Properties

Label 6048.2.a.bm.1.4
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.22896.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.65924\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.83940 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+2.83940 q^{5} -1.00000 q^{7} +2.47909 q^{11} -5.31849 q^{13} -5.78284 q^{17} +4.78284 q^{19} -3.90161 q^{23} +3.06221 q^{25} +7.10133 q^{29} -10.1013 q^{31} -2.83940 q^{35} -6.31849 q^{37} -1.41688 q^{41} -3.31849 q^{43} +5.42253 q^{47} +1.00000 q^{49} -4.00000 q^{53} +7.03912 q^{55} +5.42253 q^{59} -4.72063 q^{61} -15.1013 q^{65} -1.52656 q^{67} -11.7976 q^{71} -10.8842 q^{73} -2.47909 q^{77} -8.03912 q^{79} +14.8933 q^{83} -16.4198 q^{85} -5.41688 q^{89} +5.31849 q^{91} +13.5804 q^{95} -2.84505 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q - 2q^{5} - 4q^{7} + 2q^{11} - 8q^{17} + 4q^{19} + 2q^{23} + 8q^{25} - 8q^{29} - 4q^{31} + 2q^{35} - 4q^{37} - 2q^{41} + 8q^{43} + 12q^{47} + 4q^{49} - 16q^{53} - 4q^{55} + 12q^{59} - 8q^{61} - 24q^{65} - 8q^{67} - 18q^{71} + 8q^{73} - 2q^{77} - 8q^{85} - 18q^{89} + 10q^{95} + 8q^{97} + O(q^{100}) \)

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
\(5\) 2.83940 1.26982 0.634910 0.772586i \(-0.281037\pi\)
0.634910 + 0.772586i \(0.281037\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.47909 0.747472 0.373736 0.927535i \(-0.378077\pi\)
0.373736 + 0.927535i \(0.378077\pi\)
\(12\) 0 0
\(13\) −5.31849 −1.47508 −0.737542 0.675302i \(-0.764013\pi\)
−0.737542 + 0.675302i \(0.764013\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.78284 −1.40255 −0.701273 0.712893i \(-0.747384\pi\)
−0.701273 + 0.712893i \(0.747384\pi\)
\(18\) 0 0
\(19\) 4.78284 1.09726 0.548630 0.836065i \(-0.315150\pi\)
0.548630 + 0.836065i \(0.315150\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.90161 −0.813542 −0.406771 0.913530i \(-0.633345\pi\)
−0.406771 + 0.913530i \(0.633345\pi\)
\(24\) 0 0
\(25\) 3.06221 0.612442
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.10133 1.31868 0.659342 0.751843i \(-0.270835\pi\)
0.659342 + 0.751843i \(0.270835\pi\)
\(30\) 0 0
\(31\) −10.1013 −1.81425 −0.907126 0.420858i \(-0.861729\pi\)
−0.907126 + 0.420858i \(0.861729\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.83940 −0.479947
\(36\) 0 0
\(37\) −6.31849 −1.03875 −0.519377 0.854545i \(-0.673836\pi\)
−0.519377 + 0.854545i \(0.673836\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.41688 −0.221279 −0.110640 0.993861i \(-0.535290\pi\)
−0.110640 + 0.993861i \(0.535290\pi\)
\(42\) 0 0
\(43\) −3.31849 −0.506065 −0.253032 0.967458i \(-0.581428\pi\)
−0.253032 + 0.967458i \(0.581428\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.42253 0.790957 0.395478 0.918475i \(-0.370579\pi\)
0.395478 + 0.918475i \(0.370579\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 7.03912 0.949155
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.42253 0.705953 0.352976 0.935632i \(-0.385170\pi\)
0.352976 + 0.935632i \(0.385170\pi\)
\(60\) 0 0
\(61\) −4.72063 −0.604415 −0.302208 0.953242i \(-0.597724\pi\)
−0.302208 + 0.953242i \(0.597724\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −15.1013 −1.87309
\(66\) 0 0
\(67\) −1.52656 −0.186499 −0.0932496 0.995643i \(-0.529725\pi\)
−0.0932496 + 0.995643i \(0.529725\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.7976 −1.40011 −0.700057 0.714087i \(-0.746842\pi\)
−0.700057 + 0.714087i \(0.746842\pi\)
\(72\) 0 0
\(73\) −10.8842 −1.27390 −0.636948 0.770907i \(-0.719803\pi\)
−0.636948 + 0.770907i \(0.719803\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.47909 −0.282518
\(78\) 0 0
\(79\) −8.03912 −0.904472 −0.452236 0.891898i \(-0.649374\pi\)
−0.452236 + 0.891898i \(0.649374\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.8933 1.63475 0.817374 0.576108i \(-0.195429\pi\)
0.817374 + 0.576108i \(0.195429\pi\)
\(84\) 0 0
\(85\) −16.4198 −1.78098
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.41688 −0.574188 −0.287094 0.957902i \(-0.592689\pi\)
−0.287094 + 0.957902i \(0.592689\pi\)
\(90\) 0 0
\(91\) 5.31849 0.557529
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.5804 1.39332
\(96\) 0 0
\(97\) −2.84505 −0.288871 −0.144436 0.989514i \(-0.546137\pi\)
−0.144436 + 0.989514i \(0.546137\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −13.7828 −1.37144 −0.685722 0.727863i \(-0.740513\pi\)
−0.685722 + 0.727863i \(0.740513\pi\)
\(102\) 0 0
\(103\) 14.9464 1.47271 0.736355 0.676595i \(-0.236545\pi\)
0.736355 + 0.676595i \(0.236545\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.17533 −0.306971 −0.153485 0.988151i \(-0.549050\pi\)
−0.153485 + 0.988151i \(0.549050\pi\)
\(108\) 0 0
\(109\) 1.38070 0.132247 0.0661234 0.997811i \(-0.478937\pi\)
0.0661234 + 0.997811i \(0.478937\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.95817 −0.842714 −0.421357 0.906895i \(-0.638446\pi\)
−0.421357 + 0.906895i \(0.638446\pi\)
\(114\) 0 0
\(115\) −11.0782 −1.03305
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.78284 0.530112
\(120\) 0 0
\(121\) −4.85413 −0.441285
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.50217 −0.492129
\(126\) 0 0
\(127\) 20.6761 1.83471 0.917354 0.398073i \(-0.130321\pi\)
0.917354 + 0.398073i \(0.130321\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.78014 −0.767124 −0.383562 0.923515i \(-0.625303\pi\)
−0.383562 + 0.923515i \(0.625303\pi\)
\(132\) 0 0
\(133\) −4.78284 −0.414725
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.1726 −1.55259 −0.776296 0.630369i \(-0.782904\pi\)
−0.776296 + 0.630369i \(0.782904\pi\)
\(138\) 0 0
\(139\) 17.4820 1.48281 0.741403 0.671060i \(-0.234161\pi\)
0.741403 + 0.671060i \(0.234161\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −13.1850 −1.10258
\(144\) 0 0
\(145\) 20.1635 1.67449
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.88688 −0.809965 −0.404982 0.914324i \(-0.632722\pi\)
−0.404982 + 0.914324i \(0.632722\pi\)
\(150\) 0 0
\(151\) −6.72063 −0.546917 −0.273459 0.961884i \(-0.588168\pi\)
−0.273459 + 0.961884i \(0.588168\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −28.6817 −2.30377
\(156\) 0 0
\(157\) 19.5212 1.55796 0.778979 0.627050i \(-0.215738\pi\)
0.778979 + 0.627050i \(0.215738\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.90161 0.307490
\(162\) 0 0
\(163\) −10.2472 −0.802622 −0.401311 0.915942i \(-0.631445\pi\)
−0.401311 + 0.915942i \(0.631445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.9582 1.00273 0.501367 0.865235i \(-0.332831\pi\)
0.501367 + 0.865235i \(0.332831\pi\)
\(168\) 0 0
\(169\) 15.2863 1.17587
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.9434 −0.832014 −0.416007 0.909361i \(-0.636571\pi\)
−0.416007 + 0.909361i \(0.636571\pi\)
\(174\) 0 0
\(175\) −3.06221 −0.231481
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.10404 −0.605724 −0.302862 0.953034i \(-0.597942\pi\)
−0.302862 + 0.953034i \(0.597942\pi\)
\(180\) 0 0
\(181\) 6.28632 0.467258 0.233629 0.972326i \(-0.424940\pi\)
0.233629 + 0.972326i \(0.424940\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −17.9407 −1.31903
\(186\) 0 0
\(187\) −14.3362 −1.04836
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.1461 −1.24065 −0.620324 0.784346i \(-0.712999\pi\)
−0.620324 + 0.784346i \(0.712999\pi\)
\(192\) 0 0
\(193\) 9.79193 0.704838 0.352419 0.935842i \(-0.385359\pi\)
0.352419 + 0.935842i \(0.385359\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.8338 −0.771873 −0.385937 0.922525i \(-0.626122\pi\)
−0.385937 + 0.922525i \(0.626122\pi\)
\(198\) 0 0
\(199\) −17.6761 −1.25303 −0.626513 0.779411i \(-0.715518\pi\)
−0.626513 + 0.779411i \(0.715518\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.10133 −0.498416
\(204\) 0 0
\(205\) −4.02309 −0.280984
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.8571 0.820171
\(210\) 0 0
\(211\) 16.9233 1.16505 0.582524 0.812814i \(-0.302065\pi\)
0.582524 + 0.812814i \(0.302065\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.42253 −0.642611
\(216\) 0 0
\(217\) 10.1013 0.685723
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 30.7560 2.06887
\(222\) 0 0
\(223\) −4.31849 −0.289187 −0.144594 0.989491i \(-0.546188\pi\)
−0.144594 + 0.989491i \(0.546188\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.2445 0.879068 0.439534 0.898226i \(-0.355144\pi\)
0.439534 + 0.898226i \(0.355144\pi\)
\(228\) 0 0
\(229\) 17.9946 1.18912 0.594558 0.804053i \(-0.297327\pi\)
0.594558 + 0.804053i \(0.297327\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.80322 0.249157 0.124579 0.992210i \(-0.460242\pi\)
0.124579 + 0.992210i \(0.460242\pi\)
\(234\) 0 0
\(235\) 15.3967 1.00437
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 25.6643 1.66009 0.830043 0.557699i \(-0.188316\pi\)
0.830043 + 0.557699i \(0.188316\pi\)
\(240\) 0 0
\(241\) 22.1635 1.42768 0.713840 0.700309i \(-0.246955\pi\)
0.713840 + 0.700309i \(0.246955\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.83940 0.181403
\(246\) 0 0
\(247\) −25.4375 −1.61855
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.8396 1.56786 0.783932 0.620847i \(-0.213211\pi\)
0.783932 + 0.620847i \(0.213211\pi\)
\(252\) 0 0
\(253\) −9.67243 −0.608100
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.3660 −1.02088 −0.510440 0.859913i \(-0.670518\pi\)
−0.510440 + 0.859913i \(0.670518\pi\)
\(258\) 0 0
\(259\) 6.31849 0.392612
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −11.6935 −0.721054 −0.360527 0.932749i \(-0.617403\pi\)
−0.360527 + 0.932749i \(0.617403\pi\)
\(264\) 0 0
\(265\) −11.3576 −0.697693
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.3011 0.628066 0.314033 0.949412i \(-0.398320\pi\)
0.314033 + 0.949412i \(0.398320\pi\)
\(270\) 0 0
\(271\) 7.07129 0.429550 0.214775 0.976664i \(-0.431098\pi\)
0.214775 + 0.976664i \(0.431098\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.59148 0.457783
\(276\) 0 0
\(277\) −5.59785 −0.336342 −0.168171 0.985758i \(-0.553786\pi\)
−0.168171 + 0.985758i \(0.553786\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.9882 −0.774812 −0.387406 0.921909i \(-0.626629\pi\)
−0.387406 + 0.921909i \(0.626629\pi\)
\(282\) 0 0
\(283\) −7.35761 −0.437365 −0.218682 0.975796i \(-0.570176\pi\)
−0.218682 + 0.975796i \(0.570176\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.41688 0.0836356
\(288\) 0 0
\(289\) 16.4413 0.967133
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −28.9324 −1.69025 −0.845124 0.534571i \(-0.820473\pi\)
−0.845124 + 0.534571i \(0.820473\pi\)
\(294\) 0 0
\(295\) 15.3967 0.896433
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 20.7507 1.20004
\(300\) 0 0
\(301\) 3.31849 0.191274
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −13.4038 −0.767498
\(306\) 0 0
\(307\) −30.6155 −1.74732 −0.873660 0.486537i \(-0.838260\pi\)
−0.873660 + 0.486537i \(0.838260\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.41123 0.0800235 0.0400117 0.999199i \(-0.487260\pi\)
0.0400117 + 0.999199i \(0.487260\pi\)
\(312\) 0 0
\(313\) −19.7292 −1.11516 −0.557581 0.830123i \(-0.688270\pi\)
−0.557581 + 0.830123i \(0.688270\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.70884 0.0959783 0.0479891 0.998848i \(-0.484719\pi\)
0.0479891 + 0.998848i \(0.484719\pi\)
\(318\) 0 0
\(319\) 17.6048 0.985680
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −27.6584 −1.53896
\(324\) 0 0
\(325\) −16.2863 −0.903402
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.42253 −0.298954
\(330\) 0 0
\(331\) −4.51256 −0.248033 −0.124016 0.992280i \(-0.539578\pi\)
−0.124016 + 0.992280i \(0.539578\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.33453 −0.236820
\(336\) 0 0
\(337\) 20.9163 1.13939 0.569693 0.821858i \(-0.307062\pi\)
0.569693 + 0.821858i \(0.307062\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −25.0421 −1.35610
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.10698 −0.381523 −0.190761 0.981636i \(-0.561096\pi\)
−0.190761 + 0.981636i \(0.561096\pi\)
\(348\) 0 0
\(349\) −7.36302 −0.394134 −0.197067 0.980390i \(-0.563142\pi\)
−0.197067 + 0.980390i \(0.563142\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.1016 1.12312 0.561562 0.827435i \(-0.310201\pi\)
0.561562 + 0.827435i \(0.310201\pi\)
\(354\) 0 0
\(355\) −33.4981 −1.77789
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.7887 1.46663 0.733316 0.679887i \(-0.237971\pi\)
0.733316 + 0.679887i \(0.237971\pi\)
\(360\) 0 0
\(361\) 3.87558 0.203978
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −30.9046 −1.61762
\(366\) 0 0
\(367\) −33.4680 −1.74702 −0.873508 0.486809i \(-0.838161\pi\)
−0.873508 + 0.486809i \(0.838161\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −32.1796 −1.66620 −0.833098 0.553126i \(-0.813435\pi\)
−0.833098 + 0.553126i \(0.813435\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −37.7683 −1.94517
\(378\) 0 0
\(379\) 1.95547 0.100446 0.0502228 0.998738i \(-0.484007\pi\)
0.0502228 + 0.998738i \(0.484007\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.5544 1.20357 0.601787 0.798657i \(-0.294456\pi\)
0.601787 + 0.798657i \(0.294456\pi\)
\(384\) 0 0
\(385\) −7.03912 −0.358747
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −29.5603 −1.49877 −0.749383 0.662137i \(-0.769650\pi\)
−0.749383 + 0.662137i \(0.769650\pi\)
\(390\) 0 0
\(391\) 22.5624 1.14103
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −22.8263 −1.14852
\(396\) 0 0
\(397\) −37.3984 −1.87697 −0.938485 0.345319i \(-0.887771\pi\)
−0.938485 + 0.345319i \(0.887771\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.1083 1.00416 0.502080 0.864821i \(-0.332568\pi\)
0.502080 + 0.864821i \(0.332568\pi\)
\(402\) 0 0
\(403\) 53.7238 2.67617
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.6641 −0.776439
\(408\) 0 0
\(409\) 25.3131 1.25165 0.625826 0.779963i \(-0.284762\pi\)
0.625826 + 0.779963i \(0.284762\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.42253 −0.266825
\(414\) 0 0
\(415\) 42.2880 2.07583
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.8338 1.11550 0.557751 0.830008i \(-0.311664\pi\)
0.557751 + 0.830008i \(0.311664\pi\)
\(420\) 0 0
\(421\) 30.2949 1.47648 0.738242 0.674536i \(-0.235656\pi\)
0.738242 + 0.674536i \(0.235656\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −17.7083 −0.858977
\(426\) 0 0
\(427\) 4.72063 0.228448
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.3011 −1.45955 −0.729775 0.683687i \(-0.760375\pi\)
−0.729775 + 0.683687i \(0.760375\pi\)
\(432\) 0 0
\(433\) 38.8005 1.86463 0.932317 0.361642i \(-0.117784\pi\)
0.932317 + 0.361642i \(0.117784\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18.6608 −0.892667
\(438\) 0 0
\(439\) −31.1314 −1.48582 −0.742910 0.669392i \(-0.766555\pi\)
−0.742910 + 0.669392i \(0.766555\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.15094 0.339751 0.169876 0.985466i \(-0.445663\pi\)
0.169876 + 0.985466i \(0.445663\pi\)
\(444\) 0 0
\(445\) −15.3807 −0.729115
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.404852 −0.0191061 −0.00955307 0.999954i \(-0.503041\pi\)
−0.00955307 + 0.999954i \(0.503041\pi\)
\(450\) 0 0
\(451\) −3.51256 −0.165400
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.1013 0.707961
\(456\) 0 0
\(457\) 26.9946 1.26275 0.631377 0.775476i \(-0.282490\pi\)
0.631377 + 0.775476i \(0.282490\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5.58042 −0.259906 −0.129953 0.991520i \(-0.541483\pi\)
−0.129953 + 0.991520i \(0.541483\pi\)
\(462\) 0 0
\(463\) 8.63698 0.401394 0.200697 0.979653i \(-0.435679\pi\)
0.200697 + 0.979653i \(0.435679\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.4295 −0.852815 −0.426407 0.904531i \(-0.640221\pi\)
−0.426407 + 0.904531i \(0.640221\pi\)
\(468\) 0 0
\(469\) 1.52656 0.0704901
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.22682 −0.378269
\(474\) 0 0
\(475\) 14.6461 0.672007
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −40.6316 −1.85650 −0.928252 0.371952i \(-0.878688\pi\)
−0.928252 + 0.371952i \(0.878688\pi\)
\(480\) 0 0
\(481\) 33.6048 1.53225
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.07825 −0.366814
\(486\) 0 0
\(487\) −1.87722 −0.0850650 −0.0425325 0.999095i \(-0.513543\pi\)
−0.0425325 + 0.999095i \(0.513543\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −30.9772 −1.39798 −0.698990 0.715132i \(-0.746367\pi\)
−0.698990 + 0.715132i \(0.746367\pi\)
\(492\) 0 0
\(493\) −41.0659 −1.84951
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.7976 0.529194
\(498\) 0 0
\(499\) −18.5979 −0.832554 −0.416277 0.909238i \(-0.636665\pi\)
−0.416277 + 0.909238i \(0.636665\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.1672 −1.61262 −0.806308 0.591496i \(-0.798538\pi\)
−0.806308 + 0.591496i \(0.798538\pi\)
\(504\) 0 0
\(505\) −39.1350 −1.74149
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 29.9018 1.32538 0.662688 0.748896i \(-0.269416\pi\)
0.662688 + 0.748896i \(0.269416\pi\)
\(510\) 0 0
\(511\) 10.8842 0.481487
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 42.4388 1.87008
\(516\) 0 0
\(517\) 13.4429 0.591218
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.0866 0.923821 0.461910 0.886927i \(-0.347164\pi\)
0.461910 + 0.886927i \(0.347164\pi\)
\(522\) 0 0
\(523\) −25.9233 −1.13355 −0.566773 0.823874i \(-0.691808\pi\)
−0.566773 + 0.823874i \(0.691808\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 58.4144 2.54457
\(528\) 0 0
\(529\) −7.77743 −0.338149
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.53565 0.326405
\(534\) 0 0
\(535\) −9.01604 −0.389797
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.47909 0.106782
\(540\) 0 0
\(541\) −0.836461 −0.0359623 −0.0179811 0.999838i \(-0.505724\pi\)
−0.0179811 + 0.999838i \(0.505724\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.92035 0.167930
\(546\) 0 0
\(547\) 4.94688 0.211513 0.105757 0.994392i \(-0.466274\pi\)
0.105757 + 0.994392i \(0.466274\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 33.9645 1.44694
\(552\) 0 0
\(553\) 8.03912 0.341858
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −34.4777 −1.46087 −0.730433 0.682984i \(-0.760682\pi\)
−0.730433 + 0.682984i \(0.760682\pi\)
\(558\) 0 0
\(559\) 17.6493 0.746488
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.22575 0.388819 0.194409 0.980920i \(-0.437721\pi\)
0.194409 + 0.980920i \(0.437721\pi\)
\(564\) 0 0
\(565\) −25.4359 −1.07009
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.0895 −1.17757 −0.588787 0.808288i \(-0.700394\pi\)
−0.588787 + 0.808288i \(0.700394\pi\)
\(570\) 0 0
\(571\) 4.35066 0.182069 0.0910347 0.995848i \(-0.470983\pi\)
0.0910347 + 0.995848i \(0.470983\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −11.9475 −0.498247
\(576\) 0 0
\(577\) 10.9287 0.454968 0.227484 0.973782i \(-0.426950\pi\)
0.227484 + 0.973782i \(0.426950\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.8933 −0.617876
\(582\) 0 0
\(583\) −9.91634 −0.410693
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.87451 −0.118644 −0.0593219 0.998239i \(-0.518894\pi\)
−0.0593219 + 0.998239i \(0.518894\pi\)
\(588\) 0 0
\(589\) −48.3131 −1.99071
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.8018 1.63446 0.817232 0.576308i \(-0.195507\pi\)
0.817232 + 0.576308i \(0.195507\pi\)
\(594\) 0 0
\(595\) 16.4198 0.673147
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.09298 −0.248952 −0.124476 0.992223i \(-0.539725\pi\)
−0.124476 + 0.992223i \(0.539725\pi\)
\(600\) 0 0
\(601\) 9.31849 0.380109 0.190055 0.981774i \(-0.439134\pi\)
0.190055 + 0.981774i \(0.439134\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −13.7828 −0.560352
\(606\) 0 0
\(607\) −4.20807 −0.170800 −0.0854002 0.996347i \(-0.527217\pi\)
−0.0854002 + 0.996347i \(0.527217\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −28.8396 −1.16673
\(612\) 0 0
\(613\) −10.1013 −0.407989 −0.203994 0.978972i \(-0.565392\pi\)
−0.203994 + 0.978972i \(0.565392\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.9046 0.600035 0.300017 0.953934i \(-0.403008\pi\)
0.300017 + 0.953934i \(0.403008\pi\)
\(618\) 0 0
\(619\) 27.5126 1.10582 0.552912 0.833240i \(-0.313517\pi\)
0.552912 + 0.833240i \(0.313517\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.41688 0.217023
\(624\) 0 0
\(625\) −30.9339 −1.23736
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 36.5388 1.45690
\(630\) 0 0
\(631\) 7.24024 0.288230 0.144115 0.989561i \(-0.453967\pi\)
0.144115 + 0.989561i \(0.453967\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 58.7078 2.32975
\(636\) 0 0
\(637\) −5.31849 −0.210726
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.7625 0.622580 0.311290 0.950315i \(-0.399239\pi\)
0.311290 + 0.950315i \(0.399239\pi\)
\(642\) 0 0
\(643\) −18.0622 −0.712304 −0.356152 0.934428i \(-0.615911\pi\)
−0.356152 + 0.934428i \(0.615911\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −31.0900 −1.22228 −0.611138 0.791524i \(-0.709288\pi\)
−0.611138 + 0.791524i \(0.709288\pi\)
\(648\) 0 0
\(649\) 13.4429 0.527680
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.7088 −0.536469 −0.268234 0.963354i \(-0.586440\pi\)
−0.268234 + 0.963354i \(0.586440\pi\)
\(654\) 0 0
\(655\) −24.9303 −0.974109
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.0394 0.624805 0.312402 0.949950i \(-0.398866\pi\)
0.312402 + 0.949950i \(0.398866\pi\)
\(660\) 0 0
\(661\) −36.4906 −1.41932 −0.709660 0.704544i \(-0.751152\pi\)
−0.709660 + 0.704544i \(0.751152\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −13.5804 −0.526626
\(666\) 0 0
\(667\) −27.7066 −1.07281
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.7029 −0.451784
\(672\) 0 0
\(673\) 13.7919 0.531640 0.265820 0.964023i \(-0.414357\pi\)
0.265820 + 0.964023i \(0.414357\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.6313 −0.408595 −0.204298 0.978909i \(-0.565491\pi\)
−0.204298 + 0.978909i \(0.565491\pi\)
\(678\) 0 0
\(679\) 2.84505 0.109183
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −27.6340 −1.05739 −0.528693 0.848813i \(-0.677318\pi\)
−0.528693 + 0.848813i \(0.677318\pi\)
\(684\) 0 0
\(685\) −51.5994 −1.97151
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 21.2740 0.810473
\(690\) 0 0
\(691\) 44.1190 1.67837 0.839183 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 49.6385 1.88290
\(696\) 0 0
\(697\) 8.19358 0.310354
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5.43382 −0.205233 −0.102616 0.994721i \(-0.532721\pi\)
−0.102616 + 0.994721i \(0.532721\pi\)
\(702\) 0 0
\(703\) −30.2203 −1.13978
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.7828 0.518357
\(708\) 0 0
\(709\) 25.4771 0.956813 0.478406 0.878139i \(-0.341215\pi\)
0.478406 + 0.878139i \(0.341215\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 39.4115 1.47597
\(714\) 0 0
\(715\) −37.4375 −1.40008
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.07187 0.263736 0.131868 0.991267i \(-0.457902\pi\)
0.131868 + 0.991267i \(0.457902\pi\)
\(720\) 0 0
\(721\) −14.9464 −0.556632
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.7458 0.807617
\(726\) 0 0
\(727\) 39.7629 1.47473 0.737363 0.675497i \(-0.236071\pi\)
0.737363 + 0.675497i \(0.236071\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 19.1903 0.709779
\(732\) 0 0
\(733\) −11.9609 −0.441785 −0.220893 0.975298i \(-0.570897\pi\)
−0.220893 + 0.975298i \(0.570897\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.78448 −0.139403
\(738\) 0 0
\(739\) −16.9415 −0.623202 −0.311601 0.950213i \(-0.600865\pi\)
−0.311601 + 0.950213i \(0.600865\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32.2920 −1.18468 −0.592339 0.805689i \(-0.701795\pi\)
−0.592339 + 0.805689i \(0.701795\pi\)
\(744\) 0 0
\(745\) −28.0728 −1.02851
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.17533 0.116024
\(750\) 0 0
\(751\) 26.1635 0.954721 0.477361 0.878708i \(-0.341594\pi\)
0.477361 + 0.878708i \(0.341594\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −19.0826 −0.694486
\(756\) 0 0
\(757\) 41.9109 1.52328 0.761639 0.648001i \(-0.224395\pi\)
0.761639 + 0.648001i \(0.224395\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.1335 0.584839 0.292419 0.956290i \(-0.405540\pi\)
0.292419 + 0.956290i \(0.405540\pi\)
\(762\) 0 0
\(763\) −1.38070 −0.0499846
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28.8396 −1.04134
\(768\) 0 0
\(769\) −36.7951 −1.32687 −0.663433 0.748236i \(-0.730901\pi\)
−0.663433 + 0.748236i \(0.730901\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14.3155 0.514894 0.257447 0.966292i \(-0.417119\pi\)
0.257447 + 0.966292i \(0.417119\pi\)
\(774\) 0 0
\(775\) −30.9324 −1.11112
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.77670 −0.242801
\(780\) 0 0
\(781\) −29.2472 −1.04655
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 55.4284 1.97832
\(786\) 0 0
\(787\) −2.42890 −0.0865810 −0.0432905 0.999063i \(-0.513784\pi\)
−0.0432905 + 0.999063i \(0.513784\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.95817 0.318516
\(792\) 0 0
\(793\) 25.1066 0.891563
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −41.8035 −1.48075 −0.740377 0.672192i \(-0.765353\pi\)
−0.740377 + 0.672192i \(0.765353\pi\)
\(798\) 0 0
\(799\) −31.3576 −1.10935
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −26.9828 −0.952202
\(804\) 0 0
\(805\) 11.0782 0.390457
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.6670 0.515665 0.257832 0.966190i \(-0.416992\pi\)
0.257832 + 0.966190i \(0.416992\pi\)
\(810\) 0 0
\(811\) −14.3630 −0.504354 −0.252177 0.967681i \(-0.581147\pi\)
−0.252177 + 0.967681i \(0.581147\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −29.0959 −1.01919
\(816\) 0 0
\(817\) −15.8718 −0.555284
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 33.7496 1.17787 0.588935 0.808181i \(-0.299548\pi\)
0.588935 + 0.808181i \(0.299548\pi\)
\(822\) 0 0
\(823\) 38.4053 1.33873 0.669363 0.742936i \(-0.266567\pi\)
0.669363 + 0.742936i \(0.266567\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25.6137 −0.890674 −0.445337 0.895363i \(-0.646916\pi\)
−0.445337 + 0.895363i \(0.646916\pi\)
\(828\) 0 0
\(829\) 17.0868 0.593450 0.296725 0.954963i \(-0.404105\pi\)
0.296725 + 0.954963i \(0.404105\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.78284 −0.200364
\(834\) 0 0
\(835\) 36.7935 1.27329
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −20.8697 −0.720501 −0.360251 0.932856i \(-0.617309\pi\)
−0.360251 + 0.932856i \(0.617309\pi\)
\(840\) 0 0
\(841\) 21.4289 0.738928
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 43.4040 1.49314
\(846\) 0 0
\(847\) 4.85413 0.166790
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.6523 0.845069
\(852\) 0 0
\(853\) 10.4289 0.357079 0.178539 0.983933i \(-0.442863\pi\)
0.178539 + 0.983933i \(0.442863\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −38.8989 −1.32876 −0.664381 0.747394i \(-0.731305\pi\)
−0.664381 + 0.747394i \(0.731305\pi\)
\(858\) 0 0
\(859\) 19.3791 0.661205 0.330603 0.943770i \(-0.392748\pi\)
0.330603 + 0.943770i \(0.392748\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19.1572 −0.652117 −0.326059 0.945350i \(-0.605721\pi\)
−0.326059 + 0.945350i \(0.605721\pi\)
\(864\) 0 0
\(865\) −31.0728 −1.05651
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −19.9297 −0.676068
\(870\) 0 0
\(871\) 8.11900 0.275102
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.50217 0.186007
\(876\) 0 0
\(877\) −16.1801 −0.546362 −0.273181 0.961963i \(-0.588076\pi\)
−0.273181 + 0.961963i \(0.588076\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.95744 0.268093 0.134047 0.990975i \(-0.457203\pi\)
0.134047 + 0.990975i \(0.457203\pi\)
\(882\) 0 0
\(883\) −14.4107 −0.484960 −0.242480 0.970156i \(-0.577961\pi\)
−0.242480 + 0.970156i \(0.577961\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.8627 1.60707 0.803537 0.595255i \(-0.202949\pi\)
0.803537 + 0.595255i \(0.202949\pi\)
\(888\) 0 0
\(889\) −20.6761 −0.693454
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.9351 0.867885
\(894\) 0 0
\(895\) −23.0106 −0.769160
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −71.7329 −2.39243
\(900\) 0 0
\(901\) 23.1314 0.770618
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.8494 0.593334
\(906\) 0 0
\(907\) 26.9287 0.894153 0.447077 0.894496i \(-0.352465\pi\)
0.447077 + 0.894496i \(0.352465\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21.7833 0.721713 0.360857 0.932621i \(-0.382484\pi\)
0.360857 + 0.932621i \(0.382484\pi\)
\(912\) 0 0
\(913\) 36.9217 1.22193
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.78014 0.289946
\(918\) 0 0
\(919\) −0.535153 −0.0176531 −0.00882653 0.999961i \(-0.502810\pi\)
−0.00882653 + 0.999961i \(0.502810\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 62.7453 2.06529
\(924\) 0 0
\(925\) −19.3485 −0.636176
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.4729 −0.507651 −0.253825 0.967250i \(-0.581689\pi\)
−0.253825 + 0.967250i \(0.581689\pi\)
\(930\) 0 0
\(931\) 4.78284 0.156751
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −40.7061 −1.33123
\(936\) 0 0
\(937\) 27.4359 0.896290 0.448145 0.893961i \(-0.352085\pi\)
0.448145 + 0.893961i \(0.352085\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.79169 0.188804 0.0944018 0.995534i \(-0.469906\pi\)
0.0944018 + 0.995534i \(0.469906\pi\)
\(942\) 0 0
\(943\) 5.52810 0.180020
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.57771 −0.148756 −0.0743778 0.997230i \(-0.523697\pi\)
−0.0743778 + 0.997230i \(0.523697\pi\)
\(948\) 0 0
\(949\) 57.8874 1.87910
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36.1672 −1.17157 −0.585785 0.810466i \(-0.699214\pi\)
−0.585785 + 0.810466i \(0.699214\pi\)
\(954\) 0 0
\(955\) −48.6847 −1.57540
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.1726 0.586825
\(960\) 0 0
\(961\) 71.0369 2.29151
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 27.8032 0.895017
\(966\) 0 0
\(967\) −19.0885 −0.613844 −0.306922 0.951735i \(-0.599299\pi\)
−0.306922 + 0.951735i \(0.599299\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 44.6616 1.43326 0.716630 0.697454i \(-0.245684\pi\)
0.716630 + 0.697454i \(0.245684\pi\)
\(972\) 0 0
\(973\) −17.4820 −0.560448
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −36.9179 −1.18111 −0.590554 0.806998i \(-0.701091\pi\)
−0.590554 + 0.806998i \(0.701091\pi\)
\(978\) 0 0
\(979\) −13.4289 −0.429190
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 47.0610 1.50101 0.750507 0.660862i \(-0.229809\pi\)
0.750507 + 0.660862i \(0.229809\pi\)
\(984\) 0 0
\(985\) −30.7614 −0.980139
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.9475 0.411705
\(990\) 0 0
\(991\) −30.6525 −0.973709 −0.486855 0.873483i \(-0.661856\pi\)
−0.486855 + 0.873483i \(0.661856\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −50.1896 −1.59112
\(996\) 0 0
\(997\) −5.13137 −0.162512 −0.0812561 0.996693i \(-0.525893\pi\)
−0.0812561 + 0.996693i \(0.525893\pi\)
\(998\) 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.a.bm.1.4 4
3.2 odd 2 6048.2.a.br.1.1 yes 4
4.3 odd 2 6048.2.a.bo.1.4 yes 4
12.11 even 2 6048.2.a.bv.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bm.1.4 4 1.1 even 1 trivial
6048.2.a.bo.1.4 yes 4 4.3 odd 2
6048.2.a.br.1.1 yes 4 3.2 odd 2
6048.2.a.bv.1.1 yes 4 12.11 even 2