Properties

Label 2304.3.g.p.1279.3
Level $2304$
Weight $3$
Character 2304.1279
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1279.3
Root \(-1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1279
Dual form 2304.3.g.p.1279.4

$q$-expansion

\(f(q)\) \(=\) \(q+5.29150 q^{5} -7.48331i q^{7} +O(q^{10})\) \(q+5.29150 q^{5} -7.48331i q^{7} +5.65685i q^{11} -4.00000 q^{13} -21.1660 q^{17} -29.9333i q^{19} +22.6274i q^{23} +3.00000 q^{25} -5.29150 q^{29} +22.4499i q^{31} -39.5980i q^{35} -28.0000 q^{37} -63.4980 q^{41} -29.9333i q^{43} -67.8823i q^{47} -7.00000 q^{49} -47.6235 q^{53} +29.9333i q^{55} +101.823i q^{59} -76.0000 q^{61} -21.1660 q^{65} +59.8665i q^{67} +90.5097i q^{71} -26.0000 q^{73} +42.3320 q^{77} -127.216i q^{79} -118.794i q^{83} -112.000 q^{85} +42.3320 q^{89} +29.9333i q^{91} -158.392i q^{95} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 16q^{13} + 12q^{25} - 112q^{37} - 28q^{49} - 304q^{61} - 104q^{73} - 448q^{85} + 72q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 5.29150 1.05830 0.529150 0.848528i \(-0.322511\pi\)
0.529150 + 0.848528i \(0.322511\pi\)
\(6\) 0 0
\(7\) − 7.48331i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685i 0.514259i 0.966377 + 0.257130i \(0.0827768\pi\)
−0.966377 + 0.257130i \(0.917223\pi\)
\(12\) 0 0
\(13\) −4.00000 −0.307692 −0.153846 0.988095i \(-0.549166\pi\)
−0.153846 + 0.988095i \(0.549166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −21.1660 −1.24506 −0.622530 0.782596i \(-0.713895\pi\)
−0.622530 + 0.782596i \(0.713895\pi\)
\(18\) 0 0
\(19\) − 29.9333i − 1.57543i −0.616037 0.787717i \(-0.711263\pi\)
0.616037 0.787717i \(-0.288737\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 22.6274i 0.983801i 0.870651 + 0.491900i \(0.163698\pi\)
−0.870651 + 0.491900i \(0.836302\pi\)
\(24\) 0 0
\(25\) 3.00000 0.120000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.29150 −0.182466 −0.0912328 0.995830i \(-0.529081\pi\)
−0.0912328 + 0.995830i \(0.529081\pi\)
\(30\) 0 0
\(31\) 22.4499i 0.724192i 0.932141 + 0.362096i \(0.117939\pi\)
−0.932141 + 0.362096i \(0.882061\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 39.5980i − 1.13137i
\(36\) 0 0
\(37\) −28.0000 −0.756757 −0.378378 0.925651i \(-0.623518\pi\)
−0.378378 + 0.925651i \(0.623518\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −63.4980 −1.54873 −0.774366 0.632738i \(-0.781931\pi\)
−0.774366 + 0.632738i \(0.781931\pi\)
\(42\) 0 0
\(43\) − 29.9333i − 0.696122i −0.937472 0.348061i \(-0.886840\pi\)
0.937472 0.348061i \(-0.113160\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 67.8823i − 1.44430i −0.691735 0.722152i \(-0.743153\pi\)
0.691735 0.722152i \(-0.256847\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −47.6235 −0.898557 −0.449279 0.893392i \(-0.648319\pi\)
−0.449279 + 0.893392i \(0.648319\pi\)
\(54\) 0 0
\(55\) 29.9333i 0.544241i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 101.823i 1.72582i 0.505358 + 0.862910i \(0.331361\pi\)
−0.505358 + 0.862910i \(0.668639\pi\)
\(60\) 0 0
\(61\) −76.0000 −1.24590 −0.622951 0.782261i \(-0.714066\pi\)
−0.622951 + 0.782261i \(0.714066\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −21.1660 −0.325631
\(66\) 0 0
\(67\) 59.8665i 0.893530i 0.894651 + 0.446765i \(0.147424\pi\)
−0.894651 + 0.446765i \(0.852576\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 90.5097i 1.27478i 0.770540 + 0.637392i \(0.219987\pi\)
−0.770540 + 0.637392i \(0.780013\pi\)
\(72\) 0 0
\(73\) −26.0000 −0.356164 −0.178082 0.984016i \(-0.556989\pi\)
−0.178082 + 0.984016i \(0.556989\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 42.3320 0.549767
\(78\) 0 0
\(79\) − 127.216i − 1.61033i −0.593048 0.805167i \(-0.702076\pi\)
0.593048 0.805167i \(-0.297924\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 118.794i − 1.43125i −0.698484 0.715626i \(-0.746141\pi\)
0.698484 0.715626i \(-0.253859\pi\)
\(84\) 0 0
\(85\) −112.000 −1.31765
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 42.3320 0.475641 0.237820 0.971309i \(-0.423567\pi\)
0.237820 + 0.971309i \(0.423567\pi\)
\(90\) 0 0
\(91\) 29.9333i 0.328937i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 158.392i − 1.66728i
\(96\) 0 0
\(97\) 18.0000 0.185567 0.0927835 0.995686i \(-0.470424\pi\)
0.0927835 + 0.995686i \(0.470424\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 164.037 1.62412 0.812062 0.583571i \(-0.198345\pi\)
0.812062 + 0.583571i \(0.198345\pi\)
\(102\) 0 0
\(103\) − 22.4499i − 0.217961i −0.994044 0.108980i \(-0.965241\pi\)
0.994044 0.108980i \(-0.0347585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 101.823i 0.951620i 0.879548 + 0.475810i \(0.157845\pi\)
−0.879548 + 0.475810i \(0.842155\pi\)
\(108\) 0 0
\(109\) −148.000 −1.35780 −0.678899 0.734232i \(-0.737543\pi\)
−0.678899 + 0.734232i \(0.737543\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 42.3320 0.374620 0.187310 0.982301i \(-0.440023\pi\)
0.187310 + 0.982301i \(0.440023\pi\)
\(114\) 0 0
\(115\) 119.733i 1.04116i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 158.392i 1.33102i
\(120\) 0 0
\(121\) 89.0000 0.735537
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −116.413 −0.931304
\(126\) 0 0
\(127\) 217.016i 1.70879i 0.519625 + 0.854394i \(0.326072\pi\)
−0.519625 + 0.854394i \(0.673928\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 158.392i − 1.20910i −0.796568 0.604549i \(-0.793353\pi\)
0.796568 0.604549i \(-0.206647\pi\)
\(132\) 0 0
\(133\) −224.000 −1.68421
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 232.826 1.69946 0.849730 0.527218i \(-0.176765\pi\)
0.849730 + 0.527218i \(0.176765\pi\)
\(138\) 0 0
\(139\) 179.600i 1.29208i 0.763302 + 0.646042i \(0.223577\pi\)
−0.763302 + 0.646042i \(0.776423\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 22.6274i − 0.158234i
\(144\) 0 0
\(145\) −28.0000 −0.193103
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −248.701 −1.66913 −0.834566 0.550908i \(-0.814281\pi\)
−0.834566 + 0.550908i \(0.814281\pi\)
\(150\) 0 0
\(151\) 22.4499i 0.148675i 0.997233 + 0.0743376i \(0.0236842\pi\)
−0.997233 + 0.0743376i \(0.976316\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 118.794i 0.766413i
\(156\) 0 0
\(157\) −236.000 −1.50318 −0.751592 0.659628i \(-0.770714\pi\)
−0.751592 + 0.659628i \(0.770714\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 169.328 1.05173
\(162\) 0 0
\(163\) − 269.399i − 1.65276i −0.563115 0.826378i \(-0.690397\pi\)
0.563115 0.826378i \(-0.309603\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 113.137i 0.677468i 0.940882 + 0.338734i \(0.109999\pi\)
−0.940882 + 0.338734i \(0.890001\pi\)
\(168\) 0 0
\(169\) −153.000 −0.905325
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 89.9555 0.519974 0.259987 0.965612i \(-0.416282\pi\)
0.259987 + 0.965612i \(0.416282\pi\)
\(174\) 0 0
\(175\) − 22.4499i − 0.128285i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 33.9411i 0.189615i 0.995496 + 0.0948076i \(0.0302236\pi\)
−0.995496 + 0.0948076i \(0.969776\pi\)
\(180\) 0 0
\(181\) −228.000 −1.25967 −0.629834 0.776730i \(-0.716877\pi\)
−0.629834 + 0.776730i \(0.716877\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −148.162 −0.800876
\(186\) 0 0
\(187\) − 119.733i − 0.640284i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 22.6274i − 0.118468i −0.998244 0.0592341i \(-0.981134\pi\)
0.998244 0.0592341i \(-0.0188658\pi\)
\(192\) 0 0
\(193\) 70.0000 0.362694 0.181347 0.983419i \(-0.441954\pi\)
0.181347 + 0.983419i \(0.441954\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 121.705 0.617790 0.308895 0.951096i \(-0.400041\pi\)
0.308895 + 0.951096i \(0.400041\pi\)
\(198\) 0 0
\(199\) − 202.049i − 1.01532i −0.861556 0.507662i \(-0.830510\pi\)
0.861556 0.507662i \(-0.169490\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 39.5980i 0.195064i
\(204\) 0 0
\(205\) −336.000 −1.63902
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 169.328 0.810182
\(210\) 0 0
\(211\) − 59.8665i − 0.283728i −0.989886 0.141864i \(-0.954691\pi\)
0.989886 0.141864i \(-0.0453095\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 158.392i − 0.736707i
\(216\) 0 0
\(217\) 168.000 0.774194
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 84.6640 0.383095
\(222\) 0 0
\(223\) − 172.116i − 0.771822i −0.922536 0.385911i \(-0.873887\pi\)
0.922536 0.385911i \(-0.126113\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 96.1665i 0.423641i 0.977309 + 0.211821i \(0.0679392\pi\)
−0.977309 + 0.211821i \(0.932061\pi\)
\(228\) 0 0
\(229\) −244.000 −1.06550 −0.532751 0.846272i \(-0.678842\pi\)
−0.532751 + 0.846272i \(0.678842\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −42.3320 −0.181682 −0.0908412 0.995865i \(-0.528956\pi\)
−0.0908412 + 0.995865i \(0.528956\pi\)
\(234\) 0 0
\(235\) − 359.199i − 1.52851i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 67.8823i − 0.284026i −0.989865 0.142013i \(-0.954642\pi\)
0.989865 0.142013i \(-0.0453575\pi\)
\(240\) 0 0
\(241\) −166.000 −0.688797 −0.344398 0.938824i \(-0.611917\pi\)
−0.344398 + 0.938824i \(0.611917\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −37.0405 −0.151186
\(246\) 0 0
\(247\) 119.733i 0.484749i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 401.637i − 1.60015i −0.599903 0.800073i \(-0.704794\pi\)
0.599903 0.800073i \(-0.295206\pi\)
\(252\) 0 0
\(253\) −128.000 −0.505929
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −423.320 −1.64716 −0.823580 0.567200i \(-0.808027\pi\)
−0.823580 + 0.567200i \(0.808027\pi\)
\(258\) 0 0
\(259\) 209.533i 0.809007i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 271.529i − 1.03243i −0.856459 0.516215i \(-0.827341\pi\)
0.856459 0.516215i \(-0.172659\pi\)
\(264\) 0 0
\(265\) −252.000 −0.950943
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 37.0405 0.137697 0.0688485 0.997627i \(-0.478067\pi\)
0.0688485 + 0.997627i \(0.478067\pi\)
\(270\) 0 0
\(271\) − 52.3832i − 0.193296i −0.995319 0.0966480i \(-0.969188\pi\)
0.995319 0.0966480i \(-0.0308121\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.9706i 0.0617111i
\(276\) 0 0
\(277\) −372.000 −1.34296 −0.671480 0.741023i \(-0.734341\pi\)
−0.671480 + 0.741023i \(0.734341\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 84.6640 0.301296 0.150648 0.988588i \(-0.451864\pi\)
0.150648 + 0.988588i \(0.451864\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 475.176i 1.65566i
\(288\) 0 0
\(289\) 159.000 0.550173
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 291.033 0.993285 0.496643 0.867955i \(-0.334566\pi\)
0.496643 + 0.867955i \(0.334566\pi\)
\(294\) 0 0
\(295\) 538.799i 1.82644i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 90.5097i − 0.302708i
\(300\) 0 0
\(301\) −224.000 −0.744186
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −402.154 −1.31854
\(306\) 0 0
\(307\) − 179.600i − 0.585015i −0.956263 0.292507i \(-0.905510\pi\)
0.956263 0.292507i \(-0.0944896\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 135.765i − 0.436542i −0.975888 0.218271i \(-0.929958\pi\)
0.975888 0.218271i \(-0.0700416\pi\)
\(312\) 0 0
\(313\) 470.000 1.50160 0.750799 0.660531i \(-0.229669\pi\)
0.750799 + 0.660531i \(0.229669\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −386.280 −1.21855 −0.609274 0.792960i \(-0.708539\pi\)
−0.609274 + 0.792960i \(0.708539\pi\)
\(318\) 0 0
\(319\) − 29.9333i − 0.0938347i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 633.568i 1.96151i
\(324\) 0 0
\(325\) −12.0000 −0.0369231
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −507.984 −1.54403
\(330\) 0 0
\(331\) 119.733i 0.361731i 0.983508 + 0.180866i \(0.0578899\pi\)
−0.983508 + 0.180866i \(0.942110\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 316.784i 0.945623i
\(336\) 0 0
\(337\) −42.0000 −0.124629 −0.0623145 0.998057i \(-0.519848\pi\)
−0.0623145 + 0.998057i \(0.519848\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −126.996 −0.372422
\(342\) 0 0
\(343\) − 314.299i − 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 164.049i − 0.472763i −0.971660 0.236382i \(-0.924038\pi\)
0.971660 0.236382i \(-0.0759615\pi\)
\(348\) 0 0
\(349\) −284.000 −0.813754 −0.406877 0.913483i \(-0.633382\pi\)
−0.406877 + 0.913483i \(0.633382\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 211.660 0.599604 0.299802 0.954001i \(-0.403079\pi\)
0.299802 + 0.954001i \(0.403079\pi\)
\(354\) 0 0
\(355\) 478.932i 1.34910i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 452.548i − 1.26058i −0.776360 0.630290i \(-0.782936\pi\)
0.776360 0.630290i \(-0.217064\pi\)
\(360\) 0 0
\(361\) −535.000 −1.48199
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −137.579 −0.376929
\(366\) 0 0
\(367\) 561.249i 1.52929i 0.644453 + 0.764644i \(0.277085\pi\)
−0.644453 + 0.764644i \(0.722915\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 356.382i 0.960598i
\(372\) 0 0
\(373\) 84.0000 0.225201 0.112601 0.993640i \(-0.464082\pi\)
0.112601 + 0.993640i \(0.464082\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 21.1660 0.0561433
\(378\) 0 0
\(379\) 448.999i 1.18469i 0.805683 + 0.592347i \(0.201798\pi\)
−0.805683 + 0.592347i \(0.798202\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 158.392i 0.413556i 0.978388 + 0.206778i \(0.0662978\pi\)
−0.978388 + 0.206778i \(0.933702\pi\)
\(384\) 0 0
\(385\) 224.000 0.581818
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −714.353 −1.83638 −0.918191 0.396137i \(-0.870350\pi\)
−0.918191 + 0.396137i \(0.870350\pi\)
\(390\) 0 0
\(391\) − 478.932i − 1.22489i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 673.166i − 1.70422i
\(396\) 0 0
\(397\) 244.000 0.614610 0.307305 0.951611i \(-0.400573\pi\)
0.307305 + 0.951611i \(0.400573\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −232.826 −0.580614 −0.290307 0.956934i \(-0.593757\pi\)
−0.290307 + 0.956934i \(0.593757\pi\)
\(402\) 0 0
\(403\) − 89.7998i − 0.222828i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 158.392i − 0.389169i
\(408\) 0 0
\(409\) 418.000 1.02200 0.511002 0.859579i \(-0.329274\pi\)
0.511002 + 0.859579i \(0.329274\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 761.976 1.84498
\(414\) 0 0
\(415\) − 628.598i − 1.51470i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 152.735i − 0.364523i −0.983250 0.182261i \(-0.941658\pi\)
0.983250 0.182261i \(-0.0583417\pi\)
\(420\) 0 0
\(421\) −212.000 −0.503563 −0.251781 0.967784i \(-0.581016\pi\)
−0.251781 + 0.967784i \(0.581016\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −63.4980 −0.149407
\(426\) 0 0
\(427\) 568.732i 1.33192i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 746.705i 1.73249i 0.499616 + 0.866247i \(0.333474\pi\)
−0.499616 + 0.866247i \(0.666526\pi\)
\(432\) 0 0
\(433\) −466.000 −1.07621 −0.538106 0.842877i \(-0.680860\pi\)
−0.538106 + 0.842877i \(0.680860\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 677.312 1.54991
\(438\) 0 0
\(439\) − 381.649i − 0.869360i −0.900585 0.434680i \(-0.856861\pi\)
0.900585 0.434680i \(-0.143139\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 254.558i − 0.574624i −0.957837 0.287312i \(-0.907238\pi\)
0.957837 0.287312i \(-0.0927617\pi\)
\(444\) 0 0
\(445\) 224.000 0.503371
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 571.482 1.27279 0.636395 0.771364i \(-0.280425\pi\)
0.636395 + 0.771364i \(0.280425\pi\)
\(450\) 0 0
\(451\) − 359.199i − 0.796450i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 158.392i 0.348114i
\(456\) 0 0
\(457\) −542.000 −1.18600 −0.592998 0.805204i \(-0.702056\pi\)
−0.592998 + 0.805204i \(0.702056\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 174.620 0.378784 0.189392 0.981902i \(-0.439348\pi\)
0.189392 + 0.981902i \(0.439348\pi\)
\(462\) 0 0
\(463\) 336.749i 0.727320i 0.931532 + 0.363660i \(0.118473\pi\)
−0.931532 + 0.363660i \(0.881527\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 526.087i 1.12653i 0.826278 + 0.563263i \(0.190454\pi\)
−0.826278 + 0.563263i \(0.809546\pi\)
\(468\) 0 0
\(469\) 448.000 0.955224
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 169.328 0.357987
\(474\) 0 0
\(475\) − 89.7998i − 0.189052i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 181.019i − 0.377911i −0.981986 0.188955i \(-0.939490\pi\)
0.981986 0.188955i \(-0.0605102\pi\)
\(480\) 0 0
\(481\) 112.000 0.232848
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 95.2470 0.196386
\(486\) 0 0
\(487\) 651.048i 1.33686i 0.743777 + 0.668428i \(0.233032\pi\)
−0.743777 + 0.668428i \(0.766968\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 248.902i 0.506928i 0.967345 + 0.253464i \(0.0815699\pi\)
−0.967345 + 0.253464i \(0.918430\pi\)
\(492\) 0 0
\(493\) 112.000 0.227181
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 677.312 1.36280
\(498\) 0 0
\(499\) 239.466i 0.479892i 0.970786 + 0.239946i \(0.0771297\pi\)
−0.970786 + 0.239946i \(0.922870\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 950.352i − 1.88937i −0.327984 0.944683i \(-0.606369\pi\)
0.327984 0.944683i \(-0.393631\pi\)
\(504\) 0 0
\(505\) 868.000 1.71881
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −428.612 −0.842066 −0.421033 0.907045i \(-0.638332\pi\)
−0.421033 + 0.907045i \(0.638332\pi\)
\(510\) 0 0
\(511\) 194.566i 0.380756i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 118.794i − 0.230668i
\(516\) 0 0
\(517\) 384.000 0.742747
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 698.478 1.34065 0.670325 0.742068i \(-0.266155\pi\)
0.670325 + 0.742068i \(0.266155\pi\)
\(522\) 0 0
\(523\) − 868.065i − 1.65978i −0.557928 0.829890i \(-0.688403\pi\)
0.557928 0.829890i \(-0.311597\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 475.176i − 0.901662i
\(528\) 0 0
\(529\) 17.0000 0.0321361
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 253.992 0.476533
\(534\) 0 0
\(535\) 538.799i 1.00710i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 39.5980i − 0.0734656i
\(540\) 0 0
\(541\) 556.000 1.02773 0.513863 0.857872i \(-0.328214\pi\)
0.513863 + 0.857872i \(0.328214\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −783.142 −1.43696
\(546\) 0 0
\(547\) − 448.999i − 0.820839i −0.911897 0.410419i \(-0.865382\pi\)
0.911897 0.410419i \(-0.134618\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 158.392i 0.287463i
\(552\) 0 0
\(553\) −952.000 −1.72152
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −799.017 −1.43450 −0.717250 0.696816i \(-0.754600\pi\)
−0.717250 + 0.696816i \(0.754600\pi\)
\(558\) 0 0
\(559\) 119.733i 0.214191i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 220.617i − 0.391860i −0.980618 0.195930i \(-0.937227\pi\)
0.980618 0.195930i \(-0.0627726\pi\)
\(564\) 0 0
\(565\) 224.000 0.396460
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 529.150 0.929965 0.464983 0.885320i \(-0.346061\pi\)
0.464983 + 0.885320i \(0.346061\pi\)
\(570\) 0 0
\(571\) − 897.998i − 1.57268i −0.617797 0.786338i \(-0.711975\pi\)
0.617797 0.786338i \(-0.288025\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 67.8823i 0.118056i
\(576\) 0 0
\(577\) 586.000 1.01560 0.507799 0.861476i \(-0.330459\pi\)
0.507799 + 0.861476i \(0.330459\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −888.972 −1.53007
\(582\) 0 0
\(583\) − 269.399i − 0.462091i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.2548i 0.0770951i 0.999257 + 0.0385476i \(0.0122731\pi\)
−0.999257 + 0.0385476i \(0.987727\pi\)
\(588\) 0 0
\(589\) 672.000 1.14092
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 296.324 0.499703 0.249852 0.968284i \(-0.419618\pi\)
0.249852 + 0.968284i \(0.419618\pi\)
\(594\) 0 0
\(595\) 838.131i 1.40862i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1154.00i 1.92654i 0.268532 + 0.963271i \(0.413461\pi\)
−0.268532 + 0.963271i \(0.586539\pi\)
\(600\) 0 0
\(601\) 634.000 1.05491 0.527454 0.849583i \(-0.323147\pi\)
0.527454 + 0.849583i \(0.323147\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 470.944 0.778419
\(606\) 0 0
\(607\) − 800.715i − 1.31913i −0.751646 0.659567i \(-0.770740\pi\)
0.751646 0.659567i \(-0.229260\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 271.529i 0.444401i
\(612\) 0 0
\(613\) 1140.00 1.85971 0.929853 0.367931i \(-0.119934\pi\)
0.929853 + 0.367931i \(0.119934\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 592.648 0.960532 0.480266 0.877123i \(-0.340540\pi\)
0.480266 + 0.877123i \(0.340540\pi\)
\(618\) 0 0
\(619\) 359.199i 0.580289i 0.956983 + 0.290145i \(0.0937034\pi\)
−0.956983 + 0.290145i \(0.906297\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 316.784i − 0.508481i
\(624\) 0 0
\(625\) −691.000 −1.10560
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 592.648 0.942207
\(630\) 0 0
\(631\) 291.849i 0.462519i 0.972892 + 0.231259i \(0.0742846\pi\)
−0.972892 + 0.231259i \(0.925715\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1148.34i 1.80841i
\(636\) 0 0
\(637\) 28.0000 0.0439560
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 486.818 0.759467 0.379733 0.925096i \(-0.376016\pi\)
0.379733 + 0.925096i \(0.376016\pi\)
\(642\) 0 0
\(643\) − 808.198i − 1.25692i −0.777843 0.628459i \(-0.783686\pi\)
0.777843 0.628459i \(-0.216314\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 294.156i − 0.454647i −0.973819 0.227323i \(-0.927003\pi\)
0.973819 0.227323i \(-0.0729974\pi\)
\(648\) 0 0
\(649\) −576.000 −0.887519
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −79.3725 −0.121551 −0.0607753 0.998151i \(-0.519357\pi\)
−0.0607753 + 0.998151i \(0.519357\pi\)
\(654\) 0 0
\(655\) − 838.131i − 1.27959i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1040.86i − 1.57946i −0.613457 0.789728i \(-0.710222\pi\)
0.613457 0.789728i \(-0.289778\pi\)
\(660\) 0 0
\(661\) 548.000 0.829047 0.414523 0.910039i \(-0.363948\pi\)
0.414523 + 0.910039i \(0.363948\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1185.30 −1.78240
\(666\) 0 0
\(667\) − 119.733i − 0.179510i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 429.921i − 0.640717i
\(672\) 0 0
\(673\) 334.000 0.496285 0.248143 0.968724i \(-0.420180\pi\)
0.248143 + 0.968724i \(0.420180\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −79.3725 −0.117242 −0.0586208 0.998280i \(-0.518670\pi\)
−0.0586208 + 0.998280i \(0.518670\pi\)
\(678\) 0 0
\(679\) − 134.700i − 0.198379i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 627.911i − 0.919342i −0.888089 0.459671i \(-0.847967\pi\)
0.888089 0.459671i \(-0.152033\pi\)
\(684\) 0 0
\(685\) 1232.00 1.79854
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 190.494 0.276479
\(690\) 0 0
\(691\) − 748.331i − 1.08297i −0.840711 0.541484i \(-0.817863\pi\)
0.840711 0.541484i \(-0.182137\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 950.352i 1.36741i
\(696\) 0 0
\(697\) 1344.00 1.92826
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 121.705 0.173616 0.0868078 0.996225i \(-0.472333\pi\)
0.0868078 + 0.996225i \(0.472333\pi\)
\(702\) 0 0
\(703\) 838.131i 1.19222i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1227.54i − 1.73626i
\(708\) 0 0
\(709\) −4.00000 −0.00564175 −0.00282087 0.999996i \(-0.500898\pi\)
−0.00282087 + 0.999996i \(0.500898\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −507.984 −0.712460
\(714\) 0 0
\(715\) − 119.733i − 0.167459i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 316.784i − 0.440589i −0.975433 0.220295i \(-0.929298\pi\)
0.975433 0.220295i \(-0.0707019\pi\)
\(720\) 0 0
\(721\) −168.000 −0.233010
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −15.8745 −0.0218959
\(726\) 0 0
\(727\) − 1174.88i − 1.61607i −0.589137 0.808033i \(-0.700532\pi\)
0.589137 0.808033i \(-0.299468\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 633.568i 0.866714i
\(732\) 0 0
\(733\) 1212.00 1.65348 0.826739 0.562585i \(-0.190193\pi\)
0.826739 + 0.562585i \(0.190193\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −338.656 −0.459506
\(738\) 0 0
\(739\) 1257.20i 1.70121i 0.525802 + 0.850607i \(0.323765\pi\)
−0.525802 + 0.850607i \(0.676235\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1018.23i 1.37044i 0.728338 + 0.685218i \(0.240293\pi\)
−0.728338 + 0.685218i \(0.759707\pi\)
\(744\) 0 0
\(745\) −1316.00 −1.76644
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 761.976 1.01732
\(750\) 0 0
\(751\) 411.582i 0.548046i 0.961723 + 0.274023i \(0.0883544\pi\)
−0.961723 + 0.274023i \(0.911646\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 118.794i 0.157343i
\(756\) 0 0
\(757\) 1020.00 1.34742 0.673712 0.738994i \(-0.264699\pi\)
0.673712 + 0.738994i \(0.264699\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −529.150 −0.695335 −0.347668 0.937618i \(-0.613026\pi\)
−0.347668 + 0.937618i \(0.613026\pi\)
\(762\) 0 0
\(763\) 1107.53i 1.45155i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 407.294i − 0.531022i
\(768\) 0 0
\(769\) 262.000 0.340702 0.170351 0.985383i \(-0.445510\pi\)
0.170351 + 0.985383i \(0.445510\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −851.932 −1.10211 −0.551056 0.834469i \(-0.685775\pi\)
−0.551056 + 0.834469i \(0.685775\pi\)
\(774\) 0 0
\(775\) 67.3498i 0.0869030i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1900.70i 2.43993i
\(780\) 0 0
\(781\) −512.000 −0.655570
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1248.79 −1.59082
\(786\) 0 0
\(787\) − 568.732i − 0.722658i −0.932438 0.361329i \(-0.882323\pi\)
0.932438 0.361329i \(-0.117677\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 316.784i − 0.400485i
\(792\) 0 0
\(793\) 304.000 0.383354
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 894.264 1.12204 0.561019 0.827803i \(-0.310410\pi\)
0.561019 + 0.827803i \(0.310410\pi\)
\(798\) 0 0
\(799\) 1436.80i 1.79824i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 147.078i − 0.183161i
\(804\) 0 0
\(805\) 896.000 1.11304
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −105.830 −0.130816 −0.0654079 0.997859i \(-0.520835\pi\)
−0.0654079 + 0.997859i \(0.520835\pi\)
\(810\) 0 0
\(811\) 987.798i 1.21800i 0.793170 + 0.609000i \(0.208429\pi\)
−0.793170 + 0.609000i \(0.791571\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1425.53i − 1.74911i
\(816\) 0 0
\(817\) −896.000 −1.09670
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1010.68 1.23103 0.615516 0.788125i \(-0.288948\pi\)
0.615516 + 0.788125i \(0.288948\pi\)
\(822\) 0 0
\(823\) 830.648i 1.00929i 0.863326 + 0.504646i \(0.168377\pi\)
−0.863326 + 0.504646i \(0.831623\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 328.098i − 0.396732i −0.980128 0.198366i \(-0.936437\pi\)
0.980128 0.198366i \(-0.0635635\pi\)
\(828\) 0 0
\(829\) 1164.00 1.40410 0.702051 0.712127i \(-0.252268\pi\)
0.702051 + 0.712127i \(0.252268\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 148.162 0.177866
\(834\) 0 0
\(835\) 598.665i 0.716964i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 995.606i 1.18666i 0.804960 + 0.593329i \(0.202187\pi\)
−0.804960 + 0.593329i \(0.797813\pi\)
\(840\) 0 0
\(841\) −813.000 −0.966706
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −809.600 −0.958106
\(846\) 0 0
\(847\) − 666.015i − 0.786322i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 633.568i − 0.744498i
\(852\) 0 0
\(853\) 276.000 0.323564 0.161782 0.986827i \(-0.448276\pi\)
0.161782 + 0.986827i \(0.448276\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 275.158 0.321071 0.160536 0.987030i \(-0.448678\pi\)
0.160536 + 0.987030i \(0.448678\pi\)
\(858\) 0 0
\(859\) 628.598i 0.731779i 0.930658 + 0.365890i \(0.119235\pi\)
−0.930658 + 0.365890i \(0.880765\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 294.156i 0.340853i 0.985370 + 0.170427i \(0.0545146\pi\)
−0.985370 + 0.170427i \(0.945485\pi\)
\(864\) 0 0
\(865\) 476.000 0.550289
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 719.644 0.828129
\(870\) 0 0
\(871\) − 239.466i − 0.274932i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 871.156i 0.995606i
\(876\) 0 0
\(877\) 340.000 0.387685 0.193843 0.981033i \(-0.437905\pi\)
0.193843 + 0.981033i \(0.437905\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −761.976 −0.864899 −0.432450 0.901658i \(-0.642351\pi\)
−0.432450 + 0.901658i \(0.642351\pi\)
\(882\) 0 0
\(883\) 89.7998i 0.101699i 0.998706 + 0.0508493i \(0.0161928\pi\)
−0.998706 + 0.0508493i \(0.983807\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 859.842i 0.969382i 0.874685 + 0.484691i \(0.161068\pi\)
−0.874685 + 0.484691i \(0.838932\pi\)
\(888\) 0 0
\(889\) 1624.00 1.82677
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2031.94 −2.27541
\(894\) 0 0
\(895\) 179.600i 0.200670i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 118.794i − 0.132140i
\(900\) 0 0
\(901\) 1008.00 1.11876
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1206.46 −1.33311
\(906\) 0 0
\(907\) − 448.999i − 0.495037i −0.968883 0.247519i \(-0.920385\pi\)
0.968883 0.247519i \(-0.0796152\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 248.902i − 0.273218i −0.990625 0.136609i \(-0.956380\pi\)
0.990625 0.136609i \(-0.0436204\pi\)
\(912\) 0 0
\(913\) 672.000 0.736035
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1185.30 −1.29258
\(918\) 0 0
\(919\) 710.915i 0.773574i 0.922169 + 0.386787i \(0.126415\pi\)
−0.922169 + 0.386787i \(0.873585\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 362.039i − 0.392241i
\(924\) 0 0
\(925\) −84.0000 −0.0908108
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −910.138 −0.979697 −0.489848 0.871808i \(-0.662948\pi\)
−0.489848 + 0.871808i \(0.662948\pi\)
\(930\) 0 0
\(931\) 209.533i 0.225062i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 633.568i − 0.677612i
\(936\) 0 0
\(937\) −1166.00 −1.24440 −0.622199 0.782860i \(-0.713760\pi\)
−0.622199 + 0.782860i \(0.713760\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −418.029 −0.444239 −0.222119 0.975019i \(-0.571297\pi\)
−0.222119 + 0.975019i \(0.571297\pi\)
\(942\) 0 0
\(943\) − 1436.80i − 1.52364i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 893.783i − 0.943805i −0.881651 0.471902i \(-0.843567\pi\)
0.881651 0.471902i \(-0.156433\pi\)
\(948\) 0 0
\(949\) 104.000 0.109589
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1291.13 1.35480 0.677401 0.735614i \(-0.263106\pi\)
0.677401 + 0.735614i \(0.263106\pi\)
\(954\) 0 0
\(955\) − 119.733i − 0.125375i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1742.31i − 1.81680i
\(960\) 0 0
\(961\) 457.000 0.475546
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 370.405 0.383840
\(966\) 0 0
\(967\) − 1070.11i − 1.10663i −0.832971 0.553316i \(-0.813362\pi\)
0.832971 0.553316i \(-0.186638\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 514.774i − 0.530148i −0.964228 0.265074i \(-0.914604\pi\)
0.964228 0.265074i \(-0.0853964\pi\)
\(972\) 0 0
\(973\) 1344.00 1.38129
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1206.46 1.23486 0.617432 0.786624i \(-0.288173\pi\)
0.617432 + 0.786624i \(0.288173\pi\)
\(978\) 0 0
\(979\) 239.466i 0.244603i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 950.352i − 0.966787i −0.875403 0.483393i \(-0.839404\pi\)
0.875403 0.483393i \(-0.160596\pi\)
\(984\) 0 0
\(985\) 644.000 0.653807
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 677.312 0.684846
\(990\) 0 0
\(991\) − 486.415i − 0.490833i −0.969418 0.245416i \(-0.921075\pi\)
0.969418 0.245416i \(-0.0789247\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1069.15i − 1.07452i
\(996\) 0 0
\(997\) −1244.00 −1.24774 −0.623872 0.781527i \(-0.714441\pi\)
−0.623872 + 0.781527i \(0.714441\pi\)
\(998\) 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 2304.3.g.p.1279.3 4
3.2 odd 2 inner 2304.3.g.p.1279.1 4
4.3 odd 2 inner 2304.3.g.p.1279.4 4
8.3 odd 2 2304.3.g.w.1279.2 4
8.5 even 2 2304.3.g.w.1279.1 4
12.11 even 2 inner 2304.3.g.p.1279.2 4
16.3 odd 4 1152.3.b.i.703.1 8
16.5 even 4 1152.3.b.i.703.8 yes 8
16.11 odd 4 1152.3.b.i.703.6 yes 8
16.13 even 4 1152.3.b.i.703.3 yes 8
24.5 odd 2 2304.3.g.w.1279.3 4
24.11 even 2 2304.3.g.w.1279.4 4
48.5 odd 4 1152.3.b.i.703.4 yes 8
48.11 even 4 1152.3.b.i.703.2 yes 8
48.29 odd 4 1152.3.b.i.703.7 yes 8
48.35 even 4 1152.3.b.i.703.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.b.i.703.1 8 16.3 odd 4
1152.3.b.i.703.2 yes 8 48.11 even 4
1152.3.b.i.703.3 yes 8 16.13 even 4
1152.3.b.i.703.4 yes 8 48.5 odd 4
1152.3.b.i.703.5 yes 8 48.35 even 4
1152.3.b.i.703.6 yes 8 16.11 odd 4
1152.3.b.i.703.7 yes 8 48.29 odd 4
1152.3.b.i.703.8 yes 8 16.5 even 4
2304.3.g.p.1279.1 4 3.2 odd 2 inner
2304.3.g.p.1279.2 4 12.11 even 2 inner
2304.3.g.p.1279.3 4 1.1 even 1 trivial
2304.3.g.p.1279.4 4 4.3 odd 2 inner
2304.3.g.w.1279.1 4 8.5 even 2
2304.3.g.w.1279.2 4 8.3 odd 2
2304.3.g.w.1279.3 4 24.5 odd 2
2304.3.g.w.1279.4 4 24.11 even 2