Properties

Label 6048.2.h.i.2591.4
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 24
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.4
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.i.2591.3

$q$-expansion

\(f(q)\) \(=\) \(q+0.865222i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+0.865222i q^{5} +1.00000i q^{7} -5.82380 q^{11} +0.727273 q^{13} -7.62531i q^{17} -2.29824i q^{19} -7.50524 q^{23} +4.25139 q^{25} -0.231154i q^{29} +10.7347i q^{31} -0.865222 q^{35} +1.64759 q^{37} +0.229547i q^{41} -6.70919i q^{43} +11.6729 q^{47} -1.00000 q^{49} +2.80475i q^{53} -5.03888i q^{55} +11.0135 q^{59} -9.08990 q^{61} +0.629252i q^{65} +11.2616i q^{67} +9.76188 q^{71} +14.5795 q^{73} -5.82380i q^{77} +4.39759i q^{79} +2.47977 q^{83} +6.59759 q^{85} +3.15706i q^{89} +0.727273i q^{91} +1.98848 q^{95} +1.25455 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q + O(q^{10}) \) \( 24q - 8q^{13} - 24q^{25} + 48q^{37} - 24q^{49} - 48q^{61} - 32q^{73} - 80q^{85} + 64q^{97} + O(q^{100}) \)

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
\(5\) 0.865222i 0.386939i 0.981106 + 0.193469i \(0.0619741\pi\)
−0.981106 + 0.193469i \(0.938026\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.82380 −1.75594 −0.877971 0.478715i \(-0.841103\pi\)
−0.877971 + 0.478715i \(0.841103\pi\)
\(12\) 0 0
\(13\) 0.727273 0.201709 0.100855 0.994901i \(-0.467842\pi\)
0.100855 + 0.994901i \(0.467842\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.62531i − 1.84941i −0.380684 0.924705i \(-0.624312\pi\)
0.380684 0.924705i \(-0.375688\pi\)
\(18\) 0 0
\(19\) − 2.29824i − 0.527252i −0.964625 0.263626i \(-0.915082\pi\)
0.964625 0.263626i \(-0.0849184\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.50524 −1.56495 −0.782476 0.622681i \(-0.786043\pi\)
−0.782476 + 0.622681i \(0.786043\pi\)
\(24\) 0 0
\(25\) 4.25139 0.850278
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.231154i − 0.0429243i −0.999770 0.0214621i \(-0.993168\pi\)
0.999770 0.0214621i \(-0.00683214\pi\)
\(30\) 0 0
\(31\) 10.7347i 1.92801i 0.265885 + 0.964005i \(0.414336\pi\)
−0.265885 + 0.964005i \(0.585664\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.865222 −0.146249
\(36\) 0 0
\(37\) 1.64759 0.270862 0.135431 0.990787i \(-0.456758\pi\)
0.135431 + 0.990787i \(0.456758\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.229547i 0.0358493i 0.999839 + 0.0179246i \(0.00570590\pi\)
−0.999839 + 0.0179246i \(0.994294\pi\)
\(42\) 0 0
\(43\) − 6.70919i − 1.02314i −0.859241 0.511571i \(-0.829064\pi\)
0.859241 0.511571i \(-0.170936\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.6729 1.70266 0.851332 0.524627i \(-0.175795\pi\)
0.851332 + 0.524627i \(0.175795\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.80475i 0.385263i 0.981271 + 0.192631i \(0.0617021\pi\)
−0.981271 + 0.192631i \(0.938298\pi\)
\(54\) 0 0
\(55\) − 5.03888i − 0.679442i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.0135 1.43384 0.716920 0.697156i \(-0.245551\pi\)
0.716920 + 0.697156i \(0.245551\pi\)
\(60\) 0 0
\(61\) −9.08990 −1.16384 −0.581921 0.813245i \(-0.697699\pi\)
−0.581921 + 0.813245i \(0.697699\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.629252i 0.0780492i
\(66\) 0 0
\(67\) 11.2616i 1.37582i 0.725794 + 0.687912i \(0.241472\pi\)
−0.725794 + 0.687912i \(0.758528\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.76188 1.15852 0.579261 0.815142i \(-0.303341\pi\)
0.579261 + 0.815142i \(0.303341\pi\)
\(72\) 0 0
\(73\) 14.5795 1.70640 0.853201 0.521582i \(-0.174658\pi\)
0.853201 + 0.521582i \(0.174658\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.82380i − 0.663683i
\(78\) 0 0
\(79\) 4.39759i 0.494767i 0.968918 + 0.247384i \(0.0795708\pi\)
−0.968918 + 0.247384i \(0.920429\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.47977 0.272190 0.136095 0.990696i \(-0.456545\pi\)
0.136095 + 0.990696i \(0.456545\pi\)
\(84\) 0 0
\(85\) 6.59759 0.715609
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.15706i 0.334648i 0.985902 + 0.167324i \(0.0535126\pi\)
−0.985902 + 0.167324i \(0.946487\pi\)
\(90\) 0 0
\(91\) 0.727273i 0.0762389i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.98848 0.204014
\(96\) 0 0
\(97\) 1.25455 0.127380 0.0636899 0.997970i \(-0.479713\pi\)
0.0636899 + 0.997970i \(0.479713\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 15.8863i − 1.58075i −0.612626 0.790373i \(-0.709887\pi\)
0.612626 0.790373i \(-0.290113\pi\)
\(102\) 0 0
\(103\) − 0.689526i − 0.0679410i −0.999423 0.0339705i \(-0.989185\pi\)
0.999423 0.0339705i \(-0.0108152\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.3665 −1.38887 −0.694433 0.719557i \(-0.744345\pi\)
−0.694433 + 0.719557i \(0.744345\pi\)
\(108\) 0 0
\(109\) 9.69861 0.928958 0.464479 0.885584i \(-0.346242\pi\)
0.464479 + 0.885584i \(0.346242\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.4645i 1.54885i 0.632665 + 0.774425i \(0.281961\pi\)
−0.632665 + 0.774425i \(0.718039\pi\)
\(114\) 0 0
\(115\) − 6.49370i − 0.605541i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.62531 0.699011
\(120\) 0 0
\(121\) 22.9166 2.08333
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.00451i 0.715945i
\(126\) 0 0
\(127\) 19.0764i 1.69275i 0.532585 + 0.846377i \(0.321221\pi\)
−0.532585 + 0.846377i \(0.678779\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.57149 0.661524 0.330762 0.943714i \(-0.392694\pi\)
0.330762 + 0.943714i \(0.392694\pi\)
\(132\) 0 0
\(133\) 2.29824 0.199282
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11.8040i − 1.00849i −0.863562 0.504243i \(-0.831771\pi\)
0.863562 0.504243i \(-0.168229\pi\)
\(138\) 0 0
\(139\) 12.7285i 1.07962i 0.841788 + 0.539808i \(0.181503\pi\)
−0.841788 + 0.539808i \(0.818497\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.23549 −0.354189
\(144\) 0 0
\(145\) 0.200000 0.0166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.3820i 1.58783i 0.608028 + 0.793916i \(0.291961\pi\)
−0.608028 + 0.793916i \(0.708039\pi\)
\(150\) 0 0
\(151\) − 13.7766i − 1.12113i −0.828111 0.560564i \(-0.810585\pi\)
0.828111 0.560564i \(-0.189415\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.28790 −0.746022
\(156\) 0 0
\(157\) −9.10808 −0.726904 −0.363452 0.931613i \(-0.618402\pi\)
−0.363452 + 0.931613i \(0.618402\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 7.50524i − 0.591496i
\(162\) 0 0
\(163\) 17.1137i 1.34045i 0.742157 + 0.670226i \(0.233803\pi\)
−0.742157 + 0.670226i \(0.766197\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.37485 0.648065 0.324033 0.946046i \(-0.394961\pi\)
0.324033 + 0.946046i \(0.394961\pi\)
\(168\) 0 0
\(169\) −12.4711 −0.959313
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.6871i 1.04061i 0.853980 + 0.520306i \(0.174182\pi\)
−0.853980 + 0.520306i \(0.825818\pi\)
\(174\) 0 0
\(175\) 4.25139i 0.321375i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.23685 −0.690395 −0.345197 0.938530i \(-0.612188\pi\)
−0.345197 + 0.938530i \(0.612188\pi\)
\(180\) 0 0
\(181\) 5.93006 0.440778 0.220389 0.975412i \(-0.429267\pi\)
0.220389 + 0.975412i \(0.429267\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.42553i 0.104807i
\(186\) 0 0
\(187\) 44.4083i 3.24746i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.5565 −1.19798 −0.598992 0.800755i \(-0.704432\pi\)
−0.598992 + 0.800755i \(0.704432\pi\)
\(192\) 0 0
\(193\) −15.1697 −1.09194 −0.545969 0.837806i \(-0.683838\pi\)
−0.545969 + 0.837806i \(0.683838\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.02017i 0.500166i 0.968224 + 0.250083i \(0.0804580\pi\)
−0.968224 + 0.250083i \(0.919542\pi\)
\(198\) 0 0
\(199\) 7.43368i 0.526960i 0.964665 + 0.263480i \(0.0848703\pi\)
−0.964665 + 0.263480i \(0.915130\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.231154 0.0162239
\(204\) 0 0
\(205\) −0.198609 −0.0138715
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.3845i 0.925823i
\(210\) 0 0
\(211\) − 25.9303i − 1.78511i −0.450935 0.892557i \(-0.648910\pi\)
0.450935 0.892557i \(-0.351090\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.80494 0.395894
\(216\) 0 0
\(217\) −10.7347 −0.728719
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 5.54568i − 0.373043i
\(222\) 0 0
\(223\) 22.4305i 1.50206i 0.660269 + 0.751030i \(0.270442\pi\)
−0.660269 + 0.751030i \(0.729558\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.2642 −1.34498 −0.672491 0.740105i \(-0.734776\pi\)
−0.672491 + 0.740105i \(0.734776\pi\)
\(228\) 0 0
\(229\) 28.4103 1.87741 0.938703 0.344726i \(-0.112028\pi\)
0.938703 + 0.344726i \(0.112028\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 0.260864i − 0.0170898i −0.999963 0.00854488i \(-0.997280\pi\)
0.999963 0.00854488i \(-0.00271995\pi\)
\(234\) 0 0
\(235\) 10.0996i 0.658827i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.1228 1.17227 0.586133 0.810215i \(-0.300650\pi\)
0.586133 + 0.810215i \(0.300650\pi\)
\(240\) 0 0
\(241\) 24.2341 1.56105 0.780526 0.625123i \(-0.214951\pi\)
0.780526 + 0.625123i \(0.214951\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.865222i − 0.0552770i
\(246\) 0 0
\(247\) − 1.67145i − 0.106352i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.9471 −1.32217 −0.661083 0.750312i \(-0.729903\pi\)
−0.661083 + 0.750312i \(0.729903\pi\)
\(252\) 0 0
\(253\) 43.7090 2.74796
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.08927i 0.255082i 0.991833 + 0.127541i \(0.0407084\pi\)
−0.991833 + 0.127541i \(0.959292\pi\)
\(258\) 0 0
\(259\) 1.64759i 0.102376i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 21.5691 1.33001 0.665005 0.746839i \(-0.268429\pi\)
0.665005 + 0.746839i \(0.268429\pi\)
\(264\) 0 0
\(265\) −2.42673 −0.149073
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 18.6825i − 1.13909i −0.821959 0.569547i \(-0.807119\pi\)
0.821959 0.569547i \(-0.192881\pi\)
\(270\) 0 0
\(271\) − 4.52107i − 0.274636i −0.990527 0.137318i \(-0.956152\pi\)
0.990527 0.137318i \(-0.0438482\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.7592 −1.49304
\(276\) 0 0
\(277\) 15.0289 0.902997 0.451499 0.892272i \(-0.350890\pi\)
0.451499 + 0.892272i \(0.350890\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 21.8228i − 1.30184i −0.759147 0.650919i \(-0.774384\pi\)
0.759147 0.650919i \(-0.225616\pi\)
\(282\) 0 0
\(283\) − 11.9277i − 0.709031i −0.935050 0.354515i \(-0.884646\pi\)
0.935050 0.354515i \(-0.115354\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.229547 −0.0135498
\(288\) 0 0
\(289\) −41.1454 −2.42032
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 2.35874i − 0.137799i −0.997624 0.0688995i \(-0.978051\pi\)
0.997624 0.0688995i \(-0.0219488\pi\)
\(294\) 0 0
\(295\) 9.52915i 0.554808i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.45836 −0.315665
\(300\) 0 0
\(301\) 6.70919 0.386711
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 7.86478i − 0.450336i
\(306\) 0 0
\(307\) − 20.0568i − 1.14470i −0.820009 0.572351i \(-0.806032\pi\)
0.820009 0.572351i \(-0.193968\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.4255 0.591173 0.295587 0.955316i \(-0.404485\pi\)
0.295587 + 0.955316i \(0.404485\pi\)
\(312\) 0 0
\(313\) 16.0767 0.908710 0.454355 0.890821i \(-0.349870\pi\)
0.454355 + 0.890821i \(0.349870\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.40845i 0.303769i 0.988398 + 0.151884i \(0.0485341\pi\)
−0.988398 + 0.151884i \(0.951466\pi\)
\(318\) 0 0
\(319\) 1.34620i 0.0753725i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −17.5248 −0.975105
\(324\) 0 0
\(325\) 3.09192 0.171509
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.6729i 0.643546i
\(330\) 0 0
\(331\) 12.5645i 0.690606i 0.938491 + 0.345303i \(0.112224\pi\)
−0.938491 + 0.345303i \(0.887776\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.74379 −0.532360
\(336\) 0 0
\(337\) 29.1719 1.58910 0.794548 0.607202i \(-0.207708\pi\)
0.794548 + 0.607202i \(0.207708\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 62.5167i − 3.38547i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.5974 1.53519 0.767595 0.640935i \(-0.221453\pi\)
0.767595 + 0.640935i \(0.221453\pi\)
\(348\) 0 0
\(349\) 19.8320 1.06158 0.530792 0.847502i \(-0.321895\pi\)
0.530792 + 0.847502i \(0.321895\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.8083i 0.628494i 0.949341 + 0.314247i \(0.101752\pi\)
−0.949341 + 0.314247i \(0.898248\pi\)
\(354\) 0 0
\(355\) 8.44620i 0.448278i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.5304 0.925220 0.462610 0.886562i \(-0.346913\pi\)
0.462610 + 0.886562i \(0.346913\pi\)
\(360\) 0 0
\(361\) 13.7181 0.722006
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.6145i 0.660274i
\(366\) 0 0
\(367\) 2.40621i 0.125603i 0.998026 + 0.0628017i \(0.0200036\pi\)
−0.998026 + 0.0628017i \(0.979996\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.80475 −0.145616
\(372\) 0 0
\(373\) −18.3774 −0.951545 −0.475772 0.879568i \(-0.657831\pi\)
−0.475772 + 0.879568i \(0.657831\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 0.168112i − 0.00865823i
\(378\) 0 0
\(379\) 12.6572i 0.650155i 0.945687 + 0.325077i \(0.105390\pi\)
−0.945687 + 0.325077i \(0.894610\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.64584 0.135196 0.0675980 0.997713i \(-0.478466\pi\)
0.0675980 + 0.997713i \(0.478466\pi\)
\(384\) 0 0
\(385\) 5.03888 0.256805
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 4.84454i − 0.245628i −0.992430 0.122814i \(-0.960808\pi\)
0.992430 0.122814i \(-0.0391919\pi\)
\(390\) 0 0
\(391\) 57.2298i 2.89424i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.80489 −0.191445
\(396\) 0 0
\(397\) −19.7055 −0.988991 −0.494495 0.869180i \(-0.664647\pi\)
−0.494495 + 0.869180i \(0.664647\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 7.51371i − 0.375217i −0.982244 0.187608i \(-0.939926\pi\)
0.982244 0.187608i \(-0.0600736\pi\)
\(402\) 0 0
\(403\) 7.80706i 0.388897i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.59522 −0.475617
\(408\) 0 0
\(409\) 3.37679 0.166972 0.0834858 0.996509i \(-0.473395\pi\)
0.0834858 + 0.996509i \(0.473395\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.0135i 0.541940i
\(414\) 0 0
\(415\) 2.14555i 0.105321i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.74114 −0.231620 −0.115810 0.993271i \(-0.536946\pi\)
−0.115810 + 0.993271i \(0.536946\pi\)
\(420\) 0 0
\(421\) −2.42905 −0.118385 −0.0591924 0.998247i \(-0.518853\pi\)
−0.0591924 + 0.998247i \(0.518853\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 32.4182i − 1.57251i
\(426\) 0 0
\(427\) − 9.08990i − 0.439891i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.485660 0.0233934 0.0116967 0.999932i \(-0.496277\pi\)
0.0116967 + 0.999932i \(0.496277\pi\)
\(432\) 0 0
\(433\) 6.61560 0.317926 0.158963 0.987285i \(-0.449185\pi\)
0.158963 + 0.987285i \(0.449185\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17.2488i 0.825123i
\(438\) 0 0
\(439\) 19.2736i 0.919877i 0.887951 + 0.459939i \(0.152129\pi\)
−0.887951 + 0.459939i \(0.847871\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.680297 0.0323219 0.0161609 0.999869i \(-0.494856\pi\)
0.0161609 + 0.999869i \(0.494856\pi\)
\(444\) 0 0
\(445\) −2.73156 −0.129488
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.2474i 0.672378i 0.941795 + 0.336189i \(0.109138\pi\)
−0.941795 + 0.336189i \(0.890862\pi\)
\(450\) 0 0
\(451\) − 1.33684i − 0.0629492i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.629252 −0.0294998
\(456\) 0 0
\(457\) 17.6350 0.824930 0.412465 0.910973i \(-0.364668\pi\)
0.412465 + 0.910973i \(0.364668\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.8412i 0.737798i 0.929469 + 0.368899i \(0.120265\pi\)
−0.929469 + 0.368899i \(0.879735\pi\)
\(462\) 0 0
\(463\) − 30.1324i − 1.40037i −0.713961 0.700186i \(-0.753101\pi\)
0.713961 0.700186i \(-0.246899\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.2405 −0.982893 −0.491446 0.870908i \(-0.663532\pi\)
−0.491446 + 0.870908i \(0.663532\pi\)
\(468\) 0 0
\(469\) −11.2616 −0.520013
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 39.0730i 1.79658i
\(474\) 0 0
\(475\) − 9.77070i − 0.448311i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.8779 0.771172 0.385586 0.922672i \(-0.373999\pi\)
0.385586 + 0.922672i \(0.373999\pi\)
\(480\) 0 0
\(481\) 1.19825 0.0546353
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.08546i 0.0492882i
\(486\) 0 0
\(487\) − 34.5687i − 1.56646i −0.621735 0.783228i \(-0.713572\pi\)
0.621735 0.783228i \(-0.286428\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.298579 0.0134747 0.00673735 0.999977i \(-0.497855\pi\)
0.00673735 + 0.999977i \(0.497855\pi\)
\(492\) 0 0
\(493\) −1.76262 −0.0793846
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.76188i 0.437880i
\(498\) 0 0
\(499\) 16.8857i 0.755908i 0.925824 + 0.377954i \(0.123372\pi\)
−0.925824 + 0.377954i \(0.876628\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.33762 −0.104230 −0.0521148 0.998641i \(-0.516596\pi\)
−0.0521148 + 0.998641i \(0.516596\pi\)
\(504\) 0 0
\(505\) 13.7452 0.611652
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.1664i 0.849538i 0.905302 + 0.424769i \(0.139645\pi\)
−0.905302 + 0.424769i \(0.860355\pi\)
\(510\) 0 0
\(511\) 14.5795i 0.644959i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.596593 0.0262890
\(516\) 0 0
\(517\) −67.9805 −2.98978
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 6.95754i − 0.304815i −0.988318 0.152408i \(-0.951297\pi\)
0.988318 0.152408i \(-0.0487027\pi\)
\(522\) 0 0
\(523\) 6.39416i 0.279597i 0.990180 + 0.139798i \(0.0446455\pi\)
−0.990180 + 0.139798i \(0.955355\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 81.8555 3.56568
\(528\) 0 0
\(529\) 33.3287 1.44907
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.166944i 0.00723113i
\(534\) 0 0
\(535\) − 12.4303i − 0.537407i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.82380 0.250849
\(540\) 0 0
\(541\) 22.4961 0.967183 0.483591 0.875294i \(-0.339332\pi\)
0.483591 + 0.875294i \(0.339332\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.39145i 0.359450i
\(546\) 0 0
\(547\) − 1.17336i − 0.0501692i −0.999685 0.0250846i \(-0.992014\pi\)
0.999685 0.0250846i \(-0.00798552\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.531248 −0.0226319
\(552\) 0 0
\(553\) −4.39759 −0.187004
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.87547i 0.291323i 0.989334 + 0.145662i \(0.0465310\pi\)
−0.989334 + 0.145662i \(0.953469\pi\)
\(558\) 0 0
\(559\) − 4.87941i − 0.206377i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.4969 −0.821696 −0.410848 0.911704i \(-0.634767\pi\)
−0.410848 + 0.911704i \(0.634767\pi\)
\(564\) 0 0
\(565\) −14.2455 −0.599311
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 11.1191i − 0.466136i −0.972460 0.233068i \(-0.925124\pi\)
0.972460 0.233068i \(-0.0748765\pi\)
\(570\) 0 0
\(571\) 19.8397i 0.830267i 0.909761 + 0.415133i \(0.136265\pi\)
−0.909761 + 0.415133i \(0.863735\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −31.9077 −1.33064
\(576\) 0 0
\(577\) 19.5159 0.812457 0.406229 0.913772i \(-0.366844\pi\)
0.406229 + 0.913772i \(0.366844\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.47977i 0.102878i
\(582\) 0 0
\(583\) − 16.3343i − 0.676499i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.10003 −0.169227 −0.0846133 0.996414i \(-0.526965\pi\)
−0.0846133 + 0.996414i \(0.526965\pi\)
\(588\) 0 0
\(589\) 24.6709 1.01655
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.1396i 1.60727i 0.595122 + 0.803636i \(0.297104\pi\)
−0.595122 + 0.803636i \(0.702896\pi\)
\(594\) 0 0
\(595\) 6.59759i 0.270475i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.2716 0.909992 0.454996 0.890493i \(-0.349641\pi\)
0.454996 + 0.890493i \(0.349641\pi\)
\(600\) 0 0
\(601\) 6.57598 0.268240 0.134120 0.990965i \(-0.457179\pi\)
0.134120 + 0.990965i \(0.457179\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 19.8280i 0.806121i
\(606\) 0 0
\(607\) − 29.1564i − 1.18342i −0.806150 0.591711i \(-0.798453\pi\)
0.806150 0.591711i \(-0.201547\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.48937 0.343443
\(612\) 0 0
\(613\) 24.4722 0.988422 0.494211 0.869342i \(-0.335457\pi\)
0.494211 + 0.869342i \(0.335457\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 4.62276i − 0.186105i −0.995661 0.0930526i \(-0.970338\pi\)
0.995661 0.0930526i \(-0.0296625\pi\)
\(618\) 0 0
\(619\) − 28.6712i − 1.15239i −0.817312 0.576196i \(-0.804537\pi\)
0.817312 0.576196i \(-0.195463\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.15706 −0.126485
\(624\) 0 0
\(625\) 14.3313 0.573251
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12.5634i − 0.500935i
\(630\) 0 0
\(631\) − 2.30009i − 0.0915653i −0.998951 0.0457826i \(-0.985422\pi\)
0.998951 0.0457826i \(-0.0145782\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.5053 −0.654992
\(636\) 0 0
\(637\) −0.727273 −0.0288156
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27.1632i 1.07288i 0.843938 + 0.536440i \(0.180231\pi\)
−0.843938 + 0.536440i \(0.819769\pi\)
\(642\) 0 0
\(643\) 37.0176i 1.45983i 0.683536 + 0.729917i \(0.260441\pi\)
−0.683536 + 0.729917i \(0.739559\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.3022 0.758847 0.379423 0.925223i \(-0.376122\pi\)
0.379423 + 0.925223i \(0.376122\pi\)
\(648\) 0 0
\(649\) −64.1406 −2.51774
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.42983i 0.329885i 0.986303 + 0.164942i \(0.0527438\pi\)
−0.986303 + 0.164942i \(0.947256\pi\)
\(654\) 0 0
\(655\) 6.55102i 0.255970i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 18.8983 0.736172 0.368086 0.929792i \(-0.380013\pi\)
0.368086 + 0.929792i \(0.380013\pi\)
\(660\) 0 0
\(661\) 19.1750 0.745820 0.372910 0.927868i \(-0.378360\pi\)
0.372910 + 0.927868i \(0.378360\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.98848i 0.0771101i
\(666\) 0 0
\(667\) 1.73487i 0.0671744i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 52.9377 2.04364
\(672\) 0 0
\(673\) 14.3064 0.551471 0.275736 0.961234i \(-0.411079\pi\)
0.275736 + 0.961234i \(0.411079\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 17.1600i − 0.659512i −0.944066 0.329756i \(-0.893034\pi\)
0.944066 0.329756i \(-0.106966\pi\)
\(678\) 0 0
\(679\) 1.25455i 0.0481451i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18.4867 −0.707375 −0.353687 0.935364i \(-0.615072\pi\)
−0.353687 + 0.935364i \(0.615072\pi\)
\(684\) 0 0
\(685\) 10.2131 0.390223
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.03982i 0.0777110i
\(690\) 0 0
\(691\) − 19.4447i − 0.739709i −0.929090 0.369855i \(-0.879407\pi\)
0.929090 0.369855i \(-0.120593\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.0130 −0.417746
\(696\) 0 0
\(697\) 1.75037 0.0663000
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 28.3467i − 1.07064i −0.844649 0.535320i \(-0.820191\pi\)
0.844649 0.535320i \(-0.179809\pi\)
\(702\) 0 0
\(703\) − 3.78655i − 0.142812i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.8863 0.597466
\(708\) 0 0
\(709\) −25.4347 −0.955220 −0.477610 0.878572i \(-0.658497\pi\)
−0.477610 + 0.878572i \(0.658497\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 80.5665i − 3.01724i
\(714\) 0 0
\(715\) − 3.66464i − 0.137050i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −27.3320 −1.01931 −0.509656 0.860378i \(-0.670228\pi\)
−0.509656 + 0.860378i \(0.670228\pi\)
\(720\) 0 0
\(721\) 0.689526 0.0256793
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 0.982728i − 0.0364976i
\(726\) 0 0
\(727\) − 16.9089i − 0.627117i −0.949569 0.313559i \(-0.898479\pi\)
0.949569 0.313559i \(-0.101521\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −51.1597 −1.89221
\(732\) 0 0
\(733\) −32.8540 −1.21349 −0.606745 0.794896i \(-0.707525\pi\)
−0.606745 + 0.794896i \(0.707525\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 65.5853i − 2.41587i
\(738\) 0 0
\(739\) 35.6492i 1.31138i 0.755032 + 0.655688i \(0.227622\pi\)
−0.755032 + 0.655688i \(0.772378\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 35.9725 1.31970 0.659852 0.751396i \(-0.270619\pi\)
0.659852 + 0.751396i \(0.270619\pi\)
\(744\) 0 0
\(745\) −16.7697 −0.614394
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 14.3665i − 0.524942i
\(750\) 0 0
\(751\) 32.6600i 1.19178i 0.803065 + 0.595891i \(0.203201\pi\)
−0.803065 + 0.595891i \(0.796799\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.9199 0.433808
\(756\) 0 0
\(757\) 1.73551 0.0630783 0.0315391 0.999503i \(-0.489959\pi\)
0.0315391 + 0.999503i \(0.489959\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 5.04240i − 0.182787i −0.995815 0.0913935i \(-0.970868\pi\)
0.995815 0.0913935i \(-0.0291321\pi\)
\(762\) 0 0
\(763\) 9.69861i 0.351113i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00984 0.289219
\(768\) 0 0
\(769\) −15.5543 −0.560902 −0.280451 0.959868i \(-0.590484\pi\)
−0.280451 + 0.959868i \(0.590484\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 27.4351i − 0.986772i −0.869810 0.493386i \(-0.835759\pi\)
0.869810 0.493386i \(-0.164241\pi\)
\(774\) 0 0
\(775\) 45.6374i 1.63934i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.527554 0.0189016
\(780\) 0 0
\(781\) −56.8512 −2.03430
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 7.88051i − 0.281267i
\(786\) 0 0
\(787\) 12.8128i 0.456728i 0.973576 + 0.228364i \(0.0733376\pi\)
−0.973576 + 0.228364i \(0.926662\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −16.4645 −0.585411
\(792\) 0 0
\(793\) −6.61084 −0.234758
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.2364i 0.752232i 0.926573 + 0.376116i \(0.122741\pi\)
−0.926573 + 0.376116i \(0.877259\pi\)
\(798\) 0 0
\(799\) − 89.0093i − 3.14892i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −84.9081 −2.99634
\(804\) 0 0
\(805\) 6.49370 0.228873
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 22.0978i − 0.776916i −0.921467 0.388458i \(-0.873008\pi\)
0.921467 0.388458i \(-0.126992\pi\)
\(810\) 0 0
\(811\) 7.97299i 0.279969i 0.990154 + 0.139985i \(0.0447054\pi\)
−0.990154 + 0.139985i \(0.955295\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −14.8072 −0.518673
\(816\) 0 0
\(817\) −15.4193 −0.539454
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 28.2052i − 0.984368i −0.870491 0.492184i \(-0.836199\pi\)
0.870491 0.492184i \(-0.163801\pi\)
\(822\) 0 0
\(823\) − 19.6354i − 0.684448i −0.939618 0.342224i \(-0.888820\pi\)
0.939618 0.342224i \(-0.111180\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.34720 0.0468466 0.0234233 0.999726i \(-0.492543\pi\)
0.0234233 + 0.999726i \(0.492543\pi\)
\(828\) 0 0
\(829\) 30.0701 1.04438 0.522190 0.852829i \(-0.325115\pi\)
0.522190 + 0.852829i \(0.325115\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.62531i 0.264201i
\(834\) 0 0
\(835\) 7.24610i 0.250762i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.1288 −0.418731 −0.209366 0.977837i \(-0.567140\pi\)
−0.209366 + 0.977837i \(0.567140\pi\)
\(840\) 0 0
\(841\) 28.9466 0.998158
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 10.7902i − 0.371196i
\(846\) 0 0
\(847\) 22.9166i 0.787424i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −12.3656 −0.423886
\(852\) 0 0
\(853\) −46.5388 −1.59346 −0.796729 0.604337i \(-0.793438\pi\)
−0.796729 + 0.604337i \(0.793438\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 43.0088i 1.46915i 0.678526 + 0.734577i \(0.262619\pi\)
−0.678526 + 0.734577i \(0.737381\pi\)
\(858\) 0 0
\(859\) − 9.13230i − 0.311590i −0.987789 0.155795i \(-0.950206\pi\)
0.987789 0.155795i \(-0.0497939\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.0928 0.479723 0.239861 0.970807i \(-0.422898\pi\)
0.239861 + 0.970807i \(0.422898\pi\)
\(864\) 0 0
\(865\) −11.8424 −0.402654
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 25.6107i − 0.868782i
\(870\) 0 0
\(871\) 8.19026i 0.277516i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −8.00451 −0.270602
\(876\) 0 0
\(877\) 46.6612 1.57564 0.787818 0.615908i \(-0.211211\pi\)
0.787818 + 0.615908i \(0.211211\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 20.1212i − 0.677901i −0.940804 0.338950i \(-0.889928\pi\)
0.940804 0.338950i \(-0.110072\pi\)
\(882\) 0 0
\(883\) − 47.2857i − 1.59129i −0.605763 0.795645i \(-0.707132\pi\)
0.605763 0.795645i \(-0.292868\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.0988 0.372663 0.186331 0.982487i \(-0.440340\pi\)
0.186331 + 0.982487i \(0.440340\pi\)
\(888\) 0 0
\(889\) −19.0764 −0.639801
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 26.8270i − 0.897732i
\(894\) 0 0
\(895\) − 7.99193i − 0.267141i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.48137 0.0827584
\(900\) 0 0
\(901\) 21.3871 0.712509
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.13081i 0.170554i
\(906\) 0 0
\(907\) − 39.6177i − 1.31549i −0.753243 0.657743i \(-0.771511\pi\)
0.753243 0.657743i \(-0.228489\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 31.1527 1.03213 0.516067 0.856548i \(-0.327396\pi\)
0.516067 + 0.856548i \(0.327396\pi\)
\(912\) 0 0
\(913\) −14.4417 −0.477949
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.57149i 0.250033i
\(918\) 0 0
\(919\) 28.4960i 0.939997i 0.882667 + 0.469999i \(0.155746\pi\)
−0.882667 + 0.469999i \(0.844254\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7.09955 0.233685
\(924\) 0 0
\(925\) 7.00454 0.230308
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 36.4726i 1.19663i 0.801261 + 0.598314i \(0.204163\pi\)
−0.801261 + 0.598314i \(0.795837\pi\)
\(930\) 0 0
\(931\) 2.29824i 0.0753217i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −38.4230 −1.25657
\(936\) 0 0
\(937\) −26.6030 −0.869081 −0.434541 0.900652i \(-0.643089\pi\)
−0.434541 + 0.900652i \(0.643089\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 31.5293i 1.02783i 0.857842 + 0.513913i \(0.171805\pi\)
−0.857842 + 0.513913i \(0.828195\pi\)
\(942\) 0 0
\(943\) − 1.72281i − 0.0561024i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.7118 1.55042 0.775212 0.631701i \(-0.217643\pi\)
0.775212 + 0.631701i \(0.217643\pi\)
\(948\) 0 0
\(949\) 10.6033 0.344197
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 49.2910i − 1.59669i −0.602200 0.798345i \(-0.705709\pi\)
0.602200 0.798345i \(-0.294291\pi\)
\(954\) 0 0
\(955\) − 14.3250i − 0.463547i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.8040 0.381172
\(960\) 0 0
\(961\) −84.2338 −2.71722
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 13.1251i − 0.422513i
\(966\) 0 0
\(967\) − 40.7797i − 1.31139i −0.755027 0.655693i \(-0.772376\pi\)
0.755027 0.655693i \(-0.227624\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.18153 0.0379171 0.0189586 0.999820i \(-0.493965\pi\)
0.0189586 + 0.999820i \(0.493965\pi\)
\(972\) 0 0
\(973\) −12.7285 −0.408057
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48.6889i 1.55769i 0.627214 + 0.778847i \(0.284195\pi\)
−0.627214 + 0.778847i \(0.715805\pi\)
\(978\) 0 0
\(979\) − 18.3861i − 0.587622i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0.453964 0.0144792 0.00723960 0.999974i \(-0.497696\pi\)
0.00723960 + 0.999974i \(0.497696\pi\)
\(984\) 0 0
\(985\) −6.07401 −0.193534
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 50.3541i 1.60117i
\(990\) 0 0
\(991\) − 5.48824i − 0.174340i −0.996193 0.0871698i \(-0.972218\pi\)
0.996193 0.0871698i \(-0.0277823\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.43178 −0.203901
\(996\) 0 0
\(997\) 17.4879 0.553846 0.276923 0.960892i \(-0.410685\pi\)
0.276923 + 0.960892i \(0.410685\pi\)
\(998\) 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.h.i.2591.4 yes 24
3.2 odd 2 inner 6048.2.h.i.2591.21 yes 24
4.3 odd 2 inner 6048.2.h.i.2591.22 yes 24
12.11 even 2 inner 6048.2.h.i.2591.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.i.2591.3 24 12.11 even 2 inner
6048.2.h.i.2591.4 yes 24 1.1 even 1 trivial
6048.2.h.i.2591.21 yes 24 3.2 odd 2 inner
6048.2.h.i.2591.22 yes 24 4.3 odd 2 inner