Properties

Label 6048.2.j.c.5615.18
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 32
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(32\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.18
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.17

$q$-expansion

\(f(q)\) \(=\) \(q+0.162025 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+0.162025 q^{5} +1.00000i q^{7} -2.40352i q^{11} +4.76784i q^{13} -2.74335i q^{17} +1.86073 q^{19} +1.32970 q^{23} -4.97375 q^{25} +3.96112 q^{29} -8.01692i q^{31} +0.162025i q^{35} +6.09436i q^{37} -3.16889i q^{41} +1.70927 q^{43} +12.1502 q^{47} -1.00000 q^{49} -8.22731 q^{53} -0.389431i q^{55} -5.45449i q^{59} +4.28553i q^{61} +0.772509i q^{65} +4.61226 q^{67} +13.9767 q^{71} -6.37815 q^{73} +2.40352 q^{77} -9.75258i q^{79} +16.4983i q^{83} -0.444491i q^{85} +2.35048i q^{89} -4.76784 q^{91} +0.301485 q^{95} +14.4569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + O(q^{10}) \) \( 32q + 64q^{19} - 16q^{25} + 48q^{43} - 32q^{49} - 16q^{67} - 16q^{73} - 16q^{91} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(5\) 0.162025 0.0724598 0.0362299 0.999343i \(-0.488465\pi\)
0.0362299 + 0.999343i \(0.488465\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.40352i − 0.724689i −0.932044 0.362345i \(-0.881976\pi\)
0.932044 0.362345i \(-0.118024\pi\)
\(12\) 0 0
\(13\) 4.76784i 1.32236i 0.750227 + 0.661181i \(0.229944\pi\)
−0.750227 + 0.661181i \(0.770056\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.74335i − 0.665359i −0.943040 0.332680i \(-0.892047\pi\)
0.943040 0.332680i \(-0.107953\pi\)
\(18\) 0 0
\(19\) 1.86073 0.426881 0.213440 0.976956i \(-0.431533\pi\)
0.213440 + 0.976956i \(0.431533\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.32970 0.277262 0.138631 0.990344i \(-0.455730\pi\)
0.138631 + 0.990344i \(0.455730\pi\)
\(24\) 0 0
\(25\) −4.97375 −0.994750
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.96112 0.735561 0.367780 0.929913i \(-0.380118\pi\)
0.367780 + 0.929913i \(0.380118\pi\)
\(30\) 0 0
\(31\) − 8.01692i − 1.43988i −0.694036 0.719941i \(-0.744169\pi\)
0.694036 0.719941i \(-0.255831\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.162025i 0.0273872i
\(36\) 0 0
\(37\) 6.09436i 1.00191i 0.865474 + 0.500953i \(0.167017\pi\)
−0.865474 + 0.500953i \(0.832983\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.16889i − 0.494898i −0.968901 0.247449i \(-0.920408\pi\)
0.968901 0.247449i \(-0.0795922\pi\)
\(42\) 0 0
\(43\) 1.70927 0.260662 0.130331 0.991471i \(-0.458396\pi\)
0.130331 + 0.991471i \(0.458396\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.1502 1.77229 0.886144 0.463410i \(-0.153374\pi\)
0.886144 + 0.463410i \(0.153374\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.22731 −1.13011 −0.565054 0.825054i \(-0.691145\pi\)
−0.565054 + 0.825054i \(0.691145\pi\)
\(54\) 0 0
\(55\) − 0.389431i − 0.0525108i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5.45449i − 0.710114i −0.934845 0.355057i \(-0.884461\pi\)
0.934845 0.355057i \(-0.115539\pi\)
\(60\) 0 0
\(61\) 4.28553i 0.548706i 0.961629 + 0.274353i \(0.0884636\pi\)
−0.961629 + 0.274353i \(0.911536\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.772509i 0.0958180i
\(66\) 0 0
\(67\) 4.61226 0.563477 0.281738 0.959491i \(-0.409089\pi\)
0.281738 + 0.959491i \(0.409089\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.9767 1.65873 0.829363 0.558710i \(-0.188703\pi\)
0.829363 + 0.558710i \(0.188703\pi\)
\(72\) 0 0
\(73\) −6.37815 −0.746506 −0.373253 0.927730i \(-0.621758\pi\)
−0.373253 + 0.927730i \(0.621758\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.40352 0.273907
\(78\) 0 0
\(79\) − 9.75258i − 1.09725i −0.836068 0.548625i \(-0.815151\pi\)
0.836068 0.548625i \(-0.184849\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.4983i 1.81093i 0.424426 + 0.905463i \(0.360476\pi\)
−0.424426 + 0.905463i \(0.639524\pi\)
\(84\) 0 0
\(85\) − 0.444491i − 0.0482118i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.35048i 0.249150i 0.992210 + 0.124575i \(0.0397567\pi\)
−0.992210 + 0.124575i \(0.960243\pi\)
\(90\) 0 0
\(91\) −4.76784 −0.499806
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.301485 0.0309317
\(96\) 0 0
\(97\) 14.4569 1.46788 0.733938 0.679217i \(-0.237680\pi\)
0.733938 + 0.679217i \(0.237680\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.453215 0.0450966 0.0225483 0.999746i \(-0.492822\pi\)
0.0225483 + 0.999746i \(0.492822\pi\)
\(102\) 0 0
\(103\) − 8.93542i − 0.880434i −0.897892 0.440217i \(-0.854902\pi\)
0.897892 0.440217i \(-0.145098\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.9379i 1.44411i 0.691838 + 0.722053i \(0.256801\pi\)
−0.691838 + 0.722053i \(0.743199\pi\)
\(108\) 0 0
\(109\) − 4.33798i − 0.415503i −0.978182 0.207751i \(-0.933385\pi\)
0.978182 0.207751i \(-0.0666145\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.451095i 0.0424354i 0.999775 + 0.0212177i \(0.00675431\pi\)
−0.999775 + 0.0212177i \(0.993246\pi\)
\(114\) 0 0
\(115\) 0.215445 0.0200903
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.74335 0.251482
\(120\) 0 0
\(121\) 5.22308 0.474826
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.61600 −0.144539
\(126\) 0 0
\(127\) 0.196751i 0.0174589i 0.999962 + 0.00872943i \(0.00277870\pi\)
−0.999962 + 0.00872943i \(0.997221\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 4.69097i − 0.409852i −0.978778 0.204926i \(-0.934305\pi\)
0.978778 0.204926i \(-0.0656953\pi\)
\(132\) 0 0
\(133\) 1.86073i 0.161346i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.1485i 0.952479i 0.879316 + 0.476240i \(0.158001\pi\)
−0.879316 + 0.476240i \(0.841999\pi\)
\(138\) 0 0
\(139\) −6.93228 −0.587988 −0.293994 0.955807i \(-0.594985\pi\)
−0.293994 + 0.955807i \(0.594985\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.4596 0.958301
\(144\) 0 0
\(145\) 0.641800 0.0532986
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.2542 1.33160 0.665799 0.746132i \(-0.268091\pi\)
0.665799 + 0.746132i \(0.268091\pi\)
\(150\) 0 0
\(151\) − 0.426011i − 0.0346683i −0.999850 0.0173341i \(-0.994482\pi\)
0.999850 0.0173341i \(-0.00551791\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.29894i − 0.104333i
\(156\) 0 0
\(157\) 14.0451i 1.12092i 0.828180 + 0.560462i \(0.189376\pi\)
−0.828180 + 0.560462i \(0.810624\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.32970i 0.104795i
\(162\) 0 0
\(163\) 4.72962 0.370453 0.185226 0.982696i \(-0.440698\pi\)
0.185226 + 0.982696i \(0.440698\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.7344 −0.908036 −0.454018 0.890993i \(-0.650010\pi\)
−0.454018 + 0.890993i \(0.650010\pi\)
\(168\) 0 0
\(169\) −9.73231 −0.748639
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.2577 1.38810 0.694052 0.719925i \(-0.255824\pi\)
0.694052 + 0.719925i \(0.255824\pi\)
\(174\) 0 0
\(175\) − 4.97375i − 0.375980i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 14.5950i − 1.09088i −0.838148 0.545442i \(-0.816362\pi\)
0.838148 0.545442i \(-0.183638\pi\)
\(180\) 0 0
\(181\) − 3.82026i − 0.283958i −0.989870 0.141979i \(-0.954653\pi\)
0.989870 0.141979i \(-0.0453465\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.987438i 0.0725979i
\(186\) 0 0
\(187\) −6.59369 −0.482178
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.4447 1.55168 0.775840 0.630929i \(-0.217326\pi\)
0.775840 + 0.630929i \(0.217326\pi\)
\(192\) 0 0
\(193\) −17.5991 −1.26681 −0.633407 0.773819i \(-0.718344\pi\)
−0.633407 + 0.773819i \(0.718344\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.1329 1.36316 0.681581 0.731743i \(-0.261293\pi\)
0.681581 + 0.731743i \(0.261293\pi\)
\(198\) 0 0
\(199\) 13.3125i 0.943699i 0.881679 + 0.471850i \(0.156414\pi\)
−0.881679 + 0.471850i \(0.843586\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.96112i 0.278016i
\(204\) 0 0
\(205\) − 0.513440i − 0.0358602i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 4.47230i − 0.309356i
\(210\) 0 0
\(211\) 23.9337 1.64766 0.823830 0.566836i \(-0.191833\pi\)
0.823830 + 0.566836i \(0.191833\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.276945 0.0188875
\(216\) 0 0
\(217\) 8.01692 0.544224
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.0798 0.879845
\(222\) 0 0
\(223\) − 13.9928i − 0.937024i −0.883457 0.468512i \(-0.844790\pi\)
0.883457 0.468512i \(-0.155210\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 10.8975i − 0.723292i −0.932316 0.361646i \(-0.882215\pi\)
0.932316 0.361646i \(-0.117785\pi\)
\(228\) 0 0
\(229\) 11.1299i 0.735485i 0.929928 + 0.367743i \(0.119869\pi\)
−0.929928 + 0.367743i \(0.880131\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 13.7778i − 0.902613i −0.892369 0.451307i \(-0.850958\pi\)
0.892369 0.451307i \(-0.149042\pi\)
\(234\) 0 0
\(235\) 1.96863 0.128420
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.99259 −0.516998 −0.258499 0.966012i \(-0.583228\pi\)
−0.258499 + 0.966012i \(0.583228\pi\)
\(240\) 0 0
\(241\) −5.18318 −0.333878 −0.166939 0.985967i \(-0.553388\pi\)
−0.166939 + 0.985967i \(0.553388\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.162025 −0.0103514
\(246\) 0 0
\(247\) 8.87166i 0.564490i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 7.88280i − 0.497558i −0.968560 0.248779i \(-0.919971\pi\)
0.968560 0.248779i \(-0.0800293\pi\)
\(252\) 0 0
\(253\) − 3.19597i − 0.200929i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.1394i 1.75529i 0.479312 + 0.877645i \(0.340886\pi\)
−0.479312 + 0.877645i \(0.659114\pi\)
\(258\) 0 0
\(259\) −6.09436 −0.378685
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 30.5298 1.88255 0.941274 0.337645i \(-0.109630\pi\)
0.941274 + 0.337645i \(0.109630\pi\)
\(264\) 0 0
\(265\) −1.33303 −0.0818874
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.2495 0.807839 0.403920 0.914795i \(-0.367648\pi\)
0.403920 + 0.914795i \(0.367648\pi\)
\(270\) 0 0
\(271\) 7.06832i 0.429370i 0.976683 + 0.214685i \(0.0688725\pi\)
−0.976683 + 0.214685i \(0.931128\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.9545i 0.720884i
\(276\) 0 0
\(277\) 10.3435i 0.621481i 0.950495 + 0.310740i \(0.100577\pi\)
−0.950495 + 0.310740i \(0.899423\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 27.0638i − 1.61449i −0.590217 0.807245i \(-0.700958\pi\)
0.590217 0.807245i \(-0.299042\pi\)
\(282\) 0 0
\(283\) −17.2246 −1.02390 −0.511948 0.859017i \(-0.671076\pi\)
−0.511948 + 0.859017i \(0.671076\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.16889 0.187054
\(288\) 0 0
\(289\) 9.47405 0.557297
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.199121 −0.0116328 −0.00581639 0.999983i \(-0.501851\pi\)
−0.00581639 + 0.999983i \(0.501851\pi\)
\(294\) 0 0
\(295\) − 0.883763i − 0.0514547i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.33981i 0.366641i
\(300\) 0 0
\(301\) 1.70927i 0.0985209i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.694363i 0.0397591i
\(306\) 0 0
\(307\) −10.0504 −0.573606 −0.286803 0.957990i \(-0.592593\pi\)
−0.286803 + 0.957990i \(0.592593\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.74569 −0.552627 −0.276314 0.961068i \(-0.589113\pi\)
−0.276314 + 0.961068i \(0.589113\pi\)
\(312\) 0 0
\(313\) 3.46199 0.195683 0.0978415 0.995202i \(-0.468806\pi\)
0.0978415 + 0.995202i \(0.468806\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.5983 1.55008 0.775039 0.631914i \(-0.217730\pi\)
0.775039 + 0.631914i \(0.217730\pi\)
\(318\) 0 0
\(319\) − 9.52063i − 0.533053i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 5.10462i − 0.284029i
\(324\) 0 0
\(325\) − 23.7140i − 1.31542i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.1502i 0.669862i
\(330\) 0 0
\(331\) 13.5802 0.746434 0.373217 0.927744i \(-0.378255\pi\)
0.373217 + 0.927744i \(0.378255\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.747301 0.0408294
\(336\) 0 0
\(337\) −0.0989875 −0.00539219 −0.00269610 0.999996i \(-0.500858\pi\)
−0.00269610 + 0.999996i \(0.500858\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −19.2688 −1.04347
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.82609i 0.312761i 0.987697 + 0.156380i \(0.0499826\pi\)
−0.987697 + 0.156380i \(0.950017\pi\)
\(348\) 0 0
\(349\) 30.1726i 1.61510i 0.589798 + 0.807551i \(0.299207\pi\)
−0.589798 + 0.807551i \(0.700793\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.0333i 0.853364i 0.904402 + 0.426682i \(0.140318\pi\)
−0.904402 + 0.426682i \(0.859682\pi\)
\(354\) 0 0
\(355\) 2.26457 0.120191
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.0001 −0.844451 −0.422225 0.906491i \(-0.638751\pi\)
−0.422225 + 0.906491i \(0.638751\pi\)
\(360\) 0 0
\(361\) −15.5377 −0.817773
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.03342 −0.0540917
\(366\) 0 0
\(367\) − 13.0471i − 0.681053i −0.940235 0.340527i \(-0.889395\pi\)
0.940235 0.340527i \(-0.110605\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 8.22731i − 0.427141i
\(372\) 0 0
\(373\) − 0.319142i − 0.0165245i −0.999966 0.00826227i \(-0.997370\pi\)
0.999966 0.00826227i \(-0.00262999\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.8860i 0.972677i
\(378\) 0 0
\(379\) −13.4802 −0.692431 −0.346215 0.938155i \(-0.612533\pi\)
−0.346215 + 0.938155i \(0.612533\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.80894 0.296823 0.148411 0.988926i \(-0.452584\pi\)
0.148411 + 0.988926i \(0.452584\pi\)
\(384\) 0 0
\(385\) 0.389431 0.0198472
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.7958 1.66282 0.831408 0.555663i \(-0.187535\pi\)
0.831408 + 0.555663i \(0.187535\pi\)
\(390\) 0 0
\(391\) − 3.64783i − 0.184479i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 1.58016i − 0.0795066i
\(396\) 0 0
\(397\) − 26.2294i − 1.31641i −0.752837 0.658207i \(-0.771315\pi\)
0.752837 0.658207i \(-0.228685\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 20.9069i − 1.04404i −0.852933 0.522020i \(-0.825179\pi\)
0.852933 0.522020i \(-0.174821\pi\)
\(402\) 0 0
\(403\) 38.2234 1.90404
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.6479 0.726071
\(408\) 0 0
\(409\) 33.7856 1.67059 0.835295 0.549801i \(-0.185297\pi\)
0.835295 + 0.549801i \(0.185297\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.45449 0.268398
\(414\) 0 0
\(415\) 2.67314i 0.131219i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.4203i 0.606771i 0.952868 + 0.303385i \(0.0981170\pi\)
−0.952868 + 0.303385i \(0.901883\pi\)
\(420\) 0 0
\(421\) 21.3631i 1.04118i 0.853808 + 0.520588i \(0.174287\pi\)
−0.853808 + 0.520588i \(0.825713\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.6447i 0.661866i
\(426\) 0 0
\(427\) −4.28553 −0.207391
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −21.3649 −1.02911 −0.514557 0.857456i \(-0.672043\pi\)
−0.514557 + 0.857456i \(0.672043\pi\)
\(432\) 0 0
\(433\) 21.5834 1.03723 0.518617 0.855007i \(-0.326447\pi\)
0.518617 + 0.855007i \(0.326447\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.47422 0.118358
\(438\) 0 0
\(439\) − 13.4860i − 0.643650i −0.946799 0.321825i \(-0.895704\pi\)
0.946799 0.321825i \(-0.104296\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.29225i 0.203931i 0.994788 + 0.101965i \(0.0325131\pi\)
−0.994788 + 0.101965i \(0.967487\pi\)
\(444\) 0 0
\(445\) 0.380836i 0.0180534i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 33.7152i − 1.59112i −0.605876 0.795559i \(-0.707177\pi\)
0.605876 0.795559i \(-0.292823\pi\)
\(450\) 0 0
\(451\) −7.61650 −0.358647
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.772509 −0.0362158
\(456\) 0 0
\(457\) −31.2919 −1.46377 −0.731886 0.681427i \(-0.761360\pi\)
−0.731886 + 0.681427i \(0.761360\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.9245 1.44030 0.720149 0.693820i \(-0.244073\pi\)
0.720149 + 0.693820i \(0.244073\pi\)
\(462\) 0 0
\(463\) 12.0160i 0.558432i 0.960228 + 0.279216i \(0.0900746\pi\)
−0.960228 + 0.279216i \(0.909925\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.24380i 0.335203i 0.985855 + 0.167601i \(0.0536022\pi\)
−0.985855 + 0.167601i \(0.946398\pi\)
\(468\) 0 0
\(469\) 4.61226i 0.212974i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 4.10828i − 0.188899i
\(474\) 0 0
\(475\) −9.25480 −0.424639
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 31.8562 1.45555 0.727773 0.685818i \(-0.240555\pi\)
0.727773 + 0.685818i \(0.240555\pi\)
\(480\) 0 0
\(481\) −29.0569 −1.32488
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.34238 0.106362
\(486\) 0 0
\(487\) 37.3634i 1.69310i 0.532313 + 0.846548i \(0.321323\pi\)
−0.532313 + 0.846548i \(0.678677\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 23.8389i − 1.07583i −0.842998 0.537917i \(-0.819211\pi\)
0.842998 0.537917i \(-0.180789\pi\)
\(492\) 0 0
\(493\) − 10.8667i − 0.489412i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.9767i 0.626940i
\(498\) 0 0
\(499\) −13.0105 −0.582428 −0.291214 0.956658i \(-0.594059\pi\)
−0.291214 + 0.956658i \(0.594059\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.4474 1.53593 0.767966 0.640491i \(-0.221269\pi\)
0.767966 + 0.640491i \(0.221269\pi\)
\(504\) 0 0
\(505\) 0.0734321 0.00326769
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.29166 −0.323197 −0.161599 0.986857i \(-0.551665\pi\)
−0.161599 + 0.986857i \(0.551665\pi\)
\(510\) 0 0
\(511\) − 6.37815i − 0.282153i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1.44776i − 0.0637960i
\(516\) 0 0
\(517\) − 29.2033i − 1.28436i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.0013i 0.876274i 0.898908 + 0.438137i \(0.144362\pi\)
−0.898908 + 0.438137i \(0.855638\pi\)
\(522\) 0 0
\(523\) 14.7421 0.644629 0.322314 0.946633i \(-0.395539\pi\)
0.322314 + 0.946633i \(0.395539\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −21.9932 −0.958038
\(528\) 0 0
\(529\) −21.2319 −0.923126
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.1088 0.654433
\(534\) 0 0
\(535\) 2.42032i 0.104640i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.40352i 0.103527i
\(540\) 0 0
\(541\) − 36.6011i − 1.57360i −0.617206 0.786802i \(-0.711735\pi\)
0.617206 0.786802i \(-0.288265\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 0.702861i − 0.0301073i
\(546\) 0 0
\(547\) −41.4335 −1.77157 −0.885783 0.464099i \(-0.846378\pi\)
−0.885783 + 0.464099i \(0.846378\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.37057 0.313997
\(552\) 0 0
\(553\) 9.75258 0.414722
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −23.2128 −0.983558 −0.491779 0.870720i \(-0.663653\pi\)
−0.491779 + 0.870720i \(0.663653\pi\)
\(558\) 0 0
\(559\) 8.14954i 0.344689i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 36.2689i − 1.52855i −0.644890 0.764276i \(-0.723097\pi\)
0.644890 0.764276i \(-0.276903\pi\)
\(564\) 0 0
\(565\) 0.0730886i 0.00307486i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.0163i 0.671439i 0.941962 + 0.335719i \(0.108979\pi\)
−0.941962 + 0.335719i \(0.891021\pi\)
\(570\) 0 0
\(571\) 32.8412 1.37436 0.687180 0.726487i \(-0.258848\pi\)
0.687180 + 0.726487i \(0.258848\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.61360 −0.275806
\(576\) 0 0
\(577\) 10.0637 0.418956 0.209478 0.977813i \(-0.432823\pi\)
0.209478 + 0.977813i \(0.432823\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16.4983 −0.684465
\(582\) 0 0
\(583\) 19.7745i 0.818977i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.8278i 1.23112i 0.788088 + 0.615562i \(0.211071\pi\)
−0.788088 + 0.615562i \(0.788929\pi\)
\(588\) 0 0
\(589\) − 14.9173i − 0.614657i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.8440i 0.938091i 0.883174 + 0.469046i \(0.155402\pi\)
−0.883174 + 0.469046i \(0.844598\pi\)
\(594\) 0 0
\(595\) 0.444491 0.0182223
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.07805 −0.330060 −0.165030 0.986289i \(-0.552772\pi\)
−0.165030 + 0.986289i \(0.552772\pi\)
\(600\) 0 0
\(601\) 30.9036 1.26058 0.630292 0.776358i \(-0.282935\pi\)
0.630292 + 0.776358i \(0.282935\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.846270 0.0344058
\(606\) 0 0
\(607\) 21.2157i 0.861121i 0.902562 + 0.430560i \(0.141684\pi\)
−0.902562 + 0.430560i \(0.858316\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 57.9302i 2.34360i
\(612\) 0 0
\(613\) − 22.9612i − 0.927392i −0.885994 0.463696i \(-0.846523\pi\)
0.885994 0.463696i \(-0.153477\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 14.9091i − 0.600217i −0.953905 0.300108i \(-0.902977\pi\)
0.953905 0.300108i \(-0.0970228\pi\)
\(618\) 0 0
\(619\) 24.6832 0.992100 0.496050 0.868294i \(-0.334783\pi\)
0.496050 + 0.868294i \(0.334783\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.35048 −0.0941699
\(624\) 0 0
\(625\) 24.6069 0.984276
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.7189 0.666627
\(630\) 0 0
\(631\) 11.1068i 0.442154i 0.975256 + 0.221077i \(0.0709573\pi\)
−0.975256 + 0.221077i \(0.929043\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.0318786i 0.00126507i
\(636\) 0 0
\(637\) − 4.76784i − 0.188909i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 35.4613i − 1.40064i −0.713831 0.700318i \(-0.753041\pi\)
0.713831 0.700318i \(-0.246959\pi\)
\(642\) 0 0
\(643\) −27.6728 −1.09131 −0.545654 0.838011i \(-0.683719\pi\)
−0.545654 + 0.838011i \(0.683719\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.7035 0.499426 0.249713 0.968320i \(-0.419664\pi\)
0.249713 + 0.968320i \(0.419664\pi\)
\(648\) 0 0
\(649\) −13.1100 −0.514612
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −42.8339 −1.67622 −0.838110 0.545501i \(-0.816340\pi\)
−0.838110 + 0.545501i \(0.816340\pi\)
\(654\) 0 0
\(655\) − 0.760054i − 0.0296978i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 35.5914i − 1.38645i −0.720723 0.693223i \(-0.756190\pi\)
0.720723 0.693223i \(-0.243810\pi\)
\(660\) 0 0
\(661\) 0.388331i 0.0151043i 0.999971 + 0.00755217i \(0.00240395\pi\)
−0.999971 + 0.00755217i \(0.997596\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.301485i 0.0116911i
\(666\) 0 0
\(667\) 5.26711 0.203943
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.3004 0.397641
\(672\) 0 0
\(673\) 37.9424 1.46257 0.731286 0.682070i \(-0.238920\pi\)
0.731286 + 0.682070i \(0.238920\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 24.1764 0.929173 0.464586 0.885528i \(-0.346203\pi\)
0.464586 + 0.885528i \(0.346203\pi\)
\(678\) 0 0
\(679\) 14.4569i 0.554805i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 17.3036i − 0.662104i −0.943613 0.331052i \(-0.892596\pi\)
0.943613 0.331052i \(-0.107404\pi\)
\(684\) 0 0
\(685\) 1.80633i 0.0690164i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 39.2265i − 1.49441i
\(690\) 0 0
\(691\) −14.6123 −0.555877 −0.277938 0.960599i \(-0.589651\pi\)
−0.277938 + 0.960599i \(0.589651\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.12320 −0.0426055
\(696\) 0 0
\(697\) −8.69336 −0.329285
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −28.3532 −1.07089 −0.535443 0.844572i \(-0.679855\pi\)
−0.535443 + 0.844572i \(0.679855\pi\)
\(702\) 0 0
\(703\) 11.3400i 0.427694i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.453215i 0.0170449i
\(708\) 0 0
\(709\) 22.7293i 0.853618i 0.904342 + 0.426809i \(0.140362\pi\)
−0.904342 + 0.426809i \(0.859638\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 10.6601i − 0.399224i
\(714\) 0 0
\(715\) 1.85674 0.0694383
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −27.7959 −1.03661 −0.518306 0.855195i \(-0.673437\pi\)
−0.518306 + 0.855195i \(0.673437\pi\)
\(720\) 0 0
\(721\) 8.93542 0.332773
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −19.7016 −0.731699
\(726\) 0 0
\(727\) 24.4253i 0.905886i 0.891540 + 0.452943i \(0.149626\pi\)
−0.891540 + 0.452943i \(0.850374\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 4.68913i − 0.173434i
\(732\) 0 0
\(733\) − 17.1266i − 0.632585i −0.948662 0.316292i \(-0.897562\pi\)
0.948662 0.316292i \(-0.102438\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 11.0857i − 0.408346i
\(738\) 0 0
\(739\) 7.60957 0.279923 0.139961 0.990157i \(-0.455302\pi\)
0.139961 + 0.990157i \(0.455302\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.75152 −0.137630 −0.0688150 0.997629i \(-0.521922\pi\)
−0.0688150 + 0.997629i \(0.521922\pi\)
\(744\) 0 0
\(745\) 2.63359 0.0964872
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14.9379 −0.545821
\(750\) 0 0
\(751\) − 39.9652i − 1.45835i −0.684326 0.729176i \(-0.739904\pi\)
0.684326 0.729176i \(-0.260096\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 0.0690244i − 0.00251206i
\(756\) 0 0
\(757\) − 26.4627i − 0.961802i −0.876775 0.480901i \(-0.840310\pi\)
0.876775 0.480901i \(-0.159690\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 3.50130i − 0.126922i −0.997984 0.0634610i \(-0.979786\pi\)
0.997984 0.0634610i \(-0.0202139\pi\)
\(762\) 0 0
\(763\) 4.33798 0.157045
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 26.0061 0.939027
\(768\) 0 0
\(769\) −26.3596 −0.950551 −0.475275 0.879837i \(-0.657652\pi\)
−0.475275 + 0.879837i \(0.657652\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −22.0363 −0.792592 −0.396296 0.918123i \(-0.629705\pi\)
−0.396296 + 0.918123i \(0.629705\pi\)
\(774\) 0 0
\(775\) 39.8741i 1.43232i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 5.89645i − 0.211262i
\(780\) 0 0
\(781\) − 33.5933i − 1.20206i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.27566i 0.0812218i
\(786\) 0 0
\(787\) −44.0988 −1.57195 −0.785976 0.618257i \(-0.787839\pi\)
−0.785976 + 0.618257i \(0.787839\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.451095 −0.0160391
\(792\) 0 0
\(793\) −20.4327 −0.725587
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −17.4271 −0.617298 −0.308649 0.951176i \(-0.599877\pi\)
−0.308649 + 0.951176i \(0.599877\pi\)
\(798\) 0 0
\(799\) − 33.3322i − 1.17921i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.3300i 0.540985i
\(804\) 0 0
\(805\) 0.215445i 0.00759344i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1.23533i − 0.0434319i −0.999764 0.0217159i \(-0.993087\pi\)
0.999764 0.0217159i \(-0.00691294\pi\)
\(810\) 0 0
\(811\) −40.7818 −1.43204 −0.716021 0.698078i \(-0.754039\pi\)
−0.716021 + 0.698078i \(0.754039\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.766317 0.0268429
\(816\) 0 0
\(817\) 3.18050 0.111271
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.2037 −0.879614 −0.439807 0.898092i \(-0.644953\pi\)
−0.439807 + 0.898092i \(0.644953\pi\)
\(822\) 0 0
\(823\) 46.4360i 1.61866i 0.587356 + 0.809329i \(0.300169\pi\)
−0.587356 + 0.809329i \(0.699831\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 24.6612i − 0.857554i −0.903410 0.428777i \(-0.858945\pi\)
0.903410 0.428777i \(-0.141055\pi\)
\(828\) 0 0
\(829\) 29.3304i 1.01869i 0.860564 + 0.509343i \(0.170111\pi\)
−0.860564 + 0.509343i \(0.829889\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.74335i 0.0950513i
\(834\) 0 0
\(835\) −1.90127 −0.0657961
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10.5258 0.363390 0.181695 0.983355i \(-0.441842\pi\)
0.181695 + 0.983355i \(0.441842\pi\)
\(840\) 0 0
\(841\) −13.3096 −0.458950
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.57688 −0.0542462
\(846\) 0 0
\(847\) 5.22308i 0.179467i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.10368i 0.277791i
\(852\) 0 0
\(853\) − 36.6192i − 1.25382i −0.779093 0.626908i \(-0.784320\pi\)
0.779093 0.626908i \(-0.215680\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 9.24273i − 0.315726i −0.987461 0.157863i \(-0.949540\pi\)
0.987461 0.157863i \(-0.0504604\pi\)
\(858\) 0 0
\(859\) 49.5357 1.69014 0.845068 0.534659i \(-0.179560\pi\)
0.845068 + 0.534659i \(0.179560\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29.0407 −0.988557 −0.494278 0.869304i \(-0.664568\pi\)
−0.494278 + 0.869304i \(0.664568\pi\)
\(864\) 0 0
\(865\) 2.95820 0.100582
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −23.4405 −0.795166
\(870\) 0 0
\(871\) 21.9905i 0.745120i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 1.61600i − 0.0546307i
\(876\) 0 0
\(877\) 15.3569i 0.518564i 0.965802 + 0.259282i \(0.0834859\pi\)
−0.965802 + 0.259282i \(0.916514\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 52.3468i 1.76361i 0.471616 + 0.881804i \(0.343671\pi\)
−0.471616 + 0.881804i \(0.656329\pi\)
\(882\) 0 0
\(883\) −7.73223 −0.260210 −0.130105 0.991500i \(-0.541531\pi\)
−0.130105 + 0.991500i \(0.541531\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.43973 0.249802 0.124901 0.992169i \(-0.460139\pi\)
0.124901 + 0.992169i \(0.460139\pi\)
\(888\) 0 0
\(889\) −0.196751 −0.00659883
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 22.6082 0.756555
\(894\) 0 0
\(895\) − 2.36476i − 0.0790452i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 31.7560i − 1.05912i
\(900\) 0 0
\(901\) 22.5704i 0.751928i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 0.618978i − 0.0205755i
\(906\) 0 0
\(907\) 31.5786 1.04855 0.524276 0.851549i \(-0.324336\pi\)
0.524276 + 0.851549i \(0.324336\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.2059 0.868240 0.434120 0.900855i \(-0.357059\pi\)
0.434120 + 0.900855i \(0.357059\pi\)
\(912\) 0 0
\(913\) 39.6541 1.31236
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.69097 0.154909
\(918\) 0 0
\(919\) 37.9577i 1.25211i 0.779779 + 0.626055i \(0.215332\pi\)
−0.779779 + 0.626055i \(0.784668\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 66.6386i 2.19344i
\(924\) 0 0
\(925\) − 30.3118i − 0.996646i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 33.3343i 1.09366i 0.837243 + 0.546831i \(0.184166\pi\)
−0.837243 + 0.546831i \(0.815834\pi\)
\(930\) 0 0
\(931\) −1.86073 −0.0609829
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.06834 −0.0349385
\(936\) 0 0
\(937\) 8.18384 0.267355 0.133677 0.991025i \(-0.457321\pi\)
0.133677 + 0.991025i \(0.457321\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −43.9315 −1.43213 −0.716064 0.698035i \(-0.754058\pi\)
−0.716064 + 0.698035i \(0.754058\pi\)
\(942\) 0 0
\(943\) − 4.21368i − 0.137216i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.0323i 1.46335i 0.681652 + 0.731676i \(0.261262\pi\)
−0.681652 + 0.731676i \(0.738738\pi\)
\(948\) 0 0
\(949\) − 30.4100i − 0.987150i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 60.2109i − 1.95042i −0.221278 0.975211i \(-0.571023\pi\)
0.221278 0.975211i \(-0.428977\pi\)
\(954\) 0 0
\(955\) 3.47457 0.112434
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.1485 −0.360003
\(960\) 0 0
\(961\) −33.2710 −1.07326
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.85150 −0.0917931
\(966\) 0 0
\(967\) 1.96099i 0.0630613i 0.999503 + 0.0315307i \(0.0100382\pi\)
−0.999503 + 0.0315307i \(0.989962\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 36.5960i 1.17442i 0.809434 + 0.587210i \(0.199774\pi\)
−0.809434 + 0.587210i \(0.800226\pi\)
\(972\) 0 0
\(973\) − 6.93228i − 0.222239i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0891i 0.578721i 0.957220 + 0.289361i \(0.0934427\pi\)
−0.957220 + 0.289361i \(0.906557\pi\)
\(978\) 0 0
\(979\) 5.64942 0.180556
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12.7144 −0.405525 −0.202763 0.979228i \(-0.564992\pi\)
−0.202763 + 0.979228i \(0.564992\pi\)
\(984\) 0 0
\(985\) 3.10001 0.0987744
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.27282 0.0722716
\(990\) 0 0
\(991\) − 58.4330i − 1.85619i −0.372348 0.928093i \(-0.621447\pi\)
0.372348 0.928093i \(-0.378553\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.15696i 0.0683803i
\(996\) 0 0
\(997\) 33.7142i 1.06774i 0.845567 + 0.533869i \(0.179263\pi\)
−0.845567 + 0.533869i \(0.820737\pi\)
\(998\) 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 6048.2.j.c.5615.18 32
3.2 odd 2 inner 6048.2.j.c.5615.15 32
4.3 odd 2 1512.2.j.c.323.18 yes 32
8.3 odd 2 inner 6048.2.j.c.5615.16 32
8.5 even 2 1512.2.j.c.323.16 yes 32
12.11 even 2 1512.2.j.c.323.15 32
24.5 odd 2 1512.2.j.c.323.17 yes 32
24.11 even 2 inner 6048.2.j.c.5615.17 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.15 32 12.11 even 2
1512.2.j.c.323.16 yes 32 8.5 even 2
1512.2.j.c.323.17 yes 32 24.5 odd 2
1512.2.j.c.323.18 yes 32 4.3 odd 2
6048.2.j.c.5615.15 32 3.2 odd 2 inner
6048.2.j.c.5615.16 32 8.3 odd 2 inner
6048.2.j.c.5615.17 32 24.11 even 2 inner
6048.2.j.c.5615.18 32 1.1 even 1 trivial