Properties

Label 8550.2.a.cg.1.3
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.24482 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.24482 q^{7} -1.00000 q^{8} -1.42801 q^{11} -6.91764 q^{13} -4.24482 q^{14} +1.00000 q^{16} +5.10083 q^{17} +1.00000 q^{19} +1.42801 q^{22} -3.67282 q^{23} +6.91764 q^{26} +4.24482 q^{28} -8.10083 q^{29} -1.28797 q^{31} -1.00000 q^{32} -5.10083 q^{34} -0.856013 q^{37} -1.00000 q^{38} +8.01847 q^{41} +3.57199 q^{43} -1.42801 q^{44} +3.67282 q^{46} +3.81681 q^{47} +11.0185 q^{49} -6.91764 q^{52} -9.06163 q^{53} -4.24482 q^{56} +8.10083 q^{58} -12.3496 q^{59} +8.20561 q^{61} +1.28797 q^{62} +1.00000 q^{64} +4.38880 q^{67} +5.10083 q^{68} -11.1008 q^{71} +5.38485 q^{73} +0.856013 q^{74} +1.00000 q^{76} -6.06163 q^{77} +2.14399 q^{79} -8.01847 q^{82} -1.04316 q^{83} -3.57199 q^{86} +1.42801 q^{88} -16.7305 q^{89} -29.3641 q^{91} -3.67282 q^{92} -3.81681 q^{94} +6.81286 q^{97} -11.0185 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} - 5q^{11} - 2q^{14} + 3q^{16} + 6q^{17} + 3q^{19} + 5q^{22} - q^{23} + 2q^{28} - 15q^{29} - q^{31} - 3q^{32} - 6q^{34} - 4q^{37} - 3q^{38} - 6q^{41} + 10q^{43} - 5q^{44} + q^{46} + 3q^{49} - 5q^{53} - 2q^{56} + 15q^{58} - 12q^{59} + q^{61} + q^{62} + 3q^{64} + q^{67} + 6q^{68} - 24q^{71} + 9q^{73} + 4q^{74} + 3q^{76} + 4q^{77} + 5q^{79} + 6q^{82} - 11q^{83} - 10q^{86} + 5q^{88} - 23q^{89} - 38q^{91} - q^{92} + 14q^{97} - 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 4.24482 1.60439 0.802195 0.597062i \(-0.203665\pi\)
0.802195 + 0.597062i \(0.203665\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −1.42801 −0.430560 −0.215280 0.976552i \(-0.569066\pi\)
−0.215280 + 0.976552i \(0.569066\pi\)
\(12\) 0 0
\(13\) −6.91764 −1.91861 −0.959304 0.282375i \(-0.908878\pi\)
−0.959304 + 0.282375i \(0.908878\pi\)
\(14\) −4.24482 −1.13448
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.10083 1.23713 0.618567 0.785732i \(-0.287714\pi\)
0.618567 + 0.785732i \(0.287714\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 1.42801 0.304452
\(23\) −3.67282 −0.765837 −0.382918 0.923782i \(-0.625081\pi\)
−0.382918 + 0.923782i \(0.625081\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.91764 1.35666
\(27\) 0 0
\(28\) 4.24482 0.802195
\(29\) −8.10083 −1.50429 −0.752143 0.659000i \(-0.770980\pi\)
−0.752143 + 0.659000i \(0.770980\pi\)
\(30\) 0 0
\(31\) −1.28797 −0.231327 −0.115663 0.993288i \(-0.536899\pi\)
−0.115663 + 0.993288i \(0.536899\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.10083 −0.874785
\(35\) 0 0
\(36\) 0 0
\(37\) −0.856013 −0.140728 −0.0703639 0.997521i \(-0.522416\pi\)
−0.0703639 + 0.997521i \(0.522416\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 8.01847 1.25227 0.626137 0.779713i \(-0.284635\pi\)
0.626137 + 0.779713i \(0.284635\pi\)
\(42\) 0 0
\(43\) 3.57199 0.544724 0.272362 0.962195i \(-0.412195\pi\)
0.272362 + 0.962195i \(0.412195\pi\)
\(44\) −1.42801 −0.215280
\(45\) 0 0
\(46\) 3.67282 0.541528
\(47\) 3.81681 0.556739 0.278369 0.960474i \(-0.410206\pi\)
0.278369 + 0.960474i \(0.410206\pi\)
\(48\) 0 0
\(49\) 11.0185 1.57407
\(50\) 0 0
\(51\) 0 0
\(52\) −6.91764 −0.959304
\(53\) −9.06163 −1.24471 −0.622355 0.782735i \(-0.713824\pi\)
−0.622355 + 0.782735i \(0.713824\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.24482 −0.567238
\(57\) 0 0
\(58\) 8.10083 1.06369
\(59\) −12.3496 −1.60778 −0.803891 0.594777i \(-0.797240\pi\)
−0.803891 + 0.594777i \(0.797240\pi\)
\(60\) 0 0
\(61\) 8.20561 1.05062 0.525311 0.850911i \(-0.323949\pi\)
0.525311 + 0.850911i \(0.323949\pi\)
\(62\) 1.28797 0.163573
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.38880 0.536178 0.268089 0.963394i \(-0.413608\pi\)
0.268089 + 0.963394i \(0.413608\pi\)
\(68\) 5.10083 0.618567
\(69\) 0 0
\(70\) 0 0
\(71\) −11.1008 −1.31743 −0.658713 0.752394i \(-0.728899\pi\)
−0.658713 + 0.752394i \(0.728899\pi\)
\(72\) 0 0
\(73\) 5.38485 0.630249 0.315125 0.949050i \(-0.397954\pi\)
0.315125 + 0.949050i \(0.397954\pi\)
\(74\) 0.856013 0.0995095
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −6.06163 −0.690787
\(78\) 0 0
\(79\) 2.14399 0.241217 0.120609 0.992700i \(-0.461515\pi\)
0.120609 + 0.992700i \(0.461515\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −8.01847 −0.885492
\(83\) −1.04316 −0.114501 −0.0572506 0.998360i \(-0.518233\pi\)
−0.0572506 + 0.998360i \(0.518233\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −3.57199 −0.385178
\(87\) 0 0
\(88\) 1.42801 0.152226
\(89\) −16.7305 −1.77343 −0.886715 0.462317i \(-0.847018\pi\)
−0.886715 + 0.462317i \(0.847018\pi\)
\(90\) 0 0
\(91\) −29.3641 −3.07820
\(92\) −3.67282 −0.382918
\(93\) 0 0
\(94\) −3.81681 −0.393674
\(95\) 0 0
\(96\) 0 0
\(97\) 6.81286 0.691741 0.345870 0.938282i \(-0.387584\pi\)
0.345870 + 0.938282i \(0.387584\pi\)
\(98\) −11.0185 −1.11303
\(99\) 0 0
\(100\) 0 0
\(101\) −11.7737 −1.17152 −0.585761 0.810484i \(-0.699204\pi\)
−0.585761 + 0.810484i \(0.699204\pi\)
\(102\) 0 0
\(103\) 4.32718 0.426369 0.213185 0.977012i \(-0.431616\pi\)
0.213185 + 0.977012i \(0.431616\pi\)
\(104\) 6.91764 0.678330
\(105\) 0 0
\(106\) 9.06163 0.880143
\(107\) 4.14003 0.400232 0.200116 0.979772i \(-0.435868\pi\)
0.200116 + 0.979772i \(0.435868\pi\)
\(108\) 0 0
\(109\) −2.48963 −0.238464 −0.119232 0.992866i \(-0.538043\pi\)
−0.119232 + 0.992866i \(0.538043\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.24482 0.401098
\(113\) −11.0185 −1.03653 −0.518265 0.855220i \(-0.673422\pi\)
−0.518265 + 0.855220i \(0.673422\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.10083 −0.752143
\(117\) 0 0
\(118\) 12.3496 1.13687
\(119\) 21.6521 1.98484
\(120\) 0 0
\(121\) −8.96080 −0.814618
\(122\) −8.20561 −0.742901
\(123\) 0 0
\(124\) −1.28797 −0.115663
\(125\) 0 0
\(126\) 0 0
\(127\) 6.05767 0.537532 0.268766 0.963206i \(-0.413384\pi\)
0.268766 + 0.963206i \(0.413384\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 9.44648 0.825343 0.412671 0.910880i \(-0.364596\pi\)
0.412671 + 0.910880i \(0.364596\pi\)
\(132\) 0 0
\(133\) 4.24482 0.368072
\(134\) −4.38880 −0.379135
\(135\) 0 0
\(136\) −5.10083 −0.437393
\(137\) −4.20166 −0.358972 −0.179486 0.983761i \(-0.557443\pi\)
−0.179486 + 0.983761i \(0.557443\pi\)
\(138\) 0 0
\(139\) −5.40727 −0.458639 −0.229320 0.973351i \(-0.573650\pi\)
−0.229320 + 0.973351i \(0.573650\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 11.1008 0.931561
\(143\) 9.87844 0.826076
\(144\) 0 0
\(145\) 0 0
\(146\) −5.38485 −0.445653
\(147\) 0 0
\(148\) −0.856013 −0.0703639
\(149\) −1.51037 −0.123734 −0.0618670 0.998084i \(-0.519705\pi\)
−0.0618670 + 0.998084i \(0.519705\pi\)
\(150\) 0 0
\(151\) −15.4504 −1.25734 −0.628669 0.777673i \(-0.716400\pi\)
−0.628669 + 0.777673i \(0.716400\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 6.06163 0.488460
\(155\) 0 0
\(156\) 0 0
\(157\) −9.16246 −0.731244 −0.365622 0.930763i \(-0.619144\pi\)
−0.365622 + 0.930763i \(0.619144\pi\)
\(158\) −2.14399 −0.170566
\(159\) 0 0
\(160\) 0 0
\(161\) −15.5905 −1.22870
\(162\) 0 0
\(163\) −12.6913 −0.994059 −0.497029 0.867734i \(-0.665576\pi\)
−0.497029 + 0.867734i \(0.665576\pi\)
\(164\) 8.01847 0.626137
\(165\) 0 0
\(166\) 1.04316 0.0809646
\(167\) −16.3249 −1.26326 −0.631630 0.775270i \(-0.717614\pi\)
−0.631630 + 0.775270i \(0.717614\pi\)
\(168\) 0 0
\(169\) 34.8538 2.68106
\(170\) 0 0
\(171\) 0 0
\(172\) 3.57199 0.272362
\(173\) −3.32322 −0.252660 −0.126330 0.991988i \(-0.540320\pi\)
−0.126330 + 0.991988i \(0.540320\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.42801 −0.107640
\(177\) 0 0
\(178\) 16.7305 1.25400
\(179\) 5.54731 0.414625 0.207313 0.978275i \(-0.433528\pi\)
0.207313 + 0.978275i \(0.433528\pi\)
\(180\) 0 0
\(181\) −17.2840 −1.28471 −0.642356 0.766407i \(-0.722043\pi\)
−0.642356 + 0.766407i \(0.722043\pi\)
\(182\) 29.3641 2.17661
\(183\) 0 0
\(184\) 3.67282 0.270764
\(185\) 0 0
\(186\) 0 0
\(187\) −7.28402 −0.532660
\(188\) 3.81681 0.278369
\(189\) 0 0
\(190\) 0 0
\(191\) −22.6336 −1.63771 −0.818856 0.573999i \(-0.805391\pi\)
−0.818856 + 0.573999i \(0.805391\pi\)
\(192\) 0 0
\(193\) 4.24482 0.305549 0.152774 0.988261i \(-0.451179\pi\)
0.152774 + 0.988261i \(0.451179\pi\)
\(194\) −6.81286 −0.489135
\(195\) 0 0
\(196\) 11.0185 0.787034
\(197\) 5.00395 0.356517 0.178258 0.983984i \(-0.442954\pi\)
0.178258 + 0.983984i \(0.442954\pi\)
\(198\) 0 0
\(199\) −27.3826 −1.94110 −0.970550 0.240899i \(-0.922558\pi\)
−0.970550 + 0.240899i \(0.922558\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 11.7737 0.828391
\(203\) −34.3865 −2.41346
\(204\) 0 0
\(205\) 0 0
\(206\) −4.32718 −0.301489
\(207\) 0 0
\(208\) −6.91764 −0.479652
\(209\) −1.42801 −0.0987773
\(210\) 0 0
\(211\) −5.14003 −0.353855 −0.176927 0.984224i \(-0.556616\pi\)
−0.176927 + 0.984224i \(0.556616\pi\)
\(212\) −9.06163 −0.622355
\(213\) 0 0
\(214\) −4.14003 −0.283007
\(215\) 0 0
\(216\) 0 0
\(217\) −5.46721 −0.371138
\(218\) 2.48963 0.168619
\(219\) 0 0
\(220\) 0 0
\(221\) −35.2857 −2.37357
\(222\) 0 0
\(223\) 10.1625 0.680528 0.340264 0.940330i \(-0.389483\pi\)
0.340264 + 0.940330i \(0.389483\pi\)
\(224\) −4.24482 −0.283619
\(225\) 0 0
\(226\) 11.0185 0.732938
\(227\) 7.92159 0.525775 0.262887 0.964827i \(-0.415325\pi\)
0.262887 + 0.964827i \(0.415325\pi\)
\(228\) 0 0
\(229\) −7.44648 −0.492077 −0.246039 0.969260i \(-0.579129\pi\)
−0.246039 + 0.969260i \(0.579129\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.10083 0.531846
\(233\) 28.4834 1.86601 0.933005 0.359862i \(-0.117176\pi\)
0.933005 + 0.359862i \(0.117176\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.3496 −0.803891
\(237\) 0 0
\(238\) −21.6521 −1.40350
\(239\) −3.24086 −0.209634 −0.104817 0.994492i \(-0.533426\pi\)
−0.104817 + 0.994492i \(0.533426\pi\)
\(240\) 0 0
\(241\) 25.9938 1.67441 0.837203 0.546892i \(-0.184189\pi\)
0.837203 + 0.546892i \(0.184189\pi\)
\(242\) 8.96080 0.576022
\(243\) 0 0
\(244\) 8.20561 0.525311
\(245\) 0 0
\(246\) 0 0
\(247\) −6.91764 −0.440159
\(248\) 1.28797 0.0817864
\(249\) 0 0
\(250\) 0 0
\(251\) −3.98153 −0.251312 −0.125656 0.992074i \(-0.540104\pi\)
−0.125656 + 0.992074i \(0.540104\pi\)
\(252\) 0 0
\(253\) 5.24482 0.329739
\(254\) −6.05767 −0.380092
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.1625 0.883430 0.441715 0.897155i \(-0.354370\pi\)
0.441715 + 0.897155i \(0.354370\pi\)
\(258\) 0 0
\(259\) −3.63362 −0.225782
\(260\) 0 0
\(261\) 0 0
\(262\) −9.44648 −0.583605
\(263\) 21.3434 1.31609 0.658045 0.752979i \(-0.271384\pi\)
0.658045 + 0.752979i \(0.271384\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.24482 −0.260266
\(267\) 0 0
\(268\) 4.38880 0.268089
\(269\) −24.4112 −1.48838 −0.744189 0.667969i \(-0.767164\pi\)
−0.744189 + 0.667969i \(0.767164\pi\)
\(270\) 0 0
\(271\) −14.8129 −0.899817 −0.449909 0.893075i \(-0.648543\pi\)
−0.449909 + 0.893075i \(0.648543\pi\)
\(272\) 5.10083 0.309283
\(273\) 0 0
\(274\) 4.20166 0.253832
\(275\) 0 0
\(276\) 0 0
\(277\) 15.2633 0.917082 0.458541 0.888673i \(-0.348372\pi\)
0.458541 + 0.888673i \(0.348372\pi\)
\(278\) 5.40727 0.324307
\(279\) 0 0
\(280\) 0 0
\(281\) 2.83528 0.169139 0.0845694 0.996418i \(-0.473049\pi\)
0.0845694 + 0.996418i \(0.473049\pi\)
\(282\) 0 0
\(283\) 0.942326 0.0560154 0.0280077 0.999608i \(-0.491084\pi\)
0.0280077 + 0.999608i \(0.491084\pi\)
\(284\) −11.1008 −0.658713
\(285\) 0 0
\(286\) −9.87844 −0.584124
\(287\) 34.0369 2.00914
\(288\) 0 0
\(289\) 9.01847 0.530498
\(290\) 0 0
\(291\) 0 0
\(292\) 5.38485 0.315125
\(293\) 13.5513 0.791673 0.395837 0.918321i \(-0.370455\pi\)
0.395837 + 0.918321i \(0.370455\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.856013 0.0497548
\(297\) 0 0
\(298\) 1.51037 0.0874932
\(299\) 25.4073 1.46934
\(300\) 0 0
\(301\) 15.1625 0.873950
\(302\) 15.4504 0.889072
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −26.7921 −1.52911 −0.764554 0.644560i \(-0.777041\pi\)
−0.764554 + 0.644560i \(0.777041\pi\)
\(308\) −6.06163 −0.345393
\(309\) 0 0
\(310\) 0 0
\(311\) −5.03920 −0.285747 −0.142873 0.989741i \(-0.545634\pi\)
−0.142873 + 0.989741i \(0.545634\pi\)
\(312\) 0 0
\(313\) −15.0185 −0.848894 −0.424447 0.905453i \(-0.639532\pi\)
−0.424447 + 0.905453i \(0.639532\pi\)
\(314\) 9.16246 0.517067
\(315\) 0 0
\(316\) 2.14399 0.120609
\(317\) 20.1008 1.12898 0.564488 0.825442i \(-0.309074\pi\)
0.564488 + 0.825442i \(0.309074\pi\)
\(318\) 0 0
\(319\) 11.5680 0.647686
\(320\) 0 0
\(321\) 0 0
\(322\) 15.5905 0.868823
\(323\) 5.10083 0.283818
\(324\) 0 0
\(325\) 0 0
\(326\) 12.6913 0.702906
\(327\) 0 0
\(328\) −8.01847 −0.442746
\(329\) 16.2017 0.893226
\(330\) 0 0
\(331\) −25.5697 −1.40544 −0.702720 0.711467i \(-0.748031\pi\)
−0.702720 + 0.711467i \(0.748031\pi\)
\(332\) −1.04316 −0.0572506
\(333\) 0 0
\(334\) 16.3249 0.893260
\(335\) 0 0
\(336\) 0 0
\(337\) −13.9137 −0.757927 −0.378963 0.925412i \(-0.623719\pi\)
−0.378963 + 0.925412i \(0.623719\pi\)
\(338\) −34.8538 −1.89579
\(339\) 0 0
\(340\) 0 0
\(341\) 1.83923 0.0996001
\(342\) 0 0
\(343\) 17.0577 0.921028
\(344\) −3.57199 −0.192589
\(345\) 0 0
\(346\) 3.32322 0.178658
\(347\) 2.56804 0.137860 0.0689298 0.997622i \(-0.478042\pi\)
0.0689298 + 0.997622i \(0.478042\pi\)
\(348\) 0 0
\(349\) −5.23691 −0.280325 −0.140163 0.990128i \(-0.544763\pi\)
−0.140163 + 0.990128i \(0.544763\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.42801 0.0761130
\(353\) 12.2818 0.653692 0.326846 0.945078i \(-0.394014\pi\)
0.326846 + 0.945078i \(0.394014\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −16.7305 −0.886715
\(357\) 0 0
\(358\) −5.54731 −0.293184
\(359\) −33.7753 −1.78259 −0.891297 0.453419i \(-0.850204\pi\)
−0.891297 + 0.453419i \(0.850204\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 17.2840 0.908428
\(363\) 0 0
\(364\) −29.3641 −1.53910
\(365\) 0 0
\(366\) 0 0
\(367\) −30.3681 −1.58520 −0.792600 0.609742i \(-0.791273\pi\)
−0.792600 + 0.609742i \(0.791273\pi\)
\(368\) −3.67282 −0.191459
\(369\) 0 0
\(370\) 0 0
\(371\) −38.4649 −1.99700
\(372\) 0 0
\(373\) 25.7322 1.33236 0.666181 0.745790i \(-0.267928\pi\)
0.666181 + 0.745790i \(0.267928\pi\)
\(374\) 7.28402 0.376648
\(375\) 0 0
\(376\) −3.81681 −0.196837
\(377\) 56.0386 2.88614
\(378\) 0 0
\(379\) 37.2593 1.91388 0.956942 0.290280i \(-0.0937485\pi\)
0.956942 + 0.290280i \(0.0937485\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 22.6336 1.15804
\(383\) −31.9154 −1.63080 −0.815400 0.578898i \(-0.803483\pi\)
−0.815400 + 0.578898i \(0.803483\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.24482 −0.216055
\(387\) 0 0
\(388\) 6.81286 0.345870
\(389\) 4.48963 0.227633 0.113817 0.993502i \(-0.463692\pi\)
0.113817 + 0.993502i \(0.463692\pi\)
\(390\) 0 0
\(391\) −18.7345 −0.947442
\(392\) −11.0185 −0.556517
\(393\) 0 0
\(394\) −5.00395 −0.252096
\(395\) 0 0
\(396\) 0 0
\(397\) −34.6873 −1.74091 −0.870454 0.492250i \(-0.836175\pi\)
−0.870454 + 0.492250i \(0.836175\pi\)
\(398\) 27.3826 1.37257
\(399\) 0 0
\(400\) 0 0
\(401\) 7.03694 0.351408 0.175704 0.984443i \(-0.443780\pi\)
0.175704 + 0.984443i \(0.443780\pi\)
\(402\) 0 0
\(403\) 8.90973 0.443826
\(404\) −11.7737 −0.585761
\(405\) 0 0
\(406\) 34.3865 1.70658
\(407\) 1.22239 0.0605918
\(408\) 0 0
\(409\) 26.4112 1.30595 0.652976 0.757379i \(-0.273520\pi\)
0.652976 + 0.757379i \(0.273520\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.32718 0.213185
\(413\) −52.4218 −2.57951
\(414\) 0 0
\(415\) 0 0
\(416\) 6.91764 0.339165
\(417\) 0 0
\(418\) 1.42801 0.0698461
\(419\) 16.4896 0.805571 0.402786 0.915294i \(-0.368042\pi\)
0.402786 + 0.915294i \(0.368042\pi\)
\(420\) 0 0
\(421\) 39.0162 1.90153 0.950767 0.309907i \(-0.100298\pi\)
0.950767 + 0.309907i \(0.100298\pi\)
\(422\) 5.14003 0.250213
\(423\) 0 0
\(424\) 9.06163 0.440072
\(425\) 0 0
\(426\) 0 0
\(427\) 34.8313 1.68561
\(428\) 4.14003 0.200116
\(429\) 0 0
\(430\) 0 0
\(431\) 39.8722 1.92058 0.960289 0.279008i \(-0.0900057\pi\)
0.960289 + 0.279008i \(0.0900057\pi\)
\(432\) 0 0
\(433\) −30.3681 −1.45940 −0.729698 0.683769i \(-0.760339\pi\)
−0.729698 + 0.683769i \(0.760339\pi\)
\(434\) 5.46721 0.262434
\(435\) 0 0
\(436\) −2.48963 −0.119232
\(437\) −3.67282 −0.175695
\(438\) 0 0
\(439\) 19.4033 0.926070 0.463035 0.886340i \(-0.346760\pi\)
0.463035 + 0.886340i \(0.346760\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 35.2857 1.67837
\(443\) −20.8705 −0.991589 −0.495794 0.868440i \(-0.665123\pi\)
−0.495794 + 0.868440i \(0.665123\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −10.1625 −0.481206
\(447\) 0 0
\(448\) 4.24482 0.200549
\(449\) 30.7753 1.45238 0.726189 0.687495i \(-0.241290\pi\)
0.726189 + 0.687495i \(0.241290\pi\)
\(450\) 0 0
\(451\) −11.4504 −0.539180
\(452\) −11.0185 −0.518265
\(453\) 0 0
\(454\) −7.92159 −0.371779
\(455\) 0 0
\(456\) 0 0
\(457\) 27.5658 1.28947 0.644736 0.764405i \(-0.276967\pi\)
0.644736 + 0.764405i \(0.276967\pi\)
\(458\) 7.44648 0.347951
\(459\) 0 0
\(460\) 0 0
\(461\) −31.3042 −1.45798 −0.728991 0.684524i \(-0.760010\pi\)
−0.728991 + 0.684524i \(0.760010\pi\)
\(462\) 0 0
\(463\) 11.1440 0.517905 0.258952 0.965890i \(-0.416623\pi\)
0.258952 + 0.965890i \(0.416623\pi\)
\(464\) −8.10083 −0.376072
\(465\) 0 0
\(466\) −28.4834 −1.31947
\(467\) 36.3289 1.68110 0.840550 0.541734i \(-0.182232\pi\)
0.840550 + 0.541734i \(0.182232\pi\)
\(468\) 0 0
\(469\) 18.6297 0.860238
\(470\) 0 0
\(471\) 0 0
\(472\) 12.3496 0.568436
\(473\) −5.10083 −0.234536
\(474\) 0 0
\(475\) 0 0
\(476\) 21.6521 0.992422
\(477\) 0 0
\(478\) 3.24086 0.148234
\(479\) −17.0185 −0.777594 −0.388797 0.921323i \(-0.627109\pi\)
−0.388797 + 0.921323i \(0.627109\pi\)
\(480\) 0 0
\(481\) 5.92159 0.270001
\(482\) −25.9938 −1.18398
\(483\) 0 0
\(484\) −8.96080 −0.407309
\(485\) 0 0
\(486\) 0 0
\(487\) −27.3536 −1.23951 −0.619754 0.784796i \(-0.712768\pi\)
−0.619754 + 0.784796i \(0.712768\pi\)
\(488\) −8.20561 −0.371451
\(489\) 0 0
\(490\) 0 0
\(491\) −12.7697 −0.576289 −0.288144 0.957587i \(-0.593038\pi\)
−0.288144 + 0.957587i \(0.593038\pi\)
\(492\) 0 0
\(493\) −41.3210 −1.86100
\(494\) 6.91764 0.311239
\(495\) 0 0
\(496\) −1.28797 −0.0578317
\(497\) −47.1210 −2.11367
\(498\) 0 0
\(499\) −33.8969 −1.51743 −0.758717 0.651420i \(-0.774173\pi\)
−0.758717 + 0.651420i \(0.774173\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.98153 0.177704
\(503\) 35.6706 1.59047 0.795236 0.606300i \(-0.207347\pi\)
0.795236 + 0.606300i \(0.207347\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.24482 −0.233161
\(507\) 0 0
\(508\) 6.05767 0.268766
\(509\) −24.2347 −1.07418 −0.537091 0.843524i \(-0.680477\pi\)
−0.537091 + 0.843524i \(0.680477\pi\)
\(510\) 0 0
\(511\) 22.8577 1.01117
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −14.1625 −0.624679
\(515\) 0 0
\(516\) 0 0
\(517\) −5.45043 −0.239710
\(518\) 3.63362 0.159652
\(519\) 0 0
\(520\) 0 0
\(521\) −34.4689 −1.51011 −0.755055 0.655661i \(-0.772390\pi\)
−0.755055 + 0.655661i \(0.772390\pi\)
\(522\) 0 0
\(523\) 35.3905 1.54752 0.773759 0.633480i \(-0.218374\pi\)
0.773759 + 0.633480i \(0.218374\pi\)
\(524\) 9.44648 0.412671
\(525\) 0 0
\(526\) −21.3434 −0.930616
\(527\) −6.56973 −0.286182
\(528\) 0 0
\(529\) −9.51037 −0.413494
\(530\) 0 0
\(531\) 0 0
\(532\) 4.24482 0.184036
\(533\) −55.4689 −2.40262
\(534\) 0 0
\(535\) 0 0
\(536\) −4.38880 −0.189567
\(537\) 0 0
\(538\) 24.4112 1.05244
\(539\) −15.7345 −0.677731
\(540\) 0 0
\(541\) −21.6560 −0.931066 −0.465533 0.885031i \(-0.654137\pi\)
−0.465533 + 0.885031i \(0.654137\pi\)
\(542\) 14.8129 0.636267
\(543\) 0 0
\(544\) −5.10083 −0.218696
\(545\) 0 0
\(546\) 0 0
\(547\) −3.21844 −0.137611 −0.0688053 0.997630i \(-0.521919\pi\)
−0.0688053 + 0.997630i \(0.521919\pi\)
\(548\) −4.20166 −0.179486
\(549\) 0 0
\(550\) 0 0
\(551\) −8.10083 −0.345107
\(552\) 0 0
\(553\) 9.10083 0.387007
\(554\) −15.2633 −0.648475
\(555\) 0 0
\(556\) −5.40727 −0.229320
\(557\) −40.3865 −1.71123 −0.855616 0.517611i \(-0.826822\pi\)
−0.855616 + 0.517611i \(0.826822\pi\)
\(558\) 0 0
\(559\) −24.7098 −1.04511
\(560\) 0 0
\(561\) 0 0
\(562\) −2.83528 −0.119599
\(563\) −9.65831 −0.407049 −0.203525 0.979070i \(-0.565240\pi\)
−0.203525 + 0.979070i \(0.565240\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.942326 −0.0396089
\(567\) 0 0
\(568\) 11.1008 0.465780
\(569\) −18.6992 −0.783911 −0.391956 0.919984i \(-0.628201\pi\)
−0.391956 + 0.919984i \(0.628201\pi\)
\(570\) 0 0
\(571\) 4.34169 0.181694 0.0908471 0.995865i \(-0.471043\pi\)
0.0908471 + 0.995865i \(0.471043\pi\)
\(572\) 9.87844 0.413038
\(573\) 0 0
\(574\) −34.0369 −1.42067
\(575\) 0 0
\(576\) 0 0
\(577\) 3.03694 0.126430 0.0632148 0.998000i \(-0.479865\pi\)
0.0632148 + 0.998000i \(0.479865\pi\)
\(578\) −9.01847 −0.375119
\(579\) 0 0
\(580\) 0 0
\(581\) −4.42801 −0.183705
\(582\) 0 0
\(583\) 12.9401 0.535923
\(584\) −5.38485 −0.222827
\(585\) 0 0
\(586\) −13.5513 −0.559797
\(587\) 33.7345 1.39237 0.696185 0.717863i \(-0.254879\pi\)
0.696185 + 0.717863i \(0.254879\pi\)
\(588\) 0 0
\(589\) −1.28797 −0.0530700
\(590\) 0 0
\(591\) 0 0
\(592\) −0.856013 −0.0351819
\(593\) −34.7361 −1.42644 −0.713221 0.700939i \(-0.752764\pi\)
−0.713221 + 0.700939i \(0.752764\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.51037 −0.0618670
\(597\) 0 0
\(598\) −25.4073 −1.03898
\(599\) 6.28176 0.256666 0.128333 0.991731i \(-0.459037\pi\)
0.128333 + 0.991731i \(0.459037\pi\)
\(600\) 0 0
\(601\) 18.4544 0.752770 0.376385 0.926463i \(-0.377167\pi\)
0.376385 + 0.926463i \(0.377167\pi\)
\(602\) −15.1625 −0.617976
\(603\) 0 0
\(604\) −15.4504 −0.628669
\(605\) 0 0
\(606\) 0 0
\(607\) −20.5367 −0.833561 −0.416780 0.909007i \(-0.636842\pi\)
−0.416780 + 0.909007i \(0.636842\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −26.4033 −1.06816
\(612\) 0 0
\(613\) −11.9216 −0.481509 −0.240754 0.970586i \(-0.577395\pi\)
−0.240754 + 0.970586i \(0.577395\pi\)
\(614\) 26.7921 1.08124
\(615\) 0 0
\(616\) 6.06163 0.244230
\(617\) −17.5473 −0.706428 −0.353214 0.935543i \(-0.614911\pi\)
−0.353214 + 0.935543i \(0.614911\pi\)
\(618\) 0 0
\(619\) −33.7322 −1.35581 −0.677906 0.735149i \(-0.737112\pi\)
−0.677906 + 0.735149i \(0.737112\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5.03920 0.202054
\(623\) −71.0179 −2.84527
\(624\) 0 0
\(625\) 0 0
\(626\) 15.0185 0.600259
\(627\) 0 0
\(628\) −9.16246 −0.365622
\(629\) −4.36638 −0.174099
\(630\) 0 0
\(631\) 22.4033 0.891862 0.445931 0.895067i \(-0.352873\pi\)
0.445931 + 0.895067i \(0.352873\pi\)
\(632\) −2.14399 −0.0852832
\(633\) 0 0
\(634\) −20.1008 −0.798306
\(635\) 0 0
\(636\) 0 0
\(637\) −76.2218 −3.02002
\(638\) −11.5680 −0.457983
\(639\) 0 0
\(640\) 0 0
\(641\) 25.2778 0.998413 0.499207 0.866483i \(-0.333625\pi\)
0.499207 + 0.866483i \(0.333625\pi\)
\(642\) 0 0
\(643\) −13.4320 −0.529705 −0.264852 0.964289i \(-0.585323\pi\)
−0.264852 + 0.964289i \(0.585323\pi\)
\(644\) −15.5905 −0.614350
\(645\) 0 0
\(646\) −5.10083 −0.200689
\(647\) −5.95289 −0.234032 −0.117016 0.993130i \(-0.537333\pi\)
−0.117016 + 0.993130i \(0.537333\pi\)
\(648\) 0 0
\(649\) 17.6353 0.692247
\(650\) 0 0
\(651\) 0 0
\(652\) −12.6913 −0.497029
\(653\) −35.3826 −1.38463 −0.692314 0.721597i \(-0.743409\pi\)
−0.692314 + 0.721597i \(0.743409\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.01847 0.313069
\(657\) 0 0
\(658\) −16.2017 −0.631606
\(659\) −46.2218 −1.80055 −0.900273 0.435325i \(-0.856633\pi\)
−0.900273 + 0.435325i \(0.856633\pi\)
\(660\) 0 0
\(661\) −37.5843 −1.46186 −0.730929 0.682454i \(-0.760913\pi\)
−0.730929 + 0.682454i \(0.760913\pi\)
\(662\) 25.5697 0.993796
\(663\) 0 0
\(664\) 1.04316 0.0404823
\(665\) 0 0
\(666\) 0 0
\(667\) 29.7529 1.15204
\(668\) −16.3249 −0.631630
\(669\) 0 0
\(670\) 0 0
\(671\) −11.7177 −0.452356
\(672\) 0 0
\(673\) −43.4257 −1.67394 −0.836970 0.547249i \(-0.815675\pi\)
−0.836970 + 0.547249i \(0.815675\pi\)
\(674\) 13.9137 0.535935
\(675\) 0 0
\(676\) 34.8538 1.34053
\(677\) −25.2218 −0.969353 −0.484677 0.874693i \(-0.661063\pi\)
−0.484677 + 0.874693i \(0.661063\pi\)
\(678\) 0 0
\(679\) 28.9193 1.10982
\(680\) 0 0
\(681\) 0 0
\(682\) −1.83923 −0.0704279
\(683\) 18.0616 0.691109 0.345554 0.938399i \(-0.387691\pi\)
0.345554 + 0.938399i \(0.387691\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −17.0577 −0.651265
\(687\) 0 0
\(688\) 3.57199 0.136181
\(689\) 62.6851 2.38811
\(690\) 0 0
\(691\) −9.36243 −0.356163 −0.178082 0.984016i \(-0.556989\pi\)
−0.178082 + 0.984016i \(0.556989\pi\)
\(692\) −3.32322 −0.126330
\(693\) 0 0
\(694\) −2.56804 −0.0974815
\(695\) 0 0
\(696\) 0 0
\(697\) 40.9009 1.54923
\(698\) 5.23691 0.198220
\(699\) 0 0
\(700\) 0 0
\(701\) 52.5714 1.98560 0.992798 0.119803i \(-0.0382264\pi\)
0.992798 + 0.119803i \(0.0382264\pi\)
\(702\) 0 0
\(703\) −0.856013 −0.0322852
\(704\) −1.42801 −0.0538200
\(705\) 0 0
\(706\) −12.2818 −0.462230
\(707\) −49.9770 −1.87958
\(708\) 0 0
\(709\) 14.3025 0.537141 0.268571 0.963260i \(-0.413449\pi\)
0.268571 + 0.963260i \(0.413449\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 16.7305 0.627002
\(713\) 4.73050 0.177159
\(714\) 0 0
\(715\) 0 0
\(716\) 5.54731 0.207313
\(717\) 0 0
\(718\) 33.7753 1.26048
\(719\) −50.6706 −1.88969 −0.944847 0.327513i \(-0.893789\pi\)
−0.944847 + 0.327513i \(0.893789\pi\)
\(720\) 0 0
\(721\) 18.3681 0.684063
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −17.2840 −0.642356
\(725\) 0 0
\(726\) 0 0
\(727\) −43.0083 −1.59509 −0.797545 0.603260i \(-0.793868\pi\)
−0.797545 + 0.603260i \(0.793868\pi\)
\(728\) 29.3641 1.08831
\(729\) 0 0
\(730\) 0 0
\(731\) 18.2201 0.673896
\(732\) 0 0
\(733\) −1.77592 −0.0655949 −0.0327975 0.999462i \(-0.510442\pi\)
−0.0327975 + 0.999462i \(0.510442\pi\)
\(734\) 30.3681 1.12091
\(735\) 0 0
\(736\) 3.67282 0.135382
\(737\) −6.26724 −0.230857
\(738\) 0 0
\(739\) 5.43196 0.199818 0.0999089 0.994997i \(-0.468145\pi\)
0.0999089 + 0.994997i \(0.468145\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 38.4649 1.41209
\(743\) 0.769701 0.0282376 0.0141188 0.999900i \(-0.495506\pi\)
0.0141188 + 0.999900i \(0.495506\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −25.7322 −0.942122
\(747\) 0 0
\(748\) −7.28402 −0.266330
\(749\) 17.5737 0.642128
\(750\) 0 0
\(751\) 42.6683 1.55699 0.778494 0.627652i \(-0.215984\pi\)
0.778494 + 0.627652i \(0.215984\pi\)
\(752\) 3.81681 0.139185
\(753\) 0 0
\(754\) −56.0386 −2.04081
\(755\) 0 0
\(756\) 0 0
\(757\) −39.2448 −1.42638 −0.713189 0.700972i \(-0.752750\pi\)
−0.713189 + 0.700972i \(0.752750\pi\)
\(758\) −37.2593 −1.35332
\(759\) 0 0
\(760\) 0 0
\(761\) −1.38090 −0.0500575 −0.0250288 0.999687i \(-0.507968\pi\)
−0.0250288 + 0.999687i \(0.507968\pi\)
\(762\) 0 0
\(763\) −10.5680 −0.382589
\(764\) −22.6336 −0.818856
\(765\) 0 0
\(766\) 31.9154 1.15315
\(767\) 85.4301 3.08470
\(768\) 0 0
\(769\) 32.9691 1.18890 0.594448 0.804134i \(-0.297371\pi\)
0.594448 + 0.804134i \(0.297371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.24482 0.152774
\(773\) 32.4975 1.16886 0.584428 0.811446i \(-0.301319\pi\)
0.584428 + 0.811446i \(0.301319\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.81286 −0.244567
\(777\) 0 0
\(778\) −4.48963 −0.160961
\(779\) 8.01847 0.287292
\(780\) 0 0
\(781\) 15.8521 0.567231
\(782\) 18.7345 0.669943
\(783\) 0 0
\(784\) 11.0185 0.393517
\(785\) 0 0
\(786\) 0 0
\(787\) 0.187143 0.00667091 0.00333546 0.999994i \(-0.498938\pi\)
0.00333546 + 0.999994i \(0.498938\pi\)
\(788\) 5.00395 0.178258
\(789\) 0 0
\(790\) 0 0
\(791\) −46.7714 −1.66300
\(792\) 0 0
\(793\) −56.7635 −2.01573
\(794\) 34.6873 1.23101
\(795\) 0 0
\(796\) −27.3826 −0.970550
\(797\) 33.9691 1.20325 0.601624 0.798780i \(-0.294521\pi\)
0.601624 + 0.798780i \(0.294521\pi\)
\(798\) 0 0
\(799\) 19.4689 0.688760
\(800\) 0 0
\(801\) 0 0
\(802\) −7.03694 −0.248483
\(803\) −7.68960 −0.271360
\(804\) 0 0
\(805\) 0 0
\(806\) −8.90973 −0.313832
\(807\) 0 0
\(808\) 11.7737 0.414196
\(809\) −34.2897 −1.20556 −0.602780 0.797907i \(-0.705940\pi\)
−0.602780 + 0.797907i \(0.705940\pi\)
\(810\) 0 0
\(811\) −18.7737 −0.659232 −0.329616 0.944115i \(-0.606919\pi\)
−0.329616 + 0.944115i \(0.606919\pi\)
\(812\) −34.3865 −1.20673
\(813\) 0 0
\(814\) −1.22239 −0.0428449
\(815\) 0 0
\(816\) 0 0
\(817\) 3.57199 0.124968
\(818\) −26.4112 −0.923447
\(819\) 0 0
\(820\) 0 0
\(821\) −1.16077 −0.0405110 −0.0202555 0.999795i \(-0.506448\pi\)
−0.0202555 + 0.999795i \(0.506448\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −4.32718 −0.150744
\(825\) 0 0
\(826\) 52.4218 1.82399
\(827\) −35.4857 −1.23396 −0.616979 0.786980i \(-0.711644\pi\)
−0.616979 + 0.786980i \(0.711644\pi\)
\(828\) 0 0
\(829\) −10.9546 −0.380468 −0.190234 0.981739i \(-0.560925\pi\)
−0.190234 + 0.981739i \(0.560925\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6.91764 −0.239826
\(833\) 56.2034 1.94733
\(834\) 0 0
\(835\) 0 0
\(836\) −1.42801 −0.0493886
\(837\) 0 0
\(838\) −16.4896 −0.569625
\(839\) 0.417441 0.0144117 0.00720584 0.999974i \(-0.497706\pi\)
0.00720584 + 0.999974i \(0.497706\pi\)
\(840\) 0 0
\(841\) 36.6235 1.26288
\(842\) −39.0162 −1.34459
\(843\) 0 0
\(844\) −5.14003 −0.176927
\(845\) 0 0
\(846\) 0 0
\(847\) −38.0369 −1.30696
\(848\) −9.06163 −0.311178
\(849\) 0 0
\(850\) 0 0
\(851\) 3.14399 0.107774
\(852\) 0 0
\(853\) −44.7776 −1.53316 −0.766578 0.642151i \(-0.778042\pi\)
−0.766578 + 0.642151i \(0.778042\pi\)
\(854\) −34.8313 −1.19190
\(855\) 0 0
\(856\) −4.14003 −0.141503
\(857\) 43.3720 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(858\) 0 0
\(859\) −2.36638 −0.0807398 −0.0403699 0.999185i \(-0.512854\pi\)
−0.0403699 + 0.999185i \(0.512854\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −39.8722 −1.35805
\(863\) 19.9921 0.680539 0.340269 0.940328i \(-0.389482\pi\)
0.340269 + 0.940328i \(0.389482\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 30.3681 1.03195
\(867\) 0 0
\(868\) −5.46721 −0.185569
\(869\) −3.06163 −0.103859
\(870\) 0 0
\(871\) −30.3602 −1.02871
\(872\) 2.48963 0.0843096
\(873\) 0 0
\(874\) 3.67282 0.124235
\(875\) 0 0
\(876\) 0 0
\(877\) 19.0946 0.644779 0.322390 0.946607i \(-0.395514\pi\)
0.322390 + 0.946607i \(0.395514\pi\)
\(878\) −19.4033 −0.654830
\(879\) 0 0
\(880\) 0 0
\(881\) −18.1233 −0.610588 −0.305294 0.952258i \(-0.598755\pi\)
−0.305294 + 0.952258i \(0.598755\pi\)
\(882\) 0 0
\(883\) 19.7737 0.665436 0.332718 0.943026i \(-0.392034\pi\)
0.332718 + 0.943026i \(0.392034\pi\)
\(884\) −35.2857 −1.18679
\(885\) 0 0
\(886\) 20.8705 0.701159
\(887\) −35.5473 −1.19356 −0.596781 0.802404i \(-0.703554\pi\)
−0.596781 + 0.802404i \(0.703554\pi\)
\(888\) 0 0
\(889\) 25.7137 0.862410
\(890\) 0 0
\(891\) 0 0
\(892\) 10.1625 0.340264
\(893\) 3.81681 0.127725
\(894\) 0 0
\(895\) 0 0
\(896\) −4.24482 −0.141809
\(897\) 0 0
\(898\) −30.7753 −1.02699
\(899\) 10.4337 0.347982
\(900\) 0 0
\(901\) −46.2218 −1.53987
\(902\) 11.4504 0.381258
\(903\) 0 0
\(904\) 11.0185 0.366469
\(905\) 0 0
\(906\) 0 0
\(907\) 32.0554 1.06438 0.532191 0.846624i \(-0.321369\pi\)
0.532191 + 0.846624i \(0.321369\pi\)
\(908\) 7.92159 0.262887
\(909\) 0 0
\(910\) 0 0
\(911\) 16.9714 0.562286 0.281143 0.959666i \(-0.409286\pi\)
0.281143 + 0.959666i \(0.409286\pi\)
\(912\) 0 0
\(913\) 1.48963 0.0492997
\(914\) −27.5658 −0.911795
\(915\) 0 0
\(916\) −7.44648 −0.246039
\(917\) 40.0986 1.32417
\(918\) 0 0
\(919\) 32.4465 1.07031 0.535155 0.844754i \(-0.320253\pi\)
0.535155 + 0.844754i \(0.320253\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 31.3042 1.03095
\(923\) 76.7916 2.52762
\(924\) 0 0
\(925\) 0 0
\(926\) −11.1440 −0.366214
\(927\) 0 0
\(928\) 8.10083 0.265923
\(929\) 13.9631 0.458113 0.229057 0.973413i \(-0.426436\pi\)
0.229057 + 0.973413i \(0.426436\pi\)
\(930\) 0 0
\(931\) 11.0185 0.361116
\(932\) 28.4834 0.933005
\(933\) 0 0
\(934\) −36.3289 −1.18872
\(935\) 0 0
\(936\) 0 0
\(937\) 19.5058 0.637228 0.318614 0.947885i \(-0.396783\pi\)
0.318614 + 0.947885i \(0.396783\pi\)
\(938\) −18.6297 −0.608280
\(939\) 0 0
\(940\) 0 0
\(941\) 8.48568 0.276625 0.138313 0.990389i \(-0.455832\pi\)
0.138313 + 0.990389i \(0.455832\pi\)
\(942\) 0 0
\(943\) −29.4504 −0.959038
\(944\) −12.3496 −0.401945
\(945\) 0 0
\(946\) 5.10083 0.165842
\(947\) 5.06558 0.164609 0.0823046 0.996607i \(-0.473772\pi\)
0.0823046 + 0.996607i \(0.473772\pi\)
\(948\) 0 0
\(949\) −37.2505 −1.20920
\(950\) 0 0
\(951\) 0 0
\(952\) −21.6521 −0.701748
\(953\) −4.46890 −0.144762 −0.0723810 0.997377i \(-0.523060\pi\)
−0.0723810 + 0.997377i \(0.523060\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −3.24086 −0.104817
\(957\) 0 0
\(958\) 17.0185 0.549842
\(959\) −17.8353 −0.575931
\(960\) 0 0
\(961\) −29.3411 −0.946488
\(962\) −5.92159 −0.190920
\(963\) 0 0
\(964\) 25.9938 0.837203
\(965\) 0 0
\(966\) 0 0
\(967\) −10.9872 −0.353324 −0.176662 0.984272i \(-0.556530\pi\)
−0.176662 + 0.984272i \(0.556530\pi\)
\(968\) 8.96080 0.288011
\(969\) 0 0
\(970\) 0 0
\(971\) 11.8969 0.381790 0.190895 0.981610i \(-0.438861\pi\)
0.190895 + 0.981610i \(0.438861\pi\)
\(972\) 0 0
\(973\) −22.9529 −0.735836
\(974\) 27.3536 0.876464
\(975\) 0 0
\(976\) 8.20561 0.262655
\(977\) −29.9031 −0.956686 −0.478343 0.878173i \(-0.658762\pi\)
−0.478343 + 0.878173i \(0.658762\pi\)
\(978\) 0 0
\(979\) 23.8913 0.763568
\(980\) 0 0
\(981\) 0 0
\(982\) 12.7697 0.407498
\(983\) −6.89917 −0.220049 −0.110025 0.993929i \(-0.535093\pi\)
−0.110025 + 0.993929i \(0.535093\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 41.3210 1.31593
\(987\) 0 0
\(988\) −6.91764 −0.220079
\(989\) −13.1193 −0.417170
\(990\) 0 0
\(991\) 12.3720 0.393010 0.196505 0.980503i \(-0.437041\pi\)
0.196505 + 0.980503i \(0.437041\pi\)
\(992\) 1.28797 0.0408932
\(993\) 0 0
\(994\) 47.1210 1.49459
\(995\) 0 0
\(996\) 0 0
\(997\) −2.73671 −0.0866725 −0.0433363 0.999061i \(-0.513799\pi\)
−0.0433363 + 0.999061i \(0.513799\pi\)
\(998\) 33.8969 1.07299
\(999\) 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 8550.2.a.cg.1.3 yes 3
3.2 odd 2 8550.2.a.cs.1.3 yes 3
5.4 even 2 8550.2.a.cn.1.1 yes 3
15.14 odd 2 8550.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8550.2.a.cd.1.1 3 15.14 odd 2
8550.2.a.cg.1.3 yes 3 1.1 even 1 trivial
8550.2.a.cn.1.1 yes 3 5.4 even 2
8550.2.a.cs.1.3 yes 3 3.2 odd 2