Properties

Label 2004.2.a.b.1.3
Level $2004$
Weight $2$
Character 2004.1
Self dual yes
Analytic conductor $16.002$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0020205651\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.261082\) of defining polynomial
Character \(\chi\) \(=\) 2004.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.38924 q^{5} -0.871845 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.38924 q^{5} -0.871845 q^{7} +1.00000 q^{9} -5.74903 q^{11} +3.90849 q^{13} -1.38924 q^{15} -0.477836 q^{17} +2.16161 q^{19} -0.871845 q^{21} +3.53227 q^{23} -3.07002 q^{25} +1.00000 q^{27} -5.05174 q^{29} +3.41763 q^{31} -5.74903 q^{33} +1.21120 q^{35} -4.98967 q^{37} +3.90849 q^{39} -5.50706 q^{41} -3.83237 q^{43} -1.38924 q^{45} -13.6631 q^{47} -6.23989 q^{49} -0.477836 q^{51} -11.6543 q^{53} +7.98676 q^{55} +2.16161 q^{57} -0.528261 q^{59} +1.27251 q^{61} -0.871845 q^{63} -5.42981 q^{65} +8.13778 q^{67} +3.53227 q^{69} +4.17968 q^{71} -6.12260 q^{73} -3.07002 q^{75} +5.01226 q^{77} -15.4461 q^{79} +1.00000 q^{81} -1.01912 q^{83} +0.663827 q^{85} -5.05174 q^{87} -9.40919 q^{89} -3.40760 q^{91} +3.41763 q^{93} -3.00299 q^{95} -0.321073 q^{97} -5.74903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} - 7q^{5} - 2q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} - 7q^{5} - 2q^{7} + 5q^{9} - 5q^{11} - 8q^{13} - 7q^{15} - 7q^{17} + 2q^{19} - 2q^{21} - 13q^{23} + 2q^{25} + 5q^{27} - 11q^{29} - 12q^{31} - 5q^{33} - 12q^{35} - 7q^{37} - 8q^{39} - 12q^{41} - 7q^{45} - 19q^{47} - 9q^{49} - 7q^{51} - 21q^{53} - q^{55} + 2q^{57} - 7q^{59} - 6q^{61} - 2q^{63} + 14q^{65} + 10q^{67} - 13q^{69} - 35q^{71} - 8q^{73} + 2q^{75} - 6q^{77} + 5q^{81} - 11q^{83} + 5q^{85} - 11q^{87} - 32q^{89} + 5q^{91} - 12q^{93} - 19q^{95} + 11q^{97} - 5q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.38924 −0.621285 −0.310643 0.950527i \(-0.600544\pi\)
−0.310643 + 0.950527i \(0.600544\pi\)
\(6\) 0 0
\(7\) −0.871845 −0.329527 −0.164763 0.986333i \(-0.552686\pi\)
−0.164763 + 0.986333i \(0.552686\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.74903 −1.73340 −0.866698 0.498833i \(-0.833762\pi\)
−0.866698 + 0.498833i \(0.833762\pi\)
\(12\) 0 0
\(13\) 3.90849 1.08402 0.542009 0.840372i \(-0.317664\pi\)
0.542009 + 0.840372i \(0.317664\pi\)
\(14\) 0 0
\(15\) −1.38924 −0.358699
\(16\) 0 0
\(17\) −0.477836 −0.115892 −0.0579461 0.998320i \(-0.518455\pi\)
−0.0579461 + 0.998320i \(0.518455\pi\)
\(18\) 0 0
\(19\) 2.16161 0.495908 0.247954 0.968772i \(-0.420242\pi\)
0.247954 + 0.968772i \(0.420242\pi\)
\(20\) 0 0
\(21\) −0.871845 −0.190252
\(22\) 0 0
\(23\) 3.53227 0.736530 0.368265 0.929721i \(-0.379952\pi\)
0.368265 + 0.929721i \(0.379952\pi\)
\(24\) 0 0
\(25\) −3.07002 −0.614004
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.05174 −0.938085 −0.469042 0.883176i \(-0.655401\pi\)
−0.469042 + 0.883176i \(0.655401\pi\)
\(30\) 0 0
\(31\) 3.41763 0.613824 0.306912 0.951738i \(-0.400704\pi\)
0.306912 + 0.951738i \(0.400704\pi\)
\(32\) 0 0
\(33\) −5.74903 −1.00078
\(34\) 0 0
\(35\) 1.21120 0.204730
\(36\) 0 0
\(37\) −4.98967 −0.820297 −0.410149 0.912019i \(-0.634523\pi\)
−0.410149 + 0.912019i \(0.634523\pi\)
\(38\) 0 0
\(39\) 3.90849 0.625859
\(40\) 0 0
\(41\) −5.50706 −0.860059 −0.430029 0.902815i \(-0.641497\pi\)
−0.430029 + 0.902815i \(0.641497\pi\)
\(42\) 0 0
\(43\) −3.83237 −0.584431 −0.292215 0.956353i \(-0.594392\pi\)
−0.292215 + 0.956353i \(0.594392\pi\)
\(44\) 0 0
\(45\) −1.38924 −0.207095
\(46\) 0 0
\(47\) −13.6631 −1.99297 −0.996486 0.0837634i \(-0.973306\pi\)
−0.996486 + 0.0837634i \(0.973306\pi\)
\(48\) 0 0
\(49\) −6.23989 −0.891412
\(50\) 0 0
\(51\) −0.477836 −0.0669104
\(52\) 0 0
\(53\) −11.6543 −1.60085 −0.800423 0.599436i \(-0.795392\pi\)
−0.800423 + 0.599436i \(0.795392\pi\)
\(54\) 0 0
\(55\) 7.98676 1.07693
\(56\) 0 0
\(57\) 2.16161 0.286313
\(58\) 0 0
\(59\) −0.528261 −0.0687737 −0.0343868 0.999409i \(-0.510948\pi\)
−0.0343868 + 0.999409i \(0.510948\pi\)
\(60\) 0 0
\(61\) 1.27251 0.162929 0.0814643 0.996676i \(-0.474040\pi\)
0.0814643 + 0.996676i \(0.474040\pi\)
\(62\) 0 0
\(63\) −0.871845 −0.109842
\(64\) 0 0
\(65\) −5.42981 −0.673485
\(66\) 0 0
\(67\) 8.13778 0.994188 0.497094 0.867697i \(-0.334400\pi\)
0.497094 + 0.867697i \(0.334400\pi\)
\(68\) 0 0
\(69\) 3.53227 0.425236
\(70\) 0 0
\(71\) 4.17968 0.496036 0.248018 0.968755i \(-0.420221\pi\)
0.248018 + 0.968755i \(0.420221\pi\)
\(72\) 0 0
\(73\) −6.12260 −0.716596 −0.358298 0.933607i \(-0.616643\pi\)
−0.358298 + 0.933607i \(0.616643\pi\)
\(74\) 0 0
\(75\) −3.07002 −0.354496
\(76\) 0 0
\(77\) 5.01226 0.571200
\(78\) 0 0
\(79\) −15.4461 −1.73782 −0.868911 0.494969i \(-0.835179\pi\)
−0.868911 + 0.494969i \(0.835179\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.01912 −0.111863 −0.0559315 0.998435i \(-0.517813\pi\)
−0.0559315 + 0.998435i \(0.517813\pi\)
\(84\) 0 0
\(85\) 0.663827 0.0720022
\(86\) 0 0
\(87\) −5.05174 −0.541603
\(88\) 0 0
\(89\) −9.40919 −0.997372 −0.498686 0.866783i \(-0.666184\pi\)
−0.498686 + 0.866783i \(0.666184\pi\)
\(90\) 0 0
\(91\) −3.40760 −0.357213
\(92\) 0 0
\(93\) 3.41763 0.354391
\(94\) 0 0
\(95\) −3.00299 −0.308101
\(96\) 0 0
\(97\) −0.321073 −0.0326000 −0.0163000 0.999867i \(-0.505189\pi\)
−0.0163000 + 0.999867i \(0.505189\pi\)
\(98\) 0 0
\(99\) −5.74903 −0.577799
\(100\) 0 0
\(101\) 14.5077 1.44357 0.721784 0.692119i \(-0.243322\pi\)
0.721784 + 0.692119i \(0.243322\pi\)
\(102\) 0 0
\(103\) 6.14201 0.605190 0.302595 0.953119i \(-0.402147\pi\)
0.302595 + 0.953119i \(0.402147\pi\)
\(104\) 0 0
\(105\) 1.21120 0.118201
\(106\) 0 0
\(107\) 4.05659 0.392165 0.196083 0.980587i \(-0.437178\pi\)
0.196083 + 0.980587i \(0.437178\pi\)
\(108\) 0 0
\(109\) −17.8310 −1.70790 −0.853951 0.520353i \(-0.825800\pi\)
−0.853951 + 0.520353i \(0.825800\pi\)
\(110\) 0 0
\(111\) −4.98967 −0.473599
\(112\) 0 0
\(113\) 1.27411 0.119858 0.0599289 0.998203i \(-0.480913\pi\)
0.0599289 + 0.998203i \(0.480913\pi\)
\(114\) 0 0
\(115\) −4.90716 −0.457595
\(116\) 0 0
\(117\) 3.90849 0.361340
\(118\) 0 0
\(119\) 0.416599 0.0381896
\(120\) 0 0
\(121\) 22.0513 2.00466
\(122\) 0 0
\(123\) −5.50706 −0.496555
\(124\) 0 0
\(125\) 11.2112 1.00276
\(126\) 0 0
\(127\) −2.67900 −0.237723 −0.118862 0.992911i \(-0.537924\pi\)
−0.118862 + 0.992911i \(0.537924\pi\)
\(128\) 0 0
\(129\) −3.83237 −0.337421
\(130\) 0 0
\(131\) 5.43675 0.475011 0.237505 0.971386i \(-0.423670\pi\)
0.237505 + 0.971386i \(0.423670\pi\)
\(132\) 0 0
\(133\) −1.88459 −0.163415
\(134\) 0 0
\(135\) −1.38924 −0.119566
\(136\) 0 0
\(137\) 4.68828 0.400547 0.200273 0.979740i \(-0.435817\pi\)
0.200273 + 0.979740i \(0.435817\pi\)
\(138\) 0 0
\(139\) 14.4563 1.22617 0.613086 0.790017i \(-0.289928\pi\)
0.613086 + 0.790017i \(0.289928\pi\)
\(140\) 0 0
\(141\) −13.6631 −1.15064
\(142\) 0 0
\(143\) −22.4700 −1.87903
\(144\) 0 0
\(145\) 7.01806 0.582818
\(146\) 0 0
\(147\) −6.23989 −0.514657
\(148\) 0 0
\(149\) 3.97719 0.325824 0.162912 0.986641i \(-0.447911\pi\)
0.162912 + 0.986641i \(0.447911\pi\)
\(150\) 0 0
\(151\) 0.552927 0.0449965 0.0224983 0.999747i \(-0.492838\pi\)
0.0224983 + 0.999747i \(0.492838\pi\)
\(152\) 0 0
\(153\) −0.477836 −0.0386307
\(154\) 0 0
\(155\) −4.74789 −0.381360
\(156\) 0 0
\(157\) −8.90396 −0.710613 −0.355307 0.934750i \(-0.615624\pi\)
−0.355307 + 0.934750i \(0.615624\pi\)
\(158\) 0 0
\(159\) −11.6543 −0.924249
\(160\) 0 0
\(161\) −3.07960 −0.242706
\(162\) 0 0
\(163\) −2.32106 −0.181800 −0.0908999 0.995860i \(-0.528974\pi\)
−0.0908999 + 0.995860i \(0.528974\pi\)
\(164\) 0 0
\(165\) 7.98676 0.621768
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 2.27626 0.175097
\(170\) 0 0
\(171\) 2.16161 0.165303
\(172\) 0 0
\(173\) −6.53893 −0.497146 −0.248573 0.968613i \(-0.579962\pi\)
−0.248573 + 0.968613i \(0.579962\pi\)
\(174\) 0 0
\(175\) 2.67658 0.202331
\(176\) 0 0
\(177\) −0.528261 −0.0397065
\(178\) 0 0
\(179\) −6.03006 −0.450708 −0.225354 0.974277i \(-0.572354\pi\)
−0.225354 + 0.974277i \(0.572354\pi\)
\(180\) 0 0
\(181\) 23.9128 1.77743 0.888713 0.458464i \(-0.151600\pi\)
0.888713 + 0.458464i \(0.151600\pi\)
\(182\) 0 0
\(183\) 1.27251 0.0940669
\(184\) 0 0
\(185\) 6.93184 0.509639
\(186\) 0 0
\(187\) 2.74709 0.200887
\(188\) 0 0
\(189\) −0.871845 −0.0634174
\(190\) 0 0
\(191\) −5.15759 −0.373190 −0.186595 0.982437i \(-0.559745\pi\)
−0.186595 + 0.982437i \(0.559745\pi\)
\(192\) 0 0
\(193\) −2.38508 −0.171682 −0.0858410 0.996309i \(-0.527358\pi\)
−0.0858410 + 0.996309i \(0.527358\pi\)
\(194\) 0 0
\(195\) −5.42981 −0.388837
\(196\) 0 0
\(197\) −0.00129542 −9.22949e−5 0 −4.61475e−5 1.00000i \(-0.500015\pi\)
−4.61475e−5 1.00000i \(0.500015\pi\)
\(198\) 0 0
\(199\) 16.7180 1.18511 0.592555 0.805530i \(-0.298119\pi\)
0.592555 + 0.805530i \(0.298119\pi\)
\(200\) 0 0
\(201\) 8.13778 0.573995
\(202\) 0 0
\(203\) 4.40434 0.309124
\(204\) 0 0
\(205\) 7.65061 0.534342
\(206\) 0 0
\(207\) 3.53227 0.245510
\(208\) 0 0
\(209\) −12.4272 −0.859605
\(210\) 0 0
\(211\) −3.08873 −0.212637 −0.106319 0.994332i \(-0.533906\pi\)
−0.106319 + 0.994332i \(0.533906\pi\)
\(212\) 0 0
\(213\) 4.17968 0.286387
\(214\) 0 0
\(215\) 5.32407 0.363098
\(216\) 0 0
\(217\) −2.97964 −0.202271
\(218\) 0 0
\(219\) −6.12260 −0.413727
\(220\) 0 0
\(221\) −1.86761 −0.125629
\(222\) 0 0
\(223\) 14.4949 0.970648 0.485324 0.874334i \(-0.338702\pi\)
0.485324 + 0.874334i \(0.338702\pi\)
\(224\) 0 0
\(225\) −3.07002 −0.204668
\(226\) 0 0
\(227\) −9.04675 −0.600454 −0.300227 0.953868i \(-0.597062\pi\)
−0.300227 + 0.953868i \(0.597062\pi\)
\(228\) 0 0
\(229\) −15.9796 −1.05596 −0.527982 0.849256i \(-0.677051\pi\)
−0.527982 + 0.849256i \(0.677051\pi\)
\(230\) 0 0
\(231\) 5.01226 0.329783
\(232\) 0 0
\(233\) 1.78880 0.117188 0.0585941 0.998282i \(-0.481338\pi\)
0.0585941 + 0.998282i \(0.481338\pi\)
\(234\) 0 0
\(235\) 18.9813 1.23820
\(236\) 0 0
\(237\) −15.4461 −1.00333
\(238\) 0 0
\(239\) 4.43119 0.286630 0.143315 0.989677i \(-0.454224\pi\)
0.143315 + 0.989677i \(0.454224\pi\)
\(240\) 0 0
\(241\) 15.3349 0.987808 0.493904 0.869517i \(-0.335570\pi\)
0.493904 + 0.869517i \(0.335570\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 8.66868 0.553821
\(246\) 0 0
\(247\) 8.44863 0.537574
\(248\) 0 0
\(249\) −1.01912 −0.0645841
\(250\) 0 0
\(251\) 25.6393 1.61834 0.809168 0.587577i \(-0.199918\pi\)
0.809168 + 0.587577i \(0.199918\pi\)
\(252\) 0 0
\(253\) −20.3071 −1.27670
\(254\) 0 0
\(255\) 0.663827 0.0415705
\(256\) 0 0
\(257\) −22.6727 −1.41429 −0.707143 0.707070i \(-0.750016\pi\)
−0.707143 + 0.707070i \(0.750016\pi\)
\(258\) 0 0
\(259\) 4.35022 0.270310
\(260\) 0 0
\(261\) −5.05174 −0.312695
\(262\) 0 0
\(263\) −28.4884 −1.75667 −0.878335 0.478045i \(-0.841346\pi\)
−0.878335 + 0.478045i \(0.841346\pi\)
\(264\) 0 0
\(265\) 16.1906 0.994582
\(266\) 0 0
\(267\) −9.40919 −0.575833
\(268\) 0 0
\(269\) 13.0734 0.797100 0.398550 0.917147i \(-0.369514\pi\)
0.398550 + 0.917147i \(0.369514\pi\)
\(270\) 0 0
\(271\) −3.04487 −0.184963 −0.0924814 0.995714i \(-0.529480\pi\)
−0.0924814 + 0.995714i \(0.529480\pi\)
\(272\) 0 0
\(273\) −3.40760 −0.206237
\(274\) 0 0
\(275\) 17.6496 1.06431
\(276\) 0 0
\(277\) 4.04405 0.242984 0.121492 0.992592i \(-0.461232\pi\)
0.121492 + 0.992592i \(0.461232\pi\)
\(278\) 0 0
\(279\) 3.41763 0.204608
\(280\) 0 0
\(281\) −17.2904 −1.03146 −0.515728 0.856752i \(-0.672479\pi\)
−0.515728 + 0.856752i \(0.672479\pi\)
\(282\) 0 0
\(283\) 23.1428 1.37570 0.687849 0.725853i \(-0.258555\pi\)
0.687849 + 0.725853i \(0.258555\pi\)
\(284\) 0 0
\(285\) −3.00299 −0.177882
\(286\) 0 0
\(287\) 4.80131 0.283412
\(288\) 0 0
\(289\) −16.7717 −0.986569
\(290\) 0 0
\(291\) −0.321073 −0.0188216
\(292\) 0 0
\(293\) −28.4973 −1.66483 −0.832416 0.554152i \(-0.813043\pi\)
−0.832416 + 0.554152i \(0.813043\pi\)
\(294\) 0 0
\(295\) 0.733879 0.0427281
\(296\) 0 0
\(297\) −5.74903 −0.333592
\(298\) 0 0
\(299\) 13.8058 0.798412
\(300\) 0 0
\(301\) 3.34123 0.192585
\(302\) 0 0
\(303\) 14.5077 0.833444
\(304\) 0 0
\(305\) −1.76782 −0.101225
\(306\) 0 0
\(307\) 23.4381 1.33768 0.668842 0.743405i \(-0.266790\pi\)
0.668842 + 0.743405i \(0.266790\pi\)
\(308\) 0 0
\(309\) 6.14201 0.349407
\(310\) 0 0
\(311\) 9.33992 0.529618 0.264809 0.964301i \(-0.414691\pi\)
0.264809 + 0.964301i \(0.414691\pi\)
\(312\) 0 0
\(313\) 17.9683 1.01563 0.507815 0.861466i \(-0.330453\pi\)
0.507815 + 0.861466i \(0.330453\pi\)
\(314\) 0 0
\(315\) 1.21120 0.0682434
\(316\) 0 0
\(317\) −24.1619 −1.35706 −0.678532 0.734570i \(-0.737384\pi\)
−0.678532 + 0.734570i \(0.737384\pi\)
\(318\) 0 0
\(319\) 29.0426 1.62607
\(320\) 0 0
\(321\) 4.05659 0.226417
\(322\) 0 0
\(323\) −1.03290 −0.0574719
\(324\) 0 0
\(325\) −11.9991 −0.665592
\(326\) 0 0
\(327\) −17.8310 −0.986058
\(328\) 0 0
\(329\) 11.9121 0.656737
\(330\) 0 0
\(331\) 17.6022 0.967505 0.483753 0.875205i \(-0.339273\pi\)
0.483753 + 0.875205i \(0.339273\pi\)
\(332\) 0 0
\(333\) −4.98967 −0.273432
\(334\) 0 0
\(335\) −11.3053 −0.617674
\(336\) 0 0
\(337\) −2.14881 −0.117053 −0.0585265 0.998286i \(-0.518640\pi\)
−0.0585265 + 0.998286i \(0.518640\pi\)
\(338\) 0 0
\(339\) 1.27411 0.0691999
\(340\) 0 0
\(341\) −19.6480 −1.06400
\(342\) 0 0
\(343\) 11.5431 0.623271
\(344\) 0 0
\(345\) −4.90716 −0.264193
\(346\) 0 0
\(347\) 11.0393 0.592622 0.296311 0.955091i \(-0.404243\pi\)
0.296311 + 0.955091i \(0.404243\pi\)
\(348\) 0 0
\(349\) −0.603266 −0.0322921 −0.0161460 0.999870i \(-0.505140\pi\)
−0.0161460 + 0.999870i \(0.505140\pi\)
\(350\) 0 0
\(351\) 3.90849 0.208620
\(352\) 0 0
\(353\) 28.6385 1.52427 0.762137 0.647416i \(-0.224150\pi\)
0.762137 + 0.647416i \(0.224150\pi\)
\(354\) 0 0
\(355\) −5.80656 −0.308180
\(356\) 0 0
\(357\) 0.416599 0.0220488
\(358\) 0 0
\(359\) −28.2365 −1.49026 −0.745132 0.666918i \(-0.767613\pi\)
−0.745132 + 0.666918i \(0.767613\pi\)
\(360\) 0 0
\(361\) −14.3274 −0.754075
\(362\) 0 0
\(363\) 22.0513 1.15739
\(364\) 0 0
\(365\) 8.50574 0.445211
\(366\) 0 0
\(367\) −19.9271 −1.04019 −0.520093 0.854110i \(-0.674103\pi\)
−0.520093 + 0.854110i \(0.674103\pi\)
\(368\) 0 0
\(369\) −5.50706 −0.286686
\(370\) 0 0
\(371\) 10.1608 0.527521
\(372\) 0 0
\(373\) −23.4724 −1.21535 −0.607677 0.794184i \(-0.707899\pi\)
−0.607677 + 0.794184i \(0.707899\pi\)
\(374\) 0 0
\(375\) 11.2112 0.578942
\(376\) 0 0
\(377\) −19.7447 −1.01690
\(378\) 0 0
\(379\) −11.2367 −0.577192 −0.288596 0.957451i \(-0.593188\pi\)
−0.288596 + 0.957451i \(0.593188\pi\)
\(380\) 0 0
\(381\) −2.67900 −0.137250
\(382\) 0 0
\(383\) −34.1612 −1.74555 −0.872777 0.488119i \(-0.837683\pi\)
−0.872777 + 0.488119i \(0.837683\pi\)
\(384\) 0 0
\(385\) −6.96322 −0.354878
\(386\) 0 0
\(387\) −3.83237 −0.194810
\(388\) 0 0
\(389\) 16.6706 0.845233 0.422617 0.906309i \(-0.361112\pi\)
0.422617 + 0.906309i \(0.361112\pi\)
\(390\) 0 0
\(391\) −1.68785 −0.0853581
\(392\) 0 0
\(393\) 5.43675 0.274248
\(394\) 0 0
\(395\) 21.4583 1.07968
\(396\) 0 0
\(397\) 13.3298 0.669006 0.334503 0.942395i \(-0.391432\pi\)
0.334503 + 0.942395i \(0.391432\pi\)
\(398\) 0 0
\(399\) −1.88459 −0.0943476
\(400\) 0 0
\(401\) 24.9727 1.24708 0.623539 0.781792i \(-0.285694\pi\)
0.623539 + 0.781792i \(0.285694\pi\)
\(402\) 0 0
\(403\) 13.3577 0.665397
\(404\) 0 0
\(405\) −1.38924 −0.0690317
\(406\) 0 0
\(407\) 28.6858 1.42190
\(408\) 0 0
\(409\) −22.3474 −1.10501 −0.552504 0.833510i \(-0.686328\pi\)
−0.552504 + 0.833510i \(0.686328\pi\)
\(410\) 0 0
\(411\) 4.68828 0.231256
\(412\) 0 0
\(413\) 0.460562 0.0226628
\(414\) 0 0
\(415\) 1.41580 0.0694988
\(416\) 0 0
\(417\) 14.4563 0.707930
\(418\) 0 0
\(419\) 26.6024 1.29961 0.649805 0.760101i \(-0.274850\pi\)
0.649805 + 0.760101i \(0.274850\pi\)
\(420\) 0 0
\(421\) −23.3727 −1.13911 −0.569557 0.821952i \(-0.692885\pi\)
−0.569557 + 0.821952i \(0.692885\pi\)
\(422\) 0 0
\(423\) −13.6631 −0.664324
\(424\) 0 0
\(425\) 1.46697 0.0711583
\(426\) 0 0
\(427\) −1.10944 −0.0536893
\(428\) 0 0
\(429\) −22.4700 −1.08486
\(430\) 0 0
\(431\) −34.7703 −1.67483 −0.837414 0.546570i \(-0.815933\pi\)
−0.837414 + 0.546570i \(0.815933\pi\)
\(432\) 0 0
\(433\) −21.3677 −1.02687 −0.513433 0.858130i \(-0.671626\pi\)
−0.513433 + 0.858130i \(0.671626\pi\)
\(434\) 0 0
\(435\) 7.01806 0.336490
\(436\) 0 0
\(437\) 7.63541 0.365251
\(438\) 0 0
\(439\) 10.2625 0.489802 0.244901 0.969548i \(-0.421245\pi\)
0.244901 + 0.969548i \(0.421245\pi\)
\(440\) 0 0
\(441\) −6.23989 −0.297137
\(442\) 0 0
\(443\) 4.69185 0.222917 0.111458 0.993769i \(-0.464448\pi\)
0.111458 + 0.993769i \(0.464448\pi\)
\(444\) 0 0
\(445\) 13.0716 0.619653
\(446\) 0 0
\(447\) 3.97719 0.188115
\(448\) 0 0
\(449\) −12.2687 −0.578998 −0.289499 0.957178i \(-0.593489\pi\)
−0.289499 + 0.957178i \(0.593489\pi\)
\(450\) 0 0
\(451\) 31.6603 1.49082
\(452\) 0 0
\(453\) 0.552927 0.0259788
\(454\) 0 0
\(455\) 4.73396 0.221931
\(456\) 0 0
\(457\) 11.1775 0.522862 0.261431 0.965222i \(-0.415806\pi\)
0.261431 + 0.965222i \(0.415806\pi\)
\(458\) 0 0
\(459\) −0.477836 −0.0223035
\(460\) 0 0
\(461\) 19.2919 0.898516 0.449258 0.893402i \(-0.351688\pi\)
0.449258 + 0.893402i \(0.351688\pi\)
\(462\) 0 0
\(463\) 12.1247 0.563481 0.281741 0.959491i \(-0.409088\pi\)
0.281741 + 0.959491i \(0.409088\pi\)
\(464\) 0 0
\(465\) −4.74789 −0.220178
\(466\) 0 0
\(467\) −12.6560 −0.585650 −0.292825 0.956166i \(-0.594595\pi\)
−0.292825 + 0.956166i \(0.594595\pi\)
\(468\) 0 0
\(469\) −7.09488 −0.327611
\(470\) 0 0
\(471\) −8.90396 −0.410273
\(472\) 0 0
\(473\) 22.0324 1.01305
\(474\) 0 0
\(475\) −6.63620 −0.304490
\(476\) 0 0
\(477\) −11.6543 −0.533615
\(478\) 0 0
\(479\) 32.3944 1.48014 0.740068 0.672532i \(-0.234793\pi\)
0.740068 + 0.672532i \(0.234793\pi\)
\(480\) 0 0
\(481\) −19.5021 −0.889218
\(482\) 0 0
\(483\) −3.07960 −0.140126
\(484\) 0 0
\(485\) 0.446046 0.0202539
\(486\) 0 0
\(487\) 38.5332 1.74610 0.873052 0.487627i \(-0.162138\pi\)
0.873052 + 0.487627i \(0.162138\pi\)
\(488\) 0 0
\(489\) −2.32106 −0.104962
\(490\) 0 0
\(491\) 17.3207 0.781672 0.390836 0.920460i \(-0.372186\pi\)
0.390836 + 0.920460i \(0.372186\pi\)
\(492\) 0 0
\(493\) 2.41390 0.108717
\(494\) 0 0
\(495\) 7.98676 0.358978
\(496\) 0 0
\(497\) −3.64403 −0.163457
\(498\) 0 0
\(499\) −16.7924 −0.751729 −0.375865 0.926675i \(-0.622654\pi\)
−0.375865 + 0.926675i \(0.622654\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −3.45950 −0.154252 −0.0771258 0.997021i \(-0.524574\pi\)
−0.0771258 + 0.997021i \(0.524574\pi\)
\(504\) 0 0
\(505\) −20.1546 −0.896868
\(506\) 0 0
\(507\) 2.27626 0.101092
\(508\) 0 0
\(509\) −14.8751 −0.659326 −0.329663 0.944099i \(-0.606935\pi\)
−0.329663 + 0.944099i \(0.606935\pi\)
\(510\) 0 0
\(511\) 5.33796 0.236138
\(512\) 0 0
\(513\) 2.16161 0.0954376
\(514\) 0 0
\(515\) −8.53270 −0.375996
\(516\) 0 0
\(517\) 78.5497 3.45461
\(518\) 0 0
\(519\) −6.53893 −0.287027
\(520\) 0 0
\(521\) −6.52656 −0.285934 −0.142967 0.989727i \(-0.545664\pi\)
−0.142967 + 0.989727i \(0.545664\pi\)
\(522\) 0 0
\(523\) 32.7225 1.43086 0.715428 0.698686i \(-0.246232\pi\)
0.715428 + 0.698686i \(0.246232\pi\)
\(524\) 0 0
\(525\) 2.67658 0.116816
\(526\) 0 0
\(527\) −1.63306 −0.0711374
\(528\) 0 0
\(529\) −10.5231 −0.457524
\(530\) 0 0
\(531\) −0.528261 −0.0229246
\(532\) 0 0
\(533\) −21.5243 −0.932320
\(534\) 0 0
\(535\) −5.63556 −0.243647
\(536\) 0 0
\(537\) −6.03006 −0.260216
\(538\) 0 0
\(539\) 35.8733 1.54517
\(540\) 0 0
\(541\) −35.3810 −1.52115 −0.760573 0.649252i \(-0.775082\pi\)
−0.760573 + 0.649252i \(0.775082\pi\)
\(542\) 0 0
\(543\) 23.9128 1.02620
\(544\) 0 0
\(545\) 24.7715 1.06109
\(546\) 0 0
\(547\) 3.10909 0.132935 0.0664676 0.997789i \(-0.478827\pi\)
0.0664676 + 0.997789i \(0.478827\pi\)
\(548\) 0 0
\(549\) 1.27251 0.0543096
\(550\) 0 0
\(551\) −10.9199 −0.465204
\(552\) 0 0
\(553\) 13.4666 0.572658
\(554\) 0 0
\(555\) 6.93184 0.294240
\(556\) 0 0
\(557\) 21.3530 0.904757 0.452379 0.891826i \(-0.350576\pi\)
0.452379 + 0.891826i \(0.350576\pi\)
\(558\) 0 0
\(559\) −14.9788 −0.633534
\(560\) 0 0
\(561\) 2.74709 0.115982
\(562\) 0 0
\(563\) 14.8320 0.625093 0.312546 0.949903i \(-0.398818\pi\)
0.312546 + 0.949903i \(0.398818\pi\)
\(564\) 0 0
\(565\) −1.77003 −0.0744659
\(566\) 0 0
\(567\) −0.871845 −0.0366141
\(568\) 0 0
\(569\) 29.6942 1.24485 0.622423 0.782681i \(-0.286148\pi\)
0.622423 + 0.782681i \(0.286148\pi\)
\(570\) 0 0
\(571\) −8.20664 −0.343437 −0.171719 0.985146i \(-0.554932\pi\)
−0.171719 + 0.985146i \(0.554932\pi\)
\(572\) 0 0
\(573\) −5.15759 −0.215462
\(574\) 0 0
\(575\) −10.8442 −0.452232
\(576\) 0 0
\(577\) 34.6609 1.44295 0.721477 0.692438i \(-0.243464\pi\)
0.721477 + 0.692438i \(0.243464\pi\)
\(578\) 0 0
\(579\) −2.38508 −0.0991207
\(580\) 0 0
\(581\) 0.888515 0.0368618
\(582\) 0 0
\(583\) 67.0011 2.77490
\(584\) 0 0
\(585\) −5.42981 −0.224495
\(586\) 0 0
\(587\) 22.3087 0.920780 0.460390 0.887717i \(-0.347710\pi\)
0.460390 + 0.887717i \(0.347710\pi\)
\(588\) 0 0
\(589\) 7.38759 0.304400
\(590\) 0 0
\(591\) −0.00129542 −5.32865e−5 0
\(592\) 0 0
\(593\) 48.3835 1.98687 0.993435 0.114394i \(-0.0364926\pi\)
0.993435 + 0.114394i \(0.0364926\pi\)
\(594\) 0 0
\(595\) −0.578755 −0.0237266
\(596\) 0 0
\(597\) 16.7180 0.684224
\(598\) 0 0
\(599\) −24.7439 −1.01101 −0.505503 0.862825i \(-0.668693\pi\)
−0.505503 + 0.862825i \(0.668693\pi\)
\(600\) 0 0
\(601\) 41.4466 1.69064 0.845320 0.534260i \(-0.179410\pi\)
0.845320 + 0.534260i \(0.179410\pi\)
\(602\) 0 0
\(603\) 8.13778 0.331396
\(604\) 0 0
\(605\) −30.6345 −1.24547
\(606\) 0 0
\(607\) 8.13247 0.330087 0.165043 0.986286i \(-0.447224\pi\)
0.165043 + 0.986286i \(0.447224\pi\)
\(608\) 0 0
\(609\) 4.40434 0.178473
\(610\) 0 0
\(611\) −53.4021 −2.16042
\(612\) 0 0
\(613\) −13.6117 −0.549771 −0.274886 0.961477i \(-0.588640\pi\)
−0.274886 + 0.961477i \(0.588640\pi\)
\(614\) 0 0
\(615\) 7.65061 0.308503
\(616\) 0 0
\(617\) −2.02573 −0.0815528 −0.0407764 0.999168i \(-0.512983\pi\)
−0.0407764 + 0.999168i \(0.512983\pi\)
\(618\) 0 0
\(619\) −13.8882 −0.558213 −0.279106 0.960260i \(-0.590038\pi\)
−0.279106 + 0.960260i \(0.590038\pi\)
\(620\) 0 0
\(621\) 3.53227 0.141745
\(622\) 0 0
\(623\) 8.20336 0.328661
\(624\) 0 0
\(625\) −0.224859 −0.00899436
\(626\) 0 0
\(627\) −12.4272 −0.496293
\(628\) 0 0
\(629\) 2.38424 0.0950661
\(630\) 0 0
\(631\) 0.724370 0.0288367 0.0144184 0.999896i \(-0.495410\pi\)
0.0144184 + 0.999896i \(0.495410\pi\)
\(632\) 0 0
\(633\) −3.08873 −0.122766
\(634\) 0 0
\(635\) 3.72177 0.147694
\(636\) 0 0
\(637\) −24.3885 −0.966308
\(638\) 0 0
\(639\) 4.17968 0.165345
\(640\) 0 0
\(641\) 18.4957 0.730537 0.365269 0.930902i \(-0.380977\pi\)
0.365269 + 0.930902i \(0.380977\pi\)
\(642\) 0 0
\(643\) −7.03941 −0.277607 −0.138804 0.990320i \(-0.544326\pi\)
−0.138804 + 0.990320i \(0.544326\pi\)
\(644\) 0 0
\(645\) 5.32407 0.209635
\(646\) 0 0
\(647\) −46.7368 −1.83741 −0.918706 0.394942i \(-0.870765\pi\)
−0.918706 + 0.394942i \(0.870765\pi\)
\(648\) 0 0
\(649\) 3.03698 0.119212
\(650\) 0 0
\(651\) −2.97964 −0.116781
\(652\) 0 0
\(653\) 2.06283 0.0807247 0.0403624 0.999185i \(-0.487149\pi\)
0.0403624 + 0.999185i \(0.487149\pi\)
\(654\) 0 0
\(655\) −7.55293 −0.295117
\(656\) 0 0
\(657\) −6.12260 −0.238865
\(658\) 0 0
\(659\) 26.3610 1.02688 0.513440 0.858126i \(-0.328371\pi\)
0.513440 + 0.858126i \(0.328371\pi\)
\(660\) 0 0
\(661\) −25.1696 −0.978982 −0.489491 0.872008i \(-0.662817\pi\)
−0.489491 + 0.872008i \(0.662817\pi\)
\(662\) 0 0
\(663\) −1.86761 −0.0725321
\(664\) 0 0
\(665\) 2.61815 0.101527
\(666\) 0 0
\(667\) −17.8441 −0.690927
\(668\) 0 0
\(669\) 14.4949 0.560404
\(670\) 0 0
\(671\) −7.31572 −0.282420
\(672\) 0 0
\(673\) 14.4885 0.558491 0.279245 0.960220i \(-0.409916\pi\)
0.279245 + 0.960220i \(0.409916\pi\)
\(674\) 0 0
\(675\) −3.07002 −0.118165
\(676\) 0 0
\(677\) −43.9862 −1.69053 −0.845263 0.534350i \(-0.820556\pi\)
−0.845263 + 0.534350i \(0.820556\pi\)
\(678\) 0 0
\(679\) 0.279926 0.0107426
\(680\) 0 0
\(681\) −9.04675 −0.346672
\(682\) 0 0
\(683\) 11.4959 0.439878 0.219939 0.975514i \(-0.429414\pi\)
0.219939 + 0.975514i \(0.429414\pi\)
\(684\) 0 0
\(685\) −6.51313 −0.248854
\(686\) 0 0
\(687\) −15.9796 −0.609661
\(688\) 0 0
\(689\) −45.5508 −1.73535
\(690\) 0 0
\(691\) 47.9678 1.82478 0.912390 0.409323i \(-0.134235\pi\)
0.912390 + 0.409323i \(0.134235\pi\)
\(692\) 0 0
\(693\) 5.01226 0.190400
\(694\) 0 0
\(695\) −20.0833 −0.761802
\(696\) 0 0
\(697\) 2.63147 0.0996741
\(698\) 0 0
\(699\) 1.78880 0.0676586
\(700\) 0 0
\(701\) 18.4427 0.696571 0.348285 0.937389i \(-0.386764\pi\)
0.348285 + 0.937389i \(0.386764\pi\)
\(702\) 0 0
\(703\) −10.7857 −0.406792
\(704\) 0 0
\(705\) 18.9813 0.714878
\(706\) 0 0
\(707\) −12.6485 −0.475694
\(708\) 0 0
\(709\) 37.1171 1.39396 0.696981 0.717090i \(-0.254526\pi\)
0.696981 + 0.717090i \(0.254526\pi\)
\(710\) 0 0
\(711\) −15.4461 −0.579274
\(712\) 0 0
\(713\) 12.0720 0.452099
\(714\) 0 0
\(715\) 31.2161 1.16742
\(716\) 0 0
\(717\) 4.43119 0.165486
\(718\) 0 0
\(719\) −8.72522 −0.325396 −0.162698 0.986676i \(-0.552020\pi\)
−0.162698 + 0.986676i \(0.552020\pi\)
\(720\) 0 0
\(721\) −5.35488 −0.199426
\(722\) 0 0
\(723\) 15.3349 0.570311
\(724\) 0 0
\(725\) 15.5090 0.575988
\(726\) 0 0
\(727\) −38.9456 −1.44441 −0.722206 0.691678i \(-0.756872\pi\)
−0.722206 + 0.691678i \(0.756872\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.83124 0.0677310
\(732\) 0 0
\(733\) −11.0832 −0.409367 −0.204684 0.978828i \(-0.565617\pi\)
−0.204684 + 0.978828i \(0.565617\pi\)
\(734\) 0 0
\(735\) 8.66868 0.319749
\(736\) 0 0
\(737\) −46.7843 −1.72332
\(738\) 0 0
\(739\) −39.0330 −1.43585 −0.717927 0.696119i \(-0.754909\pi\)
−0.717927 + 0.696119i \(0.754909\pi\)
\(740\) 0 0
\(741\) 8.44863 0.310368
\(742\) 0 0
\(743\) 8.69917 0.319142 0.159571 0.987186i \(-0.448989\pi\)
0.159571 + 0.987186i \(0.448989\pi\)
\(744\) 0 0
\(745\) −5.52526 −0.202430
\(746\) 0 0
\(747\) −1.01912 −0.0372876
\(748\) 0 0
\(749\) −3.53672 −0.129229
\(750\) 0 0
\(751\) −30.1732 −1.10104 −0.550518 0.834823i \(-0.685570\pi\)
−0.550518 + 0.834823i \(0.685570\pi\)
\(752\) 0 0
\(753\) 25.6393 0.934347
\(754\) 0 0
\(755\) −0.768146 −0.0279557
\(756\) 0 0
\(757\) −1.93772 −0.0704277 −0.0352138 0.999380i \(-0.511211\pi\)
−0.0352138 + 0.999380i \(0.511211\pi\)
\(758\) 0 0
\(759\) −20.3071 −0.737102
\(760\) 0 0
\(761\) −32.4526 −1.17641 −0.588203 0.808713i \(-0.700164\pi\)
−0.588203 + 0.808713i \(0.700164\pi\)
\(762\) 0 0
\(763\) 15.5459 0.562799
\(764\) 0 0
\(765\) 0.663827 0.0240007
\(766\) 0 0
\(767\) −2.06470 −0.0745519
\(768\) 0 0
\(769\) −13.5848 −0.489882 −0.244941 0.969538i \(-0.578769\pi\)
−0.244941 + 0.969538i \(0.578769\pi\)
\(770\) 0 0
\(771\) −22.6727 −0.816539
\(772\) 0 0
\(773\) −5.72819 −0.206029 −0.103014 0.994680i \(-0.532849\pi\)
−0.103014 + 0.994680i \(0.532849\pi\)
\(774\) 0 0
\(775\) −10.4922 −0.376890
\(776\) 0 0
\(777\) 4.35022 0.156063
\(778\) 0 0
\(779\) −11.9041 −0.426510
\(780\) 0 0
\(781\) −24.0291 −0.859828
\(782\) 0 0
\(783\) −5.05174 −0.180534
\(784\) 0 0
\(785\) 12.3697 0.441494
\(786\) 0 0
\(787\) −49.9123 −1.77918 −0.889591 0.456758i \(-0.849010\pi\)
−0.889591 + 0.456758i \(0.849010\pi\)
\(788\) 0 0
\(789\) −28.4884 −1.01421
\(790\) 0 0
\(791\) −1.11082 −0.0394963
\(792\) 0 0
\(793\) 4.97360 0.176618
\(794\) 0 0
\(795\) 16.1906 0.574222
\(796\) 0 0
\(797\) 32.1371 1.13836 0.569178 0.822214i \(-0.307262\pi\)
0.569178 + 0.822214i \(0.307262\pi\)
\(798\) 0 0
\(799\) 6.52873 0.230970
\(800\) 0 0
\(801\) −9.40919 −0.332457
\(802\) 0 0
\(803\) 35.1990 1.24215
\(804\) 0 0
\(805\) 4.27829 0.150790
\(806\) 0 0
\(807\) 13.0734 0.460206
\(808\) 0 0
\(809\) 6.77641 0.238246 0.119123 0.992880i \(-0.461992\pi\)
0.119123 + 0.992880i \(0.461992\pi\)
\(810\) 0 0
\(811\) 32.4544 1.13963 0.569814 0.821774i \(-0.307015\pi\)
0.569814 + 0.821774i \(0.307015\pi\)
\(812\) 0 0
\(813\) −3.04487 −0.106788
\(814\) 0 0
\(815\) 3.22451 0.112950
\(816\) 0 0
\(817\) −8.28410 −0.289824
\(818\) 0 0
\(819\) −3.40760 −0.119071
\(820\) 0 0
\(821\) −24.8054 −0.865716 −0.432858 0.901462i \(-0.642495\pi\)
−0.432858 + 0.901462i \(0.642495\pi\)
\(822\) 0 0
\(823\) −15.2673 −0.532184 −0.266092 0.963948i \(-0.585732\pi\)
−0.266092 + 0.963948i \(0.585732\pi\)
\(824\) 0 0
\(825\) 17.6496 0.614481
\(826\) 0 0
\(827\) −13.3232 −0.463292 −0.231646 0.972800i \(-0.574411\pi\)
−0.231646 + 0.972800i \(0.574411\pi\)
\(828\) 0 0
\(829\) −49.9301 −1.73414 −0.867072 0.498183i \(-0.834001\pi\)
−0.867072 + 0.498183i \(0.834001\pi\)
\(830\) 0 0
\(831\) 4.04405 0.140287
\(832\) 0 0
\(833\) 2.98164 0.103308
\(834\) 0 0
\(835\) 1.38924 0.0480765
\(836\) 0 0
\(837\) 3.41763 0.118130
\(838\) 0 0
\(839\) 19.8275 0.684520 0.342260 0.939605i \(-0.388808\pi\)
0.342260 + 0.939605i \(0.388808\pi\)
\(840\) 0 0
\(841\) −3.47992 −0.119997
\(842\) 0 0
\(843\) −17.2904 −0.595512
\(844\) 0 0
\(845\) −3.16226 −0.108785
\(846\) 0 0
\(847\) −19.2253 −0.660590
\(848\) 0 0
\(849\) 23.1428 0.794260
\(850\) 0 0
\(851\) −17.6249 −0.604173
\(852\) 0 0
\(853\) −17.1965 −0.588798 −0.294399 0.955683i \(-0.595119\pi\)
−0.294399 + 0.955683i \(0.595119\pi\)
\(854\) 0 0
\(855\) −3.00299 −0.102700
\(856\) 0 0
\(857\) −1.21284 −0.0414299 −0.0207149 0.999785i \(-0.506594\pi\)
−0.0207149 + 0.999785i \(0.506594\pi\)
\(858\) 0 0
\(859\) 22.6809 0.773863 0.386932 0.922108i \(-0.373535\pi\)
0.386932 + 0.922108i \(0.373535\pi\)
\(860\) 0 0
\(861\) 4.80131 0.163628
\(862\) 0 0
\(863\) 9.21615 0.313721 0.156861 0.987621i \(-0.449863\pi\)
0.156861 + 0.987621i \(0.449863\pi\)
\(864\) 0 0
\(865\) 9.08412 0.308869
\(866\) 0 0
\(867\) −16.7717 −0.569596
\(868\) 0 0
\(869\) 88.8000 3.01233
\(870\) 0 0
\(871\) 31.8064 1.07772
\(872\) 0 0
\(873\) −0.321073 −0.0108667
\(874\) 0 0
\(875\) −9.77441 −0.330435
\(876\) 0 0
\(877\) −33.5985 −1.13454 −0.567270 0.823532i \(-0.692000\pi\)
−0.567270 + 0.823532i \(0.692000\pi\)
\(878\) 0 0
\(879\) −28.4973 −0.961191
\(880\) 0 0
\(881\) −49.4124 −1.66475 −0.832373 0.554216i \(-0.813018\pi\)
−0.832373 + 0.554216i \(0.813018\pi\)
\(882\) 0 0
\(883\) 27.9558 0.940786 0.470393 0.882457i \(-0.344112\pi\)
0.470393 + 0.882457i \(0.344112\pi\)
\(884\) 0 0
\(885\) 0.733879 0.0246691
\(886\) 0 0
\(887\) −2.31976 −0.0778899 −0.0389449 0.999241i \(-0.512400\pi\)
−0.0389449 + 0.999241i \(0.512400\pi\)
\(888\) 0 0
\(889\) 2.33568 0.0783361
\(890\) 0 0
\(891\) −5.74903 −0.192600
\(892\) 0 0
\(893\) −29.5344 −0.988331
\(894\) 0 0
\(895\) 8.37718 0.280018
\(896\) 0 0
\(897\) 13.8058 0.460963
\(898\) 0 0
\(899\) −17.2650 −0.575819
\(900\) 0 0
\(901\) 5.56886 0.185526
\(902\) 0 0
\(903\) 3.34123 0.111189
\(904\) 0 0
\(905\) −33.2206 −1.10429
\(906\) 0 0
\(907\) −20.0556 −0.665936 −0.332968 0.942938i \(-0.608050\pi\)
−0.332968 + 0.942938i \(0.608050\pi\)
\(908\) 0 0
\(909\) 14.5077 0.481189
\(910\) 0 0
\(911\) −18.8720 −0.625258 −0.312629 0.949875i \(-0.601210\pi\)
−0.312629 + 0.949875i \(0.601210\pi\)
\(912\) 0 0
\(913\) 5.85895 0.193903
\(914\) 0 0
\(915\) −1.76782 −0.0584424
\(916\) 0 0
\(917\) −4.74000 −0.156529
\(918\) 0 0
\(919\) 53.4602 1.76349 0.881745 0.471727i \(-0.156369\pi\)
0.881745 + 0.471727i \(0.156369\pi\)
\(920\) 0 0
\(921\) 23.4381 0.772312
\(922\) 0 0
\(923\) 16.3362 0.537713
\(924\) 0 0
\(925\) 15.3184 0.503666
\(926\) 0 0
\(927\) 6.14201 0.201730
\(928\) 0 0
\(929\) −30.7913 −1.01023 −0.505114 0.863052i \(-0.668550\pi\)
−0.505114 + 0.863052i \(0.668550\pi\)
\(930\) 0 0
\(931\) −13.4882 −0.442059
\(932\) 0 0
\(933\) 9.33992 0.305775
\(934\) 0 0
\(935\) −3.81636 −0.124808
\(936\) 0 0
\(937\) −26.5516 −0.867405 −0.433702 0.901056i \(-0.642793\pi\)
−0.433702 + 0.901056i \(0.642793\pi\)
\(938\) 0 0
\(939\) 17.9683 0.586374
\(940\) 0 0
\(941\) −46.2200 −1.50673 −0.753364 0.657603i \(-0.771570\pi\)
−0.753364 + 0.657603i \(0.771570\pi\)
\(942\) 0 0
\(943\) −19.4525 −0.633459
\(944\) 0 0
\(945\) 1.21120 0.0394003
\(946\) 0 0
\(947\) 38.8101 1.26116 0.630579 0.776125i \(-0.282817\pi\)
0.630579 + 0.776125i \(0.282817\pi\)
\(948\) 0 0
\(949\) −23.9301 −0.776804
\(950\) 0 0
\(951\) −24.1619 −0.783502
\(952\) 0 0
\(953\) 14.1074 0.456982 0.228491 0.973546i \(-0.426621\pi\)
0.228491 + 0.973546i \(0.426621\pi\)
\(954\) 0 0
\(955\) 7.16512 0.231858
\(956\) 0 0
\(957\) 29.0426 0.938814
\(958\) 0 0
\(959\) −4.08746 −0.131991
\(960\) 0 0
\(961\) −19.3198 −0.623220
\(962\) 0 0
\(963\) 4.05659 0.130722
\(964\) 0 0
\(965\) 3.31344 0.106664
\(966\) 0 0
\(967\) −9.47705 −0.304761 −0.152381 0.988322i \(-0.548694\pi\)
−0.152381 + 0.988322i \(0.548694\pi\)
\(968\) 0 0
\(969\) −1.03290 −0.0331814
\(970\) 0 0
\(971\) −6.76536 −0.217111 −0.108555 0.994090i \(-0.534622\pi\)
−0.108555 + 0.994090i \(0.534622\pi\)
\(972\) 0 0
\(973\) −12.6037 −0.404056
\(974\) 0 0
\(975\) −11.9991 −0.384280
\(976\) 0 0
\(977\) 4.48932 0.143626 0.0718131 0.997418i \(-0.477121\pi\)
0.0718131 + 0.997418i \(0.477121\pi\)
\(978\) 0 0
\(979\) 54.0937 1.72884
\(980\) 0 0
\(981\) −17.8310 −0.569301
\(982\) 0 0
\(983\) 6.64779 0.212032 0.106016 0.994364i \(-0.466191\pi\)
0.106016 + 0.994364i \(0.466191\pi\)
\(984\) 0 0
\(985\) 0.00179965 5.73415e−5 0
\(986\) 0 0
\(987\) 11.9121 0.379167
\(988\) 0 0
\(989\) −13.5370 −0.430450
\(990\) 0 0
\(991\) −27.9862 −0.889011 −0.444505 0.895776i \(-0.646621\pi\)
−0.444505 + 0.895776i \(0.646621\pi\)
\(992\) 0 0
\(993\) 17.6022 0.558589
\(994\) 0 0
\(995\) −23.2253 −0.736292
\(996\) 0 0
\(997\) 9.59766 0.303961 0.151981 0.988383i \(-0.451435\pi\)
0.151981 + 0.988383i \(0.451435\pi\)
\(998\) 0 0
\(999\) −4.98967 −0.157866
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 2004.2.a.b.1.3 5
3.2 odd 2 6012.2.a.f.1.3 5
4.3 odd 2 8016.2.a.q.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.b.1.3 5 1.1 even 1 trivial
6012.2.a.f.1.3 5 3.2 odd 2
8016.2.a.q.1.3 5 4.3 odd 2