Properties

Label 6048.2.h.i.2591.12
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 2591.12
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.i.2591.11

$q$-expansion

\(f(q)\) \(=\) \(q+0.680677i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+0.680677i q^{5} -1.00000i q^{7} +1.70297 q^{11} +5.59108 q^{13} -3.68812i q^{17} -4.46651i q^{19} +1.16396 q^{23} +4.53668 q^{25} +1.77705i q^{29} +2.58733i q^{31} +0.680677 q^{35} +1.36230 q^{37} +1.31635i q^{41} -4.71189i q^{43} -3.87479 q^{47} -1.00000 q^{49} -6.39069i q^{53} +1.15917i q^{55} -0.948207 q^{59} -1.40831 q^{61} +3.80572i q^{65} +1.28379i q^{67} -3.09744 q^{71} -5.79256 q^{73} -1.70297i q^{77} -1.72001i q^{79} -7.69122 q^{83} +2.51041 q^{85} -0.962856i q^{89} -5.59108i q^{91} +3.04025 q^{95} +12.3918 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q + O(q^{10}) \) \( 24q - 8q^{13} - 24q^{25} + 48q^{37} - 24q^{49} - 48q^{61} - 32q^{73} - 80q^{85} + 64q^{97} + O(q^{100}) \)

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
\(5\) 0.680677i 0.304408i 0.988349 + 0.152204i \(0.0486371\pi\)
−0.988349 + 0.152204i \(0.951363\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.70297 0.513464 0.256732 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(12\) 0 0
\(13\) 5.59108 1.55069 0.775343 0.631540i \(-0.217577\pi\)
0.775343 + 0.631540i \(0.217577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.68812i − 0.894500i −0.894409 0.447250i \(-0.852404\pi\)
0.894409 0.447250i \(-0.147596\pi\)
\(18\) 0 0
\(19\) − 4.46651i − 1.02469i −0.858780 0.512344i \(-0.828777\pi\)
0.858780 0.512344i \(-0.171223\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.16396 0.242703 0.121352 0.992610i \(-0.461277\pi\)
0.121352 + 0.992610i \(0.461277\pi\)
\(24\) 0 0
\(25\) 4.53668 0.907336
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.77705i 0.329990i 0.986294 + 0.164995i \(0.0527609\pi\)
−0.986294 + 0.164995i \(0.947239\pi\)
\(30\) 0 0
\(31\) 2.58733i 0.464698i 0.972633 + 0.232349i \(0.0746411\pi\)
−0.972633 + 0.232349i \(0.925359\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.680677 0.115055
\(36\) 0 0
\(37\) 1.36230 0.223961 0.111980 0.993710i \(-0.464281\pi\)
0.111980 + 0.993710i \(0.464281\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.31635i 0.205580i 0.994703 + 0.102790i \(0.0327769\pi\)
−0.994703 + 0.102790i \(0.967223\pi\)
\(42\) 0 0
\(43\) − 4.71189i − 0.718557i −0.933230 0.359279i \(-0.883023\pi\)
0.933230 0.359279i \(-0.116977\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.87479 −0.565196 −0.282598 0.959238i \(-0.591196\pi\)
−0.282598 + 0.959238i \(0.591196\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.39069i − 0.877828i −0.898529 0.438914i \(-0.855363\pi\)
0.898529 0.438914i \(-0.144637\pi\)
\(54\) 0 0
\(55\) 1.15917i 0.156303i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.948207 −0.123446 −0.0617230 0.998093i \(-0.519660\pi\)
−0.0617230 + 0.998093i \(0.519660\pi\)
\(60\) 0 0
\(61\) −1.40831 −0.180315 −0.0901575 0.995928i \(-0.528737\pi\)
−0.0901575 + 0.995928i \(0.528737\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.80572i 0.472041i
\(66\) 0 0
\(67\) 1.28379i 0.156840i 0.996920 + 0.0784202i \(0.0249876\pi\)
−0.996920 + 0.0784202i \(0.975012\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.09744 −0.367599 −0.183799 0.982964i \(-0.558840\pi\)
−0.183799 + 0.982964i \(0.558840\pi\)
\(72\) 0 0
\(73\) −5.79256 −0.677968 −0.338984 0.940792i \(-0.610083\pi\)
−0.338984 + 0.940792i \(0.610083\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.70297i − 0.194071i
\(78\) 0 0
\(79\) − 1.72001i − 0.193517i −0.995308 0.0967583i \(-0.969153\pi\)
0.995308 0.0967583i \(-0.0308474\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.69122 −0.844221 −0.422111 0.906544i \(-0.638711\pi\)
−0.422111 + 0.906544i \(0.638711\pi\)
\(84\) 0 0
\(85\) 2.51041 0.272293
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 0.962856i − 0.102063i −0.998697 0.0510313i \(-0.983749\pi\)
0.998697 0.0510313i \(-0.0162508\pi\)
\(90\) 0 0
\(91\) − 5.59108i − 0.586104i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.04025 0.311923
\(96\) 0 0
\(97\) 12.3918 1.25819 0.629096 0.777328i \(-0.283425\pi\)
0.629096 + 0.777328i \(0.283425\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 6.74056i − 0.670711i −0.942092 0.335355i \(-0.891144\pi\)
0.942092 0.335355i \(-0.108856\pi\)
\(102\) 0 0
\(103\) − 9.66964i − 0.952778i −0.879235 0.476389i \(-0.841945\pi\)
0.879235 0.476389i \(-0.158055\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.78531 0.172592 0.0862962 0.996270i \(-0.472497\pi\)
0.0862962 + 0.996270i \(0.472497\pi\)
\(108\) 0 0
\(109\) 5.61143 0.537478 0.268739 0.963213i \(-0.413393\pi\)
0.268739 + 0.963213i \(0.413393\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 2.46274i − 0.231675i −0.993268 0.115838i \(-0.963045\pi\)
0.993268 0.115838i \(-0.0369552\pi\)
\(114\) 0 0
\(115\) 0.792283i 0.0738808i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.68812 −0.338089
\(120\) 0 0
\(121\) −8.09990 −0.736354
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.49139i 0.580608i
\(126\) 0 0
\(127\) − 3.85480i − 0.342058i −0.985266 0.171029i \(-0.945291\pi\)
0.985266 0.171029i \(-0.0547092\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −11.0680 −0.967013 −0.483507 0.875341i \(-0.660637\pi\)
−0.483507 + 0.875341i \(0.660637\pi\)
\(132\) 0 0
\(133\) −4.46651 −0.387296
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.9424i 1.27661i 0.769782 + 0.638307i \(0.220365\pi\)
−0.769782 + 0.638307i \(0.779635\pi\)
\(138\) 0 0
\(139\) 5.25129i 0.445409i 0.974886 + 0.222704i \(0.0714885\pi\)
−0.974886 + 0.222704i \(0.928512\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.52143 0.796222
\(144\) 0 0
\(145\) −1.20960 −0.100452
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 16.2480i − 1.33109i −0.746357 0.665545i \(-0.768199\pi\)
0.746357 0.665545i \(-0.231801\pi\)
\(150\) 0 0
\(151\) − 9.61927i − 0.782805i −0.920220 0.391402i \(-0.871990\pi\)
0.920220 0.391402i \(-0.128010\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.76113 −0.141458
\(156\) 0 0
\(157\) 14.5745 1.16317 0.581587 0.813484i \(-0.302432\pi\)
0.581587 + 0.813484i \(0.302432\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 1.16396i − 0.0917332i
\(162\) 0 0
\(163\) − 11.6184i − 0.910021i −0.890486 0.455011i \(-0.849635\pi\)
0.890486 0.455011i \(-0.150365\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.6376 1.67437 0.837185 0.546920i \(-0.184200\pi\)
0.837185 + 0.546920i \(0.184200\pi\)
\(168\) 0 0
\(169\) 18.2602 1.40463
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.3069i 1.61994i 0.586474 + 0.809968i \(0.300515\pi\)
−0.586474 + 0.809968i \(0.699485\pi\)
\(174\) 0 0
\(175\) − 4.53668i − 0.342941i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −14.7701 −1.10397 −0.551986 0.833853i \(-0.686130\pi\)
−0.551986 + 0.833853i \(0.686130\pi\)
\(180\) 0 0
\(181\) 14.7394 1.09557 0.547786 0.836619i \(-0.315471\pi\)
0.547786 + 0.836619i \(0.315471\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.927286i 0.0681754i
\(186\) 0 0
\(187\) − 6.28074i − 0.459294i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.1122 1.59998 0.799991 0.600012i \(-0.204838\pi\)
0.799991 + 0.600012i \(0.204838\pi\)
\(192\) 0 0
\(193\) 19.6487 1.41435 0.707173 0.707041i \(-0.249970\pi\)
0.707173 + 0.707041i \(0.249970\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.6467i 1.25728i 0.777698 + 0.628638i \(0.216387\pi\)
−0.777698 + 0.628638i \(0.783613\pi\)
\(198\) 0 0
\(199\) 4.47875i 0.317490i 0.987320 + 0.158745i \(0.0507448\pi\)
−0.987320 + 0.158745i \(0.949255\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.77705 0.124725
\(204\) 0 0
\(205\) −0.896009 −0.0625800
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 7.60633i − 0.526141i
\(210\) 0 0
\(211\) − 18.6143i − 1.28146i −0.767765 0.640732i \(-0.778631\pi\)
0.767765 0.640732i \(-0.221369\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.20728 0.218734
\(216\) 0 0
\(217\) 2.58733 0.175639
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 20.6205i − 1.38709i
\(222\) 0 0
\(223\) 0.333825i 0.0223546i 0.999938 + 0.0111773i \(0.00355792\pi\)
−0.999938 + 0.0111773i \(0.996442\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.2768 −1.08033 −0.540164 0.841559i \(-0.681638\pi\)
−0.540164 + 0.841559i \(0.681638\pi\)
\(228\) 0 0
\(229\) −8.30762 −0.548983 −0.274491 0.961590i \(-0.588509\pi\)
−0.274491 + 0.961590i \(0.588509\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.9212i 1.76367i 0.471560 + 0.881834i \(0.343691\pi\)
−0.471560 + 0.881834i \(0.656309\pi\)
\(234\) 0 0
\(235\) − 2.63748i − 0.172050i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.40591 0.414364 0.207182 0.978302i \(-0.433571\pi\)
0.207182 + 0.978302i \(0.433571\pi\)
\(240\) 0 0
\(241\) 12.1800 0.784583 0.392291 0.919841i \(-0.371682\pi\)
0.392291 + 0.919841i \(0.371682\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.680677i − 0.0434868i
\(246\) 0 0
\(247\) − 24.9726i − 1.58897i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −25.3399 −1.59944 −0.799720 0.600373i \(-0.795019\pi\)
−0.799720 + 0.600373i \(0.795019\pi\)
\(252\) 0 0
\(253\) 1.98219 0.124619
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 29.8079i − 1.85937i −0.368359 0.929684i \(-0.620080\pi\)
0.368359 0.929684i \(-0.379920\pi\)
\(258\) 0 0
\(259\) − 1.36230i − 0.0846492i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0318 1.11189 0.555943 0.831221i \(-0.312357\pi\)
0.555943 + 0.831221i \(0.312357\pi\)
\(264\) 0 0
\(265\) 4.34999 0.267218
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.18664i 0.255264i 0.991822 + 0.127632i \(0.0407376\pi\)
−0.991822 + 0.127632i \(0.959262\pi\)
\(270\) 0 0
\(271\) − 22.4083i − 1.36121i −0.732653 0.680603i \(-0.761718\pi\)
0.732653 0.680603i \(-0.238282\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.72582 0.465885
\(276\) 0 0
\(277\) 12.9099 0.775680 0.387840 0.921727i \(-0.373221\pi\)
0.387840 + 0.921727i \(0.373221\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.98114i − 0.535770i −0.963451 0.267885i \(-0.913675\pi\)
0.963451 0.267885i \(-0.0863247\pi\)
\(282\) 0 0
\(283\) − 11.4072i − 0.678087i −0.940771 0.339043i \(-0.889897\pi\)
0.940771 0.339043i \(-0.110103\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.31635 0.0777018
\(288\) 0 0
\(289\) 3.39780 0.199871
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.19308i 0.537066i 0.963271 + 0.268533i \(0.0865388\pi\)
−0.963271 + 0.268533i \(0.913461\pi\)
\(294\) 0 0
\(295\) − 0.645422i − 0.0375779i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.50781 0.376357
\(300\) 0 0
\(301\) −4.71189 −0.271589
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 0.958601i − 0.0548893i
\(306\) 0 0
\(307\) − 26.3888i − 1.50609i −0.657972 0.753043i \(-0.728585\pi\)
0.657972 0.753043i \(-0.271415\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 25.0780 1.42204 0.711021 0.703171i \(-0.248233\pi\)
0.711021 + 0.703171i \(0.248233\pi\)
\(312\) 0 0
\(313\) −4.86592 −0.275038 −0.137519 0.990499i \(-0.543913\pi\)
−0.137519 + 0.990499i \(0.543913\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 11.8835i − 0.667441i −0.942672 0.333721i \(-0.891696\pi\)
0.942672 0.333721i \(-0.108304\pi\)
\(318\) 0 0
\(319\) 3.02626i 0.169438i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.4730 −0.916583
\(324\) 0 0
\(325\) 25.3649 1.40699
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.87479i 0.213624i
\(330\) 0 0
\(331\) − 19.0454i − 1.04683i −0.852079 0.523414i \(-0.824658\pi\)
0.852079 0.523414i \(-0.175342\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.873848 −0.0477434
\(336\) 0 0
\(337\) 13.2381 0.721128 0.360564 0.932734i \(-0.382584\pi\)
0.360564 + 0.932734i \(0.382584\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.40614i 0.238606i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.4049 −0.988027 −0.494014 0.869454i \(-0.664471\pi\)
−0.494014 + 0.869454i \(0.664471\pi\)
\(348\) 0 0
\(349\) −19.3574 −1.03618 −0.518090 0.855326i \(-0.673357\pi\)
−0.518090 + 0.855326i \(0.673357\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 16.4485i − 0.875464i −0.899106 0.437732i \(-0.855782\pi\)
0.899106 0.437732i \(-0.144218\pi\)
\(354\) 0 0
\(355\) − 2.10836i − 0.111900i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.6461 −1.08966 −0.544830 0.838547i \(-0.683406\pi\)
−0.544830 + 0.838547i \(0.683406\pi\)
\(360\) 0 0
\(361\) −0.949721 −0.0499853
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 3.94286i − 0.206379i
\(366\) 0 0
\(367\) − 4.24477i − 0.221575i −0.993844 0.110788i \(-0.964663\pi\)
0.993844 0.110788i \(-0.0353373\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.39069 −0.331788
\(372\) 0 0
\(373\) −1.74622 −0.0904160 −0.0452080 0.998978i \(-0.514395\pi\)
−0.0452080 + 0.998978i \(0.514395\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.93564i 0.511712i
\(378\) 0 0
\(379\) 36.6903i 1.88465i 0.334695 + 0.942326i \(0.391367\pi\)
−0.334695 + 0.942326i \(0.608633\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.2853 1.08763 0.543814 0.839206i \(-0.316980\pi\)
0.543814 + 0.839206i \(0.316980\pi\)
\(384\) 0 0
\(385\) 1.15917 0.0590768
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 30.2083i 1.53162i 0.643066 + 0.765811i \(0.277662\pi\)
−0.643066 + 0.765811i \(0.722338\pi\)
\(390\) 0 0
\(391\) − 4.29283i − 0.217098i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.17077 0.0589080
\(396\) 0 0
\(397\) 17.5830 0.882466 0.441233 0.897393i \(-0.354541\pi\)
0.441233 + 0.897393i \(0.354541\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 7.83123i − 0.391073i −0.980696 0.195536i \(-0.937355\pi\)
0.980696 0.195536i \(-0.0626448\pi\)
\(402\) 0 0
\(403\) 14.4660i 0.720600i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.31995 0.114996
\(408\) 0 0
\(409\) 20.8638 1.03165 0.515825 0.856694i \(-0.327486\pi\)
0.515825 + 0.856694i \(0.327486\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.948207i 0.0466582i
\(414\) 0 0
\(415\) − 5.23524i − 0.256988i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 30.2529 1.47795 0.738976 0.673732i \(-0.235310\pi\)
0.738976 + 0.673732i \(0.235310\pi\)
\(420\) 0 0
\(421\) 14.4874 0.706072 0.353036 0.935610i \(-0.385149\pi\)
0.353036 + 0.935610i \(0.385149\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 16.7318i − 0.811611i
\(426\) 0 0
\(427\) 1.40831i 0.0681527i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13.2713 −0.639254 −0.319627 0.947543i \(-0.603558\pi\)
−0.319627 + 0.947543i \(0.603558\pi\)
\(432\) 0 0
\(433\) −22.9913 −1.10489 −0.552446 0.833549i \(-0.686306\pi\)
−0.552446 + 0.833549i \(0.686306\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 5.19886i − 0.248695i
\(438\) 0 0
\(439\) 13.9306i 0.664871i 0.943126 + 0.332435i \(0.107870\pi\)
−0.943126 + 0.332435i \(0.892130\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.8147 −0.798892 −0.399446 0.916757i \(-0.630797\pi\)
−0.399446 + 0.916757i \(0.630797\pi\)
\(444\) 0 0
\(445\) 0.655394 0.0310686
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.6213i 0.642829i 0.946939 + 0.321415i \(0.104158\pi\)
−0.946939 + 0.321415i \(0.895842\pi\)
\(450\) 0 0
\(451\) 2.24170i 0.105558i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.80572 0.178415
\(456\) 0 0
\(457\) 5.94687 0.278183 0.139091 0.990280i \(-0.455582\pi\)
0.139091 + 0.990280i \(0.455582\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 12.9987i − 0.605410i −0.953084 0.302705i \(-0.902110\pi\)
0.953084 0.302705i \(-0.0978896\pi\)
\(462\) 0 0
\(463\) 35.3487i 1.64280i 0.570356 + 0.821398i \(0.306805\pi\)
−0.570356 + 0.821398i \(0.693195\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.9651 −0.553679 −0.276839 0.960916i \(-0.589287\pi\)
−0.276839 + 0.960916i \(0.589287\pi\)
\(468\) 0 0
\(469\) 1.28379 0.0592801
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 8.02421i − 0.368953i
\(474\) 0 0
\(475\) − 20.2631i − 0.929736i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.4802 −1.25560 −0.627801 0.778374i \(-0.716045\pi\)
−0.627801 + 0.778374i \(0.716045\pi\)
\(480\) 0 0
\(481\) 7.61673 0.347293
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.43478i 0.383004i
\(486\) 0 0
\(487\) − 5.03432i − 0.228127i −0.993473 0.114063i \(-0.963613\pi\)
0.993473 0.114063i \(-0.0363867\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.1007 0.771745 0.385873 0.922552i \(-0.373900\pi\)
0.385873 + 0.922552i \(0.373900\pi\)
\(492\) 0 0
\(493\) 6.55398 0.295176
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.09744i 0.138939i
\(498\) 0 0
\(499\) 13.4214i 0.600826i 0.953809 + 0.300413i \(0.0971245\pi\)
−0.953809 + 0.300413i \(0.902876\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.44363 0.287307 0.143654 0.989628i \(-0.454115\pi\)
0.143654 + 0.989628i \(0.454115\pi\)
\(504\) 0 0
\(505\) 4.58814 0.204170
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 3.37393i − 0.149547i −0.997201 0.0747734i \(-0.976177\pi\)
0.997201 0.0747734i \(-0.0238233\pi\)
\(510\) 0 0
\(511\) 5.79256i 0.256248i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.58190 0.290033
\(516\) 0 0
\(517\) −6.59865 −0.290208
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.7512i 1.12818i 0.825714 + 0.564090i \(0.190773\pi\)
−0.825714 + 0.564090i \(0.809227\pi\)
\(522\) 0 0
\(523\) − 4.41398i − 0.193010i −0.995333 0.0965050i \(-0.969234\pi\)
0.995333 0.0965050i \(-0.0307664\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.54236 0.415672
\(528\) 0 0
\(529\) −21.6452 −0.941095
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.35982i 0.318789i
\(534\) 0 0
\(535\) 1.21522i 0.0525385i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.70297 −0.0733520
\(540\) 0 0
\(541\) 2.91406 0.125285 0.0626425 0.998036i \(-0.480047\pi\)
0.0626425 + 0.998036i \(0.480047\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.81957i 0.163613i
\(546\) 0 0
\(547\) 20.7674i 0.887949i 0.896039 + 0.443975i \(0.146432\pi\)
−0.896039 + 0.443975i \(0.853568\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.93723 0.338137
\(552\) 0 0
\(553\) −1.72001 −0.0731424
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 2.79984i − 0.118633i −0.998239 0.0593166i \(-0.981108\pi\)
0.998239 0.0593166i \(-0.0188921\pi\)
\(558\) 0 0
\(559\) − 26.3446i − 1.11426i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.81944 0.371695 0.185847 0.982579i \(-0.440497\pi\)
0.185847 + 0.982579i \(0.440497\pi\)
\(564\) 0 0
\(565\) 1.67633 0.0705237
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.7894i 0.955382i 0.878528 + 0.477691i \(0.158526\pi\)
−0.878528 + 0.477691i \(0.841474\pi\)
\(570\) 0 0
\(571\) − 0.247117i − 0.0103415i −0.999987 0.00517075i \(-0.998354\pi\)
0.999987 0.00517075i \(-0.00164591\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.28053 0.220213
\(576\) 0 0
\(577\) −8.45713 −0.352075 −0.176037 0.984383i \(-0.556328\pi\)
−0.176037 + 0.984383i \(0.556328\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.69122i 0.319086i
\(582\) 0 0
\(583\) − 10.8831i − 0.450733i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −37.8861 −1.56373 −0.781864 0.623449i \(-0.785731\pi\)
−0.781864 + 0.623449i \(0.785731\pi\)
\(588\) 0 0
\(589\) 11.5563 0.476170
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 12.3736i − 0.508124i −0.967188 0.254062i \(-0.918233\pi\)
0.967188 0.254062i \(-0.0817667\pi\)
\(594\) 0 0
\(595\) − 2.51041i − 0.102917i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.5376 0.675706 0.337853 0.941199i \(-0.390299\pi\)
0.337853 + 0.941199i \(0.390299\pi\)
\(600\) 0 0
\(601\) −23.0968 −0.942138 −0.471069 0.882096i \(-0.656132\pi\)
−0.471069 + 0.882096i \(0.656132\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 5.51341i − 0.224152i
\(606\) 0 0
\(607\) − 35.9691i − 1.45994i −0.683478 0.729971i \(-0.739534\pi\)
0.683478 0.729971i \(-0.260466\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −21.6643 −0.876442
\(612\) 0 0
\(613\) 3.18328 0.128571 0.0642857 0.997932i \(-0.479523\pi\)
0.0642857 + 0.997932i \(0.479523\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.65916i − 0.107054i −0.998566 0.0535268i \(-0.982954\pi\)
0.998566 0.0535268i \(-0.0170463\pi\)
\(618\) 0 0
\(619\) 4.00728i 0.161066i 0.996752 + 0.0805331i \(0.0256623\pi\)
−0.996752 + 0.0805331i \(0.974338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.962856 −0.0385760
\(624\) 0 0
\(625\) 18.2649 0.730594
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 5.02432i − 0.200333i
\(630\) 0 0
\(631\) 15.9712i 0.635804i 0.948124 + 0.317902i \(0.102978\pi\)
−0.948124 + 0.317902i \(0.897022\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.62387 0.104125
\(636\) 0 0
\(637\) −5.59108 −0.221527
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21.9427i 0.866685i 0.901229 + 0.433342i \(0.142666\pi\)
−0.901229 + 0.433342i \(0.857334\pi\)
\(642\) 0 0
\(643\) − 6.00112i − 0.236661i −0.992974 0.118331i \(-0.962246\pi\)
0.992974 0.118331i \(-0.0377542\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.2522 1.26796 0.633982 0.773348i \(-0.281419\pi\)
0.633982 + 0.773348i \(0.281419\pi\)
\(648\) 0 0
\(649\) −1.61477 −0.0633851
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 38.3807i − 1.50195i −0.660328 0.750977i \(-0.729583\pi\)
0.660328 0.750977i \(-0.270417\pi\)
\(654\) 0 0
\(655\) − 7.53371i − 0.294366i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.7292 −1.43076 −0.715382 0.698733i \(-0.753748\pi\)
−0.715382 + 0.698733i \(0.753748\pi\)
\(660\) 0 0
\(661\) −2.95296 −0.114857 −0.0574284 0.998350i \(-0.518290\pi\)
−0.0574284 + 0.998350i \(0.518290\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3.04025i − 0.117896i
\(666\) 0 0
\(667\) 2.06843i 0.0800898i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.39830 −0.0925853
\(672\) 0 0
\(673\) 42.6334 1.64340 0.821698 0.569923i \(-0.193027\pi\)
0.821698 + 0.569923i \(0.193027\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.1217i 1.11924i 0.828750 + 0.559619i \(0.189053\pi\)
−0.828750 + 0.559619i \(0.810947\pi\)
\(678\) 0 0
\(679\) − 12.3918i − 0.475552i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −30.0935 −1.15150 −0.575749 0.817627i \(-0.695289\pi\)
−0.575749 + 0.817627i \(0.695289\pi\)
\(684\) 0 0
\(685\) −10.1709 −0.388611
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 35.7308i − 1.36124i
\(690\) 0 0
\(691\) − 29.8979i − 1.13737i −0.822555 0.568685i \(-0.807452\pi\)
0.822555 0.568685i \(-0.192548\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.57443 −0.135586
\(696\) 0 0
\(697\) 4.85486 0.183891
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 46.7692i − 1.76645i −0.468950 0.883225i \(-0.655367\pi\)
0.468950 0.883225i \(-0.344633\pi\)
\(702\) 0 0
\(703\) − 6.08473i − 0.229490i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.74056 −0.253505
\(708\) 0 0
\(709\) 5.29936 0.199022 0.0995108 0.995036i \(-0.468272\pi\)
0.0995108 + 0.995036i \(0.468272\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.01156i 0.112784i
\(714\) 0 0
\(715\) 6.48101i 0.242376i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12.9460 0.482805 0.241402 0.970425i \(-0.422393\pi\)
0.241402 + 0.970425i \(0.422393\pi\)
\(720\) 0 0
\(721\) −9.66964 −0.360116
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.06192i 0.299412i
\(726\) 0 0
\(727\) − 33.7044i − 1.25003i −0.780613 0.625014i \(-0.785093\pi\)
0.780613 0.625014i \(-0.214907\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −17.3780 −0.642749
\(732\) 0 0
\(733\) −24.7469 −0.914048 −0.457024 0.889454i \(-0.651085\pi\)
−0.457024 + 0.889454i \(0.651085\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.18626i 0.0805319i
\(738\) 0 0
\(739\) 6.35804i 0.233884i 0.993139 + 0.116942i \(0.0373092\pi\)
−0.993139 + 0.116942i \(0.962691\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.8301 1.27779 0.638896 0.769293i \(-0.279392\pi\)
0.638896 + 0.769293i \(0.279392\pi\)
\(744\) 0 0
\(745\) 11.0597 0.405195
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1.78531i − 0.0652338i
\(750\) 0 0
\(751\) 10.9033i 0.397867i 0.980013 + 0.198933i \(0.0637478\pi\)
−0.980013 + 0.198933i \(0.936252\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.54761 0.238292
\(756\) 0 0
\(757\) 41.2117 1.49786 0.748932 0.662647i \(-0.230567\pi\)
0.748932 + 0.662647i \(0.230567\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 14.7395i − 0.534306i −0.963654 0.267153i \(-0.913917\pi\)
0.963654 0.267153i \(-0.0860830\pi\)
\(762\) 0 0
\(763\) − 5.61143i − 0.203148i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.30150 −0.191426
\(768\) 0 0
\(769\) −39.0357 −1.40766 −0.703832 0.710367i \(-0.748529\pi\)
−0.703832 + 0.710367i \(0.748529\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 44.0345i 1.58381i 0.610643 + 0.791906i \(0.290911\pi\)
−0.610643 + 0.791906i \(0.709089\pi\)
\(774\) 0 0
\(775\) 11.7379i 0.421637i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.87950 0.210655
\(780\) 0 0
\(781\) −5.27485 −0.188749
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.92054i 0.354079i
\(786\) 0 0
\(787\) 2.77184i 0.0988055i 0.998779 + 0.0494028i \(0.0157318\pi\)
−0.998779 + 0.0494028i \(0.984268\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.46274 −0.0875650
\(792\) 0 0
\(793\) −7.87395 −0.279612
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 50.0469i − 1.77275i −0.462967 0.886375i \(-0.653215\pi\)
0.462967 0.886375i \(-0.346785\pi\)
\(798\) 0 0
\(799\) 14.2907i 0.505568i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −9.86454 −0.348112
\(804\) 0 0
\(805\) 0.792283 0.0279243
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 25.8463i − 0.908707i −0.890821 0.454354i \(-0.849870\pi\)
0.890821 0.454354i \(-0.150130\pi\)
\(810\) 0 0
\(811\) − 23.4798i − 0.824489i −0.911073 0.412244i \(-0.864745\pi\)
0.911073 0.412244i \(-0.135255\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.90836 0.277018
\(816\) 0 0
\(817\) −21.0457 −0.736297
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 20.1359i − 0.702747i −0.936235 0.351373i \(-0.885715\pi\)
0.936235 0.351373i \(-0.114285\pi\)
\(822\) 0 0
\(823\) − 1.51467i − 0.0527982i −0.999651 0.0263991i \(-0.991596\pi\)
0.999651 0.0263991i \(-0.00840407\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −17.8499 −0.620702 −0.310351 0.950622i \(-0.600447\pi\)
−0.310351 + 0.950622i \(0.600447\pi\)
\(828\) 0 0
\(829\) 10.1908 0.353942 0.176971 0.984216i \(-0.443370\pi\)
0.176971 + 0.984216i \(0.443370\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.68812i 0.127786i
\(834\) 0 0
\(835\) 14.7282i 0.509691i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.6943 0.438256 0.219128 0.975696i \(-0.429679\pi\)
0.219128 + 0.975696i \(0.429679\pi\)
\(840\) 0 0
\(841\) 25.8421 0.891106
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.4293i 0.427580i
\(846\) 0 0
\(847\) 8.09990i 0.278316i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.58567 0.0543560
\(852\) 0 0
\(853\) −24.6149 −0.842797 −0.421399 0.906876i \(-0.638461\pi\)
−0.421399 + 0.906876i \(0.638461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5.98500i 0.204444i 0.994762 + 0.102222i \(0.0325952\pi\)
−0.994762 + 0.102222i \(0.967405\pi\)
\(858\) 0 0
\(859\) 13.9197i 0.474934i 0.971396 + 0.237467i \(0.0763171\pi\)
−0.971396 + 0.237467i \(0.923683\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.3075 1.37208 0.686042 0.727562i \(-0.259347\pi\)
0.686042 + 0.727562i \(0.259347\pi\)
\(864\) 0 0
\(865\) −14.5031 −0.493121
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 2.92913i − 0.0993638i
\(870\) 0 0
\(871\) 7.17779i 0.243210i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.49139 0.219449
\(876\) 0 0
\(877\) −44.5498 −1.50434 −0.752170 0.658969i \(-0.770993\pi\)
−0.752170 + 0.658969i \(0.770993\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 21.4243i 0.721801i 0.932604 + 0.360901i \(0.117531\pi\)
−0.932604 + 0.360901i \(0.882469\pi\)
\(882\) 0 0
\(883\) 53.9222i 1.81463i 0.420455 + 0.907314i \(0.361871\pi\)
−0.420455 + 0.907314i \(0.638129\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 15.8932 0.533640 0.266820 0.963746i \(-0.414027\pi\)
0.266820 + 0.963746i \(0.414027\pi\)
\(888\) 0 0
\(889\) −3.85480 −0.129286
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 17.3068i 0.579150i
\(894\) 0 0
\(895\) − 10.0537i − 0.336058i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.59782 −0.153346
\(900\) 0 0
\(901\) −23.5696 −0.785217
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0328i 0.333501i
\(906\) 0 0
\(907\) 11.8494i 0.393452i 0.980459 + 0.196726i \(0.0630309\pi\)
−0.980459 + 0.196726i \(0.936969\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.1245 0.799280 0.399640 0.916672i \(-0.369135\pi\)
0.399640 + 0.916672i \(0.369135\pi\)
\(912\) 0 0
\(913\) −13.0979 −0.433477
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.0680i 0.365497i
\(918\) 0 0
\(919\) 44.1085i 1.45501i 0.686105 + 0.727503i \(0.259319\pi\)
−0.686105 + 0.727503i \(0.740681\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −17.3181 −0.570031
\(924\) 0 0
\(925\) 6.18032 0.203208
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6.83370i 0.224207i 0.993697 + 0.112103i \(0.0357587\pi\)
−0.993697 + 0.112103i \(0.964241\pi\)
\(930\) 0 0
\(931\) 4.46651i 0.146384i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.27516 0.139813
\(936\) 0 0
\(937\) −52.2524 −1.70701 −0.853505 0.521084i \(-0.825528\pi\)
−0.853505 + 0.521084i \(0.825528\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 7.35658i − 0.239818i −0.992785 0.119909i \(-0.961740\pi\)
0.992785 0.119909i \(-0.0382602\pi\)
\(942\) 0 0
\(943\) 1.53219i 0.0498948i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −46.7826 −1.52023 −0.760115 0.649789i \(-0.774857\pi\)
−0.760115 + 0.649789i \(0.774857\pi\)
\(948\) 0 0
\(949\) −32.3867 −1.05132
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1.44950i − 0.0469538i −0.999724 0.0234769i \(-0.992526\pi\)
0.999724 0.0234769i \(-0.00747361\pi\)
\(954\) 0 0
\(955\) 15.0512i 0.487047i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 14.9424 0.482515
\(960\) 0 0
\(961\) 24.3057 0.784056
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 13.3744i 0.430538i
\(966\) 0 0
\(967\) 47.6909i 1.53364i 0.641864 + 0.766819i \(0.278161\pi\)
−0.641864 + 0.766819i \(0.721839\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −11.5871 −0.371848 −0.185924 0.982564i \(-0.559528\pi\)
−0.185924 + 0.982564i \(0.559528\pi\)
\(972\) 0 0
\(973\) 5.25129 0.168349
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 46.6994i − 1.49405i −0.664799 0.747023i \(-0.731483\pi\)
0.664799 0.747023i \(-0.268517\pi\)
\(978\) 0 0
\(979\) − 1.63971i − 0.0524055i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.31157 −0.296993 −0.148497 0.988913i \(-0.547443\pi\)
−0.148497 + 0.988913i \(0.547443\pi\)
\(984\) 0 0
\(985\) −12.0117 −0.382725
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 5.48448i − 0.174396i
\(990\) 0 0
\(991\) 19.8616i 0.630925i 0.948938 + 0.315463i \(0.102160\pi\)
−0.948938 + 0.315463i \(0.897840\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.04858 −0.0966465
\(996\) 0 0
\(997\) −47.9244 −1.51778 −0.758890 0.651218i \(-0.774258\pi\)
−0.758890 + 0.651218i \(0.774258\pi\)
\(998\) 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.2591.12 yes 24
3.2 odd 2 inner 6048.2.h.i.2591.13 yes 24
4.3 odd 2 inner 6048.2.h.i.2591.14 yes 24
12.11 even 2 inner 6048.2.h.i.2591.11 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.i.2591.11 24 12.11 even 2 inner
6048.2.h.i.2591.12 yes 24 1.1 even 1 trivial
6048.2.h.i.2591.13 yes 24 3.2 odd 2 inner
6048.2.h.i.2591.14 yes 24 4.3 odd 2 inner