Properties

Label 2304.3.e.g.1025.2
Level $2304$
Weight $3$
Character 2304.1025
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.2
Root \(-1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1025
Dual form 2304.3.e.g.1025.4

$q$-expansion

\(f(q)\) \(=\) \(q-7.34847i q^{5} +10.3923 q^{7} +O(q^{10})\) \(q-7.34847i q^{5} +10.3923 q^{7} +8.48528i q^{11} -10.3923 q^{13} +21.2132i q^{17} -20.0000 q^{19} -14.6969i q^{23} -29.0000 q^{25} -36.7423i q^{29} +51.9615 q^{31} -76.3675i q^{35} -41.5692 q^{37} -72.1249i q^{41} +40.0000 q^{43} -73.4847i q^{47} +59.0000 q^{49} -36.7423i q^{53} +62.3538 q^{55} +33.9411i q^{59} +76.3675i q^{65} -100.000 q^{67} -73.4847i q^{71} -20.0000 q^{73} +88.1816i q^{77} -51.9615 q^{79} -127.279i q^{83} +155.885 q^{85} -12.7279i q^{89} -108.000 q^{91} +146.969i q^{95} +40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 80q^{19} - 116q^{25} + 160q^{43} + 236q^{49} - 400q^{67} - 80q^{73} - 432q^{91} + 160q^{97} + O(q^{100}) \)

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
\(5\) − 7.34847i − 1.46969i −0.678233 0.734847i \(-0.737254\pi\)
0.678233 0.734847i \(-0.262746\pi\)
\(6\) 0 0
\(7\) 10.3923 1.48461 0.742307 0.670059i \(-0.233731\pi\)
0.742307 + 0.670059i \(0.233731\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 8.48528i 0.771389i 0.922627 + 0.385695i \(0.126038\pi\)
−0.922627 + 0.385695i \(0.873962\pi\)
\(12\) 0 0
\(13\) −10.3923 −0.799408 −0.399704 0.916644i \(-0.630887\pi\)
−0.399704 + 0.916644i \(0.630887\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 21.2132i 1.24784i 0.781490 + 0.623918i \(0.214460\pi\)
−0.781490 + 0.623918i \(0.785540\pi\)
\(18\) 0 0
\(19\) −20.0000 −1.05263 −0.526316 0.850289i \(-0.676427\pi\)
−0.526316 + 0.850289i \(0.676427\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 14.6969i − 0.638997i −0.947587 0.319499i \(-0.896486\pi\)
0.947587 0.319499i \(-0.103514\pi\)
\(24\) 0 0
\(25\) −29.0000 −1.16000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 36.7423i − 1.26698i −0.773752 0.633489i \(-0.781622\pi\)
0.773752 0.633489i \(-0.218378\pi\)
\(30\) 0 0
\(31\) 51.9615 1.67618 0.838089 0.545533i \(-0.183673\pi\)
0.838089 + 0.545533i \(0.183673\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 76.3675i − 2.18193i
\(36\) 0 0
\(37\) −41.5692 −1.12349 −0.561746 0.827310i \(-0.689870\pi\)
−0.561746 + 0.827310i \(0.689870\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 72.1249i − 1.75914i −0.475766 0.879572i \(-0.657829\pi\)
0.475766 0.879572i \(-0.342171\pi\)
\(42\) 0 0
\(43\) 40.0000 0.930233 0.465116 0.885250i \(-0.346013\pi\)
0.465116 + 0.885250i \(0.346013\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 73.4847i − 1.56350i −0.623589 0.781752i \(-0.714326\pi\)
0.623589 0.781752i \(-0.285674\pi\)
\(48\) 0 0
\(49\) 59.0000 1.20408
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 36.7423i − 0.693252i −0.938003 0.346626i \(-0.887327\pi\)
0.938003 0.346626i \(-0.112673\pi\)
\(54\) 0 0
\(55\) 62.3538 1.13371
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 33.9411i 0.575273i 0.957740 + 0.287637i \(0.0928695\pi\)
−0.957740 + 0.287637i \(0.907130\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 76.3675i 1.17489i
\(66\) 0 0
\(67\) −100.000 −1.49254 −0.746269 0.665645i \(-0.768157\pi\)
−0.746269 + 0.665645i \(0.768157\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 73.4847i − 1.03500i −0.855685 0.517498i \(-0.826864\pi\)
0.855685 0.517498i \(-0.173136\pi\)
\(72\) 0 0
\(73\) −20.0000 −0.273973 −0.136986 0.990573i \(-0.543742\pi\)
−0.136986 + 0.990573i \(0.543742\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 88.1816i 1.14522i
\(78\) 0 0
\(79\) −51.9615 −0.657741 −0.328870 0.944375i \(-0.606668\pi\)
−0.328870 + 0.944375i \(0.606668\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 127.279i − 1.53348i −0.641955 0.766742i \(-0.721876\pi\)
0.641955 0.766742i \(-0.278124\pi\)
\(84\) 0 0
\(85\) 155.885 1.83394
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 12.7279i − 0.143010i −0.997440 0.0715052i \(-0.977220\pi\)
0.997440 0.0715052i \(-0.0227802\pi\)
\(90\) 0 0
\(91\) −108.000 −1.18681
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 146.969i 1.54705i
\(96\) 0 0
\(97\) 40.0000 0.412371 0.206186 0.978513i \(-0.433895\pi\)
0.206186 + 0.978513i \(0.433895\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 183.712i 1.81893i 0.415783 + 0.909464i \(0.363508\pi\)
−0.415783 + 0.909464i \(0.636492\pi\)
\(102\) 0 0
\(103\) −93.5307 −0.908065 −0.454033 0.890985i \(-0.650015\pi\)
−0.454033 + 0.890985i \(0.650015\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 169.706i − 1.58603i −0.609200 0.793017i \(-0.708509\pi\)
0.609200 0.793017i \(-0.291491\pi\)
\(108\) 0 0
\(109\) 51.9615 0.476711 0.238356 0.971178i \(-0.423392\pi\)
0.238356 + 0.971178i \(0.423392\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 63.6396i − 0.563182i −0.959534 0.281591i \(-0.909138\pi\)
0.959534 0.281591i \(-0.0908622\pi\)
\(114\) 0 0
\(115\) −108.000 −0.939130
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 220.454i 1.85256i
\(120\) 0 0
\(121\) 49.0000 0.404959
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 29.3939i 0.235151i
\(126\) 0 0
\(127\) 10.3923 0.0818292 0.0409146 0.999163i \(-0.486973\pi\)
0.0409146 + 0.999163i \(0.486973\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 50.9117i − 0.388639i −0.980938 0.194319i \(-0.937750\pi\)
0.980938 0.194319i \(-0.0622498\pi\)
\(132\) 0 0
\(133\) −207.846 −1.56275
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 106.066i 0.774205i 0.922037 + 0.387102i \(0.126524\pi\)
−0.922037 + 0.387102i \(0.873476\pi\)
\(138\) 0 0
\(139\) −172.000 −1.23741 −0.618705 0.785623i \(-0.712342\pi\)
−0.618705 + 0.785623i \(0.712342\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 88.1816i − 0.616655i
\(144\) 0 0
\(145\) −270.000 −1.86207
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 110.227i − 0.739779i −0.929076 0.369889i \(-0.879396\pi\)
0.929076 0.369889i \(-0.120604\pi\)
\(150\) 0 0
\(151\) −155.885 −1.03235 −0.516174 0.856484i \(-0.672644\pi\)
−0.516174 + 0.856484i \(0.672644\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 381.838i − 2.46347i
\(156\) 0 0
\(157\) 166.277 1.05909 0.529544 0.848282i \(-0.322363\pi\)
0.529544 + 0.848282i \(0.322363\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 152.735i − 0.948665i
\(162\) 0 0
\(163\) 160.000 0.981595 0.490798 0.871274i \(-0.336705\pi\)
0.490798 + 0.871274i \(0.336705\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 58.7878i 0.352022i 0.984388 + 0.176011i \(0.0563195\pi\)
−0.984388 + 0.176011i \(0.943680\pi\)
\(168\) 0 0
\(169\) −61.0000 −0.360947
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 36.7423i − 0.212384i −0.994346 0.106192i \(-0.966134\pi\)
0.994346 0.106192i \(-0.0338657\pi\)
\(174\) 0 0
\(175\) −301.377 −1.72215
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 50.9117i − 0.284423i −0.989836 0.142211i \(-0.954579\pi\)
0.989836 0.142211i \(-0.0454213\pi\)
\(180\) 0 0
\(181\) 259.808 1.43540 0.717701 0.696352i \(-0.245195\pi\)
0.717701 + 0.696352i \(0.245195\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 305.470i 1.65119i
\(186\) 0 0
\(187\) −180.000 −0.962567
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 293.939i 1.53895i 0.638679 + 0.769473i \(0.279481\pi\)
−0.638679 + 0.769473i \(0.720519\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.0518135 −0.0259067 0.999664i \(-0.508247\pi\)
−0.0259067 + 0.999664i \(0.508247\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 22.0454i − 0.111906i −0.998433 0.0559528i \(-0.982180\pi\)
0.998433 0.0559528i \(-0.0178196\pi\)
\(198\) 0 0
\(199\) −51.9615 −0.261113 −0.130557 0.991441i \(-0.541676\pi\)
−0.130557 + 0.991441i \(0.541676\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 381.838i − 1.88097i
\(204\) 0 0
\(205\) −530.008 −2.58540
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 169.706i − 0.811989i
\(210\) 0 0
\(211\) −172.000 −0.815166 −0.407583 0.913168i \(-0.633628\pi\)
−0.407583 + 0.913168i \(0.633628\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 293.939i − 1.36716i
\(216\) 0 0
\(217\) 540.000 2.48848
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 220.454i − 0.997530i
\(222\) 0 0
\(223\) 10.3923 0.0466023 0.0233011 0.999728i \(-0.492582\pi\)
0.0233011 + 0.999728i \(0.492582\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 42.4264i − 0.186900i −0.995624 0.0934502i \(-0.970210\pi\)
0.995624 0.0934502i \(-0.0297896\pi\)
\(228\) 0 0
\(229\) −259.808 −1.13453 −0.567266 0.823535i \(-0.691999\pi\)
−0.567266 + 0.823535i \(0.691999\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 106.066i − 0.455219i −0.973752 0.227609i \(-0.926909\pi\)
0.973752 0.227609i \(-0.0730910\pi\)
\(234\) 0 0
\(235\) −540.000 −2.29787
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 146.969i − 0.614935i −0.951559 0.307467i \(-0.900519\pi\)
0.951559 0.307467i \(-0.0994815\pi\)
\(240\) 0 0
\(241\) 140.000 0.580913 0.290456 0.956888i \(-0.406193\pi\)
0.290456 + 0.956888i \(0.406193\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 433.560i − 1.76963i
\(246\) 0 0
\(247\) 207.846 0.841482
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 246.073i − 0.980371i −0.871618 0.490186i \(-0.836929\pi\)
0.871618 0.490186i \(-0.163071\pi\)
\(252\) 0 0
\(253\) 124.708 0.492916
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 21.2132i − 0.0825416i −0.999148 0.0412708i \(-0.986859\pi\)
0.999148 0.0412708i \(-0.0131406\pi\)
\(258\) 0 0
\(259\) −432.000 −1.66795
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 88.1816i 0.335291i 0.985847 + 0.167646i \(0.0536165\pi\)
−0.985847 + 0.167646i \(0.946384\pi\)
\(264\) 0 0
\(265\) −270.000 −1.01887
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 110.227i − 0.409766i −0.978786 0.204883i \(-0.934319\pi\)
0.978786 0.204883i \(-0.0656814\pi\)
\(270\) 0 0
\(271\) 467.654 1.72566 0.862830 0.505495i \(-0.168690\pi\)
0.862830 + 0.505495i \(0.168690\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 246.073i − 0.894811i
\(276\) 0 0
\(277\) −322.161 −1.16304 −0.581519 0.813533i \(-0.697541\pi\)
−0.581519 + 0.813533i \(0.697541\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 140.007i 0.498246i 0.968472 + 0.249123i \(0.0801424\pi\)
−0.968472 + 0.249123i \(0.919858\pi\)
\(282\) 0 0
\(283\) 80.0000 0.282686 0.141343 0.989961i \(-0.454858\pi\)
0.141343 + 0.989961i \(0.454858\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 749.544i − 2.61165i
\(288\) 0 0
\(289\) −161.000 −0.557093
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 315.984i 1.07844i 0.842164 + 0.539222i \(0.181282\pi\)
−0.842164 + 0.539222i \(0.818718\pi\)
\(294\) 0 0
\(295\) 249.415 0.845476
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 152.735i 0.510820i
\(300\) 0 0
\(301\) 415.692 1.38104
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 380.000 1.23779 0.618893 0.785476i \(-0.287582\pi\)
0.618893 + 0.785476i \(0.287582\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 146.969i 0.472570i 0.971684 + 0.236285i \(0.0759300\pi\)
−0.971684 + 0.236285i \(0.924070\pi\)
\(312\) 0 0
\(313\) −310.000 −0.990415 −0.495208 0.868775i \(-0.664908\pi\)
−0.495208 + 0.868775i \(0.664908\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 242.499i 0.764983i 0.923959 + 0.382491i \(0.124934\pi\)
−0.923959 + 0.382491i \(0.875066\pi\)
\(318\) 0 0
\(319\) 311.769 0.977333
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 424.264i − 1.31351i
\(324\) 0 0
\(325\) 301.377 0.927313
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 763.675i − 2.32120i
\(330\) 0 0
\(331\) 500.000 1.51057 0.755287 0.655394i \(-0.227497\pi\)
0.755287 + 0.655394i \(0.227497\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 734.847i 2.19357i
\(336\) 0 0
\(337\) 100.000 0.296736 0.148368 0.988932i \(-0.452598\pi\)
0.148368 + 0.988932i \(0.452598\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 440.908i 1.29299i
\(342\) 0 0
\(343\) 103.923 0.302983
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 42.4264i − 0.122266i −0.998130 0.0611332i \(-0.980529\pi\)
0.998130 0.0611332i \(-0.0194714\pi\)
\(348\) 0 0
\(349\) −207.846 −0.595548 −0.297774 0.954636i \(-0.596244\pi\)
−0.297774 + 0.954636i \(0.596244\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 615.183i 1.74273i 0.490637 + 0.871364i \(0.336764\pi\)
−0.490637 + 0.871364i \(0.663236\pi\)
\(354\) 0 0
\(355\) −540.000 −1.52113
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 514.393i − 1.43285i −0.697665 0.716425i \(-0.745777\pi\)
0.697665 0.716425i \(-0.254223\pi\)
\(360\) 0 0
\(361\) 39.0000 0.108033
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 146.969i 0.402656i
\(366\) 0 0
\(367\) 218.238 0.594655 0.297328 0.954776i \(-0.403905\pi\)
0.297328 + 0.954776i \(0.403905\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 381.838i − 1.02921i
\(372\) 0 0
\(373\) 665.108 1.78313 0.891565 0.452893i \(-0.149608\pi\)
0.891565 + 0.452893i \(0.149608\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 381.838i 1.01283i
\(378\) 0 0
\(379\) −92.0000 −0.242744 −0.121372 0.992607i \(-0.538729\pi\)
−0.121372 + 0.992607i \(0.538729\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 146.969i − 0.383732i −0.981421 0.191866i \(-0.938546\pi\)
0.981421 0.191866i \(-0.0614539\pi\)
\(384\) 0 0
\(385\) 648.000 1.68312
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 257.196i − 0.661173i −0.943776 0.330587i \(-0.892753\pi\)
0.943776 0.330587i \(-0.107247\pi\)
\(390\) 0 0
\(391\) 311.769 0.797364
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 381.838i 0.966678i
\(396\) 0 0
\(397\) 41.5692 0.104708 0.0523542 0.998629i \(-0.483328\pi\)
0.0523542 + 0.998629i \(0.483328\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 224.860i 0.560748i 0.959891 + 0.280374i \(0.0904585\pi\)
−0.959891 + 0.280374i \(0.909542\pi\)
\(402\) 0 0
\(403\) −540.000 −1.33995
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 352.727i − 0.866650i
\(408\) 0 0
\(409\) −368.000 −0.899756 −0.449878 0.893090i \(-0.648532\pi\)
−0.449878 + 0.893090i \(0.648532\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 352.727i 0.854059i
\(414\) 0 0
\(415\) −935.307 −2.25375
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 432.749i − 1.03281i −0.856343 0.516407i \(-0.827269\pi\)
0.856343 0.516407i \(-0.172731\pi\)
\(420\) 0 0
\(421\) 51.9615 0.123424 0.0617120 0.998094i \(-0.480344\pi\)
0.0617120 + 0.998094i \(0.480344\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 615.183i − 1.44749i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 367.423i − 0.852491i −0.904608 0.426245i \(-0.859836\pi\)
0.904608 0.426245i \(-0.140164\pi\)
\(432\) 0 0
\(433\) −470.000 −1.08545 −0.542725 0.839910i \(-0.682607\pi\)
−0.542725 + 0.839910i \(0.682607\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 293.939i 0.672629i
\(438\) 0 0
\(439\) −155.885 −0.355090 −0.177545 0.984113i \(-0.556816\pi\)
−0.177545 + 0.984113i \(0.556816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 551.543i 1.24502i 0.782612 + 0.622509i \(0.213887\pi\)
−0.782612 + 0.622509i \(0.786113\pi\)
\(444\) 0 0
\(445\) −93.5307 −0.210181
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 691.550i 1.54020i 0.637922 + 0.770101i \(0.279794\pi\)
−0.637922 + 0.770101i \(0.720206\pi\)
\(450\) 0 0
\(451\) 612.000 1.35698
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 793.635i 1.74425i
\(456\) 0 0
\(457\) −680.000 −1.48796 −0.743982 0.668199i \(-0.767065\pi\)
−0.743982 + 0.668199i \(0.767065\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 330.681i 0.717313i 0.933470 + 0.358656i \(0.116765\pi\)
−0.933470 + 0.358656i \(0.883235\pi\)
\(462\) 0 0
\(463\) 322.161 0.695813 0.347907 0.937529i \(-0.386893\pi\)
0.347907 + 0.937529i \(0.386893\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 466.690i 0.999337i 0.866217 + 0.499669i \(0.166545\pi\)
−0.866217 + 0.499669i \(0.833455\pi\)
\(468\) 0 0
\(469\) −1039.23 −2.21584
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 339.411i 0.717571i
\(474\) 0 0
\(475\) 580.000 1.22105
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 220.454i − 0.460238i −0.973162 0.230119i \(-0.926088\pi\)
0.973162 0.230119i \(-0.0739116\pi\)
\(480\) 0 0
\(481\) 432.000 0.898129
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 293.939i − 0.606059i
\(486\) 0 0
\(487\) −633.931 −1.30171 −0.650853 0.759204i \(-0.725588\pi\)
−0.650853 + 0.759204i \(0.725588\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 220.617i 0.449322i 0.974437 + 0.224661i \(0.0721275\pi\)
−0.974437 + 0.224661i \(0.927872\pi\)
\(492\) 0 0
\(493\) 779.423 1.58098
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 763.675i − 1.53657i
\(498\) 0 0
\(499\) −640.000 −1.28257 −0.641283 0.767305i \(-0.721597\pi\)
−0.641283 + 0.767305i \(0.721597\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 73.4847i 0.146093i 0.997329 + 0.0730464i \(0.0232721\pi\)
−0.997329 + 0.0730464i \(0.976728\pi\)
\(504\) 0 0
\(505\) 1350.00 2.67327
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 624.620i − 1.22715i −0.789636 0.613576i \(-0.789731\pi\)
0.789636 0.613576i \(-0.210269\pi\)
\(510\) 0 0
\(511\) −207.846 −0.406744
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 687.308i 1.33458i
\(516\) 0 0
\(517\) 623.538 1.20607
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 861.256i − 1.65308i −0.562876 0.826541i \(-0.690305\pi\)
0.562876 0.826541i \(-0.309695\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.0382409 −0.0191205 0.999817i \(-0.506087\pi\)
−0.0191205 + 0.999817i \(0.506087\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1102.27i 2.09159i
\(528\) 0 0
\(529\) 313.000 0.591682
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 749.544i 1.40627i
\(534\) 0 0
\(535\) −1247.08 −2.33098
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 500.632i 0.928816i
\(540\) 0 0
\(541\) −363.731 −0.672330 −0.336165 0.941803i \(-0.609130\pi\)
−0.336165 + 0.941803i \(0.609130\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 381.838i − 0.700620i
\(546\) 0 0
\(547\) 760.000 1.38940 0.694698 0.719301i \(-0.255538\pi\)
0.694698 + 0.719301i \(0.255538\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 734.847i 1.33366i
\(552\) 0 0
\(553\) −540.000 −0.976492
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 992.043i − 1.78105i −0.454937 0.890524i \(-0.650338\pi\)
0.454937 0.890524i \(-0.349662\pi\)
\(558\) 0 0
\(559\) −415.692 −0.743635
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 212.132i 0.376789i 0.982093 + 0.188394i \(0.0603283\pi\)
−0.982093 + 0.188394i \(0.939672\pi\)
\(564\) 0 0
\(565\) −467.654 −0.827706
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 479.418i 0.842563i 0.906930 + 0.421282i \(0.138420\pi\)
−0.906930 + 0.421282i \(0.861580\pi\)
\(570\) 0 0
\(571\) −280.000 −0.490368 −0.245184 0.969477i \(-0.578848\pi\)
−0.245184 + 0.969477i \(0.578848\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 426.211i 0.741237i
\(576\) 0 0
\(577\) −650.000 −1.12652 −0.563258 0.826281i \(-0.690452\pi\)
−0.563258 + 0.826281i \(0.690452\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 1322.72i − 2.27663i
\(582\) 0 0
\(583\) 311.769 0.534767
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 678.823i − 1.15643i −0.815886 0.578213i \(-0.803750\pi\)
0.815886 0.578213i \(-0.196250\pi\)
\(588\) 0 0
\(589\) −1039.23 −1.76440
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 445.477i − 0.751226i −0.926777 0.375613i \(-0.877432\pi\)
0.926777 0.375613i \(-0.122568\pi\)
\(594\) 0 0
\(595\) 1620.00 2.72269
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 367.423i − 0.613395i −0.951807 0.306697i \(-0.900776\pi\)
0.951807 0.306697i \(-0.0992240\pi\)
\(600\) 0 0
\(601\) 490.000 0.815308 0.407654 0.913137i \(-0.366347\pi\)
0.407654 + 0.913137i \(0.366347\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 360.075i − 0.595165i
\(606\) 0 0
\(607\) −322.161 −0.530744 −0.265372 0.964146i \(-0.585495\pi\)
−0.265372 + 0.964146i \(0.585495\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 763.675i 1.24988i
\(612\) 0 0
\(613\) 374.123 0.610315 0.305157 0.952302i \(-0.401291\pi\)
0.305157 + 0.952302i \(0.401291\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 784.889i 1.27210i 0.771646 + 0.636052i \(0.219434\pi\)
−0.771646 + 0.636052i \(0.780566\pi\)
\(618\) 0 0
\(619\) 1112.00 1.79645 0.898223 0.439540i \(-0.144859\pi\)
0.898223 + 0.439540i \(0.144859\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 132.272i − 0.212315i
\(624\) 0 0
\(625\) −509.000 −0.814400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 881.816i − 1.40193i
\(630\) 0 0
\(631\) 987.269 1.56461 0.782305 0.622896i \(-0.214044\pi\)
0.782305 + 0.622896i \(0.214044\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 76.3675i − 0.120264i
\(636\) 0 0
\(637\) −613.146 −0.962553
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 97.5807i − 0.152232i −0.997099 0.0761160i \(-0.975748\pi\)
0.997099 0.0761160i \(-0.0242519\pi\)
\(642\) 0 0
\(643\) −80.0000 −0.124417 −0.0622084 0.998063i \(-0.519814\pi\)
−0.0622084 + 0.998063i \(0.519814\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 514.393i − 0.795043i −0.917593 0.397522i \(-0.869870\pi\)
0.917593 0.397522i \(-0.130130\pi\)
\(648\) 0 0
\(649\) −288.000 −0.443760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1080.22i 1.65425i 0.562018 + 0.827125i \(0.310025\pi\)
−0.562018 + 0.827125i \(0.689975\pi\)
\(654\) 0 0
\(655\) −374.123 −0.571180
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 712.764i 1.08158i 0.841156 + 0.540792i \(0.181875\pi\)
−0.841156 + 0.540792i \(0.818125\pi\)
\(660\) 0 0
\(661\) −1247.08 −1.88665 −0.943326 0.331868i \(-0.892321\pi\)
−0.943326 + 0.331868i \(0.892321\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1527.35i 2.29677i
\(666\) 0 0
\(667\) −540.000 −0.809595
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −50.0000 −0.0742942 −0.0371471 0.999310i \(-0.511827\pi\)
−0.0371471 + 0.999310i \(0.511827\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 169.015i 0.249653i 0.992179 + 0.124826i \(0.0398374\pi\)
−0.992179 + 0.124826i \(0.960163\pi\)
\(678\) 0 0
\(679\) 415.692 0.612212
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 42.4264i 0.0621177i 0.999518 + 0.0310589i \(0.00988793\pi\)
−0.999518 + 0.0310589i \(0.990112\pi\)
\(684\) 0 0
\(685\) 779.423 1.13784
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 381.838i 0.554191i
\(690\) 0 0
\(691\) 400.000 0.578871 0.289436 0.957197i \(-0.406532\pi\)
0.289436 + 0.957197i \(0.406532\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1263.94i 1.81861i
\(696\) 0 0
\(697\) 1530.00 2.19512
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 992.043i 1.41518i 0.706622 + 0.707592i \(0.250218\pi\)
−0.706622 + 0.707592i \(0.749782\pi\)
\(702\) 0 0
\(703\) 831.384 1.18262
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1909.19i 2.70041i
\(708\) 0 0
\(709\) 1195.12 1.68563 0.842817 0.538200i \(-0.180895\pi\)
0.842817 + 0.538200i \(0.180895\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 763.675i − 1.07107i
\(714\) 0 0
\(715\) −648.000 −0.906294
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 661.362i − 0.919836i −0.887961 0.459918i \(-0.847879\pi\)
0.887961 0.459918i \(-0.152121\pi\)
\(720\) 0 0
\(721\) −972.000 −1.34813
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1065.53i 1.46969i
\(726\) 0 0
\(727\) −93.5307 −0.128653 −0.0643265 0.997929i \(-0.520490\pi\)
−0.0643265 + 0.997929i \(0.520490\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 848.528i 1.16078i
\(732\) 0 0
\(733\) 613.146 0.836488 0.418244 0.908335i \(-0.362646\pi\)
0.418244 + 0.908335i \(0.362646\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 848.528i − 1.15133i
\(738\) 0 0
\(739\) 920.000 1.24493 0.622463 0.782649i \(-0.286132\pi\)
0.622463 + 0.782649i \(0.286132\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1175.76i − 1.58244i −0.611530 0.791221i \(-0.709446\pi\)
0.611530 0.791221i \(-0.290554\pi\)
\(744\) 0 0
\(745\) −810.000 −1.08725
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1763.63i − 2.35465i
\(750\) 0 0
\(751\) 51.9615 0.0691898 0.0345949 0.999401i \(-0.488986\pi\)
0.0345949 + 0.999401i \(0.488986\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1145.51i 1.51724i
\(756\) 0 0
\(757\) 405.300 0.535403 0.267701 0.963502i \(-0.413736\pi\)
0.267701 + 0.963502i \(0.413736\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1030.96i 1.35475i 0.735640 + 0.677373i \(0.236882\pi\)
−0.735640 + 0.677373i \(0.763118\pi\)
\(762\) 0 0
\(763\) 540.000 0.707733
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 352.727i − 0.459878i
\(768\) 0 0
\(769\) 890.000 1.15735 0.578674 0.815559i \(-0.303571\pi\)
0.578674 + 0.815559i \(0.303571\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 933.256i 1.20732i 0.797243 + 0.603658i \(0.206291\pi\)
−0.797243 + 0.603658i \(0.793709\pi\)
\(774\) 0 0
\(775\) −1506.88 −1.94437
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1442.50i 1.85173i
\(780\) 0 0
\(781\) 623.538 0.798384
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1221.88i − 1.55654i
\(786\) 0 0
\(787\) 100.000 0.127065 0.0635324 0.997980i \(-0.479763\pi\)
0.0635324 + 0.997980i \(0.479763\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 661.362i − 0.836109i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 404.166i 0.507109i 0.967321 + 0.253554i \(0.0815997\pi\)
−0.967321 + 0.253554i \(0.918400\pi\)
\(798\) 0 0
\(799\) 1558.85 1.95100
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 169.706i − 0.211340i
\(804\) 0 0
\(805\) −1122.37 −1.39425
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1090.36i − 1.34779i −0.738829 0.673893i \(-0.764621\pi\)
0.738829 0.673893i \(-0.235379\pi\)
\(810\) 0 0
\(811\) 772.000 0.951911 0.475956 0.879469i \(-0.342102\pi\)
0.475956 + 0.879469i \(0.342102\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1175.76i − 1.44264i
\(816\) 0 0
\(817\) −800.000 −0.979192
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 477.650i 0.581791i 0.956755 + 0.290896i \(0.0939532\pi\)
−0.956755 + 0.290896i \(0.906047\pi\)
\(822\) 0 0
\(823\) 1132.76 1.37638 0.688190 0.725530i \(-0.258405\pi\)
0.688190 + 0.725530i \(0.258405\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 763.675i − 0.923428i −0.887029 0.461714i \(-0.847235\pi\)
0.887029 0.461714i \(-0.152765\pi\)
\(828\) 0 0
\(829\) −363.731 −0.438758 −0.219379 0.975640i \(-0.570403\pi\)
−0.219379 + 0.975640i \(0.570403\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1251.58i 1.50250i
\(834\) 0 0
\(835\) 432.000 0.517365
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 808.332i − 0.963447i −0.876323 0.481723i \(-0.840011\pi\)
0.876323 0.481723i \(-0.159989\pi\)
\(840\) 0 0
\(841\) −509.000 −0.605232
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 448.257i 0.530481i
\(846\) 0 0
\(847\) 509.223 0.601208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 610.940i 0.717909i
\(852\) 0 0
\(853\) 997.661 1.16959 0.584796 0.811181i \(-0.301175\pi\)
0.584796 + 0.811181i \(0.301175\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 572.756i 0.668327i 0.942515 + 0.334164i \(0.108454\pi\)
−0.942515 + 0.334164i \(0.891546\pi\)
\(858\) 0 0
\(859\) 568.000 0.661234 0.330617 0.943765i \(-0.392743\pi\)
0.330617 + 0.943765i \(0.392743\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 823.029i 0.953683i 0.878989 + 0.476842i \(0.158218\pi\)
−0.878989 + 0.476842i \(0.841782\pi\)
\(864\) 0 0
\(865\) −270.000 −0.312139
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 440.908i − 0.507374i
\(870\) 0 0
\(871\) 1039.23 1.19315
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 305.470i 0.349109i
\(876\) 0 0
\(877\) 41.5692 0.0473993 0.0236997 0.999719i \(-0.492455\pi\)
0.0236997 + 0.999719i \(0.492455\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 12.7279i 0.0144471i 0.999974 + 0.00722357i \(0.00229935\pi\)
−0.999974 + 0.00722357i \(0.997701\pi\)
\(882\) 0 0
\(883\) 520.000 0.588901 0.294451 0.955667i \(-0.404863\pi\)
0.294451 + 0.955667i \(0.404863\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 88.1816i − 0.0994156i −0.998764 0.0497078i \(-0.984171\pi\)
0.998764 0.0497078i \(-0.0158290\pi\)
\(888\) 0 0
\(889\) 108.000 0.121485
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1469.69i 1.64579i
\(894\) 0 0
\(895\) −374.123 −0.418014
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1909.19i − 2.12368i
\(900\) 0 0
\(901\) 779.423 0.865064
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1909.19i − 2.10960i
\(906\) 0 0
\(907\) 40.0000 0.0441014 0.0220507 0.999757i \(-0.492980\pi\)
0.0220507 + 0.999757i \(0.492980\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1175.76i − 1.29062i −0.763921 0.645310i \(-0.776728\pi\)
0.763921 0.645310i \(-0.223272\pi\)
\(912\) 0 0
\(913\) 1080.00 1.18291
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 529.090i − 0.576979i
\(918\) 0 0
\(919\) 1714.73 1.86587 0.932933 0.360051i \(-0.117241\pi\)
0.932933 + 0.360051i \(0.117241\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 763.675i 0.827384i
\(924\) 0 0
\(925\) 1205.51 1.30325
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1412.80i 1.52077i 0.649470 + 0.760387i \(0.274991\pi\)
−0.649470 + 0.760387i \(0.725009\pi\)
\(930\) 0 0
\(931\) −1180.00 −1.26745
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1322.72i 1.41468i
\(936\) 0 0
\(937\) 470.000 0.501601 0.250800 0.968039i \(-0.419306\pi\)
0.250800 + 0.968039i \(0.419306\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 624.620i 0.663783i 0.943318 + 0.331892i \(0.107687\pi\)
−0.943318 + 0.331892i \(0.892313\pi\)
\(942\) 0 0
\(943\) −1060.02 −1.12409
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 339.411i 0.358407i 0.983812 + 0.179203i \(0.0573520\pi\)
−0.983812 + 0.179203i \(0.942648\pi\)
\(948\) 0 0
\(949\) 207.846 0.219016
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 530.330i 0.556485i 0.960511 + 0.278242i \(0.0897519\pi\)
−0.960511 + 0.278242i \(0.910248\pi\)
\(954\) 0 0
\(955\) 2160.00 2.26178
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1102.27i 1.14940i
\(960\) 0 0
\(961\) 1739.00 1.80957
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 73.4847i 0.0761499i
\(966\) 0 0
\(967\) 1028.84 1.06395 0.531974 0.846761i \(-0.321450\pi\)
0.531974 + 0.846761i \(0.321450\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1603.72i 1.65162i 0.563952 + 0.825808i \(0.309280\pi\)
−0.563952 + 0.825808i \(0.690720\pi\)
\(972\) 0 0
\(973\) −1787.48 −1.83708
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1378.86i − 1.41132i −0.708551 0.705659i \(-0.750651\pi\)
0.708551 0.705659i \(-0.249349\pi\)
\(978\) 0 0
\(979\) 108.000 0.110317
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 440.908i − 0.448533i −0.974528 0.224267i \(-0.928001\pi\)
0.974528 0.224267i \(-0.0719986\pi\)
\(984\) 0 0
\(985\) −162.000 −0.164467
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 587.878i − 0.594416i
\(990\) 0 0
\(991\) 467.654 0.471901 0.235950 0.971765i \(-0.424180\pi\)
0.235950 + 0.971765i \(0.424180\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 381.838i 0.383756i
\(996\) 0 0
\(997\) −249.415 −0.250166 −0.125083 0.992146i \(-0.539920\pi\)
−0.125083 + 0.992146i \(0.539920\pi\)
\(998\) 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.e.g.1025.2 4
3.2 odd 2 inner 2304.3.e.g.1025.4 4
4.3 odd 2 2304.3.e.j.1025.1 4
8.3 odd 2 inner 2304.3.e.g.1025.3 4
8.5 even 2 2304.3.e.j.1025.4 4
12.11 even 2 2304.3.e.j.1025.3 4
16.3 odd 4 576.3.h.b.161.3 yes 8
16.5 even 4 576.3.h.b.161.6 yes 8
16.11 odd 4 576.3.h.b.161.8 yes 8
16.13 even 4 576.3.h.b.161.1 8
24.5 odd 2 2304.3.e.j.1025.2 4
24.11 even 2 inner 2304.3.e.g.1025.1 4
48.5 odd 4 576.3.h.b.161.2 yes 8
48.11 even 4 576.3.h.b.161.4 yes 8
48.29 odd 4 576.3.h.b.161.5 yes 8
48.35 even 4 576.3.h.b.161.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.3.h.b.161.1 8 16.13 even 4
576.3.h.b.161.2 yes 8 48.5 odd 4
576.3.h.b.161.3 yes 8 16.3 odd 4
576.3.h.b.161.4 yes 8 48.11 even 4
576.3.h.b.161.5 yes 8 48.29 odd 4
576.3.h.b.161.6 yes 8 16.5 even 4
576.3.h.b.161.7 yes 8 48.35 even 4
576.3.h.b.161.8 yes 8 16.11 odd 4
2304.3.e.g.1025.1 4 24.11 even 2 inner
2304.3.e.g.1025.2 4 1.1 even 1 trivial
2304.3.e.g.1025.3 4 8.3 odd 2 inner
2304.3.e.g.1025.4 4 3.2 odd 2 inner
2304.3.e.j.1025.1 4 4.3 odd 2
2304.3.e.j.1025.2 4 24.5 odd 2
2304.3.e.j.1025.3 4 12.11 even 2
2304.3.e.j.1025.4 4 8.5 even 2