Properties

Label 5733.2.a.x.1.1
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.7782354788\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.34292 q^{2} +3.48929 q^{4} -1.34292 q^{5} -3.48929 q^{8} +O(q^{10})\) \(q-2.34292 q^{2} +3.48929 q^{4} -1.34292 q^{5} -3.48929 q^{8} +3.14637 q^{10} -1.14637 q^{11} -1.00000 q^{13} +1.19656 q^{16} +5.83221 q^{17} +3.34292 q^{19} -4.68585 q^{20} +2.68585 q^{22} +3.17513 q^{23} -3.19656 q^{25} +2.34292 q^{26} -10.4893 q^{29} -1.63565 q^{31} +4.17513 q^{32} -13.6644 q^{34} +8.51806 q^{37} -7.83221 q^{38} +4.68585 q^{40} -0.292731 q^{41} -8.15371 q^{43} -4.00000 q^{44} -7.43910 q^{46} -10.6142 q^{47} +7.48929 q^{50} -3.48929 q^{52} +0.782020 q^{53} +1.53948 q^{55} +24.5756 q^{58} +12.6430 q^{59} +2.00000 q^{61} +3.83221 q^{62} -12.1751 q^{64} +1.34292 q^{65} -6.10038 q^{67} +20.3503 q^{68} -1.53948 q^{71} +15.3001 q^{73} -19.9572 q^{74} +11.6644 q^{76} +0.882404 q^{79} -1.60688 q^{80} +0.685846 q^{82} -12.1292 q^{83} -7.83221 q^{85} +19.1035 q^{86} +4.00000 q^{88} +5.73604 q^{89} +11.0790 q^{92} +24.8683 q^{94} -4.48929 q^{95} +5.34292 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + O(q^{10}) \) \( 3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + 8q^{10} - 2q^{11} - 3q^{13} - q^{16} + 4q^{17} + 4q^{19} - 2q^{20} - 4q^{22} - 10q^{23} - 5q^{25} + q^{26} - 24q^{29} + 4q^{31} - 7q^{32} - 14q^{34} - 10q^{38} + 2q^{40} + 2q^{41} + 10q^{43} - 12q^{44} - 18q^{46} - 8q^{47} + 15q^{50} - 3q^{52} - 8q^{53} - 6q^{55} + 12q^{58} - 4q^{59} + 6q^{61} - 2q^{62} - 17q^{64} - 2q^{65} - 12q^{67} + 22q^{68} + 6q^{71} + 10q^{73} - 30q^{74} + 8q^{76} - 14q^{79} - 14q^{80} - 10q^{82} - 12q^{83} - 10q^{85} + 26q^{86} + 12q^{88} + 2q^{89} + 12q^{92} + 10q^{94} - 6q^{95} + 10q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.34292 −1.65670 −0.828348 0.560213i \(-0.810719\pi\)
−0.828348 + 0.560213i \(0.810719\pi\)
\(3\) 0 0
\(4\) 3.48929 1.74464
\(5\) −1.34292 −0.600573 −0.300287 0.953849i \(-0.597082\pi\)
−0.300287 + 0.953849i \(0.597082\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −3.48929 −1.23365
\(9\) 0 0
\(10\) 3.14637 0.994968
\(11\) −1.14637 −0.345642 −0.172821 0.984953i \(-0.555288\pi\)
−0.172821 + 0.984953i \(0.555288\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 1.19656 0.299139
\(17\) 5.83221 1.41452 0.707260 0.706954i \(-0.249931\pi\)
0.707260 + 0.706954i \(0.249931\pi\)
\(18\) 0 0
\(19\) 3.34292 0.766919 0.383460 0.923558i \(-0.374733\pi\)
0.383460 + 0.923558i \(0.374733\pi\)
\(20\) −4.68585 −1.04779
\(21\) 0 0
\(22\) 2.68585 0.572624
\(23\) 3.17513 0.662061 0.331031 0.943620i \(-0.392604\pi\)
0.331031 + 0.943620i \(0.392604\pi\)
\(24\) 0 0
\(25\) −3.19656 −0.639312
\(26\) 2.34292 0.459485
\(27\) 0 0
\(28\) 0 0
\(29\) −10.4893 −1.94781 −0.973906 0.226952i \(-0.927124\pi\)
−0.973906 + 0.226952i \(0.927124\pi\)
\(30\) 0 0
\(31\) −1.63565 −0.293772 −0.146886 0.989153i \(-0.546925\pi\)
−0.146886 + 0.989153i \(0.546925\pi\)
\(32\) 4.17513 0.738067
\(33\) 0 0
\(34\) −13.6644 −2.34343
\(35\) 0 0
\(36\) 0 0
\(37\) 8.51806 1.40036 0.700180 0.713966i \(-0.253103\pi\)
0.700180 + 0.713966i \(0.253103\pi\)
\(38\) −7.83221 −1.27055
\(39\) 0 0
\(40\) 4.68585 0.740897
\(41\) −0.292731 −0.0457169 −0.0228584 0.999739i \(-0.507277\pi\)
−0.0228584 + 0.999739i \(0.507277\pi\)
\(42\) 0 0
\(43\) −8.15371 −1.24343 −0.621715 0.783244i \(-0.713564\pi\)
−0.621715 + 0.783244i \(0.713564\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −7.43910 −1.09683
\(47\) −10.6142 −1.54824 −0.774122 0.633036i \(-0.781809\pi\)
−0.774122 + 0.633036i \(0.781809\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 7.48929 1.05915
\(51\) 0 0
\(52\) −3.48929 −0.483877
\(53\) 0.782020 0.107419 0.0537093 0.998557i \(-0.482896\pi\)
0.0537093 + 0.998557i \(0.482896\pi\)
\(54\) 0 0
\(55\) 1.53948 0.207584
\(56\) 0 0
\(57\) 0 0
\(58\) 24.5756 3.22693
\(59\) 12.6430 1.64598 0.822989 0.568057i \(-0.192305\pi\)
0.822989 + 0.568057i \(0.192305\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 3.83221 0.486691
\(63\) 0 0
\(64\) −12.1751 −1.52189
\(65\) 1.34292 0.166569
\(66\) 0 0
\(67\) −6.10038 −0.745281 −0.372640 0.927976i \(-0.621547\pi\)
−0.372640 + 0.927976i \(0.621547\pi\)
\(68\) 20.3503 2.46783
\(69\) 0 0
\(70\) 0 0
\(71\) −1.53948 −0.182703 −0.0913514 0.995819i \(-0.529119\pi\)
−0.0913514 + 0.995819i \(0.529119\pi\)
\(72\) 0 0
\(73\) 15.3001 1.79074 0.895369 0.445324i \(-0.146912\pi\)
0.895369 + 0.445324i \(0.146912\pi\)
\(74\) −19.9572 −2.31997
\(75\) 0 0
\(76\) 11.6644 1.33800
\(77\) 0 0
\(78\) 0 0
\(79\) 0.882404 0.0992782 0.0496391 0.998767i \(-0.484193\pi\)
0.0496391 + 0.998767i \(0.484193\pi\)
\(80\) −1.60688 −0.179655
\(81\) 0 0
\(82\) 0.685846 0.0757390
\(83\) −12.1292 −1.33135 −0.665674 0.746243i \(-0.731856\pi\)
−0.665674 + 0.746243i \(0.731856\pi\)
\(84\) 0 0
\(85\) −7.83221 −0.849523
\(86\) 19.1035 2.05999
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 5.73604 0.608019 0.304009 0.952669i \(-0.401675\pi\)
0.304009 + 0.952669i \(0.401675\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 11.0790 1.15506
\(93\) 0 0
\(94\) 24.8683 2.56497
\(95\) −4.48929 −0.460591
\(96\) 0 0
\(97\) 5.34292 0.542492 0.271246 0.962510i \(-0.412564\pi\)
0.271246 + 0.962510i \(0.412564\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −11.1537 −1.11537
\(101\) 11.1464 1.10910 0.554552 0.832149i \(-0.312889\pi\)
0.554552 + 0.832149i \(0.312889\pi\)
\(102\) 0 0
\(103\) 3.41454 0.336444 0.168222 0.985749i \(-0.446197\pi\)
0.168222 + 0.985749i \(0.446197\pi\)
\(104\) 3.48929 0.342153
\(105\) 0 0
\(106\) −1.83221 −0.177960
\(107\) −4.97858 −0.481297 −0.240649 0.970612i \(-0.577360\pi\)
−0.240649 + 0.970612i \(0.577360\pi\)
\(108\) 0 0
\(109\) −13.4966 −1.29274 −0.646372 0.763023i \(-0.723714\pi\)
−0.646372 + 0.763023i \(0.723714\pi\)
\(110\) −3.60688 −0.343903
\(111\) 0 0
\(112\) 0 0
\(113\) −16.4464 −1.54715 −0.773576 0.633704i \(-0.781534\pi\)
−0.773576 + 0.633704i \(0.781534\pi\)
\(114\) 0 0
\(115\) −4.26396 −0.397616
\(116\) −36.6002 −3.39824
\(117\) 0 0
\(118\) −29.6216 −2.72689
\(119\) 0 0
\(120\) 0 0
\(121\) −9.68585 −0.880531
\(122\) −4.68585 −0.424237
\(123\) 0 0
\(124\) −5.70727 −0.512528
\(125\) 11.0073 0.984527
\(126\) 0 0
\(127\) 12.0575 1.06993 0.534967 0.844873i \(-0.320324\pi\)
0.534967 + 0.844873i \(0.320324\pi\)
\(128\) 20.1751 1.78325
\(129\) 0 0
\(130\) −3.14637 −0.275955
\(131\) −3.66442 −0.320162 −0.160081 0.987104i \(-0.551176\pi\)
−0.160081 + 0.987104i \(0.551176\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 14.2927 1.23470
\(135\) 0 0
\(136\) −20.3503 −1.74502
\(137\) 13.1035 1.11951 0.559755 0.828658i \(-0.310895\pi\)
0.559755 + 0.828658i \(0.310895\pi\)
\(138\) 0 0
\(139\) −7.49663 −0.635856 −0.317928 0.948115i \(-0.602987\pi\)
−0.317928 + 0.948115i \(0.602987\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.60688 0.302683
\(143\) 1.14637 0.0958639
\(144\) 0 0
\(145\) 14.0863 1.16980
\(146\) −35.8469 −2.96671
\(147\) 0 0
\(148\) 29.7220 2.44313
\(149\) −2.16779 −0.177592 −0.0887961 0.996050i \(-0.528302\pi\)
−0.0887961 + 0.996050i \(0.528302\pi\)
\(150\) 0 0
\(151\) 14.9112 1.21345 0.606727 0.794910i \(-0.292482\pi\)
0.606727 + 0.794910i \(0.292482\pi\)
\(152\) −11.6644 −0.946110
\(153\) 0 0
\(154\) 0 0
\(155\) 2.19656 0.176432
\(156\) 0 0
\(157\) −22.8683 −1.82509 −0.912546 0.408975i \(-0.865886\pi\)
−0.912546 + 0.408975i \(0.865886\pi\)
\(158\) −2.06740 −0.164474
\(159\) 0 0
\(160\) −5.60688 −0.443263
\(161\) 0 0
\(162\) 0 0
\(163\) 7.07896 0.554467 0.277234 0.960803i \(-0.410582\pi\)
0.277234 + 0.960803i \(0.410582\pi\)
\(164\) −1.02142 −0.0797597
\(165\) 0 0
\(166\) 28.4177 2.20564
\(167\) 2.61423 0.202295 0.101148 0.994871i \(-0.467749\pi\)
0.101148 + 0.994871i \(0.467749\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 18.3503 1.40740
\(171\) 0 0
\(172\) −28.4507 −2.16934
\(173\) 11.0031 0.836553 0.418276 0.908320i \(-0.362634\pi\)
0.418276 + 0.908320i \(0.362634\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.37169 −0.103395
\(177\) 0 0
\(178\) −13.4391 −1.00730
\(179\) −23.9614 −1.79096 −0.895478 0.445105i \(-0.853166\pi\)
−0.895478 + 0.445105i \(0.853166\pi\)
\(180\) 0 0
\(181\) −6.56090 −0.487668 −0.243834 0.969817i \(-0.578405\pi\)
−0.243834 + 0.969817i \(0.578405\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −11.0790 −0.816752
\(185\) −11.4391 −0.841019
\(186\) 0 0
\(187\) −6.68585 −0.488917
\(188\) −37.0361 −2.70114
\(189\) 0 0
\(190\) 10.5181 0.763060
\(191\) 4.39312 0.317875 0.158937 0.987289i \(-0.449193\pi\)
0.158937 + 0.987289i \(0.449193\pi\)
\(192\) 0 0
\(193\) −8.29273 −0.596924 −0.298462 0.954422i \(-0.596474\pi\)
−0.298462 + 0.954422i \(0.596474\pi\)
\(194\) −12.5181 −0.898744
\(195\) 0 0
\(196\) 0 0
\(197\) 3.17092 0.225919 0.112959 0.993600i \(-0.463967\pi\)
0.112959 + 0.993600i \(0.463967\pi\)
\(198\) 0 0
\(199\) 13.5970 0.963867 0.481934 0.876208i \(-0.339935\pi\)
0.481934 + 0.876208i \(0.339935\pi\)
\(200\) 11.1537 0.788687
\(201\) 0 0
\(202\) −26.1151 −1.83745
\(203\) 0 0
\(204\) 0 0
\(205\) 0.393115 0.0274564
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −1.19656 −0.0829663
\(209\) −3.83221 −0.265080
\(210\) 0 0
\(211\) 9.27552 0.638553 0.319277 0.947662i \(-0.396560\pi\)
0.319277 + 0.947662i \(0.396560\pi\)
\(212\) 2.72869 0.187407
\(213\) 0 0
\(214\) 11.6644 0.797364
\(215\) 10.9498 0.746771
\(216\) 0 0
\(217\) 0 0
\(218\) 31.6216 2.14168
\(219\) 0 0
\(220\) 5.37169 0.362159
\(221\) −5.83221 −0.392317
\(222\) 0 0
\(223\) 19.5928 1.31203 0.656016 0.754747i \(-0.272240\pi\)
0.656016 + 0.754747i \(0.272240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 38.5328 2.56316
\(227\) −19.6644 −1.30517 −0.652587 0.757714i \(-0.726316\pi\)
−0.652587 + 0.757714i \(0.726316\pi\)
\(228\) 0 0
\(229\) −7.76481 −0.513113 −0.256556 0.966529i \(-0.582588\pi\)
−0.256556 + 0.966529i \(0.582588\pi\)
\(230\) 9.99013 0.658730
\(231\) 0 0
\(232\) 36.6002 2.40292
\(233\) −12.1966 −0.799023 −0.399512 0.916728i \(-0.630820\pi\)
−0.399512 + 0.916728i \(0.630820\pi\)
\(234\) 0 0
\(235\) 14.2541 0.929835
\(236\) 44.1151 2.87165
\(237\) 0 0
\(238\) 0 0
\(239\) 10.2927 0.665781 0.332891 0.942965i \(-0.391976\pi\)
0.332891 + 0.942965i \(0.391976\pi\)
\(240\) 0 0
\(241\) 4.02877 0.259516 0.129758 0.991546i \(-0.458580\pi\)
0.129758 + 0.991546i \(0.458580\pi\)
\(242\) 22.6932 1.45877
\(243\) 0 0
\(244\) 6.97858 0.446758
\(245\) 0 0
\(246\) 0 0
\(247\) −3.34292 −0.212705
\(248\) 5.70727 0.362412
\(249\) 0 0
\(250\) −25.7894 −1.63106
\(251\) 2.91117 0.183752 0.0918758 0.995770i \(-0.470714\pi\)
0.0918758 + 0.995770i \(0.470714\pi\)
\(252\) 0 0
\(253\) −3.63986 −0.228836
\(254\) −28.2499 −1.77256
\(255\) 0 0
\(256\) −22.9185 −1.43241
\(257\) −19.5970 −1.22243 −0.611214 0.791465i \(-0.709319\pi\)
−0.611214 + 0.791465i \(0.709319\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4.68585 0.290604
\(261\) 0 0
\(262\) 8.58546 0.530412
\(263\) −7.56825 −0.466678 −0.233339 0.972395i \(-0.574965\pi\)
−0.233339 + 0.972395i \(0.574965\pi\)
\(264\) 0 0
\(265\) −1.05019 −0.0645128
\(266\) 0 0
\(267\) 0 0
\(268\) −21.2860 −1.30025
\(269\) −9.47208 −0.577523 −0.288761 0.957401i \(-0.593243\pi\)
−0.288761 + 0.957401i \(0.593243\pi\)
\(270\) 0 0
\(271\) −29.3717 −1.78420 −0.892102 0.451835i \(-0.850770\pi\)
−0.892102 + 0.451835i \(0.850770\pi\)
\(272\) 6.97858 0.423138
\(273\) 0 0
\(274\) −30.7005 −1.85469
\(275\) 3.66442 0.220973
\(276\) 0 0
\(277\) −1.90383 −0.114390 −0.0571949 0.998363i \(-0.518216\pi\)
−0.0571949 + 0.998363i \(0.518216\pi\)
\(278\) 17.5640 1.05342
\(279\) 0 0
\(280\) 0 0
\(281\) 20.5756 1.22744 0.613719 0.789525i \(-0.289673\pi\)
0.613719 + 0.789525i \(0.289673\pi\)
\(282\) 0 0
\(283\) 26.9933 1.60458 0.802292 0.596932i \(-0.203614\pi\)
0.802292 + 0.596932i \(0.203614\pi\)
\(284\) −5.37169 −0.318751
\(285\) 0 0
\(286\) −2.68585 −0.158817
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0147 1.00086
\(290\) −33.0031 −1.93801
\(291\) 0 0
\(292\) 53.3864 3.12420
\(293\) −14.9070 −0.870874 −0.435437 0.900219i \(-0.643406\pi\)
−0.435437 + 0.900219i \(0.643406\pi\)
\(294\) 0 0
\(295\) −16.9786 −0.988531
\(296\) −29.7220 −1.72755
\(297\) 0 0
\(298\) 5.07896 0.294216
\(299\) −3.17513 −0.183623
\(300\) 0 0
\(301\) 0 0
\(302\) −34.9357 −2.01033
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −2.68585 −0.153791
\(306\) 0 0
\(307\) −26.0288 −1.48554 −0.742770 0.669546i \(-0.766488\pi\)
−0.742770 + 0.669546i \(0.766488\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5.14637 −0.292294
\(311\) 19.4966 1.10555 0.552776 0.833330i \(-0.313568\pi\)
0.552776 + 0.833330i \(0.313568\pi\)
\(312\) 0 0
\(313\) 3.48194 0.196811 0.0984055 0.995146i \(-0.468626\pi\)
0.0984055 + 0.995146i \(0.468626\pi\)
\(314\) 53.5787 3.02362
\(315\) 0 0
\(316\) 3.07896 0.173205
\(317\) −5.02142 −0.282031 −0.141016 0.990007i \(-0.545037\pi\)
−0.141016 + 0.990007i \(0.545037\pi\)
\(318\) 0 0
\(319\) 12.0246 0.673246
\(320\) 16.3503 0.914008
\(321\) 0 0
\(322\) 0 0
\(323\) 19.4966 1.08482
\(324\) 0 0
\(325\) 3.19656 0.177313
\(326\) −16.5855 −0.918584
\(327\) 0 0
\(328\) 1.02142 0.0563986
\(329\) 0 0
\(330\) 0 0
\(331\) −6.14950 −0.338007 −0.169004 0.985615i \(-0.554055\pi\)
−0.169004 + 0.985615i \(0.554055\pi\)
\(332\) −42.3221 −2.32273
\(333\) 0 0
\(334\) −6.12494 −0.335142
\(335\) 8.19235 0.447596
\(336\) 0 0
\(337\) −25.6258 −1.39593 −0.697963 0.716134i \(-0.745910\pi\)
−0.697963 + 0.716134i \(0.745910\pi\)
\(338\) −2.34292 −0.127438
\(339\) 0 0
\(340\) −27.3288 −1.48211
\(341\) 1.87506 0.101540
\(342\) 0 0
\(343\) 0 0
\(344\) 28.4507 1.53396
\(345\) 0 0
\(346\) −25.7795 −1.38591
\(347\) 16.7005 0.896532 0.448266 0.893900i \(-0.352042\pi\)
0.448266 + 0.893900i \(0.352042\pi\)
\(348\) 0 0
\(349\) 23.5500 1.26060 0.630300 0.776351i \(-0.282932\pi\)
0.630300 + 0.776351i \(0.282932\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.78623 −0.255107
\(353\) −7.64973 −0.407154 −0.203577 0.979059i \(-0.565257\pi\)
−0.203577 + 0.979059i \(0.565257\pi\)
\(354\) 0 0
\(355\) 2.06740 0.109726
\(356\) 20.0147 1.06078
\(357\) 0 0
\(358\) 56.1396 2.96707
\(359\) −18.3748 −0.969786 −0.484893 0.874573i \(-0.661142\pi\)
−0.484893 + 0.874573i \(0.661142\pi\)
\(360\) 0 0
\(361\) −7.82487 −0.411835
\(362\) 15.3717 0.807918
\(363\) 0 0
\(364\) 0 0
\(365\) −20.5468 −1.07547
\(366\) 0 0
\(367\) −5.33871 −0.278679 −0.139339 0.990245i \(-0.544498\pi\)
−0.139339 + 0.990245i \(0.544498\pi\)
\(368\) 3.79923 0.198049
\(369\) 0 0
\(370\) 26.8009 1.39331
\(371\) 0 0
\(372\) 0 0
\(373\) 21.5212 1.11433 0.557163 0.830403i \(-0.311890\pi\)
0.557163 + 0.830403i \(0.311890\pi\)
\(374\) 15.6644 0.809988
\(375\) 0 0
\(376\) 37.0361 1.90999
\(377\) 10.4893 0.540226
\(378\) 0 0
\(379\) 4.61002 0.236801 0.118400 0.992966i \(-0.462223\pi\)
0.118400 + 0.992966i \(0.462223\pi\)
\(380\) −15.6644 −0.803568
\(381\) 0 0
\(382\) −10.2927 −0.526622
\(383\) −8.33558 −0.425928 −0.212964 0.977060i \(-0.568312\pi\)
−0.212964 + 0.977060i \(0.568312\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 19.4292 0.988922
\(387\) 0 0
\(388\) 18.6430 0.946455
\(389\) 6.44223 0.326634 0.163317 0.986574i \(-0.447781\pi\)
0.163317 + 0.986574i \(0.447781\pi\)
\(390\) 0 0
\(391\) 18.5181 0.936498
\(392\) 0 0
\(393\) 0 0
\(394\) −7.42923 −0.374279
\(395\) −1.18500 −0.0596238
\(396\) 0 0
\(397\) −1.40046 −0.0702872 −0.0351436 0.999382i \(-0.511189\pi\)
−0.0351436 + 0.999382i \(0.511189\pi\)
\(398\) −31.8568 −1.59684
\(399\) 0 0
\(400\) −3.82487 −0.191243
\(401\) 6.97858 0.348494 0.174247 0.984702i \(-0.444251\pi\)
0.174247 + 0.984702i \(0.444251\pi\)
\(402\) 0 0
\(403\) 1.63565 0.0814777
\(404\) 38.8929 1.93499
\(405\) 0 0
\(406\) 0 0
\(407\) −9.76481 −0.484024
\(408\) 0 0
\(409\) 18.3790 0.908785 0.454392 0.890802i \(-0.349856\pi\)
0.454392 + 0.890802i \(0.349856\pi\)
\(410\) −0.921039 −0.0454869
\(411\) 0 0
\(412\) 11.9143 0.586976
\(413\) 0 0
\(414\) 0 0
\(415\) 16.2885 0.799572
\(416\) −4.17513 −0.204703
\(417\) 0 0
\(418\) 8.97858 0.439157
\(419\) −30.0393 −1.46751 −0.733757 0.679412i \(-0.762235\pi\)
−0.733757 + 0.679412i \(0.762235\pi\)
\(420\) 0 0
\(421\) −8.31729 −0.405360 −0.202680 0.979245i \(-0.564965\pi\)
−0.202680 + 0.979245i \(0.564965\pi\)
\(422\) −21.7318 −1.05789
\(423\) 0 0
\(424\) −2.72869 −0.132517
\(425\) −18.6430 −0.904318
\(426\) 0 0
\(427\) 0 0
\(428\) −17.3717 −0.839692
\(429\) 0 0
\(430\) −25.6546 −1.23717
\(431\) −9.64973 −0.464811 −0.232406 0.972619i \(-0.574660\pi\)
−0.232406 + 0.972619i \(0.574660\pi\)
\(432\) 0 0
\(433\) −26.3074 −1.26425 −0.632127 0.774865i \(-0.717818\pi\)
−0.632127 + 0.774865i \(0.717818\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −47.0937 −2.25538
\(437\) 10.6142 0.507748
\(438\) 0 0
\(439\) −33.8139 −1.61385 −0.806925 0.590654i \(-0.798870\pi\)
−0.806925 + 0.590654i \(0.798870\pi\)
\(440\) −5.37169 −0.256085
\(441\) 0 0
\(442\) 13.6644 0.649950
\(443\) −26.4464 −1.25651 −0.628254 0.778008i \(-0.716230\pi\)
−0.628254 + 0.778008i \(0.716230\pi\)
\(444\) 0 0
\(445\) −7.70306 −0.365160
\(446\) −45.9044 −2.17364
\(447\) 0 0
\(448\) 0 0
\(449\) −2.64300 −0.124731 −0.0623655 0.998053i \(-0.519864\pi\)
−0.0623655 + 0.998053i \(0.519864\pi\)
\(450\) 0 0
\(451\) 0.335577 0.0158017
\(452\) −57.3864 −2.69923
\(453\) 0 0
\(454\) 46.0722 2.16228
\(455\) 0 0
\(456\) 0 0
\(457\) −33.6890 −1.57590 −0.787952 0.615737i \(-0.788859\pi\)
−0.787952 + 0.615737i \(0.788859\pi\)
\(458\) 18.1923 0.850073
\(459\) 0 0
\(460\) −14.8782 −0.693699
\(461\) 33.0790 1.54064 0.770320 0.637657i \(-0.220096\pi\)
0.770320 + 0.637657i \(0.220096\pi\)
\(462\) 0 0
\(463\) −2.51806 −0.117024 −0.0585120 0.998287i \(-0.518636\pi\)
−0.0585120 + 0.998287i \(0.518636\pi\)
\(464\) −12.5510 −0.582667
\(465\) 0 0
\(466\) 28.5756 1.32374
\(467\) 2.57560 0.119184 0.0595922 0.998223i \(-0.481020\pi\)
0.0595922 + 0.998223i \(0.481020\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −33.3963 −1.54045
\(471\) 0 0
\(472\) −44.1151 −2.03056
\(473\) 9.34713 0.429782
\(474\) 0 0
\(475\) −10.6858 −0.490300
\(476\) 0 0
\(477\) 0 0
\(478\) −24.1151 −1.10300
\(479\) 0.513847 0.0234783 0.0117391 0.999931i \(-0.496263\pi\)
0.0117391 + 0.999931i \(0.496263\pi\)
\(480\) 0 0
\(481\) −8.51806 −0.388390
\(482\) −9.43910 −0.429939
\(483\) 0 0
\(484\) −33.7967 −1.53621
\(485\) −7.17513 −0.325806
\(486\) 0 0
\(487\) 36.0575 1.63392 0.816962 0.576692i \(-0.195657\pi\)
0.816962 + 0.576692i \(0.195657\pi\)
\(488\) −6.97858 −0.315905
\(489\) 0 0
\(490\) 0 0
\(491\) 9.22846 0.416475 0.208237 0.978078i \(-0.433227\pi\)
0.208237 + 0.978078i \(0.433227\pi\)
\(492\) 0 0
\(493\) −61.1758 −2.75522
\(494\) 7.83221 0.352388
\(495\) 0 0
\(496\) −1.95715 −0.0878788
\(497\) 0 0
\(498\) 0 0
\(499\) 1.00314 0.0449065 0.0224533 0.999748i \(-0.492852\pi\)
0.0224533 + 0.999748i \(0.492852\pi\)
\(500\) 38.4078 1.71765
\(501\) 0 0
\(502\) −6.82065 −0.304421
\(503\) −30.3503 −1.35325 −0.676626 0.736327i \(-0.736559\pi\)
−0.676626 + 0.736327i \(0.736559\pi\)
\(504\) 0 0
\(505\) −14.9687 −0.666099
\(506\) 8.52792 0.379112
\(507\) 0 0
\(508\) 42.0722 1.86665
\(509\) 10.5995 0.469816 0.234908 0.972018i \(-0.424521\pi\)
0.234908 + 0.972018i \(0.424521\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 13.3461 0.589818
\(513\) 0 0
\(514\) 45.9143 2.02519
\(515\) −4.58546 −0.202060
\(516\) 0 0
\(517\) 12.1678 0.535139
\(518\) 0 0
\(519\) 0 0
\(520\) −4.68585 −0.205488
\(521\) −16.2646 −0.712564 −0.356282 0.934378i \(-0.615956\pi\)
−0.356282 + 0.934378i \(0.615956\pi\)
\(522\) 0 0
\(523\) −7.22219 −0.315804 −0.157902 0.987455i \(-0.550473\pi\)
−0.157902 + 0.987455i \(0.550473\pi\)
\(524\) −12.7862 −0.558569
\(525\) 0 0
\(526\) 17.7318 0.773144
\(527\) −9.53948 −0.415546
\(528\) 0 0
\(529\) −12.9185 −0.561675
\(530\) 2.46052 0.106878
\(531\) 0 0
\(532\) 0 0
\(533\) 0.292731 0.0126796
\(534\) 0 0
\(535\) 6.68585 0.289054
\(536\) 21.2860 0.919415
\(537\) 0 0
\(538\) 22.1923 0.956780
\(539\) 0 0
\(540\) 0 0
\(541\) 17.3534 0.746081 0.373041 0.927815i \(-0.378315\pi\)
0.373041 + 0.927815i \(0.378315\pi\)
\(542\) 68.8156 2.95588
\(543\) 0 0
\(544\) 24.3503 1.04401
\(545\) 18.1249 0.776387
\(546\) 0 0
\(547\) −34.1109 −1.45848 −0.729238 0.684261i \(-0.760125\pi\)
−0.729238 + 0.684261i \(0.760125\pi\)
\(548\) 45.7220 1.95315
\(549\) 0 0
\(550\) −8.58546 −0.366085
\(551\) −35.0649 −1.49381
\(552\) 0 0
\(553\) 0 0
\(554\) 4.46052 0.189509
\(555\) 0 0
\(556\) −26.1579 −1.10934
\(557\) 13.2222 0.560242 0.280121 0.959965i \(-0.409625\pi\)
0.280121 + 0.959965i \(0.409625\pi\)
\(558\) 0 0
\(559\) 8.15371 0.344865
\(560\) 0 0
\(561\) 0 0
\(562\) −48.2070 −2.03349
\(563\) −41.3717 −1.74361 −0.871804 0.489854i \(-0.837050\pi\)
−0.871804 + 0.489854i \(0.837050\pi\)
\(564\) 0 0
\(565\) 22.0863 0.929178
\(566\) −63.2432 −2.65831
\(567\) 0 0
\(568\) 5.37169 0.225391
\(569\) 2.68164 0.112420 0.0562100 0.998419i \(-0.482098\pi\)
0.0562100 + 0.998419i \(0.482098\pi\)
\(570\) 0 0
\(571\) −28.9315 −1.21075 −0.605373 0.795942i \(-0.706976\pi\)
−0.605373 + 0.795942i \(0.706976\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) −10.1495 −0.423263
\(576\) 0 0
\(577\) −37.5296 −1.56238 −0.781189 0.624294i \(-0.785387\pi\)
−0.781189 + 0.624294i \(0.785387\pi\)
\(578\) −39.8641 −1.65813
\(579\) 0 0
\(580\) 49.1512 2.04089
\(581\) 0 0
\(582\) 0 0
\(583\) −0.896480 −0.0371284
\(584\) −53.3864 −2.20914
\(585\) 0 0
\(586\) 34.9259 1.44277
\(587\) 23.0649 0.951990 0.475995 0.879448i \(-0.342088\pi\)
0.475995 + 0.879448i \(0.342088\pi\)
\(588\) 0 0
\(589\) −5.46787 −0.225299
\(590\) 39.7795 1.63770
\(591\) 0 0
\(592\) 10.1923 0.418903
\(593\) −11.0502 −0.453777 −0.226889 0.973921i \(-0.572855\pi\)
−0.226889 + 0.973921i \(0.572855\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.56404 −0.309835
\(597\) 0 0
\(598\) 7.43910 0.304207
\(599\) 44.6044 1.82248 0.911242 0.411870i \(-0.135124\pi\)
0.911242 + 0.411870i \(0.135124\pi\)
\(600\) 0 0
\(601\) −47.8715 −1.95272 −0.976359 0.216156i \(-0.930648\pi\)
−0.976359 + 0.216156i \(0.930648\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 52.0294 2.11705
\(605\) 13.0073 0.528824
\(606\) 0 0
\(607\) −45.0691 −1.82930 −0.914649 0.404249i \(-0.867533\pi\)
−0.914649 + 0.404249i \(0.867533\pi\)
\(608\) 13.9572 0.566037
\(609\) 0 0
\(610\) 6.29273 0.254785
\(611\) 10.6142 0.429406
\(612\) 0 0
\(613\) −25.5212 −1.03079 −0.515396 0.856952i \(-0.672355\pi\)
−0.515396 + 0.856952i \(0.672355\pi\)
\(614\) 60.9834 2.46109
\(615\) 0 0
\(616\) 0 0
\(617\) −29.2432 −1.17729 −0.588643 0.808393i \(-0.700337\pi\)
−0.588643 + 0.808393i \(0.700337\pi\)
\(618\) 0 0
\(619\) −4.78623 −0.192375 −0.0961874 0.995363i \(-0.530665\pi\)
−0.0961874 + 0.995363i \(0.530665\pi\)
\(620\) 7.66442 0.307811
\(621\) 0 0
\(622\) −45.6791 −1.83157
\(623\) 0 0
\(624\) 0 0
\(625\) 1.20077 0.0480307
\(626\) −8.15792 −0.326056
\(627\) 0 0
\(628\) −79.7942 −3.18413
\(629\) 49.6791 1.98084
\(630\) 0 0
\(631\) −28.3931 −1.13031 −0.565156 0.824984i \(-0.691184\pi\)
−0.565156 + 0.824984i \(0.691184\pi\)
\(632\) −3.07896 −0.122475
\(633\) 0 0
\(634\) 11.7648 0.467240
\(635\) −16.1923 −0.642574
\(636\) 0 0
\(637\) 0 0
\(638\) −28.1726 −1.11536
\(639\) 0 0
\(640\) −27.0937 −1.07097
\(641\) −5.96137 −0.235460 −0.117730 0.993046i \(-0.537562\pi\)
−0.117730 + 0.993046i \(0.537562\pi\)
\(642\) 0 0
\(643\) −31.1940 −1.23017 −0.615086 0.788460i \(-0.710879\pi\)
−0.615086 + 0.788460i \(0.710879\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −45.6791 −1.79722
\(647\) 14.9112 0.586219 0.293109 0.956079i \(-0.405310\pi\)
0.293109 + 0.956079i \(0.405310\pi\)
\(648\) 0 0
\(649\) −14.4935 −0.568920
\(650\) −7.48929 −0.293754
\(651\) 0 0
\(652\) 24.7005 0.967348
\(653\) 3.57246 0.139801 0.0699006 0.997554i \(-0.477732\pi\)
0.0699006 + 0.997554i \(0.477732\pi\)
\(654\) 0 0
\(655\) 4.92104 0.192281
\(656\) −0.350269 −0.0136757
\(657\) 0 0
\(658\) 0 0
\(659\) 3.90383 0.152071 0.0760357 0.997105i \(-0.475774\pi\)
0.0760357 + 0.997105i \(0.475774\pi\)
\(660\) 0 0
\(661\) −13.7936 −0.536508 −0.268254 0.963348i \(-0.586447\pi\)
−0.268254 + 0.963348i \(0.586447\pi\)
\(662\) 14.4078 0.559975
\(663\) 0 0
\(664\) 42.3221 1.64242
\(665\) 0 0
\(666\) 0 0
\(667\) −33.3049 −1.28957
\(668\) 9.12181 0.352933
\(669\) 0 0
\(670\) −19.1940 −0.741530
\(671\) −2.29273 −0.0885099
\(672\) 0 0
\(673\) 5.70306 0.219837 0.109918 0.993941i \(-0.464941\pi\)
0.109918 + 0.993941i \(0.464941\pi\)
\(674\) 60.0393 2.31263
\(675\) 0 0
\(676\) 3.48929 0.134203
\(677\) 35.2614 1.35521 0.677604 0.735427i \(-0.263019\pi\)
0.677604 + 0.735427i \(0.263019\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 27.3288 1.04801
\(681\) 0 0
\(682\) −4.39312 −0.168221
\(683\) −1.03612 −0.0396459 −0.0198229 0.999804i \(-0.506310\pi\)
−0.0198229 + 0.999804i \(0.506310\pi\)
\(684\) 0 0
\(685\) −17.5970 −0.672348
\(686\) 0 0
\(687\) 0 0
\(688\) −9.75639 −0.371959
\(689\) −0.782020 −0.0297926
\(690\) 0 0
\(691\) −3.67850 −0.139937 −0.0699684 0.997549i \(-0.522290\pi\)
−0.0699684 + 0.997549i \(0.522290\pi\)
\(692\) 38.3931 1.45949
\(693\) 0 0
\(694\) −39.1281 −1.48528
\(695\) 10.0674 0.381878
\(696\) 0 0
\(697\) −1.70727 −0.0646674
\(698\) −55.1758 −2.08843
\(699\) 0 0
\(700\) 0 0
\(701\) 0.0617493 0.00233224 0.00116612 0.999999i \(-0.499629\pi\)
0.00116612 + 0.999999i \(0.499629\pi\)
\(702\) 0 0
\(703\) 28.4752 1.07396
\(704\) 13.9572 0.526030
\(705\) 0 0
\(706\) 17.9227 0.674531
\(707\) 0 0
\(708\) 0 0
\(709\) −42.9834 −1.61428 −0.807138 0.590363i \(-0.798985\pi\)
−0.807138 + 0.590363i \(0.798985\pi\)
\(710\) −4.84377 −0.181783
\(711\) 0 0
\(712\) −20.0147 −0.750082
\(713\) −5.19342 −0.194495
\(714\) 0 0
\(715\) −1.53948 −0.0575733
\(716\) −83.6081 −3.12458
\(717\) 0 0
\(718\) 43.0508 1.60664
\(719\) 17.6546 0.658404 0.329202 0.944260i \(-0.393220\pi\)
0.329202 + 0.944260i \(0.393220\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.3331 0.682286
\(723\) 0 0
\(724\) −22.8929 −0.850807
\(725\) 33.5296 1.24526
\(726\) 0 0
\(727\) 23.8077 0.882977 0.441488 0.897267i \(-0.354451\pi\)
0.441488 + 0.897267i \(0.354451\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 48.1396 1.78173
\(731\) −47.5542 −1.75885
\(732\) 0 0
\(733\) −31.3492 −1.15791 −0.578954 0.815360i \(-0.696539\pi\)
−0.578954 + 0.815360i \(0.696539\pi\)
\(734\) 12.5082 0.461686
\(735\) 0 0
\(736\) 13.2566 0.488645
\(737\) 6.99327 0.257600
\(738\) 0 0
\(739\) −24.8108 −0.912680 −0.456340 0.889806i \(-0.650840\pi\)
−0.456340 + 0.889806i \(0.650840\pi\)
\(740\) −39.9143 −1.46728
\(741\) 0 0
\(742\) 0 0
\(743\) −21.3717 −0.784051 −0.392026 0.919954i \(-0.628226\pi\)
−0.392026 + 0.919954i \(0.628226\pi\)
\(744\) 0 0
\(745\) 2.91117 0.106657
\(746\) −50.4225 −1.84610
\(747\) 0 0
\(748\) −23.3288 −0.852987
\(749\) 0 0
\(750\) 0 0
\(751\) 19.2243 0.701503 0.350751 0.936469i \(-0.385926\pi\)
0.350751 + 0.936469i \(0.385926\pi\)
\(752\) −12.7005 −0.463141
\(753\) 0 0
\(754\) −24.5756 −0.894990
\(755\) −20.0246 −0.728768
\(756\) 0 0
\(757\) −19.8610 −0.721860 −0.360930 0.932593i \(-0.617541\pi\)
−0.360930 + 0.932593i \(0.617541\pi\)
\(758\) −10.8009 −0.392307
\(759\) 0 0
\(760\) 15.6644 0.568208
\(761\) 6.12073 0.221876 0.110938 0.993827i \(-0.464614\pi\)
0.110938 + 0.993827i \(0.464614\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 15.3288 0.554578
\(765\) 0 0
\(766\) 19.5296 0.705634
\(767\) −12.6430 −0.456512
\(768\) 0 0
\(769\) −3.82800 −0.138041 −0.0690206 0.997615i \(-0.521987\pi\)
−0.0690206 + 0.997615i \(0.521987\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −28.9357 −1.04142
\(773\) −6.53635 −0.235096 −0.117548 0.993067i \(-0.537503\pi\)
−0.117548 + 0.993067i \(0.537503\pi\)
\(774\) 0 0
\(775\) 5.22846 0.187812
\(776\) −18.6430 −0.669245
\(777\) 0 0
\(778\) −15.0937 −0.541134
\(779\) −0.978577 −0.0350612
\(780\) 0 0
\(781\) 1.76481 0.0631498
\(782\) −43.3864 −1.55149
\(783\) 0 0
\(784\) 0 0
\(785\) 30.7104 1.09610
\(786\) 0 0
\(787\) −30.8066 −1.09814 −0.549068 0.835778i \(-0.685017\pi\)
−0.549068 + 0.835778i \(0.685017\pi\)
\(788\) 11.0643 0.394148
\(789\) 0 0
\(790\) 2.77636 0.0987786
\(791\) 0 0
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 3.28117 0.116444
\(795\) 0 0
\(796\) 47.4439 1.68161
\(797\) −38.8156 −1.37492 −0.687460 0.726222i \(-0.741274\pi\)
−0.687460 + 0.726222i \(0.741274\pi\)
\(798\) 0 0
\(799\) −61.9044 −2.19002
\(800\) −13.3461 −0.471854
\(801\) 0 0
\(802\) −16.3503 −0.577348
\(803\) −17.5395 −0.618955
\(804\) 0 0
\(805\) 0 0
\(806\) −3.83221 −0.134984
\(807\) 0 0
\(808\) −38.8929 −1.36825
\(809\) −1.04033 −0.0365759 −0.0182880 0.999833i \(-0.505822\pi\)
−0.0182880 + 0.999833i \(0.505822\pi\)
\(810\) 0 0
\(811\) −12.6712 −0.444944 −0.222472 0.974939i \(-0.571413\pi\)
−0.222472 + 0.974939i \(0.571413\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 22.8782 0.801880
\(815\) −9.50650 −0.332998
\(816\) 0 0
\(817\) −27.2572 −0.953610
\(818\) −43.0607 −1.50558
\(819\) 0 0
\(820\) 1.37169 0.0479016
\(821\) −27.0361 −0.943567 −0.471783 0.881714i \(-0.656390\pi\)
−0.471783 + 0.881714i \(0.656390\pi\)
\(822\) 0 0
\(823\) −31.6363 −1.10277 −0.551386 0.834251i \(-0.685901\pi\)
−0.551386 + 0.834251i \(0.685901\pi\)
\(824\) −11.9143 −0.415055
\(825\) 0 0
\(826\) 0 0
\(827\) −56.4800 −1.96400 −0.982002 0.188872i \(-0.939517\pi\)
−0.982002 + 0.188872i \(0.939517\pi\)
\(828\) 0 0
\(829\) 42.6760 1.48220 0.741099 0.671396i \(-0.234305\pi\)
0.741099 + 0.671396i \(0.234305\pi\)
\(830\) −38.1627 −1.32465
\(831\) 0 0
\(832\) 12.1751 0.422097
\(833\) 0 0
\(834\) 0 0
\(835\) −3.51071 −0.121493
\(836\) −13.3717 −0.462470
\(837\) 0 0
\(838\) 70.3797 2.43122
\(839\) −40.1642 −1.38662 −0.693311 0.720639i \(-0.743849\pi\)
−0.693311 + 0.720639i \(0.743849\pi\)
\(840\) 0 0
\(841\) 81.0252 2.79397
\(842\) 19.4868 0.671558
\(843\) 0 0
\(844\) 32.3650 1.11405
\(845\) −1.34292 −0.0461980
\(846\) 0 0
\(847\) 0 0
\(848\) 0.935731 0.0321331
\(849\) 0 0
\(850\) 43.6791 1.49818
\(851\) 27.0460 0.927124
\(852\) 0 0
\(853\) −19.6932 −0.674282 −0.337141 0.941454i \(-0.609460\pi\)
−0.337141 + 0.941454i \(0.609460\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 17.3717 0.593752
\(857\) 1.66442 0.0568556 0.0284278 0.999596i \(-0.490950\pi\)
0.0284278 + 0.999596i \(0.490950\pi\)
\(858\) 0 0
\(859\) 41.2944 1.40895 0.704474 0.709730i \(-0.251183\pi\)
0.704474 + 0.709730i \(0.251183\pi\)
\(860\) 38.2070 1.30285
\(861\) 0 0
\(862\) 22.6086 0.770051
\(863\) 31.3288 1.06645 0.533223 0.845975i \(-0.320981\pi\)
0.533223 + 0.845975i \(0.320981\pi\)
\(864\) 0 0
\(865\) −14.7764 −0.502411
\(866\) 61.6363 2.09449
\(867\) 0 0
\(868\) 0 0
\(869\) −1.01156 −0.0343147
\(870\) 0 0
\(871\) 6.10038 0.206704
\(872\) 47.0937 1.59479
\(873\) 0 0
\(874\) −24.8683 −0.841184
\(875\) 0 0
\(876\) 0 0
\(877\) −53.8041 −1.81683 −0.908417 0.418065i \(-0.862708\pi\)
−0.908417 + 0.418065i \(0.862708\pi\)
\(878\) 79.2234 2.67366
\(879\) 0 0
\(880\) 1.84208 0.0620964
\(881\) 8.09196 0.272625 0.136313 0.990666i \(-0.456475\pi\)
0.136313 + 0.990666i \(0.456475\pi\)
\(882\) 0 0
\(883\) −27.0705 −0.910996 −0.455498 0.890237i \(-0.650539\pi\)
−0.455498 + 0.890237i \(0.650539\pi\)
\(884\) −20.3503 −0.684454
\(885\) 0 0
\(886\) 61.9620 2.08165
\(887\) −38.4935 −1.29249 −0.646243 0.763132i \(-0.723661\pi\)
−0.646243 + 0.763132i \(0.723661\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 18.0477 0.604959
\(891\) 0 0
\(892\) 68.3650 2.28903
\(893\) −35.4826 −1.18738
\(894\) 0 0
\(895\) 32.1783 1.07560
\(896\) 0 0
\(897\) 0 0
\(898\) 6.19235 0.206641
\(899\) 17.1568 0.572213
\(900\) 0 0
\(901\) 4.56090 0.151946
\(902\) −0.786230 −0.0261786
\(903\) 0 0
\(904\) 57.3864 1.90864
\(905\) 8.81079 0.292881
\(906\) 0 0
\(907\) 15.7031 0.521411 0.260706 0.965418i \(-0.416045\pi\)
0.260706 + 0.965418i \(0.416045\pi\)
\(908\) −68.6148 −2.27706
\(909\) 0 0
\(910\) 0 0
\(911\) −44.9399 −1.48893 −0.744463 0.667663i \(-0.767295\pi\)
−0.744463 + 0.667663i \(0.767295\pi\)
\(912\) 0 0
\(913\) 13.9044 0.460170
\(914\) 78.9307 2.61080
\(915\) 0 0
\(916\) −27.0937 −0.895200
\(917\) 0 0
\(918\) 0 0
\(919\) −27.2432 −0.898669 −0.449334 0.893364i \(-0.648339\pi\)
−0.449334 + 0.893364i \(0.648339\pi\)
\(920\) 14.8782 0.490519
\(921\) 0 0
\(922\) −77.5015 −2.55237
\(923\) 1.53948 0.0506726
\(924\) 0 0
\(925\) −27.2285 −0.895266
\(926\) 5.89962 0.193873
\(927\) 0 0
\(928\) −43.7942 −1.43761
\(929\) −17.1422 −0.562416 −0.281208 0.959647i \(-0.590735\pi\)
−0.281208 + 0.959647i \(0.590735\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −42.5573 −1.39401
\(933\) 0 0
\(934\) −6.03442 −0.197452
\(935\) 8.97858 0.293631
\(936\) 0 0
\(937\) 51.5197 1.68308 0.841538 0.540197i \(-0.181650\pi\)
0.841538 + 0.540197i \(0.181650\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 49.7367 1.62223
\(941\) 32.7575 1.06786 0.533931 0.845528i \(-0.320714\pi\)
0.533931 + 0.845528i \(0.320714\pi\)
\(942\) 0 0
\(943\) −0.929460 −0.0302674
\(944\) 15.1281 0.492377
\(945\) 0 0
\(946\) −21.8996 −0.712018
\(947\) −20.9295 −0.680116 −0.340058 0.940404i \(-0.610447\pi\)
−0.340058 + 0.940404i \(0.610447\pi\)
\(948\) 0 0
\(949\) −15.3001 −0.496662
\(950\) 25.0361 0.812279
\(951\) 0 0
\(952\) 0 0
\(953\) −46.4120 −1.50343 −0.751716 0.659487i \(-0.770774\pi\)
−0.751716 + 0.659487i \(0.770774\pi\)
\(954\) 0 0
\(955\) −5.89962 −0.190907
\(956\) 35.9143 1.16155
\(957\) 0 0
\(958\) −1.20390 −0.0388964
\(959\) 0 0
\(960\) 0 0
\(961\) −28.3246 −0.913698
\(962\) 19.9572 0.643444
\(963\) 0 0
\(964\) 14.0575 0.452763
\(965\) 11.1365 0.358497
\(966\) 0 0
\(967\) −23.2186 −0.746660 −0.373330 0.927699i \(-0.621784\pi\)
−0.373330 + 0.927699i \(0.621784\pi\)
\(968\) 33.7967 1.08627
\(969\) 0 0
\(970\) 16.8108 0.539762
\(971\) −14.1004 −0.452503 −0.226251 0.974069i \(-0.572647\pi\)
−0.226251 + 0.974069i \(0.572647\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −84.4800 −2.70692
\(975\) 0 0
\(976\) 2.39312 0.0766018
\(977\) 2.95402 0.0945074 0.0472537 0.998883i \(-0.484953\pi\)
0.0472537 + 0.998883i \(0.484953\pi\)
\(978\) 0 0
\(979\) −6.57560 −0.210157
\(980\) 0 0
\(981\) 0 0
\(982\) −21.6216 −0.689972
\(983\) −35.0367 −1.11750 −0.558749 0.829337i \(-0.688719\pi\)
−0.558749 + 0.829337i \(0.688719\pi\)
\(984\) 0 0
\(985\) −4.25831 −0.135681
\(986\) 143.330 4.56456
\(987\) 0 0
\(988\) −11.6644 −0.371095
\(989\) −25.8891 −0.823227
\(990\) 0 0
\(991\) 47.9718 1.52388 0.761938 0.647650i \(-0.224248\pi\)
0.761938 + 0.647650i \(0.224248\pi\)
\(992\) −6.82908 −0.216823
\(993\) 0 0
\(994\) 0 0
\(995\) −18.2598 −0.578873
\(996\) 0 0
\(997\) 38.4422 1.21748 0.608739 0.793371i \(-0.291676\pi\)
0.608739 + 0.793371i \(0.291676\pi\)
\(998\) −2.35027 −0.0743965
\(999\) 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 5733.2.a.x.1.1 3
3.2 odd 2 637.2.a.j.1.3 3
7.6 odd 2 819.2.a.i.1.1 3
21.2 odd 6 637.2.e.i.508.1 6
21.5 even 6 637.2.e.j.508.1 6
21.11 odd 6 637.2.e.i.79.1 6
21.17 even 6 637.2.e.j.79.1 6
21.20 even 2 91.2.a.d.1.3 3
39.38 odd 2 8281.2.a.bg.1.1 3
84.83 odd 2 1456.2.a.t.1.2 3
105.104 even 2 2275.2.a.m.1.1 3
168.83 odd 2 5824.2.a.bs.1.2 3
168.125 even 2 5824.2.a.by.1.2 3
273.83 odd 4 1183.2.c.f.337.1 6
273.125 odd 4 1183.2.c.f.337.6 6
273.272 even 2 1183.2.a.i.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.3 3 21.20 even 2
637.2.a.j.1.3 3 3.2 odd 2
637.2.e.i.79.1 6 21.11 odd 6
637.2.e.i.508.1 6 21.2 odd 6
637.2.e.j.79.1 6 21.17 even 6
637.2.e.j.508.1 6 21.5 even 6
819.2.a.i.1.1 3 7.6 odd 2
1183.2.a.i.1.1 3 273.272 even 2
1183.2.c.f.337.1 6 273.83 odd 4
1183.2.c.f.337.6 6 273.125 odd 4
1456.2.a.t.1.2 3 84.83 odd 2
2275.2.a.m.1.1 3 105.104 even 2
5733.2.a.x.1.1 3 1.1 even 1 trivial
5824.2.a.bs.1.2 3 168.83 odd 2
5824.2.a.by.1.2 3 168.125 even 2
8281.2.a.bg.1.1 3 39.38 odd 2