Properties

Label 6048.2.c.e.3025.2
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 20
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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.2
Root \(0.885915 + 1.10234i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.e.3025.19

$q$-expansion

\(f(q)\) \(=\) \(q-3.50133i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-3.50133i q^{5} +1.00000 q^{7} -3.01331i q^{11} +3.90632i q^{13} -1.38839 q^{17} -4.79460i q^{19} -5.06853 q^{23} -7.25934 q^{25} -4.91070i q^{29} -1.13938 q^{31} -3.50133i q^{35} +9.45610i q^{37} -4.11364 q^{41} +1.51868i q^{43} -10.7641 q^{47} +1.00000 q^{49} -0.431457i q^{53} -10.5506 q^{55} +7.40936i q^{59} +12.9520i q^{61} +13.6773 q^{65} -3.36185i q^{67} -6.26892 q^{71} +10.0859 q^{73} -3.01331i q^{77} -12.9346 q^{79} -17.4139i q^{83} +4.86120i q^{85} +0.818403 q^{89} +3.90632i q^{91} -16.7875 q^{95} -11.4727 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{7} + O(q^{10}) \) \( 20q + 20q^{7} - 28q^{25} - 36q^{31} + 20q^{49} - 48q^{55} - 64q^{79} + 56q^{97} + O(q^{100}) \)

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
\(5\) − 3.50133i − 1.56584i −0.622120 0.782922i \(-0.713728\pi\)
0.622120 0.782922i \(-0.286272\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.01331i − 0.908547i −0.890862 0.454273i \(-0.849899\pi\)
0.890862 0.454273i \(-0.150101\pi\)
\(12\) 0 0
\(13\) 3.90632i 1.08342i 0.840566 + 0.541709i \(0.182223\pi\)
−0.840566 + 0.541709i \(0.817777\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.38839 −0.336733 −0.168367 0.985724i \(-0.553849\pi\)
−0.168367 + 0.985724i \(0.553849\pi\)
\(18\) 0 0
\(19\) − 4.79460i − 1.09996i −0.835179 0.549978i \(-0.814636\pi\)
0.835179 0.549978i \(-0.185364\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.06853 −1.05686 −0.528431 0.848976i \(-0.677219\pi\)
−0.528431 + 0.848976i \(0.677219\pi\)
\(24\) 0 0
\(25\) −7.25934 −1.45187
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.91070i − 0.911893i −0.890007 0.455947i \(-0.849301\pi\)
0.890007 0.455947i \(-0.150699\pi\)
\(30\) 0 0
\(31\) −1.13938 −0.204639 −0.102320 0.994752i \(-0.532626\pi\)
−0.102320 + 0.994752i \(0.532626\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.50133i − 0.591833i
\(36\) 0 0
\(37\) 9.45610i 1.55457i 0.629146 + 0.777287i \(0.283405\pi\)
−0.629146 + 0.777287i \(0.716595\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.11364 −0.642443 −0.321222 0.947004i \(-0.604093\pi\)
−0.321222 + 0.947004i \(0.604093\pi\)
\(42\) 0 0
\(43\) 1.51868i 0.231597i 0.993273 + 0.115799i \(0.0369427\pi\)
−0.993273 + 0.115799i \(0.963057\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.7641 −1.57010 −0.785051 0.619431i \(-0.787363\pi\)
−0.785051 + 0.619431i \(0.787363\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 0.431457i − 0.0592651i −0.999561 0.0296326i \(-0.990566\pi\)
0.999561 0.0296326i \(-0.00943372\pi\)
\(54\) 0 0
\(55\) −10.5506 −1.42264
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.40936i 0.964617i 0.876002 + 0.482308i \(0.160202\pi\)
−0.876002 + 0.482308i \(0.839798\pi\)
\(60\) 0 0
\(61\) 12.9520i 1.65834i 0.559000 + 0.829168i \(0.311185\pi\)
−0.559000 + 0.829168i \(0.688815\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.6773 1.69646
\(66\) 0 0
\(67\) − 3.36185i − 0.410715i −0.978687 0.205357i \(-0.934164\pi\)
0.978687 0.205357i \(-0.0658357\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.26892 −0.743984 −0.371992 0.928236i \(-0.621325\pi\)
−0.371992 + 0.928236i \(0.621325\pi\)
\(72\) 0 0
\(73\) 10.0859 1.18046 0.590230 0.807235i \(-0.299037\pi\)
0.590230 + 0.807235i \(0.299037\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.01331i − 0.343398i
\(78\) 0 0
\(79\) −12.9346 −1.45526 −0.727631 0.685969i \(-0.759378\pi\)
−0.727631 + 0.685969i \(0.759378\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 17.4139i − 1.91142i −0.294307 0.955711i \(-0.595089\pi\)
0.294307 0.955711i \(-0.404911\pi\)
\(84\) 0 0
\(85\) 4.86120i 0.527272i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.818403 0.0867506 0.0433753 0.999059i \(-0.486189\pi\)
0.0433753 + 0.999059i \(0.486189\pi\)
\(90\) 0 0
\(91\) 3.90632i 0.409494i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.7875 −1.72236
\(96\) 0 0
\(97\) −11.4727 −1.16488 −0.582441 0.812873i \(-0.697902\pi\)
−0.582441 + 0.812873i \(0.697902\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 5.38541i − 0.535868i −0.963437 0.267934i \(-0.913659\pi\)
0.963437 0.267934i \(-0.0863410\pi\)
\(102\) 0 0
\(103\) 4.82652 0.475571 0.237785 0.971318i \(-0.423579\pi\)
0.237785 + 0.971318i \(0.423579\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.92215i 0.475842i 0.971284 + 0.237921i \(0.0764660\pi\)
−0.971284 + 0.237921i \(0.923534\pi\)
\(108\) 0 0
\(109\) − 20.1342i − 1.92851i −0.264974 0.964255i \(-0.585363\pi\)
0.264974 0.964255i \(-0.414637\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.2438 1.24587 0.622937 0.782272i \(-0.285939\pi\)
0.622937 + 0.782272i \(0.285939\pi\)
\(114\) 0 0
\(115\) 17.7466i 1.65488i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.38839 −0.127273
\(120\) 0 0
\(121\) 1.91997 0.174543
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.91070i 0.707554i
\(126\) 0 0
\(127\) −10.2252 −0.907343 −0.453671 0.891169i \(-0.649886\pi\)
−0.453671 + 0.891169i \(0.649886\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1.54726i − 0.135185i −0.997713 0.0675924i \(-0.978468\pi\)
0.997713 0.0675924i \(-0.0215317\pi\)
\(132\) 0 0
\(133\) − 4.79460i − 0.415744i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.46379 −0.637674 −0.318837 0.947809i \(-0.603292\pi\)
−0.318837 + 0.947809i \(0.603292\pi\)
\(138\) 0 0
\(139\) − 16.6383i − 1.41125i −0.708588 0.705623i \(-0.750667\pi\)
0.708588 0.705623i \(-0.249333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.7709 0.984336
\(144\) 0 0
\(145\) −17.1940 −1.42788
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 21.3846i 1.75190i 0.482405 + 0.875948i \(0.339763\pi\)
−0.482405 + 0.875948i \(0.660237\pi\)
\(150\) 0 0
\(151\) −1.21341 −0.0987459 −0.0493730 0.998780i \(-0.515722\pi\)
−0.0493730 + 0.998780i \(0.515722\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.98936i 0.320433i
\(156\) 0 0
\(157\) − 8.15645i − 0.650955i −0.945550 0.325478i \(-0.894475\pi\)
0.945550 0.325478i \(-0.105525\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.06853 −0.399456
\(162\) 0 0
\(163\) 22.8278i 1.78801i 0.448055 + 0.894006i \(0.352117\pi\)
−0.448055 + 0.894006i \(0.647883\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.3575 1.34316 0.671581 0.740931i \(-0.265616\pi\)
0.671581 + 0.740931i \(0.265616\pi\)
\(168\) 0 0
\(169\) −2.25934 −0.173795
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 23.5784i 1.79263i 0.443415 + 0.896316i \(0.353767\pi\)
−0.443415 + 0.896316i \(0.646233\pi\)
\(174\) 0 0
\(175\) −7.25934 −0.548754
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 12.0965i − 0.904134i −0.891984 0.452067i \(-0.850687\pi\)
0.891984 0.452067i \(-0.149313\pi\)
\(180\) 0 0
\(181\) 3.70662i 0.275511i 0.990466 + 0.137755i \(0.0439888\pi\)
−0.990466 + 0.137755i \(0.956011\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 33.1090 2.43422
\(186\) 0 0
\(187\) 4.18364i 0.305938i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.2770 0.815977 0.407988 0.912987i \(-0.366230\pi\)
0.407988 + 0.912987i \(0.366230\pi\)
\(192\) 0 0
\(193\) −3.15188 −0.226877 −0.113439 0.993545i \(-0.536187\pi\)
−0.113439 + 0.993545i \(0.536187\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 11.4819i − 0.818052i −0.912523 0.409026i \(-0.865869\pi\)
0.912523 0.409026i \(-0.134131\pi\)
\(198\) 0 0
\(199\) 3.38405 0.239889 0.119944 0.992781i \(-0.461728\pi\)
0.119944 + 0.992781i \(0.461728\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 4.91070i − 0.344663i
\(204\) 0 0
\(205\) 14.4032i 1.00597i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −14.4476 −0.999362
\(210\) 0 0
\(211\) − 17.0774i − 1.17565i −0.808987 0.587827i \(-0.799984\pi\)
0.808987 0.587827i \(-0.200016\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.31742 0.362645
\(216\) 0 0
\(217\) −1.13938 −0.0773463
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 5.42348i − 0.364823i
\(222\) 0 0
\(223\) −24.1161 −1.61494 −0.807468 0.589912i \(-0.799163\pi\)
−0.807468 + 0.589912i \(0.799163\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.60050i 0.504463i 0.967667 + 0.252231i \(0.0811644\pi\)
−0.967667 + 0.252231i \(0.918836\pi\)
\(228\) 0 0
\(229\) 24.0017i 1.58608i 0.609171 + 0.793039i \(0.291502\pi\)
−0.609171 + 0.793039i \(0.708498\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.6276 0.696237 0.348118 0.937451i \(-0.386821\pi\)
0.348118 + 0.937451i \(0.386821\pi\)
\(234\) 0 0
\(235\) 37.6886i 2.45853i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −28.0645 −1.81534 −0.907671 0.419682i \(-0.862142\pi\)
−0.907671 + 0.419682i \(0.862142\pi\)
\(240\) 0 0
\(241\) −20.7612 −1.33734 −0.668672 0.743558i \(-0.733137\pi\)
−0.668672 + 0.743558i \(0.733137\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 3.50133i − 0.223692i
\(246\) 0 0
\(247\) 18.7292 1.19171
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.4352i 1.35298i 0.736454 + 0.676488i \(0.236499\pi\)
−0.736454 + 0.676488i \(0.763501\pi\)
\(252\) 0 0
\(253\) 15.2730i 0.960208i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.6807 0.853380 0.426690 0.904398i \(-0.359680\pi\)
0.426690 + 0.904398i \(0.359680\pi\)
\(258\) 0 0
\(259\) 9.45610i 0.587574i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −18.8063 −1.15965 −0.579823 0.814743i \(-0.696878\pi\)
−0.579823 + 0.814743i \(0.696878\pi\)
\(264\) 0 0
\(265\) −1.51067 −0.0927999
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.1654i 1.65630i 0.560504 + 0.828152i \(0.310607\pi\)
−0.560504 + 0.828152i \(0.689393\pi\)
\(270\) 0 0
\(271\) −0.618132 −0.0375488 −0.0187744 0.999824i \(-0.505976\pi\)
−0.0187744 + 0.999824i \(0.505976\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 21.8746i 1.31909i
\(276\) 0 0
\(277\) 16.4479i 0.988260i 0.869388 + 0.494130i \(0.164513\pi\)
−0.869388 + 0.494130i \(0.835487\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −26.6482 −1.58970 −0.794849 0.606807i \(-0.792450\pi\)
−0.794849 + 0.606807i \(0.792450\pi\)
\(282\) 0 0
\(283\) 5.96019i 0.354297i 0.984184 + 0.177148i \(0.0566872\pi\)
−0.984184 + 0.177148i \(0.943313\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.11364 −0.242821
\(288\) 0 0
\(289\) −15.0724 −0.886611
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 1.52262i − 0.0889522i −0.999010 0.0444761i \(-0.985838\pi\)
0.999010 0.0444761i \(-0.0141619\pi\)
\(294\) 0 0
\(295\) 25.9426 1.51044
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 19.7993i − 1.14502i
\(300\) 0 0
\(301\) 1.51868i 0.0875355i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 45.3493 2.59669
\(306\) 0 0
\(307\) 5.29191i 0.302025i 0.988532 + 0.151013i \(0.0482534\pi\)
−0.988532 + 0.151013i \(0.951747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.95423 0.394338 0.197169 0.980370i \(-0.436825\pi\)
0.197169 + 0.980370i \(0.436825\pi\)
\(312\) 0 0
\(313\) −11.7834 −0.666039 −0.333020 0.942920i \(-0.608068\pi\)
−0.333020 + 0.942920i \(0.608068\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.74196i 0.0978384i 0.998803 + 0.0489192i \(0.0155777\pi\)
−0.998803 + 0.0489192i \(0.984422\pi\)
\(318\) 0 0
\(319\) −14.7974 −0.828498
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.65676i 0.370392i
\(324\) 0 0
\(325\) − 28.3573i − 1.57298i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.7641 −0.593443
\(330\) 0 0
\(331\) − 22.6068i − 1.24258i −0.783579 0.621292i \(-0.786608\pi\)
0.783579 0.621292i \(-0.213392\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.7709 −0.643116
\(336\) 0 0
\(337\) 24.7468 1.34804 0.674021 0.738712i \(-0.264566\pi\)
0.674021 + 0.738712i \(0.264566\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.43331i 0.185924i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 3.17968i − 0.170694i −0.996351 0.0853471i \(-0.972800\pi\)
0.996351 0.0853471i \(-0.0271999\pi\)
\(348\) 0 0
\(349\) 12.0444i 0.644722i 0.946617 + 0.322361i \(0.104477\pi\)
−0.946617 + 0.322361i \(0.895523\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.3700 −1.50998 −0.754991 0.655736i \(-0.772359\pi\)
−0.754991 + 0.655736i \(0.772359\pi\)
\(354\) 0 0
\(355\) 21.9496i 1.16496i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.2701 1.38648 0.693241 0.720706i \(-0.256182\pi\)
0.693241 + 0.720706i \(0.256182\pi\)
\(360\) 0 0
\(361\) −3.98817 −0.209904
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 35.3139i − 1.84842i
\(366\) 0 0
\(367\) −15.6261 −0.815678 −0.407839 0.913054i \(-0.633717\pi\)
−0.407839 + 0.913054i \(0.633717\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 0.431457i − 0.0224001i
\(372\) 0 0
\(373\) − 3.63905i − 0.188423i −0.995552 0.0942113i \(-0.969967\pi\)
0.995552 0.0942113i \(-0.0300329\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 19.1827 0.987962
\(378\) 0 0
\(379\) − 4.66150i − 0.239445i −0.992807 0.119723i \(-0.961799\pi\)
0.992807 0.119723i \(-0.0382005\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.9040 0.608268 0.304134 0.952629i \(-0.401633\pi\)
0.304134 + 0.952629i \(0.401633\pi\)
\(384\) 0 0
\(385\) −10.5506 −0.537708
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 35.6321i − 1.80662i −0.428987 0.903311i \(-0.641129\pi\)
0.428987 0.903311i \(-0.358871\pi\)
\(390\) 0 0
\(391\) 7.03708 0.355880
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 45.2885i 2.27871i
\(396\) 0 0
\(397\) − 10.9738i − 0.550760i −0.961335 0.275380i \(-0.911196\pi\)
0.961335 0.275380i \(-0.0888037\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.65382 0.332276 0.166138 0.986103i \(-0.446870\pi\)
0.166138 + 0.986103i \(0.446870\pi\)
\(402\) 0 0
\(403\) − 4.45079i − 0.221710i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.4942 1.41240
\(408\) 0 0
\(409\) −33.8304 −1.67281 −0.836404 0.548114i \(-0.815346\pi\)
−0.836404 + 0.548114i \(0.815346\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.40936i 0.364591i
\(414\) 0 0
\(415\) −60.9718 −2.99299
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 10.6763i − 0.521571i −0.965397 0.260786i \(-0.916018\pi\)
0.965397 0.260786i \(-0.0839816\pi\)
\(420\) 0 0
\(421\) − 15.4063i − 0.750855i −0.926852 0.375427i \(-0.877496\pi\)
0.926852 0.375427i \(-0.122504\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.0788 0.488892
\(426\) 0 0
\(427\) 12.9520i 0.626792i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.7179 −1.23879 −0.619393 0.785081i \(-0.712621\pi\)
−0.619393 + 0.785081i \(0.712621\pi\)
\(432\) 0 0
\(433\) −7.26219 −0.348998 −0.174499 0.984657i \(-0.555831\pi\)
−0.174499 + 0.984657i \(0.555831\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 24.3016i 1.16250i
\(438\) 0 0
\(439\) −28.7252 −1.37098 −0.685488 0.728084i \(-0.740411\pi\)
−0.685488 + 0.728084i \(0.740411\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 5.70732i − 0.271163i −0.990766 0.135582i \(-0.956710\pi\)
0.990766 0.135582i \(-0.0432903\pi\)
\(444\) 0 0
\(445\) − 2.86550i − 0.135838i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −33.9514 −1.60227 −0.801134 0.598485i \(-0.795770\pi\)
−0.801134 + 0.598485i \(0.795770\pi\)
\(450\) 0 0
\(451\) 12.3957i 0.583690i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13.6773 0.641203
\(456\) 0 0
\(457\) 35.4661 1.65903 0.829517 0.558482i \(-0.188616\pi\)
0.829517 + 0.558482i \(0.188616\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 26.3291i − 1.22627i −0.789979 0.613134i \(-0.789908\pi\)
0.789979 0.613134i \(-0.210092\pi\)
\(462\) 0 0
\(463\) 5.45714 0.253615 0.126807 0.991927i \(-0.459527\pi\)
0.126807 + 0.991927i \(0.459527\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.77349i 0.267165i 0.991038 + 0.133583i \(0.0426482\pi\)
−0.991038 + 0.133583i \(0.957352\pi\)
\(468\) 0 0
\(469\) − 3.36185i − 0.155236i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.57626 0.210417
\(474\) 0 0
\(475\) 34.8056i 1.59699i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 31.3044 1.43033 0.715167 0.698954i \(-0.246351\pi\)
0.715167 + 0.698954i \(0.246351\pi\)
\(480\) 0 0
\(481\) −36.9386 −1.68425
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 40.1699i 1.82402i
\(486\) 0 0
\(487\) 25.7136 1.16519 0.582597 0.812761i \(-0.302037\pi\)
0.582597 + 0.812761i \(0.302037\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 12.7216i − 0.574116i −0.957913 0.287058i \(-0.907323\pi\)
0.957913 0.287058i \(-0.0926773\pi\)
\(492\) 0 0
\(493\) 6.81794i 0.307065i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.26892 −0.281199
\(498\) 0 0
\(499\) − 37.9382i − 1.69835i −0.528115 0.849173i \(-0.677101\pi\)
0.528115 0.849173i \(-0.322899\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.9704 −0.756674 −0.378337 0.925668i \(-0.623504\pi\)
−0.378337 + 0.925668i \(0.623504\pi\)
\(504\) 0 0
\(505\) −18.8561 −0.839086
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.6789i 0.517656i 0.965923 + 0.258828i \(0.0833363\pi\)
−0.965923 + 0.258828i \(0.916664\pi\)
\(510\) 0 0
\(511\) 10.0859 0.446172
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 16.8992i − 0.744670i
\(516\) 0 0
\(517\) 32.4355i 1.42651i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.82516 −0.211394 −0.105697 0.994398i \(-0.533707\pi\)
−0.105697 + 0.994398i \(0.533707\pi\)
\(522\) 0 0
\(523\) 24.7958i 1.08424i 0.840300 + 0.542121i \(0.182379\pi\)
−0.840300 + 0.542121i \(0.817621\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.58190 0.0689088
\(528\) 0 0
\(529\) 2.68998 0.116956
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 16.0692i − 0.696035i
\(534\) 0 0
\(535\) 17.2341 0.745095
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 3.01331i − 0.129792i
\(540\) 0 0
\(541\) 14.0510i 0.604100i 0.953292 + 0.302050i \(0.0976709\pi\)
−0.953292 + 0.302050i \(0.902329\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −70.4967 −3.01975
\(546\) 0 0
\(547\) − 8.56879i − 0.366375i −0.983078 0.183188i \(-0.941358\pi\)
0.983078 0.183188i \(-0.0586416\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −23.5448 −1.00304
\(552\) 0 0
\(553\) −12.9346 −0.550037
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 3.47669i − 0.147312i −0.997284 0.0736560i \(-0.976533\pi\)
0.997284 0.0736560i \(-0.0234667\pi\)
\(558\) 0 0
\(559\) −5.93247 −0.250917
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.82325i 0.287566i 0.989609 + 0.143783i \(0.0459267\pi\)
−0.989609 + 0.143783i \(0.954073\pi\)
\(564\) 0 0
\(565\) − 46.3711i − 1.95085i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −25.1322 −1.05360 −0.526799 0.849990i \(-0.676608\pi\)
−0.526799 + 0.849990i \(0.676608\pi\)
\(570\) 0 0
\(571\) 14.1758i 0.593238i 0.954996 + 0.296619i \(0.0958591\pi\)
−0.954996 + 0.296619i \(0.904141\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 36.7942 1.53442
\(576\) 0 0
\(577\) 18.3021 0.761927 0.380963 0.924590i \(-0.375592\pi\)
0.380963 + 0.924590i \(0.375592\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 17.4139i − 0.722450i
\(582\) 0 0
\(583\) −1.30011 −0.0538451
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 6.70640i − 0.276803i −0.990376 0.138401i \(-0.955804\pi\)
0.990376 0.138401i \(-0.0441964\pi\)
\(588\) 0 0
\(589\) 5.46288i 0.225094i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.9084 0.735411 0.367706 0.929942i \(-0.380143\pi\)
0.367706 + 0.929942i \(0.380143\pi\)
\(594\) 0 0
\(595\) 4.86120i 0.199290i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −23.9838 −0.979953 −0.489976 0.871736i \(-0.662995\pi\)
−0.489976 + 0.871736i \(0.662995\pi\)
\(600\) 0 0
\(601\) −17.2548 −0.703840 −0.351920 0.936030i \(-0.614471\pi\)
−0.351920 + 0.936030i \(0.614471\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 6.72246i − 0.273307i
\(606\) 0 0
\(607\) −38.0189 −1.54314 −0.771569 0.636146i \(-0.780528\pi\)
−0.771569 + 0.636146i \(0.780528\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 42.0479i − 1.70108i
\(612\) 0 0
\(613\) − 33.1990i − 1.34089i −0.741957 0.670447i \(-0.766102\pi\)
0.741957 0.670447i \(-0.233898\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.827310 −0.0333062 −0.0166531 0.999861i \(-0.505301\pi\)
−0.0166531 + 0.999861i \(0.505301\pi\)
\(618\) 0 0
\(619\) − 42.3695i − 1.70297i −0.524377 0.851486i \(-0.675702\pi\)
0.524377 0.851486i \(-0.324298\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0.818403 0.0327886
\(624\) 0 0
\(625\) −8.59870 −0.343948
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 13.1287i − 0.523477i
\(630\) 0 0
\(631\) 18.5333 0.737801 0.368901 0.929469i \(-0.379734\pi\)
0.368901 + 0.929469i \(0.379734\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 35.8020i 1.42076i
\(636\) 0 0
\(637\) 3.90632i 0.154774i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.3013 −0.525371 −0.262686 0.964881i \(-0.584608\pi\)
−0.262686 + 0.964881i \(0.584608\pi\)
\(642\) 0 0
\(643\) 12.1885i 0.480668i 0.970690 + 0.240334i \(0.0772570\pi\)
−0.970690 + 0.240334i \(0.922743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.4813 0.726576 0.363288 0.931677i \(-0.381654\pi\)
0.363288 + 0.931677i \(0.381654\pi\)
\(648\) 0 0
\(649\) 22.3267 0.876399
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 25.0787i − 0.981405i −0.871327 0.490703i \(-0.836740\pi\)
0.871327 0.490703i \(-0.163260\pi\)
\(654\) 0 0
\(655\) −5.41748 −0.211678
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 40.3331i − 1.57115i −0.618764 0.785577i \(-0.712366\pi\)
0.618764 0.785577i \(-0.287634\pi\)
\(660\) 0 0
\(661\) 21.1566i 0.822896i 0.911433 + 0.411448i \(0.134977\pi\)
−0.911433 + 0.411448i \(0.865023\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.7875 −0.650991
\(666\) 0 0
\(667\) 24.8900i 0.963745i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 39.0284 1.50667
\(672\) 0 0
\(673\) 45.8540 1.76754 0.883772 0.467918i \(-0.154996\pi\)
0.883772 + 0.467918i \(0.154996\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 41.0495i − 1.57766i −0.614612 0.788830i \(-0.710687\pi\)
0.614612 0.788830i \(-0.289313\pi\)
\(678\) 0 0
\(679\) −11.4727 −0.440284
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.87089i 0.301171i 0.988597 + 0.150586i \(0.0481159\pi\)
−0.988597 + 0.150586i \(0.951884\pi\)
\(684\) 0 0
\(685\) 26.1332i 0.998499i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.68541 0.0642089
\(690\) 0 0
\(691\) 33.4478i 1.27241i 0.771519 + 0.636207i \(0.219497\pi\)
−0.771519 + 0.636207i \(0.780503\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −58.2564 −2.20979
\(696\) 0 0
\(697\) 5.71133 0.216332
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19.9716i 0.754317i 0.926149 + 0.377158i \(0.123099\pi\)
−0.926149 + 0.377158i \(0.876901\pi\)
\(702\) 0 0
\(703\) 45.3382 1.70996
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 5.38541i − 0.202539i
\(708\) 0 0
\(709\) 38.1379i 1.43230i 0.697947 + 0.716149i \(0.254097\pi\)
−0.697947 + 0.716149i \(0.745903\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.77499 0.216275
\(714\) 0 0
\(715\) − 41.2140i − 1.54132i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.8150 1.26109 0.630543 0.776155i \(-0.282832\pi\)
0.630543 + 0.776155i \(0.282832\pi\)
\(720\) 0 0
\(721\) 4.82652 0.179749
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 35.6484i 1.32395i
\(726\) 0 0
\(727\) 38.9720 1.44539 0.722696 0.691166i \(-0.242903\pi\)
0.722696 + 0.691166i \(0.242903\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 2.10852i − 0.0779865i
\(732\) 0 0
\(733\) 13.2978i 0.491164i 0.969376 + 0.245582i \(0.0789790\pi\)
−0.969376 + 0.245582i \(0.921021\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.1303 −0.373154
\(738\) 0 0
\(739\) − 37.4030i − 1.37589i −0.725762 0.687946i \(-0.758513\pi\)
0.725762 0.687946i \(-0.241487\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.1138 −1.14145 −0.570727 0.821140i \(-0.693338\pi\)
−0.570727 + 0.821140i \(0.693338\pi\)
\(744\) 0 0
\(745\) 74.8747 2.74320
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.92215i 0.179851i
\(750\) 0 0
\(751\) −18.6412 −0.680227 −0.340114 0.940384i \(-0.610466\pi\)
−0.340114 + 0.940384i \(0.610466\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.24855i 0.154621i
\(756\) 0 0
\(757\) − 29.8972i − 1.08663i −0.839528 0.543317i \(-0.817168\pi\)
0.839528 0.543317i \(-0.182832\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −42.6967 −1.54776 −0.773878 0.633335i \(-0.781685\pi\)
−0.773878 + 0.633335i \(0.781685\pi\)
\(762\) 0 0
\(763\) − 20.1342i − 0.728909i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28.9433 −1.04508
\(768\) 0 0
\(769\) 40.5127 1.46092 0.730462 0.682953i \(-0.239305\pi\)
0.730462 + 0.682953i \(0.239305\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 14.6650i − 0.527465i −0.964596 0.263732i \(-0.915046\pi\)
0.964596 0.263732i \(-0.0849536\pi\)
\(774\) 0 0
\(775\) 8.27116 0.297109
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19.7233i 0.706660i
\(780\) 0 0
\(781\) 18.8902i 0.675944i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −28.5584 −1.01929
\(786\) 0 0
\(787\) 17.3453i 0.618292i 0.951015 + 0.309146i \(0.100043\pi\)
−0.951015 + 0.309146i \(0.899957\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13.2438 0.470896
\(792\) 0 0
\(793\) −50.5947 −1.79667
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 21.9239i − 0.776585i −0.921536 0.388292i \(-0.873065\pi\)
0.921536 0.388292i \(-0.126935\pi\)
\(798\) 0 0
\(799\) 14.9447 0.528705
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 30.3918i − 1.07250i
\(804\) 0 0
\(805\) 17.7466i 0.625486i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.2482 0.676731 0.338366 0.941015i \(-0.390126\pi\)
0.338366 + 0.941015i \(0.390126\pi\)
\(810\) 0 0
\(811\) − 12.4354i − 0.436665i −0.975874 0.218333i \(-0.929938\pi\)
0.975874 0.218333i \(-0.0700617\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 79.9278 2.79975
\(816\) 0 0
\(817\) 7.28148 0.254747
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 55.1514i − 1.92480i −0.271640 0.962399i \(-0.587566\pi\)
0.271640 0.962399i \(-0.412434\pi\)
\(822\) 0 0
\(823\) −15.6639 −0.546009 −0.273004 0.962013i \(-0.588017\pi\)
−0.273004 + 0.962013i \(0.588017\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.1178i 0.560470i 0.959931 + 0.280235i \(0.0904124\pi\)
−0.959931 + 0.280235i \(0.909588\pi\)
\(828\) 0 0
\(829\) − 42.5904i − 1.47923i −0.673032 0.739613i \(-0.735008\pi\)
0.673032 0.739613i \(-0.264992\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.38839 −0.0481047
\(834\) 0 0
\(835\) − 60.7743i − 2.10318i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 21.1741 0.731011 0.365505 0.930809i \(-0.380896\pi\)
0.365505 + 0.930809i \(0.380896\pi\)
\(840\) 0 0
\(841\) 4.88507 0.168451
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.91070i 0.272136i
\(846\) 0 0
\(847\) 1.91997 0.0659710
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 47.9285i − 1.64297i
\(852\) 0 0
\(853\) 13.2958i 0.455240i 0.973750 + 0.227620i \(0.0730944\pi\)
−0.973750 + 0.227620i \(0.926906\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26.1940 −0.894770 −0.447385 0.894341i \(-0.647645\pi\)
−0.447385 + 0.894341i \(0.647645\pi\)
\(858\) 0 0
\(859\) 5.90287i 0.201403i 0.994917 + 0.100702i \(0.0321088\pi\)
−0.994917 + 0.100702i \(0.967891\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.1022 0.411964 0.205982 0.978556i \(-0.433961\pi\)
0.205982 + 0.978556i \(0.433961\pi\)
\(864\) 0 0
\(865\) 82.5558 2.80698
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 38.9761i 1.32217i
\(870\) 0 0
\(871\) 13.1325 0.444976
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 7.91070i 0.267430i
\(876\) 0 0
\(877\) 27.9450i 0.943634i 0.881697 + 0.471817i \(0.156402\pi\)
−0.881697 + 0.471817i \(0.843598\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −26.6503 −0.897872 −0.448936 0.893564i \(-0.648197\pi\)
−0.448936 + 0.893564i \(0.648197\pi\)
\(882\) 0 0
\(883\) − 4.47024i − 0.150435i −0.997167 0.0752177i \(-0.976035\pi\)
0.997167 0.0752177i \(-0.0239652\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.53323 −0.286518 −0.143259 0.989685i \(-0.545758\pi\)
−0.143259 + 0.989685i \(0.545758\pi\)
\(888\) 0 0
\(889\) −10.2252 −0.342943
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 51.6094i 1.72704i
\(894\) 0 0
\(895\) −42.3539 −1.41573
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.59516i 0.186609i
\(900\) 0 0
\(901\) 0.599028i 0.0199565i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.9781 0.431407
\(906\) 0 0
\(907\) − 30.0969i − 0.999353i −0.866212 0.499676i \(-0.833452\pi\)
0.866212 0.499676i \(-0.166548\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.93001 −0.196470 −0.0982350 0.995163i \(-0.531320\pi\)
−0.0982350 + 0.995163i \(0.531320\pi\)
\(912\) 0 0
\(913\) −52.4734 −1.73662
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.54726i − 0.0510951i
\(918\) 0 0
\(919\) −39.3264 −1.29726 −0.648630 0.761104i \(-0.724658\pi\)
−0.648630 + 0.761104i \(0.724658\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 24.4884i − 0.806046i
\(924\) 0 0
\(925\) − 68.6450i − 2.25704i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.9339 1.17895 0.589476 0.807786i \(-0.299334\pi\)
0.589476 + 0.807786i \(0.299334\pi\)
\(930\) 0 0
\(931\) − 4.79460i − 0.157137i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14.6483 0.479051
\(936\) 0 0
\(937\) −24.6365 −0.804838 −0.402419 0.915456i \(-0.631830\pi\)
−0.402419 + 0.915456i \(0.631830\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23.2069i 0.756522i 0.925699 + 0.378261i \(0.123478\pi\)
−0.925699 + 0.378261i \(0.876522\pi\)
\(942\) 0 0
\(943\) 20.8501 0.678973
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 19.6763i − 0.639394i −0.947520 0.319697i \(-0.896419\pi\)
0.947520 0.319697i \(-0.103581\pi\)
\(948\) 0 0
\(949\) 39.3986i 1.27893i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −9.96974 −0.322952 −0.161476 0.986877i \(-0.551625\pi\)
−0.161476 + 0.986877i \(0.551625\pi\)
\(954\) 0 0
\(955\) − 39.4846i − 1.27769i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7.46379 −0.241018
\(960\) 0 0
\(961\) −29.7018 −0.958123
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.0358i 0.355254i
\(966\) 0 0
\(967\) 7.20443 0.231679 0.115839 0.993268i \(-0.463044\pi\)
0.115839 + 0.993268i \(0.463044\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 32.0495i 1.02852i 0.857635 + 0.514258i \(0.171933\pi\)
−0.857635 + 0.514258i \(0.828067\pi\)
\(972\) 0 0
\(973\) − 16.6383i − 0.533401i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.17655 −0.261591 −0.130796 0.991409i \(-0.541753\pi\)
−0.130796 + 0.991409i \(0.541753\pi\)
\(978\) 0 0
\(979\) − 2.46610i − 0.0788169i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −32.2990 −1.03018 −0.515089 0.857137i \(-0.672241\pi\)
−0.515089 + 0.857137i \(0.672241\pi\)
\(984\) 0 0
\(985\) −40.2020 −1.28094
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 7.69749i − 0.244766i
\(990\) 0 0
\(991\) 43.7283 1.38907 0.694537 0.719457i \(-0.255609\pi\)
0.694537 + 0.719457i \(0.255609\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 11.8487i − 0.375628i
\(996\) 0 0
\(997\) − 4.64199i − 0.147013i −0.997295 0.0735066i \(-0.976581\pi\)
0.997295 0.0735066i \(-0.0234190\pi\)
\(998\) 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.e.3025.2 20
3.2 odd 2 inner 6048.2.c.e.3025.20 20
4.3 odd 2 1512.2.c.e.757.6 yes 20
8.3 odd 2 1512.2.c.e.757.5 20
8.5 even 2 inner 6048.2.c.e.3025.19 20
12.11 even 2 1512.2.c.e.757.15 yes 20
24.5 odd 2 inner 6048.2.c.e.3025.1 20
24.11 even 2 1512.2.c.e.757.16 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.e.757.5 20 8.3 odd 2
1512.2.c.e.757.6 yes 20 4.3 odd 2
1512.2.c.e.757.15 yes 20 12.11 even 2
1512.2.c.e.757.16 yes 20 24.11 even 2
6048.2.c.e.3025.1 20 24.5 odd 2 inner
6048.2.c.e.3025.2 20 1.1 even 1 trivial
6048.2.c.e.3025.19 20 8.5 even 2 inner
6048.2.c.e.3025.20 20 3.2 odd 2 inner