 −0.456576
\(431\) −39.0693 −1.88190 −0.940951 0.338542i \(-0.890066\pi\)
−0.940951 + 0.338542i \(0.890066\pi\)
\(432\) 0 0
\(433\) 20.6507 0.992409 0.496205 0.868206i \(-0.334727\pi\)
0.496205 + 0.868206i \(0.334727\pi\)
\(434\) −4.77932 −0.229415
\(435\) 0 0
\(436\) 2.55692 0.122454
\(437\) −23.7476 −1.13600
\(438\) 0 0
\(439\) −36.3101 −1.73299 −0.866493 0.499189i \(-0.833631\pi\)
−0.866493 + 0.499189i \(0.833631\pi\)
\(440\) 7.02444 0.334877
\(441\) 0 0
\(442\) 19.1864 0.912606
\(443\) −3.64725 −0.173286 −0.0866430 0.996239i \(-0.527614\pi\)
−0.0866430 + 0.996239i \(0.527614\pi\)
\(444\) 0 0
\(445\) 1.00759 0.0477645
\(446\) 0.719357 0.0340625
\(447\) 0 0
\(448\) −25.3999 −1.20003
\(449\) −38.3189 −1.80838 −0.904191 0.427129i \(-0.859525\pi\)
−0.904191 + 0.427129i \(0.859525\pi\)
\(450\) 0 0
\(451\) −9.30697 −0.438248
\(452\) −1.95955 −0.0921697
\(453\) 0 0
\(454\) −7.45548 −0.349903
\(455\) −32.3167 −1.51503
\(456\) 0 0
\(457\) 10.6266 0.497093 0.248547 0.968620i \(-0.420047\pi\)
0.248547 + 0.968620i \(0.420047\pi\)
\(458\) −17.0907 −0.798597
\(459\) 0 0
\(460\) −2.56174 −0.119442
\(461\) 24.2727 1.13049 0.565246 0.824923i \(-0.308781\pi\)
0.565246 + 0.824923i \(0.308781\pi\)
\(462\) 0 0
\(463\) 5.27536 0.245167 0.122583 0.992458i \(-0.460882\pi\)
0.122583 + 0.992458i \(0.460882\pi\)
\(464\) −7.48246 −0.347365
\(465\) 0 0
\(466\) −34.8256 −1.61327
\(467\) −5.97949 −0.276698 −0.138349 0.990384i \(-0.544180\pi\)
−0.138349 + 0.990384i \(0.544180\pi\)
\(468\) 0 0
\(469\) −26.2896 −1.21394
\(470\) 3.75525 0.173217
\(471\) 0 0
\(472\) −26.5396 −1.22158
\(473\) 3.00386 0.138118
\(474\) 0 0
\(475\) −3.05375 −0.140116
\(476\) 2.06202 0.0945125
\(477\) 0 0
\(478\) −22.8680 −1.04596
\(479\) 10.7587 0.491579 0.245790 0.969323i \(-0.420953\pi\)
0.245790 + 0.969323i \(0.420953\pi\)
\(480\) 0 0
\(481\) −26.7108 −1.21791
\(482\) 27.3540 1.24594
\(483\) 0 0
\(484\) −0.228660 −0.0103936
\(485\) 12.5068 0.567907
\(486\) 0 0
\(487\) 27.3853 1.24095 0.620474 0.784227i \(-0.286940\pi\)
0.620474 + 0.784227i \(0.286940\pi\)
\(488\) 2.96616 0.134272
\(489\) 0 0
\(490\) 4.84218 0.218747
\(491\) 12.9998 0.586672 0.293336 0.956009i \(-0.405235\pi\)
0.293336 + 0.956009i \(0.405235\pi\)
\(492\) 0 0
\(493\) 6.61665 0.297999
\(494\) −31.2047 −1.40396
\(495\) 0 0
\(496\) −4.28998 −0.192626
\(497\) 43.4193 1.94762
\(498\) 0 0
\(499\) −18.5160 −0.828889 −0.414444 0.910075i \(-0.636024\pi\)
−0.414444 + 0.910075i \(0.636024\pi\)
\(500\) 2.37814 0.106354
\(501\) 0 0
\(502\) −30.1469 −1.34552
\(503\) −19.8385 −0.884556 −0.442278 0.896878i \(-0.645830\pi\)
−0.442278 + 0.896878i \(0.645830\pi\)
\(504\) 0 0
\(505\) 42.8576 1.90714
\(506\) −6.29619 −0.279900
\(507\) 0 0
\(508\) −0.112022 −0.00497017
\(509\) 9.34304 0.414123 0.207061 0.978328i \(-0.433610\pi\)
0.207061 + 0.978328i \(0.433610\pi\)
\(510\) 0 0
\(511\) −28.6320 −1.26661
\(512\) −25.1979 −1.11360
\(513\) 0 0
\(514\) 16.3601 0.721615
\(515\) 12.8072 0.564352
\(516\) 0 0
\(517\) −1.19144 −0.0523993
\(518\) 22.2380 0.977084
\(519\) 0 0
\(520\) −32.8086 −1.43875
\(521\) 18.6247 0.815961 0.407980 0.912991i \(-0.366233\pi\)
0.407980 + 0.912991i \(0.366233\pi\)
\(522\) 0 0
\(523\) 7.84831 0.343183 0.171591 0.985168i \(-0.445109\pi\)
0.171591 + 0.985168i \(0.445109\pi\)
\(524\) −0.973579 −0.0425310
\(525\) 0 0
\(526\) −15.8688 −0.691915
\(527\) 3.79357 0.165251
\(528\) 0 0
\(529\) −0.620292 −0.0269692
\(530\) 14.6972 0.638405
\(531\) 0 0
\(532\) −3.35365 −0.145399
\(533\) 43.4694 1.88287
\(534\) 0 0
\(535\) 12.2691 0.530438
\(536\) −26.6897 −1.15282
\(537\) 0 0
\(538\) 10.8623 0.468305
\(539\) −1.53629 −0.0661727
\(540\) 0 0
\(541\) −28.4103 −1.22145 −0.610726 0.791842i \(-0.709122\pi\)
−0.610726 + 0.791842i \(0.709122\pi\)
\(542\) 36.8642 1.58345
\(543\) 0 0
\(544\) 3.97203 0.170299
\(545\) −26.4816 −1.13435
\(546\) 0 0
\(547\) −30.5691 −1.30704 −0.653520 0.756909i \(-0.726708\pi\)
−0.653520 + 0.756909i \(0.726708\pi\)
\(548\) −4.77984 −0.204185
\(549\) 0 0
\(550\) −0.809639 −0.0345231
\(551\) −10.7613 −0.458445
\(552\) 0 0
\(553\) −34.9715 −1.48714
\(554\) −2.38202 −0.101202
\(555\) 0 0
\(556\) −2.33544 −0.0990449
\(557\) −28.3132 −1.19967 −0.599835 0.800124i \(-0.704767\pi\)
−0.599835 + 0.800124i \(0.704767\pi\)
\(558\) 0 0
\(559\) −14.0299 −0.593402
\(560\) 24.1505 1.02054
\(561\) 0 0
\(562\) 15.9555 0.673041
\(563\) −35.6071 −1.50066 −0.750331 0.661063i \(-0.770106\pi\)
−0.750331 + 0.661063i \(0.770106\pi\)
\(564\) 0 0
\(565\) 20.2947 0.853806
\(566\) −24.0479 −1.01081
\(567\) 0 0
\(568\) 44.0801 1.84956
\(569\) 14.0979 0.591015 0.295507 0.955340i \(-0.404511\pi\)
0.295507 + 0.955340i \(0.404511\pi\)
\(570\) 0 0
\(571\) −21.3065 −0.891648 −0.445824 0.895121i \(-0.647089\pi\)
−0.445824 + 0.895121i \(0.647089\pi\)
\(572\) 1.06799 0.0446548
\(573\) 0 0
\(574\) −36.1904 −1.51056
\(575\) 2.87785 0.120014
\(576\) 0 0
\(577\) 33.3841 1.38980 0.694900 0.719107i \(-0.255449\pi\)
0.694900 + 0.719107i \(0.255449\pi\)
\(578\) −9.94654 −0.413722
\(579\) 0 0
\(580\) −1.16085 −0.0482018
\(581\) 37.8390 1.56982
\(582\) 0 0
\(583\) −4.66301 −0.193122
\(584\) −29.0678 −1.20283
\(585\) 0 0
\(586\) −31.1554 −1.28702
\(587\) −7.08280 −0.292338 −0.146169 0.989260i \(-0.546694\pi\)
−0.146169 + 0.989260i \(0.546694\pi\)
\(588\) 0 0
\(589\) −6.16984 −0.254224
\(590\) 28.2012 1.16103
\(591\) 0 0
\(592\) 19.9612 0.820398
\(593\) 24.1694 0.992519 0.496260 0.868174i \(-0.334706\pi\)
0.496260 + 0.868174i \(0.334706\pi\)
\(594\) 0 0
\(595\) −21.3560 −0.875509
\(596\) −3.04385 −0.124681
\(597\) 0 0
\(598\) 29.4072 1.20255
\(599\) −15.8654 −0.648242 −0.324121 0.946016i \(-0.605068\pi\)
−0.324121 + 0.946016i \(0.605068\pi\)
\(600\) 0 0
\(601\) −21.4050 −0.873128 −0.436564 0.899673i \(-0.643805\pi\)
−0.436564 + 0.899673i \(0.643805\pi\)
\(602\) 11.6806 0.476065
\(603\) 0 0
\(604\) 3.23483 0.131623
\(605\) 2.36819 0.0962807
\(606\) 0 0
\(607\) 37.9016 1.53838 0.769189 0.639021i \(-0.220660\pi\)
0.769189 + 0.639021i \(0.220660\pi\)
\(608\) −6.46007 −0.261991
\(609\) 0 0
\(610\) −3.15187 −0.127615
\(611\) 5.56476 0.225126
\(612\) 0 0
\(613\) −5.28282 −0.213371 −0.106686 0.994293i \(-0.534024\pi\)
−0.106686 + 0.994293i \(0.534024\pi\)
\(614\) 11.2866 0.455489
\(615\) 0 0
\(616\) −8.66621 −0.349172
\(617\) 44.2406 1.78106 0.890529 0.454926i \(-0.150334\pi\)
0.890529 + 0.454926i \(0.150334\pi\)
\(618\) 0 0
\(619\) 23.8962 0.960468 0.480234 0.877140i \(-0.340552\pi\)
0.480234 + 0.877140i \(0.340552\pi\)
\(620\) −0.665561 −0.0267296
\(621\) 0 0
\(622\) 21.5266 0.863139
\(623\) −1.24309 −0.0498034
\(624\) 0 0
\(625\) −27.6716 −1.10686
\(626\) 0.953729 0.0381187
\(627\) 0 0
\(628\) −3.49052 −0.139287
\(629\) −17.6514 −0.703807
\(630\) 0 0
\(631\) −4.59242 −0.182821 −0.0914106 0.995813i \(-0.529138\pi\)
−0.0914106 + 0.995813i \(0.529138\pi\)
\(632\) −35.5038 −1.41226
\(633\) 0 0
\(634\) 3.00931 0.119515
\(635\) 1.16019 0.0460408
\(636\) 0 0
\(637\) 7.17544 0.284301
\(638\) −2.85313 −0.112956
\(639\) 0 0
\(640\) 21.3056 0.842179
\(641\) 20.0957 0.793731 0.396865 0.917877i \(-0.370098\pi\)
0.396865 + 0.917877i \(0.370098\pi\)
\(642\) 0 0
\(643\) 25.7989 1.01741 0.508705 0.860941i \(-0.330124\pi\)
0.508705 + 0.860941i \(0.330124\pi\)
\(644\) 3.16047 0.124540
\(645\) 0 0
\(646\) −20.6211 −0.811327
\(647\) 44.0280 1.73092 0.865460 0.500977i \(-0.167026\pi\)
0.865460 + 0.500977i \(0.167026\pi\)
\(648\) 0 0
\(649\) −8.94746 −0.351219
\(650\) 3.78152 0.148324
\(651\) 0 0
\(652\) −4.12982 −0.161736
\(653\) 5.99511 0.234607 0.117303 0.993096i \(-0.462575\pi\)
0.117303 + 0.993096i \(0.462575\pi\)
\(654\) 0 0
\(655\) 10.0832 0.393982
\(656\) −32.4850 −1.26833
\(657\) 0 0
\(658\) −4.63294 −0.180611
\(659\) −36.8221 −1.43439 −0.717193 0.696874i \(-0.754574\pi\)
−0.717193 + 0.696874i \(0.754574\pi\)
\(660\) 0 0
\(661\) 34.6404 1.34736 0.673678 0.739025i \(-0.264714\pi\)
0.673678 + 0.739025i \(0.264714\pi\)
\(662\) −15.7593 −0.612502
\(663\) 0 0
\(664\) 38.4149 1.49079
\(665\) 34.7332 1.34690
\(666\) 0 0
\(667\) 10.1414 0.392676
\(668\) −4.31616 −0.166997
\(669\) 0 0
\(670\) 28.3607 1.09567
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −35.1235 −1.35391 −0.676955 0.736024i \(-0.736701\pi\)
−0.676955 + 0.736024i \(0.736701\pi\)
\(674\) 40.5199 1.56077
\(675\) 0 0
\(676\) −2.01559 −0.0775228
\(677\) −46.6369 −1.79240 −0.896200 0.443650i \(-0.853683\pi\)
−0.896200 + 0.443650i \(0.853683\pi\)
\(678\) 0 0
\(679\) −15.4300 −0.592148
\(680\) −21.6810 −0.831429
\(681\) 0 0
\(682\) −1.63581 −0.0626382
\(683\) 26.3591 1.00860 0.504301 0.863528i \(-0.331750\pi\)
0.504301 + 0.863528i \(0.331750\pi\)
\(684\) 0 0
\(685\) 49.5039 1.89145
\(686\) 21.2458 0.811169
\(687\) 0 0
\(688\) 10.4846 0.399723
\(689\) 21.7792 0.829722
\(690\) 0 0
\(691\) −43.4335 −1.65229 −0.826145 0.563458i \(-0.809471\pi\)
−0.826145 + 0.563458i \(0.809471\pi\)
\(692\) 4.74230 0.180275
\(693\) 0 0
\(694\) −17.1743 −0.651926
\(695\) 24.1878 0.917494
\(696\) 0 0
\(697\) 28.7261 1.08808
\(698\) −24.0222 −0.909253
\(699\) 0 0
\(700\) 0.406411 0.0153609
\(701\) 29.6826 1.12110 0.560548 0.828122i \(-0.310591\pi\)
0.560548 + 0.828122i \(0.310591\pi\)
\(702\) 0 0
\(703\) 28.7081 1.08275
\(704\) −8.69354 −0.327650
\(705\) 0 0
\(706\) −15.1510 −0.570215
\(707\) −52.8744 −1.98855
\(708\) 0 0
\(709\) 32.3827 1.21616 0.608079 0.793877i \(-0.291940\pi\)
0.608079 + 0.793877i \(0.291940\pi\)
\(710\) −46.8399 −1.75787
\(711\) 0 0
\(712\) −1.26201 −0.0472959
\(713\) 5.81444 0.217752
\(714\) 0 0
\(715\) −11.0609 −0.413656
\(716\) 2.49279 0.0931600
\(717\) 0 0
\(718\) −4.68031 −0.174668
\(719\) 38.1158 1.42148 0.710741 0.703454i \(-0.248360\pi\)
0.710741 + 0.703454i \(0.248360\pi\)
\(720\) 0 0
\(721\) −15.8005 −0.588442
\(722\) 8.25059 0.307055
\(723\) 0 0
\(724\) −2.21440 −0.0822977
\(725\) 1.30410 0.0484330
\(726\) 0 0
\(727\) −32.9791 −1.22313 −0.611564 0.791195i \(-0.709459\pi\)
−0.611564 + 0.791195i \(0.709459\pi\)
\(728\) 40.4767 1.50017
\(729\) 0 0
\(730\) 30.8877 1.14320
\(731\) −9.27144 −0.342917
\(732\) 0 0
\(733\) 22.0044 0.812751 0.406376 0.913706i \(-0.366792\pi\)
0.406376 + 0.913706i \(0.366792\pi\)
\(734\) −8.60510 −0.317620
\(735\) 0 0
\(736\) 6.08796 0.224405
\(737\) −8.99807 −0.331448
\(738\) 0 0
\(739\) 2.36992 0.0871789 0.0435894 0.999050i \(-0.486121\pi\)
0.0435894 + 0.999050i \(0.486121\pi\)
\(740\) 3.09684 0.113842
\(741\) 0 0
\(742\) −18.1323 −0.665656
\(743\) −22.5722 −0.828094 −0.414047 0.910255i \(-0.635885\pi\)
−0.414047 + 0.910255i \(0.635885\pi\)
\(744\) 0 0
\(745\) 31.5246 1.15497
\(746\) 6.01092 0.220076
\(747\) 0 0
\(748\) 0.705762 0.0258052
\(749\) −15.1366 −0.553080
\(750\) 0 0
\(751\) 4.61181 0.168287 0.0841437 0.996454i \(-0.473185\pi\)
0.0841437 + 0.996454i \(0.473185\pi\)
\(752\) −4.15858 −0.151648
\(753\) 0 0
\(754\) 13.3259 0.485301
\(755\) −33.5025 −1.21928
\(756\) 0 0
\(757\) −2.35032 −0.0854237 −0.0427119 0.999087i \(-0.513600\pi\)
−0.0427119 + 0.999087i \(0.513600\pi\)
\(758\) −34.8004 −1.26401
\(759\) 0 0
\(760\) 35.2618 1.27908
\(761\) −45.2527 −1.64041 −0.820204 0.572071i \(-0.806140\pi\)
−0.820204 + 0.572071i \(0.806140\pi\)
\(762\) 0 0
\(763\) 32.6709 1.18277
\(764\) −2.47312 −0.0894744
\(765\) 0 0
\(766\) 31.0186 1.12075
\(767\) 41.7903 1.50896
\(768\) 0 0
\(769\) −19.5824 −0.706158 −0.353079 0.935594i \(-0.614865\pi\)
−0.353079 + 0.935594i \(0.614865\pi\)
\(770\) 9.20879 0.331862
\(771\) 0 0
\(772\) 1.41669 0.0509878
\(773\) 20.2345 0.727785 0.363892 0.931441i \(-0.381448\pi\)
0.363892 + 0.931441i \(0.381448\pi\)
\(774\) 0 0
\(775\) 0.747689 0.0268578
\(776\) −15.6648 −0.562335
\(777\) 0 0
\(778\) −25.7313 −0.922513
\(779\) −46.7199 −1.67391
\(780\) 0 0
\(781\) 14.8610 0.531768
\(782\) 19.4333 0.694932
\(783\) 0 0
\(784\) −5.36226 −0.191509
\(785\) 36.1507 1.29027
\(786\) 0 0
\(787\) 19.3105 0.688344 0.344172 0.938907i \(-0.388160\pi\)
0.344172 + 0.938907i \(0.388160\pi\)
\(788\) −0.262949 −0.00936716
\(789\) 0 0
\(790\) 37.7266 1.34225
\(791\) −25.0381 −0.890252
\(792\) 0 0
\(793\) −4.67063 −0.165859
\(794\) 11.9565 0.424320
\(795\) 0 0
\(796\) −3.15690 −0.111893
\(797\) −12.0619 −0.427253 −0.213627 0.976915i \(-0.568528\pi\)
−0.213627 + 0.976915i \(0.568528\pi\)
\(798\) 0 0
\(799\) 3.67738 0.130096
\(800\) 0.782861 0.0276783
\(801\) 0 0
\(802\) 8.26494 0.291845
\(803\) −9.79980 −0.345827
\(804\) 0 0
\(805\) −32.7325 −1.15367
\(806\) 7.64024 0.269116
\(807\) 0 0
\(808\) −53.6791 −1.88843
\(809\) −35.8603 −1.26078 −0.630390 0.776278i \(-0.717105\pi\)
−0.630390 + 0.776278i \(0.717105\pi\)
\(810\) 0 0
\(811\) 43.1282 1.51444 0.757218 0.653163i \(-0.226558\pi\)
0.757218 + 0.653163i \(0.226558\pi\)
\(812\) 1.43217 0.0502594
\(813\) 0 0
\(814\) 7.61136 0.266778
\(815\) 42.7718 1.49823
\(816\) 0 0
\(817\) 15.0790 0.527547
\(818\) 36.1964 1.26558
\(819\) 0 0
\(820\) −5.03983 −0.175998
\(821\) 21.6892 0.756957 0.378479 0.925610i \(-0.376447\pi\)
0.378479 + 0.925610i \(0.376447\pi\)
\(822\) 0 0
\(823\) 6.12406 0.213471 0.106736 0.994287i \(-0.465960\pi\)
0.106736 + 0.994287i \(0.465960\pi\)
\(824\) −16.0410 −0.558815
\(825\) 0 0
\(826\) −34.7925 −1.21059
\(827\) −50.6057 −1.75973 −0.879866 0.475221i \(-0.842368\pi\)
−0.879866 + 0.475221i \(0.842368\pi\)
\(828\) 0 0
\(829\) −15.7599 −0.547365 −0.273682 0.961820i \(-0.588242\pi\)
−0.273682 + 0.961820i \(0.588242\pi\)
\(830\) −40.8199 −1.41688
\(831\) 0 0
\(832\) 40.6043 1.40770
\(833\) 4.74178 0.164293
\(834\) 0 0
\(835\) 44.7016 1.54696
\(836\) −1.14785 −0.0396991
\(837\) 0 0
\(838\) −22.7821 −0.786996
\(839\) 20.8938 0.721334 0.360667 0.932695i \(-0.382549\pi\)
0.360667 + 0.932695i \(0.382549\pi\)
\(840\) 0 0
\(841\) −24.4044 −0.841532
\(842\) −21.8563 −0.753217
\(843\) 0 0
\(844\) −1.98290 −0.0682541
\(845\) 20.8751 0.718126
\(846\) 0 0
\(847\) −2.92169 −0.100391
\(848\) −16.2757 −0.558911
\(849\) 0 0
\(850\) 2.49896 0.0857136
\(851\) −27.0544 −0.927414
\(852\) 0 0
\(853\) 19.2873 0.660383 0.330192 0.943914i \(-0.392887\pi\)
0.330192 + 0.943914i \(0.392887\pi\)
\(854\) 3.88853 0.133063
\(855\) 0 0
\(856\) −15.3670 −0.525233
\(857\) −4.94606 −0.168954 −0.0844771 0.996425i \(-0.526922\pi\)
−0.0844771 + 0.996425i \(0.526922\pi\)
\(858\) 0 0
\(859\) −28.0355 −0.956558 −0.478279 0.878208i \(-0.658739\pi\)
−0.478279 + 0.878208i \(0.658739\pi\)
\(860\) 1.62662 0.0554674
\(861\) 0 0
\(862\) −51.9980 −1.77106
\(863\) 52.8568 1.79927 0.899634 0.436645i \(-0.143833\pi\)
0.899634 + 0.436645i \(0.143833\pi\)
\(864\) 0 0
\(865\) −49.1152 −1.66997
\(866\) 27.4844 0.933957
\(867\) 0 0
\(868\) 0.821118 0.0278706
\(869\) −11.9696 −0.406041
\(870\) 0 0
\(871\) 42.0267 1.42402
\(872\) 33.1682 1.12322
\(873\) 0 0
\(874\) −31.6061 −1.06909
\(875\) 30.3865 1.02725
\(876\) 0 0
\(877\) 2.79315 0.0943180 0.0471590 0.998887i \(-0.484983\pi\)
0.0471590 + 0.998887i \(0.484983\pi\)
\(878\) −48.3257 −1.63091
\(879\) 0 0
\(880\) 8.26592 0.278644
\(881\) −21.8113 −0.734843 −0.367421 0.930055i \(-0.619759\pi\)
−0.367421 + 0.930055i \(0.619759\pi\)
\(882\) 0 0
\(883\) −6.55226 −0.220501 −0.110251 0.993904i \(-0.535165\pi\)
−0.110251 + 0.993904i \(0.535165\pi\)
\(884\) −3.29635 −0.110868
\(885\) 0 0
\(886\) −4.85418 −0.163080
\(887\) −17.6112 −0.591325 −0.295662 0.955292i \(-0.595540\pi\)
−0.295662 + 0.955292i \(0.595540\pi\)
\(888\) 0 0
\(889\) −1.43135 −0.0480061
\(890\) 1.34102 0.0449512
\(891\) 0 0
\(892\) −0.123590 −0.00413811
\(893\) −5.98087 −0.200142
\(894\) 0 0
\(895\) −25.8174 −0.862980
\(896\) −26.2853 −0.878129
\(897\) 0 0
\(898\) −50.9993 −1.70187
\(899\) 2.63482 0.0878761
\(900\) 0 0
\(901\) 14.3924 0.479481
\(902\) −12.3868 −0.412435
\(903\) 0 0
\(904\) −25.4192 −0.845429
\(905\) 22.9342 0.762358
\(906\) 0 0
\(907\) −31.0718 −1.03172 −0.515861 0.856672i \(-0.672528\pi\)
−0.515861 + 0.856672i \(0.672528\pi\)
\(908\) 1.28090 0.0425082
\(909\) 0 0
\(910\) −43.0108 −1.42580
\(911\) 3.53312 0.117058 0.0585288 0.998286i \(-0.481359\pi\)
0.0585288 + 0.998286i \(0.481359\pi\)
\(912\) 0 0
\(913\) 12.9510 0.428617
\(914\) 14.1432 0.467815
\(915\) 0 0
\(916\) 2.93630 0.0970180
\(917\) −12.4399 −0.410800
\(918\) 0 0
\(919\) 39.8129 1.31331 0.656653 0.754193i \(-0.271972\pi\)
0.656653 + 0.754193i \(0.271972\pi\)
\(920\) −33.2307 −1.09558
\(921\) 0 0
\(922\) 32.3049 1.06391
\(923\) −69.4102 −2.28467
\(924\) 0 0
\(925\) −3.47898 −0.114388
\(926\) 7.02106 0.230726
\(927\) 0 0
\(928\) 2.75876 0.0905608
\(929\) −26.4621 −0.868193 −0.434097 0.900866i \(-0.642932\pi\)
−0.434097 + 0.900866i \(0.642932\pi\)
\(930\) 0 0
\(931\) −7.71199 −0.252750
\(932\) 5.98327 0.195989
\(933\) 0 0
\(934\) −7.95821 −0.260401
\(935\) −7.30945 −0.239045
\(936\) 0 0
\(937\) −5.91019 −0.193077 −0.0965387 0.995329i \(-0.530777\pi\)
−0.0965387 + 0.995329i \(0.530777\pi\)
\(938\) −34.9893 −1.14244
\(939\) 0 0
\(940\) −0.645176 −0.0210433
\(941\) 41.2125 1.34349 0.671744 0.740784i \(-0.265546\pi\)
0.671744 + 0.740784i \(0.265546\pi\)
\(942\) 0 0
\(943\) 44.0287 1.43377
\(944\) −31.2302 −1.01646
\(945\) 0 0
\(946\) 3.99788 0.129982
\(947\) −3.35032 −0.108871 −0.0544354 0.998517i \(-0.517336\pi\)
−0.0544354 + 0.998517i \(0.517336\pi\)
\(948\) 0 0
\(949\) 45.7712 1.48580
\(950\) −4.06429 −0.131863
\(951\) 0 0
\(952\) 26.7484 0.866919
\(953\) 7.53957 0.244231 0.122115 0.992516i \(-0.461032\pi\)
0.122115 + 0.992516i \(0.461032\pi\)
\(954\) 0 0
\(955\) 25.6137 0.828838
\(956\) 3.92888 0.127069
\(957\) 0 0
\(958\) 14.3190 0.462626
\(959\) −61.0741 −1.97219
\(960\) 0 0
\(961\) −29.4894 −0.951270
\(962\) −35.5498 −1.14617
\(963\) 0 0
\(964\) −4.69959 −0.151364
\(965\) −14.6724 −0.472322
\(966\) 0 0
\(967\) −36.1572 −1.16274 −0.581369 0.813640i \(-0.697483\pi\)
−0.581369 + 0.813640i \(0.697483\pi\)
\(968\) −2.96616 −0.0953360
\(969\) 0 0
\(970\) 16.6456 0.534457
\(971\) 20.8300 0.668468 0.334234 0.942490i \(-0.391522\pi\)
0.334234 + 0.942490i \(0.391522\pi\)
\(972\) 0 0
\(973\) −29.8410 −0.956659
\(974\) 36.4476 1.16786
\(975\) 0 0
\(976\) 3.49039 0.111725
\(977\) −23.9880 −0.767444 −0.383722 0.923449i \(-0.625358\pi\)
−0.383722 + 0.923449i \(0.625358\pi\)
\(978\) 0 0
\(979\) −0.425469 −0.0135981
\(980\) −0.831918 −0.0265746
\(981\) 0 0
\(982\) 17.3016 0.552117
\(983\) 1.23884 0.0395129 0.0197565 0.999805i \(-0.493711\pi\)
0.0197565 + 0.999805i \(0.493711\pi\)
\(984\) 0 0
\(985\) 2.72331 0.0867719
\(986\) 8.80621 0.280447
\(987\) 0 0
\(988\) 5.36117 0.170561
\(989\) −14.2104 −0.451865
\(990\) 0 0
\(991\) 2.76799 0.0879281 0.0439641 0.999033i \(-0.486001\pi\)
0.0439641 + 0.999033i \(0.486001\pi\)
\(992\) 1.58170 0.0502191
\(993\) 0 0
\(994\) 57.7874 1.83291
\(995\) 32.6955 1.03652
\(996\) 0 0
\(997\) 27.1886 0.861071 0.430536 0.902574i \(-0.358325\pi\)
0.430536 + 0.902574i \(0.358325\pi\)
\(998\) −24.6432 −0.780068
\(999\) 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 6039.2.a.i.1.10 13
3.2 odd 2 2013.2.a.e.1.4 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.4 13 3.2 odd 2
6039.2.a.i.1.10 13 1.1 even 1 trivial