Properties

Label 6048.2.a.bm.1.2
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 1.2
Root \(3.15690\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.766021 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-0.766021 q^{5} -1.00000 q^{7} -5.54779 q^{11} +6.31381 q^{13} -7.15032 q^{17} +6.15032 q^{19} +7.17923 q^{23} -4.41321 q^{25} -3.16349 q^{29} +0.163489 q^{31} +0.766021 q^{35} +5.31381 q^{37} -0.865425 q^{41} +8.31381 q^{43} +2.36855 q^{47} +1.00000 q^{49} -4.00000 q^{53} +4.24972 q^{55} +2.36855 q^{59} -13.5635 q^{61} -4.83651 q^{65} -7.05092 q^{67} +7.86159 q^{71} -1.98683 q^{73} +5.54779 q^{77} -5.24972 q^{79} -12.5282 q^{83} +5.47730 q^{85} -4.86542 q^{89} -6.31381 q^{91} -4.71128 q^{95} +3.26289 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q - 2q^{5} - 4q^{7} + 2q^{11} - 8q^{17} + 4q^{19} + 2q^{23} + 8q^{25} - 8q^{29} - 4q^{31} + 2q^{35} - 4q^{37} - 2q^{41} + 8q^{43} + 12q^{47} + 4q^{49} - 16q^{53} - 4q^{55} + 12q^{59} - 8q^{61} - 24q^{65} - 8q^{67} - 18q^{71} + 8q^{73} - 2q^{77} - 8q^{85} - 18q^{89} + 10q^{95} + 8q^{97} + O(q^{100}) \)

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
\(5\) −0.766021 −0.342575 −0.171288 0.985221i \(-0.554793\pi\)
−0.171288 + 0.985221i \(0.554793\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.54779 −1.67272 −0.836360 0.548180i \(-0.815321\pi\)
−0.836360 + 0.548180i \(0.815321\pi\)
\(12\) 0 0
\(13\) 6.31381 1.75114 0.875568 0.483096i \(-0.160488\pi\)
0.875568 + 0.483096i \(0.160488\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.15032 −1.73421 −0.867104 0.498128i \(-0.834021\pi\)
−0.867104 + 0.498128i \(0.834021\pi\)
\(18\) 0 0
\(19\) 6.15032 1.41098 0.705490 0.708720i \(-0.250727\pi\)
0.705490 + 0.708720i \(0.250727\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.17923 1.49697 0.748487 0.663150i \(-0.230781\pi\)
0.748487 + 0.663150i \(0.230781\pi\)
\(24\) 0 0
\(25\) −4.41321 −0.882642
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.16349 −0.587445 −0.293723 0.955891i \(-0.594894\pi\)
−0.293723 + 0.955891i \(0.594894\pi\)
\(30\) 0 0
\(31\) 0.163489 0.0293635 0.0146817 0.999892i \(-0.495326\pi\)
0.0146817 + 0.999892i \(0.495326\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.766021 0.129481
\(36\) 0 0
\(37\) 5.31381 0.873585 0.436792 0.899562i \(-0.356114\pi\)
0.436792 + 0.899562i \(0.356114\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.865425 −0.135157 −0.0675783 0.997714i \(-0.521527\pi\)
−0.0675783 + 0.997714i \(0.521527\pi\)
\(42\) 0 0
\(43\) 8.31381 1.26784 0.633922 0.773397i \(-0.281444\pi\)
0.633922 + 0.773397i \(0.281444\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.36855 0.345489 0.172745 0.984967i \(-0.444736\pi\)
0.172745 + 0.984967i \(0.444736\pi\)
\(48\) 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
\(55\) 4.24972 0.573032
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.36855 0.308359 0.154180 0.988043i \(-0.450727\pi\)
0.154180 + 0.988043i \(0.450727\pi\)
\(60\) 0 0
\(61\) −13.5635 −1.73663 −0.868316 0.496011i \(-0.834797\pi\)
−0.868316 + 0.496011i \(0.834797\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.83651 −0.599895
\(66\) 0 0
\(67\) −7.05092 −0.861406 −0.430703 0.902494i \(-0.641734\pi\)
−0.430703 + 0.902494i \(0.641734\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.86159 0.933000 0.466500 0.884521i \(-0.345515\pi\)
0.466500 + 0.884521i \(0.345515\pi\)
\(72\) 0 0
\(73\) −1.98683 −0.232541 −0.116270 0.993218i \(-0.537094\pi\)
−0.116270 + 0.993218i \(0.537094\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.54779 0.632229
\(78\) 0 0
\(79\) −5.24972 −0.590640 −0.295320 0.955398i \(-0.595426\pi\)
−0.295320 + 0.955398i \(0.595426\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.5282 −1.37515 −0.687575 0.726113i \(-0.741325\pi\)
−0.687575 + 0.726113i \(0.741325\pi\)
\(84\) 0 0
\(85\) 5.47730 0.594096
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.86542 −0.515734 −0.257867 0.966180i \(-0.583020\pi\)
−0.257867 + 0.966180i \(0.583020\pi\)
\(90\) 0 0
\(91\) −6.31381 −0.661867
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.71128 −0.483367
\(96\) 0 0
\(97\) 3.26289 0.331297 0.165648 0.986185i \(-0.447028\pi\)
0.165648 + 0.986185i \(0.447028\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.1503 −1.50751 −0.753757 0.657154i \(-0.771760\pi\)
−0.753757 + 0.657154i \(0.771760\pi\)
\(102\) 0 0
\(103\) −1.42638 −0.140546 −0.0702728 0.997528i \(-0.522387\pi\)
−0.0702728 + 0.997528i \(0.522387\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.2459 1.37720 0.688601 0.725140i \(-0.258225\pi\)
0.688601 + 0.725140i \(0.258225\pi\)
\(108\) 0 0
\(109\) −17.7270 −1.69794 −0.848970 0.528441i \(-0.822777\pi\)
−0.848970 + 0.528441i \(0.822777\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.09557 0.667495 0.333748 0.942662i \(-0.391687\pi\)
0.333748 + 0.942662i \(0.391687\pi\)
\(114\) 0 0
\(115\) −5.49945 −0.512826
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.15032 0.655469
\(120\) 0 0
\(121\) 19.7779 1.79799
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.21072 0.644946
\(126\) 0 0
\(127\) −5.37789 −0.477211 −0.238605 0.971117i \(-0.576690\pi\)
−0.238605 + 0.971117i \(0.576690\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.69553 0.759732 0.379866 0.925042i \(-0.375970\pi\)
0.379866 + 0.925042i \(0.375970\pi\)
\(132\) 0 0
\(133\) −6.15032 −0.533300
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0917 1.54568 0.772841 0.634600i \(-0.218835\pi\)
0.772841 + 0.634600i \(0.218835\pi\)
\(138\) 0 0
\(139\) −11.8905 −1.00854 −0.504270 0.863546i \(-0.668238\pi\)
−0.504270 + 0.863546i \(0.668238\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −35.0277 −2.92916
\(144\) 0 0
\(145\) 2.42330 0.201244
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −19.8327 −1.62476 −0.812378 0.583131i \(-0.801827\pi\)
−0.812378 + 0.583131i \(0.801827\pi\)
\(150\) 0 0
\(151\) −15.5635 −1.26654 −0.633271 0.773930i \(-0.718288\pi\)
−0.633271 + 0.773930i \(0.718288\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.125236 −0.0100592
\(156\) 0 0
\(157\) −12.6408 −1.00884 −0.504422 0.863457i \(-0.668294\pi\)
−0.504422 + 0.863457i \(0.668294\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.17923 −0.565803
\(162\) 0 0
\(163\) −24.6144 −1.92795 −0.963976 0.265989i \(-0.914302\pi\)
−0.963976 + 0.265989i \(0.914302\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.09557 −0.239543 −0.119771 0.992802i \(-0.538216\pi\)
−0.119771 + 0.992802i \(0.538216\pi\)
\(168\) 0 0
\(169\) 26.8642 2.06647
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.9163 −1.21010 −0.605048 0.796189i \(-0.706846\pi\)
−0.605048 + 0.796189i \(0.706846\pi\)
\(174\) 0 0
\(175\) 4.41321 0.333607
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.6824 −1.24690 −0.623449 0.781864i \(-0.714269\pi\)
−0.623449 + 0.781864i \(0.714269\pi\)
\(180\) 0 0
\(181\) 17.8642 1.32783 0.663917 0.747807i \(-0.268893\pi\)
0.663917 + 0.747807i \(0.268893\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.07049 −0.299268
\(186\) 0 0
\(187\) 39.6684 2.90084
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.58936 −0.115002 −0.0575011 0.998345i \(-0.518313\pi\)
−0.0575011 + 0.998345i \(0.518313\pi\)
\(192\) 0 0
\(193\) −7.36472 −0.530124 −0.265062 0.964231i \(-0.585392\pi\)
−0.265062 + 0.964231i \(0.585392\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.73085 −0.693294 −0.346647 0.937996i \(-0.612680\pi\)
−0.346647 + 0.937996i \(0.612680\pi\)
\(198\) 0 0
\(199\) 8.37789 0.593893 0.296947 0.954894i \(-0.404032\pi\)
0.296947 + 0.954894i \(0.404032\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.16349 0.222033
\(204\) 0 0
\(205\) 0.662934 0.0463013
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −34.1207 −2.36018
\(210\) 0 0
\(211\) 5.23655 0.360499 0.180250 0.983621i \(-0.442309\pi\)
0.180250 + 0.983621i \(0.442309\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.36855 −0.434332
\(216\) 0 0
\(217\) −0.163489 −0.0110983
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −45.1457 −3.03683
\(222\) 0 0
\(223\) 7.31381 0.489769 0.244884 0.969552i \(-0.421250\pi\)
0.244884 + 0.969552i \(0.421250\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.76860 0.581992 0.290996 0.956724i \(-0.406013\pi\)
0.290996 + 0.956724i \(0.406013\pi\)
\(228\) 0 0
\(229\) −19.6917 −1.30126 −0.650632 0.759393i \(-0.725496\pi\)
−0.650632 + 0.759393i \(0.725496\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.3585 −1.20270 −0.601352 0.798985i \(-0.705371\pi\)
−0.601352 + 0.798985i \(0.705371\pi\)
\(234\) 0 0
\(235\) −1.81436 −0.118356
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.708701 −0.0458421 −0.0229210 0.999737i \(-0.507297\pi\)
−0.0229210 + 0.999737i \(0.507297\pi\)
\(240\) 0 0
\(241\) 4.42330 0.284930 0.142465 0.989800i \(-0.454497\pi\)
0.142465 + 0.989800i \(0.454497\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.766021 −0.0489393
\(246\) 0 0
\(247\) 38.8319 2.47082
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.9546 −1.19640 −0.598202 0.801346i \(-0.704118\pi\)
−0.598202 + 0.801346i \(0.704118\pi\)
\(252\) 0 0
\(253\) −39.8288 −2.50402
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.2849 −1.14058 −0.570290 0.821443i \(-0.693169\pi\)
−0.570290 + 0.821443i \(0.693169\pi\)
\(258\) 0 0
\(259\) −5.31381 −0.330184
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.5440 1.02014 0.510072 0.860132i \(-0.329619\pi\)
0.510072 + 0.860132i \(0.329619\pi\)
\(264\) 0 0
\(265\) 3.06409 0.188225
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.852255 0.0519629 0.0259815 0.999662i \(-0.491729\pi\)
0.0259815 + 0.999662i \(0.491729\pi\)
\(270\) 0 0
\(271\) −18.9283 −1.14981 −0.574905 0.818220i \(-0.694961\pi\)
−0.574905 + 0.818220i \(0.694961\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.4836 1.47641
\(276\) 0 0
\(277\) 14.8773 0.893893 0.446946 0.894561i \(-0.352511\pi\)
0.446946 + 0.894561i \(0.352511\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.6692 −0.755781 −0.377890 0.925850i \(-0.623350\pi\)
−0.377890 + 0.925850i \(0.623350\pi\)
\(282\) 0 0
\(283\) 7.06409 0.419916 0.209958 0.977710i \(-0.432667\pi\)
0.209958 + 0.977710i \(0.432667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.865425 0.0510844
\(288\) 0 0
\(289\) 34.1271 2.00747
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.27849 0.0746901 0.0373451 0.999302i \(-0.488110\pi\)
0.0373451 + 0.999302i \(0.488110\pi\)
\(294\) 0 0
\(295\) −1.81436 −0.105636
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 45.3283 2.62140
\(300\) 0 0
\(301\) −8.31381 −0.479200
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.3900 0.594927
\(306\) 0 0
\(307\) 32.2320 1.83958 0.919788 0.392416i \(-0.128361\pi\)
0.919788 + 0.392416i \(0.128361\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.36230 0.190658 0.0953292 0.995446i \(-0.469610\pi\)
0.0953292 + 0.995446i \(0.469610\pi\)
\(312\) 0 0
\(313\) −4.72394 −0.267013 −0.133506 0.991048i \(-0.542624\pi\)
−0.133506 + 0.991048i \(0.542624\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.2327 0.574727 0.287363 0.957822i \(-0.407221\pi\)
0.287363 + 0.957822i \(0.407221\pi\)
\(318\) 0 0
\(319\) 17.5504 0.982632
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −43.9767 −2.44693
\(324\) 0 0
\(325\) −27.8642 −1.54563
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.36855 −0.130583
\(330\) 0 0
\(331\) 3.80119 0.208932 0.104466 0.994528i \(-0.466687\pi\)
0.104466 + 0.994528i \(0.466687\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.40115 0.295096
\(336\) 0 0
\(337\) −11.1911 −0.609621 −0.304810 0.952413i \(-0.598593\pi\)
−0.304810 + 0.952413i \(0.598593\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −0.907001 −0.0491169
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.66036 0.303864 0.151932 0.988391i \(-0.451451\pi\)
0.151932 + 0.988391i \(0.451451\pi\)
\(348\) 0 0
\(349\) −30.6276 −1.63946 −0.819729 0.572751i \(-0.805876\pi\)
−0.819729 + 0.572751i \(0.805876\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −29.3521 −1.56225 −0.781126 0.624373i \(-0.785355\pi\)
−0.781126 + 0.624373i \(0.785355\pi\)
\(354\) 0 0
\(355\) −6.02215 −0.319622
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −13.5351 −0.714357 −0.357178 0.934036i \(-0.616261\pi\)
−0.357178 + 0.934036i \(0.616261\pi\)
\(360\) 0 0
\(361\) 18.8264 0.990864
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.52195 0.0796627
\(366\) 0 0
\(367\) 9.74262 0.508560 0.254280 0.967131i \(-0.418161\pi\)
0.254280 + 0.967131i \(0.418161\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −16.3360 −0.845844 −0.422922 0.906166i \(-0.638996\pi\)
−0.422922 + 0.906166i \(0.638996\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −19.9737 −1.02870
\(378\) 0 0
\(379\) −32.9414 −1.69209 −0.846044 0.533114i \(-0.821022\pi\)
−0.846044 + 0.533114i \(0.821022\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 31.2944 1.59907 0.799534 0.600621i \(-0.205080\pi\)
0.799534 + 0.600621i \(0.205080\pi\)
\(384\) 0 0
\(385\) −4.24972 −0.216586
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.39106 0.273338 0.136669 0.990617i \(-0.456360\pi\)
0.136669 + 0.990617i \(0.456360\pi\)
\(390\) 0 0
\(391\) −51.3338 −2.59606
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.02140 0.202339
\(396\) 0 0
\(397\) 24.0817 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.2642 1.51132 0.755661 0.654963i \(-0.227316\pi\)
0.755661 + 0.654963i \(0.227316\pi\)
\(402\) 0 0
\(403\) 1.03224 0.0514194
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −29.4799 −1.46126
\(408\) 0 0
\(409\) −24.0055 −1.18700 −0.593498 0.804835i \(-0.702254\pi\)
−0.593498 + 0.804835i \(0.702254\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.36855 −0.116549
\(414\) 0 0
\(415\) 9.59688 0.471092
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.7308 1.06162 0.530811 0.847490i \(-0.321888\pi\)
0.530811 + 0.847490i \(0.321888\pi\)
\(420\) 0 0
\(421\) 18.0246 0.878464 0.439232 0.898374i \(-0.355251\pi\)
0.439232 + 0.898374i \(0.355251\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 31.5559 1.53068
\(426\) 0 0
\(427\) 13.5635 0.656385
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −20.8523 −1.00442 −0.502209 0.864746i \(-0.667479\pi\)
−0.502209 + 0.864746i \(0.667479\pi\)
\(432\) 0 0
\(433\) −2.20432 −0.105933 −0.0529663 0.998596i \(-0.516868\pi\)
−0.0529663 + 0.998596i \(0.516868\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 44.1546 2.11220
\(438\) 0 0
\(439\) −36.6013 −1.74688 −0.873442 0.486929i \(-0.838117\pi\)
−0.873442 + 0.486929i \(0.838117\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −28.5075 −1.35443 −0.677217 0.735784i \(-0.736814\pi\)
−0.677217 + 0.735784i \(0.736814\pi\)
\(444\) 0 0
\(445\) 3.72702 0.176678
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −39.7232 −1.87465 −0.937327 0.348452i \(-0.886707\pi\)
−0.937327 + 0.348452i \(0.886707\pi\)
\(450\) 0 0
\(451\) 4.80119 0.226079
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.83651 0.226739
\(456\) 0 0
\(457\) −10.6917 −0.500137 −0.250068 0.968228i \(-0.580453\pi\)
−0.250068 + 0.968228i \(0.580453\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.7113 0.592023 0.296012 0.955184i \(-0.404343\pi\)
0.296012 + 0.955184i \(0.404343\pi\)
\(462\) 0 0
\(463\) −14.6276 −0.679803 −0.339901 0.940461i \(-0.610394\pi\)
−0.339901 + 0.940461i \(0.610394\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −35.7963 −1.65645 −0.828227 0.560393i \(-0.810650\pi\)
−0.828227 + 0.560393i \(0.810650\pi\)
\(468\) 0 0
\(469\) 7.05092 0.325581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −46.1232 −2.12075
\(474\) 0 0
\(475\) −27.1427 −1.24539
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 20.3193 0.928413 0.464207 0.885727i \(-0.346340\pi\)
0.464207 + 0.885727i \(0.346340\pi\)
\(480\) 0 0
\(481\) 33.5504 1.52976
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.49945 −0.113494
\(486\) 0 0
\(487\) 27.4409 1.24346 0.621732 0.783230i \(-0.286429\pi\)
0.621732 + 0.783230i \(0.286429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.52564 0.204239 0.102120 0.994772i \(-0.467438\pi\)
0.102120 + 0.994772i \(0.467438\pi\)
\(492\) 0 0
\(493\) 22.6200 1.01875
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.86159 −0.352641
\(498\) 0 0
\(499\) 1.87734 0.0840412 0.0420206 0.999117i \(-0.486620\pi\)
0.0420206 + 0.999117i \(0.486620\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 37.7834 1.68468 0.842340 0.538946i \(-0.181177\pi\)
0.842340 + 0.538946i \(0.181177\pi\)
\(504\) 0 0
\(505\) 11.6055 0.516436
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.3678 −0.947111 −0.473556 0.880764i \(-0.657030\pi\)
−0.473556 + 0.880764i \(0.657030\pi\)
\(510\) 0 0
\(511\) 1.98683 0.0878922
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.09264 0.0481474
\(516\) 0 0
\(517\) −13.1402 −0.577907
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 31.8484 1.39530 0.697652 0.716436i \(-0.254228\pi\)
0.697652 + 0.716436i \(0.254228\pi\)
\(522\) 0 0
\(523\) −14.2366 −0.622521 −0.311260 0.950325i \(-0.600751\pi\)
−0.311260 + 0.950325i \(0.600751\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.16900 −0.0509223
\(528\) 0 0
\(529\) 28.5414 1.24093
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.46413 −0.236678
\(534\) 0 0
\(535\) −10.9127 −0.471795
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.54779 −0.238960
\(540\) 0 0
\(541\) −18.5767 −0.798675 −0.399337 0.916804i \(-0.630760\pi\)
−0.399337 + 0.916804i \(0.630760\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.5793 0.581672
\(546\) 0 0
\(547\) −6.10183 −0.260895 −0.130448 0.991455i \(-0.541641\pi\)
−0.130448 + 0.991455i \(0.541641\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19.4565 −0.828873
\(552\) 0 0
\(553\) 5.24972 0.223241
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.5309 −1.29364 −0.646819 0.762644i \(-0.723901\pi\)
−0.646819 + 0.762644i \(0.723901\pi\)
\(558\) 0 0
\(559\) 52.4918 2.22017
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.9899 −0.673894 −0.336947 0.941524i \(-0.609394\pi\)
−0.336947 + 0.941524i \(0.609394\pi\)
\(564\) 0 0
\(565\) −5.43536 −0.228667
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17.5057 −0.733877 −0.366939 0.930245i \(-0.619594\pi\)
−0.366939 + 0.930245i \(0.619594\pi\)
\(570\) 0 0
\(571\) −30.4918 −1.27604 −0.638021 0.770019i \(-0.720247\pi\)
−0.638021 + 0.770019i \(0.720247\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −31.6835 −1.32129
\(576\) 0 0
\(577\) 36.9283 1.53734 0.768672 0.639644i \(-0.220918\pi\)
0.768672 + 0.639644i \(0.220918\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.5282 0.519758
\(582\) 0 0
\(583\) 22.1911 0.919063
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.2867 1.86918 0.934591 0.355723i \(-0.115765\pi\)
0.934591 + 0.355723i \(0.115765\pi\)
\(588\) 0 0
\(589\) 1.00551 0.0414313
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21.7203 0.891944 0.445972 0.895047i \(-0.352858\pi\)
0.445972 + 0.895047i \(0.352858\pi\)
\(594\) 0 0
\(595\) −5.47730 −0.224547
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20.5125 0.838117 0.419058 0.907959i \(-0.362360\pi\)
0.419058 + 0.907959i \(0.362360\pi\)
\(600\) 0 0
\(601\) −2.31381 −0.0943822 −0.0471911 0.998886i \(-0.515027\pi\)
−0.0471911 + 0.998886i \(0.515027\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −15.1503 −0.615948
\(606\) 0 0
\(607\) −21.3647 −0.867167 −0.433584 0.901113i \(-0.642751\pi\)
−0.433584 + 0.901113i \(0.642751\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 14.9546 0.604998
\(612\) 0 0
\(613\) 0.163489 0.00660325 0.00330163 0.999995i \(-0.498949\pi\)
0.00330163 + 0.999995i \(0.498949\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17.5220 −0.705407 −0.352704 0.935735i \(-0.614738\pi\)
−0.352704 + 0.935735i \(0.614738\pi\)
\(618\) 0 0
\(619\) 19.1988 0.771665 0.385833 0.922569i \(-0.373914\pi\)
0.385833 + 0.922569i \(0.373914\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.86542 0.194929
\(624\) 0 0
\(625\) 16.5425 0.661700
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −37.9954 −1.51498
\(630\) 0 0
\(631\) 1.18675 0.0472437 0.0236218 0.999721i \(-0.492480\pi\)
0.0236218 + 0.999721i \(0.492480\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.11958 0.163481
\(636\) 0 0
\(637\) 6.31381 0.250162
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 40.6591 1.60594 0.802969 0.596021i \(-0.203252\pi\)
0.802969 + 0.596021i \(0.203252\pi\)
\(642\) 0 0
\(643\) −10.5868 −0.417502 −0.208751 0.977969i \(-0.566940\pi\)
−0.208751 + 0.977969i \(0.566940\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.8303 −1.01549 −0.507746 0.861507i \(-0.669521\pi\)
−0.507746 + 0.861507i \(0.669521\pi\)
\(648\) 0 0
\(649\) −13.1402 −0.515799
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22.2327 −0.870034 −0.435017 0.900422i \(-0.643258\pi\)
−0.435017 + 0.900422i \(0.643258\pi\)
\(654\) 0 0
\(655\) −6.66096 −0.260265
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −26.9388 −1.04939 −0.524694 0.851291i \(-0.675820\pi\)
−0.524694 + 0.851291i \(0.675820\pi\)
\(660\) 0 0
\(661\) 16.7301 0.650725 0.325363 0.945589i \(-0.394514\pi\)
0.325363 + 0.945589i \(0.394514\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.71128 0.182695
\(666\) 0 0
\(667\) −22.7114 −0.879390
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 75.2476 2.90490
\(672\) 0 0
\(673\) −3.36472 −0.129701 −0.0648503 0.997895i \(-0.520657\pi\)
−0.0648503 + 0.997895i \(0.520657\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.1307 0.389356 0.194678 0.980867i \(-0.437634\pi\)
0.194678 + 0.980867i \(0.437634\pi\)
\(678\) 0 0
\(679\) −3.26289 −0.125218
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −25.7151 −0.983961 −0.491981 0.870606i \(-0.663727\pi\)
−0.491981 + 0.870606i \(0.663727\pi\)
\(684\) 0 0
\(685\) −13.8587 −0.529512
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −25.2552 −0.962148
\(690\) 0 0
\(691\) −8.51812 −0.324045 −0.162022 0.986787i \(-0.551802\pi\)
−0.162022 + 0.986787i \(0.551802\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.10838 0.345501
\(696\) 0 0
\(697\) 6.18806 0.234390
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.62519 0.0991520 0.0495760 0.998770i \(-0.484213\pi\)
0.0495760 + 0.998770i \(0.484213\pi\)
\(702\) 0 0
\(703\) 32.6816 1.23261
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.1503 0.569786
\(708\) 0 0
\(709\) −36.2577 −1.36168 −0.680842 0.732430i \(-0.738386\pi\)
−0.680842 + 0.732430i \(0.738386\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.17372 0.0439563
\(714\) 0 0
\(715\) 26.8319 1.00346
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 38.8603 1.44925 0.724623 0.689145i \(-0.242014\pi\)
0.724623 + 0.689145i \(0.242014\pi\)
\(720\) 0 0
\(721\) 1.42638 0.0531212
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.9611 0.518504
\(726\) 0 0
\(727\) −15.7180 −0.582950 −0.291475 0.956578i \(-0.594146\pi\)
−0.291475 + 0.956578i \(0.594146\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −59.4464 −2.19870
\(732\) 0 0
\(733\) −14.7503 −0.544814 −0.272407 0.962182i \(-0.587820\pi\)
−0.272407 + 0.962182i \(0.587820\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.1170 1.44089
\(738\) 0 0
\(739\) 31.7935 1.16954 0.584772 0.811198i \(-0.301184\pi\)
0.584772 + 0.811198i \(0.301184\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −41.3673 −1.51762 −0.758809 0.651313i \(-0.774219\pi\)
−0.758809 + 0.651313i \(0.774219\pi\)
\(744\) 0 0
\(745\) 15.1923 0.556601
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14.2459 −0.520534
\(750\) 0 0
\(751\) 8.42330 0.307371 0.153685 0.988120i \(-0.450886\pi\)
0.153685 + 0.988120i \(0.450886\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.9220 0.433886
\(756\) 0 0
\(757\) −27.8828 −1.01342 −0.506710 0.862117i \(-0.669138\pi\)
−0.506710 + 0.862117i \(0.669138\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17.3415 −0.628628 −0.314314 0.949319i \(-0.601774\pi\)
−0.314314 + 0.949319i \(0.601774\pi\)
\(762\) 0 0
\(763\) 17.7270 0.641761
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.9546 0.539979
\(768\) 0 0
\(769\) 41.8960 1.51081 0.755404 0.655259i \(-0.227440\pi\)
0.755404 + 0.655259i \(0.227440\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.0289 0.864260 0.432130 0.901811i \(-0.357762\pi\)
0.432130 + 0.901811i \(0.357762\pi\)
\(774\) 0 0
\(775\) −0.721511 −0.0259174
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.32264 −0.190703
\(780\) 0 0
\(781\) −43.6144 −1.56065
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.68311 0.345605
\(786\) 0 0
\(787\) 37.9923 1.35428 0.677140 0.735854i \(-0.263219\pi\)
0.677140 + 0.735854i \(0.263219\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7.09557 −0.252290
\(792\) 0 0
\(793\) −85.6375 −3.04108
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.5470 0.727813 0.363907 0.931435i \(-0.381443\pi\)
0.363907 + 0.931435i \(0.381443\pi\)
\(798\) 0 0
\(799\) −16.9359 −0.599150
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.0225 0.388976
\(804\) 0 0
\(805\) 5.49945 0.193830
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.13715 0.250929 0.125464 0.992098i \(-0.459958\pi\)
0.125464 + 0.992098i \(0.459958\pi\)
\(810\) 0 0
\(811\) −37.6276 −1.32128 −0.660642 0.750701i \(-0.729716\pi\)
−0.660642 + 0.750701i \(0.729716\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 18.8552 0.660468
\(816\) 0 0
\(817\) 51.1326 1.78890
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.78485 −0.166992 −0.0834961 0.996508i \(-0.526609\pi\)
−0.0834961 + 0.996508i \(0.526609\pi\)
\(822\) 0 0
\(823\) −2.65396 −0.0925111 −0.0462555 0.998930i \(-0.514729\pi\)
−0.0462555 + 0.998930i \(0.514729\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −47.2239 −1.64214 −0.821068 0.570831i \(-0.806621\pi\)
−0.821068 + 0.570831i \(0.806621\pi\)
\(828\) 0 0
\(829\) −12.3401 −0.428591 −0.214296 0.976769i \(-0.568746\pi\)
−0.214296 + 0.976769i \(0.568746\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.15032 −0.247744
\(834\) 0 0
\(835\) 2.37127 0.0820613
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.18983 0.248220 0.124110 0.992268i \(-0.460392\pi\)
0.124110 + 0.992268i \(0.460392\pi\)
\(840\) 0 0
\(841\) −18.9923 −0.654908
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −20.5785 −0.707923
\(846\) 0 0
\(847\) −19.7779 −0.679578
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 38.1491 1.30773
\(852\) 0 0
\(853\) −29.9923 −1.02692 −0.513459 0.858114i \(-0.671636\pi\)
−0.513459 + 0.858114i \(0.671636\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.97492 −0.306577 −0.153289 0.988181i \(-0.548986\pi\)
−0.153289 + 0.988181i \(0.548986\pi\)
\(858\) 0 0
\(859\) 44.5403 1.51969 0.759847 0.650102i \(-0.225274\pi\)
0.759847 + 0.650102i \(0.225274\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −38.7842 −1.32023 −0.660115 0.751165i \(-0.729492\pi\)
−0.660115 + 0.751165i \(0.729492\pi\)
\(864\) 0 0
\(865\) 12.1923 0.414549
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 29.1243 0.987976
\(870\) 0 0
\(871\) −44.5181 −1.50844
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −7.21072 −0.243767
\(876\) 0 0
\(877\) −5.66051 −0.191142 −0.0955709 0.995423i \(-0.530468\pi\)
−0.0955709 + 0.995423i \(0.530468\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.7684 0.901852 0.450926 0.892561i \(-0.351094\pi\)
0.450926 + 0.892561i \(0.351094\pi\)
\(882\) 0 0
\(883\) −11.0377 −0.371450 −0.185725 0.982602i \(-0.559463\pi\)
−0.185725 + 0.982602i \(0.559463\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.617527 −0.0207345 −0.0103673 0.999946i \(-0.503300\pi\)
−0.0103673 + 0.999946i \(0.503300\pi\)
\(888\) 0 0
\(889\) 5.37789 0.180369
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 14.5674 0.487478
\(894\) 0 0
\(895\) 12.7790 0.427156
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.517195 −0.0172494
\(900\) 0 0
\(901\) 28.6013 0.952847
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13.6843 −0.454883
\(906\) 0 0
\(907\) 52.9283 1.75745 0.878727 0.477325i \(-0.158394\pi\)
0.878727 + 0.477325i \(0.158394\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −57.2268 −1.89601 −0.948005 0.318256i \(-0.896903\pi\)
−0.948005 + 0.318256i \(0.896903\pi\)
\(912\) 0 0
\(913\) 69.5038 2.30024
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.69553 −0.287152
\(918\) 0 0
\(919\) 17.7887 0.586794 0.293397 0.955991i \(-0.405214\pi\)
0.293397 + 0.955991i \(0.405214\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 49.6366 1.63381
\(924\) 0 0
\(925\) −23.4510 −0.771063
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.62453 −0.151726 −0.0758630 0.997118i \(-0.524171\pi\)
−0.0758630 + 0.997118i \(0.524171\pi\)
\(930\) 0 0
\(931\) 6.15032 0.201569
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −30.3869 −0.993757
\(936\) 0 0
\(937\) 7.43536 0.242903 0.121451 0.992597i \(-0.461245\pi\)
0.121451 + 0.992597i \(0.461245\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 28.8238 0.939631 0.469815 0.882765i \(-0.344321\pi\)
0.469815 + 0.882765i \(0.344321\pi\)
\(942\) 0 0
\(943\) −6.21309 −0.202326
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.5571 1.05796 0.528982 0.848633i \(-0.322574\pi\)
0.528982 + 0.848633i \(0.322574\pi\)
\(948\) 0 0
\(949\) −12.5445 −0.407211
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 37.7834 1.22393 0.611963 0.790886i \(-0.290380\pi\)
0.611963 + 0.790886i \(0.290380\pi\)
\(954\) 0 0
\(955\) 1.21749 0.0393969
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0917 −0.584213
\(960\) 0 0
\(961\) −30.9733 −0.999138
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.64153 0.181607
\(966\) 0 0
\(967\) 54.6074 1.75606 0.878028 0.478608i \(-0.158859\pi\)
0.878028 + 0.478608i \(0.158859\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −0.554552 −0.0177964 −0.00889820 0.999960i \(-0.502832\pi\)
−0.00889820 + 0.999960i \(0.502832\pi\)
\(972\) 0 0
\(973\) 11.8905 0.381192
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.4551 0.398475 0.199238 0.979951i \(-0.436153\pi\)
0.199238 + 0.979951i \(0.436153\pi\)
\(978\) 0 0
\(979\) 26.9923 0.862679
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.47694 0.110897 0.0554485 0.998462i \(-0.482341\pi\)
0.0554485 + 0.998462i \(0.482341\pi\)
\(984\) 0 0
\(985\) 7.45404 0.237505
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 59.6868 1.89793
\(990\) 0 0
\(991\) −3.96049 −0.125809 −0.0629046 0.998020i \(-0.520036\pi\)
−0.0629046 + 0.998020i \(0.520036\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.41764 −0.203453
\(996\) 0 0
\(997\) −10.6013 −0.335746 −0.167873 0.985809i \(-0.553690\pi\)
−0.167873 + 0.985809i \(0.553690\pi\)
\(998\) 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.1.2 4
3.2 odd 2 6048.2.a.br.1.3 yes 4
4.3 odd 2 6048.2.a.bo.1.2 yes 4
12.11 even 2 6048.2.a.bv.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bm.1.2 4 1.1 even 1 trivial
6048.2.a.bo.1.2 yes 4 4.3 odd 2
6048.2.a.br.1.3 yes 4 3.2 odd 2
6048.2.a.bv.1.3 yes 4 12.11 even 2