Properties

Label 6048.2.j.c.5615.25
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 32
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.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(32\)
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 5615.25
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.26

$q$-expansion

\(f(q)\) \(=\) \(q+2.09311 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+2.09311 q^{5} -1.00000i q^{7} +0.961211i q^{11} -4.60210i q^{13} +3.28066i q^{17} -3.66977 q^{19} -2.40322 q^{23} -0.618872 q^{25} +9.31771 q^{29} +1.84927i q^{31} -2.09311i q^{35} -2.29364i q^{37} -0.314406i q^{41} +8.64261 q^{43} -2.34305 q^{47} -1.00000 q^{49} +7.73354 q^{53} +2.01192i q^{55} -7.99530i q^{59} -10.6321i q^{61} -9.63272i q^{65} -4.12203 q^{67} +6.79440 q^{71} +16.0267 q^{73} +0.961211 q^{77} +1.26231i q^{79} +2.99495i q^{83} +6.86680i q^{85} -12.8990i q^{89} -4.60210 q^{91} -7.68126 q^{95} +4.98346 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + O(q^{10}) \) \( 32q + 64q^{19} - 16q^{25} + 48q^{43} - 32q^{49} - 16q^{67} - 16q^{73} - 16q^{91} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.09311 0.936069 0.468035 0.883710i \(-0.344962\pi\)
0.468035 + 0.883710i \(0.344962\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.961211i 0.289816i 0.989445 + 0.144908i \(0.0462886\pi\)
−0.989445 + 0.144908i \(0.953711\pi\)
\(12\) 0 0
\(13\) − 4.60210i − 1.27639i −0.769874 0.638196i \(-0.779681\pi\)
0.769874 0.638196i \(-0.220319\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.28066i 0.795677i 0.917455 + 0.397839i \(0.130240\pi\)
−0.917455 + 0.397839i \(0.869760\pi\)
\(18\) 0 0
\(19\) −3.66977 −0.841904 −0.420952 0.907083i \(-0.638304\pi\)
−0.420952 + 0.907083i \(0.638304\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.40322 −0.501105 −0.250553 0.968103i \(-0.580612\pi\)
−0.250553 + 0.968103i \(0.580612\pi\)
\(24\) 0 0
\(25\) −0.618872 −0.123774
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.31771 1.73026 0.865128 0.501552i \(-0.167237\pi\)
0.865128 + 0.501552i \(0.167237\pi\)
\(30\) 0 0
\(31\) 1.84927i 0.332138i 0.986114 + 0.166069i \(0.0531075\pi\)
−0.986114 + 0.166069i \(0.946892\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.09311i − 0.353801i
\(36\) 0 0
\(37\) − 2.29364i − 0.377072i −0.982066 0.188536i \(-0.939626\pi\)
0.982066 0.188536i \(-0.0603742\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 0.314406i − 0.0491020i −0.999699 0.0245510i \(-0.992184\pi\)
0.999699 0.0245510i \(-0.00781561\pi\)
\(42\) 0 0
\(43\) 8.64261 1.31799 0.658993 0.752149i \(-0.270983\pi\)
0.658993 + 0.752149i \(0.270983\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.34305 −0.341769 −0.170884 0.985291i \(-0.554662\pi\)
−0.170884 + 0.985291i \(0.554662\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.73354 1.06228 0.531142 0.847283i \(-0.321763\pi\)
0.531142 + 0.847283i \(0.321763\pi\)
\(54\) 0 0
\(55\) 2.01192i 0.271288i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 7.99530i − 1.04090i −0.853892 0.520450i \(-0.825764\pi\)
0.853892 0.520450i \(-0.174236\pi\)
\(60\) 0 0
\(61\) − 10.6321i − 1.36130i −0.732610 0.680649i \(-0.761698\pi\)
0.732610 0.680649i \(-0.238302\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 9.63272i − 1.19479i
\(66\) 0 0
\(67\) −4.12203 −0.503586 −0.251793 0.967781i \(-0.581020\pi\)
−0.251793 + 0.967781i \(0.581020\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.79440 0.806347 0.403174 0.915124i \(-0.367907\pi\)
0.403174 + 0.915124i \(0.367907\pi\)
\(72\) 0 0
\(73\) 16.0267 1.87579 0.937894 0.346922i \(-0.112773\pi\)
0.937894 + 0.346922i \(0.112773\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.961211 0.109540
\(78\) 0 0
\(79\) 1.26231i 0.142021i 0.997476 + 0.0710106i \(0.0226224\pi\)
−0.997476 + 0.0710106i \(0.977378\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.99495i 0.328738i 0.986399 + 0.164369i \(0.0525589\pi\)
−0.986399 + 0.164369i \(0.947441\pi\)
\(84\) 0 0
\(85\) 6.86680i 0.744809i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 12.8990i − 1.36729i −0.729815 0.683645i \(-0.760394\pi\)
0.729815 0.683645i \(-0.239606\pi\)
\(90\) 0 0
\(91\) −4.60210 −0.482431
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.68126 −0.788080
\(96\) 0 0
\(97\) 4.98346 0.505994 0.252997 0.967467i \(-0.418584\pi\)
0.252997 + 0.967467i \(0.418584\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.18764 −0.317182 −0.158591 0.987344i \(-0.550695\pi\)
−0.158591 + 0.987344i \(0.550695\pi\)
\(102\) 0 0
\(103\) − 0.535425i − 0.0527570i −0.999652 0.0263785i \(-0.991602\pi\)
0.999652 0.0263785i \(-0.00839750\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.70673i − 0.938385i −0.883096 0.469193i \(-0.844545\pi\)
0.883096 0.469193i \(-0.155455\pi\)
\(108\) 0 0
\(109\) − 8.03576i − 0.769686i −0.922982 0.384843i \(-0.874256\pi\)
0.922982 0.384843i \(-0.125744\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 12.7809i − 1.20233i −0.799126 0.601163i \(-0.794704\pi\)
0.799126 0.601163i \(-0.205296\pi\)
\(114\) 0 0
\(115\) −5.03021 −0.469069
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.28066 0.300738
\(120\) 0 0
\(121\) 10.0761 0.916007
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.7609 −1.05193
\(126\) 0 0
\(127\) − 16.6260i − 1.47532i −0.675175 0.737658i \(-0.735932\pi\)
0.675175 0.737658i \(-0.264068\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 14.5603i − 1.27214i −0.771632 0.636069i \(-0.780560\pi\)
0.771632 0.636069i \(-0.219440\pi\)
\(132\) 0 0
\(133\) 3.66977i 0.318210i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.2672i 1.47524i 0.675216 + 0.737620i \(0.264050\pi\)
−0.675216 + 0.737620i \(0.735950\pi\)
\(138\) 0 0
\(139\) −14.5580 −1.23480 −0.617398 0.786651i \(-0.711813\pi\)
−0.617398 + 0.786651i \(0.711813\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.42358 0.369919
\(144\) 0 0
\(145\) 19.5030 1.61964
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.1739 1.65271 0.826356 0.563148i \(-0.190410\pi\)
0.826356 + 0.563148i \(0.190410\pi\)
\(150\) 0 0
\(151\) 11.9560i 0.972966i 0.873690 + 0.486483i \(0.161720\pi\)
−0.873690 + 0.486483i \(0.838280\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.87073i 0.310905i
\(156\) 0 0
\(157\) 22.1011i 1.76386i 0.471381 + 0.881930i \(0.343756\pi\)
−0.471381 + 0.881930i \(0.656244\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.40322i 0.189400i
\(162\) 0 0
\(163\) −11.2054 −0.877674 −0.438837 0.898567i \(-0.644609\pi\)
−0.438837 + 0.898567i \(0.644609\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.76628 0.136679 0.0683395 0.997662i \(-0.478230\pi\)
0.0683395 + 0.997662i \(0.478230\pi\)
\(168\) 0 0
\(169\) −8.17930 −0.629177
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.37240 −0.332428 −0.166214 0.986090i \(-0.553154\pi\)
−0.166214 + 0.986090i \(0.553154\pi\)
\(174\) 0 0
\(175\) 0.618872i 0.0467823i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 17.0068i − 1.27115i −0.772041 0.635573i \(-0.780764\pi\)
0.772041 0.635573i \(-0.219236\pi\)
\(180\) 0 0
\(181\) − 8.50624i − 0.632264i −0.948715 0.316132i \(-0.897616\pi\)
0.948715 0.316132i \(-0.102384\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 4.80085i − 0.352965i
\(186\) 0 0
\(187\) −3.15341 −0.230600
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.725194 −0.0524732 −0.0262366 0.999656i \(-0.508352\pi\)
−0.0262366 + 0.999656i \(0.508352\pi\)
\(192\) 0 0
\(193\) −15.7648 −1.13478 −0.567388 0.823450i \(-0.692046\pi\)
−0.567388 + 0.823450i \(0.692046\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.76032 −0.410406 −0.205203 0.978719i \(-0.565785\pi\)
−0.205203 + 0.978719i \(0.565785\pi\)
\(198\) 0 0
\(199\) 11.7773i 0.834872i 0.908706 + 0.417436i \(0.137071\pi\)
−0.908706 + 0.417436i \(0.862929\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 9.31771i − 0.653975i
\(204\) 0 0
\(205\) − 0.658088i − 0.0459629i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 3.52742i − 0.243997i
\(210\) 0 0
\(211\) −19.9971 −1.37666 −0.688328 0.725399i \(-0.741655\pi\)
−0.688328 + 0.725399i \(0.741655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.0900 1.23373
\(216\) 0 0
\(217\) 1.84927 0.125537
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.0979 1.01560
\(222\) 0 0
\(223\) − 6.84047i − 0.458071i −0.973418 0.229036i \(-0.926443\pi\)
0.973418 0.229036i \(-0.0735573\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 13.9061i − 0.922981i −0.887145 0.461490i \(-0.847315\pi\)
0.887145 0.461490i \(-0.152685\pi\)
\(228\) 0 0
\(229\) 3.87934i 0.256354i 0.991751 + 0.128177i \(0.0409126\pi\)
−0.991751 + 0.128177i \(0.959087\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 21.8307i − 1.43017i −0.699035 0.715087i \(-0.746387\pi\)
0.699035 0.715087i \(-0.253613\pi\)
\(234\) 0 0
\(235\) −4.90427 −0.319919
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 27.8956 1.80441 0.902207 0.431303i \(-0.141946\pi\)
0.902207 + 0.431303i \(0.141946\pi\)
\(240\) 0 0
\(241\) −6.17606 −0.397835 −0.198918 0.980016i \(-0.563743\pi\)
−0.198918 + 0.980016i \(0.563743\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.09311 −0.133724
\(246\) 0 0
\(247\) 16.8887i 1.07460i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.6366i 1.17633i 0.808740 + 0.588167i \(0.200150\pi\)
−0.808740 + 0.588167i \(0.799850\pi\)
\(252\) 0 0
\(253\) − 2.31000i − 0.145228i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 8.25845i − 0.515148i −0.966259 0.257574i \(-0.917077\pi\)
0.966259 0.257574i \(-0.0829231\pi\)
\(258\) 0 0
\(259\) −2.29364 −0.142520
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.34158 0.576027 0.288013 0.957626i \(-0.407005\pi\)
0.288013 + 0.957626i \(0.407005\pi\)
\(264\) 0 0
\(265\) 16.1872 0.994371
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.9972 0.731484 0.365742 0.930716i \(-0.380815\pi\)
0.365742 + 0.930716i \(0.380815\pi\)
\(270\) 0 0
\(271\) 23.7030i 1.43986i 0.694049 + 0.719928i \(0.255825\pi\)
−0.694049 + 0.719928i \(0.744175\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 0.594866i − 0.0358718i
\(276\) 0 0
\(277\) − 13.7688i − 0.827284i −0.910440 0.413642i \(-0.864256\pi\)
0.910440 0.413642i \(-0.135744\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.69078i − 0.518448i −0.965817 0.259224i \(-0.916533\pi\)
0.965817 0.259224i \(-0.0834669\pi\)
\(282\) 0 0
\(283\) 32.7518 1.94689 0.973447 0.228913i \(-0.0735170\pi\)
0.973447 + 0.228913i \(0.0735170\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.314406 −0.0185588
\(288\) 0 0
\(289\) 6.23727 0.366898
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.3629 0.722249 0.361124 0.932518i \(-0.382393\pi\)
0.361124 + 0.932518i \(0.382393\pi\)
\(294\) 0 0
\(295\) − 16.7351i − 0.974354i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 11.0598i 0.639607i
\(300\) 0 0
\(301\) − 8.64261i − 0.498152i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 22.2542i − 1.27427i
\(306\) 0 0
\(307\) −15.2600 −0.870935 −0.435468 0.900204i \(-0.643417\pi\)
−0.435468 + 0.900204i \(0.643417\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.12076 0.290372 0.145186 0.989404i \(-0.453622\pi\)
0.145186 + 0.989404i \(0.453622\pi\)
\(312\) 0 0
\(313\) −11.7058 −0.661649 −0.330825 0.943692i \(-0.607327\pi\)
−0.330825 + 0.943692i \(0.607327\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.5302 −1.04076 −0.520379 0.853935i \(-0.674209\pi\)
−0.520379 + 0.853935i \(0.674209\pi\)
\(318\) 0 0
\(319\) 8.95628i 0.501456i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 12.0393i − 0.669883i
\(324\) 0 0
\(325\) 2.84811i 0.157985i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.34305i 0.129176i
\(330\) 0 0
\(331\) 20.4688 1.12507 0.562534 0.826774i \(-0.309826\pi\)
0.562534 + 0.826774i \(0.309826\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.62789 −0.471392
\(336\) 0 0
\(337\) −19.9419 −1.08631 −0.543153 0.839634i \(-0.682769\pi\)
−0.543153 + 0.839634i \(0.682769\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.77754 −0.0962590
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 10.1743i − 0.546187i −0.961987 0.273094i \(-0.911953\pi\)
0.961987 0.273094i \(-0.0880469\pi\)
\(348\) 0 0
\(349\) 31.9386i 1.70963i 0.518930 + 0.854817i \(0.326331\pi\)
−0.518930 + 0.854817i \(0.673669\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.3932i 1.29832i 0.760653 + 0.649159i \(0.224879\pi\)
−0.760653 + 0.649159i \(0.775121\pi\)
\(354\) 0 0
\(355\) 14.2215 0.754797
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.59571 0.453664 0.226832 0.973934i \(-0.427163\pi\)
0.226832 + 0.973934i \(0.427163\pi\)
\(360\) 0 0
\(361\) −5.53276 −0.291198
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 33.5458 1.75587
\(366\) 0 0
\(367\) − 29.1285i − 1.52050i −0.649632 0.760249i \(-0.725077\pi\)
0.649632 0.760249i \(-0.274923\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 7.73354i − 0.401505i
\(372\) 0 0
\(373\) 4.56533i 0.236384i 0.992991 + 0.118192i \(0.0377098\pi\)
−0.992991 + 0.118192i \(0.962290\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 42.8810i − 2.20848i
\(378\) 0 0
\(379\) −15.5673 −0.799641 −0.399820 0.916593i \(-0.630928\pi\)
−0.399820 + 0.916593i \(0.630928\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.08947 −0.157865 −0.0789323 0.996880i \(-0.525151\pi\)
−0.0789323 + 0.996880i \(0.525151\pi\)
\(384\) 0 0
\(385\) 2.01192 0.102537
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.6022 −1.55160 −0.775798 0.630982i \(-0.782652\pi\)
−0.775798 + 0.630982i \(0.782652\pi\)
\(390\) 0 0
\(391\) − 7.88413i − 0.398718i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.64216i 0.132942i
\(396\) 0 0
\(397\) − 0.403742i − 0.0202632i −0.999949 0.0101316i \(-0.996775\pi\)
0.999949 0.0101316i \(-0.00322505\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 30.3779i − 1.51700i −0.651672 0.758501i \(-0.725932\pi\)
0.651672 0.758501i \(-0.274068\pi\)
\(402\) 0 0
\(403\) 8.51051 0.423939
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.20467 0.109281
\(408\) 0 0
\(409\) −17.8549 −0.882869 −0.441434 0.897293i \(-0.645530\pi\)
−0.441434 + 0.897293i \(0.645530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.99530 −0.393423
\(414\) 0 0
\(415\) 6.26877i 0.307722i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 31.1609i − 1.52231i −0.648569 0.761156i \(-0.724632\pi\)
0.648569 0.761156i \(-0.275368\pi\)
\(420\) 0 0
\(421\) 7.19292i 0.350562i 0.984518 + 0.175281i \(0.0560833\pi\)
−0.984518 + 0.175281i \(0.943917\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 2.03031i − 0.0984844i
\(426\) 0 0
\(427\) −10.6321 −0.514522
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.7598 1.24081 0.620404 0.784283i \(-0.286969\pi\)
0.620404 + 0.784283i \(0.286969\pi\)
\(432\) 0 0
\(433\) −3.66449 −0.176104 −0.0880521 0.996116i \(-0.528064\pi\)
−0.0880521 + 0.996116i \(0.528064\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.81926 0.421882
\(438\) 0 0
\(439\) 39.4350i 1.88213i 0.338227 + 0.941064i \(0.390173\pi\)
−0.338227 + 0.941064i \(0.609827\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.1437i 1.71724i 0.512612 + 0.858620i \(0.328678\pi\)
−0.512612 + 0.858620i \(0.671322\pi\)
\(444\) 0 0
\(445\) − 26.9990i − 1.27988i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.9465i 1.17730i 0.808389 + 0.588649i \(0.200340\pi\)
−0.808389 + 0.588649i \(0.799660\pi\)
\(450\) 0 0
\(451\) 0.302210 0.0142305
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.63272 −0.451589
\(456\) 0 0
\(457\) 15.1856 0.710354 0.355177 0.934799i \(-0.384421\pi\)
0.355177 + 0.934799i \(0.384421\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.29479 −0.432902 −0.216451 0.976294i \(-0.569448\pi\)
−0.216451 + 0.976294i \(0.569448\pi\)
\(462\) 0 0
\(463\) − 2.13438i − 0.0991930i −0.998769 0.0495965i \(-0.984206\pi\)
0.998769 0.0495965i \(-0.0157935\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 41.6077i − 1.92538i −0.270612 0.962688i \(-0.587226\pi\)
0.270612 0.962688i \(-0.412774\pi\)
\(468\) 0 0
\(469\) 4.12203i 0.190338i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.30736i 0.381973i
\(474\) 0 0
\(475\) 2.27112 0.104206
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 40.9442 1.87079 0.935394 0.353608i \(-0.115045\pi\)
0.935394 + 0.353608i \(0.115045\pi\)
\(480\) 0 0
\(481\) −10.5556 −0.481292
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.4310 0.473645
\(486\) 0 0
\(487\) 14.2039i 0.643639i 0.946801 + 0.321820i \(0.104294\pi\)
−0.946801 + 0.321820i \(0.895706\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 39.3013i 1.77364i 0.462113 + 0.886821i \(0.347091\pi\)
−0.462113 + 0.886821i \(0.652909\pi\)
\(492\) 0 0
\(493\) 30.5682i 1.37672i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.79440i − 0.304771i
\(498\) 0 0
\(499\) 32.3257 1.44710 0.723548 0.690274i \(-0.242510\pi\)
0.723548 + 0.690274i \(0.242510\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19.7861 −0.882217 −0.441108 0.897454i \(-0.645415\pi\)
−0.441108 + 0.897454i \(0.645415\pi\)
\(504\) 0 0
\(505\) −6.67210 −0.296904
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −33.7174 −1.49450 −0.747248 0.664545i \(-0.768625\pi\)
−0.747248 + 0.664545i \(0.768625\pi\)
\(510\) 0 0
\(511\) − 16.0267i − 0.708981i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1.12070i − 0.0493842i
\(516\) 0 0
\(517\) − 2.25216i − 0.0990500i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 28.4958i 1.24842i 0.781255 + 0.624212i \(0.214580\pi\)
−0.781255 + 0.624212i \(0.785420\pi\)
\(522\) 0 0
\(523\) 4.42637 0.193552 0.0967758 0.995306i \(-0.469147\pi\)
0.0967758 + 0.995306i \(0.469147\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.06682 −0.264275
\(528\) 0 0
\(529\) −17.2246 −0.748894
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.44693 −0.0626734
\(534\) 0 0
\(535\) − 20.3173i − 0.878394i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 0.961211i − 0.0414023i
\(540\) 0 0
\(541\) − 33.7548i − 1.45123i −0.688101 0.725615i \(-0.741555\pi\)
0.688101 0.725615i \(-0.258445\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 16.8198i − 0.720480i
\(546\) 0 0
\(547\) 14.6641 0.626992 0.313496 0.949589i \(-0.398500\pi\)
0.313496 + 0.949589i \(0.398500\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −34.1939 −1.45671
\(552\) 0 0
\(553\) 1.26231 0.0536790
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.4373 0.654099 0.327050 0.945007i \(-0.393946\pi\)
0.327050 + 0.945007i \(0.393946\pi\)
\(558\) 0 0
\(559\) − 39.7741i − 1.68227i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 26.9472i − 1.13569i −0.823135 0.567845i \(-0.807777\pi\)
0.823135 0.567845i \(-0.192223\pi\)
\(564\) 0 0
\(565\) − 26.7519i − 1.12546i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.63013i 0.152183i 0.997101 + 0.0760915i \(0.0242441\pi\)
−0.997101 + 0.0760915i \(0.975756\pi\)
\(570\) 0 0
\(571\) 16.2293 0.679174 0.339587 0.940575i \(-0.389713\pi\)
0.339587 + 0.940575i \(0.389713\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.48728 0.0620240
\(576\) 0 0
\(577\) 42.1511 1.75477 0.877386 0.479786i \(-0.159286\pi\)
0.877386 + 0.479786i \(0.159286\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.99495 0.124251
\(582\) 0 0
\(583\) 7.43356i 0.307867i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 22.7097i − 0.937328i −0.883377 0.468664i \(-0.844736\pi\)
0.883377 0.468664i \(-0.155264\pi\)
\(588\) 0 0
\(589\) − 6.78640i − 0.279629i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.8921i 0.940068i 0.882648 + 0.470034i \(0.155758\pi\)
−0.882648 + 0.470034i \(0.844242\pi\)
\(594\) 0 0
\(595\) 6.86680 0.281511
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.0216 1.26751 0.633755 0.773534i \(-0.281513\pi\)
0.633755 + 0.773534i \(0.281513\pi\)
\(600\) 0 0
\(601\) −36.3657 −1.48339 −0.741693 0.670739i \(-0.765977\pi\)
−0.741693 + 0.670739i \(0.765977\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.0904 0.857446
\(606\) 0 0
\(607\) − 27.1019i − 1.10003i −0.835154 0.550016i \(-0.814622\pi\)
0.835154 0.550016i \(-0.185378\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.7829i 0.436231i
\(612\) 0 0
\(613\) − 21.7164i − 0.877118i −0.898702 0.438559i \(-0.855489\pi\)
0.898702 0.438559i \(-0.144511\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 15.0335i − 0.605224i −0.953114 0.302612i \(-0.902141\pi\)
0.953114 0.302612i \(-0.0978587\pi\)
\(618\) 0 0
\(619\) 12.2249 0.491359 0.245680 0.969351i \(-0.420989\pi\)
0.245680 + 0.969351i \(0.420989\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.8990 −0.516787
\(624\) 0 0
\(625\) −21.5226 −0.860906
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.52465 0.300028
\(630\) 0 0
\(631\) − 0.290118i − 0.0115494i −0.999983 0.00577472i \(-0.998162\pi\)
0.999983 0.00577472i \(-0.00183816\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 34.8000i − 1.38100i
\(636\) 0 0
\(637\) 4.60210i 0.182342i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 32.2612i 1.27424i 0.770764 + 0.637120i \(0.219875\pi\)
−0.770764 + 0.637120i \(0.780125\pi\)
\(642\) 0 0
\(643\) 25.2641 0.996317 0.498159 0.867086i \(-0.334010\pi\)
0.498159 + 0.867086i \(0.334010\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.6735 0.616188 0.308094 0.951356i \(-0.400309\pi\)
0.308094 + 0.951356i \(0.400309\pi\)
\(648\) 0 0
\(649\) 7.68517 0.301669
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −17.4580 −0.683183 −0.341591 0.939849i \(-0.610966\pi\)
−0.341591 + 0.939849i \(0.610966\pi\)
\(654\) 0 0
\(655\) − 30.4763i − 1.19081i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 34.8692i 1.35831i 0.733995 + 0.679155i \(0.237653\pi\)
−0.733995 + 0.679155i \(0.762347\pi\)
\(660\) 0 0
\(661\) 13.7607i 0.535230i 0.963526 + 0.267615i \(0.0862355\pi\)
−0.963526 + 0.267615i \(0.913765\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.68126i 0.297866i
\(666\) 0 0
\(667\) −22.3925 −0.867040
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.2197 0.394526
\(672\) 0 0
\(673\) −8.50938 −0.328013 −0.164006 0.986459i \(-0.552442\pi\)
−0.164006 + 0.986459i \(0.552442\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.14515 0.120878 0.0604390 0.998172i \(-0.480750\pi\)
0.0604390 + 0.998172i \(0.480750\pi\)
\(678\) 0 0
\(679\) − 4.98346i − 0.191248i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 9.12590i − 0.349193i −0.984640 0.174596i \(-0.944138\pi\)
0.984640 0.174596i \(-0.0558621\pi\)
\(684\) 0 0
\(685\) 36.1423i 1.38093i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 35.5905i − 1.35589i
\(690\) 0 0
\(691\) 15.6298 0.594586 0.297293 0.954786i \(-0.403916\pi\)
0.297293 + 0.954786i \(0.403916\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −30.4716 −1.15585
\(696\) 0 0
\(697\) 1.03146 0.0390693
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 50.4853 1.90680 0.953402 0.301701i \(-0.0975546\pi\)
0.953402 + 0.301701i \(0.0975546\pi\)
\(702\) 0 0
\(703\) 8.41714i 0.317458i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.18764i 0.119884i
\(708\) 0 0
\(709\) 28.7706i 1.08050i 0.841504 + 0.540251i \(0.181671\pi\)
−0.841504 + 0.540251i \(0.818329\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 4.44419i − 0.166436i
\(714\) 0 0
\(715\) 9.25907 0.346270
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.582170 0.0217113 0.0108556 0.999941i \(-0.496544\pi\)
0.0108556 + 0.999941i \(0.496544\pi\)
\(720\) 0 0
\(721\) −0.535425 −0.0199403
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.76647 −0.214161
\(726\) 0 0
\(727\) − 6.92907i − 0.256985i −0.991710 0.128492i \(-0.958986\pi\)
0.991710 0.128492i \(-0.0410138\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 28.3535i 1.04869i
\(732\) 0 0
\(733\) − 1.86770i − 0.0689850i −0.999405 0.0344925i \(-0.989019\pi\)
0.999405 0.0344925i \(-0.0109815\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 3.96214i − 0.145947i
\(738\) 0 0
\(739\) 38.7036 1.42373 0.711867 0.702314i \(-0.247850\pi\)
0.711867 + 0.702314i \(0.247850\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23.3088 0.855118 0.427559 0.903987i \(-0.359374\pi\)
0.427559 + 0.903987i \(0.359374\pi\)
\(744\) 0 0
\(745\) 42.2263 1.54705
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.70673 −0.354676
\(750\) 0 0
\(751\) − 26.6345i − 0.971908i −0.873984 0.485954i \(-0.838472\pi\)
0.873984 0.485954i \(-0.161528\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 25.0253i 0.910764i
\(756\) 0 0
\(757\) 26.4285i 0.960562i 0.877115 + 0.480281i \(0.159465\pi\)
−0.877115 + 0.480281i \(0.840535\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.2617i 0.480736i 0.970682 + 0.240368i \(0.0772681\pi\)
−0.970682 + 0.240368i \(0.922732\pi\)
\(762\) 0 0
\(763\) −8.03576 −0.290914
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −36.7951 −1.32860
\(768\) 0 0
\(769\) −4.34986 −0.156860 −0.0784300 0.996920i \(-0.524991\pi\)
−0.0784300 + 0.996920i \(0.524991\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.8861 1.25477 0.627383 0.778711i \(-0.284126\pi\)
0.627383 + 0.778711i \(0.284126\pi\)
\(774\) 0 0
\(775\) − 1.14446i − 0.0411102i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.15380i 0.0413391i
\(780\) 0 0
\(781\) 6.53085i 0.233692i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 46.2601i 1.65109i
\(786\) 0 0
\(787\) −12.9317 −0.460964 −0.230482 0.973077i \(-0.574030\pi\)
−0.230482 + 0.973077i \(0.574030\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.7809 −0.454437
\(792\) 0 0
\(793\) −48.9299 −1.73755
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.1981 0.396656 0.198328 0.980136i \(-0.436449\pi\)
0.198328 + 0.980136i \(0.436449\pi\)
\(798\) 0 0
\(799\) − 7.68675i − 0.271938i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.4051i 0.543633i
\(804\) 0 0
\(805\) 5.03021i 0.177291i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 23.6434i − 0.831258i −0.909534 0.415629i \(-0.863561\pi\)
0.909534 0.415629i \(-0.136439\pi\)
\(810\) 0 0
\(811\) −34.8213 −1.22274 −0.611371 0.791344i \(-0.709382\pi\)
−0.611371 + 0.791344i \(0.709382\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −23.4542 −0.821564
\(816\) 0 0
\(817\) −31.7164 −1.10962
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.361884 −0.0126298 −0.00631491 0.999980i \(-0.502010\pi\)
−0.00631491 + 0.999980i \(0.502010\pi\)
\(822\) 0 0
\(823\) 49.2033i 1.71512i 0.514386 + 0.857559i \(0.328020\pi\)
−0.514386 + 0.857559i \(0.671980\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 16.2920i − 0.566529i −0.959042 0.283264i \(-0.908583\pi\)
0.959042 0.283264i \(-0.0914174\pi\)
\(828\) 0 0
\(829\) 6.40526i 0.222464i 0.993794 + 0.111232i \(0.0354796\pi\)
−0.993794 + 0.111232i \(0.964520\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 3.28066i − 0.113668i
\(834\) 0 0
\(835\) 3.69703 0.127941
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23.3877 −0.807432 −0.403716 0.914884i \(-0.632282\pi\)
−0.403716 + 0.914884i \(0.632282\pi\)
\(840\) 0 0
\(841\) 57.8197 1.99378
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −17.1202 −0.588953
\(846\) 0 0
\(847\) − 10.0761i − 0.346218i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.51211i 0.188953i
\(852\) 0 0
\(853\) − 22.3667i − 0.765821i −0.923786 0.382910i \(-0.874922\pi\)
0.923786 0.382910i \(-0.125078\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 34.7703i 1.18773i 0.804564 + 0.593866i \(0.202399\pi\)
−0.804564 + 0.593866i \(0.797601\pi\)
\(858\) 0 0
\(859\) −16.1287 −0.550305 −0.275152 0.961401i \(-0.588728\pi\)
−0.275152 + 0.961401i \(0.588728\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −34.3951 −1.17082 −0.585412 0.810736i \(-0.699067\pi\)
−0.585412 + 0.810736i \(0.699067\pi\)
\(864\) 0 0
\(865\) −9.15194 −0.311175
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.21335 −0.0411600
\(870\) 0 0
\(871\) 18.9700i 0.642774i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.7609i 0.397592i
\(876\) 0 0
\(877\) − 14.0555i − 0.474620i −0.971434 0.237310i \(-0.923734\pi\)
0.971434 0.237310i \(-0.0762658\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 41.1209i − 1.38540i −0.721226 0.692700i \(-0.756421\pi\)
0.721226 0.692700i \(-0.243579\pi\)
\(882\) 0 0
\(883\) 18.8276 0.633598 0.316799 0.948493i \(-0.397392\pi\)
0.316799 + 0.948493i \(0.397392\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.8728 −0.365071 −0.182536 0.983199i \(-0.558430\pi\)
−0.182536 + 0.983199i \(0.558430\pi\)
\(888\) 0 0
\(889\) −16.6260 −0.557617
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.59846 0.287736
\(894\) 0 0
\(895\) − 35.5971i − 1.18988i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 17.2310i 0.574684i
\(900\) 0 0
\(901\) 25.3711i 0.845235i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 17.8045i − 0.591843i
\(906\) 0 0
\(907\) −15.4589 −0.513304 −0.256652 0.966504i \(-0.582619\pi\)
−0.256652 + 0.966504i \(0.582619\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.9492 −0.793473 −0.396737 0.917933i \(-0.629857\pi\)
−0.396737 + 0.917933i \(0.629857\pi\)
\(912\) 0 0
\(913\) −2.87878 −0.0952736
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −14.5603 −0.480823
\(918\) 0 0
\(919\) 24.4853i 0.807694i 0.914826 + 0.403847i \(0.132327\pi\)
−0.914826 + 0.403847i \(0.867673\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 31.2685i − 1.02922i
\(924\) 0 0
\(925\) 1.41947i 0.0466719i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.7122i 1.17168i 0.810428 + 0.585839i \(0.199235\pi\)
−0.810428 + 0.585839i \(0.800765\pi\)
\(930\) 0 0
\(931\) 3.66977 0.120272
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.60044 −0.215857
\(936\) 0 0
\(937\) 0.263959 0.00862315 0.00431157 0.999991i \(-0.498628\pi\)
0.00431157 + 0.999991i \(0.498628\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −40.9124 −1.33371 −0.666853 0.745189i \(-0.732359\pi\)
−0.666853 + 0.745189i \(0.732359\pi\)
\(942\) 0 0
\(943\) 0.755586i 0.0246053i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 43.3236i 1.40783i 0.710286 + 0.703913i \(0.248566\pi\)
−0.710286 + 0.703913i \(0.751434\pi\)
\(948\) 0 0
\(949\) − 73.7566i − 2.39424i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 52.0756i 1.68690i 0.537211 + 0.843448i \(0.319478\pi\)
−0.537211 + 0.843448i \(0.680522\pi\)
\(954\) 0 0
\(955\) −1.51791 −0.0491185
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17.2672 0.557588
\(960\) 0 0
\(961\) 27.5802 0.889684
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −32.9976 −1.06223
\(966\) 0 0
\(967\) 17.3116i 0.556705i 0.960479 + 0.278352i \(0.0897883\pi\)
−0.960479 + 0.278352i \(0.910212\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 8.86074i − 0.284355i −0.989841 0.142177i \(-0.954590\pi\)
0.989841 0.142177i \(-0.0454103\pi\)
\(972\) 0 0
\(973\) 14.5580i 0.466709i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 8.48509i − 0.271462i −0.990746 0.135731i \(-0.956662\pi\)
0.990746 0.135731i \(-0.0433383\pi\)
\(978\) 0 0
\(979\) 12.3986 0.396262
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −43.6181 −1.39120 −0.695600 0.718429i \(-0.744862\pi\)
−0.695600 + 0.718429i \(0.744862\pi\)
\(984\) 0 0
\(985\) −12.0570 −0.384168
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −20.7700 −0.660449
\(990\) 0 0
\(991\) 51.0937i 1.62304i 0.584322 + 0.811522i \(0.301361\pi\)
−0.584322 + 0.811522i \(0.698639\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 24.6513i 0.781498i
\(996\) 0 0
\(997\) 1.59455i 0.0504998i 0.999681 + 0.0252499i \(0.00803815\pi\)
−0.999681 + 0.0252499i \(0.991962\pi\)
\(998\) 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.j.c.5615.25 32
3.2 odd 2 inner 6048.2.j.c.5615.8 32
4.3 odd 2 1512.2.j.c.323.8 yes 32
8.3 odd 2 inner 6048.2.j.c.5615.7 32
8.5 even 2 1512.2.j.c.323.26 yes 32
12.11 even 2 1512.2.j.c.323.25 yes 32
24.5 odd 2 1512.2.j.c.323.7 32
24.11 even 2 inner 6048.2.j.c.5615.26 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.7 32 24.5 odd 2
1512.2.j.c.323.8 yes 32 4.3 odd 2
1512.2.j.c.323.25 yes 32 12.11 even 2
1512.2.j.c.323.26 yes 32 8.5 even 2
6048.2.j.c.5615.7 32 8.3 odd 2 inner
6048.2.j.c.5615.8 32 3.2 odd 2 inner
6048.2.j.c.5615.25 32 1.1 even 1 trivial
6048.2.j.c.5615.26 32 24.11 even 2 inner