Properties

Label 882.3.c.c.685.1
Level $882$
Weight $3$
Character 882.685
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.3.c.c.685.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} -4.89898i q^{5} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} -4.89898i q^{5} -2.82843 q^{8} +6.92820i q^{10} +16.9706 q^{11} -1.73205i q^{13} +4.00000 q^{16} -4.89898i q^{17} +29.4449i q^{19} -9.79796i q^{20} -24.0000 q^{22} +8.48528 q^{23} +1.00000 q^{25} +2.44949i q^{26} +33.9411 q^{29} +12.1244i q^{31} -5.65685 q^{32} +6.92820i q^{34} -47.0000 q^{37} -41.6413i q^{38} +13.8564i q^{40} -68.5857i q^{41} +31.0000 q^{43} +33.9411 q^{44} -12.0000 q^{46} +83.2827i q^{47} -1.41421 q^{50} -3.46410i q^{52} +76.3675 q^{53} -83.1384i q^{55} -48.0000 q^{58} -83.2827i q^{59} -83.1384i q^{61} -17.1464i q^{62} +8.00000 q^{64} -8.48528 q^{65} -31.0000 q^{67} -9.79796i q^{68} -59.3970 q^{71} +81.4064i q^{73} +66.4680 q^{74} +58.8897i q^{76} +41.0000 q^{79} -19.5959i q^{80} +96.9948i q^{82} +4.89898i q^{83} -24.0000 q^{85} -43.8406 q^{86} -48.0000 q^{88} -58.7878i q^{89} +16.9706 q^{92} -117.779i q^{94} +144.250 q^{95} +41.5692i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + O(q^{10}) \) \( 4q + 8q^{4} + 16q^{16} - 96q^{22} + 4q^{25} - 188q^{37} + 124q^{43} - 48q^{46} - 192q^{58} + 32q^{64} - 124q^{67} + 164q^{79} - 96q^{85} - 192q^{88} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) − 4.89898i − 0.979796i −0.871780 0.489898i \(-0.837034\pi\)
0.871780 0.489898i \(-0.162966\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) 6.92820i 0.692820i
\(11\) 16.9706 1.54278 0.771389 0.636364i \(-0.219562\pi\)
0.771389 + 0.636364i \(0.219562\pi\)
\(12\) 0 0
\(13\) − 1.73205i − 0.133235i −0.997779 0.0666173i \(-0.978779\pi\)
0.997779 0.0666173i \(-0.0212207\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 4.89898i − 0.288175i −0.989565 0.144088i \(-0.953975\pi\)
0.989565 0.144088i \(-0.0460247\pi\)
\(18\) 0 0
\(19\) 29.4449i 1.54973i 0.632127 + 0.774865i \(0.282182\pi\)
−0.632127 + 0.774865i \(0.717818\pi\)
\(20\) − 9.79796i − 0.489898i
\(21\) 0 0
\(22\) −24.0000 −1.09091
\(23\) 8.48528 0.368925 0.184463 0.982840i \(-0.440946\pi\)
0.184463 + 0.982840i \(0.440946\pi\)
\(24\) 0 0
\(25\) 1.00000 0.0400000
\(26\) 2.44949i 0.0942111i
\(27\) 0 0
\(28\) 0 0
\(29\) 33.9411 1.17038 0.585192 0.810895i \(-0.301019\pi\)
0.585192 + 0.810895i \(0.301019\pi\)
\(30\) 0 0
\(31\) 12.1244i 0.391108i 0.980693 + 0.195554i \(0.0626505\pi\)
−0.980693 + 0.195554i \(0.937349\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) 6.92820i 0.203771i
\(35\) 0 0
\(36\) 0 0
\(37\) −47.0000 −1.27027 −0.635135 0.772401i \(-0.719056\pi\)
−0.635135 + 0.772401i \(0.719056\pi\)
\(38\) − 41.6413i − 1.09582i
\(39\) 0 0
\(40\) 13.8564i 0.346410i
\(41\) − 68.5857i − 1.67282i −0.548103 0.836411i \(-0.684650\pi\)
0.548103 0.836411i \(-0.315350\pi\)
\(42\) 0 0
\(43\) 31.0000 0.720930 0.360465 0.932773i \(-0.382618\pi\)
0.360465 + 0.932773i \(0.382618\pi\)
\(44\) 33.9411 0.771389
\(45\) 0 0
\(46\) −12.0000 −0.260870
\(47\) 83.2827i 1.77197i 0.463713 + 0.885986i \(0.346517\pi\)
−0.463713 + 0.885986i \(0.653483\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.41421 −0.0282843
\(51\) 0 0
\(52\) − 3.46410i − 0.0666173i
\(53\) 76.3675 1.44090 0.720448 0.693509i \(-0.243936\pi\)
0.720448 + 0.693509i \(0.243936\pi\)
\(54\) 0 0
\(55\) − 83.1384i − 1.51161i
\(56\) 0 0
\(57\) 0 0
\(58\) −48.0000 −0.827586
\(59\) − 83.2827i − 1.41157i −0.708426 0.705785i \(-0.750594\pi\)
0.708426 0.705785i \(-0.249406\pi\)
\(60\) 0 0
\(61\) − 83.1384i − 1.36293i −0.731853 0.681463i \(-0.761344\pi\)
0.731853 0.681463i \(-0.238656\pi\)
\(62\) − 17.1464i − 0.276555i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) −8.48528 −0.130543
\(66\) 0 0
\(67\) −31.0000 −0.462687 −0.231343 0.972872i \(-0.574312\pi\)
−0.231343 + 0.972872i \(0.574312\pi\)
\(68\) − 9.79796i − 0.144088i
\(69\) 0 0
\(70\) 0 0
\(71\) −59.3970 −0.836577 −0.418289 0.908314i \(-0.637370\pi\)
−0.418289 + 0.908314i \(0.637370\pi\)
\(72\) 0 0
\(73\) 81.4064i 1.11516i 0.830125 + 0.557578i \(0.188269\pi\)
−0.830125 + 0.557578i \(0.811731\pi\)
\(74\) 66.4680 0.898217
\(75\) 0 0
\(76\) 58.8897i 0.774865i
\(77\) 0 0
\(78\) 0 0
\(79\) 41.0000 0.518987 0.259494 0.965745i \(-0.416444\pi\)
0.259494 + 0.965745i \(0.416444\pi\)
\(80\) − 19.5959i − 0.244949i
\(81\) 0 0
\(82\) 96.9948i 1.18286i
\(83\) 4.89898i 0.0590238i 0.999564 + 0.0295119i \(0.00939530\pi\)
−0.999564 + 0.0295119i \(0.990605\pi\)
\(84\) 0 0
\(85\) −24.0000 −0.282353
\(86\) −43.8406 −0.509775
\(87\) 0 0
\(88\) −48.0000 −0.545455
\(89\) − 58.7878i − 0.660537i −0.943887 0.330268i \(-0.892861\pi\)
0.943887 0.330268i \(-0.107139\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 16.9706 0.184463
\(93\) 0 0
\(94\) − 117.779i − 1.25297i
\(95\) 144.250 1.51842
\(96\) 0 0
\(97\) 41.5692i 0.428549i 0.976774 + 0.214274i \(0.0687387\pi\)
−0.976774 + 0.214274i \(0.931261\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.00000 0.0200000
\(101\) − 176.363i − 1.74617i −0.487567 0.873085i \(-0.662116\pi\)
0.487567 0.873085i \(-0.337884\pi\)
\(102\) 0 0
\(103\) − 29.4449i − 0.285872i −0.989732 0.142936i \(-0.954346\pi\)
0.989732 0.142936i \(-0.0456544\pi\)
\(104\) 4.89898i 0.0471056i
\(105\) 0 0
\(106\) −108.000 −1.01887
\(107\) 144.250 1.34813 0.674064 0.738673i \(-0.264547\pi\)
0.674064 + 0.738673i \(0.264547\pi\)
\(108\) 0 0
\(109\) 169.000 1.55046 0.775229 0.631680i \(-0.217634\pi\)
0.775229 + 0.631680i \(0.217634\pi\)
\(110\) 117.576i 1.06887i
\(111\) 0 0
\(112\) 0 0
\(113\) 59.3970 0.525637 0.262818 0.964845i \(-0.415348\pi\)
0.262818 + 0.964845i \(0.415348\pi\)
\(114\) 0 0
\(115\) − 41.5692i − 0.361471i
\(116\) 67.8823 0.585192
\(117\) 0 0
\(118\) 117.779i 0.998131i
\(119\) 0 0
\(120\) 0 0
\(121\) 167.000 1.38017
\(122\) 117.576i 0.963734i
\(123\) 0 0
\(124\) 24.2487i 0.195554i
\(125\) − 127.373i − 1.01899i
\(126\) 0 0
\(127\) 209.000 1.64567 0.822835 0.568281i \(-0.192391\pi\)
0.822835 + 0.568281i \(0.192391\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) 12.0000 0.0923077
\(131\) 58.7878i 0.448761i 0.974502 + 0.224381i \(0.0720359\pi\)
−0.974502 + 0.224381i \(0.927964\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 43.8406 0.327169
\(135\) 0 0
\(136\) 13.8564i 0.101885i
\(137\) −152.735 −1.11485 −0.557427 0.830226i \(-0.688211\pi\)
−0.557427 + 0.830226i \(0.688211\pi\)
\(138\) 0 0
\(139\) − 195.722i − 1.40807i −0.710165 0.704035i \(-0.751380\pi\)
0.710165 0.704035i \(-0.248620\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 84.0000 0.591549
\(143\) − 29.3939i − 0.205552i
\(144\) 0 0
\(145\) − 166.277i − 1.14674i
\(146\) − 115.126i − 0.788534i
\(147\) 0 0
\(148\) −94.0000 −0.635135
\(149\) −50.9117 −0.341689 −0.170845 0.985298i \(-0.554650\pi\)
−0.170845 + 0.985298i \(0.554650\pi\)
\(150\) 0 0
\(151\) 10.0000 0.0662252 0.0331126 0.999452i \(-0.489458\pi\)
0.0331126 + 0.999452i \(0.489458\pi\)
\(152\) − 83.2827i − 0.547912i
\(153\) 0 0
\(154\) 0 0
\(155\) 59.3970 0.383206
\(156\) 0 0
\(157\) − 41.5692i − 0.264772i −0.991198 0.132386i \(-0.957736\pi\)
0.991198 0.132386i \(-0.0422639\pi\)
\(158\) −57.9828 −0.366979
\(159\) 0 0
\(160\) 27.7128i 0.173205i
\(161\) 0 0
\(162\) 0 0
\(163\) 86.0000 0.527607 0.263804 0.964576i \(-0.415023\pi\)
0.263804 + 0.964576i \(0.415023\pi\)
\(164\) − 137.171i − 0.836411i
\(165\) 0 0
\(166\) − 6.92820i − 0.0417362i
\(167\) − 181.262i − 1.08540i −0.839926 0.542701i \(-0.817402\pi\)
0.839926 0.542701i \(-0.182598\pi\)
\(168\) 0 0
\(169\) 166.000 0.982249
\(170\) 33.9411 0.199654
\(171\) 0 0
\(172\) 62.0000 0.360465
\(173\) 44.0908i 0.254860i 0.991848 + 0.127430i \(0.0406728\pi\)
−0.991848 + 0.127430i \(0.959327\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 67.8823 0.385695
\(177\) 0 0
\(178\) 83.1384i 0.467070i
\(179\) 8.48528 0.0474038 0.0237019 0.999719i \(-0.492455\pi\)
0.0237019 + 0.999719i \(0.492455\pi\)
\(180\) 0 0
\(181\) 43.3013i 0.239234i 0.992820 + 0.119617i \(0.0381666\pi\)
−0.992820 + 0.119617i \(0.961833\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −24.0000 −0.130435
\(185\) 230.252i 1.24461i
\(186\) 0 0
\(187\) − 83.1384i − 0.444591i
\(188\) 166.565i 0.885986i
\(189\) 0 0
\(190\) −204.000 −1.07368
\(191\) −76.3675 −0.399830 −0.199915 0.979813i \(-0.564067\pi\)
−0.199915 + 0.979813i \(0.564067\pi\)
\(192\) 0 0
\(193\) −287.000 −1.48705 −0.743523 0.668710i \(-0.766847\pi\)
−0.743523 + 0.668710i \(0.766847\pi\)
\(194\) − 58.7878i − 0.303030i
\(195\) 0 0
\(196\) 0 0
\(197\) −127.279 −0.646087 −0.323044 0.946384i \(-0.604706\pi\)
−0.323044 + 0.946384i \(0.604706\pi\)
\(198\) 0 0
\(199\) − 207.846i − 1.04445i −0.852807 0.522226i \(-0.825102\pi\)
0.852807 0.522226i \(-0.174898\pi\)
\(200\) −2.82843 −0.0141421
\(201\) 0 0
\(202\) 249.415i 1.23473i
\(203\) 0 0
\(204\) 0 0
\(205\) −336.000 −1.63902
\(206\) 41.6413i 0.202142i
\(207\) 0 0
\(208\) − 6.92820i − 0.0333087i
\(209\) 499.696i 2.39089i
\(210\) 0 0
\(211\) 82.0000 0.388626 0.194313 0.980940i \(-0.437752\pi\)
0.194313 + 0.980940i \(0.437752\pi\)
\(212\) 152.735 0.720448
\(213\) 0 0
\(214\) −204.000 −0.953271
\(215\) − 151.868i − 0.706364i
\(216\) 0 0
\(217\) 0 0
\(218\) −239.002 −1.09634
\(219\) 0 0
\(220\) − 166.277i − 0.755804i
\(221\) −8.48528 −0.0383949
\(222\) 0 0
\(223\) − 41.5692i − 0.186409i −0.995647 0.0932045i \(-0.970289\pi\)
0.995647 0.0932045i \(-0.0297110\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −84.0000 −0.371681
\(227\) 382.120i 1.68335i 0.539985 + 0.841675i \(0.318430\pi\)
−0.539985 + 0.841675i \(0.681570\pi\)
\(228\) 0 0
\(229\) − 81.4064i − 0.355486i −0.984077 0.177743i \(-0.943120\pi\)
0.984077 0.177743i \(-0.0568796\pi\)
\(230\) 58.7878i 0.255599i
\(231\) 0 0
\(232\) −96.0000 −0.413793
\(233\) −229.103 −0.983273 −0.491636 0.870801i \(-0.663601\pi\)
−0.491636 + 0.870801i \(0.663601\pi\)
\(234\) 0 0
\(235\) 408.000 1.73617
\(236\) − 166.565i − 0.705785i
\(237\) 0 0
\(238\) 0 0
\(239\) 67.8823 0.284026 0.142013 0.989865i \(-0.454642\pi\)
0.142013 + 0.989865i \(0.454642\pi\)
\(240\) 0 0
\(241\) 457.261i 1.89735i 0.316252 + 0.948675i \(0.397575\pi\)
−0.316252 + 0.948675i \(0.602425\pi\)
\(242\) −236.174 −0.975924
\(243\) 0 0
\(244\) − 166.277i − 0.681463i
\(245\) 0 0
\(246\) 0 0
\(247\) 51.0000 0.206478
\(248\) − 34.2929i − 0.138278i
\(249\) 0 0
\(250\) 180.133i 0.720533i
\(251\) 347.828i 1.38577i 0.721050 + 0.692884i \(0.243660\pi\)
−0.721050 + 0.692884i \(0.756340\pi\)
\(252\) 0 0
\(253\) 144.000 0.569170
\(254\) −295.571 −1.16366
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 161.666i − 0.629052i −0.949249 0.314526i \(-0.898154\pi\)
0.949249 0.314526i \(-0.101846\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −16.9706 −0.0652714
\(261\) 0 0
\(262\) − 83.1384i − 0.317322i
\(263\) −254.558 −0.967903 −0.483951 0.875095i \(-0.660799\pi\)
−0.483951 + 0.875095i \(0.660799\pi\)
\(264\) 0 0
\(265\) − 374.123i − 1.41178i
\(266\) 0 0
\(267\) 0 0
\(268\) −62.0000 −0.231343
\(269\) 19.5959i 0.0728473i 0.999336 + 0.0364236i \(0.0115966\pi\)
−0.999336 + 0.0364236i \(0.988403\pi\)
\(270\) 0 0
\(271\) 41.5692i 0.153392i 0.997055 + 0.0766960i \(0.0244371\pi\)
−0.997055 + 0.0766960i \(0.975563\pi\)
\(272\) − 19.5959i − 0.0720438i
\(273\) 0 0
\(274\) 216.000 0.788321
\(275\) 16.9706 0.0617111
\(276\) 0 0
\(277\) −337.000 −1.21661 −0.608303 0.793705i \(-0.708150\pi\)
−0.608303 + 0.793705i \(0.708150\pi\)
\(278\) 276.792i 0.995656i
\(279\) 0 0
\(280\) 0 0
\(281\) −246.073 −0.875705 −0.437853 0.899047i \(-0.644261\pi\)
−0.437853 + 0.899047i \(0.644261\pi\)
\(282\) 0 0
\(283\) 195.722i 0.691596i 0.938309 + 0.345798i \(0.112392\pi\)
−0.938309 + 0.345798i \(0.887608\pi\)
\(284\) −118.794 −0.418289
\(285\) 0 0
\(286\) 41.5692i 0.145347i
\(287\) 0 0
\(288\) 0 0
\(289\) 265.000 0.916955
\(290\) 235.151i 0.810866i
\(291\) 0 0
\(292\) 162.813i 0.557578i
\(293\) − 97.9796i − 0.334401i −0.985923 0.167201i \(-0.946527\pi\)
0.985923 0.167201i \(-0.0534728\pi\)
\(294\) 0 0
\(295\) −408.000 −1.38305
\(296\) 132.936 0.449108
\(297\) 0 0
\(298\) 72.0000 0.241611
\(299\) − 14.6969i − 0.0491536i
\(300\) 0 0
\(301\) 0 0
\(302\) −14.1421 −0.0468283
\(303\) 0 0
\(304\) 117.779i 0.387432i
\(305\) −407.294 −1.33539
\(306\) 0 0
\(307\) − 71.0141i − 0.231316i −0.993289 0.115658i \(-0.963102\pi\)
0.993289 0.115658i \(-0.0368977\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −84.0000 −0.270968
\(311\) 215.555i 0.693103i 0.938031 + 0.346552i \(0.112647\pi\)
−0.938031 + 0.346552i \(0.887353\pi\)
\(312\) 0 0
\(313\) 292.717i 0.935197i 0.883941 + 0.467598i \(0.154881\pi\)
−0.883941 + 0.467598i \(0.845119\pi\)
\(314\) 58.7878i 0.187222i
\(315\) 0 0
\(316\) 82.0000 0.259494
\(317\) 237.588 0.749489 0.374744 0.927128i \(-0.377731\pi\)
0.374744 + 0.927128i \(0.377731\pi\)
\(318\) 0 0
\(319\) 576.000 1.80564
\(320\) − 39.1918i − 0.122474i
\(321\) 0 0
\(322\) 0 0
\(323\) 144.250 0.446594
\(324\) 0 0
\(325\) − 1.73205i − 0.00532939i
\(326\) −121.622 −0.373075
\(327\) 0 0
\(328\) 193.990i 0.591432i
\(329\) 0 0
\(330\) 0 0
\(331\) −185.000 −0.558912 −0.279456 0.960158i \(-0.590154\pi\)
−0.279456 + 0.960158i \(0.590154\pi\)
\(332\) 9.79796i 0.0295119i
\(333\) 0 0
\(334\) 256.344i 0.767496i
\(335\) 151.868i 0.453338i
\(336\) 0 0
\(337\) −359.000 −1.06528 −0.532641 0.846341i \(-0.678800\pi\)
−0.532641 + 0.846341i \(0.678800\pi\)
\(338\) −234.759 −0.694555
\(339\) 0 0
\(340\) −48.0000 −0.141176
\(341\) 205.757i 0.603393i
\(342\) 0 0
\(343\) 0 0
\(344\) −87.6812 −0.254887
\(345\) 0 0
\(346\) − 62.3538i − 0.180213i
\(347\) −466.690 −1.34493 −0.672465 0.740129i \(-0.734764\pi\)
−0.672465 + 0.740129i \(0.734764\pi\)
\(348\) 0 0
\(349\) 581.969i 1.66753i 0.552117 + 0.833767i \(0.313820\pi\)
−0.552117 + 0.833767i \(0.686180\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −96.0000 −0.272727
\(353\) 289.040i 0.818810i 0.912353 + 0.409405i \(0.134264\pi\)
−0.912353 + 0.409405i \(0.865736\pi\)
\(354\) 0 0
\(355\) 290.985i 0.819675i
\(356\) − 117.576i − 0.330268i
\(357\) 0 0
\(358\) −12.0000 −0.0335196
\(359\) −339.411 −0.945435 −0.472718 0.881214i \(-0.656727\pi\)
−0.472718 + 0.881214i \(0.656727\pi\)
\(360\) 0 0
\(361\) −506.000 −1.40166
\(362\) − 61.2372i − 0.169164i
\(363\) 0 0
\(364\) 0 0
\(365\) 398.808 1.09263
\(366\) 0 0
\(367\) − 154.153i − 0.420034i −0.977698 0.210017i \(-0.932648\pi\)
0.977698 0.210017i \(-0.0673520\pi\)
\(368\) 33.9411 0.0922313
\(369\) 0 0
\(370\) − 325.626i − 0.880069i
\(371\) 0 0
\(372\) 0 0
\(373\) −289.000 −0.774799 −0.387399 0.921912i \(-0.626627\pi\)
−0.387399 + 0.921912i \(0.626627\pi\)
\(374\) 117.576i 0.314373i
\(375\) 0 0
\(376\) − 235.559i − 0.626486i
\(377\) − 58.7878i − 0.155936i
\(378\) 0 0
\(379\) 7.00000 0.0184697 0.00923483 0.999957i \(-0.497060\pi\)
0.00923483 + 0.999957i \(0.497060\pi\)
\(380\) 288.500 0.759209
\(381\) 0 0
\(382\) 108.000 0.282723
\(383\) − 494.797i − 1.29190i −0.763381 0.645949i \(-0.776462\pi\)
0.763381 0.645949i \(-0.223538\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 405.879 1.05150
\(387\) 0 0
\(388\) 83.1384i 0.214274i
\(389\) 229.103 0.588953 0.294476 0.955659i \(-0.404855\pi\)
0.294476 + 0.955659i \(0.404855\pi\)
\(390\) 0 0
\(391\) − 41.5692i − 0.106315i
\(392\) 0 0
\(393\) 0 0
\(394\) 180.000 0.456853
\(395\) − 200.858i − 0.508502i
\(396\) 0 0
\(397\) 81.4064i 0.205054i 0.994730 + 0.102527i \(0.0326928\pi\)
−0.994730 + 0.102527i \(0.967307\pi\)
\(398\) 293.939i 0.738540i
\(399\) 0 0
\(400\) 4.00000 0.0100000
\(401\) 93.3381 0.232763 0.116382 0.993205i \(-0.462870\pi\)
0.116382 + 0.993205i \(0.462870\pi\)
\(402\) 0 0
\(403\) 21.0000 0.0521092
\(404\) − 352.727i − 0.873085i
\(405\) 0 0
\(406\) 0 0
\(407\) −797.616 −1.95975
\(408\) 0 0
\(409\) 417.424i 1.02060i 0.859997 + 0.510299i \(0.170465\pi\)
−0.859997 + 0.510299i \(0.829535\pi\)
\(410\) 475.176 1.15897
\(411\) 0 0
\(412\) − 58.8897i − 0.142936i
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 0.0578313
\(416\) 9.79796i 0.0235528i
\(417\) 0 0
\(418\) − 706.677i − 1.69061i
\(419\) − 19.5959i − 0.0467683i −0.999727 0.0233842i \(-0.992556\pi\)
0.999727 0.0233842i \(-0.00744408\pi\)
\(420\) 0 0
\(421\) 407.000 0.966746 0.483373 0.875415i \(-0.339412\pi\)
0.483373 + 0.875415i \(0.339412\pi\)
\(422\) −115.966 −0.274800
\(423\) 0 0
\(424\) −216.000 −0.509434
\(425\) − 4.89898i − 0.0115270i
\(426\) 0 0
\(427\) 0 0
\(428\) 288.500 0.674064
\(429\) 0 0
\(430\) 214.774i 0.499475i
\(431\) 161.220 0.374061 0.187031 0.982354i \(-0.440114\pi\)
0.187031 + 0.982354i \(0.440114\pi\)
\(432\) 0 0
\(433\) − 168.009i − 0.388011i −0.981000 0.194006i \(-0.937852\pi\)
0.981000 0.194006i \(-0.0621480\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 338.000 0.775229
\(437\) 249.848i 0.571734i
\(438\) 0 0
\(439\) − 540.400i − 1.23098i −0.788145 0.615490i \(-0.788958\pi\)
0.788145 0.615490i \(-0.211042\pi\)
\(440\) 235.151i 0.534434i
\(441\) 0 0
\(442\) 12.0000 0.0271493
\(443\) −127.279 −0.287312 −0.143656 0.989628i \(-0.545886\pi\)
−0.143656 + 0.989628i \(0.545886\pi\)
\(444\) 0 0
\(445\) −288.000 −0.647191
\(446\) 58.7878i 0.131811i
\(447\) 0 0
\(448\) 0 0
\(449\) −110.309 −0.245676 −0.122838 0.992427i \(-0.539200\pi\)
−0.122838 + 0.992427i \(0.539200\pi\)
\(450\) 0 0
\(451\) − 1163.94i − 2.58079i
\(452\) 118.794 0.262818
\(453\) 0 0
\(454\) − 540.400i − 1.19031i
\(455\) 0 0
\(456\) 0 0
\(457\) 25.0000 0.0547046 0.0273523 0.999626i \(-0.491292\pi\)
0.0273523 + 0.999626i \(0.491292\pi\)
\(458\) 115.126i 0.251367i
\(459\) 0 0
\(460\) − 83.1384i − 0.180736i
\(461\) 78.3837i 0.170030i 0.996380 + 0.0850148i \(0.0270938\pi\)
−0.996380 + 0.0850148i \(0.972906\pi\)
\(462\) 0 0
\(463\) 521.000 1.12527 0.562635 0.826705i \(-0.309788\pi\)
0.562635 + 0.826705i \(0.309788\pi\)
\(464\) 135.765 0.292596
\(465\) 0 0
\(466\) 324.000 0.695279
\(467\) 220.454i 0.472064i 0.971745 + 0.236032i \(0.0758471\pi\)
−0.971745 + 0.236032i \(0.924153\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −576.999 −1.22766
\(471\) 0 0
\(472\) 235.559i 0.499065i
\(473\) 526.087 1.11224
\(474\) 0 0
\(475\) 29.4449i 0.0619892i
\(476\) 0 0
\(477\) 0 0
\(478\) −96.0000 −0.200837
\(479\) − 876.917i − 1.83073i −0.402631 0.915363i \(-0.631904\pi\)
0.402631 0.915363i \(-0.368096\pi\)
\(480\) 0 0
\(481\) 81.4064i 0.169244i
\(482\) − 646.665i − 1.34163i
\(483\) 0 0
\(484\) 334.000 0.690083
\(485\) 203.647 0.419890
\(486\) 0 0
\(487\) 127.000 0.260780 0.130390 0.991463i \(-0.458377\pi\)
0.130390 + 0.991463i \(0.458377\pi\)
\(488\) 235.151i 0.481867i
\(489\) 0 0
\(490\) 0 0
\(491\) 627.911 1.27884 0.639420 0.768857i \(-0.279174\pi\)
0.639420 + 0.768857i \(0.279174\pi\)
\(492\) 0 0
\(493\) − 166.277i − 0.337276i
\(494\) −72.1249 −0.146002
\(495\) 0 0
\(496\) 48.4974i 0.0977771i
\(497\) 0 0
\(498\) 0 0
\(499\) −233.000 −0.466934 −0.233467 0.972365i \(-0.575007\pi\)
−0.233467 + 0.972365i \(0.575007\pi\)
\(500\) − 254.747i − 0.509494i
\(501\) 0 0
\(502\) − 491.902i − 0.979885i
\(503\) − 538.888i − 1.07135i −0.844425 0.535674i \(-0.820058\pi\)
0.844425 0.535674i \(-0.179942\pi\)
\(504\) 0 0
\(505\) −864.000 −1.71089
\(506\) −203.647 −0.402464
\(507\) 0 0
\(508\) 418.000 0.822835
\(509\) − 318.434i − 0.625606i −0.949818 0.312803i \(-0.898732\pi\)
0.949818 0.312803i \(-0.101268\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) 228.631i 0.444807i
\(515\) −144.250 −0.280097
\(516\) 0 0
\(517\) 1413.35i 2.73376i
\(518\) 0 0
\(519\) 0 0
\(520\) 24.0000 0.0461538
\(521\) − 568.282i − 1.09075i −0.838192 0.545376i \(-0.816387\pi\)
0.838192 0.545376i \(-0.183613\pi\)
\(522\) 0 0
\(523\) 528.275i 1.01009i 0.863094 + 0.505043i \(0.168524\pi\)
−0.863094 + 0.505043i \(0.831476\pi\)
\(524\) 117.576i 0.224381i
\(525\) 0 0
\(526\) 360.000 0.684411
\(527\) 59.3970 0.112708
\(528\) 0 0
\(529\) −457.000 −0.863894
\(530\) 529.090i 0.998283i
\(531\) 0 0
\(532\) 0 0
\(533\) −118.794 −0.222878
\(534\) 0 0
\(535\) − 706.677i − 1.32089i
\(536\) 87.6812 0.163584
\(537\) 0 0
\(538\) − 27.7128i − 0.0515108i
\(539\) 0 0
\(540\) 0 0
\(541\) 335.000 0.619224 0.309612 0.950863i \(-0.399801\pi\)
0.309612 + 0.950863i \(0.399801\pi\)
\(542\) − 58.7878i − 0.108464i
\(543\) 0 0
\(544\) 27.7128i 0.0509427i
\(545\) − 827.928i − 1.51913i
\(546\) 0 0
\(547\) −658.000 −1.20293 −0.601463 0.798901i \(-0.705415\pi\)
−0.601463 + 0.798901i \(0.705415\pi\)
\(548\) −305.470 −0.557427
\(549\) 0 0
\(550\) −24.0000 −0.0436364
\(551\) 999.392i 1.81378i
\(552\) 0 0
\(553\) 0 0
\(554\) 476.590 0.860271
\(555\) 0 0
\(556\) − 391.443i − 0.704035i
\(557\) 271.529 0.487485 0.243742 0.969840i \(-0.421625\pi\)
0.243742 + 0.969840i \(0.421625\pi\)
\(558\) 0 0
\(559\) − 53.6936i − 0.0960529i
\(560\) 0 0
\(561\) 0 0
\(562\) 348.000 0.619217
\(563\) − 14.6969i − 0.0261047i −0.999915 0.0130523i \(-0.995845\pi\)
0.999915 0.0130523i \(-0.00415481\pi\)
\(564\) 0 0
\(565\) − 290.985i − 0.515017i
\(566\) − 276.792i − 0.489032i
\(567\) 0 0
\(568\) 168.000 0.295775
\(569\) 848.528 1.49126 0.745631 0.666359i \(-0.232148\pi\)
0.745631 + 0.666359i \(0.232148\pi\)
\(570\) 0 0
\(571\) 449.000 0.786340 0.393170 0.919466i \(-0.371378\pi\)
0.393170 + 0.919466i \(0.371378\pi\)
\(572\) − 58.7878i − 0.102776i
\(573\) 0 0
\(574\) 0 0
\(575\) 8.48528 0.0147570
\(576\) 0 0
\(577\) 292.717i 0.507308i 0.967295 + 0.253654i \(0.0816324\pi\)
−0.967295 + 0.253654i \(0.918368\pi\)
\(578\) −374.767 −0.648385
\(579\) 0 0
\(580\) − 332.554i − 0.573369i
\(581\) 0 0
\(582\) 0 0
\(583\) 1296.00 2.22298
\(584\) − 230.252i − 0.394267i
\(585\) 0 0
\(586\) 138.564i 0.236457i
\(587\) 529.090i 0.901345i 0.892689 + 0.450673i \(0.148816\pi\)
−0.892689 + 0.450673i \(0.851184\pi\)
\(588\) 0 0
\(589\) −357.000 −0.606112
\(590\) 576.999 0.977965
\(591\) 0 0
\(592\) −188.000 −0.317568
\(593\) 1048.38i 1.76793i 0.467554 + 0.883964i \(0.345135\pi\)
−0.467554 + 0.883964i \(0.654865\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −101.823 −0.170845
\(597\) 0 0
\(598\) 20.7846i 0.0347569i
\(599\) 644.881 1.07660 0.538298 0.842754i \(-0.319067\pi\)
0.538298 + 0.842754i \(0.319067\pi\)
\(600\) 0 0
\(601\) 458.993i 0.763716i 0.924221 + 0.381858i \(0.124716\pi\)
−0.924221 + 0.381858i \(0.875284\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 20.0000 0.0331126
\(605\) − 818.130i − 1.35228i
\(606\) 0 0
\(607\) 1051.35i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) − 166.565i − 0.273956i
\(609\) 0 0
\(610\) 576.000 0.944262
\(611\) 144.250 0.236088
\(612\) 0 0
\(613\) 290.000 0.473083 0.236542 0.971621i \(-0.423986\pi\)
0.236542 + 0.971621i \(0.423986\pi\)
\(614\) 100.429i 0.163565i
\(615\) 0 0
\(616\) 0 0
\(617\) 729.734 1.18271 0.591357 0.806410i \(-0.298593\pi\)
0.591357 + 0.806410i \(0.298593\pi\)
\(618\) 0 0
\(619\) 819.260i 1.32352i 0.749715 + 0.661761i \(0.230191\pi\)
−0.749715 + 0.661761i \(0.769809\pi\)
\(620\) 118.794 0.191603
\(621\) 0 0
\(622\) − 304.841i − 0.490098i
\(623\) 0 0
\(624\) 0 0
\(625\) −599.000 −0.958400
\(626\) − 413.964i − 0.661284i
\(627\) 0 0
\(628\) − 83.1384i − 0.132386i
\(629\) 230.252i 0.366060i
\(630\) 0 0
\(631\) −58.0000 −0.0919176 −0.0459588 0.998943i \(-0.514634\pi\)
−0.0459588 + 0.998943i \(0.514634\pi\)
\(632\) −115.966 −0.183490
\(633\) 0 0
\(634\) −336.000 −0.529968
\(635\) − 1023.89i − 1.61242i
\(636\) 0 0
\(637\) 0 0
\(638\) −814.587 −1.27678
\(639\) 0 0
\(640\) 55.4256i 0.0866025i
\(641\) 958.837 1.49585 0.747923 0.663786i \(-0.231051\pi\)
0.747923 + 0.663786i \(0.231051\pi\)
\(642\) 0 0
\(643\) − 760.370i − 1.18254i −0.806475 0.591268i \(-0.798628\pi\)
0.806475 0.591268i \(-0.201372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −204.000 −0.315789
\(647\) 352.727i 0.545172i 0.962131 + 0.272586i \(0.0878790\pi\)
−0.962131 + 0.272586i \(0.912121\pi\)
\(648\) 0 0
\(649\) − 1413.35i − 2.17774i
\(650\) 2.44949i 0.00376845i
\(651\) 0 0
\(652\) 172.000 0.263804
\(653\) −441.235 −0.675704 −0.337852 0.941199i \(-0.609700\pi\)
−0.337852 + 0.941199i \(0.609700\pi\)
\(654\) 0 0
\(655\) 288.000 0.439695
\(656\) − 274.343i − 0.418206i
\(657\) 0 0
\(658\) 0 0
\(659\) 161.220 0.244644 0.122322 0.992490i \(-0.460966\pi\)
0.122322 + 0.992490i \(0.460966\pi\)
\(660\) 0 0
\(661\) 833.116i 1.26039i 0.776438 + 0.630194i \(0.217025\pi\)
−0.776438 + 0.630194i \(0.782975\pi\)
\(662\) 261.630 0.395211
\(663\) 0 0
\(664\) − 13.8564i − 0.0208681i
\(665\) 0 0
\(666\) 0 0
\(667\) 288.000 0.431784
\(668\) − 362.524i − 0.542701i
\(669\) 0 0
\(670\) − 214.774i − 0.320559i
\(671\) − 1410.91i − 2.10269i
\(672\) 0 0
\(673\) −263.000 −0.390788 −0.195394 0.980725i \(-0.562598\pi\)
−0.195394 + 0.980725i \(0.562598\pi\)
\(674\) 507.703 0.753268
\(675\) 0 0
\(676\) 332.000 0.491124
\(677\) 499.696i 0.738103i 0.929409 + 0.369052i \(0.120317\pi\)
−0.929409 + 0.369052i \(0.879683\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 67.8823 0.0998268
\(681\) 0 0
\(682\) − 290.985i − 0.426664i
\(683\) 958.837 1.40386 0.701930 0.712246i \(-0.252322\pi\)
0.701930 + 0.712246i \(0.252322\pi\)
\(684\) 0 0
\(685\) 748.246i 1.09233i
\(686\) 0 0
\(687\) 0 0
\(688\) 124.000 0.180233
\(689\) − 132.272i − 0.191977i
\(690\) 0 0
\(691\) 1234.95i 1.78720i 0.448868 + 0.893598i \(0.351827\pi\)
−0.448868 + 0.893598i \(0.648173\pi\)
\(692\) 88.1816i 0.127430i
\(693\) 0 0
\(694\) 660.000 0.951009
\(695\) −958.837 −1.37962
\(696\) 0 0
\(697\) −336.000 −0.482066
\(698\) − 823.029i − 1.17912i
\(699\) 0 0
\(700\) 0 0
\(701\) −975.807 −1.39202 −0.696011 0.718031i \(-0.745044\pi\)
−0.696011 + 0.718031i \(0.745044\pi\)
\(702\) 0 0
\(703\) − 1383.91i − 1.96858i
\(704\) 135.765 0.192847
\(705\) 0 0
\(706\) − 408.764i − 0.578986i
\(707\) 0 0
\(708\) 0 0
\(709\) −1106.00 −1.55994 −0.779972 0.625815i \(-0.784767\pi\)
−0.779972 + 0.625815i \(0.784767\pi\)
\(710\) − 411.514i − 0.579598i
\(711\) 0 0
\(712\) 166.277i 0.233535i
\(713\) 102.879i 0.144290i
\(714\) 0 0
\(715\) −144.000 −0.201399
\(716\) 16.9706 0.0237019
\(717\) 0 0
\(718\) 480.000 0.668524
\(719\) − 685.857i − 0.953904i −0.878929 0.476952i \(-0.841742\pi\)
0.878929 0.476952i \(-0.158258\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 715.592 0.991125
\(723\) 0 0
\(724\) 86.6025i 0.119617i
\(725\) 33.9411 0.0468153
\(726\) 0 0
\(727\) − 427.817i − 0.588468i −0.955733 0.294234i \(-0.904935\pi\)
0.955733 0.294234i \(-0.0950646\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −564.000 −0.772603
\(731\) − 151.868i − 0.207754i
\(732\) 0 0
\(733\) 39.8372i 0.0543481i 0.999631 + 0.0271741i \(0.00865084\pi\)
−0.999631 + 0.0271741i \(0.991349\pi\)
\(734\) 218.005i 0.297009i
\(735\) 0 0
\(736\) −48.0000 −0.0652174
\(737\) −526.087 −0.713823
\(738\) 0 0
\(739\) −487.000 −0.658999 −0.329499 0.944156i \(-0.606880\pi\)
−0.329499 + 0.944156i \(0.606880\pi\)
\(740\) 460.504i 0.622303i
\(741\) 0 0
\(742\) 0 0
\(743\) 509.117 0.685218 0.342609 0.939478i \(-0.388689\pi\)
0.342609 + 0.939478i \(0.388689\pi\)
\(744\) 0 0
\(745\) 249.415i 0.334786i
\(746\) 408.708 0.547866
\(747\) 0 0
\(748\) − 166.277i − 0.222295i
\(749\) 0 0
\(750\) 0 0
\(751\) −545.000 −0.725699 −0.362850 0.931848i \(-0.618196\pi\)
−0.362850 + 0.931848i \(0.618196\pi\)
\(752\) 333.131i 0.442993i
\(753\) 0 0
\(754\) 83.1384i 0.110263i
\(755\) − 48.9898i − 0.0648871i
\(756\) 0 0
\(757\) −770.000 −1.01717 −0.508587 0.861011i \(-0.669832\pi\)
−0.508587 + 0.861011i \(0.669832\pi\)
\(758\) −9.89949 −0.0130600
\(759\) 0 0
\(760\) −408.000 −0.536842
\(761\) − 171.464i − 0.225314i −0.993634 0.112657i \(-0.964064\pi\)
0.993634 0.112657i \(-0.0359362\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −152.735 −0.199915
\(765\) 0 0
\(766\) 699.749i 0.913510i
\(767\) −144.250 −0.188070
\(768\) 0 0
\(769\) 704.945i 0.916703i 0.888771 + 0.458352i \(0.151560\pi\)
−0.888771 + 0.458352i \(0.848440\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −574.000 −0.743523
\(773\) 921.008i 1.19147i 0.803180 + 0.595736i \(0.203140\pi\)
−0.803180 + 0.595736i \(0.796860\pi\)
\(774\) 0 0
\(775\) 12.1244i 0.0156443i
\(776\) − 117.576i − 0.151515i
\(777\) 0 0
\(778\) −324.000 −0.416452
\(779\) 2019.50 2.59242
\(780\) 0 0
\(781\) −1008.00 −1.29065
\(782\) 58.7878i 0.0751762i
\(783\) 0 0
\(784\) 0 0
\(785\) −203.647 −0.259423
\(786\) 0 0
\(787\) − 457.261i − 0.581018i −0.956872 0.290509i \(-0.906175\pi\)
0.956872 0.290509i \(-0.0938247\pi\)
\(788\) −254.558 −0.323044
\(789\) 0 0
\(790\) 284.056i 0.359565i
\(791\) 0 0
\(792\) 0 0
\(793\) −144.000 −0.181589
\(794\) − 115.126i − 0.144995i
\(795\) 0 0
\(796\) − 415.692i − 0.522226i
\(797\) 14.6969i 0.0184403i 0.999957 + 0.00922016i \(0.00293491\pi\)
−0.999957 + 0.00922016i \(0.997065\pi\)
\(798\) 0 0
\(799\) 408.000 0.510638
\(800\) −5.65685 −0.00707107
\(801\) 0 0
\(802\) −132.000 −0.164589
\(803\) 1381.51i 1.72044i
\(804\) 0 0
\(805\) 0 0
\(806\) −29.6985 −0.0368468
\(807\) 0 0
\(808\) 498.831i 0.617365i
\(809\) −941.866 −1.16424 −0.582118 0.813105i \(-0.697776\pi\)
−0.582118 + 0.813105i \(0.697776\pi\)
\(810\) 0 0
\(811\) − 498.831i − 0.615081i −0.951535 0.307540i \(-0.900494\pi\)
0.951535 0.307540i \(-0.0995059\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1128.00 1.38575
\(815\) − 421.312i − 0.516948i
\(816\) 0 0
\(817\) 912.791i 1.11725i
\(818\) − 590.327i − 0.721671i
\(819\) 0 0
\(820\) −672.000 −0.819512
\(821\) −602.455 −0.733806 −0.366903 0.930259i \(-0.619582\pi\)
−0.366903 + 0.930259i \(0.619582\pi\)
\(822\) 0 0
\(823\) 38.0000 0.0461725 0.0230863 0.999733i \(-0.492651\pi\)
0.0230863 + 0.999733i \(0.492651\pi\)
\(824\) 83.2827i 0.101071i
\(825\) 0 0
\(826\) 0 0
\(827\) 687.308 0.831086 0.415543 0.909574i \(-0.363592\pi\)
0.415543 + 0.909574i \(0.363592\pi\)
\(828\) 0 0
\(829\) 833.116i 1.00497i 0.864587 + 0.502483i \(0.167580\pi\)
−0.864587 + 0.502483i \(0.832420\pi\)
\(830\) −33.9411 −0.0408929
\(831\) 0 0
\(832\) − 13.8564i − 0.0166543i
\(833\) 0 0
\(834\) 0 0
\(835\) −888.000 −1.06347
\(836\) 999.392i 1.19544i
\(837\) 0 0
\(838\) 27.7128i 0.0330702i
\(839\) 244.949i 0.291953i 0.989288 + 0.145977i \(0.0466325\pi\)
−0.989288 + 0.145977i \(0.953368\pi\)
\(840\) 0 0
\(841\) 311.000 0.369798
\(842\) −575.585 −0.683593
\(843\) 0 0
\(844\) 164.000 0.194313
\(845\) − 813.231i − 0.962403i
\(846\) 0 0
\(847\) 0 0
\(848\) 305.470 0.360224
\(849\) 0 0
\(850\) 6.92820i 0.00815083i
\(851\) −398.808 −0.468635
\(852\) 0 0
\(853\) − 1245.34i − 1.45996i −0.683469 0.729979i \(-0.739530\pi\)
0.683469 0.729979i \(-0.260470\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −408.000 −0.476636
\(857\) − 1224.74i − 1.42911i −0.699581 0.714554i \(-0.746630\pi\)
0.699581 0.714554i \(-0.253370\pi\)
\(858\) 0 0
\(859\) 249.415i 0.290355i 0.989406 + 0.145178i \(0.0463754\pi\)
−0.989406 + 0.145178i \(0.953625\pi\)
\(860\) − 303.737i − 0.353182i
\(861\) 0 0
\(862\) −228.000 −0.264501
\(863\) 661.852 0.766920 0.383460 0.923557i \(-0.374732\pi\)
0.383460 + 0.923557i \(0.374732\pi\)
\(864\) 0 0
\(865\) 216.000 0.249711
\(866\) 237.601i 0.274365i
\(867\) 0 0
\(868\) 0 0
\(869\) 695.793 0.800682
\(870\) 0 0
\(871\) 53.6936i 0.0616459i
\(872\) −478.004 −0.548170
\(873\) 0 0
\(874\) − 353.338i − 0.404277i
\(875\) 0 0
\(876\) 0 0
\(877\) −574.000 −0.654504 −0.327252 0.944937i \(-0.606123\pi\)
−0.327252 + 0.944937i \(0.606123\pi\)
\(878\) 764.241i 0.870434i
\(879\) 0 0
\(880\) − 332.554i − 0.377902i
\(881\) 161.666i 0.183503i 0.995782 + 0.0917516i \(0.0292466\pi\)
−0.995782 + 0.0917516i \(0.970753\pi\)
\(882\) 0 0
\(883\) 1735.00 1.96489 0.982446 0.186546i \(-0.0597294\pi\)
0.982446 + 0.186546i \(0.0597294\pi\)
\(884\) −16.9706 −0.0191975
\(885\) 0 0
\(886\) 180.000 0.203160
\(887\) − 195.959i − 0.220924i −0.993880 0.110462i \(-0.964767\pi\)
0.993880 0.110462i \(-0.0352330\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 407.294 0.457633
\(891\) 0 0
\(892\) − 83.1384i − 0.0932045i
\(893\) −2452.25 −2.74608
\(894\) 0 0
\(895\) − 41.5692i − 0.0464461i
\(896\) 0 0
\(897\) 0 0
\(898\) 156.000 0.173719
\(899\) 411.514i 0.457747i
\(900\) 0 0
\(901\) − 374.123i − 0.415231i
\(902\) 1646.06i 1.82490i
\(903\) 0 0
\(904\) −168.000 −0.185841
\(905\) 212.132 0.234400
\(906\) 0 0
\(907\) 751.000 0.828004 0.414002 0.910276i \(-0.364131\pi\)
0.414002 + 0.910276i \(0.364131\pi\)
\(908\) 764.241i 0.841675i
\(909\) 0 0
\(910\) 0 0
\(911\) −1247.34 −1.36919 −0.684597 0.728921i \(-0.740022\pi\)
−0.684597 + 0.728921i \(0.740022\pi\)
\(912\) 0 0
\(913\) 83.1384i 0.0910607i
\(914\) −35.3553 −0.0386820
\(915\) 0 0
\(916\) − 162.813i − 0.177743i
\(917\) 0 0
\(918\) 0 0
\(919\) 1015.00 1.10446 0.552231 0.833691i \(-0.313777\pi\)
0.552231 + 0.833691i \(0.313777\pi\)
\(920\) 117.576i 0.127799i
\(921\) 0 0
\(922\) − 110.851i − 0.120229i
\(923\) 102.879i 0.111461i
\(924\) 0 0
\(925\) −47.0000 −0.0508108
\(926\) −736.805 −0.795686
\(927\) 0 0
\(928\) −192.000 −0.206897
\(929\) − 1092.47i − 1.17597i −0.808873 0.587983i \(-0.799922\pi\)
0.808873 0.587983i \(-0.200078\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −458.205 −0.491636
\(933\) 0 0
\(934\) − 311.769i − 0.333800i
\(935\) −407.294 −0.435608
\(936\) 0 0
\(937\) 1747.64i 1.86514i 0.360985 + 0.932572i \(0.382441\pi\)
−0.360985 + 0.932572i \(0.617559\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 816.000 0.868085
\(941\) 1239.44i 1.31715i 0.752513 + 0.658577i \(0.228841\pi\)
−0.752513 + 0.658577i \(0.771159\pi\)
\(942\) 0 0
\(943\) − 581.969i − 0.617146i
\(944\) − 333.131i − 0.352893i
\(945\) 0 0
\(946\) −744.000 −0.786469
\(947\) −1204.91 −1.27234 −0.636172 0.771547i \(-0.719483\pi\)
−0.636172 + 0.771547i \(0.719483\pi\)
\(948\) 0 0
\(949\) 141.000 0.148577
\(950\) − 41.6413i − 0.0438330i
\(951\) 0 0
\(952\) 0 0
\(953\) −1026.72 −1.07735 −0.538677 0.842512i \(-0.681076\pi\)
−0.538677 + 0.842512i \(0.681076\pi\)
\(954\) 0 0
\(955\) 374.123i 0.391752i
\(956\) 135.765 0.142013
\(957\) 0 0
\(958\) 1240.15i 1.29452i
\(959\) 0 0
\(960\) 0 0
\(961\) 814.000 0.847034
\(962\) − 115.126i − 0.119674i
\(963\) 0 0
\(964\) 914.523i 0.948675i
\(965\) 1406.01i 1.45700i
\(966\) 0 0
\(967\) −895.000 −0.925543 −0.462771 0.886478i \(-0.653145\pi\)
−0.462771 + 0.886478i \(0.653145\pi\)
\(968\) −472.347 −0.487962
\(969\) 0 0
\(970\) −288.000 −0.296907
\(971\) 210.656i 0.216948i 0.994099 + 0.108474i \(0.0345964\pi\)
−0.994099 + 0.108474i \(0.965404\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −179.605 −0.184400
\(975\) 0 0
\(976\) − 332.554i − 0.340731i
\(977\) 1255.82 1.28539 0.642693 0.766124i \(-0.277817\pi\)
0.642693 + 0.766124i \(0.277817\pi\)
\(978\) 0 0
\(979\) − 997.661i − 1.01906i
\(980\) 0 0
\(981\) 0 0
\(982\) −888.000 −0.904277
\(983\) 1180.65i 1.20107i 0.799598 + 0.600536i \(0.205046\pi\)
−0.799598 + 0.600536i \(0.794954\pi\)
\(984\) 0 0
\(985\) 623.538i 0.633034i
\(986\) 235.151i 0.238490i
\(987\) 0 0
\(988\) 102.000 0.103239
\(989\) 263.044 0.265969
\(990\) 0 0
\(991\) 655.000 0.660949 0.330474 0.943815i \(-0.392791\pi\)
0.330474 + 0.943815i \(0.392791\pi\)
\(992\) − 68.5857i − 0.0691388i
\(993\) 0 0
\(994\) 0 0
\(995\) −1018.23 −1.02335
\(996\) 0 0
\(997\) − 458.993i − 0.460375i −0.973146 0.230187i \(-0.926066\pi\)
0.973146 0.230187i \(-0.0739339\pi\)
\(998\) 329.512 0.330172
\(999\) 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.c.c.685.1 4
3.2 odd 2 inner 882.3.c.c.685.4 4
7.2 even 3 126.3.n.b.73.2 yes 4
7.3 odd 6 126.3.n.b.19.2 yes 4
7.4 even 3 882.3.n.c.19.2 4
7.5 odd 6 882.3.n.c.325.2 4
7.6 odd 2 inner 882.3.c.c.685.2 4
21.2 odd 6 126.3.n.b.73.1 yes 4
21.5 even 6 882.3.n.c.325.1 4
21.11 odd 6 882.3.n.c.19.1 4
21.17 even 6 126.3.n.b.19.1 4
21.20 even 2 inner 882.3.c.c.685.3 4
28.3 even 6 1008.3.cg.i.145.1 4
28.23 odd 6 1008.3.cg.i.577.1 4
84.23 even 6 1008.3.cg.i.577.2 4
84.59 odd 6 1008.3.cg.i.145.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.n.b.19.1 4 21.17 even 6
126.3.n.b.19.2 yes 4 7.3 odd 6
126.3.n.b.73.1 yes 4 21.2 odd 6
126.3.n.b.73.2 yes 4 7.2 even 3
882.3.c.c.685.1 4 1.1 even 1 trivial
882.3.c.c.685.2 4 7.6 odd 2 inner
882.3.c.c.685.3 4 21.20 even 2 inner
882.3.c.c.685.4 4 3.2 odd 2 inner
882.3.n.c.19.1 4 21.11 odd 6
882.3.n.c.19.2 4 7.4 even 3
882.3.n.c.325.1 4 21.5 even 6
882.3.n.c.325.2 4 7.5 odd 6
1008.3.cg.i.145.1 4 28.3 even 6
1008.3.cg.i.145.2 4 84.59 odd 6
1008.3.cg.i.577.1 4 28.23 odd 6
1008.3.cg.i.577.2 4 84.23 even 6