Properties

Label 576.3.e.b.449.1
Level $576$
Weight $3$
Character 576.449
Analytic conductor $15.695$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 576.449
Dual form 576.3.e.b.449.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{5} -8.00000 q^{7} +O(q^{10})\) \(q-1.41421i q^{5} -8.00000 q^{7} +11.3137i q^{11} +8.00000 q^{13} -12.7279i q^{17} +32.0000 q^{19} -33.9411i q^{23} +23.0000 q^{25} -43.8406i q^{29} -40.0000 q^{31} +11.3137i q^{35} +26.0000 q^{37} -66.4680i q^{41} +16.0000 q^{43} +11.3137i q^{47} +15.0000 q^{49} -32.5269i q^{53} +16.0000 q^{55} +22.6274i q^{59} +54.0000 q^{61} -11.3137i q^{65} -80.0000 q^{67} +79.1960i q^{71} +96.0000 q^{73} -90.5097i q^{77} +104.000 q^{79} -101.823i q^{83} -18.0000 q^{85} -77.7817i q^{89} -64.0000 q^{91} -45.2548i q^{95} -80.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{7} + O(q^{10}) \) \( 2q - 16q^{7} + 16q^{13} + 64q^{19} + 46q^{25} - 80q^{31} + 52q^{37} + 32q^{43} + 30q^{49} + 32q^{55} + 108q^{61} - 160q^{67} + 192q^{73} + 208q^{79} - 36q^{85} - 128q^{91} - 160q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 1.41421i − 0.282843i −0.989949 0.141421i \(-0.954833\pi\)
0.989949 0.141421i \(-0.0451672\pi\)
\(6\) 0 0
\(7\) −8.00000 −1.14286 −0.571429 0.820652i \(-0.693611\pi\)
−0.571429 + 0.820652i \(0.693611\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 11.3137i 1.02852i 0.857635 + 0.514259i \(0.171933\pi\)
−0.857635 + 0.514259i \(0.828067\pi\)
\(12\) 0 0
\(13\) 8.00000 0.615385 0.307692 0.951486i \(-0.400443\pi\)
0.307692 + 0.951486i \(0.400443\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 12.7279i − 0.748701i −0.927287 0.374351i \(-0.877866\pi\)
0.927287 0.374351i \(-0.122134\pi\)
\(18\) 0 0
\(19\) 32.0000 1.68421 0.842105 0.539313i \(-0.181316\pi\)
0.842105 + 0.539313i \(0.181316\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 33.9411i − 1.47570i −0.674964 0.737851i \(-0.735841\pi\)
0.674964 0.737851i \(-0.264159\pi\)
\(24\) 0 0
\(25\) 23.0000 0.920000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 43.8406i − 1.51175i −0.654719 0.755873i \(-0.727213\pi\)
0.654719 0.755873i \(-0.272787\pi\)
\(30\) 0 0
\(31\) −40.0000 −1.29032 −0.645161 0.764046i \(-0.723210\pi\)
−0.645161 + 0.764046i \(0.723210\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.3137i 0.323249i
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 66.4680i − 1.62117i −0.585620 0.810586i \(-0.699149\pi\)
0.585620 0.810586i \(-0.300851\pi\)
\(42\) 0 0
\(43\) 16.0000 0.372093 0.186047 0.982541i \(-0.440432\pi\)
0.186047 + 0.982541i \(0.440432\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.3137i 0.240717i 0.992730 + 0.120359i \(0.0384044\pi\)
−0.992730 + 0.120359i \(0.961596\pi\)
\(48\) 0 0
\(49\) 15.0000 0.306122
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 32.5269i − 0.613715i −0.951755 0.306858i \(-0.900722\pi\)
0.951755 0.306858i \(-0.0992776\pi\)
\(54\) 0 0
\(55\) 16.0000 0.290909
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 22.6274i 0.383516i 0.981442 + 0.191758i \(0.0614188\pi\)
−0.981442 + 0.191758i \(0.938581\pi\)
\(60\) 0 0
\(61\) 54.0000 0.885246 0.442623 0.896708i \(-0.354048\pi\)
0.442623 + 0.896708i \(0.354048\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 11.3137i − 0.174057i
\(66\) 0 0
\(67\) −80.0000 −1.19403 −0.597015 0.802230i \(-0.703647\pi\)
−0.597015 + 0.802230i \(0.703647\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 79.1960i 1.11544i 0.830030 + 0.557718i \(0.188323\pi\)
−0.830030 + 0.557718i \(0.811677\pi\)
\(72\) 0 0
\(73\) 96.0000 1.31507 0.657534 0.753425i \(-0.271599\pi\)
0.657534 + 0.753425i \(0.271599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 90.5097i − 1.17545i
\(78\) 0 0
\(79\) 104.000 1.31646 0.658228 0.752819i \(-0.271306\pi\)
0.658228 + 0.752819i \(0.271306\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 101.823i − 1.22679i −0.789777 0.613394i \(-0.789804\pi\)
0.789777 0.613394i \(-0.210196\pi\)
\(84\) 0 0
\(85\) −18.0000 −0.211765
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 77.7817i − 0.873952i −0.899473 0.436976i \(-0.856049\pi\)
0.899473 0.436976i \(-0.143951\pi\)
\(90\) 0 0
\(91\) −64.0000 −0.703297
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 45.2548i − 0.476367i
\(96\) 0 0
\(97\) −80.0000 −0.824742 −0.412371 0.911016i \(-0.635299\pi\)
−0.412371 + 0.911016i \(0.635299\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 123.037i 1.21818i 0.793100 + 0.609092i \(0.208466\pi\)
−0.793100 + 0.609092i \(0.791534\pi\)
\(102\) 0 0
\(103\) 72.0000 0.699029 0.349515 0.936931i \(-0.386347\pi\)
0.349515 + 0.936931i \(0.386347\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 181.019i − 1.69177i −0.533366 0.845885i \(-0.679073\pi\)
0.533366 0.845885i \(-0.320927\pi\)
\(108\) 0 0
\(109\) 88.0000 0.807339 0.403670 0.914905i \(-0.367734\pi\)
0.403670 + 0.914905i \(0.367734\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 137.179i 1.21397i 0.794713 + 0.606985i \(0.207621\pi\)
−0.794713 + 0.606985i \(0.792379\pi\)
\(114\) 0 0
\(115\) −48.0000 −0.417391
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 101.823i 0.855659i
\(120\) 0 0
\(121\) −7.00000 −0.0578512
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 67.8823i − 0.543058i
\(126\) 0 0
\(127\) 56.0000 0.440945 0.220472 0.975393i \(-0.429240\pi\)
0.220472 + 0.975393i \(0.429240\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 248.902i 1.90001i 0.312231 + 0.950006i \(0.398924\pi\)
−0.312231 + 0.950006i \(0.601076\pi\)
\(132\) 0 0
\(133\) −256.000 −1.92481
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 46.6690i − 0.340650i −0.985388 0.170325i \(-0.945518\pi\)
0.985388 0.170325i \(-0.0544818\pi\)
\(138\) 0 0
\(139\) 16.0000 0.115108 0.0575540 0.998342i \(-0.481670\pi\)
0.0575540 + 0.998342i \(0.481670\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 90.5097i 0.632935i
\(144\) 0 0
\(145\) −62.0000 −0.427586
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 182.434i 1.22439i 0.790708 + 0.612193i \(0.209712\pi\)
−0.790708 + 0.612193i \(0.790288\pi\)
\(150\) 0 0
\(151\) −168.000 −1.11258 −0.556291 0.830987i \(-0.687776\pi\)
−0.556291 + 0.830987i \(0.687776\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 56.5685i 0.364958i
\(156\) 0 0
\(157\) 10.0000 0.0636943 0.0318471 0.999493i \(-0.489861\pi\)
0.0318471 + 0.999493i \(0.489861\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 271.529i 1.68652i
\(162\) 0 0
\(163\) 80.0000 0.490798 0.245399 0.969422i \(-0.421081\pi\)
0.245399 + 0.969422i \(0.421081\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 294.156i − 1.76142i −0.473660 0.880708i \(-0.657067\pi\)
0.473660 0.880708i \(-0.342933\pi\)
\(168\) 0 0
\(169\) −105.000 −0.621302
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 55.1543i 0.318811i 0.987213 + 0.159406i \(0.0509578\pi\)
−0.987213 + 0.159406i \(0.949042\pi\)
\(174\) 0 0
\(175\) −184.000 −1.05143
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 135.765i − 0.758461i −0.925302 0.379230i \(-0.876189\pi\)
0.925302 0.379230i \(-0.123811\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.0441989 −0.0220994 0.999756i \(-0.507035\pi\)
−0.0220994 + 0.999756i \(0.507035\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 36.7696i − 0.198754i
\(186\) 0 0
\(187\) 144.000 0.770053
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 67.8823i − 0.355404i −0.984084 0.177702i \(-0.943134\pi\)
0.984084 0.177702i \(-0.0568664\pi\)
\(192\) 0 0
\(193\) −258.000 −1.33679 −0.668394 0.743808i \(-0.733018\pi\)
−0.668394 + 0.743808i \(0.733018\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 371.938i 1.88801i 0.329930 + 0.944005i \(0.392975\pi\)
−0.329930 + 0.944005i \(0.607025\pi\)
\(198\) 0 0
\(199\) −88.0000 −0.442211 −0.221106 0.975250i \(-0.570967\pi\)
−0.221106 + 0.975250i \(0.570967\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 350.725i 1.72771i
\(204\) 0 0
\(205\) −94.0000 −0.458537
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 362.039i 1.73224i
\(210\) 0 0
\(211\) 368.000 1.74408 0.872038 0.489438i \(-0.162798\pi\)
0.872038 + 0.489438i \(0.162798\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 22.6274i − 0.105244i
\(216\) 0 0
\(217\) 320.000 1.47465
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 101.823i − 0.460739i
\(222\) 0 0
\(223\) −104.000 −0.466368 −0.233184 0.972433i \(-0.574914\pi\)
−0.233184 + 0.972433i \(0.574914\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 169.706i − 0.747602i −0.927509 0.373801i \(-0.878054\pi\)
0.927509 0.373801i \(-0.121946\pi\)
\(228\) 0 0
\(229\) −344.000 −1.50218 −0.751092 0.660198i \(-0.770472\pi\)
−0.751092 + 0.660198i \(0.770472\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 80.6102i − 0.345966i −0.984925 0.172983i \(-0.944659\pi\)
0.984925 0.172983i \(-0.0553406\pi\)
\(234\) 0 0
\(235\) 16.0000 0.0680851
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 271.529i − 1.13610i −0.822992 0.568052i \(-0.807697\pi\)
0.822992 0.568052i \(-0.192303\pi\)
\(240\) 0 0
\(241\) −272.000 −1.12863 −0.564315 0.825559i \(-0.690860\pi\)
−0.564315 + 0.825559i \(0.690860\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 21.2132i − 0.0865845i
\(246\) 0 0
\(247\) 256.000 1.03644
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 33.9411i 0.135224i 0.997712 + 0.0676118i \(0.0215379\pi\)
−0.997712 + 0.0676118i \(0.978462\pi\)
\(252\) 0 0
\(253\) 384.000 1.51779
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 43.8406i 0.170586i 0.996356 + 0.0852930i \(0.0271826\pi\)
−0.996356 + 0.0852930i \(0.972817\pi\)
\(258\) 0 0
\(259\) −208.000 −0.803089
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 203.647i 0.774322i 0.922012 + 0.387161i \(0.126544\pi\)
−0.922012 + 0.387161i \(0.873456\pi\)
\(264\) 0 0
\(265\) −46.0000 −0.173585
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 46.6690i − 0.173491i −0.996231 0.0867454i \(-0.972353\pi\)
0.996231 0.0867454i \(-0.0276467\pi\)
\(270\) 0 0
\(271\) −264.000 −0.974170 −0.487085 0.873355i \(-0.661940\pi\)
−0.487085 + 0.873355i \(0.661940\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 260.215i 0.946237i
\(276\) 0 0
\(277\) −40.0000 −0.144404 −0.0722022 0.997390i \(-0.523003\pi\)
−0.0722022 + 0.997390i \(0.523003\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 190.919i − 0.679426i −0.940529 0.339713i \(-0.889670\pi\)
0.940529 0.339713i \(-0.110330\pi\)
\(282\) 0 0
\(283\) −224.000 −0.791519 −0.395760 0.918354i \(-0.629519\pi\)
−0.395760 + 0.918354i \(0.629519\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 531.744i 1.85277i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 352.139i 1.20184i 0.799309 + 0.600920i \(0.205199\pi\)
−0.799309 + 0.600920i \(0.794801\pi\)
\(294\) 0 0
\(295\) 32.0000 0.108475
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 271.529i − 0.908124i
\(300\) 0 0
\(301\) −128.000 −0.425249
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 76.3675i − 0.250385i
\(306\) 0 0
\(307\) −432.000 −1.40717 −0.703583 0.710613i \(-0.748418\pi\)
−0.703583 + 0.710613i \(0.748418\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 203.647i − 0.654813i −0.944884 0.327406i \(-0.893825\pi\)
0.944884 0.327406i \(-0.106175\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.0447284 −0.0223642 0.999750i \(-0.507119\pi\)
−0.0223642 + 0.999750i \(0.507119\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 589.727i − 1.86034i −0.367132 0.930169i \(-0.619660\pi\)
0.367132 0.930169i \(-0.380340\pi\)
\(318\) 0 0
\(319\) 496.000 1.55486
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 407.294i − 1.26097i
\(324\) 0 0
\(325\) 184.000 0.566154
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 90.5097i − 0.275105i
\(330\) 0 0
\(331\) 16.0000 0.0483384 0.0241692 0.999708i \(-0.492306\pi\)
0.0241692 + 0.999708i \(0.492306\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 113.137i 0.337723i
\(336\) 0 0
\(337\) 128.000 0.379822 0.189911 0.981801i \(-0.439180\pi\)
0.189911 + 0.981801i \(0.439180\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 452.548i − 1.32712i
\(342\) 0 0
\(343\) 272.000 0.793003
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 350.725i − 1.01073i −0.862904 0.505367i \(-0.831357\pi\)
0.862904 0.505367i \(-0.168643\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.0286533 −0.0143266 0.999897i \(-0.504560\pi\)
−0.0143266 + 0.999897i \(0.504560\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 247.487i − 0.701097i −0.936545 0.350549i \(-0.885995\pi\)
0.936545 0.350549i \(-0.114005\pi\)
\(354\) 0 0
\(355\) 112.000 0.315493
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 328.098i 0.913921i 0.889487 + 0.456960i \(0.151062\pi\)
−0.889487 + 0.456960i \(0.848938\pi\)
\(360\) 0 0
\(361\) 663.000 1.83657
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 135.765i − 0.371958i
\(366\) 0 0
\(367\) 696.000 1.89646 0.948229 0.317588i \(-0.102873\pi\)
0.948229 + 0.317588i \(0.102873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 260.215i 0.701389i
\(372\) 0 0
\(373\) 454.000 1.21716 0.608579 0.793493i \(-0.291740\pi\)
0.608579 + 0.793493i \(0.291740\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 350.725i − 0.930305i
\(378\) 0 0
\(379\) −64.0000 −0.168865 −0.0844327 0.996429i \(-0.526908\pi\)
−0.0844327 + 0.996429i \(0.526908\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 362.039i − 0.945271i −0.881258 0.472635i \(-0.843303\pi\)
0.881258 0.472635i \(-0.156697\pi\)
\(384\) 0 0
\(385\) −128.000 −0.332468
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 386.080i 0.992494i 0.868181 + 0.496247i \(0.165289\pi\)
−0.868181 + 0.496247i \(0.834711\pi\)
\(390\) 0 0
\(391\) −432.000 −1.10486
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 147.078i − 0.372350i
\(396\) 0 0
\(397\) 662.000 1.66751 0.833753 0.552137i \(-0.186188\pi\)
0.833753 + 0.552137i \(0.186188\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 171.120i − 0.426733i −0.976972 0.213366i \(-0.931557\pi\)
0.976972 0.213366i \(-0.0684428\pi\)
\(402\) 0 0
\(403\) −320.000 −0.794045
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 294.156i 0.722743i
\(408\) 0 0
\(409\) −176.000 −0.430318 −0.215159 0.976579i \(-0.569027\pi\)
−0.215159 + 0.976579i \(0.569027\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 181.019i − 0.438303i
\(414\) 0 0
\(415\) −144.000 −0.346988
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 79.1960i 0.189012i 0.995524 + 0.0945059i \(0.0301271\pi\)
−0.995524 + 0.0945059i \(0.969873\pi\)
\(420\) 0 0
\(421\) −488.000 −1.15914 −0.579572 0.814921i \(-0.696780\pi\)
−0.579572 + 0.814921i \(0.696780\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 292.742i − 0.688805i
\(426\) 0 0
\(427\) −432.000 −1.01171
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 305.470i − 0.708747i −0.935104 0.354374i \(-0.884694\pi\)
0.935104 0.354374i \(-0.115306\pi\)
\(432\) 0 0
\(433\) 478.000 1.10393 0.551963 0.833869i \(-0.313879\pi\)
0.551963 + 0.833869i \(0.313879\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1086.12i − 2.48539i
\(438\) 0 0
\(439\) −392.000 −0.892938 −0.446469 0.894799i \(-0.647319\pi\)
−0.446469 + 0.894799i \(0.647319\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 33.9411i 0.0766165i 0.999266 + 0.0383083i \(0.0121969\pi\)
−0.999266 + 0.0383083i \(0.987803\pi\)
\(444\) 0 0
\(445\) −110.000 −0.247191
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 125.865i 0.280323i 0.990129 + 0.140161i \(0.0447622\pi\)
−0.990129 + 0.140161i \(0.955238\pi\)
\(450\) 0 0
\(451\) 752.000 1.66741
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 90.5097i 0.198922i
\(456\) 0 0
\(457\) −16.0000 −0.0350109 −0.0175055 0.999847i \(-0.505572\pi\)
−0.0175055 + 0.999847i \(0.505572\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 284.257i 0.616609i 0.951288 + 0.308305i \(0.0997616\pi\)
−0.951288 + 0.308305i \(0.900238\pi\)
\(462\) 0 0
\(463\) −568.000 −1.22678 −0.613391 0.789779i \(-0.710195\pi\)
−0.613391 + 0.789779i \(0.710195\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 418.607i 0.896375i 0.893940 + 0.448188i \(0.147930\pi\)
−0.893940 + 0.448188i \(0.852070\pi\)
\(468\) 0 0
\(469\) 640.000 1.36461
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 181.019i 0.382705i
\(474\) 0 0
\(475\) 736.000 1.54947
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 33.9411i − 0.0708583i −0.999372 0.0354291i \(-0.988720\pi\)
0.999372 0.0354291i \(-0.0112798\pi\)
\(480\) 0 0
\(481\) 208.000 0.432432
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 113.137i 0.233272i
\(486\) 0 0
\(487\) 424.000 0.870637 0.435318 0.900277i \(-0.356636\pi\)
0.435318 + 0.900277i \(0.356636\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 724.077i 1.47470i 0.675511 + 0.737350i \(0.263923\pi\)
−0.675511 + 0.737350i \(0.736077\pi\)
\(492\) 0 0
\(493\) −558.000 −1.13185
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 633.568i − 1.27478i
\(498\) 0 0
\(499\) −192.000 −0.384770 −0.192385 0.981320i \(-0.561622\pi\)
−0.192385 + 0.981320i \(0.561622\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 441.235i − 0.877206i −0.898681 0.438603i \(-0.855473\pi\)
0.898681 0.438603i \(-0.144527\pi\)
\(504\) 0 0
\(505\) 174.000 0.344554
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 250.316i 0.491780i 0.969298 + 0.245890i \(0.0790801\pi\)
−0.969298 + 0.245890i \(0.920920\pi\)
\(510\) 0 0
\(511\) −768.000 −1.50294
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 101.823i − 0.197715i
\(516\) 0 0
\(517\) −128.000 −0.247582
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 156.978i 0.301301i 0.988587 + 0.150650i \(0.0481368\pi\)
−0.988587 + 0.150650i \(0.951863\pi\)
\(522\) 0 0
\(523\) 576.000 1.10134 0.550669 0.834724i \(-0.314373\pi\)
0.550669 + 0.834724i \(0.314373\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 509.117i 0.966066i
\(528\) 0 0
\(529\) −623.000 −1.17769
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 531.744i − 0.997644i
\(534\) 0 0
\(535\) −256.000 −0.478505
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 169.706i 0.314853i
\(540\) 0 0
\(541\) 536.000 0.990758 0.495379 0.868677i \(-0.335029\pi\)
0.495379 + 0.868677i \(0.335029\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 124.451i − 0.228350i
\(546\) 0 0
\(547\) 144.000 0.263254 0.131627 0.991299i \(-0.457980\pi\)
0.131627 + 0.991299i \(0.457980\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 1402.90i − 2.54610i
\(552\) 0 0
\(553\) −832.000 −1.50452
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 533.159i 0.957197i 0.878034 + 0.478598i \(0.158855\pi\)
−0.878034 + 0.478598i \(0.841145\pi\)
\(558\) 0 0
\(559\) 128.000 0.228980
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 576.999i − 1.02487i −0.858727 0.512433i \(-0.828744\pi\)
0.858727 0.512433i \(-0.171256\pi\)
\(564\) 0 0
\(565\) 194.000 0.343363
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 994.192i 1.74726i 0.486589 + 0.873631i \(0.338241\pi\)
−0.486589 + 0.873631i \(0.661759\pi\)
\(570\) 0 0
\(571\) 416.000 0.728546 0.364273 0.931292i \(-0.381317\pi\)
0.364273 + 0.931292i \(0.381317\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 780.646i − 1.35765i
\(576\) 0 0
\(577\) 834.000 1.44541 0.722704 0.691158i \(-0.242899\pi\)
0.722704 + 0.691158i \(0.242899\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 814.587i 1.40204i
\(582\) 0 0
\(583\) 368.000 0.631218
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 67.8823i 0.115643i 0.998327 + 0.0578213i \(0.0184154\pi\)
−0.998327 + 0.0578213i \(0.981585\pi\)
\(588\) 0 0
\(589\) −1280.00 −2.17317
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 224.860i 0.379190i 0.981862 + 0.189595i \(0.0607176\pi\)
−0.981862 + 0.189595i \(0.939282\pi\)
\(594\) 0 0
\(595\) 144.000 0.242017
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 690.136i 1.15215i 0.817398 + 0.576074i \(0.195416\pi\)
−0.817398 + 0.576074i \(0.804584\pi\)
\(600\) 0 0
\(601\) −626.000 −1.04160 −0.520799 0.853680i \(-0.674366\pi\)
−0.520799 + 0.853680i \(0.674366\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.89949i 0.0163628i
\(606\) 0 0
\(607\) −232.000 −0.382208 −0.191104 0.981570i \(-0.561207\pi\)
−0.191104 + 0.981570i \(0.561207\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 90.5097i 0.148134i
\(612\) 0 0
\(613\) −666.000 −1.08646 −0.543230 0.839584i \(-0.682799\pi\)
−0.543230 + 0.839584i \(0.682799\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 284.257i 0.460708i 0.973107 + 0.230354i \(0.0739884\pi\)
−0.973107 + 0.230354i \(0.926012\pi\)
\(618\) 0 0
\(619\) −768.000 −1.24071 −0.620355 0.784321i \(-0.713012\pi\)
−0.620355 + 0.784321i \(0.713012\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 622.254i 0.998803i
\(624\) 0 0
\(625\) 479.000 0.766400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 330.926i − 0.526114i
\(630\) 0 0
\(631\) −472.000 −0.748019 −0.374010 0.927425i \(-0.622017\pi\)
−0.374010 + 0.927425i \(0.622017\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 79.1960i − 0.124718i
\(636\) 0 0
\(637\) 120.000 0.188383
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 371.938i 0.580247i 0.956989 + 0.290123i \(0.0936963\pi\)
−0.956989 + 0.290123i \(0.906304\pi\)
\(642\) 0 0
\(643\) −240.000 −0.373250 −0.186625 0.982431i \(-0.559755\pi\)
−0.186625 + 0.982431i \(0.559755\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 712.764i 1.10164i 0.834623 + 0.550822i \(0.185686\pi\)
−0.834623 + 0.550822i \(0.814314\pi\)
\(648\) 0 0
\(649\) −256.000 −0.394453
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1163.90i 1.78239i 0.453625 + 0.891193i \(0.350131\pi\)
−0.453625 + 0.891193i \(0.649869\pi\)
\(654\) 0 0
\(655\) 352.000 0.537405
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 203.647i 0.309024i 0.987991 + 0.154512i \(0.0493805\pi\)
−0.987991 + 0.154512i \(0.950619\pi\)
\(660\) 0 0
\(661\) 1018.00 1.54009 0.770045 0.637989i \(-0.220234\pi\)
0.770045 + 0.637989i \(0.220234\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 362.039i 0.544419i
\(666\) 0 0
\(667\) −1488.00 −2.23088
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 610.940i 0.910492i
\(672\) 0 0
\(673\) −382.000 −0.567608 −0.283804 0.958882i \(-0.591596\pi\)
−0.283804 + 0.958882i \(0.591596\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 247.487i 0.365565i 0.983153 + 0.182782i \(0.0585104\pi\)
−0.983153 + 0.182782i \(0.941490\pi\)
\(678\) 0 0
\(679\) 640.000 0.942563
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 169.706i 0.248471i 0.992253 + 0.124235i \(0.0396478\pi\)
−0.992253 + 0.124235i \(0.960352\pi\)
\(684\) 0 0
\(685\) −66.0000 −0.0963504
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 260.215i − 0.377671i
\(690\) 0 0
\(691\) 1040.00 1.50507 0.752533 0.658555i \(-0.228832\pi\)
0.752533 + 0.658555i \(0.228832\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 22.6274i − 0.0325574i
\(696\) 0 0
\(697\) −846.000 −1.21377
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 657.609i 0.938102i 0.883171 + 0.469051i \(0.155404\pi\)
−0.883171 + 0.469051i \(0.844596\pi\)
\(702\) 0 0
\(703\) 832.000 1.18350
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 984.293i − 1.39221i
\(708\) 0 0
\(709\) −952.000 −1.34274 −0.671368 0.741124i \(-0.734293\pi\)
−0.671368 + 0.741124i \(0.734293\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1357.65i 1.90413i
\(714\) 0 0
\(715\) 128.000 0.179021
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 192.333i − 0.267501i −0.991015 0.133750i \(-0.957298\pi\)
0.991015 0.133750i \(-0.0427020\pi\)
\(720\) 0 0
\(721\) −576.000 −0.798890
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 1008.33i − 1.39081i
\(726\) 0 0
\(727\) 648.000 0.891334 0.445667 0.895199i \(-0.352967\pi\)
0.445667 + 0.895199i \(0.352967\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 203.647i − 0.278587i
\(732\) 0 0
\(733\) −1208.00 −1.64802 −0.824011 0.566574i \(-0.808269\pi\)
−0.824011 + 0.566574i \(0.808269\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 905.097i − 1.22808i
\(738\) 0 0
\(739\) −1312.00 −1.77537 −0.887686 0.460449i \(-0.847688\pi\)
−0.887686 + 0.460449i \(0.847688\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 610.940i 0.822261i 0.911576 + 0.411131i \(0.134866\pi\)
−0.911576 + 0.411131i \(0.865134\pi\)
\(744\) 0 0
\(745\) 258.000 0.346309
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1448.15i 1.93345i
\(750\) 0 0
\(751\) 632.000 0.841545 0.420772 0.907166i \(-0.361759\pi\)
0.420772 + 0.907166i \(0.361759\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 237.588i 0.314686i
\(756\) 0 0
\(757\) 840.000 1.10964 0.554822 0.831969i \(-0.312786\pi\)
0.554822 + 0.831969i \(0.312786\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 521.845i 0.685736i 0.939384 + 0.342868i \(0.111398\pi\)
−0.939384 + 0.342868i \(0.888602\pi\)
\(762\) 0 0
\(763\) −704.000 −0.922674
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 181.019i 0.236010i
\(768\) 0 0
\(769\) 130.000 0.169051 0.0845254 0.996421i \(-0.473063\pi\)
0.0845254 + 0.996421i \(0.473063\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 284.257i 0.367732i 0.982951 + 0.183866i \(0.0588613\pi\)
−0.982951 + 0.183866i \(0.941139\pi\)
\(774\) 0 0
\(775\) −920.000 −1.18710
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 2126.98i − 2.73039i
\(780\) 0 0
\(781\) −896.000 −1.14725
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 14.1421i − 0.0180155i
\(786\) 0 0
\(787\) −864.000 −1.09784 −0.548920 0.835875i \(-0.684961\pi\)
−0.548920 + 0.835875i \(0.684961\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1097.43i − 1.38740i
\(792\) 0 0
\(793\) 432.000 0.544767
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 759.433i − 0.952864i −0.879211 0.476432i \(-0.841930\pi\)
0.879211 0.476432i \(-0.158070\pi\)
\(798\) 0 0
\(799\) 144.000 0.180225
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1086.12i 1.35257i
\(804\) 0 0
\(805\) 384.000 0.477019
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 906.511i 1.12053i 0.828313 + 0.560266i \(0.189301\pi\)
−0.828313 + 0.560266i \(0.810699\pi\)
\(810\) 0 0
\(811\) 1472.00 1.81504 0.907522 0.420005i \(-0.137972\pi\)
0.907522 + 0.420005i \(0.137972\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 113.137i − 0.138819i
\(816\) 0 0
\(817\) 512.000 0.626683
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 1189.35i − 1.44866i −0.689451 0.724332i \(-0.742148\pi\)
0.689451 0.724332i \(-0.257852\pi\)
\(822\) 0 0
\(823\) 664.000 0.806804 0.403402 0.915023i \(-0.367828\pi\)
0.403402 + 0.915023i \(0.367828\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 248.902i 0.300969i 0.988612 + 0.150485i \(0.0480834\pi\)
−0.988612 + 0.150485i \(0.951917\pi\)
\(828\) 0 0
\(829\) 280.000 0.337756 0.168878 0.985637i \(-0.445986\pi\)
0.168878 + 0.985637i \(0.445986\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 190.919i − 0.229194i
\(834\) 0 0
\(835\) −416.000 −0.498204
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 554.372i 0.660753i 0.943849 + 0.330376i \(0.107176\pi\)
−0.943849 + 0.330376i \(0.892824\pi\)
\(840\) 0 0
\(841\) −1081.00 −1.28537
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 148.492i 0.175731i
\(846\) 0 0
\(847\) 56.0000 0.0661157
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 882.469i − 1.03698i
\(852\) 0 0
\(853\) 762.000 0.893318 0.446659 0.894704i \(-0.352614\pi\)
0.446659 + 0.894704i \(0.352614\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1537.25i − 1.79376i −0.442277 0.896879i \(-0.645829\pi\)
0.442277 0.896879i \(-0.354171\pi\)
\(858\) 0 0
\(859\) 1552.00 1.80675 0.903376 0.428849i \(-0.141081\pi\)
0.903376 + 0.428849i \(0.141081\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 22.6274i − 0.0262195i −0.999914 0.0131097i \(-0.995827\pi\)
0.999914 0.0131097i \(-0.00417308\pi\)
\(864\) 0 0
\(865\) 78.0000 0.0901734
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1176.63i 1.35400i
\(870\) 0 0
\(871\) −640.000 −0.734788
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 543.058i 0.620638i
\(876\) 0 0
\(877\) 86.0000 0.0980616 0.0490308 0.998797i \(-0.484387\pi\)
0.0490308 + 0.998797i \(0.484387\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 657.609i − 0.746435i −0.927744 0.373218i \(-0.878255\pi\)
0.927744 0.373218i \(-0.121745\pi\)
\(882\) 0 0
\(883\) 496.000 0.561721 0.280861 0.959749i \(-0.409380\pi\)
0.280861 + 0.959749i \(0.409380\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1018.23i 1.14795i 0.818872 + 0.573976i \(0.194600\pi\)
−0.818872 + 0.573976i \(0.805400\pi\)
\(888\) 0 0
\(889\) −448.000 −0.503937
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 362.039i 0.405418i
\(894\) 0 0
\(895\) −192.000 −0.214525
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1753.62i 1.95064i
\(900\) 0 0
\(901\) −414.000 −0.459489
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.3137i 0.0125013i
\(906\) 0 0
\(907\) −880.000 −0.970232 −0.485116 0.874450i \(-0.661223\pi\)
−0.485116 + 0.874450i \(0.661223\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1561.29i − 1.71382i −0.515464 0.856911i \(-0.672381\pi\)
0.515464 0.856911i \(-0.327619\pi\)
\(912\) 0 0
\(913\) 1152.00 1.26177
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1991.21i − 2.17144i
\(918\) 0 0
\(919\) −264.000 −0.287269 −0.143634 0.989631i \(-0.545879\pi\)
−0.143634 + 0.989631i \(0.545879\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 633.568i 0.686422i
\(924\) 0 0
\(925\) 598.000 0.646486
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 284.257i − 0.305982i −0.988228 0.152991i \(-0.951110\pi\)
0.988228 0.152991i \(-0.0488905\pi\)
\(930\) 0 0
\(931\) 480.000 0.515575
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 203.647i − 0.217804i
\(936\) 0 0
\(937\) 1070.00 1.14194 0.570971 0.820970i \(-0.306567\pi\)
0.570971 + 0.820970i \(0.306567\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 575.585i − 0.611674i −0.952084 0.305837i \(-0.901064\pi\)
0.952084 0.305837i \(-0.0989362\pi\)
\(942\) 0 0
\(943\) −2256.00 −2.39236
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 610.940i − 0.645132i −0.946547 0.322566i \(-0.895455\pi\)
0.946547 0.322566i \(-0.104545\pi\)
\(948\) 0 0
\(949\) 768.000 0.809273
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1039.45i 1.09071i 0.838205 + 0.545355i \(0.183605\pi\)
−0.838205 + 0.545355i \(0.816395\pi\)
\(954\) 0 0
\(955\) −96.0000 −0.100524
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 373.352i 0.389314i
\(960\) 0 0
\(961\) 639.000 0.664932
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 364.867i 0.378101i
\(966\) 0 0
\(967\) −696.000 −0.719752 −0.359876 0.933000i \(-0.617181\pi\)
−0.359876 + 0.933000i \(0.617181\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1572.61i 1.61957i 0.586725 + 0.809787i \(0.300417\pi\)
−0.586725 + 0.809787i \(0.699583\pi\)
\(972\) 0 0
\(973\) −128.000 −0.131552
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 507.703i − 0.519655i −0.965655 0.259827i \(-0.916334\pi\)
0.965655 0.259827i \(-0.0836657\pi\)
\(978\) 0 0
\(979\) 880.000 0.898876
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 475.176i − 0.483393i −0.970352 0.241697i \(-0.922296\pi\)
0.970352 0.241697i \(-0.0777039\pi\)
\(984\) 0 0
\(985\) 526.000 0.534010
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 543.058i − 0.549098i
\(990\) 0 0
\(991\) −520.000 −0.524723 −0.262361 0.964970i \(-0.584501\pi\)
−0.262361 + 0.964970i \(0.584501\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 124.451i 0.125076i
\(996\) 0 0
\(997\) 486.000 0.487462 0.243731 0.969843i \(-0.421629\pi\)
0.243731 + 0.969843i \(0.421629\pi\)
\(998\) 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 576.3.e.b.449.1 2
3.2 odd 2 inner 576.3.e.b.449.2 2
4.3 odd 2 576.3.e.g.449.1 2
8.3 odd 2 288.3.e.d.161.2 yes 2
8.5 even 2 288.3.e.a.161.2 yes 2
12.11 even 2 576.3.e.g.449.2 2
16.3 odd 4 2304.3.h.b.2177.2 4
16.5 even 4 2304.3.h.g.2177.3 4
16.11 odd 4 2304.3.h.b.2177.3 4
16.13 even 4 2304.3.h.g.2177.2 4
24.5 odd 2 288.3.e.a.161.1 2
24.11 even 2 288.3.e.d.161.1 yes 2
48.5 odd 4 2304.3.h.g.2177.1 4
48.11 even 4 2304.3.h.b.2177.1 4
48.29 odd 4 2304.3.h.g.2177.4 4
48.35 even 4 2304.3.h.b.2177.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.a.161.1 2 24.5 odd 2
288.3.e.a.161.2 yes 2 8.5 even 2
288.3.e.d.161.1 yes 2 24.11 even 2
288.3.e.d.161.2 yes 2 8.3 odd 2
576.3.e.b.449.1 2 1.1 even 1 trivial
576.3.e.b.449.2 2 3.2 odd 2 inner
576.3.e.g.449.1 2 4.3 odd 2
576.3.e.g.449.2 2 12.11 even 2
2304.3.h.b.2177.1 4 48.11 even 4
2304.3.h.b.2177.2 4 16.3 odd 4
2304.3.h.b.2177.3 4 16.11 odd 4
2304.3.h.b.2177.4 4 48.35 even 4
2304.3.h.g.2177.1 4 48.5 odd 4
2304.3.h.g.2177.2 4 16.13 even 4
2304.3.h.g.2177.3 4 16.5 even 4
2304.3.h.g.2177.4 4 48.29 odd 4