Properties

Label 1800.4.f.e.649.2
Level $1800$
Weight $4$
Character 1800.649
Analytic conductor $106.203$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(106.203438010\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.649
Dual form 1800.4.f.e.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+18.0000i q^{7} +O(q^{10})\) \(q+18.0000i q^{7} -34.0000 q^{11} +12.0000i q^{13} -102.000i q^{17} -164.000 q^{19} -48.0000i q^{23} +146.000 q^{29} +100.000 q^{31} -328.000i q^{37} +288.000 q^{41} +120.000i q^{43} +16.0000i q^{47} +19.0000 q^{49} +126.000i q^{53} +642.000 q^{59} +602.000 q^{61} -436.000i q^{67} -652.000 q^{71} +1062.00i q^{73} -612.000i q^{77} -388.000 q^{79} +444.000i q^{83} -820.000 q^{89} -216.000 q^{91} +766.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 68 q^{11} - 328 q^{19} + 292 q^{29} + 200 q^{31} + 576 q^{41} + 38 q^{49} + 1284 q^{59} + 1204 q^{61} - 1304 q^{71} - 776 q^{79} - 1640 q^{89} - 432 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(7\) 18.0000i 0.971909i 0.873984 + 0.485954i \(0.161528\pi\)
−0.873984 + 0.485954i \(0.838472\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −34.0000 −0.931944 −0.465972 0.884799i \(-0.654295\pi\)
−0.465972 + 0.884799i \(0.654295\pi\)
\(12\) 0 0
\(13\) 12.0000i 0.256015i 0.991773 + 0.128008i \(0.0408582\pi\)
−0.991773 + 0.128008i \(0.959142\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 102.000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) −164.000 −1.98022 −0.990110 0.140293i \(-0.955195\pi\)
−0.990110 + 0.140293i \(0.955195\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 48.0000i − 0.435161i −0.976042 0.217580i \(-0.930184\pi\)
0.976042 0.217580i \(-0.0698164\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 146.000 0.934880 0.467440 0.884025i \(-0.345176\pi\)
0.467440 + 0.884025i \(0.345176\pi\)
\(30\) 0 0
\(31\) 100.000 0.579372 0.289686 0.957122i \(-0.406449\pi\)
0.289686 + 0.957122i \(0.406449\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 328.000i − 1.45737i −0.684846 0.728687i \(-0.740131\pi\)
0.684846 0.728687i \(-0.259869\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 288.000 1.09703 0.548513 0.836142i \(-0.315194\pi\)
0.548513 + 0.836142i \(0.315194\pi\)
\(42\) 0 0
\(43\) 120.000i 0.425577i 0.977098 + 0.212789i \(0.0682546\pi\)
−0.977098 + 0.212789i \(0.931745\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.0000i 0.0496562i 0.999692 + 0.0248281i \(0.00790384\pi\)
−0.999692 + 0.0248281i \(0.992096\pi\)
\(48\) 0 0
\(49\) 19.0000 0.0553936
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 126.000i 0.326555i 0.986580 + 0.163278i \(0.0522066\pi\)
−0.986580 + 0.163278i \(0.947793\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 642.000 1.41663 0.708316 0.705896i \(-0.249455\pi\)
0.708316 + 0.705896i \(0.249455\pi\)
\(60\) 0 0
\(61\) 602.000 1.26358 0.631789 0.775141i \(-0.282321\pi\)
0.631789 + 0.775141i \(0.282321\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 436.000i − 0.795013i −0.917599 0.397507i \(-0.869876\pi\)
0.917599 0.397507i \(-0.130124\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −652.000 −1.08983 −0.544917 0.838490i \(-0.683439\pi\)
−0.544917 + 0.838490i \(0.683439\pi\)
\(72\) 0 0
\(73\) 1062.00i 1.70271i 0.524591 + 0.851354i \(0.324218\pi\)
−0.524591 + 0.851354i \(0.675782\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 612.000i − 0.905765i
\(78\) 0 0
\(79\) −388.000 −0.552575 −0.276287 0.961075i \(-0.589104\pi\)
−0.276287 + 0.961075i \(0.589104\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 444.000i 0.587173i 0.955933 + 0.293586i \(0.0948488\pi\)
−0.955933 + 0.293586i \(0.905151\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −820.000 −0.976627 −0.488314 0.872668i \(-0.662388\pi\)
−0.488314 + 0.872668i \(0.662388\pi\)
\(90\) 0 0
\(91\) −216.000 −0.248824
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 766.000i 0.801809i 0.916120 + 0.400905i \(0.131304\pi\)
−0.916120 + 0.400905i \(0.868696\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 798.000 0.786178 0.393089 0.919500i \(-0.371406\pi\)
0.393089 + 0.919500i \(0.371406\pi\)
\(102\) 0 0
\(103\) − 402.000i − 0.384565i −0.981340 0.192283i \(-0.938411\pi\)
0.981340 0.192283i \(-0.0615891\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1444.00i 1.30464i 0.757943 + 0.652321i \(0.226205\pi\)
−0.757943 + 0.652321i \(0.773795\pi\)
\(108\) 0 0
\(109\) 198.000 0.173990 0.0869952 0.996209i \(-0.472274\pi\)
0.0869952 + 0.996209i \(0.472274\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2010.00i 1.67332i 0.547724 + 0.836659i \(0.315494\pi\)
−0.547724 + 0.836659i \(0.684506\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1836.00 1.41433
\(120\) 0 0
\(121\) −175.000 −0.131480
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 866.000i 0.605079i 0.953137 + 0.302540i \(0.0978345\pi\)
−0.953137 + 0.302540i \(0.902166\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2098.00 1.39926 0.699630 0.714505i \(-0.253348\pi\)
0.699630 + 0.714505i \(0.253348\pi\)
\(132\) 0 0
\(133\) − 2952.00i − 1.92459i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 886.000i 0.552526i 0.961082 + 0.276263i \(0.0890961\pi\)
−0.961082 + 0.276263i \(0.910904\pi\)
\(138\) 0 0
\(139\) 500.000 0.305104 0.152552 0.988295i \(-0.451251\pi\)
0.152552 + 0.988295i \(0.451251\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 408.000i − 0.238592i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2302.00 1.26569 0.632843 0.774280i \(-0.281888\pi\)
0.632843 + 0.774280i \(0.281888\pi\)
\(150\) 0 0
\(151\) −2384.00 −1.28482 −0.642408 0.766363i \(-0.722064\pi\)
−0.642408 + 0.766363i \(0.722064\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 1452.00i − 0.738103i −0.929409 0.369052i \(-0.879683\pi\)
0.929409 0.369052i \(-0.120317\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 864.000 0.422936
\(162\) 0 0
\(163\) 604.000i 0.290239i 0.989414 + 0.145119i \(0.0463566\pi\)
−0.989414 + 0.145119i \(0.953643\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 664.000i − 0.307676i −0.988096 0.153838i \(-0.950837\pi\)
0.988096 0.153838i \(-0.0491634\pi\)
\(168\) 0 0
\(169\) 2053.00 0.934456
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4118.00i 1.80974i 0.425684 + 0.904872i \(0.360034\pi\)
−0.425684 + 0.904872i \(0.639966\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1746.00 0.729062 0.364531 0.931191i \(-0.381229\pi\)
0.364531 + 0.931191i \(0.381229\pi\)
\(180\) 0 0
\(181\) −1270.00 −0.521538 −0.260769 0.965401i \(-0.583976\pi\)
−0.260769 + 0.965401i \(0.583976\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3468.00i 1.35618i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2676.00 −1.01376 −0.506881 0.862016i \(-0.669202\pi\)
−0.506881 + 0.862016i \(0.669202\pi\)
\(192\) 0 0
\(193\) − 3146.00i − 1.17334i −0.809827 0.586668i \(-0.800439\pi\)
0.809827 0.586668i \(-0.199561\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3674.00i − 1.32874i −0.747404 0.664370i \(-0.768700\pi\)
0.747404 0.664370i \(-0.231300\pi\)
\(198\) 0 0
\(199\) 1392.00 0.495861 0.247930 0.968778i \(-0.420250\pi\)
0.247930 + 0.968778i \(0.420250\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2628.00i 0.908618i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5576.00 1.84545
\(210\) 0 0
\(211\) −540.000 −0.176185 −0.0880927 0.996112i \(-0.528077\pi\)
−0.0880927 + 0.996112i \(0.528077\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1800.00i 0.563097i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1224.00 0.372557
\(222\) 0 0
\(223\) 4166.00i 1.25101i 0.780219 + 0.625507i \(0.215108\pi\)
−0.780219 + 0.625507i \(0.784892\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 5024.00i − 1.46896i −0.678629 0.734481i \(-0.737425\pi\)
0.678629 0.734481i \(-0.262575\pi\)
\(228\) 0 0
\(229\) −4454.00 −1.28528 −0.642639 0.766169i \(-0.722160\pi\)
−0.642639 + 0.766169i \(0.722160\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1526.00i 0.429063i 0.976717 + 0.214531i \(0.0688224\pi\)
−0.976717 + 0.214531i \(0.931178\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6828.00 1.84798 0.923989 0.382420i \(-0.124909\pi\)
0.923989 + 0.382420i \(0.124909\pi\)
\(240\) 0 0
\(241\) 5782.00 1.54544 0.772721 0.634746i \(-0.218895\pi\)
0.772721 + 0.634746i \(0.218895\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1968.00i − 0.506967i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6394.00 1.60791 0.803956 0.594689i \(-0.202725\pi\)
0.803956 + 0.594689i \(0.202725\pi\)
\(252\) 0 0
\(253\) 1632.00i 0.405545i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1862.00i 0.451939i 0.974134 + 0.225970i \(0.0725550\pi\)
−0.974134 + 0.225970i \(0.927445\pi\)
\(258\) 0 0
\(259\) 5904.00 1.41644
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6504.00i 1.52492i 0.647036 + 0.762460i \(0.276008\pi\)
−0.647036 + 0.762460i \(0.723992\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8298.00 1.88081 0.940405 0.340056i \(-0.110446\pi\)
0.940405 + 0.340056i \(0.110446\pi\)
\(270\) 0 0
\(271\) 1848.00 0.414236 0.207118 0.978316i \(-0.433592\pi\)
0.207118 + 0.978316i \(0.433592\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2824.00i − 0.612555i −0.951942 0.306277i \(-0.900916\pi\)
0.951942 0.306277i \(-0.0990835\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1940.00 0.411853 0.205927 0.978567i \(-0.433979\pi\)
0.205927 + 0.978567i \(0.433979\pi\)
\(282\) 0 0
\(283\) 6548.00i 1.37540i 0.725995 + 0.687700i \(0.241380\pi\)
−0.725995 + 0.687700i \(0.758620\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5184.00i 1.06621i
\(288\) 0 0
\(289\) −5491.00 −1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 6566.00i − 1.30918i −0.755984 0.654590i \(-0.772841\pi\)
0.755984 0.654590i \(-0.227159\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 576.000 0.111408
\(300\) 0 0
\(301\) −2160.00 −0.413622
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 8432.00i − 1.56756i −0.621041 0.783778i \(-0.713290\pi\)
0.621041 0.783778i \(-0.286710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4916.00 0.896337 0.448168 0.893949i \(-0.352076\pi\)
0.448168 + 0.893949i \(0.352076\pi\)
\(312\) 0 0
\(313\) − 10106.0i − 1.82500i −0.409077 0.912500i \(-0.634149\pi\)
0.409077 0.912500i \(-0.365851\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3382.00i 0.599218i 0.954062 + 0.299609i \(0.0968562\pi\)
−0.954062 + 0.299609i \(0.903144\pi\)
\(318\) 0 0
\(319\) −4964.00 −0.871256
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16728.0i 2.88164i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −288.000 −0.0482613
\(330\) 0 0
\(331\) −6460.00 −1.07273 −0.536365 0.843986i \(-0.680203\pi\)
−0.536365 + 0.843986i \(0.680203\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 5294.00i − 0.855735i −0.903842 0.427867i \(-0.859265\pi\)
0.903842 0.427867i \(-0.140735\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3400.00 −0.539942
\(342\) 0 0
\(343\) 6516.00i 1.02575i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12096.0i − 1.87132i −0.352906 0.935659i \(-0.614806\pi\)
0.352906 0.935659i \(-0.385194\pi\)
\(348\) 0 0
\(349\) 862.000 0.132211 0.0661057 0.997813i \(-0.478943\pi\)
0.0661057 + 0.997813i \(0.478943\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6878.00i 1.03705i 0.855062 + 0.518525i \(0.173519\pi\)
−0.855062 + 0.518525i \(0.826481\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6216.00 0.913838 0.456919 0.889508i \(-0.348953\pi\)
0.456919 + 0.889508i \(0.348953\pi\)
\(360\) 0 0
\(361\) 20037.0 2.92127
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13274.0i 1.88800i 0.329941 + 0.944002i \(0.392971\pi\)
−0.329941 + 0.944002i \(0.607029\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2268.00 −0.317382
\(372\) 0 0
\(373\) 1300.00i 0.180460i 0.995921 + 0.0902298i \(0.0287602\pi\)
−0.995921 + 0.0902298i \(0.971240\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1752.00i 0.239344i
\(378\) 0 0
\(379\) −13324.0 −1.80583 −0.902913 0.429824i \(-0.858576\pi\)
−0.902913 + 0.429824i \(0.858576\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 6192.00i − 0.826100i −0.910708 0.413050i \(-0.864463\pi\)
0.910708 0.413050i \(-0.135537\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2022.00 0.263546 0.131773 0.991280i \(-0.457933\pi\)
0.131773 + 0.991280i \(0.457933\pi\)
\(390\) 0 0
\(391\) −4896.00 −0.633252
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 7856.00i − 0.993152i −0.867993 0.496576i \(-0.834590\pi\)
0.867993 0.496576i \(-0.165410\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1148.00 −0.142964 −0.0714818 0.997442i \(-0.522773\pi\)
−0.0714818 + 0.997442i \(0.522773\pi\)
\(402\) 0 0
\(403\) 1200.00i 0.148328i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11152.0i 1.35819i
\(408\) 0 0
\(409\) 6310.00 0.762859 0.381430 0.924398i \(-0.375432\pi\)
0.381430 + 0.924398i \(0.375432\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11556.0i 1.37684i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −13362.0 −1.55794 −0.778969 0.627062i \(-0.784257\pi\)
−0.778969 + 0.627062i \(0.784257\pi\)
\(420\) 0 0
\(421\) −5146.00 −0.595726 −0.297863 0.954609i \(-0.596274\pi\)
−0.297863 + 0.954609i \(0.596274\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10836.0i 1.22808i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6368.00 0.711684 0.355842 0.934546i \(-0.384194\pi\)
0.355842 + 0.934546i \(0.384194\pi\)
\(432\) 0 0
\(433\) − 6138.00i − 0.681232i −0.940202 0.340616i \(-0.889364\pi\)
0.940202 0.340616i \(-0.110636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7872.00i 0.861714i
\(438\) 0 0
\(439\) 4424.00 0.480970 0.240485 0.970653i \(-0.422693\pi\)
0.240485 + 0.970653i \(0.422693\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 488.000i 0.0523377i 0.999658 + 0.0261688i \(0.00833075\pi\)
−0.999658 + 0.0261688i \(0.991669\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16884.0 1.77462 0.887311 0.461172i \(-0.152571\pi\)
0.887311 + 0.461172i \(0.152571\pi\)
\(450\) 0 0
\(451\) −9792.00 −1.02237
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5398.00i 0.552533i 0.961081 + 0.276267i \(0.0890973\pi\)
−0.961081 + 0.276267i \(0.910903\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6122.00 0.618503 0.309252 0.950980i \(-0.399921\pi\)
0.309252 + 0.950980i \(0.399921\pi\)
\(462\) 0 0
\(463\) 8162.00i 0.819266i 0.912250 + 0.409633i \(0.134343\pi\)
−0.912250 + 0.409633i \(0.865657\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 2660.00i − 0.263576i −0.991278 0.131788i \(-0.957928\pi\)
0.991278 0.131788i \(-0.0420719\pi\)
\(468\) 0 0
\(469\) 7848.00 0.772680
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 4080.00i − 0.396614i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9788.00 0.933664 0.466832 0.884346i \(-0.345395\pi\)
0.466832 + 0.884346i \(0.345395\pi\)
\(480\) 0 0
\(481\) 3936.00 0.373111
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 4714.00i − 0.438628i −0.975654 0.219314i \(-0.929618\pi\)
0.975654 0.219314i \(-0.0703819\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6690.00 0.614899 0.307450 0.951564i \(-0.400524\pi\)
0.307450 + 0.951564i \(0.400524\pi\)
\(492\) 0 0
\(493\) − 14892.0i − 1.36045i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 11736.0i − 1.05922i
\(498\) 0 0
\(499\) 20636.0 1.85129 0.925646 0.378392i \(-0.123523\pi\)
0.925646 + 0.378392i \(0.123523\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 15952.0i − 1.41404i −0.707191 0.707022i \(-0.750038\pi\)
0.707191 0.707022i \(-0.249962\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2230.00 −0.194191 −0.0970953 0.995275i \(-0.530955\pi\)
−0.0970953 + 0.995275i \(0.530955\pi\)
\(510\) 0 0
\(511\) −19116.0 −1.65488
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 544.000i − 0.0462768i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1260.00 −0.105953 −0.0529766 0.998596i \(-0.516871\pi\)
−0.0529766 + 0.998596i \(0.516871\pi\)
\(522\) 0 0
\(523\) 2900.00i 0.242463i 0.992624 + 0.121231i \(0.0386843\pi\)
−0.992624 + 0.121231i \(0.961316\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 10200.0i − 0.843110i
\(528\) 0 0
\(529\) 9863.00 0.810635
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3456.00i 0.280855i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −646.000 −0.0516237
\(540\) 0 0
\(541\) 19554.0 1.55396 0.776980 0.629526i \(-0.216751\pi\)
0.776980 + 0.629526i \(0.216751\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 2664.00i − 0.208235i −0.994565 0.104117i \(-0.966798\pi\)
0.994565 0.104117i \(-0.0332018\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −23944.0 −1.85127
\(552\) 0 0
\(553\) − 6984.00i − 0.537052i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 11358.0i − 0.864011i −0.901871 0.432005i \(-0.857806\pi\)
0.901871 0.432005i \(-0.142194\pi\)
\(558\) 0 0
\(559\) −1440.00 −0.108954
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2440.00i 0.182653i 0.995821 + 0.0913266i \(0.0291107\pi\)
−0.995821 + 0.0913266i \(0.970889\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24156.0 −1.77974 −0.889870 0.456214i \(-0.849205\pi\)
−0.889870 + 0.456214i \(0.849205\pi\)
\(570\) 0 0
\(571\) −2220.00 −0.162704 −0.0813521 0.996685i \(-0.525924\pi\)
−0.0813521 + 0.996685i \(0.525924\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 5782.00i 0.417171i 0.978004 + 0.208586i \(0.0668860\pi\)
−0.978004 + 0.208586i \(0.933114\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7992.00 −0.570678
\(582\) 0 0
\(583\) − 4284.00i − 0.304331i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1684.00i 0.118409i 0.998246 + 0.0592045i \(0.0188564\pi\)
−0.998246 + 0.0592045i \(0.981144\pi\)
\(588\) 0 0
\(589\) −16400.0 −1.14728
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 15246.0i − 1.05578i −0.849312 0.527891i \(-0.822983\pi\)
0.849312 0.527891i \(-0.177017\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9016.00 0.614998 0.307499 0.951548i \(-0.400508\pi\)
0.307499 + 0.951548i \(0.400508\pi\)
\(600\) 0 0
\(601\) −18682.0 −1.26798 −0.633989 0.773342i \(-0.718584\pi\)
−0.633989 + 0.773342i \(0.718584\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 22022.0i 1.47256i 0.676676 + 0.736281i \(0.263420\pi\)
−0.676676 + 0.736281i \(0.736580\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −192.000 −0.0127127
\(612\) 0 0
\(613\) − 22808.0i − 1.50278i −0.659856 0.751392i \(-0.729383\pi\)
0.659856 0.751392i \(-0.270617\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 9422.00i − 0.614774i −0.951585 0.307387i \(-0.900545\pi\)
0.951585 0.307387i \(-0.0994546\pi\)
\(618\) 0 0
\(619\) −4172.00 −0.270900 −0.135450 0.990784i \(-0.543248\pi\)
−0.135450 + 0.990784i \(0.543248\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 14760.0i − 0.949192i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −33456.0 −2.12079
\(630\) 0 0
\(631\) −11572.0 −0.730070 −0.365035 0.930994i \(-0.618943\pi\)
−0.365035 + 0.930994i \(0.618943\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 228.000i 0.0141816i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 936.000 0.0576752 0.0288376 0.999584i \(-0.490819\pi\)
0.0288376 + 0.999584i \(0.490819\pi\)
\(642\) 0 0
\(643\) 15892.0i 0.974680i 0.873212 + 0.487340i \(0.162033\pi\)
−0.873212 + 0.487340i \(0.837967\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25056.0i 1.52249i 0.648463 + 0.761247i \(0.275412\pi\)
−0.648463 + 0.761247i \(0.724588\pi\)
\(648\) 0 0
\(649\) −21828.0 −1.32022
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 4054.00i − 0.242948i −0.992595 0.121474i \(-0.961238\pi\)
0.992595 0.121474i \(-0.0387622\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6758.00 0.399475 0.199738 0.979849i \(-0.435991\pi\)
0.199738 + 0.979849i \(0.435991\pi\)
\(660\) 0 0
\(661\) 25098.0 1.47685 0.738426 0.674335i \(-0.235569\pi\)
0.738426 + 0.674335i \(0.235569\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 7008.00i − 0.406823i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −20468.0 −1.17758
\(672\) 0 0
\(673\) 2830.00i 0.162093i 0.996710 + 0.0810464i \(0.0258262\pi\)
−0.996710 + 0.0810464i \(0.974174\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10654.0i 0.604825i 0.953177 + 0.302412i \(0.0977920\pi\)
−0.953177 + 0.302412i \(0.902208\pi\)
\(678\) 0 0
\(679\) −13788.0 −0.779286
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 17156.0i − 0.961136i −0.876957 0.480568i \(-0.840430\pi\)
0.876957 0.480568i \(-0.159570\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1512.00 −0.0836032
\(690\) 0 0
\(691\) −812.000 −0.0447032 −0.0223516 0.999750i \(-0.507115\pi\)
−0.0223516 + 0.999750i \(0.507115\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 29376.0i − 1.59641i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30270.0 1.63093 0.815465 0.578806i \(-0.196481\pi\)
0.815465 + 0.578806i \(0.196481\pi\)
\(702\) 0 0
\(703\) 53792.0i 2.88592i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14364.0i 0.764093i
\(708\) 0 0
\(709\) 394.000 0.0208702 0.0104351 0.999946i \(-0.496678\pi\)
0.0104351 + 0.999946i \(0.496678\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 4800.00i − 0.252120i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −37224.0 −1.93077 −0.965383 0.260836i \(-0.916002\pi\)
−0.965383 + 0.260836i \(0.916002\pi\)
\(720\) 0 0
\(721\) 7236.00 0.373762
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12614.0i 0.643504i 0.946824 + 0.321752i \(0.104272\pi\)
−0.946824 + 0.321752i \(0.895728\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12240.0 0.619306
\(732\) 0 0
\(733\) − 25664.0i − 1.29321i −0.762826 0.646604i \(-0.776189\pi\)
0.762826 0.646604i \(-0.223811\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14824.0i 0.740908i
\(738\) 0 0
\(739\) 18772.0 0.934424 0.467212 0.884145i \(-0.345259\pi\)
0.467212 + 0.884145i \(0.345259\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19376.0i 0.956711i 0.878166 + 0.478356i \(0.158767\pi\)
−0.878166 + 0.478356i \(0.841233\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −25992.0 −1.26799
\(750\) 0 0
\(751\) 30092.0 1.46215 0.731074 0.682299i \(-0.239020\pi\)
0.731074 + 0.682299i \(0.239020\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 18136.0i − 0.870758i −0.900247 0.435379i \(-0.856614\pi\)
0.900247 0.435379i \(-0.143386\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10948.0 −0.521504 −0.260752 0.965406i \(-0.583971\pi\)
−0.260752 + 0.965406i \(0.583971\pi\)
\(762\) 0 0
\(763\) 3564.00i 0.169103i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7704.00i 0.362680i
\(768\) 0 0
\(769\) −1422.00 −0.0666822 −0.0333411 0.999444i \(-0.510615\pi\)
−0.0333411 + 0.999444i \(0.510615\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 26142.0i − 1.21638i −0.793791 0.608190i \(-0.791896\pi\)
0.793791 0.608190i \(-0.208104\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −47232.0 −2.17235
\(780\) 0 0
\(781\) 22168.0 1.01566
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 23404.0i − 1.06005i −0.847981 0.530027i \(-0.822182\pi\)
0.847981 0.530027i \(-0.177818\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −36180.0 −1.62631
\(792\) 0 0
\(793\) 7224.00i 0.323495i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1418.00i 0.0630215i 0.999503 + 0.0315108i \(0.0100318\pi\)
−0.999503 + 0.0315108i \(0.989968\pi\)
\(798\) 0 0
\(799\) 1632.00 0.0722603
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 36108.0i − 1.58683i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −17304.0 −0.752010 −0.376005 0.926618i \(-0.622702\pi\)
−0.376005 + 0.926618i \(0.622702\pi\)
\(810\) 0 0
\(811\) −28012.0 −1.21287 −0.606433 0.795135i \(-0.707400\pi\)
−0.606433 + 0.795135i \(0.707400\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 19680.0i − 0.842737i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32266.0 −1.37161 −0.685805 0.727786i \(-0.740550\pi\)
−0.685805 + 0.727786i \(0.740550\pi\)
\(822\) 0 0
\(823\) 4962.00i 0.210163i 0.994464 + 0.105082i \(0.0335104\pi\)
−0.994464 + 0.105082i \(0.966490\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5064.00i 0.212929i 0.994316 + 0.106465i \(0.0339531\pi\)
−0.994316 + 0.106465i \(0.966047\pi\)
\(828\) 0 0
\(829\) 8174.00 0.342454 0.171227 0.985232i \(-0.445227\pi\)
0.171227 + 0.985232i \(0.445227\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1938.00i − 0.0806095i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28240.0 −1.16204 −0.581021 0.813889i \(-0.697347\pi\)
−0.581021 + 0.813889i \(0.697347\pi\)
\(840\) 0 0
\(841\) −3073.00 −0.125999
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 3150.00i − 0.127787i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −15744.0 −0.634192
\(852\) 0 0
\(853\) 10472.0i 0.420345i 0.977664 + 0.210173i \(0.0674026\pi\)
−0.977664 + 0.210173i \(0.932597\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 32102.0i − 1.27956i −0.768558 0.639780i \(-0.779025\pi\)
0.768558 0.639780i \(-0.220975\pi\)
\(858\) 0 0
\(859\) 11060.0 0.439304 0.219652 0.975578i \(-0.429508\pi\)
0.219652 + 0.975578i \(0.429508\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 36088.0i − 1.42346i −0.702451 0.711732i \(-0.747911\pi\)
0.702451 0.711732i \(-0.252089\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13192.0 0.514969
\(870\) 0 0
\(871\) 5232.00 0.203536
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34508.0i 1.32868i 0.747431 + 0.664340i \(0.231287\pi\)
−0.747431 + 0.664340i \(0.768713\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6596.00 0.252242 0.126121 0.992015i \(-0.459747\pi\)
0.126121 + 0.992015i \(0.459747\pi\)
\(882\) 0 0
\(883\) 17620.0i 0.671529i 0.941946 + 0.335765i \(0.108995\pi\)
−0.941946 + 0.335765i \(0.891005\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 50784.0i − 1.92239i −0.275870 0.961195i \(-0.588966\pi\)
0.275870 0.961195i \(-0.411034\pi\)
\(888\) 0 0
\(889\) −15588.0 −0.588082
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 2624.00i − 0.0983301i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14600.0 0.541643
\(900\) 0 0
\(901\) 12852.0 0.475208
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 16072.0i − 0.588381i −0.955747 0.294191i \(-0.904950\pi\)
0.955747 0.294191i \(-0.0950501\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 41760.0 1.51874 0.759369 0.650660i \(-0.225508\pi\)
0.759369 + 0.650660i \(0.225508\pi\)
\(912\) 0 0
\(913\) − 15096.0i − 0.547212i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 37764.0i 1.35995i
\(918\) 0 0
\(919\) −34100.0 −1.22400 −0.612000 0.790858i \(-0.709635\pi\)
−0.612000 + 0.790858i \(0.709635\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 7824.00i − 0.279014i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22812.0 −0.805638 −0.402819 0.915280i \(-0.631970\pi\)
−0.402819 + 0.915280i \(0.631970\pi\)
\(930\) 0 0
\(931\) −3116.00 −0.109691
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38982.0i 1.35911i 0.733624 + 0.679555i \(0.237827\pi\)
−0.733624 + 0.679555i \(0.762173\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −52766.0 −1.82797 −0.913986 0.405745i \(-0.867012\pi\)
−0.913986 + 0.405745i \(0.867012\pi\)
\(942\) 0 0
\(943\) − 13824.0i − 0.477382i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13608.0i 0.466949i 0.972363 + 0.233474i \(0.0750095\pi\)
−0.972363 + 0.233474i \(0.924990\pi\)
\(948\) 0 0
\(949\) −12744.0 −0.435920
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 6446.00i − 0.219104i −0.993981 0.109552i \(-0.965058\pi\)
0.993981 0.109552i \(-0.0349417\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15948.0 −0.537005
\(960\) 0 0
\(961\) −19791.0 −0.664328
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 42642.0i 1.41807i 0.705173 + 0.709035i \(0.250869\pi\)
−0.705173 + 0.709035i \(0.749131\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 19938.0 0.658950 0.329475 0.944164i \(-0.393128\pi\)
0.329475 + 0.944164i \(0.393128\pi\)
\(972\) 0 0
\(973\) 9000.00i 0.296533i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 49754.0i 1.62924i 0.579992 + 0.814622i \(0.303056\pi\)
−0.579992 + 0.814622i \(0.696944\pi\)
\(978\) 0 0
\(979\) 27880.0 0.910162
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 7936.00i − 0.257497i −0.991677 0.128748i \(-0.958904\pi\)
0.991677 0.128748i \(-0.0410959\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5760.00 0.185194
\(990\) 0 0
\(991\) −33248.0 −1.06575 −0.532875 0.846194i \(-0.678888\pi\)
−0.532875 + 0.846194i \(0.678888\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 20196.0i 0.641538i 0.947157 + 0.320769i \(0.103941\pi\)
−0.947157 + 0.320769i \(0.896059\pi\)
\(998\) 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 1800.4.f.e.649.2 2
3.2 odd 2 1800.4.f.s.649.2 2
5.2 odd 4 360.4.a.a.1.1 1
5.3 odd 4 1800.4.a.bc.1.1 1
5.4 even 2 inner 1800.4.f.e.649.1 2
15.2 even 4 360.4.a.j.1.1 yes 1
15.8 even 4 1800.4.a.be.1.1 1
15.14 odd 2 1800.4.f.s.649.1 2
20.7 even 4 720.4.a.m.1.1 1
60.47 odd 4 720.4.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.4.a.a.1.1 1 5.2 odd 4
360.4.a.j.1.1 yes 1 15.2 even 4
720.4.a.m.1.1 1 20.7 even 4
720.4.a.z.1.1 1 60.47 odd 4
1800.4.a.bc.1.1 1 5.3 odd 4
1800.4.a.be.1.1 1 15.8 even 4
1800.4.f.e.649.1 2 5.4 even 2 inner
1800.4.f.e.649.2 2 1.1 even 1 trivial
1800.4.f.s.649.1 2 15.14 odd 2
1800.4.f.s.649.2 2 3.2 odd 2