Properties

Label 882.3.b.h.197.4
Level $882$
Weight $3$
Character 882.197
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.3.b.h.197.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} +4.00000i q^{5} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} +4.00000i q^{5} -2.82843i q^{8} -5.65685 q^{10} -2.82843i q^{11} +12.7279 q^{13} +4.00000 q^{16} +4.00000i q^{17} +22.6274 q^{19} -8.00000i q^{20} +4.00000 q^{22} +36.7696i q^{23} +9.00000 q^{25} +18.0000i q^{26} +32.5269i q^{29} -50.9117 q^{31} +5.65685i q^{32} -5.65685 q^{34} -32.0000 q^{37} +32.0000i q^{38} +11.3137 q^{40} -38.0000i q^{41} +20.0000 q^{43} +5.65685i q^{44} -52.0000 q^{46} -20.0000i q^{47} +12.7279i q^{50} -25.4558 q^{52} +94.7523i q^{53} +11.3137 q^{55} -46.0000 q^{58} +4.00000i q^{59} +83.4386 q^{61} -72.0000i q^{62} -8.00000 q^{64} +50.9117i q^{65} -48.0000 q^{67} -8.00000i q^{68} +76.3675i q^{71} -120.208 q^{73} -45.2548i q^{74} -45.2548 q^{76} -148.000 q^{79} +16.0000i q^{80} +53.7401 q^{82} +80.0000i q^{83} -16.0000 q^{85} +28.2843i q^{86} -8.00000 q^{88} -106.000i q^{89} -73.5391i q^{92} +28.2843 q^{94} +90.5097i q^{95} +154.149 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + O(q^{10}) \) \( 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} + 36 q^{25} - 128 q^{37} + 80 q^{43} - 208 q^{46} - 184 q^{58} - 32 q^{64} - 192 q^{67} - 592 q^{79} - 64 q^{85} - 32 q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 4.00000i 0.800000i 0.916515 + 0.400000i \(0.130990\pi\)
−0.916515 + 0.400000i \(0.869010\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) −5.65685 −0.565685
\(11\) − 2.82843i − 0.257130i −0.991701 0.128565i \(-0.958963\pi\)
0.991701 0.128565i \(-0.0410371\pi\)
\(12\) 0 0
\(13\) 12.7279 0.979071 0.489535 0.871983i \(-0.337166\pi\)
0.489535 + 0.871983i \(0.337166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 4.00000i 0.235294i 0.993055 + 0.117647i \(0.0375352\pi\)
−0.993055 + 0.117647i \(0.962465\pi\)
\(18\) 0 0
\(19\) 22.6274 1.19092 0.595458 0.803386i \(-0.296970\pi\)
0.595458 + 0.803386i \(0.296970\pi\)
\(20\) − 8.00000i − 0.400000i
\(21\) 0 0
\(22\) 4.00000 0.181818
\(23\) 36.7696i 1.59868i 0.600882 + 0.799338i \(0.294816\pi\)
−0.600882 + 0.799338i \(0.705184\pi\)
\(24\) 0 0
\(25\) 9.00000 0.360000
\(26\) 18.0000i 0.692308i
\(27\) 0 0
\(28\) 0 0
\(29\) 32.5269i 1.12162i 0.827945 + 0.560809i \(0.189510\pi\)
−0.827945 + 0.560809i \(0.810490\pi\)
\(30\) 0 0
\(31\) −50.9117 −1.64231 −0.821156 0.570703i \(-0.806671\pi\)
−0.821156 + 0.570703i \(0.806671\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) −5.65685 −0.166378
\(35\) 0 0
\(36\) 0 0
\(37\) −32.0000 −0.864865 −0.432432 0.901666i \(-0.642345\pi\)
−0.432432 + 0.901666i \(0.642345\pi\)
\(38\) 32.0000i 0.842105i
\(39\) 0 0
\(40\) 11.3137 0.282843
\(41\) − 38.0000i − 0.926829i −0.886142 0.463415i \(-0.846624\pi\)
0.886142 0.463415i \(-0.153376\pi\)
\(42\) 0 0
\(43\) 20.0000 0.465116 0.232558 0.972582i \(-0.425290\pi\)
0.232558 + 0.972582i \(0.425290\pi\)
\(44\) 5.65685i 0.128565i
\(45\) 0 0
\(46\) −52.0000 −1.13043
\(47\) − 20.0000i − 0.425532i −0.977103 0.212766i \(-0.931753\pi\)
0.977103 0.212766i \(-0.0682472\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 12.7279i 0.254558i
\(51\) 0 0
\(52\) −25.4558 −0.489535
\(53\) 94.7523i 1.78778i 0.448287 + 0.893890i \(0.352034\pi\)
−0.448287 + 0.893890i \(0.647966\pi\)
\(54\) 0 0
\(55\) 11.3137 0.205704
\(56\) 0 0
\(57\) 0 0
\(58\) −46.0000 −0.793103
\(59\) 4.00000i 0.0677966i 0.999425 + 0.0338983i \(0.0107922\pi\)
−0.999425 + 0.0338983i \(0.989208\pi\)
\(60\) 0 0
\(61\) 83.4386 1.36785 0.683923 0.729554i \(-0.260272\pi\)
0.683923 + 0.729554i \(0.260272\pi\)
\(62\) − 72.0000i − 1.16129i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 50.9117i 0.783257i
\(66\) 0 0
\(67\) −48.0000 −0.716418 −0.358209 0.933641i \(-0.616612\pi\)
−0.358209 + 0.933641i \(0.616612\pi\)
\(68\) − 8.00000i − 0.117647i
\(69\) 0 0
\(70\) 0 0
\(71\) 76.3675i 1.07560i 0.843073 + 0.537800i \(0.180744\pi\)
−0.843073 + 0.537800i \(0.819256\pi\)
\(72\) 0 0
\(73\) −120.208 −1.64669 −0.823344 0.567543i \(-0.807894\pi\)
−0.823344 + 0.567543i \(0.807894\pi\)
\(74\) − 45.2548i − 0.611552i
\(75\) 0 0
\(76\) −45.2548 −0.595458
\(77\) 0 0
\(78\) 0 0
\(79\) −148.000 −1.87342 −0.936709 0.350109i \(-0.886144\pi\)
−0.936709 + 0.350109i \(0.886144\pi\)
\(80\) 16.0000i 0.200000i
\(81\) 0 0
\(82\) 53.7401 0.655367
\(83\) 80.0000i 0.963855i 0.876211 + 0.481928i \(0.160063\pi\)
−0.876211 + 0.481928i \(0.839937\pi\)
\(84\) 0 0
\(85\) −16.0000 −0.188235
\(86\) 28.2843i 0.328887i
\(87\) 0 0
\(88\) −8.00000 −0.0909091
\(89\) − 106.000i − 1.19101i −0.803351 0.595506i \(-0.796952\pi\)
0.803351 0.595506i \(-0.203048\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 73.5391i − 0.799338i
\(93\) 0 0
\(94\) 28.2843 0.300897
\(95\) 90.5097i 0.952733i
\(96\) 0 0
\(97\) 154.149 1.58917 0.794584 0.607154i \(-0.207689\pi\)
0.794584 + 0.607154i \(0.207689\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −18.0000 −0.180000
\(101\) 126.000i 1.24752i 0.781614 + 0.623762i \(0.214397\pi\)
−0.781614 + 0.623762i \(0.785603\pi\)
\(102\) 0 0
\(103\) −73.5391 −0.713972 −0.356986 0.934110i \(-0.616196\pi\)
−0.356986 + 0.934110i \(0.616196\pi\)
\(104\) − 36.0000i − 0.346154i
\(105\) 0 0
\(106\) −134.000 −1.26415
\(107\) 70.7107i 0.660847i 0.943833 + 0.330424i \(0.107192\pi\)
−0.943833 + 0.330424i \(0.892808\pi\)
\(108\) 0 0
\(109\) −86.0000 −0.788991 −0.394495 0.918898i \(-0.629081\pi\)
−0.394495 + 0.918898i \(0.629081\pi\)
\(110\) 16.0000i 0.145455i
\(111\) 0 0
\(112\) 0 0
\(113\) − 21.2132i − 0.187727i −0.995585 0.0938637i \(-0.970078\pi\)
0.995585 0.0938637i \(-0.0299218\pi\)
\(114\) 0 0
\(115\) −147.078 −1.27894
\(116\) − 65.0538i − 0.560809i
\(117\) 0 0
\(118\) −5.65685 −0.0479394
\(119\) 0 0
\(120\) 0 0
\(121\) 113.000 0.933884
\(122\) 118.000i 0.967213i
\(123\) 0 0
\(124\) 101.823 0.821156
\(125\) 136.000i 1.08800i
\(126\) 0 0
\(127\) 4.00000 0.0314961 0.0157480 0.999876i \(-0.494987\pi\)
0.0157480 + 0.999876i \(0.494987\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) 0 0
\(130\) −72.0000 −0.553846
\(131\) − 24.0000i − 0.183206i −0.995796 0.0916031i \(-0.970801\pi\)
0.995796 0.0916031i \(-0.0291991\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 67.8823i − 0.506584i
\(135\) 0 0
\(136\) 11.3137 0.0831890
\(137\) 72.1249i 0.526459i 0.964733 + 0.263230i \(0.0847877\pi\)
−0.964733 + 0.263230i \(0.915212\pi\)
\(138\) 0 0
\(139\) 141.421 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −108.000 −0.760563
\(143\) − 36.0000i − 0.251748i
\(144\) 0 0
\(145\) −130.108 −0.897294
\(146\) − 170.000i − 1.16438i
\(147\) 0 0
\(148\) 64.0000 0.432432
\(149\) 26.8701i 0.180336i 0.995927 + 0.0901680i \(0.0287404\pi\)
−0.995927 + 0.0901680i \(0.971260\pi\)
\(150\) 0 0
\(151\) 152.000 1.00662 0.503311 0.864105i \(-0.332115\pi\)
0.503311 + 0.864105i \(0.332115\pi\)
\(152\) − 64.0000i − 0.421053i
\(153\) 0 0
\(154\) 0 0
\(155\) − 203.647i − 1.31385i
\(156\) 0 0
\(157\) 94.7523 0.603518 0.301759 0.953384i \(-0.402426\pi\)
0.301759 + 0.953384i \(0.402426\pi\)
\(158\) − 209.304i − 1.32471i
\(159\) 0 0
\(160\) −22.6274 −0.141421
\(161\) 0 0
\(162\) 0 0
\(163\) −152.000 −0.932515 −0.466258 0.884649i \(-0.654398\pi\)
−0.466258 + 0.884649i \(0.654398\pi\)
\(164\) 76.0000i 0.463415i
\(165\) 0 0
\(166\) −113.137 −0.681549
\(167\) 116.000i 0.694611i 0.937752 + 0.347305i \(0.112903\pi\)
−0.937752 + 0.347305i \(0.887097\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.0414201
\(170\) − 22.6274i − 0.133102i
\(171\) 0 0
\(172\) −40.0000 −0.232558
\(173\) − 222.000i − 1.28324i −0.767024 0.641618i \(-0.778263\pi\)
0.767024 0.641618i \(-0.221737\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 11.3137i − 0.0642824i
\(177\) 0 0
\(178\) 149.907 0.842172
\(179\) 59.3970i 0.331827i 0.986140 + 0.165913i \(0.0530572\pi\)
−0.986140 + 0.165913i \(0.946943\pi\)
\(180\) 0 0
\(181\) 80.6102 0.445360 0.222680 0.974892i \(-0.428519\pi\)
0.222680 + 0.974892i \(0.428519\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 104.000 0.565217
\(185\) − 128.000i − 0.691892i
\(186\) 0 0
\(187\) 11.3137 0.0605011
\(188\) 40.0000i 0.212766i
\(189\) 0 0
\(190\) −128.000 −0.673684
\(191\) − 302.642i − 1.58451i −0.610189 0.792256i \(-0.708907\pi\)
0.610189 0.792256i \(-0.291093\pi\)
\(192\) 0 0
\(193\) 218.000 1.12953 0.564767 0.825251i \(-0.308966\pi\)
0.564767 + 0.825251i \(0.308966\pi\)
\(194\) 218.000i 1.12371i
\(195\) 0 0
\(196\) 0 0
\(197\) 287.085i 1.45729i 0.684894 + 0.728643i \(0.259849\pi\)
−0.684894 + 0.728643i \(0.740151\pi\)
\(198\) 0 0
\(199\) −67.8823 −0.341117 −0.170558 0.985348i \(-0.554557\pi\)
−0.170558 + 0.985348i \(0.554557\pi\)
\(200\) − 25.4558i − 0.127279i
\(201\) 0 0
\(202\) −178.191 −0.882133
\(203\) 0 0
\(204\) 0 0
\(205\) 152.000 0.741463
\(206\) − 104.000i − 0.504854i
\(207\) 0 0
\(208\) 50.9117 0.244768
\(209\) − 64.0000i − 0.306220i
\(210\) 0 0
\(211\) 132.000 0.625592 0.312796 0.949820i \(-0.398734\pi\)
0.312796 + 0.949820i \(0.398734\pi\)
\(212\) − 189.505i − 0.893890i
\(213\) 0 0
\(214\) −100.000 −0.467290
\(215\) 80.0000i 0.372093i
\(216\) 0 0
\(217\) 0 0
\(218\) − 121.622i − 0.557901i
\(219\) 0 0
\(220\) −22.6274 −0.102852
\(221\) 50.9117i 0.230370i
\(222\) 0 0
\(223\) −169.706 −0.761012 −0.380506 0.924778i \(-0.624250\pi\)
−0.380506 + 0.924778i \(0.624250\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 30.0000 0.132743
\(227\) − 300.000i − 1.32159i −0.750568 0.660793i \(-0.770220\pi\)
0.750568 0.660793i \(-0.229780\pi\)
\(228\) 0 0
\(229\) −250.316 −1.09308 −0.546541 0.837432i \(-0.684056\pi\)
−0.546541 + 0.837432i \(0.684056\pi\)
\(230\) − 208.000i − 0.904348i
\(231\) 0 0
\(232\) 92.0000 0.396552
\(233\) 29.6985i 0.127461i 0.997967 + 0.0637307i \(0.0202999\pi\)
−0.997967 + 0.0637307i \(0.979700\pi\)
\(234\) 0 0
\(235\) 80.0000 0.340426
\(236\) − 8.00000i − 0.0338983i
\(237\) 0 0
\(238\) 0 0
\(239\) 161.220i 0.674562i 0.941404 + 0.337281i \(0.109507\pi\)
−0.941404 + 0.337281i \(0.890493\pi\)
\(240\) 0 0
\(241\) 377.595 1.56678 0.783392 0.621528i \(-0.213488\pi\)
0.783392 + 0.621528i \(0.213488\pi\)
\(242\) 159.806i 0.660356i
\(243\) 0 0
\(244\) −166.877 −0.683923
\(245\) 0 0
\(246\) 0 0
\(247\) 288.000 1.16599
\(248\) 144.000i 0.580645i
\(249\) 0 0
\(250\) −192.333 −0.769332
\(251\) 476.000i 1.89641i 0.317653 + 0.948207i \(0.397105\pi\)
−0.317653 + 0.948207i \(0.602895\pi\)
\(252\) 0 0
\(253\) 104.000 0.411067
\(254\) 5.65685i 0.0222711i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 102.000i 0.396887i 0.980112 + 0.198444i \(0.0635887\pi\)
−0.980112 + 0.198444i \(0.936411\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 101.823i − 0.391628i
\(261\) 0 0
\(262\) 33.9411 0.129546
\(263\) − 364.867i − 1.38733i −0.720299 0.693664i \(-0.755995\pi\)
0.720299 0.693664i \(-0.244005\pi\)
\(264\) 0 0
\(265\) −379.009 −1.43022
\(266\) 0 0
\(267\) 0 0
\(268\) 96.0000 0.358209
\(269\) − 196.000i − 0.728625i −0.931277 0.364312i \(-0.881304\pi\)
0.931277 0.364312i \(-0.118696\pi\)
\(270\) 0 0
\(271\) 175.362 0.647094 0.323547 0.946212i \(-0.395125\pi\)
0.323547 + 0.946212i \(0.395125\pi\)
\(272\) 16.0000i 0.0588235i
\(273\) 0 0
\(274\) −102.000 −0.372263
\(275\) − 25.4558i − 0.0925667i
\(276\) 0 0
\(277\) 256.000 0.924188 0.462094 0.886831i \(-0.347098\pi\)
0.462094 + 0.886831i \(0.347098\pi\)
\(278\) 200.000i 0.719424i
\(279\) 0 0
\(280\) 0 0
\(281\) 26.8701i 0.0956230i 0.998856 + 0.0478115i \(0.0152247\pi\)
−0.998856 + 0.0478115i \(0.984775\pi\)
\(282\) 0 0
\(283\) 503.460 1.77901 0.889505 0.456925i \(-0.151049\pi\)
0.889505 + 0.456925i \(0.151049\pi\)
\(284\) − 152.735i − 0.537800i
\(285\) 0 0
\(286\) 50.9117 0.178013
\(287\) 0 0
\(288\) 0 0
\(289\) 273.000 0.944637
\(290\) − 184.000i − 0.634483i
\(291\) 0 0
\(292\) 240.416 0.823344
\(293\) − 190.000i − 0.648464i −0.945978 0.324232i \(-0.894894\pi\)
0.945978 0.324232i \(-0.105106\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.0542373
\(296\) 90.5097i 0.305776i
\(297\) 0 0
\(298\) −38.0000 −0.127517
\(299\) 468.000i 1.56522i
\(300\) 0 0
\(301\) 0 0
\(302\) 214.960i 0.711790i
\(303\) 0 0
\(304\) 90.5097 0.297729
\(305\) 333.754i 1.09428i
\(306\) 0 0
\(307\) 265.872 0.866033 0.433017 0.901386i \(-0.357449\pi\)
0.433017 + 0.901386i \(0.357449\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 288.000 0.929032
\(311\) − 532.000i − 1.71061i −0.518124 0.855305i \(-0.673369\pi\)
0.518124 0.855305i \(-0.326631\pi\)
\(312\) 0 0
\(313\) 148.492 0.474417 0.237208 0.971459i \(-0.423768\pi\)
0.237208 + 0.971459i \(0.423768\pi\)
\(314\) 134.000i 0.426752i
\(315\) 0 0
\(316\) 296.000 0.936709
\(317\) 230.517i 0.727182i 0.931559 + 0.363591i \(0.118449\pi\)
−0.931559 + 0.363591i \(0.881551\pi\)
\(318\) 0 0
\(319\) 92.0000 0.288401
\(320\) − 32.0000i − 0.100000i
\(321\) 0 0
\(322\) 0 0
\(323\) 90.5097i 0.280216i
\(324\) 0 0
\(325\) 114.551 0.352466
\(326\) − 214.960i − 0.659388i
\(327\) 0 0
\(328\) −107.480 −0.327684
\(329\) 0 0
\(330\) 0 0
\(331\) 268.000 0.809668 0.404834 0.914390i \(-0.367329\pi\)
0.404834 + 0.914390i \(0.367329\pi\)
\(332\) − 160.000i − 0.481928i
\(333\) 0 0
\(334\) −164.049 −0.491164
\(335\) − 192.000i − 0.573134i
\(336\) 0 0
\(337\) 170.000 0.504451 0.252226 0.967668i \(-0.418838\pi\)
0.252226 + 0.967668i \(0.418838\pi\)
\(338\) − 9.89949i − 0.0292884i
\(339\) 0 0
\(340\) 32.0000 0.0941176
\(341\) 144.000i 0.422287i
\(342\) 0 0
\(343\) 0 0
\(344\) − 56.5685i − 0.164443i
\(345\) 0 0
\(346\) 313.955 0.907386
\(347\) − 280.014i − 0.806958i −0.914989 0.403479i \(-0.867801\pi\)
0.914989 0.403479i \(-0.132199\pi\)
\(348\) 0 0
\(349\) −241.831 −0.692924 −0.346462 0.938064i \(-0.612617\pi\)
−0.346462 + 0.938064i \(0.612617\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.0000 0.0454545
\(353\) 532.000i 1.50708i 0.657401 + 0.753541i \(0.271656\pi\)
−0.657401 + 0.753541i \(0.728344\pi\)
\(354\) 0 0
\(355\) −305.470 −0.860479
\(356\) 212.000i 0.595506i
\(357\) 0 0
\(358\) −84.0000 −0.234637
\(359\) − 342.240i − 0.953314i −0.879089 0.476657i \(-0.841848\pi\)
0.879089 0.476657i \(-0.158152\pi\)
\(360\) 0 0
\(361\) 151.000 0.418283
\(362\) 114.000i 0.314917i
\(363\) 0 0
\(364\) 0 0
\(365\) − 480.833i − 1.31735i
\(366\) 0 0
\(367\) −684.479 −1.86507 −0.932533 0.361084i \(-0.882407\pi\)
−0.932533 + 0.361084i \(0.882407\pi\)
\(368\) 147.078i 0.399669i
\(369\) 0 0
\(370\) 181.019 0.489241
\(371\) 0 0
\(372\) 0 0
\(373\) 382.000 1.02413 0.512064 0.858947i \(-0.328881\pi\)
0.512064 + 0.858947i \(0.328881\pi\)
\(374\) 16.0000i 0.0427807i
\(375\) 0 0
\(376\) −56.5685 −0.150448
\(377\) 414.000i 1.09814i
\(378\) 0 0
\(379\) −140.000 −0.369393 −0.184697 0.982796i \(-0.559130\pi\)
−0.184697 + 0.982796i \(0.559130\pi\)
\(380\) − 181.019i − 0.476367i
\(381\) 0 0
\(382\) 428.000 1.12042
\(383\) 248.000i 0.647520i 0.946139 + 0.323760i \(0.104947\pi\)
−0.946139 + 0.323760i \(0.895053\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 308.299i 0.798701i
\(387\) 0 0
\(388\) −308.299 −0.794584
\(389\) − 340.825i − 0.876158i −0.898936 0.438079i \(-0.855659\pi\)
0.898936 0.438079i \(-0.144341\pi\)
\(390\) 0 0
\(391\) −147.078 −0.376159
\(392\) 0 0
\(393\) 0 0
\(394\) −406.000 −1.03046
\(395\) − 592.000i − 1.49873i
\(396\) 0 0
\(397\) −244.659 −0.616269 −0.308135 0.951343i \(-0.599705\pi\)
−0.308135 + 0.951343i \(0.599705\pi\)
\(398\) − 96.0000i − 0.241206i
\(399\) 0 0
\(400\) 36.0000 0.0900000
\(401\) 29.6985i 0.0740611i 0.999314 + 0.0370305i \(0.0117899\pi\)
−0.999314 + 0.0370305i \(0.988210\pi\)
\(402\) 0 0
\(403\) −648.000 −1.60794
\(404\) − 252.000i − 0.623762i
\(405\) 0 0
\(406\) 0 0
\(407\) 90.5097i 0.222382i
\(408\) 0 0
\(409\) 606.698 1.48337 0.741684 0.670749i \(-0.234027\pi\)
0.741684 + 0.670749i \(0.234027\pi\)
\(410\) 214.960i 0.524294i
\(411\) 0 0
\(412\) 147.078 0.356986
\(413\) 0 0
\(414\) 0 0
\(415\) −320.000 −0.771084
\(416\) 72.0000i 0.173077i
\(417\) 0 0
\(418\) 90.5097 0.216530
\(419\) − 692.000i − 1.65155i −0.563999 0.825776i \(-0.690738\pi\)
0.563999 0.825776i \(-0.309262\pi\)
\(420\) 0 0
\(421\) 384.000 0.912114 0.456057 0.889951i \(-0.349261\pi\)
0.456057 + 0.889951i \(0.349261\pi\)
\(422\) 186.676i 0.442361i
\(423\) 0 0
\(424\) 268.000 0.632075
\(425\) 36.0000i 0.0847059i
\(426\) 0 0
\(427\) 0 0
\(428\) − 141.421i − 0.330424i
\(429\) 0 0
\(430\) −113.137 −0.263109
\(431\) 110.309i 0.255937i 0.991778 + 0.127968i \(0.0408456\pi\)
−0.991778 + 0.127968i \(0.959154\pi\)
\(432\) 0 0
\(433\) −156.978 −0.362535 −0.181268 0.983434i \(-0.558020\pi\)
−0.181268 + 0.983434i \(0.558020\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 172.000 0.394495
\(437\) 832.000i 1.90389i
\(438\) 0 0
\(439\) 203.647 0.463888 0.231944 0.972729i \(-0.425491\pi\)
0.231944 + 0.972729i \(0.425491\pi\)
\(440\) − 32.0000i − 0.0727273i
\(441\) 0 0
\(442\) −72.0000 −0.162896
\(443\) − 811.759i − 1.83241i −0.400707 0.916206i \(-0.631236\pi\)
0.400707 0.916206i \(-0.368764\pi\)
\(444\) 0 0
\(445\) 424.000 0.952809
\(446\) − 240.000i − 0.538117i
\(447\) 0 0
\(448\) 0 0
\(449\) 284.257i 0.633089i 0.948578 + 0.316544i \(0.102523\pi\)
−0.948578 + 0.316544i \(0.897477\pi\)
\(450\) 0 0
\(451\) −107.480 −0.238315
\(452\) 42.4264i 0.0938637i
\(453\) 0 0
\(454\) 424.264 0.934502
\(455\) 0 0
\(456\) 0 0
\(457\) −288.000 −0.630197 −0.315098 0.949059i \(-0.602038\pi\)
−0.315098 + 0.949059i \(0.602038\pi\)
\(458\) − 354.000i − 0.772926i
\(459\) 0 0
\(460\) 294.156 0.639470
\(461\) − 706.000i − 1.53145i −0.643166 0.765727i \(-0.722380\pi\)
0.643166 0.765727i \(-0.277620\pi\)
\(462\) 0 0
\(463\) −356.000 −0.768898 −0.384449 0.923146i \(-0.625609\pi\)
−0.384449 + 0.923146i \(0.625609\pi\)
\(464\) 130.108i 0.280404i
\(465\) 0 0
\(466\) −42.0000 −0.0901288
\(467\) − 884.000i − 1.89293i −0.322801 0.946467i \(-0.604625\pi\)
0.322801 0.946467i \(-0.395375\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 113.137i 0.240717i
\(471\) 0 0
\(472\) 11.3137 0.0239697
\(473\) − 56.5685i − 0.119595i
\(474\) 0 0
\(475\) 203.647 0.428730
\(476\) 0 0
\(477\) 0 0
\(478\) −228.000 −0.476987
\(479\) 508.000i 1.06054i 0.847828 + 0.530271i \(0.177910\pi\)
−0.847828 + 0.530271i \(0.822090\pi\)
\(480\) 0 0
\(481\) −407.294 −0.846764
\(482\) 534.000i 1.10788i
\(483\) 0 0
\(484\) −226.000 −0.466942
\(485\) 616.597i 1.27133i
\(486\) 0 0
\(487\) −624.000 −1.28131 −0.640657 0.767827i \(-0.721338\pi\)
−0.640657 + 0.767827i \(0.721338\pi\)
\(488\) − 236.000i − 0.483607i
\(489\) 0 0
\(490\) 0 0
\(491\) − 840.043i − 1.71088i −0.517901 0.855441i \(-0.673286\pi\)
0.517901 0.855441i \(-0.326714\pi\)
\(492\) 0 0
\(493\) −130.108 −0.263910
\(494\) 407.294i 0.824481i
\(495\) 0 0
\(496\) −203.647 −0.410578
\(497\) 0 0
\(498\) 0 0
\(499\) −940.000 −1.88377 −0.941884 0.335939i \(-0.890946\pi\)
−0.941884 + 0.335939i \(0.890946\pi\)
\(500\) − 272.000i − 0.544000i
\(501\) 0 0
\(502\) −673.166 −1.34097
\(503\) 632.000i 1.25646i 0.778027 + 0.628231i \(0.216221\pi\)
−0.778027 + 0.628231i \(0.783779\pi\)
\(504\) 0 0
\(505\) −504.000 −0.998020
\(506\) 147.078i 0.290668i
\(507\) 0 0
\(508\) −8.00000 −0.0157480
\(509\) − 220.000i − 0.432220i −0.976369 0.216110i \(-0.930663\pi\)
0.976369 0.216110i \(-0.0693370\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 0.0441942i
\(513\) 0 0
\(514\) −144.250 −0.280642
\(515\) − 294.156i − 0.571178i
\(516\) 0 0
\(517\) −56.5685 −0.109417
\(518\) 0 0
\(519\) 0 0
\(520\) 144.000 0.276923
\(521\) 508.000i 0.975048i 0.873110 + 0.487524i \(0.162100\pi\)
−0.873110 + 0.487524i \(0.837900\pi\)
\(522\) 0 0
\(523\) 814.587 1.55753 0.778764 0.627317i \(-0.215847\pi\)
0.778764 + 0.627317i \(0.215847\pi\)
\(524\) 48.0000i 0.0916031i
\(525\) 0 0
\(526\) 516.000 0.980989
\(527\) − 203.647i − 0.386426i
\(528\) 0 0
\(529\) −823.000 −1.55577
\(530\) − 536.000i − 1.01132i
\(531\) 0 0
\(532\) 0 0
\(533\) − 483.661i − 0.907432i
\(534\) 0 0
\(535\) −282.843 −0.528678
\(536\) 135.765i 0.253292i
\(537\) 0 0
\(538\) 277.186 0.515215
\(539\) 0 0
\(540\) 0 0
\(541\) −464.000 −0.857671 −0.428835 0.903383i \(-0.641076\pi\)
−0.428835 + 0.903383i \(0.641076\pi\)
\(542\) 248.000i 0.457565i
\(543\) 0 0
\(544\) −22.6274 −0.0415945
\(545\) − 344.000i − 0.631193i
\(546\) 0 0
\(547\) −80.0000 −0.146252 −0.0731261 0.997323i \(-0.523298\pi\)
−0.0731261 + 0.997323i \(0.523298\pi\)
\(548\) − 144.250i − 0.263230i
\(549\) 0 0
\(550\) 36.0000 0.0654545
\(551\) 736.000i 1.33575i
\(552\) 0 0
\(553\) 0 0
\(554\) 362.039i 0.653499i
\(555\) 0 0
\(556\) −282.843 −0.508710
\(557\) 912.168i 1.63764i 0.574047 + 0.818822i \(0.305372\pi\)
−0.574047 + 0.818822i \(0.694628\pi\)
\(558\) 0 0
\(559\) 254.558 0.455382
\(560\) 0 0
\(561\) 0 0
\(562\) −38.0000 −0.0676157
\(563\) 316.000i 0.561279i 0.959813 + 0.280639i \(0.0905465\pi\)
−0.959813 + 0.280639i \(0.909453\pi\)
\(564\) 0 0
\(565\) 84.8528 0.150182
\(566\) 712.000i 1.25795i
\(567\) 0 0
\(568\) 216.000 0.380282
\(569\) 1112.99i 1.95604i 0.208514 + 0.978019i \(0.433137\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(570\) 0 0
\(571\) −440.000 −0.770578 −0.385289 0.922796i \(-0.625898\pi\)
−0.385289 + 0.922796i \(0.625898\pi\)
\(572\) 72.0000i 0.125874i
\(573\) 0 0
\(574\) 0 0
\(575\) 330.926i 0.575523i
\(576\) 0 0
\(577\) −318.198 −0.551470 −0.275735 0.961234i \(-0.588921\pi\)
−0.275735 + 0.961234i \(0.588921\pi\)
\(578\) 386.080i 0.667959i
\(579\) 0 0
\(580\) 260.215 0.448647
\(581\) 0 0
\(582\) 0 0
\(583\) 268.000 0.459691
\(584\) 340.000i 0.582192i
\(585\) 0 0
\(586\) 268.701 0.458533
\(587\) − 300.000i − 0.511073i −0.966799 0.255537i \(-0.917748\pi\)
0.966799 0.255537i \(-0.0822521\pi\)
\(588\) 0 0
\(589\) −1152.00 −1.95586
\(590\) − 22.6274i − 0.0383516i
\(591\) 0 0
\(592\) −128.000 −0.216216
\(593\) − 1036.00i − 1.74705i −0.486780 0.873524i \(-0.661829\pi\)
0.486780 0.873524i \(-0.338171\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 53.7401i − 0.0901680i
\(597\) 0 0
\(598\) −661.852 −1.10678
\(599\) − 981.464i − 1.63850i −0.573433 0.819252i \(-0.694389\pi\)
0.573433 0.819252i \(-0.305611\pi\)
\(600\) 0 0
\(601\) −352.139 −0.585922 −0.292961 0.956124i \(-0.594641\pi\)
−0.292961 + 0.956124i \(0.594641\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −304.000 −0.503311
\(605\) 452.000i 0.747107i
\(606\) 0 0
\(607\) 865.499 1.42586 0.712931 0.701234i \(-0.247367\pi\)
0.712931 + 0.701234i \(0.247367\pi\)
\(608\) 128.000i 0.210526i
\(609\) 0 0
\(610\) −472.000 −0.773770
\(611\) − 254.558i − 0.416626i
\(612\) 0 0
\(613\) −264.000 −0.430669 −0.215334 0.976540i \(-0.569084\pi\)
−0.215334 + 0.976540i \(0.569084\pi\)
\(614\) 376.000i 0.612378i
\(615\) 0 0
\(616\) 0 0
\(617\) − 15.5563i − 0.0252129i −0.999921 0.0126064i \(-0.995987\pi\)
0.999921 0.0126064i \(-0.00401286\pi\)
\(618\) 0 0
\(619\) −401.637 −0.648848 −0.324424 0.945912i \(-0.605170\pi\)
−0.324424 + 0.945912i \(0.605170\pi\)
\(620\) 407.294i 0.656925i
\(621\) 0 0
\(622\) 752.362 1.20958
\(623\) 0 0
\(624\) 0 0
\(625\) −319.000 −0.510400
\(626\) 210.000i 0.335463i
\(627\) 0 0
\(628\) −189.505 −0.301759
\(629\) − 128.000i − 0.203498i
\(630\) 0 0
\(631\) −676.000 −1.07132 −0.535658 0.844435i \(-0.679936\pi\)
−0.535658 + 0.844435i \(0.679936\pi\)
\(632\) 418.607i 0.662353i
\(633\) 0 0
\(634\) −326.000 −0.514196
\(635\) 16.0000i 0.0251969i
\(636\) 0 0
\(637\) 0 0
\(638\) 130.108i 0.203930i
\(639\) 0 0
\(640\) 45.2548 0.0707107
\(641\) − 909.339i − 1.41863i −0.704894 0.709313i \(-0.749005\pi\)
0.704894 0.709313i \(-0.250995\pi\)
\(642\) 0 0
\(643\) 480.833 0.747796 0.373898 0.927470i \(-0.378021\pi\)
0.373898 + 0.927470i \(0.378021\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −128.000 −0.198142
\(647\) − 588.000i − 0.908810i −0.890795 0.454405i \(-0.849852\pi\)
0.890795 0.454405i \(-0.150148\pi\)
\(648\) 0 0
\(649\) 11.3137 0.0174325
\(650\) 162.000i 0.249231i
\(651\) 0 0
\(652\) 304.000 0.466258
\(653\) 453.963i 0.695195i 0.937644 + 0.347598i \(0.113002\pi\)
−0.937644 + 0.347598i \(0.886998\pi\)
\(654\) 0 0
\(655\) 96.0000 0.146565
\(656\) − 152.000i − 0.231707i
\(657\) 0 0
\(658\) 0 0
\(659\) − 755.190i − 1.14596i −0.819568 0.572982i \(-0.805787\pi\)
0.819568 0.572982i \(-0.194213\pi\)
\(660\) 0 0
\(661\) 241.831 0.365856 0.182928 0.983126i \(-0.441443\pi\)
0.182928 + 0.983126i \(0.441443\pi\)
\(662\) 379.009i 0.572522i
\(663\) 0 0
\(664\) 226.274 0.340774
\(665\) 0 0
\(666\) 0 0
\(667\) −1196.00 −1.79310
\(668\) − 232.000i − 0.347305i
\(669\) 0 0
\(670\) 271.529 0.405267
\(671\) − 236.000i − 0.351714i
\(672\) 0 0
\(673\) 1176.00 1.74740 0.873700 0.486465i \(-0.161714\pi\)
0.873700 + 0.486465i \(0.161714\pi\)
\(674\) 240.416i 0.356701i
\(675\) 0 0
\(676\) 14.0000 0.0207101
\(677\) − 222.000i − 0.327917i −0.986467 0.163959i \(-0.947574\pi\)
0.986467 0.163959i \(-0.0524264\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 45.2548i 0.0665512i
\(681\) 0 0
\(682\) −203.647 −0.298602
\(683\) 517.602i 0.757836i 0.925430 + 0.378918i \(0.123704\pi\)
−0.925430 + 0.378918i \(0.876296\pi\)
\(684\) 0 0
\(685\) −288.500 −0.421167
\(686\) 0 0
\(687\) 0 0
\(688\) 80.0000 0.116279
\(689\) 1206.00i 1.75036i
\(690\) 0 0
\(691\) 644.881 0.933258 0.466629 0.884453i \(-0.345468\pi\)
0.466629 + 0.884453i \(0.345468\pi\)
\(692\) 444.000i 0.641618i
\(693\) 0 0
\(694\) 396.000 0.570605
\(695\) 565.685i 0.813936i
\(696\) 0 0
\(697\) 152.000 0.218077
\(698\) − 342.000i − 0.489971i
\(699\) 0 0
\(700\) 0 0
\(701\) 657.609i 0.938102i 0.883171 + 0.469051i \(0.155404\pi\)
−0.883171 + 0.469051i \(0.844596\pi\)
\(702\) 0 0
\(703\) −724.077 −1.02998
\(704\) 22.6274i 0.0321412i
\(705\) 0 0
\(706\) −752.362 −1.06567
\(707\) 0 0
\(708\) 0 0
\(709\) 106.000 0.149506 0.0747532 0.997202i \(-0.476183\pi\)
0.0747532 + 0.997202i \(0.476183\pi\)
\(710\) − 432.000i − 0.608451i
\(711\) 0 0
\(712\) −299.813 −0.421086
\(713\) − 1872.00i − 2.62553i
\(714\) 0 0
\(715\) 144.000 0.201399
\(716\) − 118.794i − 0.165913i
\(717\) 0 0
\(718\) 484.000 0.674095
\(719\) 1216.00i 1.69124i 0.533787 + 0.845619i \(0.320768\pi\)
−0.533787 + 0.845619i \(0.679232\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 213.546i 0.295770i
\(723\) 0 0
\(724\) −161.220 −0.222680
\(725\) 292.742i 0.403782i
\(726\) 0 0
\(727\) 73.5391 0.101154 0.0505771 0.998720i \(-0.483894\pi\)
0.0505771 + 0.998720i \(0.483894\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 680.000 0.931507
\(731\) 80.0000i 0.109439i
\(732\) 0 0
\(733\) 420.021 0.573017 0.286508 0.958078i \(-0.407505\pi\)
0.286508 + 0.958078i \(0.407505\pi\)
\(734\) − 968.000i − 1.31880i
\(735\) 0 0
\(736\) −208.000 −0.282609
\(737\) 135.765i 0.184212i
\(738\) 0 0
\(739\) −408.000 −0.552097 −0.276049 0.961144i \(-0.589025\pi\)
−0.276049 + 0.961144i \(0.589025\pi\)
\(740\) 256.000i 0.345946i
\(741\) 0 0
\(742\) 0 0
\(743\) 489.318i 0.658571i 0.944230 + 0.329285i \(0.106808\pi\)
−0.944230 + 0.329285i \(0.893192\pi\)
\(744\) 0 0
\(745\) −107.480 −0.144269
\(746\) 540.230i 0.724168i
\(747\) 0 0
\(748\) −22.6274 −0.0302506
\(749\) 0 0
\(750\) 0 0
\(751\) −136.000 −0.181092 −0.0905459 0.995892i \(-0.528861\pi\)
−0.0905459 + 0.995892i \(0.528861\pi\)
\(752\) − 80.0000i − 0.106383i
\(753\) 0 0
\(754\) −585.484 −0.776505
\(755\) 608.000i 0.805298i
\(756\) 0 0
\(757\) 758.000 1.00132 0.500661 0.865644i \(-0.333091\pi\)
0.500661 + 0.865644i \(0.333091\pi\)
\(758\) − 197.990i − 0.261200i
\(759\) 0 0
\(760\) 256.000 0.336842
\(761\) − 58.0000i − 0.0762155i −0.999274 0.0381078i \(-0.987867\pi\)
0.999274 0.0381078i \(-0.0121330\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 605.283i 0.792256i
\(765\) 0 0
\(766\) −350.725 −0.457865
\(767\) 50.9117i 0.0663777i
\(768\) 0 0
\(769\) 292.742 0.380679 0.190340 0.981718i \(-0.439041\pi\)
0.190340 + 0.981718i \(0.439041\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −436.000 −0.564767
\(773\) 302.000i 0.390686i 0.980735 + 0.195343i \(0.0625819\pi\)
−0.980735 + 0.195343i \(0.937418\pi\)
\(774\) 0 0
\(775\) −458.205 −0.591233
\(776\) − 436.000i − 0.561856i
\(777\) 0 0
\(778\) 482.000 0.619537
\(779\) − 859.842i − 1.10378i
\(780\) 0 0
\(781\) 216.000 0.276569
\(782\) − 208.000i − 0.265985i
\(783\) 0 0
\(784\) 0 0
\(785\) 379.009i 0.482814i
\(786\) 0 0
\(787\) −497.803 −0.632533 −0.316266 0.948670i \(-0.602429\pi\)
−0.316266 + 0.948670i \(0.602429\pi\)
\(788\) − 574.171i − 0.728643i
\(789\) 0 0
\(790\) 837.214 1.05977
\(791\) 0 0
\(792\) 0 0
\(793\) 1062.00 1.33922
\(794\) − 346.000i − 0.435768i
\(795\) 0 0
\(796\) 135.765 0.170558
\(797\) 30.0000i 0.0376412i 0.999823 + 0.0188206i \(0.00599113\pi\)
−0.999823 + 0.0188206i \(0.994009\pi\)
\(798\) 0 0
\(799\) 80.0000 0.100125
\(800\) 50.9117i 0.0636396i
\(801\) 0 0
\(802\) −42.0000 −0.0523691
\(803\) 340.000i 0.423412i
\(804\) 0 0
\(805\) 0 0
\(806\) − 916.410i − 1.13699i
\(807\) 0 0
\(808\) 356.382 0.441067
\(809\) 982.878i 1.21493i 0.794346 + 0.607465i \(0.207814\pi\)
−0.794346 + 0.607465i \(0.792186\pi\)
\(810\) 0 0
\(811\) −458.205 −0.564988 −0.282494 0.959269i \(-0.591162\pi\)
−0.282494 + 0.959269i \(0.591162\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −128.000 −0.157248
\(815\) − 608.000i − 0.746012i
\(816\) 0 0
\(817\) 452.548 0.553915
\(818\) 858.000i 1.04890i
\(819\) 0 0
\(820\) −304.000 −0.370732
\(821\) − 606.698i − 0.738974i −0.929236 0.369487i \(-0.879533\pi\)
0.929236 0.369487i \(-0.120467\pi\)
\(822\) 0 0
\(823\) −68.0000 −0.0826245 −0.0413123 0.999146i \(-0.513154\pi\)
−0.0413123 + 0.999146i \(0.513154\pi\)
\(824\) 208.000i 0.252427i
\(825\) 0 0
\(826\) 0 0
\(827\) 195.161i 0.235987i 0.993014 + 0.117994i \(0.0376462\pi\)
−0.993014 + 0.117994i \(0.962354\pi\)
\(828\) 0 0
\(829\) 1245.92 1.50292 0.751461 0.659778i \(-0.229350\pi\)
0.751461 + 0.659778i \(0.229350\pi\)
\(830\) − 452.548i − 0.545239i
\(831\) 0 0
\(832\) −101.823 −0.122384
\(833\) 0 0
\(834\) 0 0
\(835\) −464.000 −0.555689
\(836\) 128.000i 0.153110i
\(837\) 0 0
\(838\) 978.636 1.16782
\(839\) − 1196.00i − 1.42551i −0.701415 0.712753i \(-0.747448\pi\)
0.701415 0.712753i \(-0.252552\pi\)
\(840\) 0 0
\(841\) −217.000 −0.258026
\(842\) 543.058i 0.644962i
\(843\) 0 0
\(844\) −264.000 −0.312796
\(845\) − 28.0000i − 0.0331361i
\(846\) 0 0
\(847\) 0 0
\(848\) 379.009i 0.446945i
\(849\) 0 0
\(850\) −50.9117 −0.0598961
\(851\) − 1176.63i − 1.38264i
\(852\) 0 0
\(853\) −1217.64 −1.42748 −0.713738 0.700412i \(-0.752999\pi\)
−0.713738 + 0.700412i \(0.752999\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 200.000 0.233645
\(857\) − 634.000i − 0.739790i −0.929074 0.369895i \(-0.879394\pi\)
0.929074 0.369895i \(-0.120606\pi\)
\(858\) 0 0
\(859\) −1600.89 −1.86367 −0.931833 0.362886i \(-0.881791\pi\)
−0.931833 + 0.362886i \(0.881791\pi\)
\(860\) − 160.000i − 0.186047i
\(861\) 0 0
\(862\) −156.000 −0.180974
\(863\) 461.034i 0.534222i 0.963666 + 0.267111i \(0.0860690\pi\)
−0.963666 + 0.267111i \(0.913931\pi\)
\(864\) 0 0
\(865\) 888.000 1.02659
\(866\) − 222.000i − 0.256351i
\(867\) 0 0
\(868\) 0 0
\(869\) 418.607i 0.481711i
\(870\) 0 0
\(871\) −610.940 −0.701424
\(872\) 243.245i 0.278950i
\(873\) 0 0
\(874\) −1176.63 −1.34625
\(875\) 0 0
\(876\) 0 0
\(877\) 1256.00 1.43216 0.716078 0.698021i \(-0.245936\pi\)
0.716078 + 0.698021i \(0.245936\pi\)
\(878\) 288.000i 0.328018i
\(879\) 0 0
\(880\) 45.2548 0.0514259
\(881\) 1092.00i 1.23950i 0.784799 + 0.619750i \(0.212766\pi\)
−0.784799 + 0.619750i \(0.787234\pi\)
\(882\) 0 0
\(883\) 284.000 0.321631 0.160815 0.986985i \(-0.448588\pi\)
0.160815 + 0.986985i \(0.448588\pi\)
\(884\) − 101.823i − 0.115185i
\(885\) 0 0
\(886\) 1148.00 1.29571
\(887\) 716.000i 0.807215i 0.914932 + 0.403608i \(0.132244\pi\)
−0.914932 + 0.403608i \(0.867756\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 599.627i 0.673738i
\(891\) 0 0
\(892\) 339.411 0.380506
\(893\) − 452.548i − 0.506773i
\(894\) 0 0
\(895\) −237.588 −0.265461
\(896\) 0 0
\(897\) 0 0
\(898\) −402.000 −0.447661
\(899\) − 1656.00i − 1.84205i
\(900\) 0 0
\(901\) −379.009 −0.420654
\(902\) − 152.000i − 0.168514i
\(903\) 0 0
\(904\) −60.0000 −0.0663717
\(905\) 322.441i 0.356288i
\(906\) 0 0
\(907\) 724.000 0.798236 0.399118 0.916900i \(-0.369316\pi\)
0.399118 + 0.916900i \(0.369316\pi\)
\(908\) 600.000i 0.660793i
\(909\) 0 0
\(910\) 0 0
\(911\) − 302.642i − 0.332208i −0.986108 0.166104i \(-0.946881\pi\)
0.986108 0.166104i \(-0.0531188\pi\)
\(912\) 0 0
\(913\) 226.274 0.247836
\(914\) − 407.294i − 0.445617i
\(915\) 0 0
\(916\) 500.632 0.546541
\(917\) 0 0
\(918\) 0 0
\(919\) 376.000 0.409140 0.204570 0.978852i \(-0.434420\pi\)
0.204570 + 0.978852i \(0.434420\pi\)
\(920\) 416.000i 0.452174i
\(921\) 0 0
\(922\) 998.435 1.08290
\(923\) 972.000i 1.05309i
\(924\) 0 0
\(925\) −288.000 −0.311351
\(926\) − 503.460i − 0.543693i
\(927\) 0 0
\(928\) −184.000 −0.198276
\(929\) 1210.00i 1.30248i 0.758874 + 0.651238i \(0.225750\pi\)
−0.758874 + 0.651238i \(0.774250\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 59.3970i − 0.0637307i
\(933\) 0 0
\(934\) 1250.16 1.33851
\(935\) 45.2548i 0.0484009i
\(936\) 0 0
\(937\) 994.192 1.06104 0.530519 0.847673i \(-0.321997\pi\)
0.530519 + 0.847673i \(0.321997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −160.000 −0.170213
\(941\) − 44.0000i − 0.0467588i −0.999727 0.0233794i \(-0.992557\pi\)
0.999727 0.0233794i \(-0.00744257\pi\)
\(942\) 0 0
\(943\) 1397.24 1.48170
\(944\) 16.0000i 0.0169492i
\(945\) 0 0
\(946\) 80.0000 0.0845666
\(947\) − 1009.75i − 1.06626i −0.846033 0.533130i \(-0.821016\pi\)
0.846033 0.533130i \(-0.178984\pi\)
\(948\) 0 0
\(949\) −1530.00 −1.61222
\(950\) 288.000i 0.303158i
\(951\) 0 0
\(952\) 0 0
\(953\) − 1322.29i − 1.38750i −0.720215 0.693751i \(-0.755957\pi\)
0.720215 0.693751i \(-0.244043\pi\)
\(954\) 0 0
\(955\) 1210.57 1.26761
\(956\) − 322.441i − 0.337281i
\(957\) 0 0
\(958\) −718.420 −0.749917
\(959\) 0 0
\(960\) 0 0
\(961\) 1631.00 1.69719
\(962\) − 576.000i − 0.598753i
\(963\) 0 0
\(964\) −755.190 −0.783392
\(965\) 872.000i 0.903627i
\(966\) 0 0
\(967\) 876.000 0.905895 0.452947 0.891537i \(-0.350373\pi\)
0.452947 + 0.891537i \(0.350373\pi\)
\(968\) − 319.612i − 0.330178i
\(969\) 0 0
\(970\) −872.000 −0.898969
\(971\) − 1448.00i − 1.49125i −0.666368 0.745623i \(-0.732152\pi\)
0.666368 0.745623i \(-0.267848\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 882.469i − 0.906026i
\(975\) 0 0
\(976\) 333.754 0.341961
\(977\) 1489.17i 1.52422i 0.647445 + 0.762112i \(0.275837\pi\)
−0.647445 + 0.762112i \(0.724163\pi\)
\(978\) 0 0
\(979\) −299.813 −0.306244
\(980\) 0 0
\(981\) 0 0
\(982\) 1188.00 1.20978
\(983\) 32.0000i 0.0325534i 0.999868 + 0.0162767i \(0.00518126\pi\)
−0.999868 + 0.0162767i \(0.994819\pi\)
\(984\) 0 0
\(985\) −1148.34 −1.16583
\(986\) − 184.000i − 0.186613i
\(987\) 0 0
\(988\) −576.000 −0.582996
\(989\) 735.391i 0.743570i
\(990\) 0 0
\(991\) −280.000 −0.282543 −0.141271 0.989971i \(-0.545119\pi\)
−0.141271 + 0.989971i \(0.545119\pi\)
\(992\) − 288.000i − 0.290323i
\(993\) 0 0
\(994\) 0 0
\(995\) − 271.529i − 0.272893i
\(996\) 0 0
\(997\) 1093.19 1.09648 0.548238 0.836322i \(-0.315299\pi\)
0.548238 + 0.836322i \(0.315299\pi\)
\(998\) − 1329.36i − 1.33202i
\(999\) 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 882.3.b.h.197.4 yes 4
3.2 odd 2 inner 882.3.b.h.197.1 4
7.2 even 3 882.3.s.g.557.4 8
7.3 odd 6 882.3.s.g.863.2 8
7.4 even 3 882.3.s.g.863.1 8
7.5 odd 6 882.3.s.g.557.3 8
7.6 odd 2 inner 882.3.b.h.197.3 yes 4
21.2 odd 6 882.3.s.g.557.1 8
21.5 even 6 882.3.s.g.557.2 8
21.11 odd 6 882.3.s.g.863.4 8
21.17 even 6 882.3.s.g.863.3 8
21.20 even 2 inner 882.3.b.h.197.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.3.b.h.197.1 4 3.2 odd 2 inner
882.3.b.h.197.2 yes 4 21.20 even 2 inner
882.3.b.h.197.3 yes 4 7.6 odd 2 inner
882.3.b.h.197.4 yes 4 1.1 even 1 trivial
882.3.s.g.557.1 8 21.2 odd 6
882.3.s.g.557.2 8 21.5 even 6
882.3.s.g.557.3 8 7.5 odd 6
882.3.s.g.557.4 8 7.2 even 3
882.3.s.g.863.1 8 7.4 even 3
882.3.s.g.863.2 8 7.3 odd 6
882.3.s.g.863.3 8 21.17 even 6
882.3.s.g.863.4 8 21.11 odd 6