Properties

Label 4864.2.a.bm.1.5
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.34309996544.1
Defining polynomial: \(x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.836706\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.222191 q^{3} -1.55081 q^{5} +3.06334 q^{7} -2.95063 q^{9} +O(q^{10})\) \(q+0.222191 q^{3} -1.55081 q^{5} +3.06334 q^{7} -2.95063 q^{9} +0.950631 q^{11} -3.36113 q^{13} -0.344577 q^{15} +1.82843 q^{17} +1.00000 q^{19} +0.680647 q^{21} +3.36113 q^{23} -2.59499 q^{25} -1.32218 q^{27} -4.95873 q^{29} +3.44620 q^{31} +0.211222 q^{33} -4.75066 q^{35} +2.95889 q^{37} -0.746814 q^{39} -4.55687 q^{41} -1.69373 q^{43} +4.57587 q^{45} +3.39941 q^{47} +2.38404 q^{49} +0.406261 q^{51} -4.47142 q^{53} -1.47425 q^{55} +0.222191 q^{57} +0.395014 q^{59} -5.34158 q^{61} -9.03878 q^{63} +5.21247 q^{65} -6.85064 q^{67} +0.746814 q^{69} -5.15207 q^{71} -1.19998 q^{73} -0.576584 q^{75} +2.91210 q^{77} +14.7249 q^{79} +8.55812 q^{81} -3.35564 q^{83} -2.83554 q^{85} -1.10179 q^{87} -9.64561 q^{89} -10.2963 q^{91} +0.765715 q^{93} -1.55081 q^{95} +3.55562 q^{97} -2.80496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{9} - 20q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 24q^{33} - 12q^{35} + 8q^{41} - 28q^{43} + 8q^{49} - 12q^{51} - 4q^{57} - 36q^{59} + 8q^{65} - 28q^{67} - 8q^{73} - 68q^{75} - 32q^{81} - 40q^{83} - 8q^{89} + 12q^{91} + 40q^{97} - 60q^{99} + O(q^{100}) \)

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
\(3\) 0.222191 0.128282 0.0641411 0.997941i \(-0.479569\pi\)
0.0641411 + 0.997941i \(0.479569\pi\)
\(4\) 0 0
\(5\) −1.55081 −0.693543 −0.346772 0.937950i \(-0.612722\pi\)
−0.346772 + 0.937950i \(0.612722\pi\)
\(6\) 0 0
\(7\) 3.06334 1.15783 0.578917 0.815387i \(-0.303476\pi\)
0.578917 + 0.815387i \(0.303476\pi\)
\(8\) 0 0
\(9\) −2.95063 −0.983544
\(10\) 0 0
\(11\) 0.950631 0.286626 0.143313 0.989677i \(-0.454224\pi\)
0.143313 + 0.989677i \(0.454224\pi\)
\(12\) 0 0
\(13\) −3.36113 −0.932209 −0.466105 0.884730i \(-0.654343\pi\)
−0.466105 + 0.884730i \(0.654343\pi\)
\(14\) 0 0
\(15\) −0.344577 −0.0889693
\(16\) 0 0
\(17\) 1.82843 0.443459 0.221729 0.975108i \(-0.428830\pi\)
0.221729 + 0.975108i \(0.428830\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.680647 0.148529
\(22\) 0 0
\(23\) 3.36113 0.700844 0.350422 0.936592i \(-0.386038\pi\)
0.350422 + 0.936592i \(0.386038\pi\)
\(24\) 0 0
\(25\) −2.59499 −0.518998
\(26\) 0 0
\(27\) −1.32218 −0.254453
\(28\) 0 0
\(29\) −4.95873 −0.920812 −0.460406 0.887708i \(-0.652296\pi\)
−0.460406 + 0.887708i \(0.652296\pi\)
\(30\) 0 0
\(31\) 3.44620 0.618955 0.309478 0.950907i \(-0.399846\pi\)
0.309478 + 0.950907i \(0.399846\pi\)
\(32\) 0 0
\(33\) 0.211222 0.0367690
\(34\) 0 0
\(35\) −4.75066 −0.803007
\(36\) 0 0
\(37\) 2.95889 0.486439 0.243219 0.969971i \(-0.421797\pi\)
0.243219 + 0.969971i \(0.421797\pi\)
\(38\) 0 0
\(39\) −0.746814 −0.119586
\(40\) 0 0
\(41\) −4.55687 −0.711663 −0.355832 0.934550i \(-0.615802\pi\)
−0.355832 + 0.934550i \(0.615802\pi\)
\(42\) 0 0
\(43\) −1.69373 −0.258291 −0.129145 0.991626i \(-0.541223\pi\)
−0.129145 + 0.991626i \(0.541223\pi\)
\(44\) 0 0
\(45\) 4.57587 0.682130
\(46\) 0 0
\(47\) 3.39941 0.495855 0.247927 0.968779i \(-0.420251\pi\)
0.247927 + 0.968779i \(0.420251\pi\)
\(48\) 0 0
\(49\) 2.38404 0.340578
\(50\) 0 0
\(51\) 0.406261 0.0568879
\(52\) 0 0
\(53\) −4.47142 −0.614197 −0.307098 0.951678i \(-0.599358\pi\)
−0.307098 + 0.951678i \(0.599358\pi\)
\(54\) 0 0
\(55\) −1.47425 −0.198788
\(56\) 0 0
\(57\) 0.222191 0.0294300
\(58\) 0 0
\(59\) 0.395014 0.0514264 0.0257132 0.999669i \(-0.491814\pi\)
0.0257132 + 0.999669i \(0.491814\pi\)
\(60\) 0 0
\(61\) −5.34158 −0.683920 −0.341960 0.939715i \(-0.611091\pi\)
−0.341960 + 0.939715i \(0.611091\pi\)
\(62\) 0 0
\(63\) −9.03878 −1.13878
\(64\) 0 0
\(65\) 5.21247 0.646528
\(66\) 0 0
\(67\) −6.85064 −0.836939 −0.418470 0.908231i \(-0.637433\pi\)
−0.418470 + 0.908231i \(0.637433\pi\)
\(68\) 0 0
\(69\) 0.746814 0.0899058
\(70\) 0 0
\(71\) −5.15207 −0.611438 −0.305719 0.952122i \(-0.598897\pi\)
−0.305719 + 0.952122i \(0.598897\pi\)
\(72\) 0 0
\(73\) −1.19998 −0.140446 −0.0702232 0.997531i \(-0.522371\pi\)
−0.0702232 + 0.997531i \(0.522371\pi\)
\(74\) 0 0
\(75\) −0.576584 −0.0665782
\(76\) 0 0
\(77\) 2.91210 0.331865
\(78\) 0 0
\(79\) 14.7249 1.65669 0.828343 0.560222i \(-0.189284\pi\)
0.828343 + 0.560222i \(0.189284\pi\)
\(80\) 0 0
\(81\) 8.55812 0.950902
\(82\) 0 0
\(83\) −3.35564 −0.368330 −0.184165 0.982895i \(-0.558958\pi\)
−0.184165 + 0.982895i \(0.558958\pi\)
\(84\) 0 0
\(85\) −2.83554 −0.307558
\(86\) 0 0
\(87\) −1.10179 −0.118124
\(88\) 0 0
\(89\) −9.64561 −1.02243 −0.511216 0.859452i \(-0.670805\pi\)
−0.511216 + 0.859452i \(0.670805\pi\)
\(90\) 0 0
\(91\) −10.2963 −1.07934
\(92\) 0 0
\(93\) 0.765715 0.0794010
\(94\) 0 0
\(95\) −1.55081 −0.159110
\(96\) 0 0
\(97\) 3.55562 0.361018 0.180509 0.983573i \(-0.442225\pi\)
0.180509 + 0.983573i \(0.442225\pi\)
\(98\) 0 0
\(99\) −2.80496 −0.281909
\(100\) 0 0
\(101\) −13.5381 −1.34709 −0.673545 0.739146i \(-0.735229\pi\)
−0.673545 + 0.739146i \(0.735229\pi\)
\(102\) 0 0
\(103\) −6.64570 −0.654820 −0.327410 0.944882i \(-0.606176\pi\)
−0.327410 + 0.944882i \(0.606176\pi\)
\(104\) 0 0
\(105\) −1.05555 −0.103012
\(106\) 0 0
\(107\) −15.8519 −1.53246 −0.766230 0.642566i \(-0.777870\pi\)
−0.766230 + 0.642566i \(0.777870\pi\)
\(108\) 0 0
\(109\) 13.4213 1.28553 0.642764 0.766064i \(-0.277788\pi\)
0.642764 + 0.766064i \(0.277788\pi\)
\(110\) 0 0
\(111\) 0.657440 0.0624015
\(112\) 0 0
\(113\) 6.65810 0.626342 0.313171 0.949697i \(-0.398609\pi\)
0.313171 + 0.949697i \(0.398609\pi\)
\(114\) 0 0
\(115\) −5.21247 −0.486065
\(116\) 0 0
\(117\) 9.91745 0.916869
\(118\) 0 0
\(119\) 5.60109 0.513451
\(120\) 0 0
\(121\) −10.0963 −0.917846
\(122\) 0 0
\(123\) −1.01250 −0.0912937
\(124\) 0 0
\(125\) 11.7784 1.05349
\(126\) 0 0
\(127\) −6.54782 −0.581025 −0.290512 0.956871i \(-0.593826\pi\)
−0.290512 + 0.956871i \(0.593826\pi\)
\(128\) 0 0
\(129\) −0.376331 −0.0331341
\(130\) 0 0
\(131\) −14.2838 −1.24798 −0.623990 0.781432i \(-0.714489\pi\)
−0.623990 + 0.781432i \(0.714489\pi\)
\(132\) 0 0
\(133\) 3.06334 0.265625
\(134\) 0 0
\(135\) 2.05045 0.176474
\(136\) 0 0
\(137\) −11.9138 −1.01786 −0.508931 0.860808i \(-0.669959\pi\)
−0.508931 + 0.860808i \(0.669959\pi\)
\(138\) 0 0
\(139\) −11.1950 −0.949551 −0.474775 0.880107i \(-0.657471\pi\)
−0.474775 + 0.880107i \(0.657471\pi\)
\(140\) 0 0
\(141\) 0.755320 0.0636094
\(142\) 0 0
\(143\) −3.19519 −0.267195
\(144\) 0 0
\(145\) 7.69004 0.638623
\(146\) 0 0
\(147\) 0.529714 0.0436901
\(148\) 0 0
\(149\) 7.67749 0.628964 0.314482 0.949263i \(-0.398169\pi\)
0.314482 + 0.949263i \(0.398169\pi\)
\(150\) 0 0
\(151\) −3.10162 −0.252406 −0.126203 0.992004i \(-0.540279\pi\)
−0.126203 + 0.992004i \(0.540279\pi\)
\(152\) 0 0
\(153\) −5.39501 −0.436161
\(154\) 0 0
\(155\) −5.34440 −0.429272
\(156\) 0 0
\(157\) −14.8994 −1.18910 −0.594550 0.804059i \(-0.702670\pi\)
−0.594550 + 0.804059i \(0.702670\pi\)
\(158\) 0 0
\(159\) −0.993511 −0.0787905
\(160\) 0 0
\(161\) 10.2963 0.811460
\(162\) 0 0
\(163\) −18.5481 −1.45280 −0.726400 0.687272i \(-0.758808\pi\)
−0.726400 + 0.687272i \(0.758808\pi\)
\(164\) 0 0
\(165\) −0.327565 −0.0255009
\(166\) 0 0
\(167\) 11.4532 0.886274 0.443137 0.896454i \(-0.353865\pi\)
0.443137 + 0.896454i \(0.353865\pi\)
\(168\) 0 0
\(169\) −1.70281 −0.130986
\(170\) 0 0
\(171\) −2.95063 −0.225640
\(172\) 0 0
\(173\) 7.20956 0.548133 0.274066 0.961711i \(-0.411631\pi\)
0.274066 + 0.961711i \(0.411631\pi\)
\(174\) 0 0
\(175\) −7.94933 −0.600913
\(176\) 0 0
\(177\) 0.0877686 0.00659710
\(178\) 0 0
\(179\) 4.90251 0.366431 0.183215 0.983073i \(-0.441349\pi\)
0.183215 + 0.983073i \(0.441349\pi\)
\(180\) 0 0
\(181\) 19.1775 1.42545 0.712725 0.701444i \(-0.247461\pi\)
0.712725 + 0.701444i \(0.247461\pi\)
\(182\) 0 0
\(183\) −1.18685 −0.0877347
\(184\) 0 0
\(185\) −4.58868 −0.337366
\(186\) 0 0
\(187\) 1.73816 0.127107
\(188\) 0 0
\(189\) −4.05028 −0.294615
\(190\) 0 0
\(191\) −22.8812 −1.65563 −0.827814 0.561003i \(-0.810416\pi\)
−0.827814 + 0.561003i \(0.810416\pi\)
\(192\) 0 0
\(193\) −2.11248 −0.152060 −0.0760300 0.997106i \(-0.524224\pi\)
−0.0760300 + 0.997106i \(0.524224\pi\)
\(194\) 0 0
\(195\) 1.15817 0.0829380
\(196\) 0 0
\(197\) −6.05012 −0.431053 −0.215526 0.976498i \(-0.569147\pi\)
−0.215526 + 0.976498i \(0.569147\pi\)
\(198\) 0 0
\(199\) −4.03090 −0.285743 −0.142872 0.989741i \(-0.545634\pi\)
−0.142872 + 0.989741i \(0.545634\pi\)
\(200\) 0 0
\(201\) −1.52215 −0.107364
\(202\) 0 0
\(203\) −15.1903 −1.06615
\(204\) 0 0
\(205\) 7.06683 0.493569
\(206\) 0 0
\(207\) −9.91745 −0.689310
\(208\) 0 0
\(209\) 0.950631 0.0657565
\(210\) 0 0
\(211\) −1.66532 −0.114646 −0.0573228 0.998356i \(-0.518256\pi\)
−0.0573228 + 0.998356i \(0.518256\pi\)
\(212\) 0 0
\(213\) −1.14475 −0.0784366
\(214\) 0 0
\(215\) 2.62665 0.179136
\(216\) 0 0
\(217\) 10.5569 0.716647
\(218\) 0 0
\(219\) −0.266624 −0.0180168
\(220\) 0 0
\(221\) −6.14558 −0.413396
\(222\) 0 0
\(223\) 28.9352 1.93764 0.968821 0.247761i \(-0.0796948\pi\)
0.968821 + 0.247761i \(0.0796948\pi\)
\(224\) 0 0
\(225\) 7.65685 0.510457
\(226\) 0 0
\(227\) −20.0519 −1.33089 −0.665445 0.746447i \(-0.731758\pi\)
−0.665445 + 0.746447i \(0.731758\pi\)
\(228\) 0 0
\(229\) 14.9953 0.990919 0.495459 0.868631i \(-0.335000\pi\)
0.495459 + 0.868631i \(0.335000\pi\)
\(230\) 0 0
\(231\) 0.647045 0.0425724
\(232\) 0 0
\(233\) 11.2738 0.738570 0.369285 0.929316i \(-0.379603\pi\)
0.369285 + 0.929316i \(0.379603\pi\)
\(234\) 0 0
\(235\) −5.27184 −0.343897
\(236\) 0 0
\(237\) 3.27176 0.212523
\(238\) 0 0
\(239\) 19.9049 1.28754 0.643770 0.765219i \(-0.277369\pi\)
0.643770 + 0.765219i \(0.277369\pi\)
\(240\) 0 0
\(241\) −15.8163 −1.01882 −0.509408 0.860525i \(-0.670135\pi\)
−0.509408 + 0.860525i \(0.670135\pi\)
\(242\) 0 0
\(243\) 5.86808 0.376437
\(244\) 0 0
\(245\) −3.69720 −0.236205
\(246\) 0 0
\(247\) −3.36113 −0.213863
\(248\) 0 0
\(249\) −0.745595 −0.0472501
\(250\) 0 0
\(251\) 10.5087 0.663306 0.331653 0.943401i \(-0.392394\pi\)
0.331653 + 0.943401i \(0.392394\pi\)
\(252\) 0 0
\(253\) 3.19519 0.200880
\(254\) 0 0
\(255\) −0.630033 −0.0394542
\(256\) 0 0
\(257\) −1.70129 −0.106123 −0.0530617 0.998591i \(-0.516898\pi\)
−0.0530617 + 0.998591i \(0.516898\pi\)
\(258\) 0 0
\(259\) 9.06409 0.563215
\(260\) 0 0
\(261\) 14.6314 0.905659
\(262\) 0 0
\(263\) 13.8359 0.853157 0.426578 0.904451i \(-0.359719\pi\)
0.426578 + 0.904451i \(0.359719\pi\)
\(264\) 0 0
\(265\) 6.93432 0.425972
\(266\) 0 0
\(267\) −2.14317 −0.131160
\(268\) 0 0
\(269\) −14.9655 −0.912466 −0.456233 0.889860i \(-0.650802\pi\)
−0.456233 + 0.889860i \(0.650802\pi\)
\(270\) 0 0
\(271\) −4.12684 −0.250688 −0.125344 0.992113i \(-0.540003\pi\)
−0.125344 + 0.992113i \(0.540003\pi\)
\(272\) 0 0
\(273\) −2.28774 −0.138461
\(274\) 0 0
\(275\) −2.46688 −0.148758
\(276\) 0 0
\(277\) −25.7721 −1.54849 −0.774247 0.632884i \(-0.781871\pi\)
−0.774247 + 0.632884i \(0.781871\pi\)
\(278\) 0 0
\(279\) −10.1685 −0.608769
\(280\) 0 0
\(281\) −10.2694 −0.612621 −0.306311 0.951932i \(-0.599095\pi\)
−0.306311 + 0.951932i \(0.599095\pi\)
\(282\) 0 0
\(283\) 6.99506 0.415813 0.207907 0.978149i \(-0.433335\pi\)
0.207907 + 0.978149i \(0.433335\pi\)
\(284\) 0 0
\(285\) −0.344577 −0.0204110
\(286\) 0 0
\(287\) −13.9592 −0.823987
\(288\) 0 0
\(289\) −13.6569 −0.803344
\(290\) 0 0
\(291\) 0.790027 0.0463122
\(292\) 0 0
\(293\) −31.8434 −1.86031 −0.930157 0.367162i \(-0.880330\pi\)
−0.930157 + 0.367162i \(0.880330\pi\)
\(294\) 0 0
\(295\) −0.612591 −0.0356664
\(296\) 0 0
\(297\) −1.25690 −0.0729330
\(298\) 0 0
\(299\) −11.2972 −0.653333
\(300\) 0 0
\(301\) −5.18846 −0.299058
\(302\) 0 0
\(303\) −3.00805 −0.172808
\(304\) 0 0
\(305\) 8.28378 0.474328
\(306\) 0 0
\(307\) −6.45813 −0.368585 −0.184292 0.982871i \(-0.558999\pi\)
−0.184292 + 0.982871i \(0.558999\pi\)
\(308\) 0 0
\(309\) −1.47662 −0.0840018
\(310\) 0 0
\(311\) −28.1973 −1.59892 −0.799462 0.600716i \(-0.794882\pi\)
−0.799462 + 0.600716i \(0.794882\pi\)
\(312\) 0 0
\(313\) 17.1212 0.967746 0.483873 0.875138i \(-0.339230\pi\)
0.483873 + 0.875138i \(0.339230\pi\)
\(314\) 0 0
\(315\) 14.0174 0.789793
\(316\) 0 0
\(317\) −29.5841 −1.66161 −0.830804 0.556564i \(-0.812119\pi\)
−0.830804 + 0.556564i \(0.812119\pi\)
\(318\) 0 0
\(319\) −4.71392 −0.263929
\(320\) 0 0
\(321\) −3.52215 −0.196587
\(322\) 0 0
\(323\) 1.82843 0.101736
\(324\) 0 0
\(325\) 8.72209 0.483815
\(326\) 0 0
\(327\) 2.98210 0.164910
\(328\) 0 0
\(329\) 10.4135 0.574117
\(330\) 0 0
\(331\) −20.5519 −1.12964 −0.564818 0.825215i \(-0.691054\pi\)
−0.564818 + 0.825215i \(0.691054\pi\)
\(332\) 0 0
\(333\) −8.73060 −0.478434
\(334\) 0 0
\(335\) 10.6240 0.580454
\(336\) 0 0
\(337\) −19.1832 −1.04497 −0.522487 0.852647i \(-0.674996\pi\)
−0.522487 + 0.852647i \(0.674996\pi\)
\(338\) 0 0
\(339\) 1.47937 0.0803485
\(340\) 0 0
\(341\) 3.27606 0.177409
\(342\) 0 0
\(343\) −14.1402 −0.763501
\(344\) 0 0
\(345\) −1.15817 −0.0623536
\(346\) 0 0
\(347\) −10.5507 −0.566390 −0.283195 0.959062i \(-0.591394\pi\)
−0.283195 + 0.959062i \(0.591394\pi\)
\(348\) 0 0
\(349\) 29.4693 1.57745 0.788727 0.614744i \(-0.210741\pi\)
0.788727 + 0.614744i \(0.210741\pi\)
\(350\) 0 0
\(351\) 4.44401 0.237204
\(352\) 0 0
\(353\) −3.98903 −0.212315 −0.106157 0.994349i \(-0.533855\pi\)
−0.106157 + 0.994349i \(0.533855\pi\)
\(354\) 0 0
\(355\) 7.98988 0.424059
\(356\) 0 0
\(357\) 1.24451 0.0658667
\(358\) 0 0
\(359\) −14.9377 −0.788380 −0.394190 0.919029i \(-0.628975\pi\)
−0.394190 + 0.919029i \(0.628975\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.24331 −0.117743
\(364\) 0 0
\(365\) 1.86093 0.0974057
\(366\) 0 0
\(367\) −19.1009 −0.997059 −0.498529 0.866873i \(-0.666126\pi\)
−0.498529 + 0.866873i \(0.666126\pi\)
\(368\) 0 0
\(369\) 13.4456 0.699952
\(370\) 0 0
\(371\) −13.6975 −0.711137
\(372\) 0 0
\(373\) 0.461357 0.0238882 0.0119441 0.999929i \(-0.496198\pi\)
0.0119441 + 0.999929i \(0.496198\pi\)
\(374\) 0 0
\(375\) 2.61706 0.135144
\(376\) 0 0
\(377\) 16.6669 0.858390
\(378\) 0 0
\(379\) 29.9622 1.53906 0.769528 0.638613i \(-0.220492\pi\)
0.769528 + 0.638613i \(0.220492\pi\)
\(380\) 0 0
\(381\) −1.45487 −0.0745352
\(382\) 0 0
\(383\) 16.9077 0.863944 0.431972 0.901887i \(-0.357818\pi\)
0.431972 + 0.901887i \(0.357818\pi\)
\(384\) 0 0
\(385\) −4.51612 −0.230163
\(386\) 0 0
\(387\) 4.99756 0.254040
\(388\) 0 0
\(389\) 7.92418 0.401772 0.200886 0.979615i \(-0.435618\pi\)
0.200886 + 0.979615i \(0.435618\pi\)
\(390\) 0 0
\(391\) 6.14558 0.310795
\(392\) 0 0
\(393\) −3.17373 −0.160094
\(394\) 0 0
\(395\) −22.8356 −1.14898
\(396\) 0 0
\(397\) −13.6718 −0.686170 −0.343085 0.939304i \(-0.611472\pi\)
−0.343085 + 0.939304i \(0.611472\pi\)
\(398\) 0 0
\(399\) 0.680647 0.0340750
\(400\) 0 0
\(401\) 12.0107 0.599785 0.299893 0.953973i \(-0.403049\pi\)
0.299893 + 0.953973i \(0.403049\pi\)
\(402\) 0 0
\(403\) −11.5831 −0.576996
\(404\) 0 0
\(405\) −13.2720 −0.659492
\(406\) 0 0
\(407\) 2.81281 0.139426
\(408\) 0 0
\(409\) 9.47063 0.468292 0.234146 0.972201i \(-0.424771\pi\)
0.234146 + 0.972201i \(0.424771\pi\)
\(410\) 0 0
\(411\) −2.64713 −0.130574
\(412\) 0 0
\(413\) 1.21006 0.0595432
\(414\) 0 0
\(415\) 5.20396 0.255453
\(416\) 0 0
\(417\) −2.48744 −0.121811
\(418\) 0 0
\(419\) −28.0843 −1.37201 −0.686004 0.727598i \(-0.740637\pi\)
−0.686004 + 0.727598i \(0.740637\pi\)
\(420\) 0 0
\(421\) −9.83238 −0.479201 −0.239601 0.970872i \(-0.577016\pi\)
−0.239601 + 0.970872i \(0.577016\pi\)
\(422\) 0 0
\(423\) −10.0304 −0.487695
\(424\) 0 0
\(425\) −4.74475 −0.230154
\(426\) 0 0
\(427\) −16.3631 −0.791865
\(428\) 0 0
\(429\) −0.709944 −0.0342764
\(430\) 0 0
\(431\) 8.94284 0.430761 0.215381 0.976530i \(-0.430901\pi\)
0.215381 + 0.976530i \(0.430901\pi\)
\(432\) 0 0
\(433\) −29.5274 −1.41900 −0.709499 0.704707i \(-0.751079\pi\)
−0.709499 + 0.704707i \(0.751079\pi\)
\(434\) 0 0
\(435\) 1.70866 0.0819240
\(436\) 0 0
\(437\) 3.36113 0.160785
\(438\) 0 0
\(439\) 20.5113 0.978953 0.489477 0.872016i \(-0.337188\pi\)
0.489477 + 0.872016i \(0.337188\pi\)
\(440\) 0 0
\(441\) −7.03444 −0.334973
\(442\) 0 0
\(443\) 13.3188 0.632794 0.316397 0.948627i \(-0.397527\pi\)
0.316397 + 0.948627i \(0.397527\pi\)
\(444\) 0 0
\(445\) 14.9585 0.709101
\(446\) 0 0
\(447\) 1.70587 0.0806850
\(448\) 0 0
\(449\) 10.5138 0.496177 0.248089 0.968737i \(-0.420198\pi\)
0.248089 + 0.968737i \(0.420198\pi\)
\(450\) 0 0
\(451\) −4.33190 −0.203981
\(452\) 0 0
\(453\) −0.689153 −0.0323792
\(454\) 0 0
\(455\) 15.9676 0.748571
\(456\) 0 0
\(457\) 26.5831 1.24351 0.621753 0.783214i \(-0.286421\pi\)
0.621753 + 0.783214i \(0.286421\pi\)
\(458\) 0 0
\(459\) −2.41751 −0.112840
\(460\) 0 0
\(461\) −19.2417 −0.896175 −0.448087 0.893990i \(-0.647895\pi\)
−0.448087 + 0.893990i \(0.647895\pi\)
\(462\) 0 0
\(463\) −2.24701 −0.104427 −0.0522137 0.998636i \(-0.516628\pi\)
−0.0522137 + 0.998636i \(0.516628\pi\)
\(464\) 0 0
\(465\) −1.18748 −0.0550680
\(466\) 0 0
\(467\) −27.5682 −1.27570 −0.637852 0.770159i \(-0.720177\pi\)
−0.637852 + 0.770159i \(0.720177\pi\)
\(468\) 0 0
\(469\) −20.9858 −0.969036
\(470\) 0 0
\(471\) −3.31051 −0.152540
\(472\) 0 0
\(473\) −1.61011 −0.0740329
\(474\) 0 0
\(475\) −2.59499 −0.119066
\(476\) 0 0
\(477\) 13.1935 0.604089
\(478\) 0 0
\(479\) −10.5447 −0.481802 −0.240901 0.970550i \(-0.577443\pi\)
−0.240901 + 0.970550i \(0.577443\pi\)
\(480\) 0 0
\(481\) −9.94521 −0.453463
\(482\) 0 0
\(483\) 2.28774 0.104096
\(484\) 0 0
\(485\) −5.51409 −0.250382
\(486\) 0 0
\(487\) 39.7294 1.80031 0.900154 0.435571i \(-0.143454\pi\)
0.900154 + 0.435571i \(0.143454\pi\)
\(488\) 0 0
\(489\) −4.12123 −0.186369
\(490\) 0 0
\(491\) −31.0150 −1.39969 −0.699844 0.714296i \(-0.746747\pi\)
−0.699844 + 0.714296i \(0.746747\pi\)
\(492\) 0 0
\(493\) −9.06667 −0.408342
\(494\) 0 0
\(495\) 4.34996 0.195516
\(496\) 0 0
\(497\) −15.7825 −0.707943
\(498\) 0 0
\(499\) 4.90620 0.219632 0.109816 0.993952i \(-0.464974\pi\)
0.109816 + 0.993952i \(0.464974\pi\)
\(500\) 0 0
\(501\) 2.54480 0.113693
\(502\) 0 0
\(503\) 14.6847 0.654759 0.327380 0.944893i \(-0.393834\pi\)
0.327380 + 0.944893i \(0.393834\pi\)
\(504\) 0 0
\(505\) 20.9950 0.934265
\(506\) 0 0
\(507\) −0.378351 −0.0168031
\(508\) 0 0
\(509\) 20.3988 0.904159 0.452080 0.891978i \(-0.350682\pi\)
0.452080 + 0.891978i \(0.350682\pi\)
\(510\) 0 0
\(511\) −3.67593 −0.162614
\(512\) 0 0
\(513\) −1.32218 −0.0583756
\(514\) 0 0
\(515\) 10.3062 0.454146
\(516\) 0 0
\(517\) 3.23158 0.142125
\(518\) 0 0
\(519\) 1.60190 0.0703157
\(520\) 0 0
\(521\) 22.4356 0.982923 0.491462 0.870899i \(-0.336463\pi\)
0.491462 + 0.870899i \(0.336463\pi\)
\(522\) 0 0
\(523\) 9.71531 0.424821 0.212410 0.977181i \(-0.431869\pi\)
0.212410 + 0.977181i \(0.431869\pi\)
\(524\) 0 0
\(525\) −1.76627 −0.0770865
\(526\) 0 0
\(527\) 6.30112 0.274481
\(528\) 0 0
\(529\) −11.7028 −0.508818
\(530\) 0 0
\(531\) −1.16554 −0.0505801
\(532\) 0 0
\(533\) 15.3162 0.663419
\(534\) 0 0
\(535\) 24.5833 1.06283
\(536\) 0 0
\(537\) 1.08930 0.0470066
\(538\) 0 0
\(539\) 2.26635 0.0976185
\(540\) 0 0
\(541\) 14.7230 0.632991 0.316496 0.948594i \(-0.397494\pi\)
0.316496 + 0.948594i \(0.397494\pi\)
\(542\) 0 0
\(543\) 4.26107 0.182860
\(544\) 0 0
\(545\) −20.8139 −0.891569
\(546\) 0 0
\(547\) 10.9906 0.469922 0.234961 0.972005i \(-0.424504\pi\)
0.234961 + 0.972005i \(0.424504\pi\)
\(548\) 0 0
\(549\) 15.7610 0.672665
\(550\) 0 0
\(551\) −4.95873 −0.211249
\(552\) 0 0
\(553\) 45.1075 1.91817
\(554\) 0 0
\(555\) −1.01956 −0.0432781
\(556\) 0 0
\(557\) 25.9252 1.09849 0.549243 0.835663i \(-0.314916\pi\)
0.549243 + 0.835663i \(0.314916\pi\)
\(558\) 0 0
\(559\) 5.69283 0.240781
\(560\) 0 0
\(561\) 0.386204 0.0163055
\(562\) 0 0
\(563\) 21.0775 0.888310 0.444155 0.895950i \(-0.353504\pi\)
0.444155 + 0.895950i \(0.353504\pi\)
\(564\) 0 0
\(565\) −10.3255 −0.434395
\(566\) 0 0
\(567\) 26.2164 1.10099
\(568\) 0 0
\(569\) −11.2369 −0.471076 −0.235538 0.971865i \(-0.575685\pi\)
−0.235538 + 0.971865i \(0.575685\pi\)
\(570\) 0 0
\(571\) 42.5037 1.77872 0.889362 0.457204i \(-0.151149\pi\)
0.889362 + 0.457204i \(0.151149\pi\)
\(572\) 0 0
\(573\) −5.08401 −0.212388
\(574\) 0 0
\(575\) −8.72209 −0.363736
\(576\) 0 0
\(577\) −26.6503 −1.10947 −0.554733 0.832029i \(-0.687179\pi\)
−0.554733 + 0.832029i \(0.687179\pi\)
\(578\) 0 0
\(579\) −0.469376 −0.0195066
\(580\) 0 0
\(581\) −10.2795 −0.426464
\(582\) 0 0
\(583\) −4.25067 −0.176045
\(584\) 0 0
\(585\) −15.3801 −0.635888
\(586\) 0 0
\(587\) −33.2195 −1.37111 −0.685557 0.728019i \(-0.740441\pi\)
−0.685557 + 0.728019i \(0.740441\pi\)
\(588\) 0 0
\(589\) 3.44620 0.141998
\(590\) 0 0
\(591\) −1.34428 −0.0552964
\(592\) 0 0
\(593\) −25.4149 −1.04367 −0.521833 0.853047i \(-0.674752\pi\)
−0.521833 + 0.853047i \(0.674752\pi\)
\(594\) 0 0
\(595\) −8.68623 −0.356101
\(596\) 0 0
\(597\) −0.895632 −0.0366558
\(598\) 0 0
\(599\) −46.6431 −1.90578 −0.952892 0.303310i \(-0.901908\pi\)
−0.952892 + 0.303310i \(0.901908\pi\)
\(600\) 0 0
\(601\) 47.4644 1.93611 0.968056 0.250734i \(-0.0806718\pi\)
0.968056 + 0.250734i \(0.0806718\pi\)
\(602\) 0 0
\(603\) 20.2137 0.823166
\(604\) 0 0
\(605\) 15.6574 0.636566
\(606\) 0 0
\(607\) 24.4897 0.994006 0.497003 0.867749i \(-0.334434\pi\)
0.497003 + 0.867749i \(0.334434\pi\)
\(608\) 0 0
\(609\) −3.37514 −0.136768
\(610\) 0 0
\(611\) −11.4259 −0.462241
\(612\) 0 0
\(613\) 18.9950 0.767201 0.383600 0.923499i \(-0.374684\pi\)
0.383600 + 0.923499i \(0.374684\pi\)
\(614\) 0 0
\(615\) 1.57019 0.0633162
\(616\) 0 0
\(617\) −0.219907 −0.00885311 −0.00442656 0.999990i \(-0.501409\pi\)
−0.00442656 + 0.999990i \(0.501409\pi\)
\(618\) 0 0
\(619\) 15.9136 0.639623 0.319811 0.947481i \(-0.396380\pi\)
0.319811 + 0.947481i \(0.396380\pi\)
\(620\) 0 0
\(621\) −4.44401 −0.178332
\(622\) 0 0
\(623\) −29.5478 −1.18381
\(624\) 0 0
\(625\) −5.29109 −0.211644
\(626\) 0 0
\(627\) 0.211222 0.00843539
\(628\) 0 0
\(629\) 5.41012 0.215716
\(630\) 0 0
\(631\) 27.6207 1.09956 0.549781 0.835309i \(-0.314711\pi\)
0.549781 + 0.835309i \(0.314711\pi\)
\(632\) 0 0
\(633\) −0.370021 −0.0147070
\(634\) 0 0
\(635\) 10.1544 0.402966
\(636\) 0 0
\(637\) −8.01308 −0.317490
\(638\) 0 0
\(639\) 15.2019 0.601376
\(640\) 0 0
\(641\) −4.17066 −0.164731 −0.0823656 0.996602i \(-0.526248\pi\)
−0.0823656 + 0.996602i \(0.526248\pi\)
\(642\) 0 0
\(643\) 20.8643 0.822806 0.411403 0.911453i \(-0.365039\pi\)
0.411403 + 0.911453i \(0.365039\pi\)
\(644\) 0 0
\(645\) 0.583619 0.0229800
\(646\) 0 0
\(647\) −31.7924 −1.24989 −0.624943 0.780670i \(-0.714878\pi\)
−0.624943 + 0.780670i \(0.714878\pi\)
\(648\) 0 0
\(649\) 0.375512 0.0147401
\(650\) 0 0
\(651\) 2.34564 0.0919331
\(652\) 0 0
\(653\) −27.2099 −1.06481 −0.532403 0.846491i \(-0.678711\pi\)
−0.532403 + 0.846491i \(0.678711\pi\)
\(654\) 0 0
\(655\) 22.1514 0.865528
\(656\) 0 0
\(657\) 3.54068 0.138135
\(658\) 0 0
\(659\) 32.8090 1.27806 0.639030 0.769182i \(-0.279336\pi\)
0.639030 + 0.769182i \(0.279336\pi\)
\(660\) 0 0
\(661\) 12.5824 0.489398 0.244699 0.969599i \(-0.421311\pi\)
0.244699 + 0.969599i \(0.421311\pi\)
\(662\) 0 0
\(663\) −1.36549 −0.0530314
\(664\) 0 0
\(665\) −4.75066 −0.184223
\(666\) 0 0
\(667\) −16.6669 −0.645345
\(668\) 0 0
\(669\) 6.42915 0.248565
\(670\) 0 0
\(671\) −5.07787 −0.196029
\(672\) 0 0
\(673\) −2.58049 −0.0994704 −0.0497352 0.998762i \(-0.515838\pi\)
−0.0497352 + 0.998762i \(0.515838\pi\)
\(674\) 0 0
\(675\) 3.43104 0.132061
\(676\) 0 0
\(677\) 5.86443 0.225388 0.112694 0.993630i \(-0.464052\pi\)
0.112694 + 0.993630i \(0.464052\pi\)
\(678\) 0 0
\(679\) 10.8921 0.417999
\(680\) 0 0
\(681\) −4.45535 −0.170729
\(682\) 0 0
\(683\) 13.3162 0.509531 0.254765 0.967003i \(-0.418002\pi\)
0.254765 + 0.967003i \(0.418002\pi\)
\(684\) 0 0
\(685\) 18.4760 0.705931
\(686\) 0 0
\(687\) 3.33183 0.127117
\(688\) 0 0
\(689\) 15.0290 0.572560
\(690\) 0 0
\(691\) 34.9751 1.33051 0.665257 0.746614i \(-0.268322\pi\)
0.665257 + 0.746614i \(0.268322\pi\)
\(692\) 0 0
\(693\) −8.59255 −0.326404
\(694\) 0 0
\(695\) 17.3614 0.658555
\(696\) 0 0
\(697\) −8.33190 −0.315593
\(698\) 0 0
\(699\) 2.50494 0.0947454
\(700\) 0 0
\(701\) 4.91851 0.185769 0.0928847 0.995677i \(-0.470391\pi\)
0.0928847 + 0.995677i \(0.470391\pi\)
\(702\) 0 0
\(703\) 2.95889 0.111597
\(704\) 0 0
\(705\) −1.17136 −0.0441159
\(706\) 0 0
\(707\) −41.4717 −1.55971
\(708\) 0 0
\(709\) 27.4930 1.03252 0.516261 0.856431i \(-0.327324\pi\)
0.516261 + 0.856431i \(0.327324\pi\)
\(710\) 0 0
\(711\) −43.4479 −1.62942
\(712\) 0 0
\(713\) 11.5831 0.433791
\(714\) 0 0
\(715\) 4.95514 0.185312
\(716\) 0 0
\(717\) 4.42270 0.165169
\(718\) 0 0
\(719\) −10.8746 −0.405553 −0.202777 0.979225i \(-0.564996\pi\)
−0.202777 + 0.979225i \(0.564996\pi\)
\(720\) 0 0
\(721\) −20.3580 −0.758172
\(722\) 0 0
\(723\) −3.51424 −0.130696
\(724\) 0 0
\(725\) 12.8678 0.477899
\(726\) 0 0
\(727\) 1.13818 0.0422127 0.0211063 0.999777i \(-0.493281\pi\)
0.0211063 + 0.999777i \(0.493281\pi\)
\(728\) 0 0
\(729\) −24.3705 −0.902612
\(730\) 0 0
\(731\) −3.09686 −0.114541
\(732\) 0 0
\(733\) −27.9950 −1.03402 −0.517010 0.855980i \(-0.672955\pi\)
−0.517010 + 0.855980i \(0.672955\pi\)
\(734\) 0 0
\(735\) −0.821486 −0.0303010
\(736\) 0 0
\(737\) −6.51243 −0.239889
\(738\) 0 0
\(739\) 44.2375 1.62730 0.813652 0.581352i \(-0.197476\pi\)
0.813652 + 0.581352i \(0.197476\pi\)
\(740\) 0 0
\(741\) −0.746814 −0.0274349
\(742\) 0 0
\(743\) −49.4810 −1.81528 −0.907640 0.419749i \(-0.862118\pi\)
−0.907640 + 0.419749i \(0.862118\pi\)
\(744\) 0 0
\(745\) −11.9063 −0.436214
\(746\) 0 0
\(747\) 9.90126 0.362268
\(748\) 0 0
\(749\) −48.5597 −1.77433
\(750\) 0 0
\(751\) −25.9833 −0.948145 −0.474073 0.880486i \(-0.657217\pi\)
−0.474073 + 0.880486i \(0.657217\pi\)
\(752\) 0 0
\(753\) 2.33495 0.0850904
\(754\) 0 0
\(755\) 4.81002 0.175055
\(756\) 0 0
\(757\) 43.1605 1.56869 0.784347 0.620322i \(-0.212998\pi\)
0.784347 + 0.620322i \(0.212998\pi\)
\(758\) 0 0
\(759\) 0.709944 0.0257693
\(760\) 0 0
\(761\) 4.16498 0.150981 0.0754903 0.997147i \(-0.475948\pi\)
0.0754903 + 0.997147i \(0.475948\pi\)
\(762\) 0 0
\(763\) 41.1140 1.48843
\(764\) 0 0
\(765\) 8.36664 0.302497
\(766\) 0 0
\(767\) −1.32769 −0.0479402
\(768\) 0 0
\(769\) 5.24441 0.189118 0.0945591 0.995519i \(-0.469856\pi\)
0.0945591 + 0.995519i \(0.469856\pi\)
\(770\) 0 0
\(771\) −0.378011 −0.0136137
\(772\) 0 0
\(773\) 3.91186 0.140700 0.0703499 0.997522i \(-0.477588\pi\)
0.0703499 + 0.997522i \(0.477588\pi\)
\(774\) 0 0
\(775\) −8.94284 −0.321236
\(776\) 0 0
\(777\) 2.01396 0.0722505
\(778\) 0 0
\(779\) −4.55687 −0.163267
\(780\) 0 0
\(781\) −4.89772 −0.175254
\(782\) 0 0
\(783\) 6.55632 0.234304
\(784\) 0 0
\(785\) 23.1061 0.824692
\(786\) 0 0
\(787\) 16.8869 0.601952 0.300976 0.953632i \(-0.402688\pi\)
0.300976 + 0.953632i \(0.402688\pi\)
\(788\) 0 0
\(789\) 3.07421 0.109445
\(790\) 0 0
\(791\) 20.3960 0.725199
\(792\) 0 0
\(793\) 17.9537 0.637556
\(794\) 0 0
\(795\) 1.54075 0.0546447
\(796\) 0 0
\(797\) 25.3722 0.898729 0.449365 0.893348i \(-0.351650\pi\)
0.449365 + 0.893348i \(0.351650\pi\)
\(798\) 0 0
\(799\) 6.21557 0.219891
\(800\) 0 0
\(801\) 28.4606 1.00561
\(802\) 0 0
\(803\) −1.14073 −0.0402556
\(804\) 0 0
\(805\) −15.9676 −0.562783
\(806\) 0 0
\(807\) −3.32522 −0.117053
\(808\) 0 0
\(809\) −31.5640 −1.10973 −0.554866 0.831940i \(-0.687230\pi\)
−0.554866 + 0.831940i \(0.687230\pi\)
\(810\) 0 0
\(811\) −44.1203 −1.54927 −0.774636 0.632407i \(-0.782067\pi\)
−0.774636 + 0.632407i \(0.782067\pi\)
\(812\) 0 0
\(813\) −0.916949 −0.0321588
\(814\) 0 0
\(815\) 28.7646 1.00758
\(816\) 0 0
\(817\) −1.69373 −0.0592560
\(818\) 0 0
\(819\) 30.3805 1.06158
\(820\) 0 0
\(821\) 3.81015 0.132975 0.0664876 0.997787i \(-0.478821\pi\)
0.0664876 + 0.997787i \(0.478821\pi\)
\(822\) 0 0
\(823\) −34.3301 −1.19667 −0.598336 0.801245i \(-0.704171\pi\)
−0.598336 + 0.801245i \(0.704171\pi\)
\(824\) 0 0
\(825\) −0.548119 −0.0190830
\(826\) 0 0
\(827\) 30.7484 1.06923 0.534613 0.845097i \(-0.320457\pi\)
0.534613 + 0.845097i \(0.320457\pi\)
\(828\) 0 0
\(829\) −10.1917 −0.353971 −0.176986 0.984213i \(-0.556635\pi\)
−0.176986 + 0.984213i \(0.556635\pi\)
\(830\) 0 0
\(831\) −5.72633 −0.198644
\(832\) 0 0
\(833\) 4.35905 0.151032
\(834\) 0 0
\(835\) −17.7617 −0.614669
\(836\) 0 0
\(837\) −4.55649 −0.157495
\(838\) 0 0
\(839\) 7.81123 0.269674 0.134837 0.990868i \(-0.456949\pi\)
0.134837 + 0.990868i \(0.456949\pi\)
\(840\) 0 0
\(841\) −4.41104 −0.152105
\(842\) 0 0
\(843\) −2.28177 −0.0785884
\(844\) 0 0
\(845\) 2.64074 0.0908442
\(846\) 0 0
\(847\) −30.9284 −1.06271
\(848\) 0 0
\(849\) 1.55424 0.0533415
\(850\) 0 0
\(851\) 9.94521 0.340918
\(852\) 0 0
\(853\) −46.9706 −1.60824 −0.804122 0.594465i \(-0.797364\pi\)
−0.804122 + 0.594465i \(0.797364\pi\)
\(854\) 0 0
\(855\) 4.57587 0.156491
\(856\) 0 0
\(857\) 2.24191 0.0765821 0.0382911 0.999267i \(-0.487809\pi\)
0.0382911 + 0.999267i \(0.487809\pi\)
\(858\) 0 0
\(859\) 49.9456 1.70412 0.852061 0.523442i \(-0.175352\pi\)
0.852061 + 0.523442i \(0.175352\pi\)
\(860\) 0 0
\(861\) −3.10162 −0.105703
\(862\) 0 0
\(863\) 10.4322 0.355115 0.177557 0.984110i \(-0.443180\pi\)
0.177557 + 0.984110i \(0.443180\pi\)
\(864\) 0 0
\(865\) −11.1807 −0.380154
\(866\) 0 0
\(867\) −3.03444 −0.103055
\(868\) 0 0
\(869\) 13.9980 0.474849
\(870\) 0 0
\(871\) 23.0259 0.780203
\(872\) 0 0
\(873\) −10.4913 −0.355077
\(874\) 0 0
\(875\) 36.0812 1.21977
\(876\) 0 0
\(877\) 23.6385 0.798215 0.399107 0.916904i \(-0.369320\pi\)
0.399107 + 0.916904i \(0.369320\pi\)
\(878\) 0 0
\(879\) −7.07534 −0.238645
\(880\) 0 0
\(881\) −29.2713 −0.986175 −0.493087 0.869980i \(-0.664132\pi\)
−0.493087 + 0.869980i \(0.664132\pi\)
\(882\) 0 0
\(883\) 42.1875 1.41972 0.709862 0.704341i \(-0.248757\pi\)
0.709862 + 0.704341i \(0.248757\pi\)
\(884\) 0 0
\(885\) −0.136112 −0.00457537
\(886\) 0 0
\(887\) −1.32254 −0.0444064 −0.0222032 0.999753i \(-0.507068\pi\)
−0.0222032 + 0.999753i \(0.507068\pi\)
\(888\) 0 0
\(889\) −20.0582 −0.672730
\(890\) 0 0
\(891\) 8.13561 0.272553
\(892\) 0 0
\(893\) 3.39941 0.113757
\(894\) 0 0
\(895\) −7.60286 −0.254136
\(896\) 0 0
\(897\) −2.51014 −0.0838110
\(898\) 0 0
\(899\) −17.0887 −0.569941
\(900\) 0 0
\(901\) −8.17567 −0.272371
\(902\) 0 0
\(903\) −1.15283 −0.0383638
\(904\) 0 0
\(905\) −29.7406 −0.988611
\(906\) 0 0
\(907\) 3.58771 0.119128 0.0595639 0.998224i \(-0.481029\pi\)
0.0595639 + 0.998224i \(0.481029\pi\)
\(908\) 0 0
\(909\) 39.9459 1.32492
\(910\) 0 0
\(911\) −14.6828 −0.486464 −0.243232 0.969968i \(-0.578208\pi\)
−0.243232 + 0.969968i \(0.578208\pi\)
\(912\) 0 0
\(913\) −3.18998 −0.105573
\(914\) 0 0
\(915\) 1.84058 0.0608478
\(916\) 0 0
\(917\) −43.7561 −1.44495
\(918\) 0 0
\(919\) −38.2168 −1.26066 −0.630328 0.776329i \(-0.717080\pi\)
−0.630328 + 0.776329i \(0.717080\pi\)
\(920\) 0 0
\(921\) −1.43494 −0.0472829
\(922\) 0 0
\(923\) 17.3168 0.569988
\(924\) 0 0
\(925\) −7.67829 −0.252461
\(926\) 0 0
\(927\) 19.6090 0.644044
\(928\) 0 0
\(929\) 36.8919 1.21038 0.605192 0.796080i \(-0.293096\pi\)
0.605192 + 0.796080i \(0.293096\pi\)
\(930\) 0 0
\(931\) 2.38404 0.0781339
\(932\) 0 0
\(933\) −6.26521 −0.205114
\(934\) 0 0
\(935\) −2.69555 −0.0881541
\(936\) 0 0
\(937\) 33.7347 1.10206 0.551032 0.834484i \(-0.314234\pi\)
0.551032 + 0.834484i \(0.314234\pi\)
\(938\) 0 0
\(939\) 3.80418 0.124145
\(940\) 0 0
\(941\) 3.91029 0.127472 0.0637360 0.997967i \(-0.479698\pi\)
0.0637360 + 0.997967i \(0.479698\pi\)
\(942\) 0 0
\(943\) −15.3162 −0.498765
\(944\) 0 0
\(945\) 6.28122 0.204328
\(946\) 0 0
\(947\) 46.8518 1.52248 0.761240 0.648470i \(-0.224591\pi\)
0.761240 + 0.648470i \(0.224591\pi\)
\(948\) 0 0
\(949\) 4.03327 0.130925
\(950\) 0 0
\(951\) −6.57333 −0.213155
\(952\) 0 0
\(953\) 26.5470 0.859941 0.429971 0.902843i \(-0.358524\pi\)
0.429971 + 0.902843i \(0.358524\pi\)
\(954\) 0 0
\(955\) 35.4844 1.14825
\(956\) 0 0
\(957\) −1.04739 −0.0338574
\(958\) 0 0
\(959\) −36.4959 −1.17851
\(960\) 0 0
\(961\) −19.1237 −0.616895
\(962\) 0 0
\(963\) 46.7731 1.50724
\(964\) 0 0
\(965\) 3.27606 0.105460
\(966\) 0 0
\(967\) 6.80878 0.218956 0.109478 0.993989i \(-0.465082\pi\)
0.109478 + 0.993989i \(0.465082\pi\)
\(968\) 0 0
\(969\) 0.406261 0.0130510
\(970\) 0 0
\(971\) −48.4549 −1.55499 −0.777496 0.628887i \(-0.783511\pi\)
−0.777496 + 0.628887i \(0.783511\pi\)
\(972\) 0 0
\(973\) −34.2942 −1.09942
\(974\) 0 0
\(975\) 1.93797 0.0620648
\(976\) 0 0
\(977\) 46.0137 1.47211 0.736055 0.676922i \(-0.236686\pi\)
0.736055 + 0.676922i \(0.236686\pi\)
\(978\) 0 0
\(979\) −9.16941 −0.293056
\(980\) 0 0
\(981\) −39.6013 −1.26437
\(982\) 0 0
\(983\) 27.8031 0.886782 0.443391 0.896328i \(-0.353775\pi\)
0.443391 + 0.896328i \(0.353775\pi\)
\(984\) 0 0
\(985\) 9.38258 0.298954
\(986\) 0 0
\(987\) 2.31380 0.0736490
\(988\) 0 0
\(989\) −5.69283 −0.181022
\(990\) 0 0
\(991\) 5.16951 0.164215 0.0821074 0.996623i \(-0.473835\pi\)
0.0821074 + 0.996623i \(0.473835\pi\)
\(992\) 0 0
\(993\) −4.56646 −0.144912
\(994\) 0 0
\(995\) 6.25116 0.198175
\(996\) 0 0
\(997\) 18.7227 0.592953 0.296476 0.955040i \(-0.404188\pi\)
0.296476 + 0.955040i \(0.404188\pi\)
\(998\) 0 0
\(999\) −3.91218 −0.123776
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 4864.2.a.bm.1.5 8
4.3 odd 2 4864.2.a.br.1.3 8
8.3 odd 2 inner 4864.2.a.bm.1.6 8
8.5 even 2 4864.2.a.br.1.4 8
16.3 odd 4 2432.2.c.i.1217.8 yes 16
16.5 even 4 2432.2.c.i.1217.7 16
16.11 odd 4 2432.2.c.i.1217.9 yes 16
16.13 even 4 2432.2.c.i.1217.10 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.i.1217.7 16 16.5 even 4
2432.2.c.i.1217.8 yes 16 16.3 odd 4
2432.2.c.i.1217.9 yes 16 16.11 odd 4
2432.2.c.i.1217.10 yes 16 16.13 even 4
4864.2.a.bm.1.5 8 1.1 even 1 trivial
4864.2.a.bm.1.6 8 8.3 odd 2 inner
4864.2.a.br.1.3 8 4.3 odd 2
4864.2.a.br.1.4 8 8.5 even 2