Properties

Label 6048.2.j.a.5615.7
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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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: \(8\)
Coefficient field: 8.0.56070144.2
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.7
Root \(0.500000 + 2.19293i\)
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.a.5615.8

$q$-expansion

\(f(q)\) \(=\) \(q+2.38587 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+2.38587 q^{5} -1.00000i q^{7} -1.65382i q^{11} +0.253423i q^{13} +2.00000i q^{17} -7.03969 q^{19} -2.82481 q^{23} +0.692366 q^{25} -1.26795 q^{29} -0.746577i q^{31} -2.38587i q^{35} +6.94273i q^{37} -5.55686i q^{41} -8.67478 q^{43} -10.9679 q^{47} -1.00000 q^{49} -3.30763 q^{53} -3.94579i q^{55} +10.0397i q^{59} +4.53590i q^{61} +0.604635i q^{65} -0.253423 q^{67} -3.17519 q^{71} -3.05421 q^{73} -1.65382 q^{77} -14.5183i q^{79} +1.77480i q^{83} +4.77174i q^{85} +3.13550i q^{89} +0.253423 q^{91} -16.7958 q^{95} -5.49315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 16q^{19} - 16q^{23} + 32q^{25} - 24q^{29} - 8q^{43} + 8q^{47} - 8q^{49} - 8q^{67} - 32q^{71} + 8q^{73} + 8q^{91} - 112q^{95} - 32q^{97} + O(q^{100}) \)

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
\(5\) 2.38587 1.06699 0.533496 0.845802i \(-0.320878\pi\)
0.533496 + 0.845802i \(0.320878\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.65382i − 0.498645i −0.968421 0.249322i \(-0.919792\pi\)
0.968421 0.249322i \(-0.0802079\pi\)
\(12\) 0 0
\(13\) 0.253423i 0.0702870i 0.999382 + 0.0351435i \(0.0111888\pi\)
−0.999382 + 0.0351435i \(0.988811\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −7.03969 −1.61501 −0.807507 0.589858i \(-0.799184\pi\)
−0.807507 + 0.589858i \(0.799184\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82481 −0.589014 −0.294507 0.955649i \(-0.595155\pi\)
−0.294507 + 0.955649i \(0.595155\pi\)
\(24\) 0 0
\(25\) 0.692366 0.138473
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.26795 −0.235452 −0.117726 0.993046i \(-0.537560\pi\)
−0.117726 + 0.993046i \(0.537560\pi\)
\(30\) 0 0
\(31\) − 0.746577i − 0.134089i −0.997750 0.0670446i \(-0.978643\pi\)
0.997750 0.0670446i \(-0.0213570\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.38587i − 0.403285i
\(36\) 0 0
\(37\) 6.94273i 1.14138i 0.821166 + 0.570689i \(0.193324\pi\)
−0.821166 + 0.570689i \(0.806676\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.55686i − 0.867836i −0.900952 0.433918i \(-0.857131\pi\)
0.900952 0.433918i \(-0.142869\pi\)
\(42\) 0 0
\(43\) −8.67478 −1.32289 −0.661446 0.749993i \(-0.730057\pi\)
−0.661446 + 0.749993i \(0.730057\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.9679 −1.59983 −0.799915 0.600113i \(-0.795122\pi\)
−0.799915 + 0.600113i \(0.795122\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.30763 −0.454339 −0.227169 0.973855i \(-0.572947\pi\)
−0.227169 + 0.973855i \(0.572947\pi\)
\(54\) 0 0
\(55\) − 3.94579i − 0.532050i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0397i 1.30706i 0.756902 + 0.653528i \(0.226712\pi\)
−0.756902 + 0.653528i \(0.773288\pi\)
\(60\) 0 0
\(61\) 4.53590i 0.580762i 0.956911 + 0.290381i \(0.0937821\pi\)
−0.956911 + 0.290381i \(0.906218\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.604635i 0.0749957i
\(66\) 0 0
\(67\) −0.253423 −0.0309606 −0.0154803 0.999880i \(-0.504928\pi\)
−0.0154803 + 0.999880i \(0.504928\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.17519 −0.376826 −0.188413 0.982090i \(-0.560334\pi\)
−0.188413 + 0.982090i \(0.560334\pi\)
\(72\) 0 0
\(73\) −3.05421 −0.357468 −0.178734 0.983897i \(-0.557200\pi\)
−0.178734 + 0.983897i \(0.557200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.65382 −0.188470
\(78\) 0 0
\(79\) − 14.5183i − 1.63344i −0.577036 0.816719i \(-0.695791\pi\)
0.577036 0.816719i \(-0.304209\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.77480i 0.194809i 0.995245 + 0.0974046i \(0.0310541\pi\)
−0.995245 + 0.0974046i \(0.968946\pi\)
\(84\) 0 0
\(85\) 4.77174i 0.517567i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.13550i 0.332363i 0.986095 + 0.166181i \(0.0531437\pi\)
−0.986095 + 0.166181i \(0.946856\pi\)
\(90\) 0 0
\(91\) 0.253423 0.0265660
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.7958 −1.72321
\(96\) 0 0
\(97\) −5.49315 −0.557745 −0.278873 0.960328i \(-0.589961\pi\)
−0.278873 + 0.960328i \(0.589961\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) − 3.28247i − 0.323432i −0.986837 0.161716i \(-0.948297\pi\)
0.986837 0.161716i \(-0.0517028\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.1931i 1.27542i 0.770275 + 0.637712i \(0.220119\pi\)
−0.770275 + 0.637712i \(0.779881\pi\)
\(108\) 0 0
\(109\) 7.29311i 0.698553i 0.937020 + 0.349277i \(0.113573\pi\)
−0.937020 + 0.349277i \(0.886427\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 19.0076i − 1.78808i −0.447985 0.894041i \(-0.647858\pi\)
0.447985 0.894041i \(-0.352142\pi\)
\(114\) 0 0
\(115\) −6.73962 −0.628473
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 8.26489 0.751354
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.2774 −0.919243
\(126\) 0 0
\(127\) 5.05421i 0.448489i 0.974533 + 0.224244i \(0.0719914\pi\)
−0.974533 + 0.224244i \(0.928009\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.6893i 1.02130i 0.859789 + 0.510650i \(0.170595\pi\)
−0.859789 + 0.510650i \(0.829405\pi\)
\(132\) 0 0
\(133\) 7.03969i 0.610418i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.03211i 0.0881793i 0.999028 + 0.0440896i \(0.0140387\pi\)
−0.999028 + 0.0440896i \(0.985961\pi\)
\(138\) 0 0
\(139\) 12.0794 1.02456 0.512279 0.858819i \(-0.328801\pi\)
0.512279 + 0.858819i \(0.328801\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.419116 0.0350482
\(144\) 0 0
\(145\) −3.02516 −0.251226
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.1213 −1.48455 −0.742277 0.670093i \(-0.766254\pi\)
−0.742277 + 0.670093i \(0.766254\pi\)
\(150\) 0 0
\(151\) 19.4289i 1.58110i 0.612395 + 0.790552i \(0.290206\pi\)
−0.612395 + 0.790552i \(0.709794\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.78123i − 0.143072i
\(156\) 0 0
\(157\) − 18.2183i − 1.45397i −0.686651 0.726987i \(-0.740920\pi\)
0.686651 0.726987i \(-0.259080\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.82481i 0.222626i
\(162\) 0 0
\(163\) −3.71753 −0.291179 −0.145590 0.989345i \(-0.546508\pi\)
−0.145590 + 0.989345i \(0.546508\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.97095 −0.462046 −0.231023 0.972948i \(-0.574207\pi\)
−0.231023 + 0.972948i \(0.574207\pi\)
\(168\) 0 0
\(169\) 12.9358 0.995060
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 16.0858 1.22298 0.611491 0.791252i \(-0.290570\pi\)
0.611491 + 0.791252i \(0.290570\pi\)
\(174\) 0 0
\(175\) − 0.692366i − 0.0523379i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 13.3786i − 0.999964i −0.866036 0.499982i \(-0.833340\pi\)
0.866036 0.499982i \(-0.166660\pi\)
\(180\) 0 0
\(181\) 20.4214i 1.51791i 0.651145 + 0.758954i \(0.274289\pi\)
−0.651145 + 0.758954i \(0.725711\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.5644i 1.21784i
\(186\) 0 0
\(187\) 3.30763 0.241878
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.9889 1.44634 0.723171 0.690669i \(-0.242684\pi\)
0.723171 + 0.690669i \(0.242684\pi\)
\(192\) 0 0
\(193\) 19.0076 1.36820 0.684098 0.729391i \(-0.260196\pi\)
0.684098 + 0.729391i \(0.260196\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −16.9573 −1.20815 −0.604077 0.796926i \(-0.706458\pi\)
−0.604077 + 0.796926i \(0.706458\pi\)
\(198\) 0 0
\(199\) − 25.4541i − 1.80439i −0.431326 0.902196i \(-0.641954\pi\)
0.431326 0.902196i \(-0.358046\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.26795i 0.0889926i
\(204\) 0 0
\(205\) − 13.2579i − 0.925974i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.6424i 0.805318i
\(210\) 0 0
\(211\) 19.5435 1.34543 0.672714 0.739903i \(-0.265128\pi\)
0.672714 + 0.739903i \(0.265128\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −20.6969 −1.41152
\(216\) 0 0
\(217\) −0.746577 −0.0506809
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.506847 −0.0340942
\(222\) 0 0
\(223\) − 3.09696i − 0.207388i −0.994609 0.103694i \(-0.966934\pi\)
0.994609 0.103694i \(-0.0330662\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 23.5854i − 1.56542i −0.622388 0.782709i \(-0.713837\pi\)
0.622388 0.782709i \(-0.286163\pi\)
\(228\) 0 0
\(229\) 6.53590i 0.431904i 0.976404 + 0.215952i \(0.0692855\pi\)
−0.976404 + 0.215952i \(0.930714\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.5359i 0.690230i 0.938560 + 0.345115i \(0.112160\pi\)
−0.938560 + 0.345115i \(0.887840\pi\)
\(234\) 0 0
\(235\) −26.1679 −1.70701
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −22.5298 −1.45733 −0.728665 0.684870i \(-0.759859\pi\)
−0.728665 + 0.684870i \(0.759859\pi\)
\(240\) 0 0
\(241\) −9.53201 −0.614010 −0.307005 0.951708i \(-0.599327\pi\)
−0.307005 + 0.951708i \(0.599327\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.38587 −0.152428
\(246\) 0 0
\(247\) − 1.78402i − 0.113515i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.1931i 1.71641i 0.513305 + 0.858206i \(0.328421\pi\)
−0.513305 + 0.858206i \(0.671579\pi\)
\(252\) 0 0
\(253\) 4.67172i 0.293708i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.2568i 1.20121i 0.799547 + 0.600603i \(0.205073\pi\)
−0.799547 + 0.600603i \(0.794927\pi\)
\(258\) 0 0
\(259\) 6.94273 0.431400
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.1828 0.751221 0.375611 0.926778i \(-0.377433\pi\)
0.375611 + 0.926778i \(0.377433\pi\)
\(264\) 0 0
\(265\) −7.89158 −0.484776
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.1149 −1.59225 −0.796126 0.605132i \(-0.793121\pi\)
−0.796126 + 0.605132i \(0.793121\pi\)
\(270\) 0 0
\(271\) − 31.3999i − 1.90741i −0.300749 0.953703i \(-0.597237\pi\)
0.300749 0.953703i \(-0.402763\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 1.14505i − 0.0690489i
\(276\) 0 0
\(277\) − 12.8846i − 0.774162i −0.922046 0.387081i \(-0.873483\pi\)
0.922046 0.387081i \(-0.126517\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.35262i − 0.498276i −0.968468 0.249138i \(-0.919853\pi\)
0.968468 0.249138i \(-0.0801472\pi\)
\(282\) 0 0
\(283\) 4.45041 0.264549 0.132275 0.991213i \(-0.457772\pi\)
0.132275 + 0.991213i \(0.457772\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.55686 −0.328011
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −23.0495 −1.34657 −0.673283 0.739385i \(-0.735116\pi\)
−0.673283 + 0.739385i \(0.735116\pi\)
\(294\) 0 0
\(295\) 23.9534i 1.39462i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 0.715873i − 0.0414000i
\(300\) 0 0
\(301\) 8.67478i 0.500006i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.8221i 0.619669i
\(306\) 0 0
\(307\) 13.4610 0.768262 0.384131 0.923279i \(-0.374501\pi\)
0.384131 + 0.923279i \(0.374501\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −33.5038 −1.89983 −0.949913 0.312515i \(-0.898828\pi\)
−0.949913 + 0.312515i \(0.898828\pi\)
\(312\) 0 0
\(313\) 4.94579 0.279553 0.139776 0.990183i \(-0.455362\pi\)
0.139776 + 0.990183i \(0.455362\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.4029 −1.37061 −0.685303 0.728258i \(-0.740330\pi\)
−0.685303 + 0.728258i \(0.740330\pi\)
\(318\) 0 0
\(319\) 2.09696i 0.117407i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 14.0794i − 0.783397i
\(324\) 0 0
\(325\) 0.175462i 0.00973287i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.9679i 0.604679i
\(330\) 0 0
\(331\) 17.9648 0.987436 0.493718 0.869622i \(-0.335637\pi\)
0.493718 + 0.869622i \(0.335637\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.604635 −0.0330347
\(336\) 0 0
\(337\) −7.02905 −0.382897 −0.191448 0.981503i \(-0.561318\pi\)
−0.191448 + 0.981503i \(0.561318\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.23470 −0.0668628
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 0.473599i − 0.0254241i −0.999919 0.0127121i \(-0.995954\pi\)
0.999919 0.0127121i \(-0.00404648\pi\)
\(348\) 0 0
\(349\) 10.5007i 0.562091i 0.959694 + 0.281045i \(0.0906812\pi\)
−0.959694 + 0.281045i \(0.909319\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 4.48506i − 0.238716i −0.992851 0.119358i \(-0.961916\pi\)
0.992851 0.119358i \(-0.0380836\pi\)
\(354\) 0 0
\(355\) −7.57558 −0.402070
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.60770 −0.0848509 −0.0424255 0.999100i \(-0.513508\pi\)
−0.0424255 + 0.999100i \(0.513508\pi\)
\(360\) 0 0
\(361\) 30.5572 1.60827
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7.28694 −0.381416
\(366\) 0 0
\(367\) − 18.4968i − 0.965527i −0.875751 0.482763i \(-0.839633\pi\)
0.875751 0.482763i \(-0.160367\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.30763i 0.171724i
\(372\) 0 0
\(373\) − 26.1358i − 1.35326i −0.736322 0.676631i \(-0.763439\pi\)
0.736322 0.676631i \(-0.236561\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 0.321328i − 0.0165492i
\(378\) 0 0
\(379\) −15.1816 −0.779828 −0.389914 0.920851i \(-0.627495\pi\)
−0.389914 + 0.920851i \(0.627495\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.47780 −0.126609 −0.0633047 0.997994i \(-0.520164\pi\)
−0.0633047 + 0.997994i \(0.520164\pi\)
\(384\) 0 0
\(385\) −3.94579 −0.201096
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −11.4932 −0.582726 −0.291363 0.956613i \(-0.594109\pi\)
−0.291363 + 0.956613i \(0.594109\pi\)
\(390\) 0 0
\(391\) − 5.64962i − 0.285714i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 34.6388i − 1.74287i
\(396\) 0 0
\(397\) − 5.57252i − 0.279677i −0.990174 0.139838i \(-0.955342\pi\)
0.990174 0.139838i \(-0.0446583\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 24.0045i − 1.19873i −0.800477 0.599364i \(-0.795420\pi\)
0.800477 0.599364i \(-0.204580\pi\)
\(402\) 0 0
\(403\) 0.189200 0.00942472
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.4820 0.569142
\(408\) 0 0
\(409\) 6.24503 0.308797 0.154398 0.988009i \(-0.450656\pi\)
0.154398 + 0.988009i \(0.450656\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.0397 0.494021
\(414\) 0 0
\(415\) 4.23443i 0.207860i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 2.80079i − 0.136827i −0.997657 0.0684137i \(-0.978206\pi\)
0.997657 0.0684137i \(-0.0217938\pi\)
\(420\) 0 0
\(421\) − 10.8634i − 0.529448i −0.964324 0.264724i \(-0.914719\pi\)
0.964324 0.264724i \(-0.0852808\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.38473i 0.0671693i
\(426\) 0 0
\(427\) 4.53590 0.219508
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.87513 −0.427500 −0.213750 0.976888i \(-0.568568\pi\)
−0.213750 + 0.976888i \(0.568568\pi\)
\(432\) 0 0
\(433\) 10.0115 0.481120 0.240560 0.970634i \(-0.422669\pi\)
0.240560 + 0.970634i \(0.422669\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 19.8858 0.951266
\(438\) 0 0
\(439\) 31.3999i 1.49863i 0.662211 + 0.749317i \(0.269618\pi\)
−0.662211 + 0.749317i \(0.730382\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.7751i 1.69973i 0.527003 + 0.849863i \(0.323316\pi\)
−0.527003 + 0.849863i \(0.676684\pi\)
\(444\) 0 0
\(445\) 7.48090i 0.354629i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.3443i 0.960105i 0.877240 + 0.480052i \(0.159382\pi\)
−0.877240 + 0.480052i \(0.840618\pi\)
\(450\) 0 0
\(451\) −9.19003 −0.432742
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.604635 0.0283457
\(456\) 0 0
\(457\) −24.3205 −1.13767 −0.568833 0.822453i \(-0.692605\pi\)
−0.568833 + 0.822453i \(0.692605\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.5423 1.42250 0.711249 0.702940i \(-0.248130\pi\)
0.711249 + 0.702940i \(0.248130\pi\)
\(462\) 0 0
\(463\) 1.71612i 0.0797547i 0.999205 + 0.0398774i \(0.0126967\pi\)
−0.999205 + 0.0398774i \(0.987303\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 3.60994i − 0.167048i −0.996506 0.0835239i \(-0.973382\pi\)
0.996506 0.0835239i \(-0.0266175\pi\)
\(468\) 0 0
\(469\) 0.253423i 0.0117020i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14.3465i 0.659653i
\(474\) 0 0
\(475\) −4.87404 −0.223636
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.62057 0.165428 0.0827140 0.996573i \(-0.473641\pi\)
0.0827140 + 0.996573i \(0.473641\pi\)
\(480\) 0 0
\(481\) −1.75945 −0.0802240
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −13.1059 −0.595110
\(486\) 0 0
\(487\) 10.3808i 0.470401i 0.971947 + 0.235200i \(0.0755746\pi\)
−0.971947 + 0.235200i \(0.924425\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.66139i 0.120107i 0.998195 + 0.0600534i \(0.0191271\pi\)
−0.998195 + 0.0600534i \(0.980873\pi\)
\(492\) 0 0
\(493\) − 2.53590i − 0.114211i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.17519i 0.142427i
\(498\) 0 0
\(499\) −33.1755 −1.48514 −0.742570 0.669769i \(-0.766393\pi\)
−0.742570 + 0.669769i \(0.766393\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.2679 0.502413 0.251207 0.967934i \(-0.419173\pi\)
0.251207 + 0.967934i \(0.419173\pi\)
\(504\) 0 0
\(505\) −4.77174 −0.212339
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −38.9221 −1.72519 −0.862595 0.505894i \(-0.831163\pi\)
−0.862595 + 0.505894i \(0.831163\pi\)
\(510\) 0 0
\(511\) 3.05421i 0.135110i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 7.83155i − 0.345099i
\(516\) 0 0
\(517\) 18.1389i 0.797747i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 37.1797i − 1.62887i −0.580253 0.814436i \(-0.697046\pi\)
0.580253 0.814436i \(-0.302954\pi\)
\(522\) 0 0
\(523\) 5.58846 0.244366 0.122183 0.992508i \(-0.461010\pi\)
0.122183 + 0.992508i \(0.461010\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.49315 0.0650428
\(528\) 0 0
\(529\) −15.0204 −0.653063
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.40824 0.0609976
\(534\) 0 0
\(535\) 31.4770i 1.36087i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.65382i 0.0712349i
\(540\) 0 0
\(541\) 31.8358i 1.36873i 0.729141 + 0.684363i \(0.239920\pi\)
−0.729141 + 0.684363i \(0.760080\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 17.4004i 0.745351i
\(546\) 0 0
\(547\) 29.9648 1.28120 0.640602 0.767873i \(-0.278685\pi\)
0.640602 + 0.767873i \(0.278685\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.92596 0.380259
\(552\) 0 0
\(553\) −14.5183 −0.617381
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.9573 −0.887987 −0.443994 0.896030i \(-0.646439\pi\)
−0.443994 + 0.896030i \(0.646439\pi\)
\(558\) 0 0
\(559\) − 2.19839i − 0.0929821i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 5.56801i − 0.234664i −0.993093 0.117332i \(-0.962566\pi\)
0.993093 0.117332i \(-0.0374341\pi\)
\(564\) 0 0
\(565\) − 45.3496i − 1.90787i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 24.3571i − 1.02110i −0.859847 0.510552i \(-0.829441\pi\)
0.859847 0.510552i \(-0.170559\pi\)
\(570\) 0 0
\(571\) −29.3496 −1.22824 −0.614120 0.789212i \(-0.710489\pi\)
−0.614120 + 0.789212i \(0.710489\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.95580 −0.0815626
\(576\) 0 0
\(577\) −32.8992 −1.36961 −0.684805 0.728727i \(-0.740113\pi\)
−0.684805 + 0.728727i \(0.740113\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.77480 0.0736309
\(582\) 0 0
\(583\) 5.47022i 0.226553i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.9939i 0.742687i 0.928496 + 0.371343i \(0.121103\pi\)
−0.928496 + 0.371343i \(0.878897\pi\)
\(588\) 0 0
\(589\) 5.25566i 0.216556i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.72702i 0.276246i 0.990415 + 0.138123i \(0.0441069\pi\)
−0.990415 + 0.138123i \(0.955893\pi\)
\(594\) 0 0
\(595\) 4.77174 0.195622
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10.9752 −0.448433 −0.224216 0.974539i \(-0.571982\pi\)
−0.224216 + 0.974539i \(0.571982\pi\)
\(600\) 0 0
\(601\) −13.0542 −0.532492 −0.266246 0.963905i \(-0.585783\pi\)
−0.266246 + 0.963905i \(0.585783\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 19.7189 0.801689
\(606\) 0 0
\(607\) 23.2350i 0.943080i 0.881845 + 0.471540i \(0.156302\pi\)
−0.881845 + 0.471540i \(0.843698\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.77952i − 0.112447i
\(612\) 0 0
\(613\) − 31.6854i − 1.27976i −0.768474 0.639881i \(-0.778984\pi\)
0.768474 0.639881i \(-0.221016\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 31.7312i − 1.27745i −0.769435 0.638726i \(-0.779462\pi\)
0.769435 0.638726i \(-0.220538\pi\)
\(618\) 0 0
\(619\) 48.4602 1.94778 0.973890 0.227019i \(-0.0728979\pi\)
0.973890 + 0.227019i \(0.0728979\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.13550 0.125621
\(624\) 0 0
\(625\) −27.9825 −1.11930
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −13.8855 −0.553649
\(630\) 0 0
\(631\) 23.9472i 0.953325i 0.879086 + 0.476662i \(0.158154\pi\)
−0.879086 + 0.476662i \(0.841846\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0587i 0.478534i
\(636\) 0 0
\(637\) − 0.253423i − 0.0100410i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 3.24953i − 0.128349i −0.997939 0.0641744i \(-0.979559\pi\)
0.997939 0.0641744i \(-0.0204414\pi\)
\(642\) 0 0
\(643\) 24.2808 0.957542 0.478771 0.877940i \(-0.341082\pi\)
0.478771 + 0.877940i \(0.341082\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −36.6388 −1.44042 −0.720209 0.693757i \(-0.755954\pi\)
−0.720209 + 0.693757i \(0.755954\pi\)
\(648\) 0 0
\(649\) 16.6038 0.651756
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.8748 −0.621230 −0.310615 0.950536i \(-0.600535\pi\)
−0.310615 + 0.950536i \(0.600535\pi\)
\(654\) 0 0
\(655\) 27.8891i 1.08972i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 23.4538i − 0.913630i −0.889562 0.456815i \(-0.848990\pi\)
0.889562 0.456815i \(-0.151010\pi\)
\(660\) 0 0
\(661\) − 25.0090i − 0.972737i −0.873754 0.486368i \(-0.838321\pi\)
0.873754 0.486368i \(-0.161679\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.7958i 0.651312i
\(666\) 0 0
\(667\) 3.58172 0.138685
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.50155 0.289594
\(672\) 0 0
\(673\) −22.8137 −0.879402 −0.439701 0.898144i \(-0.644916\pi\)
−0.439701 + 0.898144i \(0.644916\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.6851 1.14089 0.570446 0.821335i \(-0.306770\pi\)
0.570446 + 0.821335i \(0.306770\pi\)
\(678\) 0 0
\(679\) 5.49315i 0.210808i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.8469i 1.40991i 0.709253 + 0.704954i \(0.249032\pi\)
−0.709253 + 0.704954i \(0.750968\pi\)
\(684\) 0 0
\(685\) 2.46248i 0.0940866i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 0.838232i − 0.0319341i
\(690\) 0 0
\(691\) −30.7556 −1.17000 −0.584998 0.811035i \(-0.698905\pi\)
−0.584998 + 0.811035i \(0.698905\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 28.8198 1.09320
\(696\) 0 0
\(697\) 11.1137 0.420962
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.88852 0.184637 0.0923184 0.995730i \(-0.470572\pi\)
0.0923184 + 0.995730i \(0.470572\pi\)
\(702\) 0 0
\(703\) − 48.8746i − 1.84334i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.00000i 0.0752177i
\(708\) 0 0
\(709\) − 27.1221i − 1.01859i −0.860591 0.509296i \(-0.829906\pi\)
0.860591 0.509296i \(-0.170094\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.10894i 0.0789803i
\(714\) 0 0
\(715\) 0.999956 0.0373962
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.95501 0.110203 0.0551017 0.998481i \(-0.482452\pi\)
0.0551017 + 0.998481i \(0.482452\pi\)
\(720\) 0 0
\(721\) −3.28247 −0.122246
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.877885 −0.0326038
\(726\) 0 0
\(727\) 18.3923i 0.682133i 0.940039 + 0.341066i \(0.110788\pi\)
−0.940039 + 0.341066i \(0.889212\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 17.3496i − 0.641697i
\(732\) 0 0
\(733\) − 19.6885i − 0.727210i −0.931553 0.363605i \(-0.881546\pi\)
0.931553 0.363605i \(-0.118454\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.419116i 0.0154383i
\(738\) 0 0
\(739\) −8.03517 −0.295579 −0.147789 0.989019i \(-0.547216\pi\)
−0.147789 + 0.989019i \(0.547216\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −29.4820 −1.08159 −0.540795 0.841154i \(-0.681876\pi\)
−0.540795 + 0.841154i \(0.681876\pi\)
\(744\) 0 0
\(745\) −43.2350 −1.58401
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.1931 0.482065
\(750\) 0 0
\(751\) 0.103077i 0.00376132i 0.999998 + 0.00188066i \(0.000598633\pi\)
−0.999998 + 0.00188066i \(0.999401\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 46.3549i 1.68703i
\(756\) 0 0
\(757\) 32.1006i 1.16672i 0.812215 + 0.583359i \(0.198262\pi\)
−0.812215 + 0.583359i \(0.801738\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 39.3228i 1.42545i 0.701444 + 0.712725i \(0.252539\pi\)
−0.701444 + 0.712725i \(0.747461\pi\)
\(762\) 0 0
\(763\) 7.29311 0.264028
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.54429 −0.0918690
\(768\) 0 0
\(769\) −38.1199 −1.37464 −0.687319 0.726356i \(-0.741213\pi\)
−0.687319 + 0.726356i \(0.741213\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.6583 0.922866 0.461433 0.887175i \(-0.347335\pi\)
0.461433 + 0.887175i \(0.347335\pi\)
\(774\) 0 0
\(775\) − 0.516904i − 0.0185677i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 39.1186i 1.40157i
\(780\) 0 0
\(781\) 5.25118i 0.187902i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 43.4663i − 1.55138i
\(786\) 0 0
\(787\) −22.7785 −0.811965 −0.405983 0.913881i \(-0.633071\pi\)
−0.405983 + 0.913881i \(0.633071\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −19.0076 −0.675832
\(792\) 0 0
\(793\) −1.14950 −0.0408201
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.04306 −0.249478 −0.124739 0.992190i \(-0.539809\pi\)
−0.124739 + 0.992190i \(0.539809\pi\)
\(798\) 0 0
\(799\) − 21.9358i − 0.776032i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.05111i 0.178250i
\(804\) 0 0
\(805\) 6.73962i 0.237541i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 36.1772i − 1.27192i −0.771722 0.635961i \(-0.780604\pi\)
0.771722 0.635961i \(-0.219396\pi\)
\(810\) 0 0
\(811\) 7.84047 0.275316 0.137658 0.990480i \(-0.456042\pi\)
0.137658 + 0.990480i \(0.456042\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.86952 −0.310686
\(816\) 0 0
\(817\) 61.0677 2.13649
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 52.1342 1.81950 0.909748 0.415161i \(-0.136275\pi\)
0.909748 + 0.415161i \(0.136275\pi\)
\(822\) 0 0
\(823\) 48.1923i 1.67988i 0.542682 + 0.839939i \(0.317409\pi\)
−0.542682 + 0.839939i \(0.682591\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 34.6966i − 1.20652i −0.797545 0.603259i \(-0.793869\pi\)
0.797545 0.603259i \(-0.206131\pi\)
\(828\) 0 0
\(829\) − 12.5848i − 0.437088i −0.975827 0.218544i \(-0.929869\pi\)
0.975827 0.218544i \(-0.0701308\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.00000i − 0.0692959i
\(834\) 0 0
\(835\) −14.2459 −0.493000
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.30987 0.321413 0.160706 0.987002i \(-0.448623\pi\)
0.160706 + 0.987002i \(0.448623\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 30.8631 1.06172
\(846\) 0 0
\(847\) − 8.26489i − 0.283985i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 19.6119i − 0.672287i
\(852\) 0 0
\(853\) − 31.3099i − 1.07203i −0.844208 0.536015i \(-0.819929\pi\)
0.844208 0.536015i \(-0.180071\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 7.14838i − 0.244184i −0.992519 0.122092i \(-0.961040\pi\)
0.992519 0.122092i \(-0.0389603\pi\)
\(858\) 0 0
\(859\) 14.7022 0.501632 0.250816 0.968035i \(-0.419301\pi\)
0.250816 + 0.968035i \(0.419301\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.8640 −1.05062 −0.525311 0.850910i \(-0.676051\pi\)
−0.525311 + 0.850910i \(0.676051\pi\)
\(864\) 0 0
\(865\) 38.3786 1.30491
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −24.0106 −0.814505
\(870\) 0 0
\(871\) − 0.0642234i − 0.00217613i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.2774i 0.347441i
\(876\) 0 0
\(877\) − 43.9570i − 1.48432i −0.670221 0.742162i \(-0.733801\pi\)
0.670221 0.742162i \(-0.266199\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 10.7432i − 0.361948i −0.983488 0.180974i \(-0.942075\pi\)
0.983488 0.180974i \(-0.0579249\pi\)
\(882\) 0 0
\(883\) 36.8640 1.24057 0.620286 0.784376i \(-0.287017\pi\)
0.620286 + 0.784376i \(0.287017\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.0710 1.07684 0.538420 0.842677i \(-0.319022\pi\)
0.538420 + 0.842677i \(0.319022\pi\)
\(888\) 0 0
\(889\) 5.05421 0.169513
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 77.2105 2.58375
\(894\) 0 0
\(895\) − 31.9196i − 1.06695i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.946621i 0.0315716i
\(900\) 0 0
\(901\) − 6.61527i − 0.220387i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 48.7227i 1.61960i
\(906\) 0 0
\(907\) −12.2733 −0.407528 −0.203764 0.979020i \(-0.565317\pi\)
−0.203764 + 0.979020i \(0.565317\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 39.9487 1.32356 0.661779 0.749699i \(-0.269802\pi\)
0.661779 + 0.749699i \(0.269802\pi\)
\(912\) 0 0
\(913\) 2.93519 0.0971406
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.6893 0.386015
\(918\) 0 0
\(919\) 27.5374i 0.908373i 0.890907 + 0.454187i \(0.150070\pi\)
−0.890907 + 0.454187i \(0.849930\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 0.804668i − 0.0264860i
\(924\) 0 0
\(925\) 4.80691i 0.158050i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 43.6938i 1.43355i 0.697306 + 0.716774i \(0.254382\pi\)
−0.697306 + 0.716774i \(0.745618\pi\)
\(930\) 0 0
\(931\) 7.03969 0.230716
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.89158 0.258082
\(936\) 0 0
\(937\) −32.3923 −1.05821 −0.529105 0.848556i \(-0.677472\pi\)
−0.529105 + 0.848556i \(0.677472\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −23.4806 −0.765446 −0.382723 0.923863i \(-0.625014\pi\)
−0.382723 + 0.923863i \(0.625014\pi\)
\(942\) 0 0
\(943\) 15.6971i 0.511167i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.76079i 0.284687i 0.989817 + 0.142344i \(0.0454638\pi\)
−0.989817 + 0.142344i \(0.954536\pi\)
\(948\) 0 0
\(949\) − 0.774009i − 0.0251254i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 42.8617i − 1.38843i −0.719769 0.694214i \(-0.755752\pi\)
0.719769 0.694214i \(-0.244248\pi\)
\(954\) 0 0
\(955\) 47.6908 1.54324
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.03211 0.0333286
\(960\) 0 0
\(961\) 30.4426 0.982020
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 45.3496 1.45985
\(966\) 0 0
\(967\) − 31.8855i − 1.02537i −0.858578 0.512684i \(-0.828651\pi\)
0.858578 0.512684i \(-0.171349\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 19.2518i 0.617819i 0.951091 + 0.308909i \(0.0999640\pi\)
−0.951091 + 0.308909i \(0.900036\pi\)
\(972\) 0 0
\(973\) − 12.0794i − 0.387247i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 54.2449i − 1.73545i −0.497047 0.867723i \(-0.665582\pi\)
0.497047 0.867723i \(-0.334418\pi\)
\(978\) 0 0
\(979\) 5.18555 0.165731
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17.3764 0.554220 0.277110 0.960838i \(-0.410623\pi\)
0.277110 + 0.960838i \(0.410623\pi\)
\(984\) 0 0
\(985\) −40.4578 −1.28909
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.5046 0.779201
\(990\) 0 0
\(991\) 13.3107i 0.422829i 0.977397 + 0.211414i \(0.0678069\pi\)
−0.977397 + 0.211414i \(0.932193\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 60.7301i − 1.92527i
\(996\) 0 0
\(997\) 49.6564i 1.57263i 0.617824 + 0.786317i \(0.288014\pi\)
−0.617824 + 0.786317i \(0.711986\pi\)
\(998\) 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.a.5615.7 8
3.2 odd 2 6048.2.j.b.5615.1 8
4.3 odd 2 1512.2.j.a.323.6 8
8.3 odd 2 6048.2.j.b.5615.2 8
8.5 even 2 1512.2.j.b.323.1 yes 8
12.11 even 2 1512.2.j.b.323.3 yes 8
24.5 odd 2 1512.2.j.a.323.8 yes 8
24.11 even 2 inner 6048.2.j.a.5615.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.a.323.6 8 4.3 odd 2
1512.2.j.a.323.8 yes 8 24.5 odd 2
1512.2.j.b.323.1 yes 8 8.5 even 2
1512.2.j.b.323.3 yes 8 12.11 even 2
6048.2.j.a.5615.7 8 1.1 even 1 trivial
6048.2.j.a.5615.8 8 24.11 even 2 inner
6048.2.j.b.5615.1 8 3.2 odd 2
6048.2.j.b.5615.2 8 8.3 odd 2