Properties

Label 4864.2.a.bo.1.6
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 68 x^{3} - 62 x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.25259\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.91886 q^{3} +1.51356 q^{5} -0.580162 q^{7} +0.682013 q^{9} +O(q^{10})\) \(q+1.91886 q^{3} +1.51356 q^{5} -0.580162 q^{7} +0.682013 q^{9} -1.31799 q^{11} -3.89230 q^{13} +2.90431 q^{15} -1.20142 q^{17} -1.00000 q^{19} -1.11325 q^{21} -5.85527 q^{23} -2.70913 q^{25} -4.44789 q^{27} -1.29188 q^{29} +2.96413 q^{31} -2.52903 q^{33} -0.878110 q^{35} -1.18418 q^{37} -7.46877 q^{39} +9.04577 q^{41} +8.38816 q^{43} +1.03227 q^{45} -12.8560 q^{47} -6.66341 q^{49} -2.30536 q^{51} -3.07183 q^{53} -1.99485 q^{55} -1.91886 q^{57} +0.258163 q^{59} -14.7200 q^{61} -0.395678 q^{63} -5.89123 q^{65} +9.54884 q^{67} -11.2354 q^{69} +6.93697 q^{71} -15.2934 q^{73} -5.19844 q^{75} +0.764646 q^{77} -13.1332 q^{79} -10.5809 q^{81} +3.70615 q^{83} -1.81843 q^{85} -2.47893 q^{87} +8.36653 q^{89} +2.25816 q^{91} +5.68774 q^{93} -1.51356 q^{95} +17.0442 q^{97} -0.898884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 4q^{7} + 12q^{9} - 4q^{11} - 8q^{13} - 4q^{17} - 8q^{19} - 16q^{21} + 12q^{25} - 28q^{29} + 8q^{31} + 12q^{35} - 4q^{37} - 4q^{39} - 8q^{41} + 4q^{43} - 24q^{45} + 12q^{47} + 12q^{49} - 12q^{51} - 32q^{53} - 8q^{55} - 12q^{59} - 8q^{61} - 16q^{63} + 8q^{65} + 4q^{67} - 28q^{69} - 24q^{71} - 24q^{77} - 24q^{79} - 8q^{81} - 40q^{83} - 24q^{85} + 24q^{87} + 8q^{89} + 4q^{91} - 32q^{93} + 8q^{95} + 16q^{97} + 76q^{99} + O(q^{100}) \)

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
\(3\) 1.91886 1.10785 0.553926 0.832566i \(-0.313129\pi\)
0.553926 + 0.832566i \(0.313129\pi\)
\(4\) 0 0
\(5\) 1.51356 0.676885 0.338442 0.940987i \(-0.390100\pi\)
0.338442 + 0.940987i \(0.390100\pi\)
\(6\) 0 0
\(7\) −0.580162 −0.219281 −0.109640 0.993971i \(-0.534970\pi\)
−0.109640 + 0.993971i \(0.534970\pi\)
\(8\) 0 0
\(9\) 0.682013 0.227338
\(10\) 0 0
\(11\) −1.31799 −0.397388 −0.198694 0.980062i \(-0.563670\pi\)
−0.198694 + 0.980062i \(0.563670\pi\)
\(12\) 0 0
\(13\) −3.89230 −1.07953 −0.539765 0.841816i \(-0.681487\pi\)
−0.539765 + 0.841816i \(0.681487\pi\)
\(14\) 0 0
\(15\) 2.90431 0.749889
\(16\) 0 0
\(17\) −1.20142 −0.291388 −0.145694 0.989330i \(-0.546541\pi\)
−0.145694 + 0.989330i \(0.546541\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.11325 −0.242931
\(22\) 0 0
\(23\) −5.85527 −1.22091 −0.610454 0.792052i \(-0.709013\pi\)
−0.610454 + 0.792052i \(0.709013\pi\)
\(24\) 0 0
\(25\) −2.70913 −0.541827
\(26\) 0 0
\(27\) −4.44789 −0.855996
\(28\) 0 0
\(29\) −1.29188 −0.239896 −0.119948 0.992780i \(-0.538273\pi\)
−0.119948 + 0.992780i \(0.538273\pi\)
\(30\) 0 0
\(31\) 2.96413 0.532374 0.266187 0.963921i \(-0.414236\pi\)
0.266187 + 0.963921i \(0.414236\pi\)
\(32\) 0 0
\(33\) −2.52903 −0.440248
\(34\) 0 0
\(35\) −0.878110 −0.148428
\(36\) 0 0
\(37\) −1.18418 −0.194677 −0.0973387 0.995251i \(-0.531033\pi\)
−0.0973387 + 0.995251i \(0.531033\pi\)
\(38\) 0 0
\(39\) −7.46877 −1.19596
\(40\) 0 0
\(41\) 9.04577 1.41271 0.706356 0.707856i \(-0.250338\pi\)
0.706356 + 0.707856i \(0.250338\pi\)
\(42\) 0 0
\(43\) 8.38816 1.27918 0.639592 0.768715i \(-0.279104\pi\)
0.639592 + 0.768715i \(0.279104\pi\)
\(44\) 0 0
\(45\) 1.03227 0.153881
\(46\) 0 0
\(47\) −12.8560 −1.87524 −0.937620 0.347661i \(-0.886976\pi\)
−0.937620 + 0.347661i \(0.886976\pi\)
\(48\) 0 0
\(49\) −6.66341 −0.951916
\(50\) 0 0
\(51\) −2.30536 −0.322815
\(52\) 0 0
\(53\) −3.07183 −0.421949 −0.210974 0.977492i \(-0.567664\pi\)
−0.210974 + 0.977492i \(0.567664\pi\)
\(54\) 0 0
\(55\) −1.99485 −0.268986
\(56\) 0 0
\(57\) −1.91886 −0.254159
\(58\) 0 0
\(59\) 0.258163 0.0336100 0.0168050 0.999859i \(-0.494651\pi\)
0.0168050 + 0.999859i \(0.494651\pi\)
\(60\) 0 0
\(61\) −14.7200 −1.88470 −0.942352 0.334623i \(-0.891391\pi\)
−0.942352 + 0.334623i \(0.891391\pi\)
\(62\) 0 0
\(63\) −0.395678 −0.0498507
\(64\) 0 0
\(65\) −5.89123 −0.730717
\(66\) 0 0
\(67\) 9.54884 1.16658 0.583288 0.812265i \(-0.301766\pi\)
0.583288 + 0.812265i \(0.301766\pi\)
\(68\) 0 0
\(69\) −11.2354 −1.35259
\(70\) 0 0
\(71\) 6.93697 0.823267 0.411634 0.911349i \(-0.364958\pi\)
0.411634 + 0.911349i \(0.364958\pi\)
\(72\) 0 0
\(73\) −15.2934 −1.78996 −0.894978 0.446110i \(-0.852809\pi\)
−0.894978 + 0.446110i \(0.852809\pi\)
\(74\) 0 0
\(75\) −5.19844 −0.600264
\(76\) 0 0
\(77\) 0.764646 0.0871395
\(78\) 0 0
\(79\) −13.1332 −1.47760 −0.738798 0.673927i \(-0.764606\pi\)
−0.738798 + 0.673927i \(0.764606\pi\)
\(80\) 0 0
\(81\) −10.5809 −1.17566
\(82\) 0 0
\(83\) 3.70615 0.406803 0.203402 0.979095i \(-0.434800\pi\)
0.203402 + 0.979095i \(0.434800\pi\)
\(84\) 0 0
\(85\) −1.81843 −0.197236
\(86\) 0 0
\(87\) −2.47893 −0.265769
\(88\) 0 0
\(89\) 8.36653 0.886850 0.443425 0.896311i \(-0.353763\pi\)
0.443425 + 0.896311i \(0.353763\pi\)
\(90\) 0 0
\(91\) 2.25816 0.236720
\(92\) 0 0
\(93\) 5.68774 0.589792
\(94\) 0 0
\(95\) −1.51356 −0.155288
\(96\) 0 0
\(97\) 17.0442 1.73057 0.865286 0.501278i \(-0.167137\pi\)
0.865286 + 0.501278i \(0.167137\pi\)
\(98\) 0 0
\(99\) −0.898884 −0.0903412
\(100\) 0 0
\(101\) −1.97721 −0.196739 −0.0983697 0.995150i \(-0.531363\pi\)
−0.0983697 + 0.995150i \(0.531363\pi\)
\(102\) 0 0
\(103\) 2.30853 0.227466 0.113733 0.993511i \(-0.463719\pi\)
0.113733 + 0.993511i \(0.463719\pi\)
\(104\) 0 0
\(105\) −1.68497 −0.164436
\(106\) 0 0
\(107\) −15.0396 −1.45393 −0.726965 0.686675i \(-0.759070\pi\)
−0.726965 + 0.686675i \(0.759070\pi\)
\(108\) 0 0
\(109\) 13.4002 1.28351 0.641753 0.766911i \(-0.278207\pi\)
0.641753 + 0.766911i \(0.278207\pi\)
\(110\) 0 0
\(111\) −2.27227 −0.215674
\(112\) 0 0
\(113\) 4.89717 0.460687 0.230343 0.973109i \(-0.426015\pi\)
0.230343 + 0.973109i \(0.426015\pi\)
\(114\) 0 0
\(115\) −8.86230 −0.826414
\(116\) 0 0
\(117\) −2.65460 −0.245418
\(118\) 0 0
\(119\) 0.697020 0.0638957
\(120\) 0 0
\(121\) −9.26291 −0.842083
\(122\) 0 0
\(123\) 17.3575 1.56508
\(124\) 0 0
\(125\) −11.6682 −1.04364
\(126\) 0 0
\(127\) −2.61625 −0.232155 −0.116078 0.993240i \(-0.537032\pi\)
−0.116078 + 0.993240i \(0.537032\pi\)
\(128\) 0 0
\(129\) 16.0957 1.41715
\(130\) 0 0
\(131\) −2.64498 −0.231093 −0.115547 0.993302i \(-0.536862\pi\)
−0.115547 + 0.993302i \(0.536862\pi\)
\(132\) 0 0
\(133\) 0.580162 0.0503064
\(134\) 0 0
\(135\) −6.73215 −0.579411
\(136\) 0 0
\(137\) −4.88035 −0.416957 −0.208478 0.978027i \(-0.566851\pi\)
−0.208478 + 0.978027i \(0.566851\pi\)
\(138\) 0 0
\(139\) 10.3334 0.876468 0.438234 0.898861i \(-0.355604\pi\)
0.438234 + 0.898861i \(0.355604\pi\)
\(140\) 0 0
\(141\) −24.6688 −2.07749
\(142\) 0 0
\(143\) 5.13000 0.428992
\(144\) 0 0
\(145\) −1.95533 −0.162382
\(146\) 0 0
\(147\) −12.7861 −1.05458
\(148\) 0 0
\(149\) −8.35324 −0.684324 −0.342162 0.939641i \(-0.611159\pi\)
−0.342162 + 0.939641i \(0.611159\pi\)
\(150\) 0 0
\(151\) 5.84941 0.476019 0.238009 0.971263i \(-0.423505\pi\)
0.238009 + 0.971263i \(0.423505\pi\)
\(152\) 0 0
\(153\) −0.819386 −0.0662434
\(154\) 0 0
\(155\) 4.48639 0.360356
\(156\) 0 0
\(157\) 0.361645 0.0288624 0.0144312 0.999896i \(-0.495406\pi\)
0.0144312 + 0.999896i \(0.495406\pi\)
\(158\) 0 0
\(159\) −5.89441 −0.467457
\(160\) 0 0
\(161\) 3.39700 0.267721
\(162\) 0 0
\(163\) −0.146029 −0.0114379 −0.00571894 0.999984i \(-0.501820\pi\)
−0.00571894 + 0.999984i \(0.501820\pi\)
\(164\) 0 0
\(165\) −3.82784 −0.297997
\(166\) 0 0
\(167\) −0.406603 −0.0314639 −0.0157319 0.999876i \(-0.505008\pi\)
−0.0157319 + 0.999876i \(0.505008\pi\)
\(168\) 0 0
\(169\) 2.14999 0.165384
\(170\) 0 0
\(171\) −0.682013 −0.0521548
\(172\) 0 0
\(173\) −12.6466 −0.961507 −0.480753 0.876856i \(-0.659637\pi\)
−0.480753 + 0.876856i \(0.659637\pi\)
\(174\) 0 0
\(175\) 1.57174 0.118812
\(176\) 0 0
\(177\) 0.495378 0.0372349
\(178\) 0 0
\(179\) 5.55598 0.415274 0.207637 0.978206i \(-0.433423\pi\)
0.207637 + 0.978206i \(0.433423\pi\)
\(180\) 0 0
\(181\) −9.13401 −0.678926 −0.339463 0.940619i \(-0.610245\pi\)
−0.339463 + 0.940619i \(0.610245\pi\)
\(182\) 0 0
\(183\) −28.2456 −2.08797
\(184\) 0 0
\(185\) −1.79232 −0.131774
\(186\) 0 0
\(187\) 1.58346 0.115794
\(188\) 0 0
\(189\) 2.58049 0.187703
\(190\) 0 0
\(191\) 5.47532 0.396180 0.198090 0.980184i \(-0.436526\pi\)
0.198090 + 0.980184i \(0.436526\pi\)
\(192\) 0 0
\(193\) 16.9697 1.22151 0.610753 0.791821i \(-0.290867\pi\)
0.610753 + 0.791821i \(0.290867\pi\)
\(194\) 0 0
\(195\) −11.3044 −0.809527
\(196\) 0 0
\(197\) −23.7727 −1.69374 −0.846868 0.531803i \(-0.821515\pi\)
−0.846868 + 0.531803i \(0.821515\pi\)
\(198\) 0 0
\(199\) 17.5747 1.24584 0.622918 0.782287i \(-0.285947\pi\)
0.622918 + 0.782287i \(0.285947\pi\)
\(200\) 0 0
\(201\) 18.3229 1.29239
\(202\) 0 0
\(203\) 0.749498 0.0526044
\(204\) 0 0
\(205\) 13.6913 0.956244
\(206\) 0 0
\(207\) −3.99337 −0.277558
\(208\) 0 0
\(209\) 1.31799 0.0911671
\(210\) 0 0
\(211\) −17.2804 −1.18963 −0.594815 0.803863i \(-0.702775\pi\)
−0.594815 + 0.803863i \(0.702775\pi\)
\(212\) 0 0
\(213\) 13.3111 0.912059
\(214\) 0 0
\(215\) 12.6960 0.865860
\(216\) 0 0
\(217\) −1.71968 −0.116739
\(218\) 0 0
\(219\) −29.3458 −1.98301
\(220\) 0 0
\(221\) 4.67630 0.314562
\(222\) 0 0
\(223\) 11.6027 0.776973 0.388487 0.921454i \(-0.372998\pi\)
0.388487 + 0.921454i \(0.372998\pi\)
\(224\) 0 0
\(225\) −1.84766 −0.123178
\(226\) 0 0
\(227\) 27.4484 1.82182 0.910908 0.412609i \(-0.135382\pi\)
0.910908 + 0.412609i \(0.135382\pi\)
\(228\) 0 0
\(229\) −21.8680 −1.44508 −0.722540 0.691329i \(-0.757025\pi\)
−0.722540 + 0.691329i \(0.757025\pi\)
\(230\) 0 0
\(231\) 1.46725 0.0965377
\(232\) 0 0
\(233\) 22.1975 1.45421 0.727103 0.686528i \(-0.240866\pi\)
0.727103 + 0.686528i \(0.240866\pi\)
\(234\) 0 0
\(235\) −19.4583 −1.26932
\(236\) 0 0
\(237\) −25.2007 −1.63696
\(238\) 0 0
\(239\) 21.1350 1.36711 0.683555 0.729899i \(-0.260433\pi\)
0.683555 + 0.729899i \(0.260433\pi\)
\(240\) 0 0
\(241\) −17.2437 −1.11076 −0.555382 0.831595i \(-0.687428\pi\)
−0.555382 + 0.831595i \(0.687428\pi\)
\(242\) 0 0
\(243\) −6.95957 −0.446457
\(244\) 0 0
\(245\) −10.0855 −0.644338
\(246\) 0 0
\(247\) 3.89230 0.247661
\(248\) 0 0
\(249\) 7.11158 0.450678
\(250\) 0 0
\(251\) −6.08851 −0.384303 −0.192152 0.981365i \(-0.561547\pi\)
−0.192152 + 0.981365i \(0.561547\pi\)
\(252\) 0 0
\(253\) 7.71717 0.485174
\(254\) 0 0
\(255\) −3.48930 −0.218509
\(256\) 0 0
\(257\) 10.1741 0.634643 0.317322 0.948318i \(-0.397217\pi\)
0.317322 + 0.948318i \(0.397217\pi\)
\(258\) 0 0
\(259\) 0.687014 0.0426890
\(260\) 0 0
\(261\) −0.881077 −0.0545373
\(262\) 0 0
\(263\) 15.8414 0.976822 0.488411 0.872614i \(-0.337577\pi\)
0.488411 + 0.872614i \(0.337577\pi\)
\(264\) 0 0
\(265\) −4.64940 −0.285611
\(266\) 0 0
\(267\) 16.0542 0.982500
\(268\) 0 0
\(269\) −22.8533 −1.39339 −0.696695 0.717367i \(-0.745347\pi\)
−0.696695 + 0.717367i \(0.745347\pi\)
\(270\) 0 0
\(271\) 12.2830 0.746142 0.373071 0.927803i \(-0.378305\pi\)
0.373071 + 0.927803i \(0.378305\pi\)
\(272\) 0 0
\(273\) 4.33309 0.262251
\(274\) 0 0
\(275\) 3.57060 0.215316
\(276\) 0 0
\(277\) 0.643776 0.0386808 0.0193404 0.999813i \(-0.493843\pi\)
0.0193404 + 0.999813i \(0.493843\pi\)
\(278\) 0 0
\(279\) 2.02157 0.121029
\(280\) 0 0
\(281\) 8.93027 0.532735 0.266368 0.963872i \(-0.414177\pi\)
0.266368 + 0.963872i \(0.414177\pi\)
\(282\) 0 0
\(283\) 14.7397 0.876187 0.438094 0.898929i \(-0.355654\pi\)
0.438094 + 0.898929i \(0.355654\pi\)
\(284\) 0 0
\(285\) −2.90431 −0.172036
\(286\) 0 0
\(287\) −5.24801 −0.309780
\(288\) 0 0
\(289\) −15.5566 −0.915093
\(290\) 0 0
\(291\) 32.7053 1.91722
\(292\) 0 0
\(293\) 23.9816 1.40102 0.700509 0.713643i \(-0.252956\pi\)
0.700509 + 0.713643i \(0.252956\pi\)
\(294\) 0 0
\(295\) 0.390746 0.0227501
\(296\) 0 0
\(297\) 5.86226 0.340163
\(298\) 0 0
\(299\) 22.7905 1.31801
\(300\) 0 0
\(301\) −4.86649 −0.280500
\(302\) 0 0
\(303\) −3.79398 −0.217958
\(304\) 0 0
\(305\) −22.2796 −1.27573
\(306\) 0 0
\(307\) 1.82132 0.103948 0.0519740 0.998648i \(-0.483449\pi\)
0.0519740 + 0.998648i \(0.483449\pi\)
\(308\) 0 0
\(309\) 4.42974 0.251999
\(310\) 0 0
\(311\) 4.53302 0.257044 0.128522 0.991707i \(-0.458977\pi\)
0.128522 + 0.991707i \(0.458977\pi\)
\(312\) 0 0
\(313\) −12.4149 −0.701730 −0.350865 0.936426i \(-0.614112\pi\)
−0.350865 + 0.936426i \(0.614112\pi\)
\(314\) 0 0
\(315\) −0.598882 −0.0337432
\(316\) 0 0
\(317\) −23.0948 −1.29713 −0.648567 0.761157i \(-0.724632\pi\)
−0.648567 + 0.761157i \(0.724632\pi\)
\(318\) 0 0
\(319\) 1.70268 0.0953317
\(320\) 0 0
\(321\) −28.8588 −1.61074
\(322\) 0 0
\(323\) 1.20142 0.0668490
\(324\) 0 0
\(325\) 10.5448 0.584918
\(326\) 0 0
\(327\) 25.7131 1.42194
\(328\) 0 0
\(329\) 7.45856 0.411204
\(330\) 0 0
\(331\) −26.3743 −1.44966 −0.724831 0.688927i \(-0.758082\pi\)
−0.724831 + 0.688927i \(0.758082\pi\)
\(332\) 0 0
\(333\) −0.807623 −0.0442575
\(334\) 0 0
\(335\) 14.4527 0.789638
\(336\) 0 0
\(337\) 3.82102 0.208144 0.104072 0.994570i \(-0.466813\pi\)
0.104072 + 0.994570i \(0.466813\pi\)
\(338\) 0 0
\(339\) 9.39697 0.510373
\(340\) 0 0
\(341\) −3.90669 −0.211559
\(342\) 0 0
\(343\) 7.92699 0.428017
\(344\) 0 0
\(345\) −17.0055 −0.915545
\(346\) 0 0
\(347\) 29.8808 1.60408 0.802042 0.597268i \(-0.203747\pi\)
0.802042 + 0.597268i \(0.203747\pi\)
\(348\) 0 0
\(349\) −12.7016 −0.679902 −0.339951 0.940443i \(-0.610411\pi\)
−0.339951 + 0.940443i \(0.610411\pi\)
\(350\) 0 0
\(351\) 17.3125 0.924073
\(352\) 0 0
\(353\) 22.8398 1.21564 0.607820 0.794074i \(-0.292044\pi\)
0.607820 + 0.794074i \(0.292044\pi\)
\(354\) 0 0
\(355\) 10.4995 0.557257
\(356\) 0 0
\(357\) 1.33748 0.0707870
\(358\) 0 0
\(359\) −35.4003 −1.86835 −0.934177 0.356810i \(-0.883864\pi\)
−0.934177 + 0.356810i \(0.883864\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −17.7742 −0.932904
\(364\) 0 0
\(365\) −23.1475 −1.21159
\(366\) 0 0
\(367\) −17.2148 −0.898608 −0.449304 0.893379i \(-0.648328\pi\)
−0.449304 + 0.893379i \(0.648328\pi\)
\(368\) 0 0
\(369\) 6.16933 0.321163
\(370\) 0 0
\(371\) 1.78216 0.0925251
\(372\) 0 0
\(373\) −20.3506 −1.05372 −0.526858 0.849953i \(-0.676630\pi\)
−0.526858 + 0.849953i \(0.676630\pi\)
\(374\) 0 0
\(375\) −22.3897 −1.15620
\(376\) 0 0
\(377\) 5.02837 0.258974
\(378\) 0 0
\(379\) 4.22765 0.217160 0.108580 0.994088i \(-0.465370\pi\)
0.108580 + 0.994088i \(0.465370\pi\)
\(380\) 0 0
\(381\) −5.02022 −0.257194
\(382\) 0 0
\(383\) 9.71281 0.496301 0.248151 0.968721i \(-0.420177\pi\)
0.248151 + 0.968721i \(0.420177\pi\)
\(384\) 0 0
\(385\) 1.15734 0.0589834
\(386\) 0 0
\(387\) 5.72083 0.290806
\(388\) 0 0
\(389\) −22.3291 −1.13213 −0.566064 0.824361i \(-0.691534\pi\)
−0.566064 + 0.824361i \(0.691534\pi\)
\(390\) 0 0
\(391\) 7.03466 0.355758
\(392\) 0 0
\(393\) −5.07534 −0.256017
\(394\) 0 0
\(395\) −19.8778 −1.00016
\(396\) 0 0
\(397\) 23.9210 1.20056 0.600281 0.799789i \(-0.295055\pi\)
0.600281 + 0.799789i \(0.295055\pi\)
\(398\) 0 0
\(399\) 1.11325 0.0557321
\(400\) 0 0
\(401\) −16.5237 −0.825154 −0.412577 0.910923i \(-0.635371\pi\)
−0.412577 + 0.910923i \(0.635371\pi\)
\(402\) 0 0
\(403\) −11.5373 −0.574713
\(404\) 0 0
\(405\) −16.0148 −0.795783
\(406\) 0 0
\(407\) 1.56073 0.0773625
\(408\) 0 0
\(409\) −16.0228 −0.792278 −0.396139 0.918191i \(-0.629650\pi\)
−0.396139 + 0.918191i \(0.629650\pi\)
\(410\) 0 0
\(411\) −9.36470 −0.461926
\(412\) 0 0
\(413\) −0.149776 −0.00737002
\(414\) 0 0
\(415\) 5.60949 0.275359
\(416\) 0 0
\(417\) 19.8283 0.970998
\(418\) 0 0
\(419\) −12.7496 −0.622858 −0.311429 0.950269i \(-0.600808\pi\)
−0.311429 + 0.950269i \(0.600808\pi\)
\(420\) 0 0
\(421\) 1.89293 0.0922560 0.0461280 0.998936i \(-0.485312\pi\)
0.0461280 + 0.998936i \(0.485312\pi\)
\(422\) 0 0
\(423\) −8.76796 −0.426313
\(424\) 0 0
\(425\) 3.25482 0.157882
\(426\) 0 0
\(427\) 8.53999 0.413279
\(428\) 0 0
\(429\) 9.84374 0.475260
\(430\) 0 0
\(431\) 37.2166 1.79266 0.896331 0.443386i \(-0.146223\pi\)
0.896331 + 0.443386i \(0.146223\pi\)
\(432\) 0 0
\(433\) −36.0314 −1.73156 −0.865780 0.500425i \(-0.833177\pi\)
−0.865780 + 0.500425i \(0.833177\pi\)
\(434\) 0 0
\(435\) −3.75201 −0.179895
\(436\) 0 0
\(437\) 5.85527 0.280095
\(438\) 0 0
\(439\) 18.6188 0.888629 0.444315 0.895871i \(-0.353447\pi\)
0.444315 + 0.895871i \(0.353447\pi\)
\(440\) 0 0
\(441\) −4.54453 −0.216406
\(442\) 0 0
\(443\) 2.26887 0.107797 0.0538987 0.998546i \(-0.482835\pi\)
0.0538987 + 0.998546i \(0.482835\pi\)
\(444\) 0 0
\(445\) 12.6633 0.600296
\(446\) 0 0
\(447\) −16.0287 −0.758130
\(448\) 0 0
\(449\) 0.248909 0.0117467 0.00587337 0.999983i \(-0.498130\pi\)
0.00587337 + 0.999983i \(0.498130\pi\)
\(450\) 0 0
\(451\) −11.9222 −0.561395
\(452\) 0 0
\(453\) 11.2242 0.527358
\(454\) 0 0
\(455\) 3.41787 0.160232
\(456\) 0 0
\(457\) 6.98767 0.326869 0.163435 0.986554i \(-0.447743\pi\)
0.163435 + 0.986554i \(0.447743\pi\)
\(458\) 0 0
\(459\) 5.34379 0.249427
\(460\) 0 0
\(461\) 34.1624 1.59110 0.795550 0.605888i \(-0.207182\pi\)
0.795550 + 0.605888i \(0.207182\pi\)
\(462\) 0 0
\(463\) 34.5311 1.60480 0.802399 0.596788i \(-0.203556\pi\)
0.802399 + 0.596788i \(0.203556\pi\)
\(464\) 0 0
\(465\) 8.60875 0.399221
\(466\) 0 0
\(467\) −39.6496 −1.83476 −0.917382 0.398008i \(-0.869702\pi\)
−0.917382 + 0.398008i \(0.869702\pi\)
\(468\) 0 0
\(469\) −5.53987 −0.255808
\(470\) 0 0
\(471\) 0.693945 0.0319753
\(472\) 0 0
\(473\) −11.0555 −0.508332
\(474\) 0 0
\(475\) 2.70913 0.124304
\(476\) 0 0
\(477\) −2.09503 −0.0959247
\(478\) 0 0
\(479\) −24.8669 −1.13620 −0.568098 0.822961i \(-0.692321\pi\)
−0.568098 + 0.822961i \(0.692321\pi\)
\(480\) 0 0
\(481\) 4.60917 0.210160
\(482\) 0 0
\(483\) 6.51836 0.296596
\(484\) 0 0
\(485\) 25.7974 1.17140
\(486\) 0 0
\(487\) 39.7320 1.80043 0.900214 0.435448i \(-0.143410\pi\)
0.900214 + 0.435448i \(0.143410\pi\)
\(488\) 0 0
\(489\) −0.280209 −0.0126715
\(490\) 0 0
\(491\) −3.45561 −0.155949 −0.0779747 0.996955i \(-0.524845\pi\)
−0.0779747 + 0.996955i \(0.524845\pi\)
\(492\) 0 0
\(493\) 1.55209 0.0699027
\(494\) 0 0
\(495\) −1.36052 −0.0611506
\(496\) 0 0
\(497\) −4.02457 −0.180527
\(498\) 0 0
\(499\) −17.5837 −0.787156 −0.393578 0.919291i \(-0.628763\pi\)
−0.393578 + 0.919291i \(0.628763\pi\)
\(500\) 0 0
\(501\) −0.780212 −0.0348573
\(502\) 0 0
\(503\) −2.72712 −0.121596 −0.0607981 0.998150i \(-0.519365\pi\)
−0.0607981 + 0.998150i \(0.519365\pi\)
\(504\) 0 0
\(505\) −2.99262 −0.133170
\(506\) 0 0
\(507\) 4.12553 0.183221
\(508\) 0 0
\(509\) 6.73123 0.298356 0.149178 0.988810i \(-0.452337\pi\)
0.149178 + 0.988810i \(0.452337\pi\)
\(510\) 0 0
\(511\) 8.87264 0.392503
\(512\) 0 0
\(513\) 4.44789 0.196379
\(514\) 0 0
\(515\) 3.49410 0.153968
\(516\) 0 0
\(517\) 16.9441 0.745198
\(518\) 0 0
\(519\) −24.2671 −1.06521
\(520\) 0 0
\(521\) −38.2190 −1.67440 −0.837202 0.546894i \(-0.815810\pi\)
−0.837202 + 0.546894i \(0.815810\pi\)
\(522\) 0 0
\(523\) −5.37182 −0.234893 −0.117447 0.993079i \(-0.537471\pi\)
−0.117447 + 0.993079i \(0.537471\pi\)
\(524\) 0 0
\(525\) 3.01594 0.131626
\(526\) 0 0
\(527\) −3.56118 −0.155127
\(528\) 0 0
\(529\) 11.2842 0.490616
\(530\) 0 0
\(531\) 0.176071 0.00764081
\(532\) 0 0
\(533\) −35.2089 −1.52506
\(534\) 0 0
\(535\) −22.7633 −0.984143
\(536\) 0 0
\(537\) 10.6611 0.460062
\(538\) 0 0
\(539\) 8.78229 0.378280
\(540\) 0 0
\(541\) −34.3497 −1.47681 −0.738404 0.674359i \(-0.764420\pi\)
−0.738404 + 0.674359i \(0.764420\pi\)
\(542\) 0 0
\(543\) −17.5269 −0.752150
\(544\) 0 0
\(545\) 20.2820 0.868786
\(546\) 0 0
\(547\) 2.58358 0.110466 0.0552330 0.998473i \(-0.482410\pi\)
0.0552330 + 0.998473i \(0.482410\pi\)
\(548\) 0 0
\(549\) −10.0392 −0.428464
\(550\) 0 0
\(551\) 1.29188 0.0550358
\(552\) 0 0
\(553\) 7.61936 0.324008
\(554\) 0 0
\(555\) −3.43921 −0.145986
\(556\) 0 0
\(557\) 0.498941 0.0211408 0.0105704 0.999944i \(-0.496635\pi\)
0.0105704 + 0.999944i \(0.496635\pi\)
\(558\) 0 0
\(559\) −32.6492 −1.38092
\(560\) 0 0
\(561\) 3.03843 0.128283
\(562\) 0 0
\(563\) −32.5553 −1.37204 −0.686021 0.727581i \(-0.740644\pi\)
−0.686021 + 0.727581i \(0.740644\pi\)
\(564\) 0 0
\(565\) 7.41216 0.311832
\(566\) 0 0
\(567\) 6.13863 0.257798
\(568\) 0 0
\(569\) −13.6834 −0.573636 −0.286818 0.957985i \(-0.592598\pi\)
−0.286818 + 0.957985i \(0.592598\pi\)
\(570\) 0 0
\(571\) 12.2520 0.512732 0.256366 0.966580i \(-0.417475\pi\)
0.256366 + 0.966580i \(0.417475\pi\)
\(572\) 0 0
\(573\) 10.5064 0.438909
\(574\) 0 0
\(575\) 15.8627 0.661521
\(576\) 0 0
\(577\) 14.6672 0.610605 0.305302 0.952255i \(-0.401242\pi\)
0.305302 + 0.952255i \(0.401242\pi\)
\(578\) 0 0
\(579\) 32.5624 1.35325
\(580\) 0 0
\(581\) −2.15017 −0.0892040
\(582\) 0 0
\(583\) 4.04864 0.167677
\(584\) 0 0
\(585\) −4.01789 −0.166119
\(586\) 0 0
\(587\) −5.53206 −0.228333 −0.114166 0.993462i \(-0.536420\pi\)
−0.114166 + 0.993462i \(0.536420\pi\)
\(588\) 0 0
\(589\) −2.96413 −0.122135
\(590\) 0 0
\(591\) −45.6165 −1.87641
\(592\) 0 0
\(593\) 21.5870 0.886470 0.443235 0.896405i \(-0.353831\pi\)
0.443235 + 0.896405i \(0.353831\pi\)
\(594\) 0 0
\(595\) 1.05498 0.0432500
\(596\) 0 0
\(597\) 33.7233 1.38020
\(598\) 0 0
\(599\) 28.3513 1.15840 0.579202 0.815184i \(-0.303364\pi\)
0.579202 + 0.815184i \(0.303364\pi\)
\(600\) 0 0
\(601\) 21.9758 0.896410 0.448205 0.893931i \(-0.352064\pi\)
0.448205 + 0.893931i \(0.352064\pi\)
\(602\) 0 0
\(603\) 6.51243 0.265207
\(604\) 0 0
\(605\) −14.0200 −0.569993
\(606\) 0 0
\(607\) −7.89484 −0.320441 −0.160221 0.987081i \(-0.551221\pi\)
−0.160221 + 0.987081i \(0.551221\pi\)
\(608\) 0 0
\(609\) 1.43818 0.0582780
\(610\) 0 0
\(611\) 50.0394 2.02438
\(612\) 0 0
\(613\) −29.5174 −1.19220 −0.596098 0.802912i \(-0.703283\pi\)
−0.596098 + 0.802912i \(0.703283\pi\)
\(614\) 0 0
\(615\) 26.2717 1.05938
\(616\) 0 0
\(617\) −3.08379 −0.124149 −0.0620744 0.998072i \(-0.519772\pi\)
−0.0620744 + 0.998072i \(0.519772\pi\)
\(618\) 0 0
\(619\) 10.9487 0.440066 0.220033 0.975492i \(-0.429384\pi\)
0.220033 + 0.975492i \(0.429384\pi\)
\(620\) 0 0
\(621\) 26.0436 1.04509
\(622\) 0 0
\(623\) −4.85394 −0.194469
\(624\) 0 0
\(625\) −4.11492 −0.164597
\(626\) 0 0
\(627\) 2.52903 0.101000
\(628\) 0 0
\(629\) 1.42270 0.0567267
\(630\) 0 0
\(631\) 20.7432 0.825773 0.412886 0.910783i \(-0.364521\pi\)
0.412886 + 0.910783i \(0.364521\pi\)
\(632\) 0 0
\(633\) −33.1586 −1.31794
\(634\) 0 0
\(635\) −3.95986 −0.157142
\(636\) 0 0
\(637\) 25.9360 1.02762
\(638\) 0 0
\(639\) 4.73110 0.187160
\(640\) 0 0
\(641\) −34.5310 −1.36389 −0.681946 0.731403i \(-0.738866\pi\)
−0.681946 + 0.731403i \(0.738866\pi\)
\(642\) 0 0
\(643\) 21.5300 0.849062 0.424531 0.905413i \(-0.360439\pi\)
0.424531 + 0.905413i \(0.360439\pi\)
\(644\) 0 0
\(645\) 24.3618 0.959245
\(646\) 0 0
\(647\) −22.7527 −0.894502 −0.447251 0.894409i \(-0.647597\pi\)
−0.447251 + 0.894409i \(0.647597\pi\)
\(648\) 0 0
\(649\) −0.340256 −0.0133562
\(650\) 0 0
\(651\) −3.29981 −0.129330
\(652\) 0 0
\(653\) −35.6805 −1.39629 −0.698143 0.715958i \(-0.745990\pi\)
−0.698143 + 0.715958i \(0.745990\pi\)
\(654\) 0 0
\(655\) −4.00334 −0.156423
\(656\) 0 0
\(657\) −10.4303 −0.406924
\(658\) 0 0
\(659\) −35.8589 −1.39686 −0.698432 0.715676i \(-0.746119\pi\)
−0.698432 + 0.715676i \(0.746119\pi\)
\(660\) 0 0
\(661\) 25.4409 0.989537 0.494769 0.869025i \(-0.335253\pi\)
0.494769 + 0.869025i \(0.335253\pi\)
\(662\) 0 0
\(663\) 8.97315 0.348488
\(664\) 0 0
\(665\) 0.878110 0.0340516
\(666\) 0 0
\(667\) 7.56429 0.292890
\(668\) 0 0
\(669\) 22.2639 0.860772
\(670\) 0 0
\(671\) 19.4008 0.748959
\(672\) 0 0
\(673\) 34.2142 1.31886 0.659431 0.751765i \(-0.270797\pi\)
0.659431 + 0.751765i \(0.270797\pi\)
\(674\) 0 0
\(675\) 12.0499 0.463802
\(676\) 0 0
\(677\) −37.9103 −1.45701 −0.728505 0.685040i \(-0.759785\pi\)
−0.728505 + 0.685040i \(0.759785\pi\)
\(678\) 0 0
\(679\) −9.88837 −0.379481
\(680\) 0 0
\(681\) 52.6696 2.01830
\(682\) 0 0
\(683\) −30.3542 −1.16147 −0.580736 0.814092i \(-0.697235\pi\)
−0.580736 + 0.814092i \(0.697235\pi\)
\(684\) 0 0
\(685\) −7.38671 −0.282232
\(686\) 0 0
\(687\) −41.9616 −1.60094
\(688\) 0 0
\(689\) 11.9565 0.455506
\(690\) 0 0
\(691\) 3.50694 0.133410 0.0667052 0.997773i \(-0.478751\pi\)
0.0667052 + 0.997773i \(0.478751\pi\)
\(692\) 0 0
\(693\) 0.521498 0.0198101
\(694\) 0 0
\(695\) 15.6402 0.593268
\(696\) 0 0
\(697\) −10.8678 −0.411647
\(698\) 0 0
\(699\) 42.5938 1.61105
\(700\) 0 0
\(701\) 13.2168 0.499190 0.249595 0.968350i \(-0.419703\pi\)
0.249595 + 0.968350i \(0.419703\pi\)
\(702\) 0 0
\(703\) 1.18418 0.0446621
\(704\) 0 0
\(705\) −37.3378 −1.40622
\(706\) 0 0
\(707\) 1.14710 0.0431411
\(708\) 0 0
\(709\) 23.2882 0.874605 0.437302 0.899314i \(-0.355934\pi\)
0.437302 + 0.899314i \(0.355934\pi\)
\(710\) 0 0
\(711\) −8.95699 −0.335913
\(712\) 0 0
\(713\) −17.3558 −0.649979
\(714\) 0 0
\(715\) 7.76457 0.290378
\(716\) 0 0
\(717\) 40.5551 1.51456
\(718\) 0 0
\(719\) 0.192843 0.00719184 0.00359592 0.999994i \(-0.498855\pi\)
0.00359592 + 0.999994i \(0.498855\pi\)
\(720\) 0 0
\(721\) −1.33932 −0.0498789
\(722\) 0 0
\(723\) −33.0882 −1.23056
\(724\) 0 0
\(725\) 3.49987 0.129982
\(726\) 0 0
\(727\) −14.5604 −0.540014 −0.270007 0.962858i \(-0.587026\pi\)
−0.270007 + 0.962858i \(0.587026\pi\)
\(728\) 0 0
\(729\) 18.3883 0.681047
\(730\) 0 0
\(731\) −10.0777 −0.372739
\(732\) 0 0
\(733\) 29.0856 1.07430 0.537150 0.843487i \(-0.319501\pi\)
0.537150 + 0.843487i \(0.319501\pi\)
\(734\) 0 0
\(735\) −19.3526 −0.713831
\(736\) 0 0
\(737\) −12.5853 −0.463584
\(738\) 0 0
\(739\) −28.8090 −1.05975 −0.529877 0.848074i \(-0.677762\pi\)
−0.529877 + 0.848074i \(0.677762\pi\)
\(740\) 0 0
\(741\) 7.46877 0.274372
\(742\) 0 0
\(743\) −42.2025 −1.54826 −0.774129 0.633027i \(-0.781812\pi\)
−0.774129 + 0.633027i \(0.781812\pi\)
\(744\) 0 0
\(745\) −12.6431 −0.463208
\(746\) 0 0
\(747\) 2.52764 0.0924816
\(748\) 0 0
\(749\) 8.72538 0.318818
\(750\) 0 0
\(751\) 2.64036 0.0963483 0.0481741 0.998839i \(-0.484660\pi\)
0.0481741 + 0.998839i \(0.484660\pi\)
\(752\) 0 0
\(753\) −11.6830 −0.425751
\(754\) 0 0
\(755\) 8.85344 0.322210
\(756\) 0 0
\(757\) 4.51764 0.164197 0.0820983 0.996624i \(-0.473838\pi\)
0.0820983 + 0.996624i \(0.473838\pi\)
\(758\) 0 0
\(759\) 14.8081 0.537502
\(760\) 0 0
\(761\) 23.5124 0.852323 0.426162 0.904647i \(-0.359866\pi\)
0.426162 + 0.904647i \(0.359866\pi\)
\(762\) 0 0
\(763\) −7.77428 −0.281448
\(764\) 0 0
\(765\) −1.24019 −0.0448392
\(766\) 0 0
\(767\) −1.00485 −0.0362830
\(768\) 0 0
\(769\) −29.6727 −1.07002 −0.535012 0.844845i \(-0.679693\pi\)
−0.535012 + 0.844845i \(0.679693\pi\)
\(770\) 0 0
\(771\) 19.5227 0.703091
\(772\) 0 0
\(773\) −10.1647 −0.365598 −0.182799 0.983150i \(-0.558516\pi\)
−0.182799 + 0.983150i \(0.558516\pi\)
\(774\) 0 0
\(775\) −8.03023 −0.288454
\(776\) 0 0
\(777\) 1.31828 0.0472931
\(778\) 0 0
\(779\) −9.04577 −0.324098
\(780\) 0 0
\(781\) −9.14285 −0.327157
\(782\) 0 0
\(783\) 5.74612 0.205350
\(784\) 0 0
\(785\) 0.547372 0.0195365
\(786\) 0 0
\(787\) −18.8481 −0.671863 −0.335931 0.941886i \(-0.609051\pi\)
−0.335931 + 0.941886i \(0.609051\pi\)
\(788\) 0 0
\(789\) 30.3974 1.08217
\(790\) 0 0
\(791\) −2.84115 −0.101020
\(792\) 0 0
\(793\) 57.2947 2.03459
\(794\) 0 0
\(795\) −8.92154 −0.316414
\(796\) 0 0
\(797\) 13.9098 0.492709 0.246355 0.969180i \(-0.420767\pi\)
0.246355 + 0.969180i \(0.420767\pi\)
\(798\) 0 0
\(799\) 15.4455 0.546423
\(800\) 0 0
\(801\) 5.70608 0.201614
\(802\) 0 0
\(803\) 20.1565 0.711308
\(804\) 0 0
\(805\) 5.14157 0.181217
\(806\) 0 0
\(807\) −43.8522 −1.54367
\(808\) 0 0
\(809\) −19.7630 −0.694831 −0.347415 0.937711i \(-0.612941\pi\)
−0.347415 + 0.937711i \(0.612941\pi\)
\(810\) 0 0
\(811\) −30.1099 −1.05730 −0.528651 0.848839i \(-0.677302\pi\)
−0.528651 + 0.848839i \(0.677302\pi\)
\(812\) 0 0
\(813\) 23.5694 0.826615
\(814\) 0 0
\(815\) −0.221024 −0.00774213
\(816\) 0 0
\(817\) −8.38816 −0.293465
\(818\) 0 0
\(819\) 1.54010 0.0538153
\(820\) 0 0
\(821\) −26.1788 −0.913645 −0.456823 0.889558i \(-0.651013\pi\)
−0.456823 + 0.889558i \(0.651013\pi\)
\(822\) 0 0
\(823\) −41.9817 −1.46339 −0.731696 0.681631i \(-0.761271\pi\)
−0.731696 + 0.681631i \(0.761271\pi\)
\(824\) 0 0
\(825\) 6.85148 0.238538
\(826\) 0 0
\(827\) 20.6394 0.717703 0.358851 0.933395i \(-0.383168\pi\)
0.358851 + 0.933395i \(0.383168\pi\)
\(828\) 0 0
\(829\) −36.6292 −1.27218 −0.636092 0.771613i \(-0.719450\pi\)
−0.636092 + 0.771613i \(0.719450\pi\)
\(830\) 0 0
\(831\) 1.23531 0.0428526
\(832\) 0 0
\(833\) 8.00558 0.277377
\(834\) 0 0
\(835\) −0.615418 −0.0212974
\(836\) 0 0
\(837\) −13.1841 −0.455710
\(838\) 0 0
\(839\) −3.71855 −0.128379 −0.0641893 0.997938i \(-0.520446\pi\)
−0.0641893 + 0.997938i \(0.520446\pi\)
\(840\) 0 0
\(841\) −27.3311 −0.942450
\(842\) 0 0
\(843\) 17.1359 0.590192
\(844\) 0 0
\(845\) 3.25414 0.111946
\(846\) 0 0
\(847\) 5.37399 0.184652
\(848\) 0 0
\(849\) 28.2835 0.970686
\(850\) 0 0
\(851\) 6.93367 0.237683
\(852\) 0 0
\(853\) 49.7965 1.70500 0.852501 0.522726i \(-0.175085\pi\)
0.852501 + 0.522726i \(0.175085\pi\)
\(854\) 0 0
\(855\) −1.03227 −0.0353028
\(856\) 0 0
\(857\) −35.6409 −1.21747 −0.608735 0.793374i \(-0.708323\pi\)
−0.608735 + 0.793374i \(0.708323\pi\)
\(858\) 0 0
\(859\) −43.5752 −1.48677 −0.743383 0.668866i \(-0.766780\pi\)
−0.743383 + 0.668866i \(0.766780\pi\)
\(860\) 0 0
\(861\) −10.0702 −0.343191
\(862\) 0 0
\(863\) −43.6579 −1.48613 −0.743067 0.669217i \(-0.766630\pi\)
−0.743067 + 0.669217i \(0.766630\pi\)
\(864\) 0 0
\(865\) −19.1415 −0.650829
\(866\) 0 0
\(867\) −29.8509 −1.01379
\(868\) 0 0
\(869\) 17.3093 0.587179
\(870\) 0 0
\(871\) −37.1669 −1.25935
\(872\) 0 0
\(873\) 11.6243 0.393424
\(874\) 0 0
\(875\) 6.76947 0.228850
\(876\) 0 0
\(877\) 5.23078 0.176631 0.0883154 0.996093i \(-0.471852\pi\)
0.0883154 + 0.996093i \(0.471852\pi\)
\(878\) 0 0
\(879\) 46.0172 1.55212
\(880\) 0 0
\(881\) −22.0358 −0.742405 −0.371203 0.928552i \(-0.621054\pi\)
−0.371203 + 0.928552i \(0.621054\pi\)
\(882\) 0 0
\(883\) 56.9658 1.91705 0.958525 0.285008i \(-0.0919962\pi\)
0.958525 + 0.285008i \(0.0919962\pi\)
\(884\) 0 0
\(885\) 0.749785 0.0252038
\(886\) 0 0
\(887\) −3.28694 −0.110365 −0.0551823 0.998476i \(-0.517574\pi\)
−0.0551823 + 0.998476i \(0.517574\pi\)
\(888\) 0 0
\(889\) 1.51785 0.0509071
\(890\) 0 0
\(891\) 13.9455 0.467191
\(892\) 0 0
\(893\) 12.8560 0.430210
\(894\) 0 0
\(895\) 8.40932 0.281092
\(896\) 0 0
\(897\) 43.7316 1.46016
\(898\) 0 0
\(899\) −3.82929 −0.127714
\(900\) 0 0
\(901\) 3.69057 0.122951
\(902\) 0 0
\(903\) −9.33810 −0.310753
\(904\) 0 0
\(905\) −13.8249 −0.459555
\(906\) 0 0
\(907\) 0.0491571 0.00163223 0.000816117 1.00000i \(-0.499740\pi\)
0.000816117 1.00000i \(0.499740\pi\)
\(908\) 0 0
\(909\) −1.34848 −0.0447263
\(910\) 0 0
\(911\) 32.4990 1.07674 0.538370 0.842708i \(-0.319040\pi\)
0.538370 + 0.842708i \(0.319040\pi\)
\(912\) 0 0
\(913\) −4.88466 −0.161659
\(914\) 0 0
\(915\) −42.7514 −1.41332
\(916\) 0 0
\(917\) 1.53452 0.0506742
\(918\) 0 0
\(919\) 19.8145 0.653619 0.326810 0.945090i \(-0.394026\pi\)
0.326810 + 0.945090i \(0.394026\pi\)
\(920\) 0 0
\(921\) 3.49485 0.115159
\(922\) 0 0
\(923\) −27.0008 −0.888742
\(924\) 0 0
\(925\) 3.20809 0.105481
\(926\) 0 0
\(927\) 1.57445 0.0517116
\(928\) 0 0
\(929\) 43.9293 1.44127 0.720637 0.693313i \(-0.243850\pi\)
0.720637 + 0.693313i \(0.243850\pi\)
\(930\) 0 0
\(931\) 6.66341 0.218385
\(932\) 0 0
\(933\) 8.69822 0.284767
\(934\) 0 0
\(935\) 2.39666 0.0783793
\(936\) 0 0
\(937\) 23.9026 0.780863 0.390431 0.920632i \(-0.372326\pi\)
0.390431 + 0.920632i \(0.372326\pi\)
\(938\) 0 0
\(939\) −23.8224 −0.777413
\(940\) 0 0
\(941\) −20.4109 −0.665375 −0.332688 0.943037i \(-0.607955\pi\)
−0.332688 + 0.943037i \(0.607955\pi\)
\(942\) 0 0
\(943\) −52.9654 −1.72479
\(944\) 0 0
\(945\) 3.90573 0.127054
\(946\) 0 0
\(947\) −2.72706 −0.0886176 −0.0443088 0.999018i \(-0.514109\pi\)
−0.0443088 + 0.999018i \(0.514109\pi\)
\(948\) 0 0
\(949\) 59.5265 1.93231
\(950\) 0 0
\(951\) −44.3157 −1.43703
\(952\) 0 0
\(953\) 32.7679 1.06146 0.530729 0.847542i \(-0.321918\pi\)
0.530729 + 0.847542i \(0.321918\pi\)
\(954\) 0 0
\(955\) 8.28723 0.268168
\(956\) 0 0
\(957\) 3.26720 0.105613
\(958\) 0 0
\(959\) 2.83139 0.0914305
\(960\) 0 0
\(961\) −22.2139 −0.716578
\(962\) 0 0
\(963\) −10.2572 −0.330533
\(964\) 0 0
\(965\) 25.6847 0.826818
\(966\) 0 0
\(967\) −36.4163 −1.17107 −0.585535 0.810647i \(-0.699115\pi\)
−0.585535 + 0.810647i \(0.699115\pi\)
\(968\) 0 0
\(969\) 2.30536 0.0740588
\(970\) 0 0
\(971\) 6.40329 0.205491 0.102746 0.994708i \(-0.467237\pi\)
0.102746 + 0.994708i \(0.467237\pi\)
\(972\) 0 0
\(973\) −5.99505 −0.192192
\(974\) 0 0
\(975\) 20.2339 0.648003
\(976\) 0 0
\(977\) 11.6473 0.372629 0.186315 0.982490i \(-0.440346\pi\)
0.186315 + 0.982490i \(0.440346\pi\)
\(978\) 0 0
\(979\) −11.0270 −0.352424
\(980\) 0 0
\(981\) 9.13910 0.291789
\(982\) 0 0
\(983\) 34.7061 1.10695 0.553477 0.832864i \(-0.313301\pi\)
0.553477 + 0.832864i \(0.313301\pi\)
\(984\) 0 0
\(985\) −35.9815 −1.14646
\(986\) 0 0
\(987\) 14.3119 0.455553
\(988\) 0 0
\(989\) −49.1150 −1.56176
\(990\) 0 0
\(991\) 3.30503 0.104988 0.0524939 0.998621i \(-0.483283\pi\)
0.0524939 + 0.998621i \(0.483283\pi\)
\(992\) 0 0
\(993\) −50.6085 −1.60601
\(994\) 0 0
\(995\) 26.6003 0.843288
\(996\) 0 0
\(997\) 15.9009 0.503586 0.251793 0.967781i \(-0.418980\pi\)
0.251793 + 0.967781i \(0.418980\pi\)
\(998\) 0 0
\(999\) 5.26708 0.166643
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.bo.1.6 8
4.3 odd 2 4864.2.a.bn.1.3 8
8.3 odd 2 4864.2.a.bp.1.6 8
8.5 even 2 4864.2.a.bq.1.3 8
16.3 odd 4 608.2.c.b.305.5 16
16.5 even 4 152.2.c.b.77.11 16
16.11 odd 4 608.2.c.b.305.12 16
16.13 even 4 152.2.c.b.77.12 yes 16
48.5 odd 4 1368.2.g.b.685.6 16
48.11 even 4 5472.2.g.b.2737.7 16
48.29 odd 4 1368.2.g.b.685.5 16
48.35 even 4 5472.2.g.b.2737.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.11 16 16.5 even 4
152.2.c.b.77.12 yes 16 16.13 even 4
608.2.c.b.305.5 16 16.3 odd 4
608.2.c.b.305.12 16 16.11 odd 4
1368.2.g.b.685.5 16 48.29 odd 4
1368.2.g.b.685.6 16 48.5 odd 4
4864.2.a.bn.1.3 8 4.3 odd 2
4864.2.a.bo.1.6 8 1.1 even 1 trivial
4864.2.a.bp.1.6 8 8.3 odd 2
4864.2.a.bq.1.3 8 8.5 even 2
5472.2.g.b.2737.7 16 48.11 even 4
5472.2.g.b.2737.10 16 48.35 even 4