Properties

Label 900.3.b.b.449.8
Level $900$
Weight $3$
Character 900.449
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.8
Root \(-1.14412 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 900.449
Dual form 900.3.b.b.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+13.4868i q^{7} +O(q^{10})\) \(q+13.4868i q^{7} +17.6590i q^{11} -7.48683i q^{13} -16.9706 q^{17} +10.9737 q^{19} -21.9017 q^{23} -47.3575i q^{29} -16.9737 q^{31} +5.53950i q^{37} -66.3936i q^{41} +38.9737i q^{43} -32.5642 q^{47} -132.895 q^{49} +11.2392 q^{53} -31.8757i q^{59} +46.9210 q^{61} +76.0000i q^{67} +77.7445i q^{71} +94.9210i q^{73} -238.165 q^{77} +6.92100 q^{79} -62.1509 q^{83} -62.2626i q^{89} +100.974 q^{91} +124.974i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{19} + 16 q^{31} - 456 q^{49} - 80 q^{61} - 400 q^{79} + 656 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 13.4868i 1.92669i 0.268265 + 0.963345i \(0.413550\pi\)
−0.268265 + 0.963345i \(0.586450\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 17.6590i 1.60537i 0.596405 + 0.802684i \(0.296595\pi\)
−0.596405 + 0.802684i \(0.703405\pi\)
\(12\) 0 0
\(13\) − 7.48683i − 0.575910i −0.957644 0.287955i \(-0.907025\pi\)
0.957644 0.287955i \(-0.0929754\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −16.9706 −0.998268 −0.499134 0.866525i \(-0.666349\pi\)
−0.499134 + 0.866525i \(0.666349\pi\)
\(18\) 0 0
\(19\) 10.9737 0.577561 0.288781 0.957395i \(-0.406750\pi\)
0.288781 + 0.957395i \(0.406750\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −21.9017 −0.952247 −0.476124 0.879378i \(-0.657959\pi\)
−0.476124 + 0.879378i \(0.657959\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 47.3575i − 1.63302i −0.577332 0.816509i \(-0.695906\pi\)
0.577332 0.816509i \(-0.304094\pi\)
\(30\) 0 0
\(31\) −16.9737 −0.547538 −0.273769 0.961796i \(-0.588270\pi\)
−0.273769 + 0.961796i \(0.588270\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.53950i 0.149716i 0.997194 + 0.0748581i \(0.0238504\pi\)
−0.997194 + 0.0748581i \(0.976150\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 66.3936i − 1.61935i −0.586875 0.809677i \(-0.699642\pi\)
0.586875 0.809677i \(-0.300358\pi\)
\(42\) 0 0
\(43\) 38.9737i 0.906364i 0.891418 + 0.453182i \(0.149711\pi\)
−0.891418 + 0.453182i \(0.850289\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −32.5642 −0.692854 −0.346427 0.938077i \(-0.612605\pi\)
−0.346427 + 0.938077i \(0.612605\pi\)
\(48\) 0 0
\(49\) −132.895 −2.71214
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.2392 0.212061 0.106030 0.994363i \(-0.466186\pi\)
0.106030 + 0.994363i \(0.466186\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 31.8757i − 0.540266i −0.962823 0.270133i \(-0.912932\pi\)
0.962823 0.270133i \(-0.0870676\pi\)
\(60\) 0 0
\(61\) 46.9210 0.769197 0.384598 0.923084i \(-0.374340\pi\)
0.384598 + 0.923084i \(0.374340\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 76.0000i 1.13433i 0.823605 + 0.567164i \(0.191960\pi\)
−0.823605 + 0.567164i \(0.808040\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 77.7445i 1.09499i 0.836808 + 0.547497i \(0.184419\pi\)
−0.836808 + 0.547497i \(0.815581\pi\)
\(72\) 0 0
\(73\) 94.9210i 1.30029i 0.759811 + 0.650144i \(0.225291\pi\)
−0.759811 + 0.650144i \(0.774709\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −238.165 −3.09305
\(78\) 0 0
\(79\) 6.92100 0.0876076 0.0438038 0.999040i \(-0.486052\pi\)
0.0438038 + 0.999040i \(0.486052\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −62.1509 −0.748806 −0.374403 0.927266i \(-0.622152\pi\)
−0.374403 + 0.927266i \(0.622152\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 62.2626i − 0.699580i −0.936828 0.349790i \(-0.886253\pi\)
0.936828 0.349790i \(-0.113747\pi\)
\(90\) 0 0
\(91\) 100.974 1.10960
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 124.974i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 112.262i 1.11151i 0.831347 + 0.555754i \(0.187571\pi\)
−0.831347 + 0.555754i \(0.812429\pi\)
\(102\) 0 0
\(103\) − 170.302i − 1.65342i −0.562627 0.826711i \(-0.690209\pi\)
0.562627 0.826711i \(-0.309791\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −93.3381 −0.872319 −0.436159 0.899869i \(-0.643662\pi\)
−0.436159 + 0.899869i \(0.643662\pi\)
\(108\) 0 0
\(109\) 68.8683 0.631820 0.315910 0.948789i \(-0.397690\pi\)
0.315910 + 0.948789i \(0.397690\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −110.309 −0.976183 −0.488091 0.872793i \(-0.662307\pi\)
−0.488091 + 0.872793i \(0.662307\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 228.879i − 1.92335i
\(120\) 0 0
\(121\) −190.842 −1.57721
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 65.3815i 0.514815i 0.966303 + 0.257407i \(0.0828683\pi\)
−0.966303 + 0.257407i \(0.917132\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 47.2458i − 0.360655i −0.983607 0.180328i \(-0.942284\pi\)
0.983607 0.180328i \(-0.0577158\pi\)
\(132\) 0 0
\(133\) 148.000i 1.11278i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 216.039 1.57693 0.788465 0.615079i \(-0.210876\pi\)
0.788465 + 0.615079i \(0.210876\pi\)
\(138\) 0 0
\(139\) −211.842 −1.52404 −0.762022 0.647552i \(-0.775793\pi\)
−0.762022 + 0.647552i \(0.775793\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 132.210 0.924548
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 92.5379i 0.621060i 0.950564 + 0.310530i \(0.100506\pi\)
−0.950564 + 0.310530i \(0.899494\pi\)
\(150\) 0 0
\(151\) −123.842 −0.820146 −0.410073 0.912053i \(-0.634497\pi\)
−0.410073 + 0.912053i \(0.634497\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 31.4868i 0.200553i 0.994960 + 0.100277i \(0.0319727\pi\)
−0.994960 + 0.100277i \(0.968027\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 295.384i − 1.83469i
\(162\) 0 0
\(163\) − 101.132i − 0.620440i −0.950665 0.310220i \(-0.899597\pi\)
0.950665 0.310220i \(-0.100403\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 147.804 0.885054 0.442527 0.896755i \(-0.354082\pi\)
0.442527 + 0.896755i \(0.354082\pi\)
\(168\) 0 0
\(169\) 112.947 0.668327
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −332.079 −1.91953 −0.959767 0.280797i \(-0.909401\pi\)
−0.959767 + 0.280797i \(0.909401\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 45.8688i 0.256250i 0.991758 + 0.128125i \(0.0408959\pi\)
−0.991758 + 0.128125i \(0.959104\pi\)
\(180\) 0 0
\(181\) 132.868 0.734079 0.367040 0.930205i \(-0.380371\pi\)
0.367040 + 0.930205i \(0.380371\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 299.684i − 1.60259i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 163.974i 0.858504i 0.903185 + 0.429252i \(0.141223\pi\)
−0.903185 + 0.429252i \(0.858777\pi\)
\(192\) 0 0
\(193\) 110.000i 0.569948i 0.958535 + 0.284974i \(0.0919850\pi\)
−0.958535 + 0.284974i \(0.908015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −211.108 −1.07162 −0.535808 0.844340i \(-0.679993\pi\)
−0.535808 + 0.844340i \(0.679993\pi\)
\(198\) 0 0
\(199\) 18.1053 0.0909816 0.0454908 0.998965i \(-0.485515\pi\)
0.0454908 + 0.998965i \(0.485515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 638.703 3.14632
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 193.785i 0.927199i
\(210\) 0 0
\(211\) 341.579 1.61886 0.809428 0.587219i \(-0.199777\pi\)
0.809428 + 0.587219i \(0.199777\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 228.921i − 1.05494i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 127.056i 0.574913i
\(222\) 0 0
\(223\) 59.3288i 0.266049i 0.991113 + 0.133024i \(0.0424688\pi\)
−0.991113 + 0.133024i \(0.957531\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −115.593 −0.509221 −0.254610 0.967044i \(-0.581947\pi\)
−0.254610 + 0.967044i \(0.581947\pi\)
\(228\) 0 0
\(229\) 253.895 1.10871 0.554355 0.832280i \(-0.312965\pi\)
0.554355 + 0.832280i \(0.312965\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −206.401 −0.885840 −0.442920 0.896561i \(-0.646057\pi\)
−0.442920 + 0.896561i \(0.646057\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 346.296i 1.44894i 0.689308 + 0.724469i \(0.257915\pi\)
−0.689308 + 0.724469i \(0.742085\pi\)
\(240\) 0 0
\(241\) −182.053 −0.755405 −0.377703 0.925927i \(-0.623286\pi\)
−0.377703 + 0.925927i \(0.623286\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 82.1580i − 0.332623i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 140.807i 0.560985i 0.959856 + 0.280493i \(0.0904979\pi\)
−0.959856 + 0.280493i \(0.909502\pi\)
\(252\) 0 0
\(253\) − 386.763i − 1.52871i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 292.630 1.13864 0.569320 0.822116i \(-0.307207\pi\)
0.569320 + 0.822116i \(0.307207\pi\)
\(258\) 0 0
\(259\) −74.7103 −0.288457
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 270.952 1.03024 0.515118 0.857119i \(-0.327748\pi\)
0.515118 + 0.857119i \(0.327748\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 275.083i 1.02261i 0.859398 + 0.511307i \(0.170838\pi\)
−0.859398 + 0.511307i \(0.829162\pi\)
\(270\) 0 0
\(271\) 248.158 0.915712 0.457856 0.889026i \(-0.348617\pi\)
0.457856 + 0.889026i \(0.348617\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 342.039i 1.23480i 0.786650 + 0.617399i \(0.211814\pi\)
−0.786650 + 0.617399i \(0.788186\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 352.139i − 1.25316i −0.779355 0.626582i \(-0.784453\pi\)
0.779355 0.626582i \(-0.215547\pi\)
\(282\) 0 0
\(283\) 249.631i 0.882089i 0.897485 + 0.441045i \(0.145392\pi\)
−0.897485 + 0.441045i \(0.854608\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 895.439 3.12000
\(288\) 0 0
\(289\) −1.00000 −0.00346021
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −37.4953 −0.127970 −0.0639851 0.997951i \(-0.520381\pi\)
−0.0639851 + 0.997951i \(0.520381\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 163.974i 0.548409i
\(300\) 0 0
\(301\) −525.631 −1.74628
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 457.842i − 1.49134i −0.666314 0.745671i \(-0.732129\pi\)
0.666314 0.745671i \(-0.267871\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.48528i 0.0272839i 0.999907 + 0.0136419i \(0.00434250\pi\)
−0.999907 + 0.0136419i \(0.995658\pi\)
\(312\) 0 0
\(313\) − 1.57866i − 0.00504363i −0.999997 0.00252181i \(-0.999197\pi\)
0.999997 0.00252181i \(-0.000802719\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 529.195 1.66938 0.834692 0.550717i \(-0.185646\pi\)
0.834692 + 0.550717i \(0.185646\pi\)
\(318\) 0 0
\(319\) 836.289 2.62160
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −186.229 −0.576561
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 439.187i − 1.33492i
\(330\) 0 0
\(331\) −390.763 −1.18055 −0.590276 0.807201i \(-0.700981\pi\)
−0.590276 + 0.807201i \(0.700981\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 414.710i − 1.23059i −0.788295 0.615297i \(-0.789036\pi\)
0.788295 0.615297i \(-0.210964\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 299.739i − 0.878999i
\(342\) 0 0
\(343\) − 1131.47i − 3.29876i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 128.656 0.370767 0.185384 0.982666i \(-0.440647\pi\)
0.185384 + 0.982666i \(0.440647\pi\)
\(348\) 0 0
\(349\) 37.0790 0.106244 0.0531218 0.998588i \(-0.483083\pi\)
0.0531218 + 0.998588i \(0.483083\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 190.584 0.539897 0.269949 0.962875i \(-0.412993\pi\)
0.269949 + 0.962875i \(0.412993\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 254.782i 0.709699i 0.934923 + 0.354849i \(0.115468\pi\)
−0.934923 + 0.354849i \(0.884532\pi\)
\(360\) 0 0
\(361\) −240.579 −0.666423
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 300.460i − 0.818693i −0.912379 0.409347i \(-0.865757\pi\)
0.912379 0.409347i \(-0.134243\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 151.582i 0.408576i
\(372\) 0 0
\(373\) 704.092i 1.88765i 0.330452 + 0.943823i \(0.392799\pi\)
−0.330452 + 0.943823i \(0.607201\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −354.558 −0.940472
\(378\) 0 0
\(379\) −333.789 −0.880711 −0.440355 0.897824i \(-0.645148\pi\)
−0.440355 + 0.897824i \(0.645148\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −260.996 −0.681453 −0.340726 0.940163i \(-0.610673\pi\)
−0.340726 + 0.940163i \(0.610673\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 35.8949i 0.0922747i 0.998935 + 0.0461373i \(0.0146912\pi\)
−0.998935 + 0.0461373i \(0.985309\pi\)
\(390\) 0 0
\(391\) 371.684 0.950598
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 90.3552i 0.227595i 0.993504 + 0.113797i \(0.0363015\pi\)
−0.993504 + 0.113797i \(0.963699\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 43.6917i 0.108957i 0.998515 + 0.0544784i \(0.0173496\pi\)
−0.998515 + 0.0544784i \(0.982650\pi\)
\(402\) 0 0
\(403\) 127.079i 0.315333i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −97.8223 −0.240350
\(408\) 0 0
\(409\) 260.053 0.635826 0.317913 0.948120i \(-0.397018\pi\)
0.317913 + 0.948120i \(0.397018\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 429.902 1.04092
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 263.285i 0.628366i 0.949362 + 0.314183i \(0.101731\pi\)
−0.949362 + 0.314183i \(0.898269\pi\)
\(420\) 0 0
\(421\) 367.210 0.872233 0.436116 0.899890i \(-0.356354\pi\)
0.436116 + 0.899890i \(0.356354\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 632.816i 1.48200i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 690.768i 1.60271i 0.598189 + 0.801355i \(0.295887\pi\)
−0.598189 + 0.801355i \(0.704113\pi\)
\(432\) 0 0
\(433\) − 117.500i − 0.271363i −0.990752 0.135682i \(-0.956678\pi\)
0.990752 0.135682i \(-0.0433224\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −240.342 −0.549981
\(438\) 0 0
\(439\) −50.0000 −0.113895 −0.0569476 0.998377i \(-0.518137\pi\)
−0.0569476 + 0.998377i \(0.518137\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 526.794 1.18915 0.594576 0.804040i \(-0.297320\pi\)
0.594576 + 0.804040i \(0.297320\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 321.882i 0.716887i 0.933552 + 0.358443i \(0.116692\pi\)
−0.933552 + 0.358443i \(0.883308\pi\)
\(450\) 0 0
\(451\) 1172.45 2.59966
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 433.079i 0.947656i 0.880617 + 0.473828i \(0.157128\pi\)
−0.880617 + 0.473828i \(0.842872\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 486.768i − 1.05590i −0.849277 0.527948i \(-0.822961\pi\)
0.849277 0.527948i \(-0.177039\pi\)
\(462\) 0 0
\(463\) 423.223i 0.914090i 0.889444 + 0.457045i \(0.151092\pi\)
−0.889444 + 0.457045i \(0.848908\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −295.831 −0.633472 −0.316736 0.948514i \(-0.602587\pi\)
−0.316736 + 0.948514i \(0.602587\pi\)
\(468\) 0 0
\(469\) −1025.00 −2.18550
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −688.238 −1.45505
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 266.728i 0.556843i 0.960459 + 0.278421i \(0.0898112\pi\)
−0.960459 + 0.278421i \(0.910189\pi\)
\(480\) 0 0
\(481\) 41.4733 0.0862231
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 247.750i − 0.508727i −0.967109 0.254364i \(-0.918134\pi\)
0.967109 0.254364i \(-0.0818660\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 170.394i 0.347035i 0.984831 + 0.173517i \(0.0555133\pi\)
−0.984831 + 0.173517i \(0.944487\pi\)
\(492\) 0 0
\(493\) 803.684i 1.63019i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1048.53 −2.10971
\(498\) 0 0
\(499\) −47.3950 −0.0949800 −0.0474900 0.998872i \(-0.515122\pi\)
−0.0474900 + 0.998872i \(0.515122\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 321.064 0.638298 0.319149 0.947705i \(-0.396603\pi\)
0.319149 + 0.947705i \(0.396603\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 122.125i − 0.239931i −0.992778 0.119965i \(-0.961722\pi\)
0.992778 0.119965i \(-0.0382783\pi\)
\(510\) 0 0
\(511\) −1280.18 −2.50525
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 575.052i − 1.11229i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 79.4567i 0.152508i 0.997088 + 0.0762540i \(0.0242960\pi\)
−0.997088 + 0.0762540i \(0.975704\pi\)
\(522\) 0 0
\(523\) − 864.605i − 1.65316i −0.562816 0.826582i \(-0.690282\pi\)
0.562816 0.826582i \(-0.309718\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 288.053 0.546589
\(528\) 0 0
\(529\) −49.3160 −0.0932250
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −497.077 −0.932603
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2346.79i − 4.35398i
\(540\) 0 0
\(541\) 33.6057 0.0621177 0.0310589 0.999518i \(-0.490112\pi\)
0.0310589 + 0.999518i \(0.490112\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 559.079i 1.02208i 0.859556 + 0.511041i \(0.170740\pi\)
−0.859556 + 0.511041i \(0.829260\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 519.686i − 0.943168i
\(552\) 0 0
\(553\) 93.3423i 0.168793i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 310.308 0.557105 0.278553 0.960421i \(-0.410145\pi\)
0.278553 + 0.960421i \(0.410145\pi\)
\(558\) 0 0
\(559\) 291.789 0.521984
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −566.690 −1.00655 −0.503277 0.864125i \(-0.667872\pi\)
−0.503277 + 0.864125i \(0.667872\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 134.052i 0.235593i 0.993038 + 0.117796i \(0.0375830\pi\)
−0.993038 + 0.117796i \(0.962417\pi\)
\(570\) 0 0
\(571\) −661.920 −1.15923 −0.579615 0.814890i \(-0.696797\pi\)
−0.579615 + 0.814890i \(0.696797\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 76.7630i − 0.133038i −0.997785 0.0665191i \(-0.978811\pi\)
0.997785 0.0665191i \(-0.0211893\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 838.219i − 1.44272i
\(582\) 0 0
\(583\) 198.474i 0.340436i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −684.554 −1.16619 −0.583095 0.812404i \(-0.698159\pi\)
−0.583095 + 0.812404i \(0.698159\pi\)
\(588\) 0 0
\(589\) −186.263 −0.316237
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −592.146 −0.998560 −0.499280 0.866441i \(-0.666402\pi\)
−0.499280 + 0.866441i \(0.666402\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1012.76i − 1.69075i −0.534169 0.845377i \(-0.679376\pi\)
0.534169 0.845377i \(-0.320624\pi\)
\(600\) 0 0
\(601\) 323.579 0.538400 0.269200 0.963084i \(-0.413241\pi\)
0.269200 + 0.963084i \(0.413241\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 799.828i 1.31767i 0.752285 + 0.658837i \(0.228951\pi\)
−0.752285 + 0.658837i \(0.771049\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 243.802i 0.399022i
\(612\) 0 0
\(613\) − 680.302i − 1.10979i −0.831920 0.554896i \(-0.812758\pi\)
0.831920 0.554896i \(-0.187242\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −264.868 −0.429283 −0.214641 0.976693i \(-0.568858\pi\)
−0.214641 + 0.976693i \(0.568858\pi\)
\(618\) 0 0
\(619\) −535.842 −0.865658 −0.432829 0.901476i \(-0.642485\pi\)
−0.432829 + 0.901476i \(0.642485\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 839.726 1.34787
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 94.0085i − 0.149457i
\(630\) 0 0
\(631\) 307.026 0.486571 0.243286 0.969955i \(-0.421775\pi\)
0.243286 + 0.969955i \(0.421775\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 994.960i 1.56195i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1246.33i 1.94435i 0.234248 + 0.972177i \(0.424737\pi\)
−0.234248 + 0.972177i \(0.575263\pi\)
\(642\) 0 0
\(643\) 155.395i 0.241672i 0.992672 + 0.120836i \(0.0385575\pi\)
−0.992672 + 0.120836i \(0.961443\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 142.649 0.220478 0.110239 0.993905i \(-0.464838\pi\)
0.110239 + 0.993905i \(0.464838\pi\)
\(648\) 0 0
\(649\) 562.894 0.867325
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −924.579 −1.41589 −0.707947 0.706266i \(-0.750378\pi\)
−0.707947 + 0.706266i \(0.750378\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 352.046i 0.534212i 0.963667 + 0.267106i \(0.0860674\pi\)
−0.963667 + 0.267106i \(0.913933\pi\)
\(660\) 0 0
\(661\) −642.921 −0.972649 −0.486325 0.873778i \(-0.661663\pi\)
−0.486325 + 0.873778i \(0.661663\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1037.21i 1.55504i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 828.580i 1.23484i
\(672\) 0 0
\(673\) 982.605i 1.46004i 0.683427 + 0.730019i \(0.260489\pi\)
−0.683427 + 0.730019i \(0.739511\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −386.769 −0.571298 −0.285649 0.958334i \(-0.592209\pi\)
−0.285649 + 0.958334i \(0.592209\pi\)
\(678\) 0 0
\(679\) −1685.50 −2.48233
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −504.539 −0.738710 −0.369355 0.929288i \(-0.620421\pi\)
−0.369355 + 0.929288i \(0.620421\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 84.1462i − 0.122128i
\(690\) 0 0
\(691\) −271.079 −0.392300 −0.196150 0.980574i \(-0.562844\pi\)
−0.196150 + 0.980574i \(0.562844\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1126.74i 1.61655i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 358.559i 0.511496i 0.966743 + 0.255748i \(0.0823218\pi\)
−0.966743 + 0.255748i \(0.917678\pi\)
\(702\) 0 0
\(703\) 60.7886i 0.0864703i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1514.06 −2.14153
\(708\) 0 0
\(709\) 733.579 1.03467 0.517333 0.855784i \(-0.326925\pi\)
0.517333 + 0.855784i \(0.326925\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 371.752 0.521391
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1108.37i 1.54155i 0.637110 + 0.770773i \(0.280130\pi\)
−0.637110 + 0.770773i \(0.719870\pi\)
\(720\) 0 0
\(721\) 2296.84 3.18563
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 1265.04i − 1.74008i −0.492981 0.870040i \(-0.664093\pi\)
0.492981 0.870040i \(-0.335907\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 661.405i − 0.904795i
\(732\) 0 0
\(733\) − 586.749i − 0.800477i −0.916411 0.400238i \(-0.868927\pi\)
0.916411 0.400238i \(-0.131073\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1342.09 −1.82101
\(738\) 0 0
\(739\) 215.973 0.292250 0.146125 0.989266i \(-0.453320\pi\)
0.146125 + 0.989266i \(0.453320\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 599.961 0.807484 0.403742 0.914873i \(-0.367709\pi\)
0.403742 + 0.914873i \(0.367709\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1258.84i − 1.68069i
\(750\) 0 0
\(751\) 813.657 1.08343 0.541716 0.840562i \(-0.317775\pi\)
0.541716 + 0.840562i \(0.317775\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 1070.43i − 1.41405i −0.707190 0.707023i \(-0.750038\pi\)
0.707190 0.707023i \(-0.249962\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 654.892i 0.860567i 0.902694 + 0.430284i \(0.141586\pi\)
−0.902694 + 0.430284i \(0.858414\pi\)
\(762\) 0 0
\(763\) 928.816i 1.21732i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −238.648 −0.311144
\(768\) 0 0
\(769\) −682.631 −0.887686 −0.443843 0.896105i \(-0.646385\pi\)
−0.443843 + 0.896105i \(0.646385\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 770.690 0.997012 0.498506 0.866886i \(-0.333882\pi\)
0.498506 + 0.866886i \(0.333882\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 728.581i − 0.935277i
\(780\) 0 0
\(781\) −1372.89 −1.75787
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 253.869i 0.322578i 0.986907 + 0.161289i \(0.0515652\pi\)
−0.986907 + 0.161289i \(0.948435\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1487.71i − 1.88080i
\(792\) 0 0
\(793\) − 351.290i − 0.442988i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1055.86 1.32479 0.662396 0.749154i \(-0.269540\pi\)
0.662396 + 0.749154i \(0.269540\pi\)
\(798\) 0 0
\(799\) 552.632 0.691655
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1676.21 −2.08744
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 506.698i 0.626326i 0.949699 + 0.313163i \(0.101389\pi\)
−0.949699 + 0.313163i \(0.898611\pi\)
\(810\) 0 0
\(811\) 537.473 0.662729 0.331365 0.943503i \(-0.392491\pi\)
0.331365 + 0.943503i \(0.392491\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 427.684i 0.523481i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 257.219i − 0.313299i −0.987654 0.156650i \(-0.949931\pi\)
0.987654 0.156650i \(-0.0500694\pi\)
\(822\) 0 0
\(823\) − 367.013i − 0.445945i −0.974825 0.222973i \(-0.928424\pi\)
0.974825 0.222973i \(-0.0715760\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 122.254 0.147829 0.0739144 0.997265i \(-0.476451\pi\)
0.0739144 + 0.997265i \(0.476451\pi\)
\(828\) 0 0
\(829\) −335.237 −0.404387 −0.202194 0.979346i \(-0.564807\pi\)
−0.202194 + 0.979346i \(0.564807\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2255.30 2.70744
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 774.021i 0.922552i 0.887257 + 0.461276i \(0.152608\pi\)
−0.887257 + 0.461276i \(0.847392\pi\)
\(840\) 0 0
\(841\) −1401.74 −1.66675
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2573.85i − 3.03879i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 121.324i − 0.142567i
\(852\) 0 0
\(853\) 764.591i 0.896356i 0.893944 + 0.448178i \(0.147927\pi\)
−0.893944 + 0.448178i \(0.852073\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1516.15 1.76913 0.884567 0.466413i \(-0.154454\pi\)
0.884567 + 0.466413i \(0.154454\pi\)
\(858\) 0 0
\(859\) −433.684 −0.504871 −0.252435 0.967614i \(-0.581232\pi\)
−0.252435 + 0.967614i \(0.581232\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1255.50 −1.45481 −0.727407 0.686206i \(-0.759275\pi\)
−0.727407 + 0.686206i \(0.759275\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 122.218i 0.140642i
\(870\) 0 0
\(871\) 568.999 0.653271
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 1003.59i − 1.14435i −0.820133 0.572173i \(-0.806100\pi\)
0.820133 0.572173i \(-0.193900\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 737.214i − 0.836792i −0.908265 0.418396i \(-0.862592\pi\)
0.908265 0.418396i \(-0.137408\pi\)
\(882\) 0 0
\(883\) 336.394i 0.380968i 0.981690 + 0.190484i \(0.0610057\pi\)
−0.981690 + 0.190484i \(0.938994\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 793.132 0.894174 0.447087 0.894491i \(-0.352461\pi\)
0.447087 + 0.894491i \(0.352461\pi\)
\(888\) 0 0
\(889\) −881.789 −0.991889
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −357.348 −0.400166
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 803.831i 0.894139i
\(900\) 0 0
\(901\) −190.736 −0.211694
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1079.66i 1.19036i 0.803592 + 0.595180i \(0.202919\pi\)
−0.803592 + 0.595180i \(0.797081\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1473.69i − 1.61766i −0.588045 0.808828i \(-0.700102\pi\)
0.588045 0.808828i \(-0.299898\pi\)
\(912\) 0 0
\(913\) − 1097.53i − 1.20211i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 637.196 0.694871
\(918\) 0 0
\(919\) 1602.92 1.74420 0.872101 0.489327i \(-0.162757\pi\)
0.872101 + 0.489327i \(0.162757\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 582.060 0.630618
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1033.98i − 1.11300i −0.830848 0.556499i \(-0.812144\pi\)
0.830848 0.556499i \(-0.187856\pi\)
\(930\) 0 0
\(931\) −1458.34 −1.56642
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1258.05i 1.34264i 0.741169 + 0.671319i \(0.234272\pi\)
−0.741169 + 0.671319i \(0.765728\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 762.875i 0.810707i 0.914160 + 0.405353i \(0.132852\pi\)
−0.914160 + 0.405353i \(0.867148\pi\)
\(942\) 0 0
\(943\) 1454.13i 1.54203i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1023.55 1.08084 0.540419 0.841396i \(-0.318266\pi\)
0.540419 + 0.841396i \(0.318266\pi\)
\(948\) 0 0
\(949\) 710.658 0.748849
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 63.7513 0.0668954 0.0334477 0.999440i \(-0.489351\pi\)
0.0334477 + 0.999440i \(0.489351\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2913.69i 3.03826i
\(960\) 0 0
\(961\) −672.895 −0.700203
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 505.986i 0.523254i 0.965169 + 0.261627i \(0.0842590\pi\)
−0.965169 + 0.261627i \(0.915741\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1803.51i 1.85738i 0.370862 + 0.928688i \(0.379062\pi\)
−0.370862 + 0.928688i \(0.620938\pi\)
\(972\) 0 0
\(973\) − 2857.08i − 2.93636i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1844.54 1.88797 0.943983 0.329994i \(-0.107047\pi\)
0.943983 + 0.329994i \(0.107047\pi\)
\(978\) 0 0
\(979\) 1099.50 1.12308
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 592.146 0.602386 0.301193 0.953563i \(-0.402615\pi\)
0.301193 + 0.953563i \(0.402615\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 853.589i − 0.863083i
\(990\) 0 0
\(991\) 961.684 0.970418 0.485209 0.874398i \(-0.338744\pi\)
0.485209 + 0.874398i \(0.338744\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 899.670i − 0.902378i −0.892429 0.451189i \(-0.851000\pi\)
0.892429 0.451189i \(-0.149000\pi\)
\(998\) 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 900.3.b.b.449.8 8
3.2 odd 2 inner 900.3.b.b.449.7 8
4.3 odd 2 3600.3.c.k.449.1 8
5.2 odd 4 180.3.g.a.161.3 yes 4
5.3 odd 4 900.3.g.d.701.4 4
5.4 even 2 inner 900.3.b.b.449.2 8
12.11 even 2 3600.3.c.k.449.2 8
15.2 even 4 180.3.g.a.161.1 4
15.8 even 4 900.3.g.d.701.3 4
15.14 odd 2 inner 900.3.b.b.449.1 8
20.3 even 4 3600.3.l.n.1601.1 4
20.7 even 4 720.3.l.c.161.4 4
20.19 odd 2 3600.3.c.k.449.7 8
40.27 even 4 2880.3.l.f.1601.2 4
40.37 odd 4 2880.3.l.b.1601.1 4
45.2 even 12 1620.3.o.f.701.4 8
45.7 odd 12 1620.3.o.f.701.2 8
45.22 odd 12 1620.3.o.f.1241.4 8
45.32 even 12 1620.3.o.f.1241.2 8
60.23 odd 4 3600.3.l.n.1601.2 4
60.47 odd 4 720.3.l.c.161.2 4
60.59 even 2 3600.3.c.k.449.8 8
120.77 even 4 2880.3.l.b.1601.3 4
120.107 odd 4 2880.3.l.f.1601.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.g.a.161.1 4 15.2 even 4
180.3.g.a.161.3 yes 4 5.2 odd 4
720.3.l.c.161.2 4 60.47 odd 4
720.3.l.c.161.4 4 20.7 even 4
900.3.b.b.449.1 8 15.14 odd 2 inner
900.3.b.b.449.2 8 5.4 even 2 inner
900.3.b.b.449.7 8 3.2 odd 2 inner
900.3.b.b.449.8 8 1.1 even 1 trivial
900.3.g.d.701.3 4 15.8 even 4
900.3.g.d.701.4 4 5.3 odd 4
1620.3.o.f.701.2 8 45.7 odd 12
1620.3.o.f.701.4 8 45.2 even 12
1620.3.o.f.1241.2 8 45.32 even 12
1620.3.o.f.1241.4 8 45.22 odd 12
2880.3.l.b.1601.1 4 40.37 odd 4
2880.3.l.b.1601.3 4 120.77 even 4
2880.3.l.f.1601.2 4 40.27 even 4
2880.3.l.f.1601.4 4 120.107 odd 4
3600.3.c.k.449.1 8 4.3 odd 2
3600.3.c.k.449.2 8 12.11 even 2
3600.3.c.k.449.7 8 20.19 odd 2
3600.3.c.k.449.8 8 60.59 even 2
3600.3.l.n.1601.1 4 20.3 even 4
3600.3.l.n.1601.2 4 60.23 odd 4