Properties

Label 6048.2.h.b.2591.3
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.3
Root \(-0.258819 - 0.965926i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.b.2591.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.96713i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.96713i q^{5} -1.00000i q^{7} -4.76028 q^{11} -1.26795 q^{13} +0.378937i q^{17} -5.00000i q^{19} +1.93185 q^{23} -3.80385 q^{25} -4.24264i q^{29} -10.4641i q^{31} -2.96713 q^{35} +2.46410 q^{37} +10.6945i q^{41} -4.73205i q^{43} +3.20736 q^{47} -1.00000 q^{49} +8.76268i q^{53} +14.1244i q^{55} -3.48477 q^{59} -4.92820 q^{61} +3.76217i q^{65} -7.66025i q^{67} +0.240237 q^{71} -0.196152 q^{73} +4.76028i q^{77} -1.80385i q^{79} -7.82894 q^{83} +1.12436 q^{85} -1.93185i q^{89} +1.26795i q^{91} -14.8356 q^{95} -12.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{13} - 72q^{25} - 8q^{37} - 8q^{49} + 16q^{61} + 40q^{73} - 88q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) − 2.96713i − 1.32694i −0.748203 0.663470i \(-0.769083\pi\)
0.748203 0.663470i \(-0.230917\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.76028 −1.43528 −0.717639 0.696415i \(-0.754777\pi\)
−0.717639 + 0.696415i \(0.754777\pi\)
\(12\) 0 0
\(13\) −1.26795 −0.351666 −0.175833 0.984420i \(-0.556262\pi\)
−0.175833 + 0.984420i \(0.556262\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.378937i 0.0919058i 0.998944 + 0.0459529i \(0.0146324\pi\)
−0.998944 + 0.0459529i \(0.985368\pi\)
\(18\) 0 0
\(19\) − 5.00000i − 1.14708i −0.819178 0.573539i \(-0.805570\pi\)
0.819178 0.573539i \(-0.194430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.93185 0.402819 0.201409 0.979507i \(-0.435448\pi\)
0.201409 + 0.979507i \(0.435448\pi\)
\(24\) 0 0
\(25\) −3.80385 −0.760770
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.24264i − 0.787839i −0.919145 0.393919i \(-0.871119\pi\)
0.919145 0.393919i \(-0.128881\pi\)
\(30\) 0 0
\(31\) − 10.4641i − 1.87941i −0.341989 0.939704i \(-0.611100\pi\)
0.341989 0.939704i \(-0.388900\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.96713 −0.501536
\(36\) 0 0
\(37\) 2.46410 0.405096 0.202548 0.979272i \(-0.435078\pi\)
0.202548 + 0.979272i \(0.435078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.6945i 1.67021i 0.550094 + 0.835103i \(0.314592\pi\)
−0.550094 + 0.835103i \(0.685408\pi\)
\(42\) 0 0
\(43\) − 4.73205i − 0.721631i −0.932637 0.360815i \(-0.882498\pi\)
0.932637 0.360815i \(-0.117502\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.20736 0.467842 0.233921 0.972256i \(-0.424844\pi\)
0.233921 + 0.972256i \(0.424844\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.76268i 1.20365i 0.798629 + 0.601824i \(0.205559\pi\)
−0.798629 + 0.601824i \(0.794441\pi\)
\(54\) 0 0
\(55\) 14.1244i 1.90453i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.48477 −0.453678 −0.226839 0.973932i \(-0.572839\pi\)
−0.226839 + 0.973932i \(0.572839\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.76217i 0.466639i
\(66\) 0 0
\(67\) − 7.66025i − 0.935849i −0.883768 0.467924i \(-0.845002\pi\)
0.883768 0.467924i \(-0.154998\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.240237 0.0285108 0.0142554 0.999898i \(-0.495462\pi\)
0.0142554 + 0.999898i \(0.495462\pi\)
\(72\) 0 0
\(73\) −0.196152 −0.0229579 −0.0114790 0.999934i \(-0.503654\pi\)
−0.0114790 + 0.999934i \(0.503654\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.76028i 0.542484i
\(78\) 0 0
\(79\) − 1.80385i − 0.202949i −0.994838 0.101474i \(-0.967644\pi\)
0.994838 0.101474i \(-0.0323560\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.82894 −0.859338 −0.429669 0.902986i \(-0.641370\pi\)
−0.429669 + 0.902986i \(0.641370\pi\)
\(84\) 0 0
\(85\) 1.12436 0.121953
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.93185i − 0.204776i −0.994745 0.102388i \(-0.967352\pi\)
0.994745 0.102388i \(-0.0326483\pi\)
\(90\) 0 0
\(91\) 1.26795i 0.132917i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −14.8356 −1.52210
\(96\) 0 0
\(97\) −12.9282 −1.31266 −0.656330 0.754474i \(-0.727892\pi\)
−0.656330 + 0.754474i \(0.727892\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 4.52004i − 0.449761i −0.974386 0.224880i \(-0.927801\pi\)
0.974386 0.224880i \(-0.0721992\pi\)
\(102\) 0 0
\(103\) − 3.53590i − 0.348402i −0.984710 0.174201i \(-0.944266\pi\)
0.984710 0.174201i \(-0.0557343\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.86422 −0.856936 −0.428468 0.903557i \(-0.640947\pi\)
−0.428468 + 0.903557i \(0.640947\pi\)
\(108\) 0 0
\(109\) −17.5885 −1.68467 −0.842334 0.538955i \(-0.818819\pi\)
−0.842334 + 0.538955i \(0.818819\pi\)
\(110\) 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
\(115\) − 5.73205i − 0.534516i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.378937 0.0347371
\(120\) 0 0
\(121\) 11.6603 1.06002
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 3.54914i − 0.317444i
\(126\) 0 0
\(127\) 17.6603i 1.56709i 0.621332 + 0.783547i \(0.286592\pi\)
−0.621332 + 0.783547i \(0.713408\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.41421 −0.123560 −0.0617802 0.998090i \(-0.519678\pi\)
−0.0617802 + 0.998090i \(0.519678\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.48477i 0.297724i 0.988858 + 0.148862i \(0.0475610\pi\)
−0.988858 + 0.148862i \(0.952439\pi\)
\(138\) 0 0
\(139\) − 11.3205i − 0.960193i −0.877216 0.480096i \(-0.840602\pi\)
0.877216 0.480096i \(-0.159398\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.03579 0.504738
\(144\) 0 0
\(145\) −12.5885 −1.04541
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.7990i 1.62200i 0.585049 + 0.810998i \(0.301075\pi\)
−0.585049 + 0.810998i \(0.698925\pi\)
\(150\) 0 0
\(151\) 14.9282i 1.21484i 0.794381 + 0.607420i \(0.207795\pi\)
−0.794381 + 0.607420i \(0.792205\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −31.0483 −2.49386
\(156\) 0 0
\(157\) 6.73205 0.537276 0.268638 0.963241i \(-0.413426\pi\)
0.268638 + 0.963241i \(0.413426\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 1.93185i − 0.152251i
\(162\) 0 0
\(163\) 18.0526i 1.41399i 0.707221 + 0.706993i \(0.249949\pi\)
−0.707221 + 0.706993i \(0.750051\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.9048 1.77243 0.886214 0.463276i \(-0.153326\pi\)
0.886214 + 0.463276i \(0.153326\pi\)
\(168\) 0 0
\(169\) −11.3923 −0.876331
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.3156i 0.784280i 0.919905 + 0.392140i \(0.128265\pi\)
−0.919905 + 0.392140i \(0.871735\pi\)
\(174\) 0 0
\(175\) 3.80385i 0.287544i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.9700 0.894683 0.447342 0.894363i \(-0.352371\pi\)
0.447342 + 0.894363i \(0.352371\pi\)
\(180\) 0 0
\(181\) 24.7846 1.84223 0.921113 0.389296i \(-0.127282\pi\)
0.921113 + 0.389296i \(0.127282\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 7.31130i − 0.537538i
\(186\) 0 0
\(187\) − 1.80385i − 0.131910i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.03768 −0.364514 −0.182257 0.983251i \(-0.558340\pi\)
−0.182257 + 0.983251i \(0.558340\pi\)
\(192\) 0 0
\(193\) 9.32051 0.670905 0.335452 0.942057i \(-0.391111\pi\)
0.335452 + 0.942057i \(0.391111\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.0764i 1.43038i 0.698928 + 0.715192i \(0.253661\pi\)
−0.698928 + 0.715192i \(0.746339\pi\)
\(198\) 0 0
\(199\) − 14.2679i − 1.01143i −0.862701 0.505714i \(-0.831229\pi\)
0.862701 0.505714i \(-0.168771\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.24264 −0.297775
\(204\) 0 0
\(205\) 31.7321 2.21626
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 23.8014i 1.64638i
\(210\) 0 0
\(211\) 14.5359i 1.00069i 0.865825 + 0.500346i \(0.166794\pi\)
−0.865825 + 0.500346i \(0.833206\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −14.0406 −0.957561
\(216\) 0 0
\(217\) −10.4641 −0.710350
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 0.480473i − 0.0323201i
\(222\) 0 0
\(223\) 24.1244i 1.61549i 0.589535 + 0.807743i \(0.299311\pi\)
−0.589535 + 0.807743i \(0.700689\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −22.6274 −1.50183 −0.750917 0.660396i \(-0.770388\pi\)
−0.750917 + 0.660396i \(0.770388\pi\)
\(228\) 0 0
\(229\) 7.46410 0.493242 0.246621 0.969112i \(-0.420680\pi\)
0.246621 + 0.969112i \(0.420680\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 10.5558i − 0.691536i −0.938320 0.345768i \(-0.887618\pi\)
0.938320 0.345768i \(-0.112382\pi\)
\(234\) 0 0
\(235\) − 9.51666i − 0.620798i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −21.7680 −1.40806 −0.704028 0.710173i \(-0.748617\pi\)
−0.704028 + 0.710173i \(0.748617\pi\)
\(240\) 0 0
\(241\) 17.8038 1.14685 0.573423 0.819259i \(-0.305615\pi\)
0.573423 + 0.819259i \(0.305615\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.96713i 0.189563i
\(246\) 0 0
\(247\) 6.33975i 0.403388i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.04008 −0.570605 −0.285303 0.958438i \(-0.592094\pi\)
−0.285303 + 0.958438i \(0.592094\pi\)
\(252\) 0 0
\(253\) −9.19615 −0.578157
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.8714i 1.30192i 0.759110 + 0.650962i \(0.225634\pi\)
−0.759110 + 0.650962i \(0.774366\pi\)
\(258\) 0 0
\(259\) − 2.46410i − 0.153112i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.4171 −0.642348 −0.321174 0.947020i \(-0.604077\pi\)
−0.321174 + 0.947020i \(0.604077\pi\)
\(264\) 0 0
\(265\) 26.0000 1.59717
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.2146i 0.927649i 0.885927 + 0.463825i \(0.153523\pi\)
−0.885927 + 0.463825i \(0.846477\pi\)
\(270\) 0 0
\(271\) − 11.8564i − 0.720225i −0.932909 0.360113i \(-0.882738\pi\)
0.932909 0.360113i \(-0.117262\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 18.1074 1.09192
\(276\) 0 0
\(277\) −7.92820 −0.476360 −0.238180 0.971221i \(-0.576551\pi\)
−0.238180 + 0.971221i \(0.576551\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 29.6985i − 1.77166i −0.464007 0.885832i \(-0.653589\pi\)
0.464007 0.885832i \(-0.346411\pi\)
\(282\) 0 0
\(283\) − 31.4641i − 1.87035i −0.354190 0.935173i \(-0.615244\pi\)
0.354190 0.935173i \(-0.384756\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.6945 0.631278
\(288\) 0 0
\(289\) 16.8564 0.991553
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.62587i 0.445508i 0.974875 + 0.222754i \(0.0715047\pi\)
−0.974875 + 0.222754i \(0.928495\pi\)
\(294\) 0 0
\(295\) 10.3397i 0.602003i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.44949 −0.141658
\(300\) 0 0
\(301\) −4.73205 −0.272751
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 14.6226i 0.837288i
\(306\) 0 0
\(307\) − 9.00000i − 0.513657i −0.966457 0.256829i \(-0.917322\pi\)
0.966457 0.256829i \(-0.0826776\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.378937 −0.0214876 −0.0107438 0.999942i \(-0.503420\pi\)
−0.0107438 + 0.999942i \(0.503420\pi\)
\(312\) 0 0
\(313\) 30.0526 1.69867 0.849336 0.527853i \(-0.177003\pi\)
0.849336 + 0.527853i \(0.177003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 16.8690i − 0.947459i −0.880670 0.473729i \(-0.842907\pi\)
0.880670 0.473729i \(-0.157093\pi\)
\(318\) 0 0
\(319\) 20.1962i 1.13077i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.89469 0.105423
\(324\) 0 0
\(325\) 4.82309 0.267537
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 3.20736i − 0.176828i
\(330\) 0 0
\(331\) − 24.7846i − 1.36229i −0.732151 0.681143i \(-0.761483\pi\)
0.732151 0.681143i \(-0.238517\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −22.7290 −1.24182
\(336\) 0 0
\(337\) −12.4641 −0.678963 −0.339481 0.940613i \(-0.610252\pi\)
−0.339481 + 0.940613i \(0.610252\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 49.8120i 2.69747i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.1136 1.07975 0.539876 0.841744i \(-0.318471\pi\)
0.539876 + 0.841744i \(0.318471\pi\)
\(348\) 0 0
\(349\) −27.4641 −1.47012 −0.735060 0.678002i \(-0.762846\pi\)
−0.735060 + 0.678002i \(0.762846\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.4219i 0.980501i 0.871582 + 0.490250i \(0.163095\pi\)
−0.871582 + 0.490250i \(0.836905\pi\)
\(354\) 0 0
\(355\) − 0.712813i − 0.0378322i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −13.2827 −0.701035 −0.350518 0.936556i \(-0.613994\pi\)
−0.350518 + 0.936556i \(0.613994\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.582009i 0.0304638i
\(366\) 0 0
\(367\) 14.5167i 0.757764i 0.925445 + 0.378882i \(0.123691\pi\)
−0.925445 + 0.378882i \(0.876309\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.76268 0.454936
\(372\) 0 0
\(373\) −12.6603 −0.655523 −0.327762 0.944760i \(-0.606294\pi\)
−0.327762 + 0.944760i \(0.606294\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.37945i 0.277056i
\(378\) 0 0
\(379\) 17.8038i 0.914522i 0.889332 + 0.457261i \(0.151170\pi\)
−0.889332 + 0.457261i \(0.848830\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.82843 −0.144526 −0.0722629 0.997386i \(-0.523022\pi\)
−0.0722629 + 0.997386i \(0.523022\pi\)
\(384\) 0 0
\(385\) 14.1244 0.719844
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 29.3195i − 1.48656i −0.668980 0.743280i \(-0.733269\pi\)
0.668980 0.743280i \(-0.266731\pi\)
\(390\) 0 0
\(391\) 0.732051i 0.0370214i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.35225 −0.269301
\(396\) 0 0
\(397\) −8.92820 −0.448094 −0.224047 0.974578i \(-0.571927\pi\)
−0.224047 + 0.974578i \(0.571927\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.0759i 0.752853i 0.926446 + 0.376427i \(0.122847\pi\)
−0.926446 + 0.376427i \(0.877153\pi\)
\(402\) 0 0
\(403\) 13.2679i 0.660924i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.7298 −0.581425
\(408\) 0 0
\(409\) 34.5885 1.71029 0.855145 0.518390i \(-0.173468\pi\)
0.855145 + 0.518390i \(0.173468\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.48477i 0.171474i
\(414\) 0 0
\(415\) 23.2295i 1.14029i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −20.9086 −1.02145 −0.510726 0.859744i \(-0.670623\pi\)
−0.510726 + 0.859744i \(0.670623\pi\)
\(420\) 0 0
\(421\) 14.3205 0.697939 0.348969 0.937134i \(-0.386532\pi\)
0.348969 + 0.937134i \(0.386532\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 1.44142i − 0.0699191i
\(426\) 0 0
\(427\) 4.92820i 0.238492i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −29.7356 −1.43232 −0.716158 0.697938i \(-0.754101\pi\)
−0.716158 + 0.697938i \(0.754101\pi\)
\(432\) 0 0
\(433\) −25.2679 −1.21430 −0.607150 0.794587i \(-0.707687\pi\)
−0.607150 + 0.794587i \(0.707687\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 9.65926i − 0.462065i
\(438\) 0 0
\(439\) 10.2487i 0.489144i 0.969631 + 0.244572i \(0.0786475\pi\)
−0.969631 + 0.244572i \(0.921352\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.93666 −0.472105 −0.236052 0.971740i \(-0.575854\pi\)
−0.236052 + 0.971740i \(0.575854\pi\)
\(444\) 0 0
\(445\) −5.73205 −0.271725
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 34.7733i − 1.64105i −0.571607 0.820527i \(-0.693680\pi\)
0.571607 0.820527i \(-0.306320\pi\)
\(450\) 0 0
\(451\) − 50.9090i − 2.39721i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.76217 0.176373
\(456\) 0 0
\(457\) −26.1769 −1.22450 −0.612252 0.790663i \(-0.709736\pi\)
−0.612252 + 0.790663i \(0.709736\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 20.3166i − 0.946240i −0.880998 0.473120i \(-0.843128\pi\)
0.880998 0.473120i \(-0.156872\pi\)
\(462\) 0 0
\(463\) 12.2487i 0.569246i 0.958640 + 0.284623i \(0.0918684\pi\)
−0.958640 + 0.284623i \(0.908132\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.1817 0.841349 0.420674 0.907212i \(-0.361793\pi\)
0.420674 + 0.907212i \(0.361793\pi\)
\(468\) 0 0
\(469\) −7.66025 −0.353718
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 22.5259i 1.03574i
\(474\) 0 0
\(475\) 19.0192i 0.872662i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.9096 −1.41230 −0.706148 0.708064i \(-0.749569\pi\)
−0.706148 + 0.708064i \(0.749569\pi\)
\(480\) 0 0
\(481\) −3.12436 −0.142458
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 38.3596i 1.74182i
\(486\) 0 0
\(487\) 16.5885i 0.751695i 0.926682 + 0.375847i \(0.122648\pi\)
−0.926682 + 0.375847i \(0.877352\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13.4214 −0.605700 −0.302850 0.953038i \(-0.597938\pi\)
−0.302850 + 0.953038i \(0.597938\pi\)
\(492\) 0 0
\(493\) 1.60770 0.0724069
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 0.240237i − 0.0107761i
\(498\) 0 0
\(499\) − 33.6603i − 1.50684i −0.657540 0.753420i \(-0.728403\pi\)
0.657540 0.753420i \(-0.271597\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19.4201 0.865897 0.432949 0.901419i \(-0.357473\pi\)
0.432949 + 0.901419i \(0.357473\pi\)
\(504\) 0 0
\(505\) −13.4115 −0.596806
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.00429i 0.133163i 0.997781 + 0.0665815i \(0.0212092\pi\)
−0.997781 + 0.0665815i \(0.978791\pi\)
\(510\) 0 0
\(511\) 0.196152i 0.00867727i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.4915 −0.462309
\(516\) 0 0
\(517\) −15.2679 −0.671484
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.0793i 1.27399i 0.770869 + 0.636994i \(0.219822\pi\)
−0.770869 + 0.636994i \(0.780178\pi\)
\(522\) 0 0
\(523\) − 28.6603i − 1.25323i −0.779331 0.626613i \(-0.784441\pi\)
0.779331 0.626613i \(-0.215559\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.96524 0.172729
\(528\) 0 0
\(529\) −19.2679 −0.837737
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 13.5601i − 0.587354i
\(534\) 0 0
\(535\) 26.3013i 1.13710i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.76028 0.205040
\(540\) 0 0
\(541\) 33.1051 1.42330 0.711650 0.702534i \(-0.247948\pi\)
0.711650 + 0.702534i \(0.247948\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 52.1872i 2.23545i
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −21.2132 −0.903713
\(552\) 0 0
\(553\) −1.80385 −0.0767074
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 35.5312i − 1.50551i −0.658303 0.752753i \(-0.728726\pi\)
0.658303 0.752753i \(-0.271274\pi\)
\(558\) 0 0
\(559\) 6.00000i 0.253773i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.4944 −0.821590 −0.410795 0.911728i \(-0.634749\pi\)
−0.410795 + 0.911728i \(0.634749\pi\)
\(564\) 0 0
\(565\) 25.1769 1.05920
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 20.2795i − 0.850159i −0.905156 0.425080i \(-0.860246\pi\)
0.905156 0.425080i \(-0.139754\pi\)
\(570\) 0 0
\(571\) 2.24871i 0.0941056i 0.998892 + 0.0470528i \(0.0149829\pi\)
−0.998892 + 0.0470528i \(0.985017\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.34847 −0.306452
\(576\) 0 0
\(577\) −26.3923 −1.09873 −0.549363 0.835584i \(-0.685130\pi\)
−0.549363 + 0.835584i \(0.685130\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.82894i 0.324799i
\(582\) 0 0
\(583\) − 41.7128i − 1.72757i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.9802 1.36124 0.680619 0.732638i \(-0.261711\pi\)
0.680619 + 0.732638i \(0.261711\pi\)
\(588\) 0 0
\(589\) −52.3205 −2.15583
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 22.3872i − 0.919331i −0.888092 0.459666i \(-0.847969\pi\)
0.888092 0.459666i \(-0.152031\pi\)
\(594\) 0 0
\(595\) − 1.12436i − 0.0460941i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.14351 −0.332735 −0.166367 0.986064i \(-0.553204\pi\)
−0.166367 + 0.986064i \(0.553204\pi\)
\(600\) 0 0
\(601\) −47.6603 −1.94410 −0.972051 0.234769i \(-0.924567\pi\)
−0.972051 + 0.234769i \(0.924567\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 34.5975i − 1.40659i
\(606\) 0 0
\(607\) 23.1769i 0.940722i 0.882474 + 0.470361i \(0.155876\pi\)
−0.882474 + 0.470361i \(0.844124\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.06678 −0.164524
\(612\) 0 0
\(613\) 1.58846 0.0641572 0.0320786 0.999485i \(-0.489787\pi\)
0.0320786 + 0.999485i \(0.489787\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 10.4543i − 0.420874i −0.977607 0.210437i \(-0.932511\pi\)
0.977607 0.210437i \(-0.0674887\pi\)
\(618\) 0 0
\(619\) − 11.3397i − 0.455783i −0.973687 0.227891i \(-0.926817\pi\)
0.973687 0.227891i \(-0.0731831\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.93185 −0.0773980
\(624\) 0 0
\(625\) −29.5500 −1.18200
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.933740i 0.0372307i
\(630\) 0 0
\(631\) − 22.0526i − 0.877899i −0.898512 0.438949i \(-0.855351\pi\)
0.898512 0.438949i \(-0.144649\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 52.4002 2.07944
\(636\) 0 0
\(637\) 1.26795 0.0502380
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 7.72741i − 0.305214i −0.988287 0.152607i \(-0.951233\pi\)
0.988287 0.152607i \(-0.0487669\pi\)
\(642\) 0 0
\(643\) 4.85641i 0.191518i 0.995405 + 0.0957590i \(0.0305278\pi\)
−0.995405 + 0.0957590i \(0.969472\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.7290 −0.893567 −0.446784 0.894642i \(-0.647431\pi\)
−0.446784 + 0.894642i \(0.647431\pi\)
\(648\) 0 0
\(649\) 16.5885 0.651154
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.7738i 1.55647i 0.627973 + 0.778235i \(0.283885\pi\)
−0.627973 + 0.778235i \(0.716115\pi\)
\(654\) 0 0
\(655\) 4.19615i 0.163957i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0072 0.467735 0.233867 0.972269i \(-0.424862\pi\)
0.233867 + 0.972269i \(0.424862\pi\)
\(660\) 0 0
\(661\) −8.73205 −0.339637 −0.169819 0.985475i \(-0.554318\pi\)
−0.169819 + 0.985475i \(0.554318\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.8356i 0.575301i
\(666\) 0 0
\(667\) − 8.19615i − 0.317356i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 23.4596 0.905649
\(672\) 0 0
\(673\) 0.928203 0.0357796 0.0178898 0.999840i \(-0.494305\pi\)
0.0178898 + 0.999840i \(0.494305\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 12.2846i − 0.472136i −0.971737 0.236068i \(-0.924141\pi\)
0.971737 0.236068i \(-0.0758588\pi\)
\(678\) 0 0
\(679\) 12.9282i 0.496139i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.83083 0.261374 0.130687 0.991424i \(-0.458282\pi\)
0.130687 + 0.991424i \(0.458282\pi\)
\(684\) 0 0
\(685\) 10.3397 0.395061
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 11.1106i − 0.423282i
\(690\) 0 0
\(691\) − 39.3205i − 1.49582i −0.663799 0.747911i \(-0.731057\pi\)
0.663799 0.747911i \(-0.268943\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −33.5894 −1.27412
\(696\) 0 0
\(697\) −4.05256 −0.153502
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 27.1475i − 1.02535i −0.858584 0.512673i \(-0.828655\pi\)
0.858584 0.512673i \(-0.171345\pi\)
\(702\) 0 0
\(703\) − 12.3205i − 0.464677i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.52004 −0.169994
\(708\) 0 0
\(709\) −0.660254 −0.0247964 −0.0123982 0.999923i \(-0.503947\pi\)
−0.0123982 + 0.999923i \(0.503947\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 20.2151i − 0.757061i
\(714\) 0 0
\(715\) − 17.9090i − 0.669757i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 35.9845 1.34199 0.670997 0.741460i \(-0.265866\pi\)
0.670997 + 0.741460i \(0.265866\pi\)
\(720\) 0 0
\(721\) −3.53590 −0.131684
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 16.1384i 0.599364i
\(726\) 0 0
\(727\) − 18.9282i − 0.702008i −0.936374 0.351004i \(-0.885840\pi\)
0.936374 0.351004i \(-0.114160\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.79315 0.0663221
\(732\) 0 0
\(733\) −36.1962 −1.33694 −0.668468 0.743741i \(-0.733050\pi\)
−0.668468 + 0.743741i \(0.733050\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.4649i 1.34320i
\(738\) 0 0
\(739\) 1.07180i 0.0394267i 0.999806 + 0.0197133i \(0.00627536\pi\)
−0.999806 + 0.0197133i \(0.993725\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.0430 0.661934 0.330967 0.943642i \(-0.392625\pi\)
0.330967 + 0.943642i \(0.392625\pi\)
\(744\) 0 0
\(745\) 58.7461 2.15229
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.86422i 0.323892i
\(750\) 0 0
\(751\) 3.66025i 0.133565i 0.997768 + 0.0667823i \(0.0212733\pi\)
−0.997768 + 0.0667823i \(0.978727\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 44.2939 1.61202
\(756\) 0 0
\(757\) 52.6410 1.91327 0.956635 0.291289i \(-0.0940841\pi\)
0.956635 + 0.291289i \(0.0940841\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.2533i 1.09668i 0.836255 + 0.548340i \(0.184740\pi\)
−0.836255 + 0.548340i \(0.815260\pi\)
\(762\) 0 0
\(763\) 17.5885i 0.636745i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.41851 0.159543
\(768\) 0 0
\(769\) 0.339746 0.0122516 0.00612578 0.999981i \(-0.498050\pi\)
0.00612578 + 0.999981i \(0.498050\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 22.5630i − 0.811536i −0.913976 0.405768i \(-0.867004\pi\)
0.913976 0.405768i \(-0.132996\pi\)
\(774\) 0 0
\(775\) 39.8038i 1.42980i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 53.4727 1.91586
\(780\) 0 0
\(781\) −1.14359 −0.0409210
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 19.9749i − 0.712933i
\(786\) 0 0
\(787\) 27.3205i 0.973871i 0.873438 + 0.486935i \(0.161885\pi\)
−0.873438 + 0.486935i \(0.838115\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.48528 0.301702
\(792\) 0 0
\(793\) 6.24871 0.221898
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 45.9483i − 1.62757i −0.581164 0.813787i \(-0.697402\pi\)
0.581164 0.813787i \(-0.302598\pi\)
\(798\) 0 0
\(799\) 1.21539i 0.0429974i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.933740 0.0329510
\(804\) 0 0
\(805\) −5.73205 −0.202028
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.58682i 0.301896i 0.988542 + 0.150948i \(0.0482327\pi\)
−0.988542 + 0.150948i \(0.951767\pi\)
\(810\) 0 0
\(811\) − 27.3397i − 0.960028i −0.877261 0.480014i \(-0.840632\pi\)
0.877261 0.480014i \(-0.159368\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 53.5642 1.87627
\(816\) 0 0
\(817\) −23.6603 −0.827768
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 44.4698i 1.55201i 0.630730 + 0.776003i \(0.282756\pi\)
−0.630730 + 0.776003i \(0.717244\pi\)
\(822\) 0 0
\(823\) − 20.9282i − 0.729511i −0.931103 0.364756i \(-0.881153\pi\)
0.931103 0.364756i \(-0.118847\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −53.2968 −1.85331 −0.926656 0.375911i \(-0.877330\pi\)
−0.926656 + 0.375911i \(0.877330\pi\)
\(828\) 0 0
\(829\) 25.8038 0.896205 0.448102 0.893982i \(-0.352100\pi\)
0.448102 + 0.893982i \(0.352100\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 0.378937i − 0.0131294i
\(834\) 0 0
\(835\) − 67.9615i − 2.35191i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.0797 −0.451560 −0.225780 0.974178i \(-0.572493\pi\)
−0.225780 + 0.974178i \(0.572493\pi\)
\(840\) 0 0
\(841\) 11.0000 0.379310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 33.8024i 1.16284i
\(846\) 0 0
\(847\) − 11.6603i − 0.400651i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.76028 0.163180
\(852\) 0 0
\(853\) 18.0000 0.616308 0.308154 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 14.3552i − 0.490363i −0.969477 0.245182i \(-0.921152\pi\)
0.969477 0.245182i \(-0.0788476\pi\)
\(858\) 0 0
\(859\) 21.1051i 0.720097i 0.932934 + 0.360049i \(0.117240\pi\)
−0.932934 + 0.360049i \(0.882760\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.0512 1.36336 0.681680 0.731650i \(-0.261250\pi\)
0.681680 + 0.731650i \(0.261250\pi\)
\(864\) 0 0
\(865\) 30.6077 1.04069
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.58682i 0.291288i
\(870\) 0 0
\(871\) 9.71281i 0.329106i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.54914 −0.119983
\(876\) 0 0
\(877\) −45.8564 −1.54846 −0.774230 0.632904i \(-0.781863\pi\)
−0.774230 + 0.632904i \(0.781863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 43.3973i 1.46209i 0.682328 + 0.731046i \(0.260968\pi\)
−0.682328 + 0.731046i \(0.739032\pi\)
\(882\) 0 0
\(883\) − 10.3923i − 0.349729i −0.984593 0.174864i \(-0.944051\pi\)
0.984593 0.174864i \(-0.0559487\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.5574 0.858133 0.429066 0.903273i \(-0.358843\pi\)
0.429066 + 0.903273i \(0.358843\pi\)
\(888\) 0 0
\(889\) 17.6603 0.592306
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 16.0368i − 0.536652i
\(894\) 0 0
\(895\) − 35.5167i − 1.18719i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −44.3954 −1.48067
\(900\) 0 0
\(901\) −3.32051 −0.110622
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 73.5391i − 2.44452i
\(906\) 0 0
\(907\) 15.0718i 0.500451i 0.968188 + 0.250225i \(0.0805047\pi\)
−0.968188 + 0.250225i \(0.919495\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −42.6023 −1.41148 −0.705738 0.708473i \(-0.749384\pi\)
−0.705738 + 0.708473i \(0.749384\pi\)
\(912\) 0 0
\(913\) 37.2679 1.23339
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.41421i 0.0467014i
\(918\) 0 0
\(919\) 25.7128i 0.848187i 0.905618 + 0.424094i \(0.139407\pi\)
−0.905618 + 0.424094i \(0.860593\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.304608 −0.0100263
\(924\) 0 0
\(925\) −9.37307 −0.308185
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 40.0512i − 1.31404i −0.753874 0.657019i \(-0.771817\pi\)
0.753874 0.657019i \(-0.228183\pi\)
\(930\) 0 0
\(931\) 5.00000i 0.163868i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.35225 −0.175037
\(936\) 0 0
\(937\) 31.5692 1.03132 0.515661 0.856793i \(-0.327547\pi\)
0.515661 + 0.856793i \(0.327547\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6.55343i 0.213636i 0.994279 + 0.106818i \(0.0340662\pi\)
−0.994279 + 0.106818i \(0.965934\pi\)
\(942\) 0 0
\(943\) 20.6603i 0.672790i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.2765 −0.366438 −0.183219 0.983072i \(-0.558652\pi\)
−0.183219 + 0.983072i \(0.558652\pi\)
\(948\) 0 0
\(949\) 0.248711 0.00807351
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 59.5728i − 1.92975i −0.262703 0.964877i \(-0.584614\pi\)
0.262703 0.964877i \(-0.415386\pi\)
\(954\) 0 0
\(955\) 14.9474i 0.483688i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.48477 0.112529
\(960\) 0 0
\(961\) −78.4974 −2.53217
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 27.6551i − 0.890250i
\(966\) 0 0
\(967\) 24.2487i 0.779786i 0.920860 + 0.389893i \(0.127488\pi\)
−0.920860 + 0.389893i \(0.872512\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 16.0368 0.514646 0.257323 0.966325i \(-0.417160\pi\)
0.257323 + 0.966325i \(0.417160\pi\)
\(972\) 0 0
\(973\) −11.3205 −0.362919
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 41.2624i − 1.32010i −0.751221 0.660050i \(-0.770535\pi\)
0.751221 0.660050i \(-0.229465\pi\)
\(978\) 0 0
\(979\) 9.19615i 0.293910i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −61.1629 −1.95079 −0.975397 0.220456i \(-0.929246\pi\)
−0.975397 + 0.220456i \(0.929246\pi\)
\(984\) 0 0
\(985\) 59.5692 1.89803
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 9.14162i − 0.290687i
\(990\) 0 0
\(991\) − 32.8372i − 1.04311i −0.853219 0.521554i \(-0.825353\pi\)
0.853219 0.521554i \(-0.174647\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −42.3348 −1.34210
\(996\) 0 0
\(997\) −15.0718 −0.477329 −0.238664 0.971102i \(-0.576710\pi\)
−0.238664 + 0.971102i \(0.576710\pi\)
\(998\) 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.b.2591.3 8
3.2 odd 2 inner 6048.2.h.b.2591.5 yes 8
4.3 odd 2 inner 6048.2.h.b.2591.4 yes 8
12.11 even 2 inner 6048.2.h.b.2591.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.b.2591.3 8 1.1 even 1 trivial
6048.2.h.b.2591.4 yes 8 4.3 odd 2 inner
6048.2.h.b.2591.5 yes 8 3.2 odd 2 inner
6048.2.h.b.2591.6 yes 8 12.11 even 2 inner