Properties

Label 6048.2.h.e.2591.2
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.2
Root \(-0.965926 + 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.e.2591.8

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.82843i q^{5} -1.00000i q^{7} +0.635674 q^{11} +0.550510 q^{13} +3.14626i q^{17} +4.00000i q^{19} +3.14626 q^{23} -3.00000 q^{25} -5.19615i q^{29} -10.7980i q^{31} -2.82843 q^{35} -8.89898 q^{37} -3.46410i q^{41} -0.550510i q^{43} +1.27135 q^{47} -1.00000 q^{49} -2.36773i q^{53} -1.79796i q^{55} +11.4887 q^{59} +3.10102 q^{61} -1.55708i q^{65} -1.44949i q^{67} +5.97469 q^{71} -4.00000 q^{73} -0.635674i q^{77} -10.0000i q^{79} -3.46410 q^{83} +8.89898 q^{85} -13.5386i q^{89} -0.550510i q^{91} +11.3137 q^{95} +3.79796 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 24q^{13} - 24q^{25} - 32q^{37} - 8q^{49} + 64q^{61} - 32q^{73} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) − 2.82843i − 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.635674 0.191663 0.0958315 0.995398i \(-0.469449\pi\)
0.0958315 + 0.995398i \(0.469449\pi\)
\(12\) 0 0
\(13\) 0.550510 0.152684 0.0763420 0.997082i \(-0.475676\pi\)
0.0763420 + 0.997082i \(0.475676\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.14626i 0.763081i 0.924352 + 0.381541i \(0.124606\pi\)
−0.924352 + 0.381541i \(0.875394\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.14626 0.656041 0.328021 0.944671i \(-0.393618\pi\)
0.328021 + 0.944671i \(0.393618\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5.19615i − 0.964901i −0.875923 0.482451i \(-0.839747\pi\)
0.875923 0.482451i \(-0.160253\pi\)
\(30\) 0 0
\(31\) − 10.7980i − 1.93937i −0.244357 0.969685i \(-0.578577\pi\)
0.244357 0.969685i \(-0.421423\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −8.89898 −1.46298 −0.731492 0.681850i \(-0.761175\pi\)
−0.731492 + 0.681850i \(0.761175\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.46410i − 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) − 0.550510i − 0.0839520i −0.999119 0.0419760i \(-0.986635\pi\)
0.999119 0.0419760i \(-0.0133653\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.27135 0.185445 0.0927227 0.995692i \(-0.470443\pi\)
0.0927227 + 0.995692i \(0.470443\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.36773i − 0.325232i −0.986689 0.162616i \(-0.948007\pi\)
0.986689 0.162616i \(-0.0519932\pi\)
\(54\) 0 0
\(55\) − 1.79796i − 0.242437i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.4887 1.49570 0.747849 0.663868i \(-0.231086\pi\)
0.747849 + 0.663868i \(0.231086\pi\)
\(60\) 0 0
\(61\) 3.10102 0.397045 0.198522 0.980096i \(-0.436386\pi\)
0.198522 + 0.980096i \(0.436386\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.55708i − 0.193132i
\(66\) 0 0
\(67\) − 1.44949i − 0.177083i −0.996072 0.0885417i \(-0.971779\pi\)
0.996072 0.0885417i \(-0.0282206\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.97469 0.709065 0.354533 0.935044i \(-0.384640\pi\)
0.354533 + 0.935044i \(0.384640\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.635674i − 0.0724418i
\(78\) 0 0
\(79\) − 10.0000i − 1.12509i −0.826767 0.562544i \(-0.809823\pi\)
0.826767 0.562544i \(-0.190177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) 8.89898 0.965230
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 13.5386i − 1.43509i −0.696514 0.717543i \(-0.745267\pi\)
0.696514 0.717543i \(-0.254733\pi\)
\(90\) 0 0
\(91\) − 0.550510i − 0.0577092i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3137 1.16076
\(96\) 0 0
\(97\) 3.79796 0.385624 0.192812 0.981236i \(-0.438239\pi\)
0.192812 + 0.981236i \(0.438239\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.65685i 0.562878i 0.959579 + 0.281439i \(0.0908117\pi\)
−0.959579 + 0.281439i \(0.909188\pi\)
\(102\) 0 0
\(103\) − 5.89898i − 0.581244i −0.956838 0.290622i \(-0.906138\pi\)
0.956838 0.290622i \(-0.0938622\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.38551 0.423963 0.211981 0.977274i \(-0.432008\pi\)
0.211981 + 0.977274i \(0.432008\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.48528i 0.798228i 0.916901 + 0.399114i \(0.130682\pi\)
−0.916901 + 0.399114i \(0.869318\pi\)
\(114\) 0 0
\(115\) − 8.89898i − 0.829834i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.14626 0.288418
\(120\) 0 0
\(121\) −10.5959 −0.963265
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) 1.10102i 0.0976998i 0.998806 + 0.0488499i \(0.0155556\pi\)
−0.998806 + 0.0488499i \(0.984444\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.02458 −0.701111 −0.350555 0.936542i \(-0.614007\pi\)
−0.350555 + 0.936542i \(0.614007\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 8.19955i − 0.700535i −0.936650 0.350268i \(-0.886091\pi\)
0.936650 0.350268i \(-0.113909\pi\)
\(138\) 0 0
\(139\) 3.79796i 0.322139i 0.986943 + 0.161069i \(0.0514943\pi\)
−0.986943 + 0.161069i \(0.948506\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.349945 0.0292639
\(144\) 0 0
\(145\) −14.6969 −1.22051
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.36773i 0.193972i 0.995286 + 0.0969858i \(0.0309201\pi\)
−0.995286 + 0.0969858i \(0.969080\pi\)
\(150\) 0 0
\(151\) − 13.7980i − 1.12286i −0.827524 0.561431i \(-0.810251\pi\)
0.827524 0.561431i \(-0.189749\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −30.5412 −2.45313
\(156\) 0 0
\(157\) −18.1464 −1.44824 −0.724121 0.689673i \(-0.757754\pi\)
−0.724121 + 0.689673i \(0.757754\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 3.14626i − 0.247960i
\(162\) 0 0
\(163\) 7.44949i 0.583489i 0.956496 + 0.291745i \(0.0942357\pi\)
−0.956496 + 0.291745i \(0.905764\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.73545 −0.366440 −0.183220 0.983072i \(-0.558652\pi\)
−0.183220 + 0.983072i \(0.558652\pi\)
\(168\) 0 0
\(169\) −12.6969 −0.976688
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.7060i 1.65028i 0.564930 + 0.825139i \(0.308903\pi\)
−0.564930 + 0.825139i \(0.691097\pi\)
\(174\) 0 0
\(175\) 3.00000i 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.6848 −1.24708 −0.623542 0.781790i \(-0.714307\pi\)
−0.623542 + 0.781790i \(0.714307\pi\)
\(180\) 0 0
\(181\) −24.3485 −1.80981 −0.904904 0.425616i \(-0.860057\pi\)
−0.904904 + 0.425616i \(0.860057\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 25.1701i 1.85054i
\(186\) 0 0
\(187\) 2.00000i 0.146254i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.6848 1.20727 0.603636 0.797260i \(-0.293718\pi\)
0.603636 + 0.797260i \(0.293718\pi\)
\(192\) 0 0
\(193\) 5.89898 0.424618 0.212309 0.977203i \(-0.431902\pi\)
0.212309 + 0.977203i \(0.431902\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 16.9706i − 1.20910i −0.796566 0.604551i \(-0.793352\pi\)
0.796566 0.604551i \(-0.206648\pi\)
\(198\) 0 0
\(199\) 1.20204i 0.0852104i 0.999092 + 0.0426052i \(0.0135658\pi\)
−0.999092 + 0.0426052i \(0.986434\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.19615 −0.364698
\(204\) 0 0
\(205\) −9.79796 −0.684319
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.54270i 0.175882i
\(210\) 0 0
\(211\) − 20.1464i − 1.38694i −0.720487 0.693469i \(-0.756082\pi\)
0.720487 0.693469i \(-0.243918\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.55708 −0.106192
\(216\) 0 0
\(217\) −10.7980 −0.733013
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.73205i 0.116510i
\(222\) 0 0
\(223\) − 9.79796i − 0.656120i −0.944657 0.328060i \(-0.893605\pi\)
0.944657 0.328060i \(-0.106395\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.8530 0.720339 0.360170 0.932887i \(-0.382719\pi\)
0.360170 + 0.932887i \(0.382719\pi\)
\(228\) 0 0
\(229\) −12.8990 −0.852389 −0.426194 0.904632i \(-0.640146\pi\)
−0.426194 + 0.904632i \(0.640146\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.4203i 1.40329i 0.712527 + 0.701645i \(0.247551\pi\)
−0.712527 + 0.701645i \(0.752449\pi\)
\(234\) 0 0
\(235\) − 3.59592i − 0.234572i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.82843 0.182956 0.0914779 0.995807i \(-0.470841\pi\)
0.0914779 + 0.995807i \(0.470841\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.82843i 0.180702i
\(246\) 0 0
\(247\) 2.20204i 0.140113i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.0492 −1.01301 −0.506507 0.862236i \(-0.669064\pi\)
−0.506507 + 0.862236i \(0.669064\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.1633i 1.19537i 0.801729 + 0.597687i \(0.203914\pi\)
−0.801729 + 0.597687i \(0.796086\pi\)
\(258\) 0 0
\(259\) 8.89898i 0.552956i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.1168 −1.24046 −0.620228 0.784421i \(-0.712960\pi\)
−0.620228 + 0.784421i \(0.712960\pi\)
\(264\) 0 0
\(265\) −6.69694 −0.411390
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.1489i 1.22850i 0.789111 + 0.614251i \(0.210542\pi\)
−0.789111 + 0.614251i \(0.789458\pi\)
\(270\) 0 0
\(271\) − 20.1010i − 1.22105i −0.791997 0.610525i \(-0.790958\pi\)
0.791997 0.610525i \(-0.209042\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.90702 −0.114998
\(276\) 0 0
\(277\) 21.1010 1.26784 0.633919 0.773400i \(-0.281445\pi\)
0.633919 + 0.773400i \(0.281445\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 27.3629i − 1.63233i −0.577818 0.816166i \(-0.696096\pi\)
0.577818 0.816166i \(-0.303904\pi\)
\(282\) 0 0
\(283\) − 13.7980i − 0.820204i −0.912040 0.410102i \(-0.865493\pi\)
0.912040 0.410102i \(-0.134507\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.46410 −0.204479
\(288\) 0 0
\(289\) 7.10102 0.417707
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 7.56388i − 0.441886i −0.975287 0.220943i \(-0.929086\pi\)
0.975287 0.220943i \(-0.0709136\pi\)
\(294\) 0 0
\(295\) − 32.4949i − 1.89193i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.73205 0.100167
\(300\) 0 0
\(301\) −0.550510 −0.0317309
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 8.77101i − 0.502227i
\(306\) 0 0
\(307\) − 2.69694i − 0.153922i −0.997034 0.0769612i \(-0.975478\pi\)
0.997034 0.0769612i \(-0.0245218\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.0771 1.53540 0.767702 0.640807i \(-0.221400\pi\)
0.767702 + 0.640807i \(0.221400\pi\)
\(312\) 0 0
\(313\) −23.5959 −1.33372 −0.666860 0.745183i \(-0.732362\pi\)
−0.666860 + 0.745183i \(0.732362\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.27135i 0.0714061i 0.999362 + 0.0357030i \(0.0113670\pi\)
−0.999362 + 0.0357030i \(0.988633\pi\)
\(318\) 0 0
\(319\) − 3.30306i − 0.184936i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.5851 −0.700251
\(324\) 0 0
\(325\) −1.65153 −0.0916104
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 1.27135i − 0.0700917i
\(330\) 0 0
\(331\) − 20.3485i − 1.11845i −0.829015 0.559227i \(-0.811098\pi\)
0.829015 0.559227i \(-0.188902\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.09978 −0.223995
\(336\) 0 0
\(337\) −10.1010 −0.550238 −0.275119 0.961410i \(-0.588717\pi\)
−0.275119 + 0.961410i \(0.588717\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 6.86399i − 0.371706i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.4921 −0.777976 −0.388988 0.921243i \(-0.627175\pi\)
−0.388988 + 0.921243i \(0.627175\pi\)
\(348\) 0 0
\(349\) −34.8434 −1.86512 −0.932561 0.361012i \(-0.882431\pi\)
−0.932561 + 0.361012i \(0.882431\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 15.3814i − 0.818668i −0.912385 0.409334i \(-0.865761\pi\)
0.912385 0.409334i \(-0.134239\pi\)
\(354\) 0 0
\(355\) − 16.8990i − 0.896905i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.78194 0.199603 0.0998015 0.995007i \(-0.468179\pi\)
0.0998015 + 0.995007i \(0.468179\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.3137i 0.592187i
\(366\) 0 0
\(367\) 15.2020i 0.793540i 0.917918 + 0.396770i \(0.129869\pi\)
−0.917918 + 0.396770i \(0.870131\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.36773 −0.122926
\(372\) 0 0
\(373\) 36.2929 1.87917 0.939586 0.342312i \(-0.111210\pi\)
0.939586 + 0.342312i \(0.111210\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.86054i − 0.147325i
\(378\) 0 0
\(379\) − 10.0000i − 0.513665i −0.966456 0.256833i \(-0.917321\pi\)
0.966456 0.256833i \(-0.0826790\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 38.6766 1.97628 0.988140 0.153553i \(-0.0490717\pi\)
0.988140 + 0.153553i \(0.0490717\pi\)
\(384\) 0 0
\(385\) −1.79796 −0.0916325
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.77101i 0.444708i 0.974966 + 0.222354i \(0.0713741\pi\)
−0.974966 + 0.222354i \(0.928626\pi\)
\(390\) 0 0
\(391\) 9.89898i 0.500613i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −28.2843 −1.42314
\(396\) 0 0
\(397\) −24.8990 −1.24964 −0.624822 0.780767i \(-0.714828\pi\)
−0.624822 + 0.780767i \(0.714828\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.9774i 1.14743i 0.819053 + 0.573717i \(0.194499\pi\)
−0.819053 + 0.573717i \(0.805501\pi\)
\(402\) 0 0
\(403\) − 5.94439i − 0.296111i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.65685 −0.280400
\(408\) 0 0
\(409\) 0.696938 0.0344614 0.0172307 0.999852i \(-0.494515\pi\)
0.0172307 + 0.999852i \(0.494515\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 11.4887i − 0.565321i
\(414\) 0 0
\(415\) 9.79796i 0.480963i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.2597 0.989751 0.494875 0.868964i \(-0.335214\pi\)
0.494875 + 0.868964i \(0.335214\pi\)
\(420\) 0 0
\(421\) 7.59592 0.370202 0.185101 0.982719i \(-0.440739\pi\)
0.185101 + 0.982719i \(0.440739\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 9.43879i − 0.457849i
\(426\) 0 0
\(427\) − 3.10102i − 0.150069i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.8708 −0.619964 −0.309982 0.950742i \(-0.600323\pi\)
−0.309982 + 0.950742i \(0.600323\pi\)
\(432\) 0 0
\(433\) 19.5959 0.941720 0.470860 0.882208i \(-0.343944\pi\)
0.470860 + 0.882208i \(0.343944\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.5851i 0.602025i
\(438\) 0 0
\(439\) 37.4949i 1.78953i 0.446534 + 0.894767i \(0.352658\pi\)
−0.446534 + 0.894767i \(0.647342\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −27.9985 −1.33025 −0.665125 0.746732i \(-0.731622\pi\)
−0.665125 + 0.746732i \(0.731622\pi\)
\(444\) 0 0
\(445\) −38.2929 −1.81526
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 12.2993i − 0.580441i −0.956960 0.290221i \(-0.906271\pi\)
0.956960 0.290221i \(-0.0937287\pi\)
\(450\) 0 0
\(451\) − 2.20204i − 0.103690i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.55708 −0.0729969
\(456\) 0 0
\(457\) −39.6969 −1.85694 −0.928472 0.371402i \(-0.878877\pi\)
−0.928472 + 0.371402i \(0.878877\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 6.92820i − 0.322679i −0.986899 0.161339i \(-0.948419\pi\)
0.986899 0.161339i \(-0.0515813\pi\)
\(462\) 0 0
\(463\) − 7.59592i − 0.353012i −0.984300 0.176506i \(-0.943520\pi\)
0.984300 0.176506i \(-0.0564795\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14.7778 −0.683835 −0.341918 0.939730i \(-0.611076\pi\)
−0.341918 + 0.939730i \(0.611076\pi\)
\(468\) 0 0
\(469\) −1.44949 −0.0669312
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 0.349945i − 0.0160905i
\(474\) 0 0
\(475\) − 12.0000i − 0.550598i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.6554 1.53775 0.768877 0.639397i \(-0.220816\pi\)
0.768877 + 0.639397i \(0.220816\pi\)
\(480\) 0 0
\(481\) −4.89898 −0.223374
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 10.7423i − 0.487780i
\(486\) 0 0
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.74983 0.169227 0.0846137 0.996414i \(-0.473034\pi\)
0.0846137 + 0.996414i \(0.473034\pi\)
\(492\) 0 0
\(493\) 16.3485 0.736298
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5.97469i − 0.268002i
\(498\) 0 0
\(499\) 16.2020i 0.725303i 0.931925 + 0.362651i \(0.118128\pi\)
−0.931925 + 0.362651i \(0.881872\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.3417 0.996167 0.498083 0.867129i \(-0.334037\pi\)
0.498083 + 0.867129i \(0.334037\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 27.9985i 1.24101i 0.784201 + 0.620507i \(0.213073\pi\)
−0.784201 + 0.620507i \(0.786927\pi\)
\(510\) 0 0
\(511\) 4.00000i 0.176950i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.6848 −0.735222
\(516\) 0 0
\(517\) 0.808164 0.0355430
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 8.16744i − 0.357822i −0.983865 0.178911i \(-0.942743\pi\)
0.983865 0.178911i \(-0.0572575\pi\)
\(522\) 0 0
\(523\) 0.202041i 0.00883464i 0.999990 + 0.00441732i \(0.00140608\pi\)
−0.999990 + 0.00441732i \(0.998594\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 33.9732 1.47990
\(528\) 0 0
\(529\) −13.1010 −0.569610
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 1.90702i − 0.0826024i
\(534\) 0 0
\(535\) − 12.4041i − 0.536275i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.635674 −0.0273804
\(540\) 0 0
\(541\) −8.49490 −0.365224 −0.182612 0.983185i \(-0.558455\pi\)
−0.182612 + 0.983185i \(0.558455\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 22.6274i 0.969252i
\(546\) 0 0
\(547\) − 43.3939i − 1.85539i −0.373341 0.927694i \(-0.621788\pi\)
0.373341 0.927694i \(-0.378212\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.7846 0.885454
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 38.8515i − 1.64619i −0.567902 0.823096i \(-0.692245\pi\)
0.567902 0.823096i \(-0.307755\pi\)
\(558\) 0 0
\(559\) − 0.303062i − 0.0128181i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.6028 0.615436 0.307718 0.951478i \(-0.400435\pi\)
0.307718 + 0.951478i \(0.400435\pi\)
\(564\) 0 0
\(565\) 24.0000 1.00969
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 9.12096i − 0.382370i −0.981554 0.191185i \(-0.938767\pi\)
0.981554 0.191185i \(-0.0612331\pi\)
\(570\) 0 0
\(571\) − 13.2474i − 0.554388i −0.960814 0.277194i \(-0.910595\pi\)
0.960814 0.277194i \(-0.0894046\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.43879 −0.393625
\(576\) 0 0
\(577\) 34.6969 1.44445 0.722226 0.691657i \(-0.243119\pi\)
0.722226 + 0.691657i \(0.243119\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.46410i 0.143715i
\(582\) 0 0
\(583\) − 1.50510i − 0.0623350i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.7026 −0.771939 −0.385970 0.922511i \(-0.626133\pi\)
−0.385970 + 0.922511i \(0.626133\pi\)
\(588\) 0 0
\(589\) 43.1918 1.77969
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.1633i 0.786943i 0.919337 + 0.393472i \(0.128726\pi\)
−0.919337 + 0.393472i \(0.871274\pi\)
\(594\) 0 0
\(595\) − 8.89898i − 0.364823i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20.4026 0.833626 0.416813 0.908992i \(-0.363147\pi\)
0.416813 + 0.908992i \(0.363147\pi\)
\(600\) 0 0
\(601\) −7.10102 −0.289657 −0.144828 0.989457i \(-0.546263\pi\)
−0.144828 + 0.989457i \(0.546263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 29.9698i 1.21844i
\(606\) 0 0
\(607\) 19.0000i 0.771186i 0.922669 + 0.385593i \(0.126003\pi\)
−0.922669 + 0.385593i \(0.873997\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.699891 0.0283145
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.2993i − 0.495152i −0.968868 0.247576i \(-0.920366\pi\)
0.968868 0.247576i \(-0.0796340\pi\)
\(618\) 0 0
\(619\) − 9.59592i − 0.385693i −0.981229 0.192846i \(-0.938228\pi\)
0.981229 0.192846i \(-0.0617719\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.5386 −0.542411
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 27.9985i − 1.11637i
\(630\) 0 0
\(631\) − 12.4949i − 0.497414i −0.968579 0.248707i \(-0.919994\pi\)
0.968579 0.248707i \(-0.0800056\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.11416 0.123582
\(636\) 0 0
\(637\) −0.550510 −0.0218120
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 14.4921i − 0.572403i −0.958169 0.286201i \(-0.907607\pi\)
0.958169 0.286201i \(-0.0923926\pi\)
\(642\) 0 0
\(643\) 0.494897i 0.0195168i 0.999952 + 0.00975842i \(0.00310625\pi\)
−0.999952 + 0.00975842i \(0.996894\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.6554 1.32313 0.661565 0.749888i \(-0.269893\pi\)
0.661565 + 0.749888i \(0.269893\pi\)
\(648\) 0 0
\(649\) 7.30306 0.286670
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 19.6882i − 0.770460i −0.922821 0.385230i \(-0.874122\pi\)
0.922821 0.385230i \(-0.125878\pi\)
\(654\) 0 0
\(655\) 22.6969i 0.886843i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.7846 0.809653 0.404827 0.914393i \(-0.367332\pi\)
0.404827 + 0.914393i \(0.367332\pi\)
\(660\) 0 0
\(661\) 3.10102 0.120616 0.0603079 0.998180i \(-0.480792\pi\)
0.0603079 + 0.998180i \(0.480792\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 11.3137i − 0.438727i
\(666\) 0 0
\(667\) − 16.3485i − 0.633015i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.97124 0.0760989
\(672\) 0 0
\(673\) 10.5959 0.408443 0.204221 0.978925i \(-0.434534\pi\)
0.204221 + 0.978925i \(0.434534\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 34.0053i − 1.30693i −0.756956 0.653466i \(-0.773314\pi\)
0.756956 0.653466i \(-0.226686\pi\)
\(678\) 0 0
\(679\) − 3.79796i − 0.145752i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.1845 0.925394 0.462697 0.886516i \(-0.346882\pi\)
0.462697 + 0.886516i \(0.346882\pi\)
\(684\) 0 0
\(685\) −23.1918 −0.886115
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1.30346i − 0.0496578i
\(690\) 0 0
\(691\) 47.7980i 1.81832i 0.416447 + 0.909160i \(0.363275\pi\)
−0.416447 + 0.909160i \(0.636725\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.7423 0.407477
\(696\) 0 0
\(697\) 10.8990 0.412828
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 5.65685i − 0.213656i −0.994277 0.106828i \(-0.965931\pi\)
0.994277 0.106828i \(-0.0340695\pi\)
\(702\) 0 0
\(703\) − 35.5959i − 1.34253i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.65685 0.212748
\(708\) 0 0
\(709\) −2.40408 −0.0902872 −0.0451436 0.998981i \(-0.514375\pi\)
−0.0451436 + 0.998981i \(0.514375\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 33.9732i − 1.27231i
\(714\) 0 0
\(715\) − 0.989795i − 0.0370162i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −23.2631 −0.867567 −0.433783 0.901017i \(-0.642822\pi\)
−0.433783 + 0.901017i \(0.642822\pi\)
\(720\) 0 0
\(721\) −5.89898 −0.219689
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.5885i 0.578941i
\(726\) 0 0
\(727\) 39.4949i 1.46478i 0.680883 + 0.732392i \(0.261596\pi\)
−0.680883 + 0.732392i \(0.738404\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.73205 0.0640622
\(732\) 0 0
\(733\) −14.3485 −0.529973 −0.264986 0.964252i \(-0.585367\pi\)
−0.264986 + 0.964252i \(0.585367\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 0.921404i − 0.0339403i
\(738\) 0 0
\(739\) − 24.2020i − 0.890286i −0.895459 0.445143i \(-0.853153\pi\)
0.895459 0.445143i \(-0.146847\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9.08885 −0.333437 −0.166719 0.986004i \(-0.553317\pi\)
−0.166719 + 0.986004i \(0.553317\pi\)
\(744\) 0 0
\(745\) 6.69694 0.245357
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 4.38551i − 0.160243i
\(750\) 0 0
\(751\) 35.1010i 1.28085i 0.768019 + 0.640427i \(0.221243\pi\)
−0.768019 + 0.640427i \(0.778757\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −39.0265 −1.42032
\(756\) 0 0
\(757\) 45.1918 1.64253 0.821263 0.570550i \(-0.193270\pi\)
0.821263 + 0.570550i \(0.193270\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 16.9385i − 0.614019i −0.951707 0.307009i \(-0.900672\pi\)
0.951707 0.307009i \(-0.0993283\pi\)
\(762\) 0 0
\(763\) 8.00000i 0.289619i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.32464 0.228369
\(768\) 0 0
\(769\) 4.89898 0.176662 0.0883309 0.996091i \(-0.471847\pi\)
0.0883309 + 0.996091i \(0.471847\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 3.17837i − 0.114318i −0.998365 0.0571591i \(-0.981796\pi\)
0.998365 0.0571591i \(-0.0182042\pi\)
\(774\) 0 0
\(775\) 32.3939i 1.16362i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.8564 0.496457
\(780\) 0 0
\(781\) 3.79796 0.135902
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 51.3258i 1.83190i
\(786\) 0 0
\(787\) 17.5959i 0.627227i 0.949551 + 0.313613i \(0.101540\pi\)
−0.949551 + 0.313613i \(0.898460\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.48528 0.301702
\(792\) 0 0
\(793\) 1.70714 0.0606224
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.4558i 0.901692i 0.892602 + 0.450846i \(0.148878\pi\)
−0.892602 + 0.450846i \(0.851122\pi\)
\(798\) 0 0
\(799\) 4.00000i 0.141510i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.54270 −0.0897299
\(804\) 0 0
\(805\) −8.89898 −0.313648
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 34.5768i − 1.21566i −0.794069 0.607828i \(-0.792041\pi\)
0.794069 0.607828i \(-0.207959\pi\)
\(810\) 0 0
\(811\) 24.4949i 0.860132i 0.902797 + 0.430066i \(0.141510\pi\)
−0.902797 + 0.430066i \(0.858490\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21.0703 0.738062
\(816\) 0 0
\(817\) 2.20204 0.0770397
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 42.6656i − 1.48904i −0.667600 0.744520i \(-0.732679\pi\)
0.667600 0.744520i \(-0.267321\pi\)
\(822\) 0 0
\(823\) − 30.8990i − 1.07707i −0.842603 0.538535i \(-0.818978\pi\)
0.842603 0.538535i \(-0.181022\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.81405 −0.132627 −0.0663137 0.997799i \(-0.521124\pi\)
−0.0663137 + 0.997799i \(0.521124\pi\)
\(828\) 0 0
\(829\) −17.3031 −0.600960 −0.300480 0.953788i \(-0.597147\pi\)
−0.300480 + 0.953788i \(0.597147\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 3.14626i − 0.109012i
\(834\) 0 0
\(835\) 13.3939i 0.463514i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.46410 0.119594 0.0597970 0.998211i \(-0.480955\pi\)
0.0597970 + 0.998211i \(0.480955\pi\)
\(840\) 0 0
\(841\) 2.00000 0.0689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 35.9124i 1.23542i
\(846\) 0 0
\(847\) 10.5959i 0.364080i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27.9985 −0.959778
\(852\) 0 0
\(853\) −46.3485 −1.58694 −0.793471 0.608608i \(-0.791728\pi\)
−0.793471 + 0.608608i \(0.791728\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 13.5386i − 0.462469i −0.972898 0.231234i \(-0.925724\pi\)
0.972898 0.231234i \(-0.0742764\pi\)
\(858\) 0 0
\(859\) 26.2929i 0.897101i 0.893758 + 0.448550i \(0.148060\pi\)
−0.893758 + 0.448550i \(0.851940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −34.2590 −1.16619 −0.583094 0.812404i \(-0.698158\pi\)
−0.583094 + 0.812404i \(0.698158\pi\)
\(864\) 0 0
\(865\) 61.3939 2.08745
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 6.35674i − 0.215638i
\(870\) 0 0
\(871\) − 0.797959i − 0.0270378i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.65685 −0.191237
\(876\) 0 0
\(877\) 38.8990 1.31353 0.656763 0.754097i \(-0.271925\pi\)
0.656763 + 0.754097i \(0.271925\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10.4244i 0.351207i 0.984461 + 0.175604i \(0.0561877\pi\)
−0.984461 + 0.175604i \(0.943812\pi\)
\(882\) 0 0
\(883\) − 16.3485i − 0.550170i −0.961420 0.275085i \(-0.911294\pi\)
0.961420 0.275085i \(-0.0887060\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 39.0265 1.31038 0.655191 0.755463i \(-0.272588\pi\)
0.655191 + 0.755463i \(0.272588\pi\)
\(888\) 0 0
\(889\) 1.10102 0.0369270
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.08540i 0.170176i
\(894\) 0 0
\(895\) 47.1918i 1.57745i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −56.1078 −1.87130
\(900\) 0 0
\(901\) 7.44949 0.248178
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 68.8679i 2.28925i
\(906\) 0 0
\(907\) 37.1918i 1.23493i 0.786597 + 0.617467i \(0.211841\pi\)
−0.786597 + 0.617467i \(0.788159\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.1981 1.19930 0.599648 0.800264i \(-0.295307\pi\)
0.599648 + 0.800264i \(0.295307\pi\)
\(912\) 0 0
\(913\) −2.20204 −0.0728769
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.02458i 0.264995i
\(918\) 0 0
\(919\) 17.5959i 0.580436i 0.956961 + 0.290218i \(0.0937278\pi\)
−0.956961 + 0.290218i \(0.906272\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.28913 0.108263
\(924\) 0 0
\(925\) 26.6969 0.877790
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 22.2775i − 0.730900i −0.930831 0.365450i \(-0.880915\pi\)
0.930831 0.365450i \(-0.119085\pi\)
\(930\) 0 0
\(931\) − 4.00000i − 0.131095i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.65685 0.184999
\(936\) 0 0
\(937\) 46.6969 1.52552 0.762761 0.646680i \(-0.223843\pi\)
0.762761 + 0.646680i \(0.223843\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.0771i 0.882689i 0.897338 + 0.441345i \(0.145498\pi\)
−0.897338 + 0.441345i \(0.854502\pi\)
\(942\) 0 0
\(943\) − 10.8990i − 0.354920i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.2419 −0.592782 −0.296391 0.955067i \(-0.595783\pi\)
−0.296391 + 0.955067i \(0.595783\pi\)
\(948\) 0 0
\(949\) −2.20204 −0.0714813
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 20.4347i − 0.661944i −0.943641 0.330972i \(-0.892623\pi\)
0.943641 0.330972i \(-0.107377\pi\)
\(954\) 0 0
\(955\) − 47.1918i − 1.52709i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.19955 −0.264777
\(960\) 0 0
\(961\) −85.5959 −2.76116
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 16.6848i − 0.537104i
\(966\) 0 0
\(967\) − 28.8990i − 0.929329i −0.885487 0.464664i \(-0.846175\pi\)
0.885487 0.464664i \(-0.153825\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.92480 0.125953 0.0629765 0.998015i \(-0.479941\pi\)
0.0629765 + 0.998015i \(0.479941\pi\)
\(972\) 0 0
\(973\) 3.79796 0.121757
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26.4415i 0.845937i 0.906144 + 0.422969i \(0.139012\pi\)
−0.906144 + 0.422969i \(0.860988\pi\)
\(978\) 0 0
\(979\) − 8.60612i − 0.275053i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.75663 −0.311188 −0.155594 0.987821i \(-0.549729\pi\)
−0.155594 + 0.987821i \(0.549729\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1.73205i − 0.0550760i
\(990\) 0 0
\(991\) − 2.49490i − 0.0792530i −0.999215 0.0396265i \(-0.987383\pi\)
0.999215 0.0396265i \(-0.0126168\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.39989 0.107784
\(996\) 0 0
\(997\) 26.5505 0.840863 0.420432 0.907324i \(-0.361879\pi\)
0.420432 + 0.907324i \(0.361879\pi\)
\(998\) 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.h.e.2591.2 8
3.2 odd 2 inner 6048.2.h.e.2591.5 yes 8
4.3 odd 2 inner 6048.2.h.e.2591.3 yes 8
12.11 even 2 inner 6048.2.h.e.2591.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.e.2591.2 8 1.1 even 1 trivial
6048.2.h.e.2591.3 yes 8 4.3 odd 2 inner
6048.2.h.e.2591.5 yes 8 3.2 odd 2 inner
6048.2.h.e.2591.8 yes 8 12.11 even 2 inner