Properties

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

$q$-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 \(0.262746\pi\)
−0.678233 + 0.734847i \(0.737254\pi\)
\(6\) 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 \(-0.873962\pi\)
0.922627 0.385695i \(-0.126038\pi\)
\(12\) 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 \(-0.785540\pi\)
0.781490 0.623918i \(-0.214460\pi\)
\(18\) 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 \(0.103514\pi\)
−0.947587 + 0.319499i \(0.896486\pi\)
\(24\) 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 \(0.218378\pi\)
−0.773752 + 0.633489i \(0.781622\pi\)
\(30\) 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 \(0.342171\pi\)
−0.475766 + 0.879572i \(0.657829\pi\)
\(42\) 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 \(0.285674\pi\)
−0.623589 + 0.781752i \(0.714326\pi\)
\(48\) 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 \(0.112673\pi\)
−0.938003 + 0.346626i \(0.887327\pi\)
\(54\) 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 \(-0.907130\pi\)
0.957740 0.287637i \(-0.0928695\pi\)
\(60\) 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 \(0.173136\pi\)
−0.855685 + 0.517498i \(0.826864\pi\)
\(72\) 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 \(0.278124\pi\)
−0.641955 + 0.766742i \(0.721876\pi\)
\(84\) 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 \(0.0227802\pi\)
−0.997440 + 0.0715052i \(0.977220\pi\)
\(90\) 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 \(-0.636492\pi\)
0.415783 0.909464i \(-0.363508\pi\)
\(102\) 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 \(0.291491\pi\)
−0.609200 + 0.793017i \(0.708509\pi\)
\(108\) 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 \(0.0908622\pi\)
−0.959534 + 0.281591i \(0.909138\pi\)
\(114\) 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 \(0.0622498\pi\)
−0.980938 + 0.194319i \(0.937750\pi\)
\(132\) 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 \(-0.873476\pi\)
0.922037 0.387102i \(-0.126524\pi\)
\(138\) 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 \(0.120604\pi\)
−0.929076 + 0.369889i \(0.879396\pi\)
\(150\) 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 \(-0.943680\pi\)
0.984388 0.176011i \(-0.0563195\pi\)
\(168\) 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 \(0.0338657\pi\)
−0.994346 + 0.106192i \(0.966134\pi\)
\(174\) 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 \(0.0454213\pi\)
−0.989836 + 0.142211i \(0.954579\pi\)
\(180\) 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 \(-0.720519\pi\)
0.638679 0.769473i \(-0.279481\pi\)
\(192\) 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 \(0.0178196\pi\)
−0.998433 + 0.0559528i \(0.982180\pi\)
\(198\) 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 \(0.0297896\pi\)
−0.995624 + 0.0934502i \(0.970210\pi\)
\(228\) 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 \(0.0730910\pi\)
−0.973752 + 0.227609i \(0.926909\pi\)
\(234\) 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 \(0.0994815\pi\)
−0.951559 + 0.307467i \(0.900519\pi\)
\(240\) 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 \(0.163071\pi\)
−0.871618 + 0.490186i \(0.836929\pi\)
\(252\) 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 \(0.0131406\pi\)
−0.999148 + 0.0412708i \(0.986859\pi\)
\(258\) 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 \(-0.946384\pi\)
0.985847 0.167646i \(-0.0536165\pi\)
\(264\) 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 \(0.0656814\pi\)
−0.978786 + 0.204883i \(0.934319\pi\)
\(270\) 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 \(-0.919858\pi\)
0.968472 0.249123i \(-0.0801424\pi\)
\(282\) 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 \(-0.818718\pi\)
0.842164 0.539222i \(-0.181282\pi\)
\(294\) 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 \(-0.924070\pi\)
0.971684 0.236285i \(-0.0759300\pi\)
\(312\) 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 \(-0.875066\pi\)
0.923959 0.382491i \(-0.124934\pi\)
\(318\) 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 \(0.0194714\pi\)
−0.998130 + 0.0611332i \(0.980529\pi\)
\(348\) 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 \(-0.663236\pi\)
0.490637 0.871364i \(-0.336764\pi\)
\(354\) 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 \(0.254223\pi\)
−0.697665 + 0.716425i \(0.745777\pi\)
\(360\) 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 \(0.0614539\pi\)
−0.981421 + 0.191866i \(0.938546\pi\)
\(384\) 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 \(0.107247\pi\)
−0.943776 + 0.330587i \(0.892753\pi\)
\(390\) 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 \(-0.909542\pi\)
0.959891 0.280374i \(-0.0904585\pi\)
\(402\) 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 \(0.172731\pi\)
−0.856343 + 0.516407i \(0.827269\pi\)
\(420\) 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 \(0.140164\pi\)
−0.904608 + 0.426245i \(0.859836\pi\)
\(432\) 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 \(-0.786113\pi\)
0.782612 0.622509i \(-0.213887\pi\)
\(444\) 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 \(-0.720206\pi\)
0.637922 0.770101i \(-0.279794\pi\)
\(450\) 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 \(-0.883235\pi\)
0.933470 0.358656i \(-0.116765\pi\)
\(462\) 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 \(-0.833455\pi\)
0.866217 0.499669i \(-0.166545\pi\)
\(468\) 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 \(0.0739116\pi\)
−0.973162 + 0.230119i \(0.926088\pi\)
\(480\) 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 \(-0.927872\pi\)
0.974437 0.224661i \(-0.0721275\pi\)
\(492\) 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 \(-0.976728\pi\)
0.997329 0.0730464i \(-0.0232721\pi\)
\(504\) 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 \(0.210269\pi\)
−0.789636 + 0.613576i \(0.789731\pi\)
\(510\) 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 \(0.309695\pi\)
−0.562876 + 0.826541i \(0.690305\pi\)
\(522\) 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 \(0.349662\pi\)
−0.454937 + 0.890524i \(0.650338\pi\)
\(558\) 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 \(-0.939672\pi\)
0.982093 0.188394i \(-0.0603283\pi\)
\(564\) 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 \(-0.861580\pi\)
0.906930 0.421282i \(-0.138420\pi\)
\(570\) 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 \(0.196250\pi\)
−0.815886 + 0.578213i \(0.803750\pi\)
\(588\) 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 \(0.122568\pi\)
−0.926777 + 0.375613i \(0.877432\pi\)
\(594\) 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 \(0.0992240\pi\)
−0.951807 + 0.306697i \(0.900776\pi\)
\(600\) 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 \(-0.780566\pi\)
0.771646 0.636052i \(-0.219434\pi\)
\(618\) 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 \(0.0242519\pi\)
−0.997099 + 0.0761160i \(0.975748\pi\)
\(642\) 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 \(0.130130\pi\)
−0.917593 + 0.397522i \(0.869870\pi\)
\(648\) 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 \(-0.689975\pi\)
0.562018 0.827125i \(-0.310025\pi\)
\(654\) 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 \(-0.818125\pi\)
0.841156 0.540792i \(-0.181875\pi\)
\(660\) 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 \(-0.960163\pi\)
0.992179 0.124826i \(-0.0398374\pi\)
\(678\) 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 \(-0.990112\pi\)
0.999518 0.0310589i \(-0.00988793\pi\)
\(684\) 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 \(-0.749782\pi\)
0.706622 0.707592i \(-0.250218\pi\)
\(702\) 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 \(0.152121\pi\)
−0.887961 + 0.459918i \(0.847879\pi\)
\(720\) 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 \(0.290554\pi\)
−0.611530 + 0.791221i \(0.709446\pi\)
\(744\) 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 \(-0.763118\pi\)
0.735640 0.677373i \(-0.236882\pi\)
\(762\) 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 \(-0.793709\pi\)
0.797243 0.603658i \(-0.206291\pi\)
\(774\) 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 \(-0.918400\pi\)
0.967321 0.253554i \(-0.0815997\pi\)
\(798\) 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 \(0.235379\pi\)
−0.738829 + 0.673893i \(0.764621\pi\)
\(810\) 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 \(-0.906047\pi\)
0.956755 0.290896i \(-0.0939532\pi\)
\(822\) 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 \(0.152765\pi\)
−0.887029 + 0.461714i \(0.847235\pi\)
\(828\) 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 \(0.159989\pi\)
−0.876323 + 0.481723i \(0.840011\pi\)
\(840\) 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 \(-0.891546\pi\)
0.942515 0.334164i \(-0.108454\pi\)
\(858\) 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 \(-0.841782\pi\)
0.878989 0.476842i \(-0.158218\pi\)
\(864\) 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 \(-0.997701\pi\)
0.999974 0.00722357i \(-0.00229935\pi\)
\(882\) 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 \(0.0158290\pi\)
−0.998764 + 0.0497078i \(0.984171\pi\)
\(888\) 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 \(0.223272\pi\)
−0.763921 + 0.645310i \(0.776728\pi\)
\(912\) 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 \(-0.725009\pi\)
0.649470 0.760387i \(-0.274991\pi\)
\(930\) 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 \(-0.892313\pi\)
0.943318 0.331892i \(-0.107687\pi\)
\(942\) 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 \(-0.942648\pi\)
0.983812 0.179203i \(-0.0573520\pi\)
\(948\) 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 \(-0.910248\pi\)
0.960511 0.278242i \(-0.0897519\pi\)
\(954\) 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 \(-0.690720\pi\)
0.563952 0.825808i \(-0.309280\pi\)
\(972\) 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 \(0.249349\pi\)
−0.708551 + 0.705659i \(0.750651\pi\)
\(978\) 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 \(0.0719986\pi\)
−0.974528 + 0.224267i \(0.928001\pi\)
\(984\) 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.1025.4 4
3.2 odd 2 inner 2304.3.e.g.1025.2 4
4.3 odd 2 2304.3.e.j.1025.3 4
8.3 odd 2 inner 2304.3.e.g.1025.1 4
8.5 even 2 2304.3.e.j.1025.2 4
12.11 even 2 2304.3.e.j.1025.1 4
16.3 odd 4 576.3.h.b.161.7 yes 8
16.5 even 4 576.3.h.b.161.2 yes 8
16.11 odd 4 576.3.h.b.161.4 yes 8
16.13 even 4 576.3.h.b.161.5 yes 8
24.5 odd 2 2304.3.e.j.1025.4 4
24.11 even 2 inner 2304.3.e.g.1025.3 4
48.5 odd 4 576.3.h.b.161.6 yes 8
48.11 even 4 576.3.h.b.161.8 yes 8
48.29 odd 4 576.3.h.b.161.1 8
48.35 even 4 576.3.h.b.161.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.3.h.b.161.1 8 48.29 odd 4
576.3.h.b.161.2 yes 8 16.5 even 4
576.3.h.b.161.3 yes 8 48.35 even 4
576.3.h.b.161.4 yes 8 16.11 odd 4
576.3.h.b.161.5 yes 8 16.13 even 4
576.3.h.b.161.6 yes 8 48.5 odd 4
576.3.h.b.161.7 yes 8 16.3 odd 4
576.3.h.b.161.8 yes 8 48.11 even 4
2304.3.e.g.1025.1 4 8.3 odd 2 inner
2304.3.e.g.1025.2 4 3.2 odd 2 inner
2304.3.e.g.1025.3 4 24.11 even 2 inner
2304.3.e.g.1025.4 4 1.1 even 1 trivial
2304.3.e.j.1025.1 4 12.11 even 2
2304.3.e.j.1025.2 4 8.5 even 2
2304.3.e.j.1025.3 4 4.3 odd 2
2304.3.e.j.1025.4 4 24.5 odd 2