Properties

Label 6048.2.h.e.2591.1
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 2591.1
Root \(0.258819 - 0.965926i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.e.2591.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.82843i q^{5} -1.00000i q^{7} -6.29253 q^{11} +5.44949 q^{13} -0.317837i q^{17} +4.00000i q^{19} -0.317837 q^{23} -3.00000 q^{25} +5.19615i q^{29} +8.79796i q^{31} -2.82843 q^{35} +0.898979 q^{37} +3.46410i q^{41} -5.44949i q^{43} -12.5851 q^{47} -1.00000 q^{49} +8.02458i q^{53} +17.7980i q^{55} -5.83183 q^{59} +12.8990 q^{61} -15.4135i q^{65} +3.44949i q^{67} +2.51059 q^{71} -4.00000 q^{73} +6.29253i q^{77} -10.0000i q^{79} +3.46410 q^{83} -0.898979 q^{85} +10.7101i q^{89} -5.44949i q^{91} +11.3137 q^{95} -15.7980 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 24q^{13} - 24q^{25} - 32q^{37} - 8q^{49} + 64q^{61} - 32q^{73} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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
\(11\) −6.29253 −1.89727 −0.948634 0.316374i \(-0.897534\pi\)
−0.948634 + 0.316374i \(0.897534\pi\)
\(12\) 0 0
\(13\) 5.44949 1.51142 0.755708 0.654908i \(-0.227293\pi\)
0.755708 + 0.654908i \(0.227293\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.317837i − 0.0770869i −0.999257 0.0385434i \(-0.987728\pi\)
0.999257 0.0385434i \(-0.0122718\pi\)
\(18\) 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
\(23\) −0.317837 −0.0662736 −0.0331368 0.999451i \(-0.510550\pi\)
−0.0331368 + 0.999451i \(0.510550\pi\)
\(24\) 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 \(0.160253\pi\)
−0.875923 + 0.482451i \(0.839747\pi\)
\(30\) 0 0
\(31\) 8.79796i 1.58016i 0.613004 + 0.790080i \(0.289961\pi\)
−0.613004 + 0.790080i \(0.710039\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) 0.898979 0.147791 0.0738957 0.997266i \(-0.476457\pi\)
0.0738957 + 0.997266i \(0.476457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410i 0.541002i 0.962720 + 0.270501i \(0.0871893\pi\)
−0.962720 + 0.270501i \(0.912811\pi\)
\(42\) 0 0
\(43\) − 5.44949i − 0.831039i −0.909584 0.415520i \(-0.863600\pi\)
0.909584 0.415520i \(-0.136400\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.5851 −1.83572 −0.917860 0.396905i \(-0.870084\pi\)
−0.917860 + 0.396905i \(0.870084\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.02458i 1.10226i 0.834419 + 0.551130i \(0.185803\pi\)
−0.834419 + 0.551130i \(0.814197\pi\)
\(54\) 0 0
\(55\) 17.7980i 2.39988i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.83183 −0.759239 −0.379620 0.925143i \(-0.623945\pi\)
−0.379620 + 0.925143i \(0.623945\pi\)
\(60\) 0 0
\(61\) 12.8990 1.65155 0.825773 0.564003i \(-0.190739\pi\)
0.825773 + 0.564003i \(0.190739\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 15.4135i − 1.91181i
\(66\) 0 0
\(67\) 3.44949i 0.421422i 0.977548 + 0.210711i \(0.0675779\pi\)
−0.977548 + 0.210711i \(0.932422\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.51059 0.297952 0.148976 0.988841i \(-0.452402\pi\)
0.148976 + 0.988841i \(0.452402\pi\)
\(72\) 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
\(77\) 6.29253i 0.717100i
\(78\) 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 \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 0 0
\(85\) −0.898979 −0.0975080
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.7101i 1.13527i 0.823279 + 0.567636i \(0.192142\pi\)
−0.823279 + 0.567636i \(0.807858\pi\)
\(90\) 0 0
\(91\) − 5.44949i − 0.571262i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3137 1.16076
\(96\) 0 0
\(97\) −15.7980 −1.60404 −0.802020 0.597297i \(-0.796241\pi\)
−0.802020 + 0.597297i \(0.796241\pi\)
\(98\) 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
\(103\) 3.89898i 0.384178i 0.981378 + 0.192089i \(0.0615262\pi\)
−0.981378 + 0.192089i \(0.938474\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.2419 1.76351 0.881756 0.471706i \(-0.156362\pi\)
0.881756 + 0.471706i \(0.156362\pi\)
\(108\) 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
\(115\) 0.898979i 0.0838303i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.317837 −0.0291361
\(120\) 0 0
\(121\) 28.5959 2.59963
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) 10.8990i 0.967128i 0.875309 + 0.483564i \(0.160658\pi\)
−0.875309 + 0.483564i \(0.839342\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.36773 0.206869 0.103435 0.994636i \(-0.467017\pi\)
0.103435 + 0.994636i \(0.467017\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.5133i 1.66713i 0.552421 + 0.833565i \(0.313704\pi\)
−0.552421 + 0.833565i \(0.686296\pi\)
\(138\) 0 0
\(139\) − 15.7980i − 1.33997i −0.742377 0.669983i \(-0.766302\pi\)
0.742377 0.669983i \(-0.233698\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −34.2911 −2.86756
\(144\) 0 0
\(145\) 14.6969 1.22051
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 8.02458i − 0.657399i −0.944435 0.328700i \(-0.893390\pi\)
0.944435 0.328700i \(-0.106610\pi\)
\(150\) 0 0
\(151\) 5.79796i 0.471831i 0.971774 + 0.235916i \(0.0758089\pi\)
−0.971774 + 0.235916i \(0.924191\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 24.8844 1.99876
\(156\) 0 0
\(157\) 16.1464 1.28863 0.644313 0.764762i \(-0.277144\pi\)
0.644313 + 0.764762i \(0.277144\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.317837i 0.0250491i
\(162\) 0 0
\(163\) 2.55051i 0.199771i 0.994999 + 0.0998857i \(0.0318477\pi\)
−0.994999 + 0.0998857i \(0.968152\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0492 1.24192 0.620961 0.783842i \(-0.286743\pi\)
0.620961 + 0.783842i \(0.286743\pi\)
\(168\) 0 0
\(169\) 16.6969 1.28438
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.921404i 0.0700530i 0.999386 + 0.0350265i \(0.0111516\pi\)
−0.999386 + 0.0350265i \(0.988848\pi\)
\(174\) 0 0
\(175\) 3.00000i 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.0280 0.824270 0.412135 0.911123i \(-0.364783\pi\)
0.412135 + 0.911123i \(0.364783\pi\)
\(180\) 0 0
\(181\) −9.65153 −0.717393 −0.358696 0.933454i \(-0.616779\pi\)
−0.358696 + 0.933454i \(0.616779\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 2.54270i − 0.186943i
\(186\) 0 0
\(187\) 2.00000i 0.146254i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.0280 −0.797957 −0.398978 0.916960i \(-0.630635\pi\)
−0.398978 + 0.916960i \(0.630635\pi\)
\(192\) 0 0
\(193\) −3.89898 −0.280655 −0.140327 0.990105i \(-0.544815\pi\)
−0.140327 + 0.990105i \(0.544815\pi\)
\(194\) 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
\(199\) 20.7980i 1.47433i 0.675714 + 0.737164i \(0.263836\pi\)
−0.675714 + 0.737164i \(0.736164\pi\)
\(200\) 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
\(209\) − 25.1701i − 1.74105i
\(210\) 0 0
\(211\) 14.1464i 0.973880i 0.873435 + 0.486940i \(0.161887\pi\)
−0.873435 + 0.486940i \(0.838113\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −15.4135 −1.05119
\(216\) 0 0
\(217\) 8.79796 0.597244
\(218\) 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 \(0.106395\pi\)
−0.944657 + 0.328060i \(0.893605\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.460702 0.0305779 0.0152889 0.999883i \(-0.495133\pi\)
0.0152889 + 0.999883i \(0.495133\pi\)
\(228\) 0 0
\(229\) −3.10102 −0.204921 −0.102461 0.994737i \(-0.532672\pi\)
−0.102461 + 0.994737i \(0.532672\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 27.0771i − 1.77388i −0.461882 0.886941i \(-0.652826\pi\)
0.461882 0.886941i \(-0.347174\pi\)
\(234\) 0 0
\(235\) 35.5959i 2.32202i
\(236\) 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
\(247\) 21.7980i 1.38697i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.73545 0.298899 0.149449 0.988769i \(-0.452250\pi\)
0.149449 + 0.988769i \(0.452250\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.0915i 1.62754i 0.581184 + 0.813772i \(0.302590\pi\)
−0.581184 + 0.813772i \(0.697410\pi\)
\(258\) 0 0
\(259\) − 0.898979i − 0.0558599i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −16.6527 −1.02685 −0.513426 0.858134i \(-0.671624\pi\)
−0.513426 + 0.858134i \(0.671624\pi\)
\(264\) 0 0
\(265\) 22.6969 1.39426
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 14.4921i − 0.883598i −0.897114 0.441799i \(-0.854340\pi\)
0.897114 0.441799i \(-0.145660\pi\)
\(270\) 0 0
\(271\) − 29.8990i − 1.81623i −0.418717 0.908117i \(-0.637520\pi\)
0.418717 0.908117i \(-0.362480\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 18.8776 1.13836
\(276\) 0 0
\(277\) 30.8990 1.85654 0.928270 0.371907i \(-0.121296\pi\)
0.928270 + 0.371907i \(0.121296\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 6.57826i − 0.392426i −0.980561 0.196213i \(-0.937136\pi\)
0.980561 0.196213i \(-0.0628644\pi\)
\(282\) 0 0
\(283\) 5.79796i 0.344653i 0.985040 + 0.172326i \(0.0551284\pi\)
−0.985040 + 0.172326i \(0.944872\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.46410 0.204479
\(288\) 0 0
\(289\) 16.8990 0.994058
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.2207i 0.772363i 0.922423 + 0.386182i \(0.126206\pi\)
−0.922423 + 0.386182i \(0.873794\pi\)
\(294\) 0 0
\(295\) 16.4949i 0.960370i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.73205 −0.100167
\(300\) 0 0
\(301\) −5.44949 −0.314103
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 36.4838i − 2.08906i
\(306\) 0 0
\(307\) 26.6969i 1.52367i 0.647768 + 0.761837i \(0.275702\pi\)
−0.647768 + 0.761837i \(0.724298\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −21.4203 −1.21463 −0.607316 0.794460i \(-0.707754\pi\)
−0.607316 + 0.794460i \(0.707754\pi\)
\(312\) 0 0
\(313\) 15.5959 0.881533 0.440767 0.897622i \(-0.354707\pi\)
0.440767 + 0.897622i \(0.354707\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 12.5851i − 0.706847i −0.935463 0.353424i \(-0.885017\pi\)
0.935463 0.353424i \(-0.114983\pi\)
\(318\) 0 0
\(319\) − 32.6969i − 1.83068i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.27135 0.0707397
\(324\) 0 0
\(325\) −16.3485 −0.906850
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.5851i 0.693837i
\(330\) 0 0
\(331\) − 5.65153i − 0.310636i −0.987865 0.155318i \(-0.950360\pi\)
0.987865 0.155318i \(-0.0496403\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.75663 0.533062
\(336\) 0 0
\(337\) −19.8990 −1.08397 −0.541983 0.840389i \(-0.682326\pi\)
−0.541983 + 0.840389i \(0.682326\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 55.3614i − 2.99799i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.1489 1.08165 0.540826 0.841135i \(-0.318112\pi\)
0.540826 + 0.841135i \(0.318112\pi\)
\(348\) 0 0
\(349\) 28.8434 1.54395 0.771975 0.635653i \(-0.219269\pi\)
0.771975 + 0.635653i \(0.219269\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 32.7019i − 1.74055i −0.492570 0.870273i \(-0.663942\pi\)
0.492570 0.870273i \(-0.336058\pi\)
\(354\) 0 0
\(355\) − 7.10102i − 0.376883i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.61037 −0.348882 −0.174441 0.984668i \(-0.555812\pi\)
−0.174441 + 0.984668i \(0.555812\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.3137i 0.592187i
\(366\) 0 0
\(367\) 34.7980i 1.81644i 0.418494 + 0.908219i \(0.362558\pi\)
−0.418494 + 0.908219i \(0.637442\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.02458 0.416615
\(372\) 0 0
\(373\) −32.2929 −1.67206 −0.836030 0.548683i \(-0.815129\pi\)
−0.836030 + 0.548683i \(0.815129\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 28.3164i 1.45837i
\(378\) 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
\(383\) 17.8920 0.914237 0.457118 0.889406i \(-0.348882\pi\)
0.457118 + 0.889406i \(0.348882\pi\)
\(384\) 0 0
\(385\) 17.7980 0.907068
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 36.4838i 1.84980i 0.380206 + 0.924902i \(0.375853\pi\)
−0.380206 + 0.924902i \(0.624147\pi\)
\(390\) 0 0
\(391\) 0.101021i 0.00510883i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −28.2843 −1.42314
\(396\) 0 0
\(397\) −15.1010 −0.757898 −0.378949 0.925417i \(-0.623715\pi\)
−0.378949 + 0.925417i \(0.623715\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 11.6637i − 0.582455i −0.956654 0.291228i \(-0.905936\pi\)
0.956654 0.291228i \(-0.0940637\pi\)
\(402\) 0 0
\(403\) 47.9444i 2.38828i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.65685 −0.280400
\(408\) 0 0
\(409\) −28.6969 −1.41897 −0.709486 0.704719i \(-0.751073\pi\)
−0.709486 + 0.704719i \(0.751073\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.83183i 0.286965i
\(414\) 0 0
\(415\) − 9.79796i − 0.480963i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 30.6520 1.49745 0.748724 0.662882i \(-0.230667\pi\)
0.748724 + 0.662882i \(0.230667\pi\)
\(420\) 0 0
\(421\) −31.5959 −1.53989 −0.769945 0.638110i \(-0.779717\pi\)
−0.769945 + 0.638110i \(0.779717\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.953512i 0.0462521i
\(426\) 0 0
\(427\) − 12.8990i − 0.624225i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.7272 −1.28740 −0.643702 0.765276i \(-0.722602\pi\)
−0.643702 + 0.765276i \(0.722602\pi\)
\(432\) 0 0
\(433\) −19.5959 −0.941720 −0.470860 0.882208i \(-0.656056\pi\)
−0.470860 + 0.882208i \(0.656056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.27135i − 0.0608169i
\(438\) 0 0
\(439\) − 11.4949i − 0.548622i −0.961641 0.274311i \(-0.911550\pi\)
0.961641 0.274311i \(-0.0884497\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.285729 −0.0135754 −0.00678770 0.999977i \(-0.502161\pi\)
−0.00678770 + 0.999977i \(0.502161\pi\)
\(444\) 0 0
\(445\) 30.2929 1.43602
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 29.2699i 1.38133i 0.723174 + 0.690666i \(0.242682\pi\)
−0.723174 + 0.690666i \(0.757318\pi\)
\(450\) 0 0
\(451\) − 21.7980i − 1.02643i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −15.4135 −0.722595
\(456\) 0 0
\(457\) −10.3031 −0.481957 −0.240978 0.970530i \(-0.577468\pi\)
−0.240978 + 0.970530i \(0.577468\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.92820i 0.322679i 0.986899 + 0.161339i \(0.0515813\pi\)
−0.986899 + 0.161339i \(0.948419\pi\)
\(462\) 0 0
\(463\) 31.5959i 1.46839i 0.678940 + 0.734193i \(0.262439\pi\)
−0.678940 + 0.734193i \(0.737561\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.84961 −0.363236 −0.181618 0.983369i \(-0.558134\pi\)
−0.181618 + 0.983369i \(0.558134\pi\)
\(468\) 0 0
\(469\) 3.44949 0.159283
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 34.2911i 1.57671i
\(474\) 0 0
\(475\) − 12.0000i − 0.550598i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.94258 0.271524 0.135762 0.990742i \(-0.456652\pi\)
0.135762 + 0.990742i \(0.456652\pi\)
\(480\) 0 0
\(481\) 4.89898 0.223374
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 44.6834i 2.02897i
\(486\) 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
\(491\) 24.5344 1.10722 0.553612 0.832775i \(-0.313249\pi\)
0.553612 + 0.832775i \(0.313249\pi\)
\(492\) 0 0
\(493\) 1.65153 0.0743812
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.51059i − 0.112615i
\(498\) 0 0
\(499\) 35.7980i 1.60254i 0.598305 + 0.801268i \(0.295841\pi\)
−0.598305 + 0.801268i \(0.704159\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.37113 −0.239487 −0.119743 0.992805i \(-0.538207\pi\)
−0.119743 + 0.992805i \(0.538207\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.285729i 0.0126647i 0.999980 + 0.00633236i \(0.00201567\pi\)
−0.999980 + 0.00633236i \(0.997984\pi\)
\(510\) 0 0
\(511\) 4.00000i 0.176950i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.0280 0.485951
\(516\) 0 0
\(517\) 79.1918 3.48285
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 11.6315i − 0.509587i −0.966995 0.254794i \(-0.917992\pi\)
0.966995 0.254794i \(-0.0820075\pi\)
\(522\) 0 0
\(523\) 19.7980i 0.865704i 0.901465 + 0.432852i \(0.142493\pi\)
−0.901465 + 0.432852i \(0.857507\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.79632 0.121810
\(528\) 0 0
\(529\) −22.8990 −0.995608
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 18.8776i 0.817679i
\(534\) 0 0
\(535\) − 51.5959i − 2.23069i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.29253 0.271038
\(540\) 0 0
\(541\) 40.4949 1.74101 0.870506 0.492158i \(-0.163792\pi\)
0.870506 + 0.492158i \(0.163792\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 22.6274i 0.969252i
\(546\) 0 0
\(547\) 15.3939i 0.658195i 0.944296 + 0.329097i \(0.106744\pi\)
−0.944296 + 0.329097i \(0.893256\pi\)
\(548\) 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
\(557\) − 0.746431i − 0.0316273i −0.999875 0.0158136i \(-0.994966\pi\)
0.999875 0.0158136i \(-0.00503385\pi\)
\(558\) 0 0
\(559\) − 29.6969i − 1.25605i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.9951 1.05342 0.526710 0.850045i \(-0.323425\pi\)
0.526710 + 0.850045i \(0.323425\pi\)
\(564\) 0 0
\(565\) 24.0000 1.00969
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 2.19275i − 0.0919250i −0.998943 0.0459625i \(-0.985365\pi\)
0.998943 0.0459625i \(-0.0146355\pi\)
\(570\) 0 0
\(571\) 11.2474i 0.470691i 0.971912 + 0.235346i \(0.0756222\pi\)
−0.971912 + 0.235346i \(0.924378\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.953512 0.0397642
\(576\) 0 0
\(577\) 5.30306 0.220769 0.110385 0.993889i \(-0.464792\pi\)
0.110385 + 0.993889i \(0.464792\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 3.46410i − 0.143715i
\(582\) 0 0
\(583\) − 50.4949i − 2.09128i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.2385 −0.628961 −0.314480 0.949264i \(-0.601830\pi\)
−0.314480 + 0.949264i \(0.601830\pi\)
\(588\) 0 0
\(589\) −35.1918 −1.45005
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 26.0915i 1.07145i 0.844392 + 0.535725i \(0.179962\pi\)
−0.844392 + 0.535725i \(0.820038\pi\)
\(594\) 0 0
\(595\) 0.898979i 0.0368546i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 44.6513 1.82440 0.912201 0.409744i \(-0.134382\pi\)
0.912201 + 0.409744i \(0.134382\pi\)
\(600\) 0 0
\(601\) −16.8990 −0.689324 −0.344662 0.938727i \(-0.612006\pi\)
−0.344662 + 0.938727i \(0.612006\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 80.8815i − 3.28830i
\(606\) 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
\(611\) −68.5821 −2.77454
\(612\) 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
\(617\) 29.2699i 1.17836i 0.808001 + 0.589181i \(0.200549\pi\)
−0.808001 + 0.589181i \(0.799451\pi\)
\(618\) 0 0
\(619\) 29.5959i 1.18956i 0.803888 + 0.594780i \(0.202761\pi\)
−0.803888 + 0.594780i \(0.797239\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.7101 0.429093
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 0.285729i − 0.0113928i
\(630\) 0 0
\(631\) 36.4949i 1.45284i 0.687252 + 0.726419i \(0.258817\pi\)
−0.687252 + 0.726419i \(0.741183\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 30.8270 1.22333
\(636\) 0 0
\(637\) −5.44949 −0.215917
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.1489i 0.795835i 0.917421 + 0.397918i \(0.130267\pi\)
−0.917421 + 0.397918i \(0.869733\pi\)
\(642\) 0 0
\(643\) − 48.4949i − 1.91245i −0.292629 0.956226i \(-0.594530\pi\)
0.292629 0.956226i \(-0.405470\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.94258 0.233627 0.116814 0.993154i \(-0.462732\pi\)
0.116814 + 0.993154i \(0.462732\pi\)
\(648\) 0 0
\(649\) 36.6969 1.44048
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.3451i 0.991830i 0.868371 + 0.495915i \(0.165167\pi\)
−0.868371 + 0.495915i \(0.834833\pi\)
\(654\) 0 0
\(655\) − 6.69694i − 0.261671i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −20.7846 −0.809653 −0.404827 0.914393i \(-0.632668\pi\)
−0.404827 + 0.914393i \(0.632668\pi\)
\(660\) 0 0
\(661\) 12.8990 0.501712 0.250856 0.968024i \(-0.419288\pi\)
0.250856 + 0.968024i \(0.419288\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 11.3137i − 0.438727i
\(666\) 0 0
\(667\) − 1.65153i − 0.0639475i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −81.1672 −3.13342
\(672\) 0 0
\(673\) −28.5959 −1.10229 −0.551146 0.834409i \(-0.685809\pi\)
−0.551146 + 0.834409i \(0.685809\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.3485i 1.08952i 0.838592 + 0.544760i \(0.183379\pi\)
−0.838592 + 0.544760i \(0.816621\pi\)
\(678\) 0 0
\(679\) 15.7980i 0.606270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.0409 1.45559 0.727797 0.685792i \(-0.240544\pi\)
0.727797 + 0.685792i \(0.240544\pi\)
\(684\) 0 0
\(685\) 55.1918 2.10877
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 43.7299i 1.66598i
\(690\) 0 0
\(691\) 28.2020i 1.07286i 0.843946 + 0.536428i \(0.180227\pi\)
−0.843946 + 0.536428i \(0.819773\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −44.6834 −1.69494
\(696\) 0 0
\(697\) 1.10102 0.0417041
\(698\) 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
\(703\) 3.59592i 0.135623i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.65685 0.212748
\(708\) 0 0
\(709\) −41.5959 −1.56217 −0.781084 0.624426i \(-0.785333\pi\)
−0.781084 + 0.624426i \(0.785333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 2.79632i − 0.104723i
\(714\) 0 0
\(715\) 96.9898i 3.62721i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.3349 −0.609189 −0.304594 0.952482i \(-0.598521\pi\)
−0.304594 + 0.952482i \(0.598521\pi\)
\(720\) 0 0
\(721\) 3.89898 0.145206
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 15.5885i − 0.578941i
\(726\) 0 0
\(727\) − 9.49490i − 0.352146i −0.984377 0.176073i \(-0.943660\pi\)
0.984377 0.176073i \(-0.0563395\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.73205 −0.0640622
\(732\) 0 0
\(733\) 0.348469 0.0128710 0.00643550 0.999979i \(-0.497952\pi\)
0.00643550 + 0.999979i \(0.497952\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 21.7060i − 0.799551i
\(738\) 0 0
\(739\) − 43.7980i − 1.61113i −0.592505 0.805567i \(-0.701861\pi\)
0.592505 0.805567i \(-0.298139\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −33.3376 −1.22304 −0.611518 0.791230i \(-0.709441\pi\)
−0.611518 + 0.791230i \(0.709441\pi\)
\(744\) 0 0
\(745\) −22.6969 −0.831551
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 18.2419i − 0.666545i
\(750\) 0 0
\(751\) 44.8990i 1.63839i 0.573517 + 0.819194i \(0.305579\pi\)
−0.573517 + 0.819194i \(0.694421\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.3991 0.596825
\(756\) 0 0
\(757\) −33.1918 −1.20638 −0.603189 0.797598i \(-0.706103\pi\)
−0.603189 + 0.797598i \(0.706103\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 48.1154i − 1.74418i −0.489345 0.872090i \(-0.662764\pi\)
0.489345 0.872090i \(-0.337236\pi\)
\(762\) 0 0
\(763\) 8.00000i 0.289619i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −31.7805 −1.14753
\(768\) 0 0
\(769\) −4.89898 −0.176662 −0.0883309 0.996091i \(-0.528153\pi\)
−0.0883309 + 0.996091i \(0.528153\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 31.4626i 1.13163i 0.824531 + 0.565816i \(0.191439\pi\)
−0.824531 + 0.565816i \(0.808561\pi\)
\(774\) 0 0
\(775\) − 26.3939i − 0.948096i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.8564 −0.496457
\(780\) 0 0
\(781\) −15.7980 −0.565295
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 45.6690i − 1.63000i
\(786\) 0 0
\(787\) − 21.5959i − 0.769811i −0.922956 0.384906i \(-0.874234\pi\)
0.922956 0.384906i \(-0.125766\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.48528 0.301702
\(792\) 0 0
\(793\) 70.2929 2.49617
\(794\) 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
\(803\) 25.1701 0.888234
\(804\) 0 0
\(805\) 0.898979 0.0316849
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 27.6486i − 0.972073i −0.873939 0.486036i \(-0.838442\pi\)
0.873939 0.486036i \(-0.161558\pi\)
\(810\) 0 0
\(811\) − 24.4949i − 0.860132i −0.902797 0.430066i \(-0.858490\pi\)
0.902797 0.430066i \(-0.141510\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.21393 0.252693
\(816\) 0 0
\(817\) 21.7980 0.762614
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 37.0087i 1.29161i 0.763501 + 0.645807i \(0.223479\pi\)
−0.763501 + 0.645807i \(0.776521\pi\)
\(822\) 0 0
\(823\) − 21.1010i − 0.735535i −0.929918 0.367768i \(-0.880122\pi\)
0.929918 0.367768i \(-0.119878\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.7552 1.31288 0.656438 0.754380i \(-0.272062\pi\)
0.656438 + 0.754380i \(0.272062\pi\)
\(828\) 0 0
\(829\) −46.6969 −1.62185 −0.810926 0.585149i \(-0.801036\pi\)
−0.810926 + 0.585149i \(0.801036\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.317837i 0.0110124i
\(834\) 0 0
\(835\) − 45.3939i − 1.57092i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.46410 −0.119594 −0.0597970 0.998211i \(-0.519045\pi\)
−0.0597970 + 0.998211i \(0.519045\pi\)
\(840\) 0 0
\(841\) 2.00000 0.0689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 47.2261i − 1.62463i
\(846\) 0 0
\(847\) − 28.5959i − 0.982567i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −0.285729 −0.00979467
\(852\) 0 0
\(853\) −31.6515 −1.08373 −0.541864 0.840466i \(-0.682281\pi\)
−0.541864 + 0.840466i \(0.682281\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.7101i 0.365851i 0.983127 + 0.182926i \(0.0585568\pi\)
−0.983127 + 0.182926i \(0.941443\pi\)
\(858\) 0 0
\(859\) − 42.2929i − 1.44301i −0.692407 0.721507i \(-0.743450\pi\)
0.692407 0.721507i \(-0.256550\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.7949 −1.04827 −0.524135 0.851635i \(-0.675611\pi\)
−0.524135 + 0.851635i \(0.675611\pi\)
\(864\) 0 0
\(865\) 2.60612 0.0886108
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 62.9253i 2.13459i
\(870\) 0 0
\(871\) 18.7980i 0.636945i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.65685 −0.191237
\(876\) 0 0
\(877\) 29.1010 0.982672 0.491336 0.870970i \(-0.336509\pi\)
0.491336 + 0.870970i \(0.336509\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 41.5371i − 1.39942i −0.714427 0.699710i \(-0.753312\pi\)
0.714427 0.699710i \(-0.246688\pi\)
\(882\) 0 0
\(883\) − 1.65153i − 0.0555784i −0.999614 0.0277892i \(-0.991153\pi\)
0.999614 0.0277892i \(-0.00884672\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.3991 −0.550628 −0.275314 0.961354i \(-0.588782\pi\)
−0.275314 + 0.961354i \(0.588782\pi\)
\(888\) 0 0
\(889\) 10.8990 0.365540
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 50.3402i − 1.68457i
\(894\) 0 0
\(895\) − 31.1918i − 1.04263i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −45.7155 −1.52470
\(900\) 0 0
\(901\) 2.55051 0.0849698
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 27.2987i 0.907438i
\(906\) 0 0
\(907\) − 41.1918i − 1.36775i −0.729598 0.683876i \(-0.760293\pi\)
0.729598 0.683876i \(-0.239707\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −19.2275 −0.637037 −0.318518 0.947917i \(-0.603185\pi\)
−0.318518 + 0.947917i \(0.603185\pi\)
\(912\) 0 0
\(913\) −21.7980 −0.721407
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2.36773i − 0.0781892i
\(918\) 0 0
\(919\) − 21.5959i − 0.712384i −0.934413 0.356192i \(-0.884075\pi\)
0.934413 0.356192i \(-0.115925\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13.6814 0.450330
\(924\) 0 0
\(925\) −2.69694 −0.0886748
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 56.9185i − 1.86744i −0.358011 0.933718i \(-0.616545\pi\)
0.358011 0.933718i \(-0.383455\pi\)
\(930\) 0 0
\(931\) − 4.00000i − 0.131095i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.65685 0.184999
\(936\) 0 0
\(937\) 17.3031 0.565266 0.282633 0.959228i \(-0.408792\pi\)
0.282633 + 0.959228i \(0.408792\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 21.4203i − 0.698281i −0.937070 0.349141i \(-0.886474\pi\)
0.937070 0.349141i \(-0.113526\pi\)
\(942\) 0 0
\(943\) − 1.10102i − 0.0358542i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.38551 −0.142510 −0.0712549 0.997458i \(-0.522700\pi\)
−0.0712549 + 0.997458i \(0.522700\pi\)
\(948\) 0 0
\(949\) −21.7980 −0.707592
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 13.5065i − 0.437517i −0.975779 0.218759i \(-0.929799\pi\)
0.975779 0.218759i \(-0.0702007\pi\)
\(954\) 0 0
\(955\) 31.1918i 1.00934i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 19.5133 0.630116
\(960\) 0 0
\(961\) −46.4041 −1.49691
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.0280i 0.355003i
\(966\) 0 0
\(967\) − 19.1010i − 0.614247i −0.951670 0.307124i \(-0.900633\pi\)
0.951670 0.307124i \(-0.0993665\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.38891 0.237121 0.118561 0.992947i \(-0.462172\pi\)
0.118561 + 0.992947i \(0.462172\pi\)
\(972\) 0 0
\(973\) −15.7980 −0.506459
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 15.1278i − 0.483980i −0.970279 0.241990i \(-0.922200\pi\)
0.970279 0.241990i \(-0.0778001\pi\)
\(978\) 0 0
\(979\) − 67.3939i − 2.15392i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.09978 0.130763 0.0653813 0.997860i \(-0.479174\pi\)
0.0653813 + 0.997860i \(0.479174\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.73205i 0.0550760i
\(990\) 0 0
\(991\) 46.4949i 1.47696i 0.674276 + 0.738480i \(0.264456\pi\)
−0.674276 + 0.738480i \(0.735544\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 58.8255 1.86489
\(996\) 0 0
\(997\) 31.4495 0.996015 0.498008 0.867173i \(-0.334065\pi\)
0.498008 + 0.867173i \(0.334065\pi\)
\(998\) 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.2591.1 8
3.2 odd 2 inner 6048.2.h.e.2591.6 yes 8
4.3 odd 2 inner 6048.2.h.e.2591.4 yes 8
12.11 even 2 inner 6048.2.h.e.2591.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.e.2591.1 8 1.1 even 1 trivial
6048.2.h.e.2591.4 yes 8 4.3 odd 2 inner
6048.2.h.e.2591.6 yes 8 3.2 odd 2 inner
6048.2.h.e.2591.7 yes 8 12.11 even 2 inner