Properties

Label 882.3.b.f.197.1
Level $882$
Weight $3$
Character 882.197
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.1
Root \(2.57794i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.3.b.f.197.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -8.89753i q^{5} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} -8.89753i q^{5} +2.82843i q^{8} -12.5830 q^{10} +17.7951i q^{11} -2.58301 q^{13} +4.00000 q^{16} +25.8681i q^{17} -20.0000 q^{19} +17.7951i q^{20} +25.1660 q^{22} +17.7951i q^{23} -54.1660 q^{25} +3.65292i q^{26} +11.9034i q^{29} +17.1660 q^{31} -5.65685i q^{32} +36.5830 q^{34} +38.0000 q^{37} +28.2843i q^{38} +25.1660 q^{40} +15.7338i q^{41} -43.4980 q^{43} -35.5901i q^{44} +25.1660 q^{46} +16.9706i q^{47} +76.6023i q^{50} +5.16601 q^{52} -85.5571i q^{53} +158.332 q^{55} +16.8340 q^{58} -1.64899i q^{59} -100.332 q^{61} -24.2764i q^{62} -8.00000 q^{64} +22.9824i q^{65} +36.6640 q^{67} -51.7362i q^{68} +17.7951i q^{71} -28.9150 q^{73} -53.7401i q^{74} +40.0000 q^{76} +118.332 q^{79} -35.5901i q^{80} +22.2510 q^{82} +120.443i q^{83} +230.162 q^{85} +61.5155i q^{86} -50.3320 q^{88} +139.475i q^{89} -35.5901i q^{92} +24.0000 q^{94} +177.951i q^{95} -44.4131 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} - 8q^{10} + 32q^{13} + 16q^{16} - 80q^{19} + 16q^{22} - 132q^{25} - 16q^{31} + 104q^{34} + 152q^{37} + 16q^{40} + 80q^{43} + 16q^{46} - 64q^{52} + 464q^{55} + 152q^{58} - 232q^{61} - 32q^{64} - 192q^{67} + 96q^{73} + 160q^{76} + 304q^{79} + 216q^{82} + 328q^{85} - 32q^{88} + 96q^{94} + 288q^{97} + O(q^{100}) \)

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
\(5\) − 8.89753i − 1.77951i −0.456443 0.889753i \(-0.650877\pi\)
0.456443 0.889753i \(-0.349123\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) −12.5830 −1.25830
\(11\) 17.7951i 1.61773i 0.587993 + 0.808866i \(0.299918\pi\)
−0.587993 + 0.808866i \(0.700082\pi\)
\(12\) 0 0
\(13\) −2.58301 −0.198693 −0.0993464 0.995053i \(-0.531675\pi\)
−0.0993464 + 0.995053i \(0.531675\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 25.8681i 1.52165i 0.648956 + 0.760826i \(0.275206\pi\)
−0.648956 + 0.760826i \(0.724794\pi\)
\(18\) 0 0
\(19\) −20.0000 −1.05263 −0.526316 0.850289i \(-0.676427\pi\)
−0.526316 + 0.850289i \(0.676427\pi\)
\(20\) 17.7951i 0.889753i
\(21\) 0 0
\(22\) 25.1660 1.14391
\(23\) 17.7951i 0.773698i 0.922143 + 0.386849i \(0.126437\pi\)
−0.922143 + 0.386849i \(0.873563\pi\)
\(24\) 0 0
\(25\) −54.1660 −2.16664
\(26\) 3.65292i 0.140497i
\(27\) 0 0
\(28\) 0 0
\(29\) 11.9034i 0.410463i 0.978713 + 0.205232i \(0.0657947\pi\)
−0.978713 + 0.205232i \(0.934205\pi\)
\(30\) 0 0
\(31\) 17.1660 0.553742 0.276871 0.960907i \(-0.410702\pi\)
0.276871 + 0.960907i \(0.410702\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) 36.5830 1.07597
\(35\) 0 0
\(36\) 0 0
\(37\) 38.0000 1.02703 0.513514 0.858082i \(-0.328344\pi\)
0.513514 + 0.858082i \(0.328344\pi\)
\(38\) 28.2843i 0.744323i
\(39\) 0 0
\(40\) 25.1660 0.629150
\(41\) 15.7338i 0.383752i 0.981419 + 0.191876i \(0.0614571\pi\)
−0.981419 + 0.191876i \(0.938543\pi\)
\(42\) 0 0
\(43\) −43.4980 −1.01158 −0.505791 0.862656i \(-0.668799\pi\)
−0.505791 + 0.862656i \(0.668799\pi\)
\(44\) − 35.5901i − 0.808866i
\(45\) 0 0
\(46\) 25.1660 0.547087
\(47\) 16.9706i 0.361076i 0.983568 + 0.180538i \(0.0577838\pi\)
−0.983568 + 0.180538i \(0.942216\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 76.6023i 1.53205i
\(51\) 0 0
\(52\) 5.16601 0.0993464
\(53\) − 85.5571i − 1.61429i −0.590356 0.807143i \(-0.701013\pi\)
0.590356 0.807143i \(-0.298987\pi\)
\(54\) 0 0
\(55\) 158.332 2.87876
\(56\) 0 0
\(57\) 0 0
\(58\) 16.8340 0.290241
\(59\) − 1.64899i − 0.0279489i −0.999902 0.0139745i \(-0.995552\pi\)
0.999902 0.0139745i \(-0.00444836\pi\)
\(60\) 0 0
\(61\) −100.332 −1.64479 −0.822394 0.568919i \(-0.807362\pi\)
−0.822394 + 0.568919i \(0.807362\pi\)
\(62\) − 24.2764i − 0.391555i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 22.9824i 0.353575i
\(66\) 0 0
\(67\) 36.6640 0.547225 0.273612 0.961840i \(-0.411781\pi\)
0.273612 + 0.961840i \(0.411781\pi\)
\(68\) − 51.7362i − 0.760826i
\(69\) 0 0
\(70\) 0 0
\(71\) 17.7951i 0.250635i 0.992117 + 0.125317i \(0.0399949\pi\)
−0.992117 + 0.125317i \(0.960005\pi\)
\(72\) 0 0
\(73\) −28.9150 −0.396096 −0.198048 0.980192i \(-0.563460\pi\)
−0.198048 + 0.980192i \(0.563460\pi\)
\(74\) − 53.7401i − 0.726218i
\(75\) 0 0
\(76\) 40.0000 0.526316
\(77\) 0 0
\(78\) 0 0
\(79\) 118.332 1.49787 0.748937 0.662641i \(-0.230565\pi\)
0.748937 + 0.662641i \(0.230565\pi\)
\(80\) − 35.5901i − 0.444876i
\(81\) 0 0
\(82\) 22.2510 0.271353
\(83\) 120.443i 1.45112i 0.688159 + 0.725560i \(0.258419\pi\)
−0.688159 + 0.725560i \(0.741581\pi\)
\(84\) 0 0
\(85\) 230.162 2.70779
\(86\) 61.5155i 0.715297i
\(87\) 0 0
\(88\) −50.3320 −0.571955
\(89\) 139.475i 1.56713i 0.621309 + 0.783566i \(0.286601\pi\)
−0.621309 + 0.783566i \(0.713399\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 35.5901i − 0.386849i
\(93\) 0 0
\(94\) 24.0000 0.255319
\(95\) 177.951i 1.87316i
\(96\) 0 0
\(97\) −44.4131 −0.457867 −0.228933 0.973442i \(-0.573524\pi\)
−0.228933 + 0.973442i \(0.573524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 108.332 1.08332
\(101\) − 31.8799i − 0.315642i −0.987468 0.157821i \(-0.949553\pi\)
0.987468 0.157821i \(-0.0504470\pi\)
\(102\) 0 0
\(103\) −4.50197 −0.0437084 −0.0218542 0.999761i \(-0.506957\pi\)
−0.0218542 + 0.999761i \(0.506957\pi\)
\(104\) − 7.30584i − 0.0702485i
\(105\) 0 0
\(106\) −120.996 −1.14147
\(107\) 172.179i 1.60915i 0.593851 + 0.804575i \(0.297607\pi\)
−0.593851 + 0.804575i \(0.702393\pi\)
\(108\) 0 0
\(109\) −177.830 −1.63147 −0.815734 0.578427i \(-0.803667\pi\)
−0.815734 + 0.578427i \(0.803667\pi\)
\(110\) − 223.915i − 2.03559i
\(111\) 0 0
\(112\) 0 0
\(113\) 31.3475i 0.277411i 0.990334 + 0.138706i \(0.0442942\pi\)
−0.990334 + 0.138706i \(0.955706\pi\)
\(114\) 0 0
\(115\) 158.332 1.37680
\(116\) − 23.8069i − 0.205232i
\(117\) 0 0
\(118\) −2.33202 −0.0197629
\(119\) 0 0
\(120\) 0 0
\(121\) −195.664 −1.61706
\(122\) 141.891i 1.16304i
\(123\) 0 0
\(124\) −34.3320 −0.276871
\(125\) 259.505i 2.07604i
\(126\) 0 0
\(127\) 214.332 1.68765 0.843827 0.536616i \(-0.180297\pi\)
0.843827 + 0.536616i \(0.180297\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) 32.5020 0.250015
\(131\) − 91.4488i − 0.698082i −0.937107 0.349041i \(-0.886507\pi\)
0.937107 0.349041i \(-0.113493\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 51.8508i − 0.386946i
\(135\) 0 0
\(136\) −73.1660 −0.537985
\(137\) − 106.891i − 0.780223i −0.920768 0.390111i \(-0.872437\pi\)
0.920768 0.390111i \(-0.127563\pi\)
\(138\) 0 0
\(139\) 121.328 0.872864 0.436432 0.899737i \(-0.356242\pi\)
0.436432 + 0.899737i \(0.356242\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 25.1660 0.177225
\(143\) − 45.9647i − 0.321432i
\(144\) 0 0
\(145\) 105.911 0.730421
\(146\) 40.8920i 0.280082i
\(147\) 0 0
\(148\) −76.0000 −0.513514
\(149\) − 17.6749i − 0.118623i −0.998240 0.0593117i \(-0.981109\pi\)
0.998240 0.0593117i \(-0.0188906\pi\)
\(150\) 0 0
\(151\) −50.8340 −0.336649 −0.168324 0.985732i \(-0.553836\pi\)
−0.168324 + 0.985732i \(0.553836\pi\)
\(152\) − 56.5685i − 0.372161i
\(153\) 0 0
\(154\) 0 0
\(155\) − 152.735i − 0.985388i
\(156\) 0 0
\(157\) −68.9961 −0.439465 −0.219733 0.975560i \(-0.570519\pi\)
−0.219733 + 0.975560i \(0.570519\pi\)
\(158\) − 167.347i − 1.05916i
\(159\) 0 0
\(160\) −50.3320 −0.314575
\(161\) 0 0
\(162\) 0 0
\(163\) −166.996 −1.02452 −0.512258 0.858832i \(-0.671191\pi\)
−0.512258 + 0.858832i \(0.671191\pi\)
\(164\) − 31.4676i − 0.191876i
\(165\) 0 0
\(166\) 170.332 1.02610
\(167\) − 120.443i − 0.721215i −0.932718 0.360608i \(-0.882569\pi\)
0.932718 0.360608i \(-0.117431\pi\)
\(168\) 0 0
\(169\) −162.328 −0.960521
\(170\) − 325.498i − 1.91470i
\(171\) 0 0
\(172\) 86.9961 0.505791
\(173\) 91.8610i 0.530989i 0.964112 + 0.265494i \(0.0855351\pi\)
−0.964112 + 0.265494i \(0.914465\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 71.1802i 0.404433i
\(177\) 0 0
\(178\) 197.247 1.10813
\(179\) 133.291i 0.744643i 0.928104 + 0.372321i \(0.121438\pi\)
−0.928104 + 0.372321i \(0.878562\pi\)
\(180\) 0 0
\(181\) −83.0850 −0.459033 −0.229517 0.973305i \(-0.573714\pi\)
−0.229517 + 0.973305i \(0.573714\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −50.3320 −0.273544
\(185\) − 338.106i − 1.82760i
\(186\) 0 0
\(187\) −460.324 −2.46163
\(188\) − 33.9411i − 0.180538i
\(189\) 0 0
\(190\) 251.660 1.32453
\(191\) 41.3616i 0.216553i 0.994121 + 0.108276i \(0.0345331\pi\)
−0.994121 + 0.108276i \(0.965467\pi\)
\(192\) 0 0
\(193\) 134.000 0.694301 0.347150 0.937810i \(-0.387149\pi\)
0.347150 + 0.937810i \(0.387149\pi\)
\(194\) 62.8095i 0.323761i
\(195\) 0 0
\(196\) 0 0
\(197\) − 68.8269i − 0.349375i −0.984624 0.174688i \(-0.944108\pi\)
0.984624 0.174688i \(-0.0558916\pi\)
\(198\) 0 0
\(199\) 278.494 1.39947 0.699734 0.714404i \(-0.253302\pi\)
0.699734 + 0.714404i \(0.253302\pi\)
\(200\) − 153.205i − 0.766023i
\(201\) 0 0
\(202\) −45.0850 −0.223193
\(203\) 0 0
\(204\) 0 0
\(205\) 139.992 0.682888
\(206\) 6.36674i 0.0309065i
\(207\) 0 0
\(208\) −10.3320 −0.0496732
\(209\) − 355.901i − 1.70288i
\(210\) 0 0
\(211\) −211.498 −1.00236 −0.501180 0.865343i \(-0.667101\pi\)
−0.501180 + 0.865343i \(0.667101\pi\)
\(212\) 171.114i 0.807143i
\(213\) 0 0
\(214\) 243.498 1.13784
\(215\) 387.025i 1.80012i
\(216\) 0 0
\(217\) 0 0
\(218\) 251.490i 1.15362i
\(219\) 0 0
\(220\) −316.664 −1.43938
\(221\) − 66.8174i − 0.302341i
\(222\) 0 0
\(223\) −222.494 −0.997731 −0.498866 0.866679i \(-0.666250\pi\)
−0.498866 + 0.866679i \(0.666250\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 44.3320 0.196159
\(227\) − 101.823i − 0.448561i −0.974525 0.224281i \(-0.927997\pi\)
0.974525 0.224281i \(-0.0720032\pi\)
\(228\) 0 0
\(229\) 163.085 0.712161 0.356081 0.934455i \(-0.384113\pi\)
0.356081 + 0.934455i \(0.384113\pi\)
\(230\) − 223.915i − 0.973545i
\(231\) 0 0
\(232\) −33.6680 −0.145121
\(233\) 362.858i 1.55733i 0.627441 + 0.778664i \(0.284102\pi\)
−0.627441 + 0.778664i \(0.715898\pi\)
\(234\) 0 0
\(235\) 150.996 0.642536
\(236\) 3.29798i 0.0139745i
\(237\) 0 0
\(238\) 0 0
\(239\) − 177.126i − 0.741113i −0.928810 0.370557i \(-0.879167\pi\)
0.928810 0.370557i \(-0.120833\pi\)
\(240\) 0 0
\(241\) −152.753 −0.633830 −0.316915 0.948454i \(-0.602647\pi\)
−0.316915 + 0.948454i \(0.602647\pi\)
\(242\) 276.711i 1.14343i
\(243\) 0 0
\(244\) 200.664 0.822394
\(245\) 0 0
\(246\) 0 0
\(247\) 51.6601 0.209150
\(248\) 48.5528i 0.195777i
\(249\) 0 0
\(250\) 366.996 1.46798
\(251\) − 356.382i − 1.41985i −0.704278 0.709924i \(-0.748729\pi\)
0.704278 0.709924i \(-0.251271\pi\)
\(252\) 0 0
\(253\) −316.664 −1.25164
\(254\) − 303.111i − 1.19335i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 59.5689i − 0.231786i −0.993262 0.115893i \(-0.963027\pi\)
0.993262 0.115893i \(-0.0369729\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 45.9647i − 0.176787i
\(261\) 0 0
\(262\) −129.328 −0.493619
\(263\) − 7.42045i − 0.0282146i −0.999900 0.0141073i \(-0.995509\pi\)
0.999900 0.0141073i \(-0.00449065\pi\)
\(264\) 0 0
\(265\) −761.247 −2.87263
\(266\) 0 0
\(267\) 0 0
\(268\) −73.3281 −0.273612
\(269\) 430.207i 1.59928i 0.600477 + 0.799642i \(0.294977\pi\)
−0.600477 + 0.799642i \(0.705023\pi\)
\(270\) 0 0
\(271\) 41.1660 0.151904 0.0759520 0.997111i \(-0.475800\pi\)
0.0759520 + 0.997111i \(0.475800\pi\)
\(272\) 103.472i 0.380413i
\(273\) 0 0
\(274\) −151.166 −0.551701
\(275\) − 963.887i − 3.50504i
\(276\) 0 0
\(277\) 32.0000 0.115523 0.0577617 0.998330i \(-0.481604\pi\)
0.0577617 + 0.998330i \(0.481604\pi\)
\(278\) − 171.584i − 0.617208i
\(279\) 0 0
\(280\) 0 0
\(281\) 17.0907i 0.0608211i 0.999537 + 0.0304106i \(0.00968147\pi\)
−0.999537 + 0.0304106i \(0.990319\pi\)
\(282\) 0 0
\(283\) −439.660 −1.55357 −0.776785 0.629766i \(-0.783151\pi\)
−0.776785 + 0.629766i \(0.783151\pi\)
\(284\) − 35.5901i − 0.125317i
\(285\) 0 0
\(286\) −65.0039 −0.227286
\(287\) 0 0
\(288\) 0 0
\(289\) −380.158 −1.31543
\(290\) − 149.781i − 0.516486i
\(291\) 0 0
\(292\) 57.8301 0.198048
\(293\) 394.377i 1.34600i 0.739644 + 0.672998i \(0.234994\pi\)
−0.739644 + 0.672998i \(0.765006\pi\)
\(294\) 0 0
\(295\) −14.6719 −0.0497353
\(296\) 107.480i 0.363109i
\(297\) 0 0
\(298\) −24.9961 −0.0838794
\(299\) − 45.9647i − 0.153728i
\(300\) 0 0
\(301\) 0 0
\(302\) 71.8901i 0.238047i
\(303\) 0 0
\(304\) −80.0000 −0.263158
\(305\) 892.707i 2.92691i
\(306\) 0 0
\(307\) 23.3360 0.0760129 0.0380064 0.999277i \(-0.487899\pi\)
0.0380064 + 0.999277i \(0.487899\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −216.000 −0.696774
\(311\) 527.256i 1.69536i 0.530511 + 0.847678i \(0.322000\pi\)
−0.530511 + 0.847678i \(0.678000\pi\)
\(312\) 0 0
\(313\) 295.328 0.943540 0.471770 0.881722i \(-0.343615\pi\)
0.471770 + 0.881722i \(0.343615\pi\)
\(314\) 97.5752i 0.310749i
\(315\) 0 0
\(316\) −236.664 −0.748937
\(317\) 107.475i 0.339037i 0.985527 + 0.169518i \(0.0542212\pi\)
−0.985527 + 0.169518i \(0.945779\pi\)
\(318\) 0 0
\(319\) −211.822 −0.664019
\(320\) 71.1802i 0.222438i
\(321\) 0 0
\(322\) 0 0
\(323\) − 517.362i − 1.60174i
\(324\) 0 0
\(325\) 139.911 0.430496
\(326\) 236.168i 0.724442i
\(327\) 0 0
\(328\) −44.5020 −0.135677
\(329\) 0 0
\(330\) 0 0
\(331\) −273.490 −0.826254 −0.413127 0.910673i \(-0.635563\pi\)
−0.413127 + 0.910673i \(0.635563\pi\)
\(332\) − 240.886i − 0.725560i
\(333\) 0 0
\(334\) −170.332 −0.509976
\(335\) − 326.219i − 0.973789i
\(336\) 0 0
\(337\) −341.166 −1.01236 −0.506181 0.862427i \(-0.668943\pi\)
−0.506181 + 0.862427i \(0.668943\pi\)
\(338\) 229.567i 0.679191i
\(339\) 0 0
\(340\) −460.324 −1.35389
\(341\) 305.470i 0.895807i
\(342\) 0 0
\(343\) 0 0
\(344\) − 123.031i − 0.357648i
\(345\) 0 0
\(346\) 129.911 0.375466
\(347\) 116.320i 0.335217i 0.985854 + 0.167609i \(0.0536045\pi\)
−0.985854 + 0.167609i \(0.946395\pi\)
\(348\) 0 0
\(349\) 158.324 0.453651 0.226825 0.973935i \(-0.427165\pi\)
0.226825 + 0.973935i \(0.427165\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 100.664 0.285977
\(353\) 230.339i 0.652519i 0.945280 + 0.326260i \(0.105788\pi\)
−0.945280 + 0.326260i \(0.894212\pi\)
\(354\) 0 0
\(355\) 158.332 0.446006
\(356\) − 278.949i − 0.783566i
\(357\) 0 0
\(358\) 188.502 0.526542
\(359\) 171.698i 0.478269i 0.970987 + 0.239134i \(0.0768636\pi\)
−0.970987 + 0.239134i \(0.923136\pi\)
\(360\) 0 0
\(361\) 39.0000 0.108033
\(362\) 117.500i 0.324585i
\(363\) 0 0
\(364\) 0 0
\(365\) 257.272i 0.704856i
\(366\) 0 0
\(367\) −517.490 −1.41005 −0.705027 0.709180i \(-0.749065\pi\)
−0.705027 + 0.709180i \(0.749065\pi\)
\(368\) 71.1802i 0.193425i
\(369\) 0 0
\(370\) −478.154 −1.29231
\(371\) 0 0
\(372\) 0 0
\(373\) −233.336 −0.625566 −0.312783 0.949825i \(-0.601261\pi\)
−0.312783 + 0.949825i \(0.601261\pi\)
\(374\) 650.997i 1.74063i
\(375\) 0 0
\(376\) −48.0000 −0.127660
\(377\) − 30.7466i − 0.0815560i
\(378\) 0 0
\(379\) 441.166 1.16403 0.582013 0.813179i \(-0.302265\pi\)
0.582013 + 0.813179i \(0.302265\pi\)
\(380\) − 355.901i − 0.936582i
\(381\) 0 0
\(382\) 58.4941 0.153126
\(383\) 213.060i 0.556292i 0.960539 + 0.278146i \(0.0897200\pi\)
−0.960539 + 0.278146i \(0.910280\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 189.505i − 0.490945i
\(387\) 0 0
\(388\) 88.8261 0.228933
\(389\) − 565.096i − 1.45269i −0.687331 0.726344i \(-0.741218\pi\)
0.687331 0.726344i \(-0.258782\pi\)
\(390\) 0 0
\(391\) −460.324 −1.17730
\(392\) 0 0
\(393\) 0 0
\(394\) −97.3360 −0.247046
\(395\) − 1052.86i − 2.66547i
\(396\) 0 0
\(397\) −498.324 −1.25522 −0.627612 0.778526i \(-0.715968\pi\)
−0.627612 + 0.778526i \(0.715968\pi\)
\(398\) − 393.850i − 0.989573i
\(399\) 0 0
\(400\) −216.664 −0.541660
\(401\) − 193.392i − 0.482275i −0.970491 0.241138i \(-0.922480\pi\)
0.970491 0.241138i \(-0.0775205\pi\)
\(402\) 0 0
\(403\) −44.3399 −0.110025
\(404\) 63.7598i 0.157821i
\(405\) 0 0
\(406\) 0 0
\(407\) 676.212i 1.66145i
\(408\) 0 0
\(409\) 454.243 1.11062 0.555309 0.831644i \(-0.312600\pi\)
0.555309 + 0.831644i \(0.312600\pi\)
\(410\) − 197.979i − 0.482875i
\(411\) 0 0
\(412\) 9.00394 0.0218542
\(413\) 0 0
\(414\) 0 0
\(415\) 1071.64 2.58228
\(416\) 14.6117i 0.0351242i
\(417\) 0 0
\(418\) −503.320 −1.20412
\(419\) − 339.411i − 0.810051i −0.914305 0.405025i \(-0.867263\pi\)
0.914305 0.405025i \(-0.132737\pi\)
\(420\) 0 0
\(421\) 247.320 0.587459 0.293729 0.955889i \(-0.405104\pi\)
0.293729 + 0.955889i \(0.405104\pi\)
\(422\) 299.103i 0.708776i
\(423\) 0 0
\(424\) 241.992 0.570736
\(425\) − 1401.17i − 3.29687i
\(426\) 0 0
\(427\) 0 0
\(428\) − 344.358i − 0.804575i
\(429\) 0 0
\(430\) 547.336 1.27287
\(431\) − 456.419i − 1.05898i −0.848317 0.529489i \(-0.822384\pi\)
0.848317 0.529489i \(-0.177616\pi\)
\(432\) 0 0
\(433\) 637.984 1.47340 0.736702 0.676217i \(-0.236382\pi\)
0.736702 + 0.676217i \(0.236382\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 355.660 0.815734
\(437\) − 355.901i − 0.814419i
\(438\) 0 0
\(439\) 784.146 1.78621 0.893105 0.449848i \(-0.148522\pi\)
0.893105 + 0.449848i \(0.148522\pi\)
\(440\) 447.831i 1.01780i
\(441\) 0 0
\(442\) −94.4941 −0.213788
\(443\) − 472.222i − 1.06596i −0.846127 0.532981i \(-0.821072\pi\)
0.846127 0.532981i \(-0.178928\pi\)
\(444\) 0 0
\(445\) 1240.98 2.78872
\(446\) 314.654i 0.705503i
\(447\) 0 0
\(448\) 0 0
\(449\) 739.852i 1.64778i 0.566752 + 0.823888i \(0.308200\pi\)
−0.566752 + 0.823888i \(0.691800\pi\)
\(450\) 0 0
\(451\) −279.984 −0.620808
\(452\) − 62.6949i − 0.138706i
\(453\) 0 0
\(454\) −144.000 −0.317181
\(455\) 0 0
\(456\) 0 0
\(457\) −248.324 −0.543379 −0.271689 0.962385i \(-0.587582\pi\)
−0.271689 + 0.962385i \(0.587582\pi\)
\(458\) − 230.637i − 0.503574i
\(459\) 0 0
\(460\) −316.664 −0.688400
\(461\) 355.970i 0.772168i 0.922464 + 0.386084i \(0.126173\pi\)
−0.922464 + 0.386084i \(0.873827\pi\)
\(462\) 0 0
\(463\) −6.33202 −0.0136761 −0.00683804 0.999977i \(-0.502177\pi\)
−0.00683804 + 0.999977i \(0.502177\pi\)
\(464\) 47.6137i 0.102616i
\(465\) 0 0
\(466\) 513.158 1.10120
\(467\) 878.691i 1.88156i 0.339011 + 0.940782i \(0.389907\pi\)
−0.339011 + 0.940782i \(0.610093\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 213.541i − 0.454342i
\(471\) 0 0
\(472\) 4.66404 0.00988144
\(473\) − 774.050i − 1.63647i
\(474\) 0 0
\(475\) 1083.32 2.28067
\(476\) 0 0
\(477\) 0 0
\(478\) −250.494 −0.524046
\(479\) 224.396i 0.468468i 0.972180 + 0.234234i \(0.0752581\pi\)
−0.972180 + 0.234234i \(0.924742\pi\)
\(480\) 0 0
\(481\) −98.1542 −0.204063
\(482\) 216.025i 0.448185i
\(483\) 0 0
\(484\) 391.328 0.808529
\(485\) 395.166i 0.814776i
\(486\) 0 0
\(487\) −717.490 −1.47329 −0.736643 0.676282i \(-0.763590\pi\)
−0.736643 + 0.676282i \(0.763590\pi\)
\(488\) − 283.782i − 0.581520i
\(489\) 0 0
\(490\) 0 0
\(491\) − 274.002i − 0.558050i −0.960284 0.279025i \(-0.909989\pi\)
0.960284 0.279025i \(-0.0900112\pi\)
\(492\) 0 0
\(493\) −307.919 −0.624582
\(494\) − 73.0584i − 0.147892i
\(495\) 0 0
\(496\) 68.6640 0.138436
\(497\) 0 0
\(498\) 0 0
\(499\) −728.810 −1.46054 −0.730271 0.683158i \(-0.760606\pi\)
−0.730271 + 0.683158i \(0.760606\pi\)
\(500\) − 519.011i − 1.03802i
\(501\) 0 0
\(502\) −504.000 −1.00398
\(503\) − 594.657i − 1.18222i −0.806590 0.591111i \(-0.798690\pi\)
0.806590 0.591111i \(-0.201310\pi\)
\(504\) 0 0
\(505\) −283.652 −0.561688
\(506\) 447.831i 0.885041i
\(507\) 0 0
\(508\) −428.664 −0.843827
\(509\) − 994.015i − 1.95288i −0.215795 0.976439i \(-0.569234\pi\)
0.215795 0.976439i \(-0.430766\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) −84.2431 −0.163897
\(515\) 40.0564i 0.0777794i
\(516\) 0 0
\(517\) −301.992 −0.584124
\(518\) 0 0
\(519\) 0 0
\(520\) −65.0039 −0.125008
\(521\) − 40.8459i − 0.0783990i −0.999231 0.0391995i \(-0.987519\pi\)
0.999231 0.0391995i \(-0.0124808\pi\)
\(522\) 0 0
\(523\) 232.000 0.443595 0.221797 0.975093i \(-0.428808\pi\)
0.221797 + 0.975093i \(0.428808\pi\)
\(524\) 182.898i 0.349041i
\(525\) 0 0
\(526\) −10.4941 −0.0199507
\(527\) 444.052i 0.842603i
\(528\) 0 0
\(529\) 212.336 0.401391
\(530\) 1076.57i 2.03126i
\(531\) 0 0
\(532\) 0 0
\(533\) − 40.6405i − 0.0762487i
\(534\) 0 0
\(535\) 1531.97 2.86349
\(536\) 103.702i 0.193473i
\(537\) 0 0
\(538\) 608.405 1.13086
\(539\) 0 0
\(540\) 0 0
\(541\) 250.332 0.462721 0.231360 0.972868i \(-0.425682\pi\)
0.231360 + 0.972868i \(0.425682\pi\)
\(542\) − 58.2175i − 0.107412i
\(543\) 0 0
\(544\) 146.332 0.268993
\(545\) 1582.25i 2.90321i
\(546\) 0 0
\(547\) 888.324 1.62399 0.811996 0.583662i \(-0.198381\pi\)
0.811996 + 0.583662i \(0.198381\pi\)
\(548\) 213.781i 0.390111i
\(549\) 0 0
\(550\) −1363.14 −2.47844
\(551\) − 238.069i − 0.432066i
\(552\) 0 0
\(553\) 0 0
\(554\) − 45.2548i − 0.0816874i
\(555\) 0 0
\(556\) −242.656 −0.436432
\(557\) 316.309i 0.567879i 0.958842 + 0.283940i \(0.0916415\pi\)
−0.958842 + 0.283940i \(0.908358\pi\)
\(558\) 0 0
\(559\) 112.356 0.200994
\(560\) 0 0
\(561\) 0 0
\(562\) 24.1699 0.0430070
\(563\) − 58.4690i − 0.103853i −0.998651 0.0519263i \(-0.983464\pi\)
0.998651 0.0519263i \(-0.0165361\pi\)
\(564\) 0 0
\(565\) 278.915 0.493655
\(566\) 621.773i 1.09854i
\(567\) 0 0
\(568\) −50.3320 −0.0886127
\(569\) 221.665i 0.389570i 0.980846 + 0.194785i \(0.0624009\pi\)
−0.980846 + 0.194785i \(0.937599\pi\)
\(570\) 0 0
\(571\) −487.644 −0.854018 −0.427009 0.904247i \(-0.640433\pi\)
−0.427009 + 0.904247i \(0.640433\pi\)
\(572\) 91.9294i 0.160716i
\(573\) 0 0
\(574\) 0 0
\(575\) − 963.887i − 1.67633i
\(576\) 0 0
\(577\) 487.328 0.844589 0.422295 0.906459i \(-0.361225\pi\)
0.422295 + 0.906459i \(0.361225\pi\)
\(578\) 537.625i 0.930147i
\(579\) 0 0
\(580\) −211.822 −0.365211
\(581\) 0 0
\(582\) 0 0
\(583\) 1522.49 2.61148
\(584\) − 81.7840i − 0.140041i
\(585\) 0 0
\(586\) 557.733 0.951763
\(587\) 445.701i 0.759286i 0.925133 + 0.379643i \(0.123953\pi\)
−0.925133 + 0.379643i \(0.876047\pi\)
\(588\) 0 0
\(589\) −343.320 −0.582887
\(590\) 20.7492i 0.0351682i
\(591\) 0 0
\(592\) 152.000 0.256757
\(593\) 276.648i 0.466523i 0.972414 + 0.233261i \(0.0749397\pi\)
−0.972414 + 0.233261i \(0.925060\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 35.3498i 0.0593117i
\(597\) 0 0
\(598\) −65.0039 −0.108702
\(599\) − 82.3793i − 0.137528i −0.997633 0.0687641i \(-0.978094\pi\)
0.997633 0.0687641i \(-0.0219056\pi\)
\(600\) 0 0
\(601\) 418.000 0.695507 0.347754 0.937586i \(-0.386945\pi\)
0.347754 + 0.937586i \(0.386945\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 101.668 0.168324
\(605\) 1740.93i 2.87756i
\(606\) 0 0
\(607\) −76.8419 −0.126593 −0.0632964 0.997995i \(-0.520161\pi\)
−0.0632964 + 0.997995i \(0.520161\pi\)
\(608\) 113.137i 0.186081i
\(609\) 0 0
\(610\) 1262.48 2.06964
\(611\) − 43.8351i − 0.0717431i
\(612\) 0 0
\(613\) 59.3281 0.0967832 0.0483916 0.998828i \(-0.484590\pi\)
0.0483916 + 0.998828i \(0.484590\pi\)
\(614\) − 33.0020i − 0.0537492i
\(615\) 0 0
\(616\) 0 0
\(617\) 29.1143i 0.0471869i 0.999722 + 0.0235935i \(0.00751073\pi\)
−0.999722 + 0.0235935i \(0.992489\pi\)
\(618\) 0 0
\(619\) −455.644 −0.736098 −0.368049 0.929806i \(-0.619974\pi\)
−0.368049 + 0.929806i \(0.619974\pi\)
\(620\) 305.470i 0.492694i
\(621\) 0 0
\(622\) 745.652 1.19880
\(623\) 0 0
\(624\) 0 0
\(625\) 954.806 1.52769
\(626\) − 417.657i − 0.667184i
\(627\) 0 0
\(628\) 137.992 0.219733
\(629\) 982.987i 1.56278i
\(630\) 0 0
\(631\) −45.0039 −0.0713216 −0.0356608 0.999364i \(-0.511354\pi\)
−0.0356608 + 0.999364i \(0.511354\pi\)
\(632\) 334.693i 0.529578i
\(633\) 0 0
\(634\) 151.992 0.239735
\(635\) − 1907.03i − 3.00319i
\(636\) 0 0
\(637\) 0 0
\(638\) 299.562i 0.469533i
\(639\) 0 0
\(640\) 100.664 0.157288
\(641\) − 641.223i − 1.00035i −0.865925 0.500174i \(-0.833269\pi\)
0.865925 0.500174i \(-0.166731\pi\)
\(642\) 0 0
\(643\) 604.000 0.939347 0.469673 0.882840i \(-0.344372\pi\)
0.469673 + 0.882840i \(0.344372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −731.660 −1.13260
\(647\) 179.600i 0.277588i 0.990321 + 0.138794i \(0.0443226\pi\)
−0.990321 + 0.138794i \(0.955677\pi\)
\(648\) 0 0
\(649\) 29.3438 0.0452139
\(650\) − 197.864i − 0.304406i
\(651\) 0 0
\(652\) 333.992 0.512258
\(653\) 392.092i 0.600447i 0.953869 + 0.300224i \(0.0970613\pi\)
−0.953869 + 0.300224i \(0.902939\pi\)
\(654\) 0 0
\(655\) −813.668 −1.24224
\(656\) 62.9353i 0.0959379i
\(657\) 0 0
\(658\) 0 0
\(659\) 1266.54i 1.92191i 0.276701 + 0.960956i \(0.410759\pi\)
−0.276701 + 0.960956i \(0.589241\pi\)
\(660\) 0 0
\(661\) −917.644 −1.38827 −0.694133 0.719846i \(-0.744212\pi\)
−0.694133 + 0.719846i \(0.744212\pi\)
\(662\) 386.773i 0.584250i
\(663\) 0 0
\(664\) −340.664 −0.513048
\(665\) 0 0
\(666\) 0 0
\(667\) −211.822 −0.317574
\(668\) 240.886i 0.360608i
\(669\) 0 0
\(670\) −461.344 −0.688573
\(671\) − 1785.41i − 2.66083i
\(672\) 0 0
\(673\) −152.008 −0.225866 −0.112933 0.993603i \(-0.536025\pi\)
−0.112933 + 0.993603i \(0.536025\pi\)
\(674\) 482.482i 0.715848i
\(675\) 0 0
\(676\) 324.656 0.480261
\(677\) − 163.178i − 0.241031i −0.992711 0.120516i \(-0.961545\pi\)
0.992711 0.120516i \(-0.0384548\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 650.997i 0.957348i
\(681\) 0 0
\(682\) 432.000 0.633431
\(683\) − 324.914i − 0.475716i −0.971300 0.237858i \(-0.923555\pi\)
0.971300 0.237858i \(-0.0764453\pi\)
\(684\) 0 0
\(685\) −951.061 −1.38841
\(686\) 0 0
\(687\) 0 0
\(688\) −173.992 −0.252896
\(689\) 220.995i 0.320747i
\(690\) 0 0
\(691\) −1218.98 −1.76408 −0.882041 0.471173i \(-0.843831\pi\)
−0.882041 + 0.471173i \(0.843831\pi\)
\(692\) − 183.722i − 0.265494i
\(693\) 0 0
\(694\) 164.502 0.237035
\(695\) − 1079.52i − 1.55327i
\(696\) 0 0
\(697\) −407.004 −0.583937
\(698\) − 223.904i − 0.320780i
\(699\) 0 0
\(700\) 0 0
\(701\) − 427.202i − 0.609417i −0.952446 0.304709i \(-0.901441\pi\)
0.952446 0.304709i \(-0.0985591\pi\)
\(702\) 0 0
\(703\) −760.000 −1.08108
\(704\) − 142.360i − 0.202217i
\(705\) 0 0
\(706\) 325.749 0.461401
\(707\) 0 0
\(708\) 0 0
\(709\) 71.4980 0.100843 0.0504217 0.998728i \(-0.483943\pi\)
0.0504217 + 0.998728i \(0.483943\pi\)
\(710\) − 223.915i − 0.315374i
\(711\) 0 0
\(712\) −394.494 −0.554065
\(713\) 305.470i 0.428429i
\(714\) 0 0
\(715\) −408.972 −0.571989
\(716\) − 266.582i − 0.372321i
\(717\) 0 0
\(718\) 242.818 0.338187
\(719\) 111.030i 0.154422i 0.997015 + 0.0772112i \(0.0246016\pi\)
−0.997015 + 0.0772112i \(0.975398\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 55.1543i − 0.0763910i
\(723\) 0 0
\(724\) 166.170 0.229517
\(725\) − 644.761i − 0.889326i
\(726\) 0 0
\(727\) 1338.82 1.84157 0.920783 0.390076i \(-0.127551\pi\)
0.920783 + 0.390076i \(0.127551\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 363.838 0.498408
\(731\) − 1125.21i − 1.53928i
\(732\) 0 0
\(733\) −49.0771 −0.0669538 −0.0334769 0.999439i \(-0.510658\pi\)
−0.0334769 + 0.999439i \(0.510658\pi\)
\(734\) 731.842i 0.997059i
\(735\) 0 0
\(736\) 100.664 0.136772
\(737\) 652.439i 0.885263i
\(738\) 0 0
\(739\) 1430.32 1.93548 0.967738 0.251960i \(-0.0810751\pi\)
0.967738 + 0.251960i \(0.0810751\pi\)
\(740\) 676.212i 0.913800i
\(741\) 0 0
\(742\) 0 0
\(743\) 875.736i 1.17865i 0.807896 + 0.589325i \(0.200606\pi\)
−0.807896 + 0.589325i \(0.799394\pi\)
\(744\) 0 0
\(745\) −157.263 −0.211091
\(746\) 329.987i 0.442342i
\(747\) 0 0
\(748\) 920.648 1.23081
\(749\) 0 0
\(750\) 0 0
\(751\) 320.826 0.427199 0.213599 0.976921i \(-0.431481\pi\)
0.213599 + 0.976921i \(0.431481\pi\)
\(752\) 67.8823i 0.0902690i
\(753\) 0 0
\(754\) −43.4823 −0.0576688
\(755\) 452.297i 0.599069i
\(756\) 0 0
\(757\) 289.830 0.382867 0.191433 0.981506i \(-0.438686\pi\)
0.191433 + 0.981506i \(0.438686\pi\)
\(758\) − 623.903i − 0.823091i
\(759\) 0 0
\(760\) −503.320 −0.662263
\(761\) 704.657i 0.925962i 0.886368 + 0.462981i \(0.153220\pi\)
−0.886368 + 0.462981i \(0.846780\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 82.7231i − 0.108276i
\(765\) 0 0
\(766\) 301.312 0.393358
\(767\) 4.25934i 0.00555325i
\(768\) 0 0
\(769\) 117.320 0.152562 0.0762810 0.997086i \(-0.475695\pi\)
0.0762810 + 0.997086i \(0.475695\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −268.000 −0.347150
\(773\) − 658.005i − 0.851235i −0.904903 0.425618i \(-0.860057\pi\)
0.904903 0.425618i \(-0.139943\pi\)
\(774\) 0 0
\(775\) −929.814 −1.19976
\(776\) − 125.619i − 0.161880i
\(777\) 0 0
\(778\) −799.166 −1.02721
\(779\) − 314.676i − 0.403949i
\(780\) 0 0
\(781\) −316.664 −0.405460
\(782\) 650.997i 0.832477i
\(783\) 0 0
\(784\) 0 0
\(785\) 613.894i 0.782031i
\(786\) 0 0
\(787\) 1200.65 1.52560 0.762801 0.646634i \(-0.223824\pi\)
0.762801 + 0.646634i \(0.223824\pi\)
\(788\) 137.654i 0.174688i
\(789\) 0 0
\(790\) −1488.97 −1.88478
\(791\) 0 0
\(792\) 0 0
\(793\) 259.158 0.326807
\(794\) 704.737i 0.887578i
\(795\) 0 0
\(796\) −556.988 −0.699734
\(797\) − 797.411i − 1.00052i −0.865876 0.500258i \(-0.833239\pi\)
0.865876 0.500258i \(-0.166761\pi\)
\(798\) 0 0
\(799\) −438.996 −0.549432
\(800\) 306.409i 0.383012i
\(801\) 0 0
\(802\) −273.498 −0.341020
\(803\) − 514.545i − 0.640778i
\(804\) 0 0
\(805\) 0 0
\(806\) 62.7061i 0.0777991i
\(807\) 0 0
\(808\) 90.1699 0.111596
\(809\) − 156.016i − 0.192851i −0.995340 0.0964254i \(-0.969259\pi\)
0.995340 0.0964254i \(-0.0307409\pi\)
\(810\) 0 0
\(811\) 598.316 0.737751 0.368876 0.929479i \(-0.379743\pi\)
0.368876 + 0.929479i \(0.379743\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 956.308 1.17483
\(815\) 1485.85i 1.82313i
\(816\) 0 0
\(817\) 869.961 1.06482
\(818\) − 642.397i − 0.785326i
\(819\) 0 0
\(820\) −279.984 −0.341444
\(821\) − 980.978i − 1.19486i −0.801922 0.597429i \(-0.796189\pi\)
0.801922 0.597429i \(-0.203811\pi\)
\(822\) 0 0
\(823\) −431.336 −0.524102 −0.262051 0.965054i \(-0.584399\pi\)
−0.262051 + 0.965054i \(0.584399\pi\)
\(824\) − 12.7335i − 0.0154533i
\(825\) 0 0
\(826\) 0 0
\(827\) − 1219.41i − 1.47449i −0.675623 0.737247i \(-0.736125\pi\)
0.675623 0.737247i \(-0.263875\pi\)
\(828\) 0 0
\(829\) 770.081 0.928928 0.464464 0.885592i \(-0.346247\pi\)
0.464464 + 0.885592i \(0.346247\pi\)
\(830\) − 1515.53i − 1.82594i
\(831\) 0 0
\(832\) 20.6640 0.0248366
\(833\) 0 0
\(834\) 0 0
\(835\) −1071.64 −1.28341
\(836\) 711.802i 0.851438i
\(837\) 0 0
\(838\) −480.000 −0.572792
\(839\) − 1310.03i − 1.56142i −0.624894 0.780710i \(-0.714858\pi\)
0.624894 0.780710i \(-0.285142\pi\)
\(840\) 0 0
\(841\) 699.308 0.831520
\(842\) − 349.764i − 0.415396i
\(843\) 0 0
\(844\) 422.996 0.501180
\(845\) 1444.32i 1.70925i
\(846\) 0 0
\(847\) 0 0
\(848\) − 342.229i − 0.403571i
\(849\) 0 0
\(850\) −1981.56 −2.33124
\(851\) 676.212i 0.794609i
\(852\) 0 0
\(853\) −898.988 −1.05391 −0.526957 0.849892i \(-0.676667\pi\)
−0.526957 + 0.849892i \(0.676667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −486.996 −0.568921
\(857\) 746.156i 0.870660i 0.900271 + 0.435330i \(0.143368\pi\)
−0.900271 + 0.435330i \(0.856632\pi\)
\(858\) 0 0
\(859\) 991.984 1.15481 0.577406 0.816457i \(-0.304065\pi\)
0.577406 + 0.816457i \(0.304065\pi\)
\(860\) − 774.050i − 0.900058i
\(861\) 0 0
\(862\) −645.474 −0.748810
\(863\) − 209.418i − 0.242663i −0.992612 0.121332i \(-0.961284\pi\)
0.992612 0.121332i \(-0.0387164\pi\)
\(864\) 0 0
\(865\) 817.336 0.944897
\(866\) − 902.246i − 1.04185i
\(867\) 0 0
\(868\) 0 0
\(869\) 2105.73i 2.42316i
\(870\) 0 0
\(871\) −94.7034 −0.108730
\(872\) − 502.979i − 0.576811i
\(873\) 0 0
\(874\) −503.320 −0.575881
\(875\) 0 0
\(876\) 0 0
\(877\) −865.304 −0.986664 −0.493332 0.869841i \(-0.664221\pi\)
−0.493332 + 0.869841i \(0.664221\pi\)
\(878\) − 1108.95i − 1.26304i
\(879\) 0 0
\(880\) 633.328 0.719691
\(881\) 995.046i 1.12945i 0.825279 + 0.564725i \(0.191018\pi\)
−0.825279 + 0.564725i \(0.808982\pi\)
\(882\) 0 0
\(883\) 101.474 0.114920 0.0574600 0.998348i \(-0.481700\pi\)
0.0574600 + 0.998348i \(0.481700\pi\)
\(884\) 133.635i 0.151171i
\(885\) 0 0
\(886\) −667.822 −0.753750
\(887\) 1074.09i 1.21093i 0.795873 + 0.605464i \(0.207012\pi\)
−0.795873 + 0.605464i \(0.792988\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 1755.01i − 1.97192i
\(891\) 0 0
\(892\) 444.988 0.498866
\(893\) − 339.411i − 0.380080i
\(894\) 0 0
\(895\) 1185.96 1.32510
\(896\) 0 0
\(897\) 0 0
\(898\) 1046.31 1.16515
\(899\) 204.334i 0.227291i
\(900\) 0 0
\(901\) 2213.20 2.45638
\(902\) 395.958i 0.438977i
\(903\) 0 0
\(904\) −88.6640 −0.0980797
\(905\) 739.251i 0.816852i
\(906\) 0 0
\(907\) −432.162 −0.476474 −0.238237 0.971207i \(-0.576570\pi\)
−0.238237 + 0.971207i \(0.576570\pi\)
\(908\) 203.647i 0.224281i
\(909\) 0 0
\(910\) 0 0
\(911\) 104.984i 0.115241i 0.998339 + 0.0576205i \(0.0183513\pi\)
−0.998339 + 0.0576205i \(0.981649\pi\)
\(912\) 0 0
\(913\) −2143.29 −2.34752
\(914\) 351.183i 0.384227i
\(915\) 0 0
\(916\) −326.170 −0.356081
\(917\) 0 0
\(918\) 0 0
\(919\) −91.8379 −0.0999325 −0.0499662 0.998751i \(-0.515911\pi\)
−0.0499662 + 0.998751i \(0.515911\pi\)
\(920\) 447.831i 0.486772i
\(921\) 0 0
\(922\) 503.417 0.546005
\(923\) − 45.9647i − 0.0497993i
\(924\) 0 0
\(925\) −2058.31 −2.22520
\(926\) 8.95483i 0.00967044i
\(927\) 0 0
\(928\) 67.3360 0.0725603
\(929\) − 1309.00i − 1.40904i −0.709682 0.704522i \(-0.751162\pi\)
0.709682 0.704522i \(-0.248838\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 725.715i − 0.778664i
\(933\) 0 0
\(934\) 1242.66 1.33047
\(935\) 4095.75i 4.38048i
\(936\) 0 0
\(937\) −1262.00 −1.34685 −0.673426 0.739255i \(-0.735178\pi\)
−0.673426 + 0.739255i \(0.735178\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −301.992 −0.321268
\(941\) − 1315.97i − 1.39849i −0.714884 0.699243i \(-0.753521\pi\)
0.714884 0.699243i \(-0.246479\pi\)
\(942\) 0 0
\(943\) −279.984 −0.296908
\(944\) − 6.59595i − 0.00698724i
\(945\) 0 0
\(946\) −1094.67 −1.15716
\(947\) − 486.582i − 0.513814i −0.966436 0.256907i \(-0.917297\pi\)
0.966436 0.256907i \(-0.0827034\pi\)
\(948\) 0 0
\(949\) 74.6877 0.0787014
\(950\) − 1532.05i − 1.61268i
\(951\) 0 0
\(952\) 0 0
\(953\) − 43.3711i − 0.0455100i −0.999741 0.0227550i \(-0.992756\pi\)
0.999741 0.0227550i \(-0.00724377\pi\)
\(954\) 0 0
\(955\) 368.016 0.385357
\(956\) 354.252i 0.370557i
\(957\) 0 0
\(958\) 317.344 0.331257
\(959\) 0 0
\(960\) 0 0
\(961\) −666.328 −0.693369
\(962\) 138.811i 0.144294i
\(963\) 0 0
\(964\) 305.506 0.316915
\(965\) − 1192.27i − 1.23551i
\(966\) 0 0
\(967\) 1648.99 1.70526 0.852631 0.522514i \(-0.175006\pi\)
0.852631 + 0.522514i \(0.175006\pi\)
\(968\) − 553.421i − 0.571716i
\(969\) 0 0
\(970\) 558.850 0.576134
\(971\) 518.323i 0.533803i 0.963724 + 0.266902i \(0.0859999\pi\)
−0.963724 + 0.266902i \(0.914000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1014.68i 1.04177i
\(975\) 0 0
\(976\) −401.328 −0.411197
\(977\) − 109.948i − 0.112536i −0.998416 0.0562682i \(-0.982080\pi\)
0.998416 0.0562682i \(-0.0179202\pi\)
\(978\) 0 0
\(979\) −2481.96 −2.53520
\(980\) 0 0
\(981\) 0 0
\(982\) −387.498 −0.394601
\(983\) − 589.710i − 0.599909i −0.953954 0.299954i \(-0.903029\pi\)
0.953954 0.299954i \(-0.0969715\pi\)
\(984\) 0 0
\(985\) −612.389 −0.621715
\(986\) 435.463i 0.441646i
\(987\) 0 0
\(988\) −103.320 −0.104575
\(989\) − 774.050i − 0.782659i
\(990\) 0 0
\(991\) 713.474 0.719954 0.359977 0.932961i \(-0.382785\pi\)
0.359977 + 0.932961i \(0.382785\pi\)
\(992\) − 97.1056i − 0.0978887i
\(993\) 0 0
\(994\) 0 0
\(995\) − 2477.91i − 2.49036i
\(996\) 0 0
\(997\) −1222.99 −1.22667 −0.613334 0.789824i \(-0.710172\pi\)
−0.613334 + 0.789824i \(0.710172\pi\)
\(998\) 1030.69i 1.03276i
\(999\) 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.f.197.1 4
3.2 odd 2 inner 882.3.b.f.197.4 4
7.2 even 3 882.3.s.i.557.1 8
7.3 odd 6 882.3.s.e.863.3 8
7.4 even 3 882.3.s.i.863.4 8
7.5 odd 6 882.3.s.e.557.2 8
7.6 odd 2 126.3.b.a.71.2 4
21.2 odd 6 882.3.s.i.557.4 8
21.5 even 6 882.3.s.e.557.3 8
21.11 odd 6 882.3.s.i.863.1 8
21.17 even 6 882.3.s.e.863.2 8
21.20 even 2 126.3.b.a.71.3 yes 4
28.27 even 2 1008.3.d.a.449.4 4
35.13 even 4 3150.3.c.b.449.2 8
35.27 even 4 3150.3.c.b.449.8 8
35.34 odd 2 3150.3.e.e.701.3 4
56.13 odd 2 4032.3.d.i.449.1 4
56.27 even 2 4032.3.d.j.449.1 4
63.13 odd 6 1134.3.q.c.1079.4 8
63.20 even 6 1134.3.q.c.701.4 8
63.34 odd 6 1134.3.q.c.701.1 8
63.41 even 6 1134.3.q.c.1079.1 8
84.83 odd 2 1008.3.d.a.449.1 4
105.62 odd 4 3150.3.c.b.449.3 8
105.83 odd 4 3150.3.c.b.449.5 8
105.104 even 2 3150.3.e.e.701.1 4
168.83 odd 2 4032.3.d.j.449.4 4
168.125 even 2 4032.3.d.i.449.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.2 4 7.6 odd 2
126.3.b.a.71.3 yes 4 21.20 even 2
882.3.b.f.197.1 4 1.1 even 1 trivial
882.3.b.f.197.4 4 3.2 odd 2 inner
882.3.s.e.557.2 8 7.5 odd 6
882.3.s.e.557.3 8 21.5 even 6
882.3.s.e.863.2 8 21.17 even 6
882.3.s.e.863.3 8 7.3 odd 6
882.3.s.i.557.1 8 7.2 even 3
882.3.s.i.557.4 8 21.2 odd 6
882.3.s.i.863.1 8 21.11 odd 6
882.3.s.i.863.4 8 7.4 even 3
1008.3.d.a.449.1 4 84.83 odd 2
1008.3.d.a.449.4 4 28.27 even 2
1134.3.q.c.701.1 8 63.34 odd 6
1134.3.q.c.701.4 8 63.20 even 6
1134.3.q.c.1079.1 8 63.41 even 6
1134.3.q.c.1079.4 8 63.13 odd 6
3150.3.c.b.449.2 8 35.13 even 4
3150.3.c.b.449.3 8 105.62 odd 4
3150.3.c.b.449.5 8 105.83 odd 4
3150.3.c.b.449.8 8 35.27 even 4
3150.3.e.e.701.1 4 105.104 even 2
3150.3.e.e.701.3 4 35.34 odd 2
4032.3.d.i.449.1 4 56.13 odd 2
4032.3.d.i.449.4 4 168.125 even 2
4032.3.d.j.449.1 4 56.27 even 2
4032.3.d.j.449.4 4 168.83 odd 2