Properties

Label 6048.2.c.f.3025.24
Level 6048
Weight 2
Character 6048.3025
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
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 3025.24
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.f.3025.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.46300i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+3.46300i q^{5} -1.00000 q^{7} +3.31902i q^{11} +3.10840i q^{13} -1.40277 q^{17} -4.80674i q^{19} -8.79960 q^{23} -6.99238 q^{25} -9.87397i q^{29} +7.83557 q^{31} -3.46300i q^{35} +5.42317i q^{37} -11.5710 q^{41} -7.03957i q^{43} -11.2916 q^{47} +1.00000 q^{49} +6.51655i q^{53} -11.4938 q^{55} +3.89654i q^{59} -9.42334i q^{61} -10.7644 q^{65} +0.909712i q^{67} +6.42314 q^{71} +1.56257 q^{73} -3.31902i q^{77} +11.1619 q^{79} +0.370241i q^{83} -4.85778i q^{85} +4.85047 q^{89} -3.10840i q^{91} +16.6457 q^{95} +1.63283 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} - 8q^{31} + 24q^{49} - 16q^{55} - 8q^{79} + O(q^{100}) \)

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
\(5\) 3.46300i 1.54870i 0.632757 + 0.774351i \(0.281923\pi\)
−0.632757 + 0.774351i \(0.718077\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.31902i 1.00072i 0.865817 + 0.500361i \(0.166799\pi\)
−0.865817 + 0.500361i \(0.833201\pi\)
\(12\) 0 0
\(13\) 3.10840i 0.862116i 0.902324 + 0.431058i \(0.141860\pi\)
−0.902324 + 0.431058i \(0.858140\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.40277 −0.340221 −0.170110 0.985425i \(-0.554412\pi\)
−0.170110 + 0.985425i \(0.554412\pi\)
\(18\) 0 0
\(19\) − 4.80674i − 1.10274i −0.834260 0.551371i \(-0.814105\pi\)
0.834260 0.551371i \(-0.185895\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.79960 −1.83484 −0.917421 0.397917i \(-0.869733\pi\)
−0.917421 + 0.397917i \(0.869733\pi\)
\(24\) 0 0
\(25\) −6.99238 −1.39848
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 9.87397i − 1.83355i −0.399404 0.916775i \(-0.630783\pi\)
0.399404 0.916775i \(-0.369217\pi\)
\(30\) 0 0
\(31\) 7.83557 1.40731 0.703655 0.710541i \(-0.251550\pi\)
0.703655 + 0.710541i \(0.251550\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.46300i − 0.585354i
\(36\) 0 0
\(37\) 5.42317i 0.891564i 0.895142 + 0.445782i \(0.147074\pi\)
−0.895142 + 0.445782i \(0.852926\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.5710 −1.80708 −0.903542 0.428499i \(-0.859043\pi\)
−0.903542 + 0.428499i \(0.859043\pi\)
\(42\) 0 0
\(43\) − 7.03957i − 1.07352i −0.843733 0.536762i \(-0.819647\pi\)
0.843733 0.536762i \(-0.180353\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.2916 −1.64704 −0.823522 0.567284i \(-0.807994\pi\)
−0.823522 + 0.567284i \(0.807994\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.51655i 0.895117i 0.894255 + 0.447559i \(0.147706\pi\)
−0.894255 + 0.447559i \(0.852294\pi\)
\(54\) 0 0
\(55\) −11.4938 −1.54982
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.89654i 0.507287i 0.967298 + 0.253643i \(0.0816290\pi\)
−0.967298 + 0.253643i \(0.918371\pi\)
\(60\) 0 0
\(61\) − 9.42334i − 1.20653i −0.797539 0.603267i \(-0.793865\pi\)
0.797539 0.603267i \(-0.206135\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.7644 −1.33516
\(66\) 0 0
\(67\) 0.909712i 0.111139i 0.998455 + 0.0555695i \(0.0176974\pi\)
−0.998455 + 0.0555695i \(0.982303\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.42314 0.762286 0.381143 0.924516i \(-0.375531\pi\)
0.381143 + 0.924516i \(0.375531\pi\)
\(72\) 0 0
\(73\) 1.56257 0.182884 0.0914422 0.995810i \(-0.470852\pi\)
0.0914422 + 0.995810i \(0.470852\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.31902i − 0.378238i
\(78\) 0 0
\(79\) 11.1619 1.25582 0.627908 0.778288i \(-0.283912\pi\)
0.627908 + 0.778288i \(0.283912\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.370241i 0.0406392i 0.999794 + 0.0203196i \(0.00646837\pi\)
−0.999794 + 0.0203196i \(0.993532\pi\)
\(84\) 0 0
\(85\) − 4.85778i − 0.526900i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.85047 0.514149 0.257075 0.966392i \(-0.417241\pi\)
0.257075 + 0.966392i \(0.417241\pi\)
\(90\) 0 0
\(91\) − 3.10840i − 0.325849i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.6457 1.70782
\(96\) 0 0
\(97\) 1.63283 0.165789 0.0828946 0.996558i \(-0.473584\pi\)
0.0828946 + 0.996558i \(0.473584\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 13.1650i − 1.30997i −0.755643 0.654983i \(-0.772676\pi\)
0.755643 0.654983i \(-0.227324\pi\)
\(102\) 0 0
\(103\) −15.6983 −1.54680 −0.773399 0.633920i \(-0.781445\pi\)
−0.773399 + 0.633920i \(0.781445\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.37257i 0.906081i 0.891490 + 0.453040i \(0.149661\pi\)
−0.891490 + 0.453040i \(0.850339\pi\)
\(108\) 0 0
\(109\) − 7.31490i − 0.700640i −0.936630 0.350320i \(-0.886073\pi\)
0.936630 0.350320i \(-0.113927\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.9153 1.02683 0.513414 0.858141i \(-0.328380\pi\)
0.513414 + 0.858141i \(0.328380\pi\)
\(114\) 0 0
\(115\) − 30.4730i − 2.84162i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.40277 0.128591
\(120\) 0 0
\(121\) −0.0159016 −0.00144560
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 6.89962i − 0.617121i
\(126\) 0 0
\(127\) −3.37545 −0.299523 −0.149761 0.988722i \(-0.547851\pi\)
−0.149761 + 0.988722i \(0.547851\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.15460i 0.625100i 0.949901 + 0.312550i \(0.101183\pi\)
−0.949901 + 0.312550i \(0.898817\pi\)
\(132\) 0 0
\(133\) 4.80674i 0.416797i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.22854 0.361269 0.180634 0.983550i \(-0.442185\pi\)
0.180634 + 0.983550i \(0.442185\pi\)
\(138\) 0 0
\(139\) − 3.31473i − 0.281152i −0.990070 0.140576i \(-0.955105\pi\)
0.990070 0.140576i \(-0.0448954\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.3169 −0.862739
\(144\) 0 0
\(145\) 34.1936 2.83962
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 7.08433i − 0.580371i −0.956970 0.290185i \(-0.906283\pi\)
0.956970 0.290185i \(-0.0937169\pi\)
\(150\) 0 0
\(151\) −0.252443 −0.0205436 −0.0102718 0.999947i \(-0.503270\pi\)
−0.0102718 + 0.999947i \(0.503270\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 27.1346i 2.17950i
\(156\) 0 0
\(157\) 3.11021i 0.248222i 0.992268 + 0.124111i \(0.0396078\pi\)
−0.992268 + 0.124111i \(0.960392\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.79960 0.693505
\(162\) 0 0
\(163\) 18.5595i 1.45369i 0.686800 + 0.726847i \(0.259015\pi\)
−0.686800 + 0.726847i \(0.740985\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.3221 1.10828 0.554140 0.832424i \(-0.313047\pi\)
0.554140 + 0.832424i \(0.313047\pi\)
\(168\) 0 0
\(169\) 3.33782 0.256755
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.40183i 0.106579i 0.998579 + 0.0532896i \(0.0169707\pi\)
−0.998579 + 0.0532896i \(0.983029\pi\)
\(174\) 0 0
\(175\) 6.99238 0.528574
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3.46849i − 0.259247i −0.991563 0.129624i \(-0.958623\pi\)
0.991563 0.129624i \(-0.0413769\pi\)
\(180\) 0 0
\(181\) − 19.3992i − 1.44193i −0.692972 0.720965i \(-0.743699\pi\)
0.692972 0.720965i \(-0.256301\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −18.7805 −1.38077
\(186\) 0 0
\(187\) − 4.65581i − 0.340466i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −23.4610 −1.69758 −0.848790 0.528729i \(-0.822669\pi\)
−0.848790 + 0.528729i \(0.822669\pi\)
\(192\) 0 0
\(193\) −15.5374 −1.11840 −0.559202 0.829031i \(-0.688893\pi\)
−0.559202 + 0.829031i \(0.688893\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 14.8278i − 1.05643i −0.849109 0.528217i \(-0.822861\pi\)
0.849109 0.528217i \(-0.177139\pi\)
\(198\) 0 0
\(199\) −13.2095 −0.936397 −0.468199 0.883623i \(-0.655097\pi\)
−0.468199 + 0.883623i \(0.655097\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.87397i 0.693017i
\(204\) 0 0
\(205\) − 40.0703i − 2.79863i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.9537 1.10354
\(210\) 0 0
\(211\) − 12.3790i − 0.852206i −0.904675 0.426103i \(-0.859886\pi\)
0.904675 0.426103i \(-0.140114\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 24.3781 1.66257
\(216\) 0 0
\(217\) −7.83557 −0.531913
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 4.36036i − 0.293310i
\(222\) 0 0
\(223\) −2.91170 −0.194982 −0.0974910 0.995236i \(-0.531082\pi\)
−0.0974910 + 0.995236i \(0.531082\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.3550i 0.753658i 0.926283 + 0.376829i \(0.122986\pi\)
−0.926283 + 0.376829i \(0.877014\pi\)
\(228\) 0 0
\(229\) 17.4776i 1.15495i 0.816407 + 0.577477i \(0.195963\pi\)
−0.816407 + 0.577477i \(0.804037\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.98827 0.130256 0.0651279 0.997877i \(-0.479254\pi\)
0.0651279 + 0.997877i \(0.479254\pi\)
\(234\) 0 0
\(235\) − 39.1027i − 2.55078i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.3430 1.38056 0.690282 0.723540i \(-0.257486\pi\)
0.690282 + 0.723540i \(0.257486\pi\)
\(240\) 0 0
\(241\) −13.3714 −0.861330 −0.430665 0.902512i \(-0.641721\pi\)
−0.430665 + 0.902512i \(0.641721\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.46300i 0.221243i
\(246\) 0 0
\(247\) 14.9413 0.950691
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 1.92814i − 0.121703i −0.998147 0.0608514i \(-0.980618\pi\)
0.998147 0.0608514i \(-0.0193816\pi\)
\(252\) 0 0
\(253\) − 29.2060i − 1.83617i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.22415 −0.325873 −0.162937 0.986637i \(-0.552097\pi\)
−0.162937 + 0.986637i \(0.552097\pi\)
\(258\) 0 0
\(259\) − 5.42317i − 0.336980i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.69624 0.289583 0.144791 0.989462i \(-0.453749\pi\)
0.144791 + 0.989462i \(0.453749\pi\)
\(264\) 0 0
\(265\) −22.5668 −1.38627
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 26.9906i − 1.64565i −0.568297 0.822824i \(-0.692397\pi\)
0.568297 0.822824i \(-0.307603\pi\)
\(270\) 0 0
\(271\) −2.75175 −0.167157 −0.0835784 0.996501i \(-0.526635\pi\)
−0.0835784 + 0.996501i \(0.526635\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 23.2079i − 1.39949i
\(276\) 0 0
\(277\) − 12.0526i − 0.724173i −0.932145 0.362086i \(-0.882065\pi\)
0.932145 0.362086i \(-0.117935\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.74449 −0.402342 −0.201171 0.979556i \(-0.564475\pi\)
−0.201171 + 0.979556i \(0.564475\pi\)
\(282\) 0 0
\(283\) 8.53083i 0.507105i 0.967322 + 0.253552i \(0.0815991\pi\)
−0.967322 + 0.253552i \(0.918401\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.5710 0.683014
\(288\) 0 0
\(289\) −15.0322 −0.884250
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 11.2926i − 0.659718i −0.944030 0.329859i \(-0.892999\pi\)
0.944030 0.329859i \(-0.107001\pi\)
\(294\) 0 0
\(295\) −13.4937 −0.785636
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 27.3527i − 1.58185i
\(300\) 0 0
\(301\) 7.03957i 0.405754i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 32.6330 1.86856
\(306\) 0 0
\(307\) 9.15810i 0.522680i 0.965247 + 0.261340i \(0.0841644\pi\)
−0.965247 + 0.261340i \(0.915836\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.0122 −0.567741 −0.283870 0.958863i \(-0.591619\pi\)
−0.283870 + 0.958863i \(0.591619\pi\)
\(312\) 0 0
\(313\) −10.3334 −0.584076 −0.292038 0.956407i \(-0.594333\pi\)
−0.292038 + 0.956407i \(0.594333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.9339i 1.96208i 0.193794 + 0.981042i \(0.437921\pi\)
−0.193794 + 0.981042i \(0.562079\pi\)
\(318\) 0 0
\(319\) 32.7719 1.83487
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.74273i 0.375175i
\(324\) 0 0
\(325\) − 21.7352i − 1.20565i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.2916 0.622524
\(330\) 0 0
\(331\) − 8.71488i − 0.479013i −0.970895 0.239507i \(-0.923014\pi\)
0.970895 0.239507i \(-0.0769857\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.15034 −0.172121
\(336\) 0 0
\(337\) −8.63703 −0.470489 −0.235244 0.971936i \(-0.575589\pi\)
−0.235244 + 0.971936i \(0.575589\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 26.0064i 1.40833i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 28.2190i − 1.51488i −0.652907 0.757438i \(-0.726451\pi\)
0.652907 0.757438i \(-0.273549\pi\)
\(348\) 0 0
\(349\) 13.6266i 0.729416i 0.931122 + 0.364708i \(0.118831\pi\)
−0.931122 + 0.364708i \(0.881169\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −32.4282 −1.72598 −0.862990 0.505220i \(-0.831411\pi\)
−0.862990 + 0.505220i \(0.831411\pi\)
\(354\) 0 0
\(355\) 22.2433i 1.18055i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.9995 0.686087 0.343044 0.939319i \(-0.388542\pi\)
0.343044 + 0.939319i \(0.388542\pi\)
\(360\) 0 0
\(361\) −4.10474 −0.216039
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.41117i 0.283233i
\(366\) 0 0
\(367\) 7.55576 0.394408 0.197204 0.980363i \(-0.436814\pi\)
0.197204 + 0.980363i \(0.436814\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 6.51655i − 0.338323i
\(372\) 0 0
\(373\) − 34.6651i − 1.79489i −0.441124 0.897446i \(-0.645420\pi\)
0.441124 0.897446i \(-0.354580\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 30.6923 1.58073
\(378\) 0 0
\(379\) − 0.308373i − 0.0158401i −0.999969 0.00792004i \(-0.997479\pi\)
0.999969 0.00792004i \(-0.00252105\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.36024 −0.222798 −0.111399 0.993776i \(-0.535533\pi\)
−0.111399 + 0.993776i \(0.535533\pi\)
\(384\) 0 0
\(385\) 11.4938 0.585777
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.33391i 0.0676317i 0.999428 + 0.0338159i \(0.0107660\pi\)
−0.999428 + 0.0338159i \(0.989234\pi\)
\(390\) 0 0
\(391\) 12.3438 0.624251
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 38.6538i 1.94488i
\(396\) 0 0
\(397\) 6.92952i 0.347783i 0.984765 + 0.173891i \(0.0556342\pi\)
−0.984765 + 0.173891i \(0.944366\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −23.9669 −1.19685 −0.598426 0.801178i \(-0.704207\pi\)
−0.598426 + 0.801178i \(0.704207\pi\)
\(402\) 0 0
\(403\) 24.3561i 1.21327i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.9996 −0.892208
\(408\) 0 0
\(409\) 21.6936 1.07268 0.536340 0.844002i \(-0.319807\pi\)
0.536340 + 0.844002i \(0.319807\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 3.89654i − 0.191736i
\(414\) 0 0
\(415\) −1.28214 −0.0629380
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 27.5910i 1.34791i 0.738772 + 0.673955i \(0.235406\pi\)
−0.738772 + 0.673955i \(0.764594\pi\)
\(420\) 0 0
\(421\) − 38.3335i − 1.86826i −0.356929 0.934131i \(-0.616176\pi\)
0.356929 0.934131i \(-0.383824\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.80867 0.475790
\(426\) 0 0
\(427\) 9.42334i 0.456027i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.22510 −0.107179 −0.0535896 0.998563i \(-0.517066\pi\)
−0.0535896 + 0.998563i \(0.517066\pi\)
\(432\) 0 0
\(433\) 28.2235 1.35633 0.678167 0.734908i \(-0.262775\pi\)
0.678167 + 0.734908i \(0.262775\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 42.2974i 2.02336i
\(438\) 0 0
\(439\) 5.63936 0.269152 0.134576 0.990903i \(-0.457033\pi\)
0.134576 + 0.990903i \(0.457033\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 5.81672i − 0.276361i −0.990407 0.138180i \(-0.955875\pi\)
0.990407 0.138180i \(-0.0441254\pi\)
\(444\) 0 0
\(445\) 16.7972i 0.796264i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.66921 −0.0787748 −0.0393874 0.999224i \(-0.512541\pi\)
−0.0393874 + 0.999224i \(0.512541\pi\)
\(450\) 0 0
\(451\) − 38.4043i − 1.80839i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10.7644 0.504643
\(456\) 0 0
\(457\) −16.4606 −0.769994 −0.384997 0.922918i \(-0.625797\pi\)
−0.384997 + 0.922918i \(0.625797\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.7915i 1.06151i 0.847527 + 0.530753i \(0.178091\pi\)
−0.847527 + 0.530753i \(0.821909\pi\)
\(462\) 0 0
\(463\) 16.3165 0.758293 0.379146 0.925337i \(-0.376218\pi\)
0.379146 + 0.925337i \(0.376218\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 33.3073i − 1.54128i −0.637271 0.770639i \(-0.719937\pi\)
0.637271 0.770639i \(-0.280063\pi\)
\(468\) 0 0
\(469\) − 0.909712i − 0.0420066i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23.3645 1.07430
\(474\) 0 0
\(475\) 33.6105i 1.54216i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −28.5717 −1.30547 −0.652737 0.757584i \(-0.726379\pi\)
−0.652737 + 0.757584i \(0.726379\pi\)
\(480\) 0 0
\(481\) −16.8574 −0.768632
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.65451i 0.256758i
\(486\) 0 0
\(487\) −39.0149 −1.76793 −0.883967 0.467550i \(-0.845137\pi\)
−0.883967 + 0.467550i \(0.845137\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 12.5799i − 0.567722i −0.958866 0.283861i \(-0.908385\pi\)
0.958866 0.283861i \(-0.0916153\pi\)
\(492\) 0 0
\(493\) 13.8509i 0.623811i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.42314 −0.288117
\(498\) 0 0
\(499\) − 17.3054i − 0.774697i −0.921933 0.387349i \(-0.873391\pi\)
0.921933 0.387349i \(-0.126609\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.1289 −0.808326 −0.404163 0.914687i \(-0.632437\pi\)
−0.404163 + 0.914687i \(0.632437\pi\)
\(504\) 0 0
\(505\) 45.5904 2.02875
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 17.3445i − 0.768780i −0.923171 0.384390i \(-0.874412\pi\)
0.923171 0.384390i \(-0.125588\pi\)
\(510\) 0 0
\(511\) −1.56257 −0.0691238
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 54.3632i − 2.39553i
\(516\) 0 0
\(517\) − 37.4770i − 1.64823i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −16.7507 −0.733861 −0.366930 0.930248i \(-0.619591\pi\)
−0.366930 + 0.930248i \(0.619591\pi\)
\(522\) 0 0
\(523\) − 20.8692i − 0.912545i −0.889840 0.456273i \(-0.849184\pi\)
0.889840 0.456273i \(-0.150816\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.9915 −0.478796
\(528\) 0 0
\(529\) 54.4329 2.36665
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 35.9673i − 1.55792i
\(534\) 0 0
\(535\) −32.4572 −1.40325
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.31902i 0.142960i
\(540\) 0 0
\(541\) 37.7498i 1.62299i 0.584359 + 0.811495i \(0.301346\pi\)
−0.584359 + 0.811495i \(0.698654\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 25.3315 1.08508
\(546\) 0 0
\(547\) − 34.4520i − 1.47306i −0.676404 0.736531i \(-0.736463\pi\)
0.676404 0.736531i \(-0.263537\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −47.4616 −2.02193
\(552\) 0 0
\(553\) −11.1619 −0.474653
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.7136i 1.17426i 0.809491 + 0.587132i \(0.199743\pi\)
−0.809491 + 0.587132i \(0.800257\pi\)
\(558\) 0 0
\(559\) 21.8818 0.925503
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 0.570384i − 0.0240388i −0.999928 0.0120194i \(-0.996174\pi\)
0.999928 0.0120194i \(-0.00382599\pi\)
\(564\) 0 0
\(565\) 37.7998i 1.59025i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.4090 −0.771745 −0.385872 0.922552i \(-0.626099\pi\)
−0.385872 + 0.922552i \(0.626099\pi\)
\(570\) 0 0
\(571\) − 43.3534i − 1.81428i −0.420824 0.907142i \(-0.638259\pi\)
0.420824 0.907142i \(-0.361741\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 61.5301 2.56598
\(576\) 0 0
\(577\) −39.6209 −1.64944 −0.824720 0.565541i \(-0.808667\pi\)
−0.824720 + 0.565541i \(0.808667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 0.370241i − 0.0153602i
\(582\) 0 0
\(583\) −21.6286 −0.895764
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.4388i 0.554677i 0.960772 + 0.277338i \(0.0894523\pi\)
−0.960772 + 0.277338i \(0.910548\pi\)
\(588\) 0 0
\(589\) − 37.6636i − 1.55190i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.3157 −0.916395 −0.458198 0.888850i \(-0.651505\pi\)
−0.458198 + 0.888850i \(0.651505\pi\)
\(594\) 0 0
\(595\) 4.85778i 0.199150i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.2622 −0.582740 −0.291370 0.956610i \(-0.594111\pi\)
−0.291370 + 0.956610i \(0.594111\pi\)
\(600\) 0 0
\(601\) −2.40215 −0.0979858 −0.0489929 0.998799i \(-0.515601\pi\)
−0.0489929 + 0.998799i \(0.515601\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 0.0550671i − 0.00223880i
\(606\) 0 0
\(607\) 10.0506 0.407940 0.203970 0.978977i \(-0.434615\pi\)
0.203970 + 0.978977i \(0.434615\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 35.0988i − 1.41994i
\(612\) 0 0
\(613\) − 2.57394i − 0.103961i −0.998648 0.0519803i \(-0.983447\pi\)
0.998648 0.0519803i \(-0.0165533\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 36.3495 1.46337 0.731687 0.681640i \(-0.238733\pi\)
0.731687 + 0.681640i \(0.238733\pi\)
\(618\) 0 0
\(619\) 16.6498i 0.669211i 0.942358 + 0.334606i \(0.108603\pi\)
−0.942358 + 0.334606i \(0.891397\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.85047 −0.194330
\(624\) 0 0
\(625\) −11.0685 −0.442740
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 7.60744i − 0.303328i
\(630\) 0 0
\(631\) −14.7898 −0.588773 −0.294387 0.955686i \(-0.595115\pi\)
−0.294387 + 0.955686i \(0.595115\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 11.6892i − 0.463871i
\(636\) 0 0
\(637\) 3.10840i 0.123159i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −32.8085 −1.29586 −0.647928 0.761701i \(-0.724364\pi\)
−0.647928 + 0.761701i \(0.724364\pi\)
\(642\) 0 0
\(643\) − 15.3479i − 0.605263i −0.953108 0.302631i \(-0.902135\pi\)
0.953108 0.302631i \(-0.0978651\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.19107 −0.243396 −0.121698 0.992567i \(-0.538834\pi\)
−0.121698 + 0.992567i \(0.538834\pi\)
\(648\) 0 0
\(649\) −12.9327 −0.507653
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 17.4848i − 0.684234i −0.939657 0.342117i \(-0.888856\pi\)
0.939657 0.342117i \(-0.111144\pi\)
\(654\) 0 0
\(655\) −24.7764 −0.968093
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.8987i 0.541416i 0.962661 + 0.270708i \(0.0872578\pi\)
−0.962661 + 0.270708i \(0.912742\pi\)
\(660\) 0 0
\(661\) − 1.69978i − 0.0661137i −0.999453 0.0330568i \(-0.989476\pi\)
0.999453 0.0330568i \(-0.0105242\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.6457 −0.645494
\(666\) 0 0
\(667\) 86.8869i 3.36428i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 31.2762 1.20741
\(672\) 0 0
\(673\) −22.7847 −0.878284 −0.439142 0.898418i \(-0.644718\pi\)
−0.439142 + 0.898418i \(0.644718\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.2963i 1.31811i 0.752093 + 0.659057i \(0.229044\pi\)
−0.752093 + 0.659057i \(0.770956\pi\)
\(678\) 0 0
\(679\) −1.63283 −0.0626624
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.5590i 0.404029i 0.979383 + 0.202014i \(0.0647487\pi\)
−0.979383 + 0.202014i \(0.935251\pi\)
\(684\) 0 0
\(685\) 14.6434i 0.559497i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −20.2561 −0.771695
\(690\) 0 0
\(691\) − 14.3879i − 0.547342i −0.961823 0.273671i \(-0.911762\pi\)
0.961823 0.273671i \(-0.0882380\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11.4789 0.435421
\(696\) 0 0
\(697\) 16.2314 0.614807
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.8913i 0.675746i 0.941192 + 0.337873i \(0.109708\pi\)
−0.941192 + 0.337873i \(0.890292\pi\)
\(702\) 0 0
\(703\) 26.0678 0.983165
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.1650i 0.495121i
\(708\) 0 0
\(709\) 22.0614i 0.828535i 0.910155 + 0.414267i \(0.135962\pi\)
−0.910155 + 0.414267i \(0.864038\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −68.9499 −2.58219
\(714\) 0 0
\(715\) − 35.7273i − 1.33613i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.8888 −0.592552 −0.296276 0.955102i \(-0.595745\pi\)
−0.296276 + 0.955102i \(0.595745\pi\)
\(720\) 0 0
\(721\) 15.6983 0.584634
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 69.0425i 2.56418i
\(726\) 0 0
\(727\) 26.2510 0.973595 0.486798 0.873515i \(-0.338165\pi\)
0.486798 + 0.873515i \(0.338165\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.87487i 0.365235i
\(732\) 0 0
\(733\) 23.1441i 0.854848i 0.904051 + 0.427424i \(0.140579\pi\)
−0.904051 + 0.427424i \(0.859421\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.01935 −0.111219
\(738\) 0 0
\(739\) − 11.2767i − 0.414819i −0.978254 0.207409i \(-0.933497\pi\)
0.978254 0.207409i \(-0.0665032\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.16580 −0.189515 −0.0947575 0.995500i \(-0.530208\pi\)
−0.0947575 + 0.995500i \(0.530208\pi\)
\(744\) 0 0
\(745\) 24.5330 0.898821
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 9.37257i − 0.342466i
\(750\) 0 0
\(751\) −35.7670 −1.30516 −0.652578 0.757722i \(-0.726313\pi\)
−0.652578 + 0.757722i \(0.726313\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 0.874212i − 0.0318158i
\(756\) 0 0
\(757\) 31.6174i 1.14916i 0.818450 + 0.574578i \(0.194834\pi\)
−0.818450 + 0.574578i \(0.805166\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.7856 −0.463477 −0.231738 0.972778i \(-0.574441\pi\)
−0.231738 + 0.972778i \(0.574441\pi\)
\(762\) 0 0
\(763\) 7.31490i 0.264817i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.1120 −0.437340
\(768\) 0 0
\(769\) 6.55622 0.236423 0.118212 0.992988i \(-0.462284\pi\)
0.118212 + 0.992988i \(0.462284\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 27.7250i − 0.997197i −0.866833 0.498599i \(-0.833848\pi\)
0.866833 0.498599i \(-0.166152\pi\)
\(774\) 0 0
\(775\) −54.7893 −1.96809
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 55.6187i 1.99275i
\(780\) 0 0
\(781\) 21.3185i 0.762837i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.7707 −0.384421
\(786\) 0 0
\(787\) − 25.9751i − 0.925911i −0.886382 0.462955i \(-0.846789\pi\)
0.886382 0.462955i \(-0.153211\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −10.9153 −0.388105
\(792\) 0 0
\(793\) 29.2915 1.04017
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 45.5049i − 1.61187i −0.592006 0.805934i \(-0.701664\pi\)
0.592006 0.805934i \(-0.298336\pi\)
\(798\) 0 0
\(799\) 15.8394 0.560358
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.18619i 0.183017i
\(804\) 0 0
\(805\) 30.4730i 1.07403i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.8396 0.732682 0.366341 0.930481i \(-0.380610\pi\)
0.366341 + 0.930481i \(0.380610\pi\)
\(810\) 0 0
\(811\) 52.1223i 1.83026i 0.403158 + 0.915130i \(0.367912\pi\)
−0.403158 + 0.915130i \(0.632088\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −64.2716 −2.25134
\(816\) 0 0
\(817\) −33.8374 −1.18382
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 13.9160i − 0.485673i −0.970067 0.242837i \(-0.921922\pi\)
0.970067 0.242837i \(-0.0780779\pi\)
\(822\) 0 0
\(823\) −1.31771 −0.0459323 −0.0229662 0.999736i \(-0.507311\pi\)
−0.0229662 + 0.999736i \(0.507311\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 21.9111i − 0.761925i −0.924591 0.380962i \(-0.875593\pi\)
0.924591 0.380962i \(-0.124407\pi\)
\(828\) 0 0
\(829\) − 33.0419i − 1.14759i −0.818999 0.573795i \(-0.805470\pi\)
0.818999 0.573795i \(-0.194530\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.40277 −0.0486029
\(834\) 0 0
\(835\) 49.5976i 1.71639i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.2548 −0.561177 −0.280589 0.959828i \(-0.590530\pi\)
−0.280589 + 0.959828i \(0.590530\pi\)
\(840\) 0 0
\(841\) −68.4952 −2.36190
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11.5589i 0.397638i
\(846\) 0 0
\(847\) 0.0159016 0.000546384 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 47.7217i − 1.63588i
\(852\) 0 0
\(853\) 43.5342i 1.49058i 0.666738 + 0.745292i \(0.267690\pi\)
−0.666738 + 0.745292i \(0.732310\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.968943 −0.0330985 −0.0165492 0.999863i \(-0.505268\pi\)
−0.0165492 + 0.999863i \(0.505268\pi\)
\(858\) 0 0
\(859\) − 11.9386i − 0.407341i −0.979040 0.203670i \(-0.934713\pi\)
0.979040 0.203670i \(-0.0652871\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.95989 0.270958 0.135479 0.990780i \(-0.456743\pi\)
0.135479 + 0.990780i \(0.456743\pi\)
\(864\) 0 0
\(865\) −4.85455 −0.165059
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 37.0467i 1.25672i
\(870\) 0 0
\(871\) −2.82775 −0.0958148
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.89962i 0.233250i
\(876\) 0 0
\(877\) − 26.3998i − 0.891459i −0.895168 0.445730i \(-0.852944\pi\)
0.895168 0.445730i \(-0.147056\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 32.4550 1.09344 0.546718 0.837317i \(-0.315877\pi\)
0.546718 + 0.837317i \(0.315877\pi\)
\(882\) 0 0
\(883\) 3.46140i 0.116486i 0.998302 + 0.0582428i \(0.0185497\pi\)
−0.998302 + 0.0582428i \(0.981450\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.9279 −0.568385 −0.284192 0.958767i \(-0.591725\pi\)
−0.284192 + 0.958767i \(0.591725\pi\)
\(888\) 0 0
\(889\) 3.37545 0.113209
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 54.2756i 1.81626i
\(894\) 0 0
\(895\) 12.0114 0.401497
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 77.3682i − 2.58037i
\(900\) 0 0
\(901\) − 9.14120i − 0.304537i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 67.1794 2.23312
\(906\) 0 0
\(907\) − 29.1236i − 0.967034i −0.875335 0.483517i \(-0.839359\pi\)
0.875335 0.483517i \(-0.160641\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.85198 −0.227016 −0.113508 0.993537i \(-0.536209\pi\)
−0.113508 + 0.993537i \(0.536209\pi\)
\(912\) 0 0
\(913\) −1.22884 −0.0406685
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 7.15460i − 0.236266i
\(918\) 0 0
\(919\) 37.3468 1.23196 0.615979 0.787763i \(-0.288761\pi\)
0.615979 + 0.787763i \(0.288761\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 19.9657i 0.657179i
\(924\) 0 0
\(925\) − 37.9209i − 1.24683i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −38.8771 −1.27552 −0.637758 0.770236i \(-0.720138\pi\)
−0.637758 + 0.770236i \(0.720138\pi\)
\(930\) 0 0
\(931\) − 4.80674i − 0.157534i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16.1231 0.527281
\(936\) 0 0
\(937\) −12.3583 −0.403727 −0.201863 0.979414i \(-0.564700\pi\)
−0.201863 + 0.979414i \(0.564700\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 40.1798i − 1.30982i −0.755705 0.654912i \(-0.772706\pi\)
0.755705 0.654912i \(-0.227294\pi\)
\(942\) 0 0
\(943\) 101.820 3.31572
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 14.7512i − 0.479351i −0.970853 0.239676i \(-0.922959\pi\)
0.970853 0.239676i \(-0.0770411\pi\)
\(948\) 0 0
\(949\) 4.85708i 0.157668i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36.2336 1.17372 0.586861 0.809687i \(-0.300363\pi\)
0.586861 + 0.809687i \(0.300363\pi\)
\(954\) 0 0
\(955\) − 81.2456i − 2.62905i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.22854 −0.136547
\(960\) 0 0
\(961\) 30.3962 0.980523
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 53.8060i − 1.73208i
\(966\) 0 0
\(967\) 32.6101 1.04867 0.524335 0.851512i \(-0.324314\pi\)
0.524335 + 0.851512i \(0.324314\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 37.3897i 1.19989i 0.800040 + 0.599947i \(0.204812\pi\)
−0.800040 + 0.599947i \(0.795188\pi\)
\(972\) 0 0
\(973\) 3.31473i 0.106265i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.76353 0.120406 0.0602029 0.998186i \(-0.480825\pi\)
0.0602029 + 0.998186i \(0.480825\pi\)
\(978\) 0 0
\(979\) 16.0988i 0.514521i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 36.8112 1.17410 0.587048 0.809552i \(-0.300290\pi\)
0.587048 + 0.809552i \(0.300290\pi\)
\(984\) 0 0
\(985\) 51.3485 1.63610
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 61.9454i 1.96975i
\(990\) 0 0
\(991\) 53.0994 1.68676 0.843379 0.537319i \(-0.180563\pi\)
0.843379 + 0.537319i \(0.180563\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 45.7445i − 1.45020i
\(996\) 0 0
\(997\) 29.5497i 0.935847i 0.883769 + 0.467923i \(0.154998\pi\)
−0.883769 + 0.467923i \(0.845002\pi\)
\(998\) 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.c.f.3025.24 24
3.2 odd 2 inner 6048.2.c.f.3025.2 24
4.3 odd 2 1512.2.c.g.757.24 yes 24
8.3 odd 2 1512.2.c.g.757.23 yes 24
8.5 even 2 inner 6048.2.c.f.3025.1 24
12.11 even 2 1512.2.c.g.757.1 24
24.5 odd 2 inner 6048.2.c.f.3025.23 24
24.11 even 2 1512.2.c.g.757.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.g.757.1 24 12.11 even 2
1512.2.c.g.757.2 yes 24 24.11 even 2
1512.2.c.g.757.23 yes 24 8.3 odd 2
1512.2.c.g.757.24 yes 24 4.3 odd 2
6048.2.c.f.3025.1 24 8.5 even 2 inner
6048.2.c.f.3025.2 24 3.2 odd 2 inner
6048.2.c.f.3025.23 24 24.5 odd 2 inner
6048.2.c.f.3025.24 24 1.1 even 1 trivial