Properties

Label 9216.2.a.bn.1.3
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.74912\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.47363 q^{5} +2.55765 q^{7} +O(q^{10})\) \(q+2.47363 q^{5} +2.55765 q^{7} +0.669808 q^{11} -4.08402 q^{13} -6.44549 q^{17} -6.44549 q^{19} +2.82843 q^{23} +1.11882 q^{25} +4.35480 q^{29} -6.55765 q^{31} +6.32666 q^{35} +3.85970 q^{37} -0.788632 q^{41} +0.550984 q^{43} -2.82843 q^{47} -0.458440 q^{49} +3.64520 q^{53} +1.65685 q^{55} -5.65685 q^{59} -6.20285 q^{61} -10.1023 q^{65} +2.99647 q^{67} +5.11529 q^{71} -14.7721 q^{73} +1.71313 q^{77} -6.32000 q^{79} -0.907457 q^{83} -15.9437 q^{85} +6.31724 q^{89} -10.4455 q^{91} -15.9437 q^{95} +12.6533 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 4q^{5} - 4q^{7} - 8q^{13} + 4q^{25} + 12q^{29} - 12q^{31} - 16q^{37} + 4q^{49} + 20q^{53} - 16q^{55} - 16q^{61} + 8q^{65} - 16q^{67} - 8q^{71} - 8q^{73} + 24q^{77} - 12q^{79} - 24q^{85} + 8q^{89} - 16q^{91} - 24q^{95} + O(q^{100}) \)

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 0
\(4\) 0 0
\(5\) 2.47363 1.10624 0.553120 0.833102i \(-0.313437\pi\)
0.553120 + 0.833102i \(0.313437\pi\)
\(6\) 0 0
\(7\) 2.55765 0.966700 0.483350 0.875427i \(-0.339420\pi\)
0.483350 + 0.875427i \(0.339420\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.669808 0.201955 0.100977 0.994889i \(-0.467803\pi\)
0.100977 + 0.994889i \(0.467803\pi\)
\(12\) 0 0
\(13\) −4.08402 −1.13270 −0.566352 0.824164i \(-0.691646\pi\)
−0.566352 + 0.824164i \(0.691646\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.44549 −1.56326 −0.781630 0.623742i \(-0.785611\pi\)
−0.781630 + 0.623742i \(0.785611\pi\)
\(18\) 0 0
\(19\) −6.44549 −1.47870 −0.739348 0.673323i \(-0.764866\pi\)
−0.739348 + 0.673323i \(0.764866\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 1.11882 0.223765
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.35480 0.808666 0.404333 0.914612i \(-0.367504\pi\)
0.404333 + 0.914612i \(0.367504\pi\)
\(30\) 0 0
\(31\) −6.55765 −1.17779 −0.588894 0.808210i \(-0.700437\pi\)
−0.588894 + 0.808210i \(0.700437\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.32666 1.06940
\(36\) 0 0
\(37\) 3.85970 0.634531 0.317265 0.948337i \(-0.397235\pi\)
0.317265 + 0.948337i \(0.397235\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.788632 −0.123164 −0.0615818 0.998102i \(-0.519615\pi\)
−0.0615818 + 0.998102i \(0.519615\pi\)
\(42\) 0 0
\(43\) 0.550984 0.0840242 0.0420121 0.999117i \(-0.486623\pi\)
0.0420121 + 0.999117i \(0.486623\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −0.458440 −0.0654915
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.64520 0.500707 0.250353 0.968155i \(-0.419453\pi\)
0.250353 + 0.968155i \(0.419453\pi\)
\(54\) 0 0
\(55\) 1.65685 0.223410
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.65685 −0.736460 −0.368230 0.929735i \(-0.620036\pi\)
−0.368230 + 0.929735i \(0.620036\pi\)
\(60\) 0 0
\(61\) −6.20285 −0.794193 −0.397097 0.917777i \(-0.629982\pi\)
−0.397097 + 0.917777i \(0.629982\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.1023 −1.25304
\(66\) 0 0
\(67\) 2.99647 0.366077 0.183039 0.983106i \(-0.441407\pi\)
0.183039 + 0.983106i \(0.441407\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.11529 0.607074 0.303537 0.952820i \(-0.401832\pi\)
0.303537 + 0.952820i \(0.401832\pi\)
\(72\) 0 0
\(73\) −14.7721 −1.72895 −0.864475 0.502676i \(-0.832349\pi\)
−0.864475 + 0.502676i \(0.832349\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.71313 0.195230
\(78\) 0 0
\(79\) −6.32000 −0.711055 −0.355528 0.934666i \(-0.615699\pi\)
−0.355528 + 0.934666i \(0.615699\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.907457 −0.0996063 −0.0498032 0.998759i \(-0.515859\pi\)
−0.0498032 + 0.998759i \(0.515859\pi\)
\(84\) 0 0
\(85\) −15.9437 −1.72934
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.31724 0.669626 0.334813 0.942285i \(-0.391327\pi\)
0.334813 + 0.942285i \(0.391327\pi\)
\(90\) 0 0
\(91\) −10.4455 −1.09498
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15.9437 −1.63579
\(96\) 0 0
\(97\) 12.6533 1.28475 0.642375 0.766390i \(-0.277949\pi\)
0.642375 + 0.766390i \(0.277949\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.6417 1.05889 0.529443 0.848346i \(-0.322401\pi\)
0.529443 + 0.848346i \(0.322401\pi\)
\(102\) 0 0
\(103\) −3.33686 −0.328790 −0.164395 0.986395i \(-0.552567\pi\)
−0.164395 + 0.986395i \(0.552567\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −19.8874 −1.92259 −0.961296 0.275518i \(-0.911151\pi\)
−0.961296 + 0.275518i \(0.911151\pi\)
\(108\) 0 0
\(109\) −3.91598 −0.375083 −0.187541 0.982257i \(-0.560052\pi\)
−0.187541 + 0.982257i \(0.560052\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.23765 0.210500 0.105250 0.994446i \(-0.466436\pi\)
0.105250 + 0.994446i \(0.466436\pi\)
\(114\) 0 0
\(115\) 6.99647 0.652424
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.4853 −1.51120
\(120\) 0 0
\(121\) −10.5514 −0.959214
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.60058 −0.858702
\(126\) 0 0
\(127\) −12.2145 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.33962 −0.466524 −0.233262 0.972414i \(-0.574940\pi\)
−0.233262 + 0.972414i \(0.574940\pi\)
\(132\) 0 0
\(133\) −16.4853 −1.42946
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.10587 0.436224 0.218112 0.975924i \(-0.430010\pi\)
0.218112 + 0.975924i \(0.430010\pi\)
\(138\) 0 0
\(139\) −16.6533 −1.41252 −0.706258 0.707954i \(-0.749618\pi\)
−0.706258 + 0.707954i \(0.749618\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.73551 −0.228755
\(144\) 0 0
\(145\) 10.7721 0.894578
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.1832 0.916166 0.458083 0.888909i \(-0.348536\pi\)
0.458083 + 0.888909i \(0.348536\pi\)
\(150\) 0 0
\(151\) −14.6506 −1.19225 −0.596123 0.802893i \(-0.703293\pi\)
−0.596123 + 0.802893i \(0.703293\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.2212 −1.30292
\(156\) 0 0
\(157\) −4.45754 −0.355750 −0.177875 0.984053i \(-0.556922\pi\)
−0.177875 + 0.984053i \(0.556922\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.23412 0.570128
\(162\) 0 0
\(163\) −7.78510 −0.609776 −0.304888 0.952388i \(-0.598619\pi\)
−0.304888 + 0.952388i \(0.598619\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −20.1814 −1.56168 −0.780841 0.624730i \(-0.785209\pi\)
−0.780841 + 0.624730i \(0.785209\pi\)
\(168\) 0 0
\(169\) 3.67923 0.283018
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.15639 0.468061 0.234031 0.972229i \(-0.424808\pi\)
0.234031 + 0.972229i \(0.424808\pi\)
\(174\) 0 0
\(175\) 2.86156 0.216313
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.7855 1.40409 0.702046 0.712131i \(-0.252270\pi\)
0.702046 + 0.712131i \(0.252270\pi\)
\(180\) 0 0
\(181\) 8.97499 0.667106 0.333553 0.942731i \(-0.391752\pi\)
0.333553 + 0.942731i \(0.391752\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.54745 0.701943
\(186\) 0 0
\(187\) −4.31724 −0.315708
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.60058 0.405243 0.202622 0.979257i \(-0.435054\pi\)
0.202622 + 0.979257i \(0.435054\pi\)
\(192\) 0 0
\(193\) −19.4514 −1.40014 −0.700071 0.714074i \(-0.746848\pi\)
−0.700071 + 0.714074i \(0.746848\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.75070 0.124732 0.0623659 0.998053i \(-0.480135\pi\)
0.0623659 + 0.998053i \(0.480135\pi\)
\(198\) 0 0
\(199\) 0.993710 0.0704422 0.0352211 0.999380i \(-0.488786\pi\)
0.0352211 + 0.999380i \(0.488786\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 11.1380 0.781738
\(204\) 0 0
\(205\) −1.95078 −0.136248
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.31724 −0.298630
\(210\) 0 0
\(211\) 5.97409 0.411273 0.205637 0.978628i \(-0.434073\pi\)
0.205637 + 0.978628i \(0.434073\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.36293 0.0929509
\(216\) 0 0
\(217\) −16.7721 −1.13857
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 26.3235 1.77071
\(222\) 0 0
\(223\) 23.7659 1.59148 0.795740 0.605639i \(-0.207082\pi\)
0.795740 + 0.605639i \(0.207082\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.907457 −0.0602300 −0.0301150 0.999546i \(-0.509587\pi\)
−0.0301150 + 0.999546i \(0.509587\pi\)
\(228\) 0 0
\(229\) −7.55579 −0.499301 −0.249650 0.968336i \(-0.580316\pi\)
−0.249650 + 0.968336i \(0.580316\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.2271 1.52166 0.760828 0.648954i \(-0.224793\pi\)
0.760828 + 0.648954i \(0.224793\pi\)
\(234\) 0 0
\(235\) −6.99647 −0.456399
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −26.9213 −1.74140 −0.870698 0.491817i \(-0.836333\pi\)
−0.870698 + 0.491817i \(0.836333\pi\)
\(240\) 0 0
\(241\) 10.3494 0.666664 0.333332 0.942809i \(-0.391827\pi\)
0.333332 + 0.942809i \(0.391827\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.13401 −0.0724492
\(246\) 0 0
\(247\) 26.3235 1.67492
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.7984 0.870949 0.435475 0.900201i \(-0.356581\pi\)
0.435475 + 0.900201i \(0.356581\pi\)
\(252\) 0 0
\(253\) 1.89450 0.119106
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.9965 −1.06021 −0.530105 0.847932i \(-0.677848\pi\)
−0.530105 + 0.847932i \(0.677848\pi\)
\(258\) 0 0
\(259\) 9.87175 0.613401
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −29.9929 −1.84944 −0.924722 0.380643i \(-0.875703\pi\)
−0.924722 + 0.380643i \(0.875703\pi\)
\(264\) 0 0
\(265\) 9.01686 0.553901
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 29.1332 1.77628 0.888142 0.459569i \(-0.151996\pi\)
0.888142 + 0.459569i \(0.151996\pi\)
\(270\) 0 0
\(271\) −26.6506 −1.61891 −0.809453 0.587184i \(-0.800236\pi\)
−0.809453 + 0.587184i \(0.800236\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.749397 0.0451904
\(276\) 0 0
\(277\) 17.1430 1.03003 0.515013 0.857183i \(-0.327787\pi\)
0.515013 + 0.857183i \(0.327787\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.76588 −0.164999 −0.0824993 0.996591i \(-0.526290\pi\)
−0.0824993 + 0.996591i \(0.526290\pi\)
\(282\) 0 0
\(283\) 6.34315 0.377061 0.188530 0.982067i \(-0.439628\pi\)
0.188530 + 0.982067i \(0.439628\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.01704 −0.119062
\(288\) 0 0
\(289\) 24.5443 1.44378
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −11.6078 −0.678133 −0.339067 0.940762i \(-0.610111\pi\)
−0.339067 + 0.940762i \(0.610111\pi\)
\(294\) 0 0
\(295\) −13.9929 −0.814700
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.5514 −0.668032
\(300\) 0 0
\(301\) 1.40922 0.0812262
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.3435 −0.878567
\(306\) 0 0
\(307\) 14.7855 0.843852 0.421926 0.906630i \(-0.361354\pi\)
0.421926 + 0.906630i \(0.361354\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.0761 −0.854885 −0.427442 0.904043i \(-0.640585\pi\)
−0.427442 + 0.904043i \(0.640585\pi\)
\(312\) 0 0
\(313\) 23.0027 1.30019 0.650096 0.759852i \(-0.274729\pi\)
0.650096 + 0.759852i \(0.274729\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.55855 0.536862 0.268431 0.963299i \(-0.413495\pi\)
0.268431 + 0.963299i \(0.413495\pi\)
\(318\) 0 0
\(319\) 2.91688 0.163314
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 41.5443 2.31159
\(324\) 0 0
\(325\) −4.56930 −0.253459
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.23412 −0.398830
\(330\) 0 0
\(331\) 27.8079 1.52846 0.764229 0.644945i \(-0.223120\pi\)
0.764229 + 0.644945i \(0.223120\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.41215 0.404969
\(336\) 0 0
\(337\) −3.00980 −0.163954 −0.0819771 0.996634i \(-0.526123\pi\)
−0.0819771 + 0.996634i \(0.526123\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.39236 −0.237860
\(342\) 0 0
\(343\) −19.0761 −1.03001
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.87449 −0.476408 −0.238204 0.971215i \(-0.576559\pi\)
−0.238204 + 0.971215i \(0.576559\pi\)
\(348\) 0 0
\(349\) −6.70698 −0.359016 −0.179508 0.983757i \(-0.557451\pi\)
−0.179508 + 0.983757i \(0.557451\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.75882 0.466185 0.233093 0.972455i \(-0.425116\pi\)
0.233093 + 0.972455i \(0.425116\pi\)
\(354\) 0 0
\(355\) 12.6533 0.671569
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 32.7917 1.73068 0.865341 0.501184i \(-0.167102\pi\)
0.865341 + 0.501184i \(0.167102\pi\)
\(360\) 0 0
\(361\) 22.5443 1.18654
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −36.5408 −1.91263
\(366\) 0 0
\(367\) 20.6435 1.07758 0.538791 0.842439i \(-0.318881\pi\)
0.538791 + 0.842439i \(0.318881\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.32313 0.484033
\(372\) 0 0
\(373\) −23.4995 −1.21676 −0.608379 0.793646i \(-0.708180\pi\)
−0.608379 + 0.793646i \(0.708180\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −17.7851 −0.915979
\(378\) 0 0
\(379\) 11.0004 0.565051 0.282526 0.959260i \(-0.408828\pi\)
0.282526 + 0.959260i \(0.408828\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.2037 −0.879070 −0.439535 0.898225i \(-0.644857\pi\)
−0.439535 + 0.898225i \(0.644857\pi\)
\(384\) 0 0
\(385\) 4.23765 0.215971
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −33.7311 −1.71023 −0.855116 0.518436i \(-0.826514\pi\)
−0.855116 + 0.518436i \(0.826514\pi\)
\(390\) 0 0
\(391\) −18.2306 −0.921961
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −15.6333 −0.786597
\(396\) 0 0
\(397\) 14.5201 0.728742 0.364371 0.931254i \(-0.381284\pi\)
0.364371 + 0.931254i \(0.381284\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.2274 1.60936 0.804681 0.593708i \(-0.202337\pi\)
0.804681 + 0.593708i \(0.202337\pi\)
\(402\) 0 0
\(403\) 26.7816 1.33409
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.58526 0.128146
\(408\) 0 0
\(409\) −11.5702 −0.572110 −0.286055 0.958213i \(-0.592344\pi\)
−0.286055 + 0.958213i \(0.592344\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −14.4682 −0.711935
\(414\) 0 0
\(415\) −2.24471 −0.110188
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.54193 0.466154 0.233077 0.972458i \(-0.425121\pi\)
0.233077 + 0.972458i \(0.425121\pi\)
\(420\) 0 0
\(421\) 24.3583 1.18715 0.593576 0.804778i \(-0.297716\pi\)
0.593576 + 0.804778i \(0.297716\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.21137 −0.349803
\(426\) 0 0
\(427\) −15.8647 −0.767746
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −40.7088 −1.96087 −0.980437 0.196832i \(-0.936935\pi\)
−0.980437 + 0.196832i \(0.936935\pi\)
\(432\) 0 0
\(433\) −7.31371 −0.351474 −0.175737 0.984437i \(-0.556231\pi\)
−0.175737 + 0.984437i \(0.556231\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18.2306 −0.872087
\(438\) 0 0
\(439\) −17.7122 −0.845356 −0.422678 0.906280i \(-0.638910\pi\)
−0.422678 + 0.906280i \(0.638910\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −22.1953 −1.05453 −0.527264 0.849701i \(-0.676782\pi\)
−0.527264 + 0.849701i \(0.676782\pi\)
\(444\) 0 0
\(445\) 15.6265 0.740766
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.3400 1.33745 0.668723 0.743511i \(-0.266841\pi\)
0.668723 + 0.743511i \(0.266841\pi\)
\(450\) 0 0
\(451\) −0.528232 −0.0248735
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −25.8382 −1.21131
\(456\) 0 0
\(457\) −17.3396 −0.811113 −0.405557 0.914070i \(-0.632922\pi\)
−0.405557 + 0.914070i \(0.632922\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.39404 −0.111501 −0.0557507 0.998445i \(-0.517755\pi\)
−0.0557507 + 0.998445i \(0.517755\pi\)
\(462\) 0 0
\(463\) 2.70238 0.125590 0.0627951 0.998026i \(-0.479999\pi\)
0.0627951 + 0.998026i \(0.479999\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.2023 1.11995 0.559975 0.828510i \(-0.310811\pi\)
0.559975 + 0.828510i \(0.310811\pi\)
\(468\) 0 0
\(469\) 7.66391 0.353887
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.369053 0.0169691
\(474\) 0 0
\(475\) −7.21137 −0.330880
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −22.2251 −1.01549 −0.507745 0.861508i \(-0.669521\pi\)
−0.507745 + 0.861508i \(0.669521\pi\)
\(480\) 0 0
\(481\) −15.7631 −0.718735
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 31.2996 1.42124
\(486\) 0 0
\(487\) −13.9839 −0.633672 −0.316836 0.948480i \(-0.602620\pi\)
−0.316836 + 0.948480i \(0.602620\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.2306 0.461700 0.230850 0.972989i \(-0.425849\pi\)
0.230850 + 0.972989i \(0.425849\pi\)
\(492\) 0 0
\(493\) −28.0688 −1.26416
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.0831 0.586858
\(498\) 0 0
\(499\) 3.66391 0.164019 0.0820097 0.996632i \(-0.473866\pi\)
0.0820097 + 0.996632i \(0.473866\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −39.6443 −1.76765 −0.883825 0.467817i \(-0.845041\pi\)
−0.883825 + 0.467817i \(0.845041\pi\)
\(504\) 0 0
\(505\) 26.3235 1.17138
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −28.6909 −1.27170 −0.635851 0.771812i \(-0.719351\pi\)
−0.635851 + 0.771812i \(0.719351\pi\)
\(510\) 0 0
\(511\) −37.7819 −1.67137
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.25413 −0.363721
\(516\) 0 0
\(517\) −1.89450 −0.0833201
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −23.1784 −1.01546 −0.507732 0.861515i \(-0.669516\pi\)
−0.507732 + 0.861515i \(0.669516\pi\)
\(522\) 0 0
\(523\) 8.18193 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.2672 1.84119
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.22079 0.139508
\(534\) 0 0
\(535\) −49.1941 −2.12685
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.307067 −0.0132263
\(540\) 0 0
\(541\) 6.43715 0.276755 0.138377 0.990380i \(-0.455811\pi\)
0.138377 + 0.990380i \(0.455811\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.68667 −0.414931
\(546\) 0 0
\(547\) 39.2239 1.67709 0.838546 0.544830i \(-0.183406\pi\)
0.838546 + 0.544830i \(0.183406\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −28.0688 −1.19577
\(552\) 0 0
\(553\) −16.1643 −0.687377
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.66224 −0.0704315 −0.0352157 0.999380i \(-0.511212\pi\)
−0.0352157 + 0.999380i \(0.511212\pi\)
\(558\) 0 0
\(559\) −2.25023 −0.0951745
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 40.6368 1.71264 0.856319 0.516447i \(-0.172746\pi\)
0.856319 + 0.516447i \(0.172746\pi\)
\(564\) 0 0
\(565\) 5.53511 0.232864
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.0004 1.13191 0.565957 0.824435i \(-0.308507\pi\)
0.565957 + 0.824435i \(0.308507\pi\)
\(570\) 0 0
\(571\) −20.9706 −0.877591 −0.438795 0.898587i \(-0.644595\pi\)
−0.438795 + 0.898587i \(0.644595\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.16451 0.131969
\(576\) 0 0
\(577\) −37.6372 −1.56686 −0.783429 0.621481i \(-0.786531\pi\)
−0.783429 + 0.621481i \(0.786531\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.32095 −0.0962894
\(582\) 0 0
\(583\) 2.44158 0.101120
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −44.2047 −1.82452 −0.912261 0.409609i \(-0.865665\pi\)
−0.912261 + 0.409609i \(0.865665\pi\)
\(588\) 0 0
\(589\) 42.2672 1.74159
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.59611 −0.147675 −0.0738373 0.997270i \(-0.523525\pi\)
−0.0738373 + 0.997270i \(0.523525\pi\)
\(594\) 0 0
\(595\) −40.7784 −1.67175
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.0296 0.900104 0.450052 0.893002i \(-0.351405\pi\)
0.450052 + 0.893002i \(0.351405\pi\)
\(600\) 0 0
\(601\) 10.7721 0.439405 0.219703 0.975567i \(-0.429491\pi\)
0.219703 + 0.975567i \(0.429491\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −26.1001 −1.06112
\(606\) 0 0
\(607\) −5.47453 −0.222204 −0.111102 0.993809i \(-0.535438\pi\)
−0.111102 + 0.993809i \(0.535438\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11.5514 0.467318
\(612\) 0 0
\(613\) 14.8562 0.600035 0.300018 0.953934i \(-0.403007\pi\)
0.300018 + 0.953934i \(0.403007\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.2235 −0.894686 −0.447343 0.894363i \(-0.647630\pi\)
−0.447343 + 0.894363i \(0.647630\pi\)
\(618\) 0 0
\(619\) −16.4612 −0.661631 −0.330815 0.943696i \(-0.607324\pi\)
−0.330815 + 0.943696i \(0.607324\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.1573 0.647327
\(624\) 0 0
\(625\) −29.3424 −1.17369
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.8776 −0.991937
\(630\) 0 0
\(631\) 4.06977 0.162015 0.0810075 0.996713i \(-0.474186\pi\)
0.0810075 + 0.996713i \(0.474186\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −30.2141 −1.19901
\(636\) 0 0
\(637\) 1.87228 0.0741824
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.41958 0.332553 0.166277 0.986079i \(-0.446826\pi\)
0.166277 + 0.986079i \(0.446826\pi\)
\(642\) 0 0
\(643\) 10.4266 0.411186 0.205593 0.978638i \(-0.434088\pi\)
0.205593 + 0.978638i \(0.434088\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.6132 0.456560 0.228280 0.973595i \(-0.426690\pi\)
0.228280 + 0.973595i \(0.426690\pi\)
\(648\) 0 0
\(649\) −3.78901 −0.148731
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.73012 −0.106838 −0.0534190 0.998572i \(-0.517012\pi\)
−0.0534190 + 0.998572i \(0.517012\pi\)
\(654\) 0 0
\(655\) −13.2082 −0.516088
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −31.5514 −1.22907 −0.614533 0.788891i \(-0.710656\pi\)
−0.614533 + 0.788891i \(0.710656\pi\)
\(660\) 0 0
\(661\) 15.1368 0.588752 0.294376 0.955690i \(-0.404888\pi\)
0.294376 + 0.955690i \(0.404888\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −40.7784 −1.58132
\(666\) 0 0
\(667\) 12.3172 0.476925
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.15472 −0.160391
\(672\) 0 0
\(673\) −20.6345 −0.795401 −0.397700 0.917515i \(-0.630192\pi\)
−0.397700 + 0.917515i \(0.630192\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 37.9357 1.45799 0.728994 0.684520i \(-0.239988\pi\)
0.728994 + 0.684520i \(0.239988\pi\)
\(678\) 0 0
\(679\) 32.3627 1.24197
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.2471 0.698205 0.349102 0.937085i \(-0.386487\pi\)
0.349102 + 0.937085i \(0.386487\pi\)
\(684\) 0 0
\(685\) 12.6300 0.482568
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14.8871 −0.567152
\(690\) 0 0
\(691\) −30.2533 −1.15089 −0.575446 0.817840i \(-0.695171\pi\)
−0.575446 + 0.817840i \(0.695171\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −41.1941 −1.56258
\(696\) 0 0
\(697\) 5.08312 0.192537
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20.0875 −0.758696 −0.379348 0.925254i \(-0.623852\pi\)
−0.379348 + 0.925254i \(0.623852\pi\)
\(702\) 0 0
\(703\) −24.8776 −0.938278
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.2176 1.02362
\(708\) 0 0
\(709\) 41.7864 1.56932 0.784660 0.619926i \(-0.212837\pi\)
0.784660 + 0.619926i \(0.212837\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.5478 −0.694622
\(714\) 0 0
\(715\) −6.76663 −0.253058
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28.3683 −1.05796 −0.528979 0.848635i \(-0.677425\pi\)
−0.528979 + 0.848635i \(0.677425\pi\)
\(720\) 0 0
\(721\) −8.53450 −0.317841
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.87226 0.180951
\(726\) 0 0
\(727\) 20.4843 0.759722 0.379861 0.925044i \(-0.375972\pi\)
0.379861 + 0.925044i \(0.375972\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.55136 −0.131352
\(732\) 0 0
\(733\) 48.0777 1.77579 0.887895 0.460045i \(-0.152167\pi\)
0.887895 + 0.460045i \(0.152167\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.00706 0.0739310
\(738\) 0 0
\(739\) −21.4459 −0.788899 −0.394449 0.918918i \(-0.629065\pi\)
−0.394449 + 0.918918i \(0.629065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.17431 0.0797677 0.0398839 0.999204i \(-0.487301\pi\)
0.0398839 + 0.999204i \(0.487301\pi\)
\(744\) 0 0
\(745\) 27.6631 1.01350
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −50.8651 −1.85857
\(750\) 0 0
\(751\) 29.8980 1.09099 0.545497 0.838113i \(-0.316341\pi\)
0.545497 + 0.838113i \(0.316341\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −36.2400 −1.31891
\(756\) 0 0
\(757\) −21.6791 −0.787939 −0.393970 0.919123i \(-0.628899\pi\)
−0.393970 + 0.919123i \(0.628899\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.29449 −0.155675 −0.0778375 0.996966i \(-0.524802\pi\)
−0.0778375 + 0.996966i \(0.524802\pi\)
\(762\) 0 0
\(763\) −10.0157 −0.362592
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.1027 0.834191
\(768\) 0 0
\(769\) 33.8819 1.22181 0.610907 0.791703i \(-0.290805\pi\)
0.610907 + 0.791703i \(0.290805\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −49.5300 −1.78147 −0.890736 0.454521i \(-0.849810\pi\)
−0.890736 + 0.454521i \(0.849810\pi\)
\(774\) 0 0
\(775\) −7.33686 −0.263548
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.08312 0.182122
\(780\) 0 0
\(781\) 3.42627 0.122601
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11.0263 −0.393545
\(786\) 0 0
\(787\) 34.0953 1.21537 0.607683 0.794180i \(-0.292099\pi\)
0.607683 + 0.794180i \(0.292099\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.72312 0.203491
\(792\) 0 0
\(793\) 25.3326 0.899585
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40.6901 1.44132 0.720659 0.693290i \(-0.243840\pi\)
0.720659 + 0.693290i \(0.243840\pi\)
\(798\) 0 0
\(799\) 18.2306 0.644952
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −9.89450 −0.349169
\(804\) 0 0
\(805\) 17.8945 0.630698
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.9926 0.386478 0.193239 0.981152i \(-0.438101\pi\)
0.193239 + 0.981152i \(0.438101\pi\)
\(810\) 0 0
\(811\) 21.2498 0.746182 0.373091 0.927795i \(-0.378298\pi\)
0.373091 + 0.927795i \(0.378298\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −19.2574 −0.674558
\(816\) 0 0
\(817\) −3.55136 −0.124246
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0572 1.04900 0.524501 0.851410i \(-0.324252\pi\)
0.524501 + 0.851410i \(0.324252\pi\)
\(822\) 0 0
\(823\) 55.0851 1.92015 0.960073 0.279751i \(-0.0902518\pi\)
0.960073 + 0.279751i \(0.0902518\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 34.5478 1.20135 0.600673 0.799495i \(-0.294899\pi\)
0.600673 + 0.799495i \(0.294899\pi\)
\(828\) 0 0
\(829\) −31.0046 −1.07683 −0.538417 0.842679i \(-0.680977\pi\)
−0.538417 + 0.842679i \(0.680977\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.95487 0.102380
\(834\) 0 0
\(835\) −49.9212 −1.72759
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.14195 0.177520 0.0887599 0.996053i \(-0.471710\pi\)
0.0887599 + 0.996053i \(0.471710\pi\)
\(840\) 0 0
\(841\) −10.0357 −0.346059
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.10104 0.313085
\(846\) 0 0
\(847\) −26.9867 −0.927272
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.9169 0.374226
\(852\) 0 0
\(853\) 18.5060 0.633632 0.316816 0.948487i \(-0.397386\pi\)
0.316816 + 0.948487i \(0.397386\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.8878 −0.781833 −0.390916 0.920426i \(-0.627842\pi\)
−0.390916 + 0.920426i \(0.627842\pi\)
\(858\) 0 0
\(859\) −35.5286 −1.21222 −0.606110 0.795381i \(-0.707271\pi\)
−0.606110 + 0.795381i \(0.707271\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −43.9296 −1.49538 −0.747691 0.664047i \(-0.768837\pi\)
−0.747691 + 0.664047i \(0.768837\pi\)
\(864\) 0 0
\(865\) 15.2286 0.517788
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.23319 −0.143601
\(870\) 0 0
\(871\) −12.2376 −0.414657
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −24.5549 −0.830107
\(876\) 0 0
\(877\) −21.5773 −0.728614 −0.364307 0.931279i \(-0.618694\pi\)
−0.364307 + 0.931279i \(0.618694\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 21.6686 0.730035 0.365018 0.931001i \(-0.381063\pi\)
0.365018 + 0.931001i \(0.381063\pi\)
\(882\) 0 0
\(883\) −0.0834930 −0.00280976 −0.00140488 0.999999i \(-0.500447\pi\)
−0.00140488 + 0.999999i \(0.500447\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 30.8043 1.03431 0.517154 0.855892i \(-0.326991\pi\)
0.517154 + 0.855892i \(0.326991\pi\)
\(888\) 0 0
\(889\) −31.2404 −1.04777
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.2306 0.610063
\(894\) 0 0
\(895\) 46.4682 1.55326
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −28.5573 −0.952438
\(900\) 0 0
\(901\) −23.4951 −0.782735
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 22.2008 0.737979
\(906\) 0 0
\(907\) −49.5215 −1.64434 −0.822168 0.569245i \(-0.807236\pi\)
−0.822168 + 0.569245i \(0.807236\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.0829331 −0.00274770 −0.00137385 0.999999i \(-0.500437\pi\)
−0.00137385 + 0.999999i \(0.500437\pi\)
\(912\) 0 0
\(913\) −0.607822 −0.0201160
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13.6569 −0.450989
\(918\) 0 0
\(919\) −20.1161 −0.663568 −0.331784 0.943355i \(-0.607650\pi\)
−0.331784 + 0.943355i \(0.607650\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −20.8910 −0.687635
\(924\) 0 0
\(925\) 4.31833 0.141986
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −8.55098 −0.280549 −0.140274 0.990113i \(-0.544798\pi\)
−0.140274 + 0.990113i \(0.544798\pi\)
\(930\) 0 0
\(931\) 2.95487 0.0968420
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −10.6792 −0.349248
\(936\) 0 0
\(937\) 33.5780 1.09695 0.548473 0.836168i \(-0.315209\pi\)
0.548473 + 0.836168i \(0.315209\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.9991 −0.391159 −0.195579 0.980688i \(-0.562659\pi\)
−0.195579 + 0.980688i \(0.562659\pi\)
\(942\) 0 0
\(943\) −2.23059 −0.0726380
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25.2537 −0.820635 −0.410318 0.911943i \(-0.634582\pi\)
−0.410318 + 0.911943i \(0.634582\pi\)
\(948\) 0 0
\(949\) 60.3298 1.95839
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.86469 −0.125190 −0.0625948 0.998039i \(-0.519938\pi\)
−0.0625948 + 0.998039i \(0.519938\pi\)
\(954\) 0 0
\(955\) 13.8537 0.448296
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 13.0590 0.421698
\(960\) 0 0
\(961\) 12.0027 0.387185
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −48.1154 −1.54889
\(966\) 0 0
\(967\) 37.8714 1.21786 0.608930 0.793224i \(-0.291599\pi\)
0.608930 + 0.793224i \(0.291599\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.74587 0.120211 0.0601053 0.998192i \(-0.480856\pi\)
0.0601053 + 0.998192i \(0.480856\pi\)
\(972\) 0 0
\(973\) −42.5933 −1.36548
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 17.6530 0.564768 0.282384 0.959301i \(-0.408875\pi\)
0.282384 + 0.959301i \(0.408875\pi\)
\(978\) 0 0
\(979\) 4.23134 0.135234
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22.3557 0.713035 0.356518 0.934289i \(-0.383964\pi\)
0.356518 + 0.934289i \(0.383964\pi\)
\(984\) 0 0
\(985\) 4.33057 0.137983
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.55842 0.0495548
\(990\) 0 0
\(991\) −17.8769 −0.567878 −0.283939 0.958842i \(-0.591641\pi\)
−0.283939 + 0.958842i \(0.591641\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.45807 0.0779260
\(996\) 0 0
\(997\) −6.06146 −0.191968 −0.0959841 0.995383i \(-0.530600\pi\)
−0.0959841 + 0.995383i \(0.530600\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9216.2.a.bn.1.3 4
3.2 odd 2 3072.2.a.o.1.2 4
4.3 odd 2 9216.2.a.bo.1.3 4
8.3 odd 2 9216.2.a.y.1.2 4
8.5 even 2 9216.2.a.x.1.2 4
12.11 even 2 3072.2.a.i.1.2 4
24.5 odd 2 3072.2.a.n.1.3 4
24.11 even 2 3072.2.a.t.1.3 4
32.3 odd 8 144.2.k.b.109.3 8
32.5 even 8 1152.2.k.f.865.4 8
32.11 odd 8 144.2.k.b.37.3 8
32.13 even 8 1152.2.k.f.289.4 8
32.19 odd 8 1152.2.k.c.289.4 8
32.21 even 8 576.2.k.b.433.1 8
32.27 odd 8 1152.2.k.c.865.4 8
32.29 even 8 576.2.k.b.145.1 8
48.5 odd 4 3072.2.d.i.1537.2 8
48.11 even 4 3072.2.d.f.1537.6 8
48.29 odd 4 3072.2.d.i.1537.7 8
48.35 even 4 3072.2.d.f.1537.3 8
96.5 odd 8 384.2.j.a.97.1 8
96.11 even 8 48.2.j.a.37.2 yes 8
96.29 odd 8 192.2.j.a.145.4 8
96.35 even 8 48.2.j.a.13.2 8
96.53 odd 8 192.2.j.a.49.4 8
96.59 even 8 384.2.j.b.97.3 8
96.77 odd 8 384.2.j.a.289.1 8
96.83 even 8 384.2.j.b.289.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.2 8 96.35 even 8
48.2.j.a.37.2 yes 8 96.11 even 8
144.2.k.b.37.3 8 32.11 odd 8
144.2.k.b.109.3 8 32.3 odd 8
192.2.j.a.49.4 8 96.53 odd 8
192.2.j.a.145.4 8 96.29 odd 8
384.2.j.a.97.1 8 96.5 odd 8
384.2.j.a.289.1 8 96.77 odd 8
384.2.j.b.97.3 8 96.59 even 8
384.2.j.b.289.3 8 96.83 even 8
576.2.k.b.145.1 8 32.29 even 8
576.2.k.b.433.1 8 32.21 even 8
1152.2.k.c.289.4 8 32.19 odd 8
1152.2.k.c.865.4 8 32.27 odd 8
1152.2.k.f.289.4 8 32.13 even 8
1152.2.k.f.865.4 8 32.5 even 8
3072.2.a.i.1.2 4 12.11 even 2
3072.2.a.n.1.3 4 24.5 odd 2
3072.2.a.o.1.2 4 3.2 odd 2
3072.2.a.t.1.3 4 24.11 even 2
3072.2.d.f.1537.3 8 48.35 even 4
3072.2.d.f.1537.6 8 48.11 even 4
3072.2.d.i.1537.2 8 48.5 odd 4
3072.2.d.i.1537.7 8 48.29 odd 4
9216.2.a.x.1.2 4 8.5 even 2
9216.2.a.y.1.2 4 8.3 odd 2
9216.2.a.bn.1.3 4 1.1 even 1 trivial
9216.2.a.bo.1.3 4 4.3 odd 2