Properties

Label 456.2.a.f.1.1
Level $456$
Weight $2$
Character 456.1
Self dual yes
Analytic conductor $3.641$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 456.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +5.56155 q^{11} +3.12311 q^{13} -1.56155 q^{15} +6.68466 q^{17} +1.00000 q^{19} -1.56155 q^{21} +9.12311 q^{23} -2.56155 q^{25} +1.00000 q^{27} -8.24621 q^{29} -2.00000 q^{31} +5.56155 q^{33} +2.43845 q^{35} -8.00000 q^{37} +3.12311 q^{39} -5.12311 q^{41} -1.56155 q^{43} -1.56155 q^{45} -6.68466 q^{47} -4.56155 q^{49} +6.68466 q^{51} +4.24621 q^{53} -8.68466 q^{55} +1.00000 q^{57} +12.0000 q^{59} +6.68466 q^{61} -1.56155 q^{63} -4.87689 q^{65} +6.24621 q^{67} +9.12311 q^{69} -16.9309 q^{73} -2.56155 q^{75} -8.68466 q^{77} +11.3693 q^{79} +1.00000 q^{81} +4.00000 q^{83} -10.4384 q^{85} -8.24621 q^{87} -6.00000 q^{89} -4.87689 q^{91} -2.00000 q^{93} -1.56155 q^{95} +4.24621 q^{97} +5.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9} + 7 q^{11} - 2 q^{13} + q^{15} + q^{17} + 2 q^{19} + q^{21} + 10 q^{23} - q^{25} + 2 q^{27} - 4 q^{31} + 7 q^{33} + 9 q^{35} - 16 q^{37} - 2 q^{39} - 2 q^{41} + q^{43} + q^{45} - q^{47} - 5 q^{49} + q^{51} - 8 q^{53} - 5 q^{55} + 2 q^{57} + 24 q^{59} + q^{61} + q^{63} - 18 q^{65} - 4 q^{67} + 10 q^{69} - 5 q^{73} - q^{75} - 5 q^{77} - 2 q^{79} + 2 q^{81} + 8 q^{83} - 25 q^{85} - 12 q^{89} - 18 q^{91} - 4 q^{93} + q^{95} - 8 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.56155 −0.698348 −0.349174 0.937058i \(-0.613538\pi\)
−0.349174 + 0.937058i \(0.613538\pi\)
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.56155 1.67687 0.838436 0.545001i \(-0.183471\pi\)
0.838436 + 0.545001i \(0.183471\pi\)
\(12\) 0 0
\(13\) 3.12311 0.866194 0.433097 0.901347i \(-0.357421\pi\)
0.433097 + 0.901347i \(0.357421\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) 6.68466 1.62127 0.810634 0.585553i \(-0.199123\pi\)
0.810634 + 0.585553i \(0.199123\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.56155 −0.340759
\(22\) 0 0
\(23\) 9.12311 1.90230 0.951150 0.308731i \(-0.0999042\pi\)
0.951150 + 0.308731i \(0.0999042\pi\)
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.24621 −1.53128 −0.765641 0.643268i \(-0.777578\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 5.56155 0.968142
\(34\) 0 0
\(35\) 2.43845 0.412173
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 3.12311 0.500097
\(40\) 0 0
\(41\) −5.12311 −0.800095 −0.400047 0.916494i \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 0 0
\(43\) −1.56155 −0.238135 −0.119067 0.992886i \(-0.537990\pi\)
−0.119067 + 0.992886i \(0.537990\pi\)
\(44\) 0 0
\(45\) −1.56155 −0.232783
\(46\) 0 0
\(47\) −6.68466 −0.975058 −0.487529 0.873107i \(-0.662102\pi\)
−0.487529 + 0.873107i \(0.662102\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) 6.68466 0.936039
\(52\) 0 0
\(53\) 4.24621 0.583262 0.291631 0.956531i \(-0.405802\pi\)
0.291631 + 0.956531i \(0.405802\pi\)
\(54\) 0 0
\(55\) −8.68466 −1.17104
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 6.68466 0.855883 0.427941 0.903806i \(-0.359239\pi\)
0.427941 + 0.903806i \(0.359239\pi\)
\(62\) 0 0
\(63\) −1.56155 −0.196737
\(64\) 0 0
\(65\) −4.87689 −0.604904
\(66\) 0 0
\(67\) 6.24621 0.763096 0.381548 0.924349i \(-0.375391\pi\)
0.381548 + 0.924349i \(0.375391\pi\)
\(68\) 0 0
\(69\) 9.12311 1.09829
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.9309 −1.98161 −0.990804 0.135303i \(-0.956799\pi\)
−0.990804 + 0.135303i \(0.956799\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) −8.68466 −0.989709
\(78\) 0 0
\(79\) 11.3693 1.27915 0.639574 0.768729i \(-0.279111\pi\)
0.639574 + 0.768729i \(0.279111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −10.4384 −1.13221
\(86\) 0 0
\(87\) −8.24621 −0.884087
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −4.87689 −0.511237
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −1.56155 −0.160212
\(96\) 0 0
\(97\) 4.24621 0.431137 0.215569 0.976489i \(-0.430839\pi\)
0.215569 + 0.976489i \(0.430839\pi\)
\(98\) 0 0
\(99\) 5.56155 0.558957
\(100\) 0 0
\(101\) 7.12311 0.708776 0.354388 0.935099i \(-0.384689\pi\)
0.354388 + 0.935099i \(0.384689\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 2.43845 0.237968
\(106\) 0 0
\(107\) −13.3693 −1.29246 −0.646230 0.763142i \(-0.723655\pi\)
−0.646230 + 0.763142i \(0.723655\pi\)
\(108\) 0 0
\(109\) −2.24621 −0.215148 −0.107574 0.994197i \(-0.534308\pi\)
−0.107574 + 0.994197i \(0.534308\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −14.2462 −1.32847
\(116\) 0 0
\(117\) 3.12311 0.288731
\(118\) 0 0
\(119\) −10.4384 −0.956891
\(120\) 0 0
\(121\) 19.9309 1.81190
\(122\) 0 0
\(123\) −5.12311 −0.461935
\(124\) 0 0
\(125\) 11.8078 1.05612
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) −1.56155 −0.137487
\(130\) 0 0
\(131\) 20.6847 1.80723 0.903613 0.428349i \(-0.140905\pi\)
0.903613 + 0.428349i \(0.140905\pi\)
\(132\) 0 0
\(133\) −1.56155 −0.135404
\(134\) 0 0
\(135\) −1.56155 −0.134397
\(136\) 0 0
\(137\) −13.8078 −1.17968 −0.589838 0.807521i \(-0.700809\pi\)
−0.589838 + 0.807521i \(0.700809\pi\)
\(138\) 0 0
\(139\) −7.80776 −0.662246 −0.331123 0.943588i \(-0.607427\pi\)
−0.331123 + 0.943588i \(0.607427\pi\)
\(140\) 0 0
\(141\) −6.68466 −0.562950
\(142\) 0 0
\(143\) 17.3693 1.45250
\(144\) 0 0
\(145\) 12.8769 1.06937
\(146\) 0 0
\(147\) −4.56155 −0.376231
\(148\) 0 0
\(149\) 3.80776 0.311944 0.155972 0.987761i \(-0.450149\pi\)
0.155972 + 0.987761i \(0.450149\pi\)
\(150\) 0 0
\(151\) −17.1231 −1.39346 −0.696729 0.717334i \(-0.745362\pi\)
−0.696729 + 0.717334i \(0.745362\pi\)
\(152\) 0 0
\(153\) 6.68466 0.540423
\(154\) 0 0
\(155\) 3.12311 0.250854
\(156\) 0 0
\(157\) −12.2462 −0.977354 −0.488677 0.872465i \(-0.662520\pi\)
−0.488677 + 0.872465i \(0.662520\pi\)
\(158\) 0 0
\(159\) 4.24621 0.336746
\(160\) 0 0
\(161\) −14.2462 −1.12276
\(162\) 0 0
\(163\) −2.24621 −0.175937 −0.0879684 0.996123i \(-0.528037\pi\)
−0.0879684 + 0.996123i \(0.528037\pi\)
\(164\) 0 0
\(165\) −8.68466 −0.676100
\(166\) 0 0
\(167\) −8.87689 −0.686915 −0.343457 0.939168i \(-0.611598\pi\)
−0.343457 + 0.939168i \(0.611598\pi\)
\(168\) 0 0
\(169\) −3.24621 −0.249709
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −4.24621 −0.322833 −0.161417 0.986886i \(-0.551606\pi\)
−0.161417 + 0.986886i \(0.551606\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −15.1231 −1.13035 −0.565177 0.824970i \(-0.691192\pi\)
−0.565177 + 0.824970i \(0.691192\pi\)
\(180\) 0 0
\(181\) 15.1231 1.12409 0.562046 0.827106i \(-0.310014\pi\)
0.562046 + 0.827106i \(0.310014\pi\)
\(182\) 0 0
\(183\) 6.68466 0.494144
\(184\) 0 0
\(185\) 12.4924 0.918461
\(186\) 0 0
\(187\) 37.1771 2.71866
\(188\) 0 0
\(189\) −1.56155 −0.113586
\(190\) 0 0
\(191\) −1.31534 −0.0951748 −0.0475874 0.998867i \(-0.515153\pi\)
−0.0475874 + 0.998867i \(0.515153\pi\)
\(192\) 0 0
\(193\) 18.8769 1.35879 0.679394 0.733773i \(-0.262243\pi\)
0.679394 + 0.733773i \(0.262243\pi\)
\(194\) 0 0
\(195\) −4.87689 −0.349242
\(196\) 0 0
\(197\) −1.36932 −0.0975598 −0.0487799 0.998810i \(-0.515533\pi\)
−0.0487799 + 0.998810i \(0.515533\pi\)
\(198\) 0 0
\(199\) 2.93087 0.207764 0.103882 0.994590i \(-0.466874\pi\)
0.103882 + 0.994590i \(0.466874\pi\)
\(200\) 0 0
\(201\) 6.24621 0.440574
\(202\) 0 0
\(203\) 12.8769 0.903781
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) 9.12311 0.634100
\(208\) 0 0
\(209\) 5.56155 0.384701
\(210\) 0 0
\(211\) 5.75379 0.396107 0.198054 0.980191i \(-0.436538\pi\)
0.198054 + 0.980191i \(0.436538\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.43845 0.166301
\(216\) 0 0
\(217\) 3.12311 0.212010
\(218\) 0 0
\(219\) −16.9309 −1.14408
\(220\) 0 0
\(221\) 20.8769 1.40433
\(222\) 0 0
\(223\) −11.3693 −0.761346 −0.380673 0.924710i \(-0.624308\pi\)
−0.380673 + 0.924710i \(0.624308\pi\)
\(224\) 0 0
\(225\) −2.56155 −0.170770
\(226\) 0 0
\(227\) −8.87689 −0.589180 −0.294590 0.955624i \(-0.595183\pi\)
−0.294590 + 0.955624i \(0.595183\pi\)
\(228\) 0 0
\(229\) 3.56155 0.235354 0.117677 0.993052i \(-0.462455\pi\)
0.117677 + 0.993052i \(0.462455\pi\)
\(230\) 0 0
\(231\) −8.68466 −0.571409
\(232\) 0 0
\(233\) −14.6847 −0.962024 −0.481012 0.876714i \(-0.659731\pi\)
−0.481012 + 0.876714i \(0.659731\pi\)
\(234\) 0 0
\(235\) 10.4384 0.680929
\(236\) 0 0
\(237\) 11.3693 0.738516
\(238\) 0 0
\(239\) 6.19224 0.400542 0.200271 0.979740i \(-0.435818\pi\)
0.200271 + 0.979740i \(0.435818\pi\)
\(240\) 0 0
\(241\) −11.3693 −0.732362 −0.366181 0.930544i \(-0.619335\pi\)
−0.366181 + 0.930544i \(0.619335\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 7.12311 0.455079
\(246\) 0 0
\(247\) 3.12311 0.198718
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −7.80776 −0.492822 −0.246411 0.969165i \(-0.579251\pi\)
−0.246411 + 0.969165i \(0.579251\pi\)
\(252\) 0 0
\(253\) 50.7386 3.18991
\(254\) 0 0
\(255\) −10.4384 −0.653681
\(256\) 0 0
\(257\) 13.1231 0.818597 0.409298 0.912401i \(-0.365773\pi\)
0.409298 + 0.912401i \(0.365773\pi\)
\(258\) 0 0
\(259\) 12.4924 0.776241
\(260\) 0 0
\(261\) −8.24621 −0.510428
\(262\) 0 0
\(263\) −14.6847 −0.905495 −0.452747 0.891639i \(-0.649556\pi\)
−0.452747 + 0.891639i \(0.649556\pi\)
\(264\) 0 0
\(265\) −6.63068 −0.407320
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) −4.87689 −0.295163
\(274\) 0 0
\(275\) −14.2462 −0.859079
\(276\) 0 0
\(277\) −18.6847 −1.12265 −0.561326 0.827595i \(-0.689709\pi\)
−0.561326 + 0.827595i \(0.689709\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −24.2462 −1.44641 −0.723204 0.690635i \(-0.757331\pi\)
−0.723204 + 0.690635i \(0.757331\pi\)
\(282\) 0 0
\(283\) 28.6847 1.70513 0.852563 0.522625i \(-0.175047\pi\)
0.852563 + 0.522625i \(0.175047\pi\)
\(284\) 0 0
\(285\) −1.56155 −0.0924984
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) 27.6847 1.62851
\(290\) 0 0
\(291\) 4.24621 0.248917
\(292\) 0 0
\(293\) −1.12311 −0.0656125 −0.0328063 0.999462i \(-0.510444\pi\)
−0.0328063 + 0.999462i \(0.510444\pi\)
\(294\) 0 0
\(295\) −18.7386 −1.09101
\(296\) 0 0
\(297\) 5.56155 0.322714
\(298\) 0 0
\(299\) 28.4924 1.64776
\(300\) 0 0
\(301\) 2.43845 0.140550
\(302\) 0 0
\(303\) 7.12311 0.409212
\(304\) 0 0
\(305\) −10.4384 −0.597704
\(306\) 0 0
\(307\) −4.49242 −0.256396 −0.128198 0.991749i \(-0.540919\pi\)
−0.128198 + 0.991749i \(0.540919\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −20.9309 −1.18688 −0.593440 0.804878i \(-0.702231\pi\)
−0.593440 + 0.804878i \(0.702231\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 2.43845 0.137391
\(316\) 0 0
\(317\) −18.8769 −1.06023 −0.530116 0.847925i \(-0.677852\pi\)
−0.530116 + 0.847925i \(0.677852\pi\)
\(318\) 0 0
\(319\) −45.8617 −2.56776
\(320\) 0 0
\(321\) −13.3693 −0.746203
\(322\) 0 0
\(323\) 6.68466 0.371944
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) −2.24621 −0.124216
\(328\) 0 0
\(329\) 10.4384 0.575490
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) −9.75379 −0.532906
\(336\) 0 0
\(337\) 20.7386 1.12971 0.564853 0.825192i \(-0.308933\pi\)
0.564853 + 0.825192i \(0.308933\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −11.1231 −0.602350
\(342\) 0 0
\(343\) 18.0540 0.974823
\(344\) 0 0
\(345\) −14.2462 −0.766990
\(346\) 0 0
\(347\) 27.4233 1.47216 0.736080 0.676895i \(-0.236675\pi\)
0.736080 + 0.676895i \(0.236675\pi\)
\(348\) 0 0
\(349\) 17.3153 0.926869 0.463434 0.886131i \(-0.346617\pi\)
0.463434 + 0.886131i \(0.346617\pi\)
\(350\) 0 0
\(351\) 3.12311 0.166699
\(352\) 0 0
\(353\) −28.2462 −1.50339 −0.751697 0.659509i \(-0.770764\pi\)
−0.751697 + 0.659509i \(0.770764\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.4384 −0.552461
\(358\) 0 0
\(359\) 0.438447 0.0231404 0.0115702 0.999933i \(-0.496317\pi\)
0.0115702 + 0.999933i \(0.496317\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 19.9309 1.04610
\(364\) 0 0
\(365\) 26.4384 1.38385
\(366\) 0 0
\(367\) 2.24621 0.117251 0.0586256 0.998280i \(-0.481328\pi\)
0.0586256 + 0.998280i \(0.481328\pi\)
\(368\) 0 0
\(369\) −5.12311 −0.266698
\(370\) 0 0
\(371\) −6.63068 −0.344248
\(372\) 0 0
\(373\) −36.4924 −1.88951 −0.944753 0.327783i \(-0.893698\pi\)
−0.944753 + 0.327783i \(0.893698\pi\)
\(374\) 0 0
\(375\) 11.8078 0.609750
\(376\) 0 0
\(377\) −25.7538 −1.32639
\(378\) 0 0
\(379\) −20.8769 −1.07237 −0.536187 0.844099i \(-0.680136\pi\)
−0.536187 + 0.844099i \(0.680136\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 12.8769 0.657979 0.328989 0.944334i \(-0.393292\pi\)
0.328989 + 0.944334i \(0.393292\pi\)
\(384\) 0 0
\(385\) 13.5616 0.691161
\(386\) 0 0
\(387\) −1.56155 −0.0793782
\(388\) 0 0
\(389\) 18.9309 0.959833 0.479917 0.877314i \(-0.340667\pi\)
0.479917 + 0.877314i \(0.340667\pi\)
\(390\) 0 0
\(391\) 60.9848 3.08414
\(392\) 0 0
\(393\) 20.6847 1.04340
\(394\) 0 0
\(395\) −17.7538 −0.893290
\(396\) 0 0
\(397\) −22.6847 −1.13851 −0.569255 0.822161i \(-0.692768\pi\)
−0.569255 + 0.822161i \(0.692768\pi\)
\(398\) 0 0
\(399\) −1.56155 −0.0781754
\(400\) 0 0
\(401\) −0.630683 −0.0314948 −0.0157474 0.999876i \(-0.505013\pi\)
−0.0157474 + 0.999876i \(0.505013\pi\)
\(402\) 0 0
\(403\) −6.24621 −0.311146
\(404\) 0 0
\(405\) −1.56155 −0.0775942
\(406\) 0 0
\(407\) −44.4924 −2.20541
\(408\) 0 0
\(409\) −0.246211 −0.0121744 −0.00608718 0.999981i \(-0.501938\pi\)
−0.00608718 + 0.999981i \(0.501938\pi\)
\(410\) 0 0
\(411\) −13.8078 −0.681087
\(412\) 0 0
\(413\) −18.7386 −0.922068
\(414\) 0 0
\(415\) −6.24621 −0.306614
\(416\) 0 0
\(417\) −7.80776 −0.382348
\(418\) 0 0
\(419\) −26.7386 −1.30627 −0.653134 0.757242i \(-0.726546\pi\)
−0.653134 + 0.757242i \(0.726546\pi\)
\(420\) 0 0
\(421\) 4.87689 0.237685 0.118843 0.992913i \(-0.462082\pi\)
0.118843 + 0.992913i \(0.462082\pi\)
\(422\) 0 0
\(423\) −6.68466 −0.325019
\(424\) 0 0
\(425\) −17.1231 −0.830593
\(426\) 0 0
\(427\) −10.4384 −0.505152
\(428\) 0 0
\(429\) 17.3693 0.838599
\(430\) 0 0
\(431\) −11.6155 −0.559500 −0.279750 0.960073i \(-0.590252\pi\)
−0.279750 + 0.960073i \(0.590252\pi\)
\(432\) 0 0
\(433\) −8.24621 −0.396288 −0.198144 0.980173i \(-0.563491\pi\)
−0.198144 + 0.980173i \(0.563491\pi\)
\(434\) 0 0
\(435\) 12.8769 0.617400
\(436\) 0 0
\(437\) 9.12311 0.436417
\(438\) 0 0
\(439\) 28.2462 1.34812 0.674059 0.738677i \(-0.264549\pi\)
0.674059 + 0.738677i \(0.264549\pi\)
\(440\) 0 0
\(441\) −4.56155 −0.217217
\(442\) 0 0
\(443\) 12.1922 0.579271 0.289635 0.957137i \(-0.406466\pi\)
0.289635 + 0.957137i \(0.406466\pi\)
\(444\) 0 0
\(445\) 9.36932 0.444148
\(446\) 0 0
\(447\) 3.80776 0.180101
\(448\) 0 0
\(449\) 35.3693 1.66918 0.834591 0.550871i \(-0.185704\pi\)
0.834591 + 0.550871i \(0.185704\pi\)
\(450\) 0 0
\(451\) −28.4924 −1.34166
\(452\) 0 0
\(453\) −17.1231 −0.804514
\(454\) 0 0
\(455\) 7.61553 0.357021
\(456\) 0 0
\(457\) −40.0540 −1.87365 −0.936823 0.349804i \(-0.886248\pi\)
−0.936823 + 0.349804i \(0.886248\pi\)
\(458\) 0 0
\(459\) 6.68466 0.312013
\(460\) 0 0
\(461\) 32.3002 1.50437 0.752185 0.658952i \(-0.229000\pi\)
0.752185 + 0.658952i \(0.229000\pi\)
\(462\) 0 0
\(463\) 31.8078 1.47823 0.739116 0.673578i \(-0.235243\pi\)
0.739116 + 0.673578i \(0.235243\pi\)
\(464\) 0 0
\(465\) 3.12311 0.144831
\(466\) 0 0
\(467\) 10.9309 0.505820 0.252910 0.967490i \(-0.418612\pi\)
0.252910 + 0.967490i \(0.418612\pi\)
\(468\) 0 0
\(469\) −9.75379 −0.450388
\(470\) 0 0
\(471\) −12.2462 −0.564276
\(472\) 0 0
\(473\) −8.68466 −0.399321
\(474\) 0 0
\(475\) −2.56155 −0.117532
\(476\) 0 0
\(477\) 4.24621 0.194421
\(478\) 0 0
\(479\) 15.3693 0.702242 0.351121 0.936330i \(-0.385801\pi\)
0.351121 + 0.936330i \(0.385801\pi\)
\(480\) 0 0
\(481\) −24.9848 −1.13921
\(482\) 0 0
\(483\) −14.2462 −0.648225
\(484\) 0 0
\(485\) −6.63068 −0.301084
\(486\) 0 0
\(487\) −33.1231 −1.50095 −0.750476 0.660898i \(-0.770176\pi\)
−0.750476 + 0.660898i \(0.770176\pi\)
\(488\) 0 0
\(489\) −2.24621 −0.101577
\(490\) 0 0
\(491\) −14.7386 −0.665145 −0.332573 0.943078i \(-0.607917\pi\)
−0.332573 + 0.943078i \(0.607917\pi\)
\(492\) 0 0
\(493\) −55.1231 −2.48262
\(494\) 0 0
\(495\) −8.68466 −0.390346
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9.17708 −0.410823 −0.205411 0.978676i \(-0.565853\pi\)
−0.205411 + 0.978676i \(0.565853\pi\)
\(500\) 0 0
\(501\) −8.87689 −0.396590
\(502\) 0 0
\(503\) 25.6155 1.14214 0.571070 0.820901i \(-0.306528\pi\)
0.571070 + 0.820901i \(0.306528\pi\)
\(504\) 0 0
\(505\) −11.1231 −0.494972
\(506\) 0 0
\(507\) −3.24621 −0.144169
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) 26.4384 1.16957
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 9.36932 0.412861
\(516\) 0 0
\(517\) −37.1771 −1.63505
\(518\) 0 0
\(519\) −4.24621 −0.186388
\(520\) 0 0
\(521\) 9.61553 0.421264 0.210632 0.977565i \(-0.432448\pi\)
0.210632 + 0.977565i \(0.432448\pi\)
\(522\) 0 0
\(523\) 17.3693 0.759507 0.379754 0.925088i \(-0.376009\pi\)
0.379754 + 0.925088i \(0.376009\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −13.3693 −0.582377
\(528\) 0 0
\(529\) 60.2311 2.61874
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −16.0000 −0.693037
\(534\) 0 0
\(535\) 20.8769 0.902587
\(536\) 0 0
\(537\) −15.1231 −0.652610
\(538\) 0 0
\(539\) −25.3693 −1.09273
\(540\) 0 0
\(541\) −17.8078 −0.765616 −0.382808 0.923828i \(-0.625043\pi\)
−0.382808 + 0.923828i \(0.625043\pi\)
\(542\) 0 0
\(543\) 15.1231 0.648995
\(544\) 0 0
\(545\) 3.50758 0.150248
\(546\) 0 0
\(547\) −0.384472 −0.0164388 −0.00821942 0.999966i \(-0.502616\pi\)
−0.00821942 + 0.999966i \(0.502616\pi\)
\(548\) 0 0
\(549\) 6.68466 0.285294
\(550\) 0 0
\(551\) −8.24621 −0.351300
\(552\) 0 0
\(553\) −17.7538 −0.754968
\(554\) 0 0
\(555\) 12.4924 0.530274
\(556\) 0 0
\(557\) 45.5616 1.93050 0.965252 0.261319i \(-0.0841575\pi\)
0.965252 + 0.261319i \(0.0841575\pi\)
\(558\) 0 0
\(559\) −4.87689 −0.206271
\(560\) 0 0
\(561\) 37.1771 1.56962
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 3.12311 0.131390
\(566\) 0 0
\(567\) −1.56155 −0.0655791
\(568\) 0 0
\(569\) −18.4924 −0.775243 −0.387621 0.921819i \(-0.626703\pi\)
−0.387621 + 0.921819i \(0.626703\pi\)
\(570\) 0 0
\(571\) 6.73863 0.282003 0.141002 0.990009i \(-0.454968\pi\)
0.141002 + 0.990009i \(0.454968\pi\)
\(572\) 0 0
\(573\) −1.31534 −0.0549492
\(574\) 0 0
\(575\) −23.3693 −0.974568
\(576\) 0 0
\(577\) −11.5616 −0.481314 −0.240657 0.970610i \(-0.577363\pi\)
−0.240657 + 0.970610i \(0.577363\pi\)
\(578\) 0 0
\(579\) 18.8769 0.784497
\(580\) 0 0
\(581\) −6.24621 −0.259137
\(582\) 0 0
\(583\) 23.6155 0.978055
\(584\) 0 0
\(585\) −4.87689 −0.201635
\(586\) 0 0
\(587\) −18.0540 −0.745167 −0.372584 0.927999i \(-0.621528\pi\)
−0.372584 + 0.927999i \(0.621528\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −1.36932 −0.0563262
\(592\) 0 0
\(593\) 38.4924 1.58069 0.790347 0.612659i \(-0.209900\pi\)
0.790347 + 0.612659i \(0.209900\pi\)
\(594\) 0 0
\(595\) 16.3002 0.668242
\(596\) 0 0
\(597\) 2.93087 0.119953
\(598\) 0 0
\(599\) 15.5076 0.633622 0.316811 0.948489i \(-0.397388\pi\)
0.316811 + 0.948489i \(0.397388\pi\)
\(600\) 0 0
\(601\) 39.8617 1.62599 0.812997 0.582268i \(-0.197834\pi\)
0.812997 + 0.582268i \(0.197834\pi\)
\(602\) 0 0
\(603\) 6.24621 0.254365
\(604\) 0 0
\(605\) −31.1231 −1.26533
\(606\) 0 0
\(607\) −22.9848 −0.932926 −0.466463 0.884541i \(-0.654472\pi\)
−0.466463 + 0.884541i \(0.654472\pi\)
\(608\) 0 0
\(609\) 12.8769 0.521798
\(610\) 0 0
\(611\) −20.8769 −0.844589
\(612\) 0 0
\(613\) 38.7926 1.56682 0.783409 0.621506i \(-0.213479\pi\)
0.783409 + 0.621506i \(0.213479\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −14.3002 −0.575704 −0.287852 0.957675i \(-0.592941\pi\)
−0.287852 + 0.957675i \(0.592941\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 9.12311 0.366098
\(622\) 0 0
\(623\) 9.36932 0.375374
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 0 0
\(627\) 5.56155 0.222107
\(628\) 0 0
\(629\) −53.4773 −2.13228
\(630\) 0 0
\(631\) 22.9309 0.912864 0.456432 0.889758i \(-0.349127\pi\)
0.456432 + 0.889758i \(0.349127\pi\)
\(632\) 0 0
\(633\) 5.75379 0.228693
\(634\) 0 0
\(635\) 28.1080 1.11543
\(636\) 0 0
\(637\) −14.2462 −0.564455
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −40.7386 −1.60908 −0.804540 0.593899i \(-0.797588\pi\)
−0.804540 + 0.593899i \(0.797588\pi\)
\(642\) 0 0
\(643\) 9.94602 0.392233 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(644\) 0 0
\(645\) 2.43845 0.0960138
\(646\) 0 0
\(647\) −21.8078 −0.857352 −0.428676 0.903458i \(-0.641020\pi\)
−0.428676 + 0.903458i \(0.641020\pi\)
\(648\) 0 0
\(649\) 66.7386 2.61972
\(650\) 0 0
\(651\) 3.12311 0.122404
\(652\) 0 0
\(653\) 40.3002 1.57707 0.788534 0.614991i \(-0.210840\pi\)
0.788534 + 0.614991i \(0.210840\pi\)
\(654\) 0 0
\(655\) −32.3002 −1.26207
\(656\) 0 0
\(657\) −16.9309 −0.660536
\(658\) 0 0
\(659\) 21.3693 0.832430 0.416215 0.909266i \(-0.363356\pi\)
0.416215 + 0.909266i \(0.363356\pi\)
\(660\) 0 0
\(661\) −42.7386 −1.66234 −0.831170 0.556018i \(-0.812328\pi\)
−0.831170 + 0.556018i \(0.812328\pi\)
\(662\) 0 0
\(663\) 20.8769 0.810791
\(664\) 0 0
\(665\) 2.43845 0.0945589
\(666\) 0 0
\(667\) −75.2311 −2.91296
\(668\) 0 0
\(669\) −11.3693 −0.439563
\(670\) 0 0
\(671\) 37.1771 1.43521
\(672\) 0 0
\(673\) −31.8617 −1.22818 −0.614090 0.789236i \(-0.710477\pi\)
−0.614090 + 0.789236i \(0.710477\pi\)
\(674\) 0 0
\(675\) −2.56155 −0.0985942
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −6.63068 −0.254462
\(680\) 0 0
\(681\) −8.87689 −0.340163
\(682\) 0 0
\(683\) −9.86174 −0.377349 −0.188674 0.982040i \(-0.560419\pi\)
−0.188674 + 0.982040i \(0.560419\pi\)
\(684\) 0 0
\(685\) 21.5616 0.823825
\(686\) 0 0
\(687\) 3.56155 0.135882
\(688\) 0 0
\(689\) 13.2614 0.505218
\(690\) 0 0
\(691\) 21.0691 0.801507 0.400754 0.916186i \(-0.368748\pi\)
0.400754 + 0.916186i \(0.368748\pi\)
\(692\) 0 0
\(693\) −8.68466 −0.329903
\(694\) 0 0
\(695\) 12.1922 0.462478
\(696\) 0 0
\(697\) −34.2462 −1.29717
\(698\) 0 0
\(699\) −14.6847 −0.555425
\(700\) 0 0
\(701\) 23.1231 0.873348 0.436674 0.899620i \(-0.356156\pi\)
0.436674 + 0.899620i \(0.356156\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 10.4384 0.393135
\(706\) 0 0
\(707\) −11.1231 −0.418327
\(708\) 0 0
\(709\) 9.50758 0.357065 0.178532 0.983934i \(-0.442865\pi\)
0.178532 + 0.983934i \(0.442865\pi\)
\(710\) 0 0
\(711\) 11.3693 0.426383
\(712\) 0 0
\(713\) −18.2462 −0.683326
\(714\) 0 0
\(715\) −27.1231 −1.01435
\(716\) 0 0
\(717\) 6.19224 0.231253
\(718\) 0 0
\(719\) 37.4233 1.39565 0.697827 0.716267i \(-0.254151\pi\)
0.697827 + 0.716267i \(0.254151\pi\)
\(720\) 0 0
\(721\) 9.36932 0.348932
\(722\) 0 0
\(723\) −11.3693 −0.422829
\(724\) 0 0
\(725\) 21.1231 0.784492
\(726\) 0 0
\(727\) 13.5616 0.502970 0.251485 0.967861i \(-0.419081\pi\)
0.251485 + 0.967861i \(0.419081\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.4384 −0.386080
\(732\) 0 0
\(733\) −18.4924 −0.683033 −0.341517 0.939876i \(-0.610941\pi\)
−0.341517 + 0.939876i \(0.610941\pi\)
\(734\) 0 0
\(735\) 7.12311 0.262740
\(736\) 0 0
\(737\) 34.7386 1.27961
\(738\) 0 0
\(739\) 25.1771 0.926154 0.463077 0.886318i \(-0.346745\pi\)
0.463077 + 0.886318i \(0.346745\pi\)
\(740\) 0 0
\(741\) 3.12311 0.114730
\(742\) 0 0
\(743\) 40.4924 1.48552 0.742761 0.669556i \(-0.233516\pi\)
0.742761 + 0.669556i \(0.233516\pi\)
\(744\) 0 0
\(745\) −5.94602 −0.217845
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 20.8769 0.762825
\(750\) 0 0
\(751\) 33.6155 1.22665 0.613324 0.789831i \(-0.289832\pi\)
0.613324 + 0.789831i \(0.289832\pi\)
\(752\) 0 0
\(753\) −7.80776 −0.284531
\(754\) 0 0
\(755\) 26.7386 0.973119
\(756\) 0 0
\(757\) −32.0540 −1.16502 −0.582511 0.812823i \(-0.697930\pi\)
−0.582511 + 0.812823i \(0.697930\pi\)
\(758\) 0 0
\(759\) 50.7386 1.84170
\(760\) 0 0
\(761\) 47.6695 1.72802 0.864009 0.503476i \(-0.167946\pi\)
0.864009 + 0.503476i \(0.167946\pi\)
\(762\) 0 0
\(763\) 3.50758 0.126983
\(764\) 0 0
\(765\) −10.4384 −0.377403
\(766\) 0 0
\(767\) 37.4773 1.35323
\(768\) 0 0
\(769\) −34.3002 −1.23690 −0.618448 0.785826i \(-0.712238\pi\)
−0.618448 + 0.785826i \(0.712238\pi\)
\(770\) 0 0
\(771\) 13.1231 0.472617
\(772\) 0 0
\(773\) 17.6155 0.633587 0.316793 0.948495i \(-0.397394\pi\)
0.316793 + 0.948495i \(0.397394\pi\)
\(774\) 0 0
\(775\) 5.12311 0.184027
\(776\) 0 0
\(777\) 12.4924 0.448163
\(778\) 0 0
\(779\) −5.12311 −0.183554
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.24621 −0.294696
\(784\) 0 0
\(785\) 19.1231 0.682533
\(786\) 0 0
\(787\) 10.2462 0.365238 0.182619 0.983184i \(-0.441543\pi\)
0.182619 + 0.983184i \(0.441543\pi\)
\(788\) 0 0
\(789\) −14.6847 −0.522788
\(790\) 0 0
\(791\) 3.12311 0.111045
\(792\) 0 0
\(793\) 20.8769 0.741360
\(794\) 0 0
\(795\) −6.63068 −0.235166
\(796\) 0 0
\(797\) 41.6155 1.47410 0.737049 0.675840i \(-0.236219\pi\)
0.737049 + 0.675840i \(0.236219\pi\)
\(798\) 0 0
\(799\) −44.6847 −1.58083
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −94.1619 −3.32290
\(804\) 0 0
\(805\) 22.2462 0.784076
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −7.94602 −0.279367 −0.139684 0.990196i \(-0.544609\pi\)
−0.139684 + 0.990196i \(0.544609\pi\)
\(810\) 0 0
\(811\) −35.1231 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) 3.50758 0.122865
\(816\) 0 0
\(817\) −1.56155 −0.0546318
\(818\) 0 0
\(819\) −4.87689 −0.170412
\(820\) 0 0
\(821\) −1.94602 −0.0679167 −0.0339584 0.999423i \(-0.510811\pi\)
−0.0339584 + 0.999423i \(0.510811\pi\)
\(822\) 0 0
\(823\) −40.3002 −1.40478 −0.702388 0.711794i \(-0.747883\pi\)
−0.702388 + 0.711794i \(0.747883\pi\)
\(824\) 0 0
\(825\) −14.2462 −0.495989
\(826\) 0 0
\(827\) −14.7386 −0.512513 −0.256256 0.966609i \(-0.582489\pi\)
−0.256256 + 0.966609i \(0.582489\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −18.6847 −0.648164
\(832\) 0 0
\(833\) −30.4924 −1.05650
\(834\) 0 0
\(835\) 13.8617 0.479705
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −49.8617 −1.72142 −0.860709 0.509097i \(-0.829979\pi\)
−0.860709 + 0.509097i \(0.829979\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 0 0
\(843\) −24.2462 −0.835084
\(844\) 0 0
\(845\) 5.06913 0.174383
\(846\) 0 0
\(847\) −31.1231 −1.06940
\(848\) 0 0
\(849\) 28.6847 0.984455
\(850\) 0 0
\(851\) −72.9848 −2.50189
\(852\) 0 0
\(853\) −7.26137 −0.248624 −0.124312 0.992243i \(-0.539672\pi\)
−0.124312 + 0.992243i \(0.539672\pi\)
\(854\) 0 0
\(855\) −1.56155 −0.0534040
\(856\) 0 0
\(857\) 15.8617 0.541827 0.270913 0.962604i \(-0.412674\pi\)
0.270913 + 0.962604i \(0.412674\pi\)
\(858\) 0 0
\(859\) −28.3002 −0.965590 −0.482795 0.875733i \(-0.660378\pi\)
−0.482795 + 0.875733i \(0.660378\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 6.63068 0.225450
\(866\) 0 0
\(867\) 27.6847 0.940220
\(868\) 0 0
\(869\) 63.2311 2.14497
\(870\) 0 0
\(871\) 19.5076 0.660989
\(872\) 0 0
\(873\) 4.24621 0.143712
\(874\) 0 0
\(875\) −18.4384 −0.623333
\(876\) 0 0
\(877\) −22.6307 −0.764184 −0.382092 0.924124i \(-0.624796\pi\)
−0.382092 + 0.924124i \(0.624796\pi\)
\(878\) 0 0
\(879\) −1.12311 −0.0378814
\(880\) 0 0
\(881\) −30.3002 −1.02084 −0.510420 0.859925i \(-0.670510\pi\)
−0.510420 + 0.859925i \(0.670510\pi\)
\(882\) 0 0
\(883\) −35.3153 −1.18846 −0.594228 0.804297i \(-0.702542\pi\)
−0.594228 + 0.804297i \(0.702542\pi\)
\(884\) 0 0
\(885\) −18.7386 −0.629892
\(886\) 0 0
\(887\) 51.2311 1.72017 0.860085 0.510150i \(-0.170410\pi\)
0.860085 + 0.510150i \(0.170410\pi\)
\(888\) 0 0
\(889\) 28.1080 0.942710
\(890\) 0 0
\(891\) 5.56155 0.186319
\(892\) 0 0
\(893\) −6.68466 −0.223694
\(894\) 0 0
\(895\) 23.6155 0.789380
\(896\) 0 0
\(897\) 28.4924 0.951334
\(898\) 0 0
\(899\) 16.4924 0.550053
\(900\) 0 0
\(901\) 28.3845 0.945624
\(902\) 0 0
\(903\) 2.43845 0.0811464
\(904\) 0 0
\(905\) −23.6155 −0.785007
\(906\) 0 0
\(907\) −43.1231 −1.43188 −0.715940 0.698162i \(-0.754001\pi\)
−0.715940 + 0.698162i \(0.754001\pi\)
\(908\) 0 0
\(909\) 7.12311 0.236259
\(910\) 0 0
\(911\) −7.12311 −0.235999 −0.118000 0.993014i \(-0.537648\pi\)
−0.118000 + 0.993014i \(0.537648\pi\)
\(912\) 0 0
\(913\) 22.2462 0.736242
\(914\) 0 0
\(915\) −10.4384 −0.345084
\(916\) 0 0
\(917\) −32.3002 −1.06665
\(918\) 0 0
\(919\) 11.5076 0.379600 0.189800 0.981823i \(-0.439216\pi\)
0.189800 + 0.981823i \(0.439216\pi\)
\(920\) 0 0
\(921\) −4.49242 −0.148030
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 20.4924 0.673787
\(926\) 0 0
\(927\) −6.00000 −0.197066
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −4.56155 −0.149499
\(932\) 0 0
\(933\) −20.9309 −0.685246
\(934\) 0 0
\(935\) −58.0540 −1.89857
\(936\) 0 0
\(937\) −12.4384 −0.406346 −0.203173 0.979143i \(-0.565125\pi\)
−0.203173 + 0.979143i \(0.565125\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −46.7386 −1.52202
\(944\) 0 0
\(945\) 2.43845 0.0793227
\(946\) 0 0
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) −52.8769 −1.71646
\(950\) 0 0
\(951\) −18.8769 −0.612125
\(952\) 0 0
\(953\) 29.6155 0.959341 0.479671 0.877449i \(-0.340756\pi\)
0.479671 + 0.877449i \(0.340756\pi\)
\(954\) 0 0
\(955\) 2.05398 0.0664651
\(956\) 0 0
\(957\) −45.8617 −1.48250
\(958\) 0 0
\(959\) 21.5616 0.696259
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −13.3693 −0.430820
\(964\) 0 0
\(965\) −29.4773 −0.948907
\(966\) 0 0
\(967\) 22.7386 0.731225 0.365613 0.930767i \(-0.380860\pi\)
0.365613 + 0.930767i \(0.380860\pi\)
\(968\) 0 0
\(969\) 6.68466 0.214742
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 12.1922 0.390865
\(974\) 0 0
\(975\) −8.00000 −0.256205
\(976\) 0 0
\(977\) −58.9848 −1.88709 −0.943546 0.331241i \(-0.892533\pi\)
−0.943546 + 0.331241i \(0.892533\pi\)
\(978\) 0 0
\(979\) −33.3693 −1.06649
\(980\) 0 0
\(981\) −2.24621 −0.0717160
\(982\) 0 0
\(983\) −18.7386 −0.597670 −0.298835 0.954305i \(-0.596598\pi\)
−0.298835 + 0.954305i \(0.596598\pi\)
\(984\) 0 0
\(985\) 2.13826 0.0681306
\(986\) 0 0
\(987\) 10.4384 0.332259
\(988\) 0 0
\(989\) −14.2462 −0.453003
\(990\) 0 0
\(991\) 35.8617 1.13919 0.569593 0.821927i \(-0.307101\pi\)
0.569593 + 0.821927i \(0.307101\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.57671 −0.145091
\(996\) 0 0
\(997\) 30.7926 0.975212 0.487606 0.873064i \(-0.337870\pi\)
0.487606 + 0.873064i \(0.337870\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 456.2.a.f.1.1 2
3.2 odd 2 1368.2.a.k.1.2 2
4.3 odd 2 912.2.a.m.1.1 2
8.3 odd 2 3648.2.a.br.1.2 2
8.5 even 2 3648.2.a.bl.1.2 2
12.11 even 2 2736.2.a.z.1.2 2
19.18 odd 2 8664.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.a.f.1.1 2 1.1 even 1 trivial
912.2.a.m.1.1 2 4.3 odd 2
1368.2.a.k.1.2 2 3.2 odd 2
2736.2.a.z.1.2 2 12.11 even 2
3648.2.a.bl.1.2 2 8.5 even 2
3648.2.a.br.1.2 2 8.3 odd 2
8664.2.a.r.1.1 2 19.18 odd 2