Properties

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

$q$-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 \(0.349123\pi\)
−0.456443 + 0.889753i \(0.650877\pi\)
\(6\) 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 \(-0.700082\pi\)
0.587993 0.808866i \(-0.299918\pi\)
\(12\) 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 \(-0.724794\pi\)
0.648956 0.760826i \(-0.275206\pi\)
\(18\) 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 \(-0.873563\pi\)
0.922143 0.386849i \(-0.126437\pi\)
\(24\) 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 \(-0.934205\pi\)
0.978713 0.205232i \(-0.0657947\pi\)
\(30\) 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 \(-0.938543\pi\)
0.981419 0.191876i \(-0.0614571\pi\)
\(42\) 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 \(-0.942216\pi\)
0.983568 0.180538i \(-0.0577838\pi\)
\(48\) 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 \(0.298987\pi\)
−0.590356 + 0.807143i \(0.701013\pi\)
\(54\) 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 \(0.00444836\pi\)
−0.999902 + 0.0139745i \(0.995552\pi\)
\(60\) 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 \(-0.960005\pi\)
0.992117 0.125317i \(-0.0399949\pi\)
\(72\) 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 \(-0.741581\pi\)
0.688159 0.725560i \(-0.258419\pi\)
\(84\) 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 \(-0.713399\pi\)
0.621309 0.783566i \(-0.286601\pi\)
\(90\) 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 \(0.0504470\pi\)
−0.987468 + 0.157821i \(0.949553\pi\)
\(102\) 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 \(-0.702393\pi\)
0.593851 0.804575i \(-0.297607\pi\)
\(108\) 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 \(-0.955706\pi\)
0.990334 0.138706i \(-0.0442942\pi\)
\(114\) 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 \(0.113493\pi\)
−0.937107 + 0.349041i \(0.886507\pi\)
\(132\) 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 \(0.127563\pi\)
−0.920768 + 0.390111i \(0.872437\pi\)
\(138\) 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 \(0.0188906\pi\)
−0.998240 + 0.0593117i \(0.981109\pi\)
\(150\) 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 \(0.117431\pi\)
−0.932718 + 0.360608i \(0.882569\pi\)
\(168\) 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 \(-0.914465\pi\)
0.964112 0.265494i \(-0.0855351\pi\)
\(174\) 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 \(-0.878562\pi\)
0.928104 0.372321i \(-0.121438\pi\)
\(180\) 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 \(-0.965467\pi\)
0.994121 0.108276i \(-0.0345331\pi\)
\(192\) 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 \(0.0558916\pi\)
−0.984624 + 0.174688i \(0.944108\pi\)
\(198\) 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 \(0.0720032\pi\)
−0.974525 + 0.224281i \(0.927997\pi\)
\(228\) 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 \(-0.715898\pi\)
0.627441 0.778664i \(-0.284102\pi\)
\(234\) 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 \(0.120833\pi\)
−0.928810 + 0.370557i \(0.879167\pi\)
\(240\) 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 \(0.251271\pi\)
−0.704278 + 0.709924i \(0.748729\pi\)
\(252\) 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 \(0.0369729\pi\)
−0.993262 + 0.115893i \(0.963027\pi\)
\(258\) 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 \(0.00449065\pi\)
−0.999900 + 0.0141073i \(0.995509\pi\)
\(264\) 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 \(-0.705023\pi\)
0.600477 0.799642i \(-0.294977\pi\)
\(270\) 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 \(-0.990319\pi\)
0.999537 0.0304106i \(-0.00968147\pi\)
\(282\) 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 \(-0.765006\pi\)
0.739644 0.672998i \(-0.234994\pi\)
\(294\) 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 \(-0.678000\pi\)
0.530511 0.847678i \(-0.322000\pi\)
\(312\) 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 \(-0.945779\pi\)
0.985527 0.169518i \(-0.0542212\pi\)
\(318\) 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 \(-0.946395\pi\)
0.985854 0.167609i \(-0.0536045\pi\)
\(348\) 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 \(-0.894212\pi\)
0.945280 0.326260i \(-0.105788\pi\)
\(354\) 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 \(-0.923136\pi\)
0.970987 0.239134i \(-0.0768636\pi\)
\(360\) 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 \(-0.910280\pi\)
0.960539 0.278146i \(-0.0897200\pi\)
\(384\) 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 \(0.258782\pi\)
−0.687331 + 0.726344i \(0.741218\pi\)
\(390\) 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 \(0.0775205\pi\)
−0.970491 + 0.241138i \(0.922480\pi\)
\(402\) 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 \(0.132737\pi\)
−0.914305 + 0.405025i \(0.867263\pi\)
\(420\) 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 \(0.177616\pi\)
−0.848317 + 0.529489i \(0.822384\pi\)
\(432\) 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 \(0.178928\pi\)
−0.846127 + 0.532981i \(0.821072\pi\)
\(444\) 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 \(-0.691800\pi\)
0.566752 0.823888i \(-0.308200\pi\)
\(450\) 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 \(-0.873827\pi\)
0.922464 0.386084i \(-0.126173\pi\)
\(462\) 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 \(-0.610093\pi\)
0.339011 0.940782i \(-0.389907\pi\)
\(468\) 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 \(-0.924742\pi\)
0.972180 0.234234i \(-0.0752581\pi\)
\(480\) 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 \(0.0900112\pi\)
−0.960284 + 0.279025i \(0.909989\pi\)
\(492\) 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 \(0.201310\pi\)
−0.806590 + 0.591111i \(0.798690\pi\)
\(504\) 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 \(0.430766\pi\)
−0.215795 + 0.976439i \(0.569234\pi\)
\(510\) 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 \(0.0124808\pi\)
−0.999231 + 0.0391995i \(0.987519\pi\)
\(522\) 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 \(-0.908358\pi\)
0.958842 0.283940i \(-0.0916415\pi\)
\(558\) 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 \(0.0165361\pi\)
−0.998651 + 0.0519263i \(0.983464\pi\)
\(564\) 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 \(-0.937599\pi\)
0.980846 0.194785i \(-0.0624009\pi\)
\(570\) 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 \(-0.876047\pi\)
0.925133 0.379643i \(-0.123953\pi\)
\(588\) 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 \(-0.925060\pi\)
0.972414 0.233261i \(-0.0749397\pi\)
\(594\) 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 \(0.0219056\pi\)
−0.997633 + 0.0687641i \(0.978094\pi\)
\(600\) 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 \(-0.992489\pi\)
0.999722 0.0235935i \(-0.00751073\pi\)
\(618\) 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 \(0.166731\pi\)
−0.865925 + 0.500174i \(0.833269\pi\)
\(642\) 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 \(-0.955677\pi\)
0.990321 0.138794i \(-0.0443226\pi\)
\(648\) 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 \(-0.902939\pi\)
0.953869 0.300224i \(-0.0970613\pi\)
\(654\) 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 \(-0.589241\pi\)
0.276701 0.960956i \(-0.410759\pi\)
\(660\) 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 \(0.0384548\pi\)
−0.992711 + 0.120516i \(0.961545\pi\)
\(678\) 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 \(0.0764453\pi\)
−0.971300 + 0.237858i \(0.923555\pi\)
\(684\) 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 \(0.0985591\pi\)
−0.952446 + 0.304709i \(0.901441\pi\)
\(702\) 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 \(-0.975398\pi\)
0.997015 0.0772112i \(-0.0246016\pi\)
\(720\) 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 \(-0.799394\pi\)
0.807896 0.589325i \(-0.200606\pi\)
\(744\) 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 \(-0.846780\pi\)
0.886368 0.462981i \(-0.153220\pi\)
\(762\) 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 \(0.139943\pi\)
−0.904903 + 0.425618i \(0.860057\pi\)
\(774\) 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 \(0.166761\pi\)
−0.865876 + 0.500258i \(0.833239\pi\)
\(798\) 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 \(0.0307409\pi\)
−0.995340 + 0.0964254i \(0.969259\pi\)
\(810\) 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 \(0.203811\pi\)
−0.801922 + 0.597429i \(0.796189\pi\)
\(822\) 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 \(0.263875\pi\)
−0.675623 + 0.737247i \(0.736125\pi\)
\(828\) 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 \(0.285142\pi\)
−0.624894 + 0.780710i \(0.714858\pi\)
\(840\) 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 \(-0.856632\pi\)
0.900271 0.435330i \(-0.143368\pi\)
\(858\) 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 \(0.0387164\pi\)
−0.992612 + 0.121332i \(0.961284\pi\)
\(864\) 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 \(-0.808982\pi\)
0.825279 0.564725i \(-0.191018\pi\)
\(882\) 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 \(-0.792988\pi\)
0.795873 0.605464i \(-0.207012\pi\)
\(888\) 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 \(-0.981649\pi\)
0.998339 0.0576205i \(-0.0183513\pi\)
\(912\) 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 \(0.248838\pi\)
−0.709682 + 0.704522i \(0.751162\pi\)
\(930\) 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 \(0.246479\pi\)
−0.714884 + 0.699243i \(0.753521\pi\)
\(942\) 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 \(0.0827034\pi\)
−0.966436 + 0.256907i \(0.917297\pi\)
\(948\) 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 \(0.00724377\pi\)
−0.999741 + 0.0227550i \(0.992756\pi\)
\(954\) 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 \(-0.914000\pi\)
0.963724 0.266902i \(-0.0859999\pi\)
\(972\) 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 \(0.0179202\pi\)
−0.998416 + 0.0562682i \(0.982080\pi\)
\(978\) 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 \(0.0969715\pi\)
−0.953954 + 0.299954i \(0.903029\pi\)
\(984\) 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.197.4 4
3.2 odd 2 inner 882.3.b.f.197.1 4
7.2 even 3 882.3.s.i.557.4 8
7.3 odd 6 882.3.s.e.863.2 8
7.4 even 3 882.3.s.i.863.1 8
7.5 odd 6 882.3.s.e.557.3 8
7.6 odd 2 126.3.b.a.71.3 yes 4
21.2 odd 6 882.3.s.i.557.1 8
21.5 even 6 882.3.s.e.557.2 8
21.11 odd 6 882.3.s.i.863.4 8
21.17 even 6 882.3.s.e.863.3 8
21.20 even 2 126.3.b.a.71.2 4
28.27 even 2 1008.3.d.a.449.1 4
35.13 even 4 3150.3.c.b.449.5 8
35.27 even 4 3150.3.c.b.449.3 8
35.34 odd 2 3150.3.e.e.701.1 4
56.13 odd 2 4032.3.d.i.449.4 4
56.27 even 2 4032.3.d.j.449.4 4
63.13 odd 6 1134.3.q.c.1079.1 8
63.20 even 6 1134.3.q.c.701.1 8
63.34 odd 6 1134.3.q.c.701.4 8
63.41 even 6 1134.3.q.c.1079.4 8
84.83 odd 2 1008.3.d.a.449.4 4
105.62 odd 4 3150.3.c.b.449.8 8
105.83 odd 4 3150.3.c.b.449.2 8
105.104 even 2 3150.3.e.e.701.3 4
168.83 odd 2 4032.3.d.j.449.1 4
168.125 even 2 4032.3.d.i.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.2 4 21.20 even 2
126.3.b.a.71.3 yes 4 7.6 odd 2
882.3.b.f.197.1 4 3.2 odd 2 inner
882.3.b.f.197.4 4 1.1 even 1 trivial
882.3.s.e.557.2 8 21.5 even 6
882.3.s.e.557.3 8 7.5 odd 6
882.3.s.e.863.2 8 7.3 odd 6
882.3.s.e.863.3 8 21.17 even 6
882.3.s.i.557.1 8 21.2 odd 6
882.3.s.i.557.4 8 7.2 even 3
882.3.s.i.863.1 8 7.4 even 3
882.3.s.i.863.4 8 21.11 odd 6
1008.3.d.a.449.1 4 28.27 even 2
1008.3.d.a.449.4 4 84.83 odd 2
1134.3.q.c.701.1 8 63.20 even 6
1134.3.q.c.701.4 8 63.34 odd 6
1134.3.q.c.1079.1 8 63.13 odd 6
1134.3.q.c.1079.4 8 63.41 even 6
3150.3.c.b.449.2 8 105.83 odd 4
3150.3.c.b.449.3 8 35.27 even 4
3150.3.c.b.449.5 8 35.13 even 4
3150.3.c.b.449.8 8 105.62 odd 4
3150.3.e.e.701.1 4 35.34 odd 2
3150.3.e.e.701.3 4 105.104 even 2
4032.3.d.i.449.1 4 168.125 even 2
4032.3.d.i.449.4 4 56.13 odd 2
4032.3.d.j.449.1 4 168.83 odd 2
4032.3.d.j.449.4 4 56.27 even 2