Properties

Label 2160.4.a.bt.1.3
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.697.1
Defining polynomial: \( x^{3} - 7x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.16601\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} +21.1457 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} +21.1457 q^{7} -67.4294 q^{11} +2.55493 q^{13} -126.114 q^{17} +103.539 q^{19} -200.434 q^{23} +25.0000 q^{25} -71.4215 q^{29} +158.130 q^{31} +105.729 q^{35} +7.18163 q^{37} +347.052 q^{41} -189.318 q^{43} +585.443 q^{47} +104.142 q^{49} -77.2791 q^{53} -337.147 q^{55} +200.790 q^{59} +681.137 q^{61} +12.7746 q^{65} +810.751 q^{67} +515.476 q^{71} +385.577 q^{73} -1425.84 q^{77} +209.405 q^{79} +887.189 q^{83} -630.572 q^{85} +548.890 q^{89} +54.0258 q^{91} +517.696 q^{95} -1523.96 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 24 q^{7} - 6 q^{11} + 48 q^{13} - 27 q^{17} + 195 q^{19} + 27 q^{23} + 75 q^{25} + 60 q^{29} + 279 q^{31} + 120 q^{35} - 138 q^{37} + 66 q^{41} - 222 q^{43} + 264 q^{47} - 237 q^{49} - 507 q^{53} - 30 q^{55} + 960 q^{59} + 543 q^{61} + 240 q^{65} + 1086 q^{67} + 1818 q^{71} + 1362 q^{73} - 1776 q^{77} + 129 q^{79} + 1569 q^{83} - 135 q^{85} - 1770 q^{89} - 1488 q^{91} + 975 q^{95} - 336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 21.1457 1.14176 0.570881 0.821033i \(-0.306602\pi\)
0.570881 + 0.821033i \(0.306602\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −67.4294 −1.84825 −0.924124 0.382093i \(-0.875203\pi\)
−0.924124 + 0.382093i \(0.875203\pi\)
\(12\) 0 0
\(13\) 2.55493 0.0545084 0.0272542 0.999629i \(-0.491324\pi\)
0.0272542 + 0.999629i \(0.491324\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −126.114 −1.79925 −0.899624 0.436665i \(-0.856160\pi\)
−0.899624 + 0.436665i \(0.856160\pi\)
\(18\) 0 0
\(19\) 103.539 1.25019 0.625093 0.780550i \(-0.285061\pi\)
0.625093 + 0.780550i \(0.285061\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −200.434 −1.81710 −0.908551 0.417774i \(-0.862810\pi\)
−0.908551 + 0.417774i \(0.862810\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −71.4215 −0.457333 −0.228666 0.973505i \(-0.573436\pi\)
−0.228666 + 0.973505i \(0.573436\pi\)
\(30\) 0 0
\(31\) 158.130 0.916161 0.458081 0.888911i \(-0.348537\pi\)
0.458081 + 0.888911i \(0.348537\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 105.729 0.510612
\(36\) 0 0
\(37\) 7.18163 0.0319095 0.0159548 0.999873i \(-0.494921\pi\)
0.0159548 + 0.999873i \(0.494921\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 347.052 1.32196 0.660980 0.750404i \(-0.270141\pi\)
0.660980 + 0.750404i \(0.270141\pi\)
\(42\) 0 0
\(43\) −189.318 −0.671413 −0.335707 0.941967i \(-0.608975\pi\)
−0.335707 + 0.941967i \(0.608975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 585.443 1.81693 0.908464 0.417963i \(-0.137256\pi\)
0.908464 + 0.417963i \(0.137256\pi\)
\(48\) 0 0
\(49\) 104.142 0.303622
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −77.2791 −0.200285 −0.100143 0.994973i \(-0.531930\pi\)
−0.100143 + 0.994973i \(0.531930\pi\)
\(54\) 0 0
\(55\) −337.147 −0.826561
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 200.790 0.443062 0.221531 0.975153i \(-0.428895\pi\)
0.221531 + 0.975153i \(0.428895\pi\)
\(60\) 0 0
\(61\) 681.137 1.42968 0.714841 0.699287i \(-0.246499\pi\)
0.714841 + 0.699287i \(0.246499\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.7746 0.0243769
\(66\) 0 0
\(67\) 810.751 1.47834 0.739172 0.673517i \(-0.235217\pi\)
0.739172 + 0.673517i \(0.235217\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 515.476 0.861631 0.430816 0.902440i \(-0.358226\pi\)
0.430816 + 0.902440i \(0.358226\pi\)
\(72\) 0 0
\(73\) 385.577 0.618197 0.309099 0.951030i \(-0.399973\pi\)
0.309099 + 0.951030i \(0.399973\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1425.84 −2.11026
\(78\) 0 0
\(79\) 209.405 0.298226 0.149113 0.988820i \(-0.452358\pi\)
0.149113 + 0.988820i \(0.452358\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 887.189 1.17327 0.586637 0.809850i \(-0.300452\pi\)
0.586637 + 0.809850i \(0.300452\pi\)
\(84\) 0 0
\(85\) −630.572 −0.804648
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 548.890 0.653733 0.326867 0.945071i \(-0.394007\pi\)
0.326867 + 0.945071i \(0.394007\pi\)
\(90\) 0 0
\(91\) 54.0258 0.0622357
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 517.696 0.559100
\(96\) 0 0
\(97\) −1523.96 −1.59520 −0.797600 0.603186i \(-0.793897\pi\)
−0.797600 + 0.603186i \(0.793897\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −347.825 −0.342672 −0.171336 0.985213i \(-0.554808\pi\)
−0.171336 + 0.985213i \(0.554808\pi\)
\(102\) 0 0
\(103\) 1955.17 1.87037 0.935187 0.354155i \(-0.115232\pi\)
0.935187 + 0.354155i \(0.115232\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1576.16 1.42405 0.712026 0.702153i \(-0.247778\pi\)
0.712026 + 0.702153i \(0.247778\pi\)
\(108\) 0 0
\(109\) −158.232 −0.139045 −0.0695224 0.997580i \(-0.522148\pi\)
−0.0695224 + 0.997580i \(0.522148\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −583.856 −0.486058 −0.243029 0.970019i \(-0.578141\pi\)
−0.243029 + 0.970019i \(0.578141\pi\)
\(114\) 0 0
\(115\) −1002.17 −0.812633
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2666.78 −2.05431
\(120\) 0 0
\(121\) 3215.72 2.41602
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2572.68 −1.79754 −0.898772 0.438416i \(-0.855540\pi\)
−0.898772 + 0.438416i \(0.855540\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1793.93 1.19646 0.598229 0.801325i \(-0.295871\pi\)
0.598229 + 0.801325i \(0.295871\pi\)
\(132\) 0 0
\(133\) 2189.41 1.42742
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1729.59 1.07860 0.539302 0.842112i \(-0.318688\pi\)
0.539302 + 0.842112i \(0.318688\pi\)
\(138\) 0 0
\(139\) −690.713 −0.421479 −0.210739 0.977542i \(-0.567587\pi\)
−0.210739 + 0.977542i \(0.567587\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −172.277 −0.100745
\(144\) 0 0
\(145\) −357.108 −0.204525
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2876.27 −1.58143 −0.790715 0.612185i \(-0.790291\pi\)
−0.790715 + 0.612185i \(0.790291\pi\)
\(150\) 0 0
\(151\) −1204.10 −0.648930 −0.324465 0.945898i \(-0.605184\pi\)
−0.324465 + 0.945898i \(0.605184\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 790.650 0.409720
\(156\) 0 0
\(157\) −790.488 −0.401833 −0.200917 0.979608i \(-0.564392\pi\)
−0.200917 + 0.979608i \(0.564392\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4238.32 −2.07470
\(162\) 0 0
\(163\) 2202.61 1.05842 0.529209 0.848492i \(-0.322489\pi\)
0.529209 + 0.848492i \(0.322489\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1833.20 −0.849445 −0.424722 0.905324i \(-0.639628\pi\)
−0.424722 + 0.905324i \(0.639628\pi\)
\(168\) 0 0
\(169\) −2190.47 −0.997029
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2686.62 −1.18069 −0.590347 0.807150i \(-0.701009\pi\)
−0.590347 + 0.807150i \(0.701009\pi\)
\(174\) 0 0
\(175\) 528.644 0.228353
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4500.41 1.87920 0.939598 0.342280i \(-0.111199\pi\)
0.939598 + 0.342280i \(0.111199\pi\)
\(180\) 0 0
\(181\) −1600.82 −0.657390 −0.328695 0.944436i \(-0.606609\pi\)
−0.328695 + 0.944436i \(0.606609\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 35.9081 0.0142704
\(186\) 0 0
\(187\) 8503.81 3.32546
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 862.280 0.326662 0.163331 0.986571i \(-0.447776\pi\)
0.163331 + 0.986571i \(0.447776\pi\)
\(192\) 0 0
\(193\) −768.319 −0.286554 −0.143277 0.989683i \(-0.545764\pi\)
−0.143277 + 0.989683i \(0.545764\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3090.06 1.11755 0.558776 0.829318i \(-0.311271\pi\)
0.558776 + 0.829318i \(0.311271\pi\)
\(198\) 0 0
\(199\) −3255.41 −1.15965 −0.579824 0.814742i \(-0.696879\pi\)
−0.579824 + 0.814742i \(0.696879\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1510.26 −0.522165
\(204\) 0 0
\(205\) 1735.26 0.591198
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6981.59 −2.31065
\(210\) 0 0
\(211\) 2891.42 0.943383 0.471691 0.881764i \(-0.343644\pi\)
0.471691 + 0.881764i \(0.343644\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −946.592 −0.300265
\(216\) 0 0
\(217\) 3343.78 1.04604
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −322.213 −0.0980742
\(222\) 0 0
\(223\) 1345.31 0.403983 0.201992 0.979387i \(-0.435259\pi\)
0.201992 + 0.979387i \(0.435259\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3643.36 1.06528 0.532639 0.846343i \(-0.321200\pi\)
0.532639 + 0.846343i \(0.321200\pi\)
\(228\) 0 0
\(229\) 314.732 0.0908213 0.0454106 0.998968i \(-0.485540\pi\)
0.0454106 + 0.998968i \(0.485540\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1962.28 −0.551732 −0.275866 0.961196i \(-0.588965\pi\)
−0.275866 + 0.961196i \(0.588965\pi\)
\(234\) 0 0
\(235\) 2927.21 0.812555
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3128.78 0.846795 0.423397 0.905944i \(-0.360837\pi\)
0.423397 + 0.905944i \(0.360837\pi\)
\(240\) 0 0
\(241\) 1538.75 0.411284 0.205642 0.978627i \(-0.434072\pi\)
0.205642 + 0.978627i \(0.434072\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 520.712 0.135784
\(246\) 0 0
\(247\) 264.535 0.0681456
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5699.82 −1.43334 −0.716672 0.697410i \(-0.754336\pi\)
−0.716672 + 0.697410i \(0.754336\pi\)
\(252\) 0 0
\(253\) 13515.1 3.35845
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1406.28 −0.341327 −0.170664 0.985329i \(-0.554591\pi\)
−0.170664 + 0.985329i \(0.554591\pi\)
\(258\) 0 0
\(259\) 151.861 0.0364331
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7658.46 1.79559 0.897796 0.440412i \(-0.145167\pi\)
0.897796 + 0.440412i \(0.145167\pi\)
\(264\) 0 0
\(265\) −386.396 −0.0895702
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −808.242 −0.183195 −0.0915974 0.995796i \(-0.529197\pi\)
−0.0915974 + 0.995796i \(0.529197\pi\)
\(270\) 0 0
\(271\) 4146.22 0.929390 0.464695 0.885471i \(-0.346164\pi\)
0.464695 + 0.885471i \(0.346164\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1685.73 −0.369649
\(276\) 0 0
\(277\) 1948.21 0.422588 0.211294 0.977423i \(-0.432232\pi\)
0.211294 + 0.977423i \(0.432232\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5888.41 1.25008 0.625041 0.780592i \(-0.285082\pi\)
0.625041 + 0.780592i \(0.285082\pi\)
\(282\) 0 0
\(283\) −826.285 −0.173560 −0.0867801 0.996227i \(-0.527658\pi\)
−0.0867801 + 0.996227i \(0.527658\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7338.66 1.50936
\(288\) 0 0
\(289\) 10991.8 2.23729
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1817.15 −0.362318 −0.181159 0.983454i \(-0.557985\pi\)
−0.181159 + 0.983454i \(0.557985\pi\)
\(294\) 0 0
\(295\) 1003.95 0.198143
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −512.094 −0.0990474
\(300\) 0 0
\(301\) −4003.28 −0.766595
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3405.68 0.639373
\(306\) 0 0
\(307\) 7055.67 1.31169 0.655845 0.754896i \(-0.272313\pi\)
0.655845 + 0.754896i \(0.272313\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1144.46 −0.208669 −0.104335 0.994542i \(-0.533271\pi\)
−0.104335 + 0.994542i \(0.533271\pi\)
\(312\) 0 0
\(313\) −5868.73 −1.05981 −0.529905 0.848057i \(-0.677772\pi\)
−0.529905 + 0.848057i \(0.677772\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9215.18 1.63273 0.816366 0.577534i \(-0.195985\pi\)
0.816366 + 0.577534i \(0.195985\pi\)
\(318\) 0 0
\(319\) 4815.91 0.845264
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −13057.8 −2.24939
\(324\) 0 0
\(325\) 63.8732 0.0109017
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12379.6 2.07450
\(330\) 0 0
\(331\) −4947.10 −0.821502 −0.410751 0.911748i \(-0.634733\pi\)
−0.410751 + 0.911748i \(0.634733\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4053.76 0.661135
\(336\) 0 0
\(337\) 1879.56 0.303816 0.151908 0.988395i \(-0.451458\pi\)
0.151908 + 0.988395i \(0.451458\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10662.6 −1.69329
\(342\) 0 0
\(343\) −5050.82 −0.795098
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2086.38 −0.322775 −0.161388 0.986891i \(-0.551597\pi\)
−0.161388 + 0.986891i \(0.551597\pi\)
\(348\) 0 0
\(349\) 7986.00 1.22487 0.612437 0.790519i \(-0.290189\pi\)
0.612437 + 0.790519i \(0.290189\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1443.39 −0.217631 −0.108815 0.994062i \(-0.534706\pi\)
−0.108815 + 0.994062i \(0.534706\pi\)
\(354\) 0 0
\(355\) 2577.38 0.385333
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9598.40 −1.41110 −0.705549 0.708661i \(-0.749300\pi\)
−0.705549 + 0.708661i \(0.749300\pi\)
\(360\) 0 0
\(361\) 3861.37 0.562964
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1927.89 0.276466
\(366\) 0 0
\(367\) −1237.82 −0.176059 −0.0880296 0.996118i \(-0.528057\pi\)
−0.0880296 + 0.996118i \(0.528057\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1634.12 −0.228678
\(372\) 0 0
\(373\) −3832.31 −0.531982 −0.265991 0.963976i \(-0.585699\pi\)
−0.265991 + 0.963976i \(0.585699\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −182.477 −0.0249285
\(378\) 0 0
\(379\) −8252.24 −1.11844 −0.559220 0.829019i \(-0.688899\pi\)
−0.559220 + 0.829019i \(0.688899\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5080.20 0.677771 0.338885 0.940828i \(-0.389950\pi\)
0.338885 + 0.940828i \(0.389950\pi\)
\(384\) 0 0
\(385\) −7129.22 −0.943737
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7382.79 0.962268 0.481134 0.876647i \(-0.340225\pi\)
0.481134 + 0.876647i \(0.340225\pi\)
\(390\) 0 0
\(391\) 25277.6 3.26942
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1047.02 0.133371
\(396\) 0 0
\(397\) 7108.65 0.898673 0.449336 0.893363i \(-0.351660\pi\)
0.449336 + 0.893363i \(0.351660\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5347.35 0.665920 0.332960 0.942941i \(-0.391953\pi\)
0.332960 + 0.942941i \(0.391953\pi\)
\(402\) 0 0
\(403\) 404.011 0.0499385
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −484.253 −0.0589767
\(408\) 0 0
\(409\) −223.839 −0.0270615 −0.0135307 0.999908i \(-0.504307\pi\)
−0.0135307 + 0.999908i \(0.504307\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4245.86 0.505872
\(414\) 0 0
\(415\) 4435.95 0.524704
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −384.631 −0.0448459 −0.0224230 0.999749i \(-0.507138\pi\)
−0.0224230 + 0.999749i \(0.507138\pi\)
\(420\) 0 0
\(421\) 4125.76 0.477618 0.238809 0.971067i \(-0.423243\pi\)
0.238809 + 0.971067i \(0.423243\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3152.86 −0.359850
\(426\) 0 0
\(427\) 14403.1 1.63236
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −457.020 −0.0510762 −0.0255381 0.999674i \(-0.508130\pi\)
−0.0255381 + 0.999674i \(0.508130\pi\)
\(432\) 0 0
\(433\) −11085.8 −1.23037 −0.615187 0.788382i \(-0.710919\pi\)
−0.615187 + 0.788382i \(0.710919\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20752.8 −2.27172
\(438\) 0 0
\(439\) 4082.70 0.443865 0.221933 0.975062i \(-0.428764\pi\)
0.221933 + 0.975062i \(0.428764\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14975.7 −1.60613 −0.803067 0.595888i \(-0.796800\pi\)
−0.803067 + 0.595888i \(0.796800\pi\)
\(444\) 0 0
\(445\) 2744.45 0.292358
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1863.53 −0.195870 −0.0979349 0.995193i \(-0.531224\pi\)
−0.0979349 + 0.995193i \(0.531224\pi\)
\(450\) 0 0
\(451\) −23401.5 −2.44331
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 270.129 0.0278326
\(456\) 0 0
\(457\) −4029.90 −0.412496 −0.206248 0.978500i \(-0.566125\pi\)
−0.206248 + 0.978500i \(0.566125\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9320.82 0.941679 0.470839 0.882219i \(-0.343951\pi\)
0.470839 + 0.882219i \(0.343951\pi\)
\(462\) 0 0
\(463\) −9948.81 −0.998619 −0.499309 0.866424i \(-0.666413\pi\)
−0.499309 + 0.866424i \(0.666413\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7946.35 −0.787395 −0.393697 0.919240i \(-0.628804\pi\)
−0.393697 + 0.919240i \(0.628804\pi\)
\(468\) 0 0
\(469\) 17143.9 1.68792
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12765.6 1.24094
\(474\) 0 0
\(475\) 2588.48 0.250037
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4166.01 −0.397390 −0.198695 0.980061i \(-0.563670\pi\)
−0.198695 + 0.980061i \(0.563670\pi\)
\(480\) 0 0
\(481\) 18.3485 0.00173934
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7619.79 −0.713395
\(486\) 0 0
\(487\) −4662.27 −0.433815 −0.216907 0.976192i \(-0.569597\pi\)
−0.216907 + 0.976192i \(0.569597\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9609.89 0.883275 0.441638 0.897194i \(-0.354398\pi\)
0.441638 + 0.897194i \(0.354398\pi\)
\(492\) 0 0
\(493\) 9007.28 0.822855
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10900.1 0.983778
\(498\) 0 0
\(499\) 5439.00 0.487942 0.243971 0.969782i \(-0.421550\pi\)
0.243971 + 0.969782i \(0.421550\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10407.2 −0.922537 −0.461268 0.887261i \(-0.652605\pi\)
−0.461268 + 0.887261i \(0.652605\pi\)
\(504\) 0 0
\(505\) −1739.13 −0.153248
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8982.15 0.782174 0.391087 0.920354i \(-0.372099\pi\)
0.391087 + 0.920354i \(0.372099\pi\)
\(510\) 0 0
\(511\) 8153.32 0.705835
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9775.84 0.836456
\(516\) 0 0
\(517\) −39476.0 −3.35813
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 871.580 0.0732910 0.0366455 0.999328i \(-0.488333\pi\)
0.0366455 + 0.999328i \(0.488333\pi\)
\(522\) 0 0
\(523\) −16215.9 −1.35578 −0.677888 0.735165i \(-0.737105\pi\)
−0.677888 + 0.735165i \(0.737105\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19942.5 −1.64840
\(528\) 0 0
\(529\) 28006.7 2.30186
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 886.691 0.0720579
\(534\) 0 0
\(535\) 7880.82 0.636856
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7022.26 −0.561169
\(540\) 0 0
\(541\) 7195.89 0.571859 0.285929 0.958251i \(-0.407698\pi\)
0.285929 + 0.958251i \(0.407698\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −791.161 −0.0621828
\(546\) 0 0
\(547\) 12847.2 1.00422 0.502109 0.864804i \(-0.332558\pi\)
0.502109 + 0.864804i \(0.332558\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7394.93 −0.571751
\(552\) 0 0
\(553\) 4428.02 0.340504
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1916.53 0.145792 0.0728958 0.997340i \(-0.476776\pi\)
0.0728958 + 0.997340i \(0.476776\pi\)
\(558\) 0 0
\(559\) −483.695 −0.0365977
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5124.14 −0.383582 −0.191791 0.981436i \(-0.561430\pi\)
−0.191791 + 0.981436i \(0.561430\pi\)
\(564\) 0 0
\(565\) −2919.28 −0.217372
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17498.6 1.28924 0.644620 0.764503i \(-0.277015\pi\)
0.644620 + 0.764503i \(0.277015\pi\)
\(570\) 0 0
\(571\) −9253.34 −0.678179 −0.339090 0.940754i \(-0.610119\pi\)
−0.339090 + 0.940754i \(0.610119\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5010.85 −0.363420
\(576\) 0 0
\(577\) −17420.7 −1.25690 −0.628451 0.777849i \(-0.716311\pi\)
−0.628451 + 0.777849i \(0.716311\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 18760.3 1.33960
\(582\) 0 0
\(583\) 5210.88 0.370176
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7839.38 0.551219 0.275610 0.961270i \(-0.411120\pi\)
0.275610 + 0.961270i \(0.411120\pi\)
\(588\) 0 0
\(589\) 16372.7 1.14537
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1611.03 −0.111564 −0.0557818 0.998443i \(-0.517765\pi\)
−0.0557818 + 0.998443i \(0.517765\pi\)
\(594\) 0 0
\(595\) −13333.9 −0.918717
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 17274.0 1.17829 0.589146 0.808026i \(-0.299464\pi\)
0.589146 + 0.808026i \(0.299464\pi\)
\(600\) 0 0
\(601\) −26908.5 −1.82632 −0.913161 0.407599i \(-0.866366\pi\)
−0.913161 + 0.407599i \(0.866366\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16078.6 1.08048
\(606\) 0 0
\(607\) 5500.73 0.367822 0.183911 0.982943i \(-0.441124\pi\)
0.183911 + 0.982943i \(0.441124\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1495.76 0.0990379
\(612\) 0 0
\(613\) 17362.0 1.14396 0.571978 0.820269i \(-0.306176\pi\)
0.571978 + 0.820269i \(0.306176\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2946.26 −0.192240 −0.0961198 0.995370i \(-0.530643\pi\)
−0.0961198 + 0.995370i \(0.530643\pi\)
\(618\) 0 0
\(619\) 15311.6 0.994223 0.497112 0.867687i \(-0.334394\pi\)
0.497112 + 0.867687i \(0.334394\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11606.7 0.746408
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −905.706 −0.0574132
\(630\) 0 0
\(631\) −1067.12 −0.0673241 −0.0336621 0.999433i \(-0.510717\pi\)
−0.0336621 + 0.999433i \(0.510717\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12863.4 −0.803886
\(636\) 0 0
\(637\) 266.076 0.0165500
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8977.71 −0.553195 −0.276598 0.960986i \(-0.589207\pi\)
−0.276598 + 0.960986i \(0.589207\pi\)
\(642\) 0 0
\(643\) 15935.5 0.977346 0.488673 0.872467i \(-0.337481\pi\)
0.488673 + 0.872467i \(0.337481\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21284.1 −1.29330 −0.646648 0.762789i \(-0.723830\pi\)
−0.646648 + 0.762789i \(0.723830\pi\)
\(648\) 0 0
\(649\) −13539.2 −0.818889
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8173.21 −0.489805 −0.244902 0.969548i \(-0.578756\pi\)
−0.244902 + 0.969548i \(0.578756\pi\)
\(654\) 0 0
\(655\) 8969.63 0.535072
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32350.6 1.91229 0.956145 0.292895i \(-0.0946186\pi\)
0.956145 + 0.292895i \(0.0946186\pi\)
\(660\) 0 0
\(661\) 26775.6 1.57557 0.787783 0.615953i \(-0.211229\pi\)
0.787783 + 0.615953i \(0.211229\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10947.1 0.638360
\(666\) 0 0
\(667\) 14315.3 0.831020
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −45928.6 −2.64241
\(672\) 0 0
\(673\) 13448.7 0.770297 0.385149 0.922855i \(-0.374150\pi\)
0.385149 + 0.922855i \(0.374150\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20767.3 1.17896 0.589478 0.807784i \(-0.299333\pi\)
0.589478 + 0.807784i \(0.299333\pi\)
\(678\) 0 0
\(679\) −32225.2 −1.82134
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11894.6 0.666373 0.333187 0.942861i \(-0.391876\pi\)
0.333187 + 0.942861i \(0.391876\pi\)
\(684\) 0 0
\(685\) 8647.94 0.482366
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −197.443 −0.0109172
\(690\) 0 0
\(691\) −14871.9 −0.818747 −0.409374 0.912367i \(-0.634253\pi\)
−0.409374 + 0.912367i \(0.634253\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3453.57 −0.188491
\(696\) 0 0
\(697\) −43768.2 −2.37853
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2423.29 −0.130566 −0.0652828 0.997867i \(-0.520795\pi\)
−0.0652828 + 0.997867i \(0.520795\pi\)
\(702\) 0 0
\(703\) 743.580 0.0398928
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7355.02 −0.391250
\(708\) 0 0
\(709\) 8971.88 0.475241 0.237621 0.971358i \(-0.423632\pi\)
0.237621 + 0.971358i \(0.423632\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −31694.6 −1.66476
\(714\) 0 0
\(715\) −861.386 −0.0450545
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18757.9 0.972951 0.486475 0.873694i \(-0.338282\pi\)
0.486475 + 0.873694i \(0.338282\pi\)
\(720\) 0 0
\(721\) 41343.5 2.13552
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1785.54 −0.0914665
\(726\) 0 0
\(727\) 14205.6 0.724700 0.362350 0.932042i \(-0.381974\pi\)
0.362350 + 0.932042i \(0.381974\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 23875.8 1.20804
\(732\) 0 0
\(733\) 23776.1 1.19808 0.599038 0.800720i \(-0.295550\pi\)
0.599038 + 0.800720i \(0.295550\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −54668.4 −2.73234
\(738\) 0 0
\(739\) −14175.5 −0.705623 −0.352811 0.935694i \(-0.614774\pi\)
−0.352811 + 0.935694i \(0.614774\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7564.68 −0.373514 −0.186757 0.982406i \(-0.559798\pi\)
−0.186757 + 0.982406i \(0.559798\pi\)
\(744\) 0 0
\(745\) −14381.3 −0.707237
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 33329.2 1.62593
\(750\) 0 0
\(751\) −18602.1 −0.903863 −0.451932 0.892053i \(-0.649265\pi\)
−0.451932 + 0.892053i \(0.649265\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6020.51 −0.290210
\(756\) 0 0
\(757\) −4665.12 −0.223985 −0.111993 0.993709i \(-0.535723\pi\)
−0.111993 + 0.993709i \(0.535723\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36063.6 −1.71788 −0.858938 0.512079i \(-0.828875\pi\)
−0.858938 + 0.512079i \(0.828875\pi\)
\(762\) 0 0
\(763\) −3345.94 −0.158756
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 513.005 0.0241506
\(768\) 0 0
\(769\) −28921.5 −1.35623 −0.678113 0.734958i \(-0.737202\pi\)
−0.678113 + 0.734958i \(0.737202\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −25489.9 −1.18604 −0.593019 0.805188i \(-0.702064\pi\)
−0.593019 + 0.805188i \(0.702064\pi\)
\(774\) 0 0
\(775\) 3953.25 0.183232
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 35933.5 1.65270
\(780\) 0 0
\(781\) −34758.3 −1.59251
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3952.44 −0.179705
\(786\) 0 0
\(787\) −3627.06 −0.164283 −0.0821415 0.996621i \(-0.526176\pi\)
−0.0821415 + 0.996621i \(0.526176\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12346.1 −0.554962
\(792\) 0 0
\(793\) 1740.26 0.0779297
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −15571.1 −0.692041 −0.346021 0.938227i \(-0.612467\pi\)
−0.346021 + 0.938227i \(0.612467\pi\)
\(798\) 0 0
\(799\) −73832.7 −3.26910
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −25999.2 −1.14258
\(804\) 0 0
\(805\) −21191.6 −0.927834
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −7352.63 −0.319536 −0.159768 0.987155i \(-0.551075\pi\)
−0.159768 + 0.987155i \(0.551075\pi\)
\(810\) 0 0
\(811\) −19743.8 −0.854869 −0.427434 0.904046i \(-0.640583\pi\)
−0.427434 + 0.904046i \(0.640583\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11013.1 0.473339
\(816\) 0 0
\(817\) −19601.9 −0.839392
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −43315.1 −1.84130 −0.920651 0.390387i \(-0.872341\pi\)
−0.920651 + 0.390387i \(0.872341\pi\)
\(822\) 0 0
\(823\) 23632.2 1.00093 0.500466 0.865756i \(-0.333162\pi\)
0.500466 + 0.865756i \(0.333162\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −24109.3 −1.01374 −0.506869 0.862023i \(-0.669197\pi\)
−0.506869 + 0.862023i \(0.669197\pi\)
\(828\) 0 0
\(829\) 1330.02 0.0557218 0.0278609 0.999612i \(-0.491130\pi\)
0.0278609 + 0.999612i \(0.491130\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −13133.8 −0.546292
\(834\) 0 0
\(835\) −9166.00 −0.379883
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 16823.3 0.692258 0.346129 0.938187i \(-0.387496\pi\)
0.346129 + 0.938187i \(0.387496\pi\)
\(840\) 0 0
\(841\) −19288.0 −0.790847
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10952.4 −0.445885
\(846\) 0 0
\(847\) 67998.8 2.75852
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1439.44 −0.0579829
\(852\) 0 0
\(853\) −11732.7 −0.470951 −0.235475 0.971880i \(-0.575665\pi\)
−0.235475 + 0.971880i \(0.575665\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25843.7 1.03011 0.515055 0.857157i \(-0.327771\pi\)
0.515055 + 0.857157i \(0.327771\pi\)
\(858\) 0 0
\(859\) −21463.9 −0.852549 −0.426275 0.904594i \(-0.640174\pi\)
−0.426275 + 0.904594i \(0.640174\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11980.8 0.472575 0.236287 0.971683i \(-0.424069\pi\)
0.236287 + 0.971683i \(0.424069\pi\)
\(864\) 0 0
\(865\) −13433.1 −0.528022
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −14120.0 −0.551196
\(870\) 0 0
\(871\) 2071.41 0.0805822
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2643.22 0.102122
\(876\) 0 0
\(877\) 3662.20 0.141008 0.0705038 0.997512i \(-0.477539\pi\)
0.0705038 + 0.997512i \(0.477539\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 27016.4 1.03315 0.516575 0.856242i \(-0.327207\pi\)
0.516575 + 0.856242i \(0.327207\pi\)
\(882\) 0 0
\(883\) −24437.6 −0.931359 −0.465679 0.884954i \(-0.654190\pi\)
−0.465679 + 0.884954i \(0.654190\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 35067.7 1.32746 0.663731 0.747971i \(-0.268972\pi\)
0.663731 + 0.747971i \(0.268972\pi\)
\(888\) 0 0
\(889\) −54401.2 −2.05237
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 60616.3 2.27150
\(894\) 0 0
\(895\) 22502.0 0.840402
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11293.9 −0.418990
\(900\) 0 0
\(901\) 9746.01 0.360363
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8004.08 −0.293994
\(906\) 0 0
\(907\) 29567.8 1.08245 0.541225 0.840878i \(-0.317961\pi\)
0.541225 + 0.840878i \(0.317961\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 37564.2 1.36614 0.683072 0.730351i \(-0.260644\pi\)
0.683072 + 0.730351i \(0.260644\pi\)
\(912\) 0 0
\(913\) −59822.6 −2.16850
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 37933.9 1.36607
\(918\) 0 0
\(919\) −42968.8 −1.54234 −0.771170 0.636629i \(-0.780328\pi\)
−0.771170 + 0.636629i \(0.780328\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1317.00 0.0469661
\(924\) 0 0
\(925\) 179.541 0.00638191
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 25931.4 0.915804 0.457902 0.889003i \(-0.348601\pi\)
0.457902 + 0.889003i \(0.348601\pi\)
\(930\) 0 0
\(931\) 10782.8 0.379584
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 42519.1 1.48719
\(936\) 0 0
\(937\) −20940.0 −0.730075 −0.365037 0.930993i \(-0.618944\pi\)
−0.365037 + 0.930993i \(0.618944\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8528.77 −0.295462 −0.147731 0.989028i \(-0.547197\pi\)
−0.147731 + 0.989028i \(0.547197\pi\)
\(942\) 0 0
\(943\) −69560.9 −2.40214
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18566.9 0.637112 0.318556 0.947904i \(-0.396802\pi\)
0.318556 + 0.947904i \(0.396802\pi\)
\(948\) 0 0
\(949\) 985.122 0.0336970
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −8866.54 −0.301380 −0.150690 0.988581i \(-0.548150\pi\)
−0.150690 + 0.988581i \(0.548150\pi\)
\(954\) 0 0
\(955\) 4311.40 0.146088
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36573.4 1.23151
\(960\) 0 0
\(961\) −4785.89 −0.160649
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3841.60 −0.128151
\(966\) 0 0
\(967\) −38880.2 −1.29297 −0.646485 0.762927i \(-0.723762\pi\)
−0.646485 + 0.762927i \(0.723762\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33655.7 1.11232 0.556161 0.831075i \(-0.312274\pi\)
0.556161 + 0.831075i \(0.312274\pi\)
\(972\) 0 0
\(973\) −14605.6 −0.481228
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6269.02 0.205285 0.102643 0.994718i \(-0.467270\pi\)
0.102643 + 0.994718i \(0.467270\pi\)
\(978\) 0 0
\(979\) −37011.3 −1.20826
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 39772.2 1.29048 0.645238 0.763982i \(-0.276758\pi\)
0.645238 + 0.763982i \(0.276758\pi\)
\(984\) 0 0
\(985\) 15450.3 0.499785
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 37945.8 1.22003
\(990\) 0 0
\(991\) 8799.28 0.282057 0.141028 0.990006i \(-0.454959\pi\)
0.141028 + 0.990006i \(0.454959\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16277.1 −0.518610
\(996\) 0 0
\(997\) 42122.5 1.33805 0.669024 0.743241i \(-0.266713\pi\)
0.669024 + 0.743241i \(0.266713\pi\)
\(998\) 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 2160.4.a.bt.1.3 3
3.2 odd 2 2160.4.a.bl.1.3 3
4.3 odd 2 1080.4.a.i.1.1 yes 3
12.11 even 2 1080.4.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.c.1.1 3 12.11 even 2
1080.4.a.i.1.1 yes 3 4.3 odd 2
2160.4.a.bl.1.3 3 3.2 odd 2
2160.4.a.bt.1.3 3 1.1 even 1 trivial