Properties

Label 882.3.b.e.197.1
Level $882$
Weight $3$
Character 882.197
Analytic conductor $24.033$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.1
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.3.b.e.197.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{5} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{5} +2.82843i q^{8} +6.00000 q^{10} +12.7279i q^{11} +1.00000 q^{13} +4.00000 q^{16} -16.9706i q^{17} -23.0000 q^{19} -8.48528i q^{20} +18.0000 q^{22} +16.9706i q^{23} +7.00000 q^{25} -1.41421i q^{26} -33.9411i q^{29} -47.0000 q^{31} -5.65685i q^{32} -24.0000 q^{34} -55.0000 q^{37} +32.5269i q^{38} -12.0000 q^{40} -46.6690i q^{41} +23.0000 q^{43} -25.4558i q^{44} +24.0000 q^{46} -4.24264i q^{47} -9.89949i q^{50} -2.00000 q^{52} +50.9117i q^{53} -54.0000 q^{55} -48.0000 q^{58} +84.8528i q^{59} -104.000 q^{61} +66.4680i q^{62} -8.00000 q^{64} +4.24264i q^{65} -97.0000 q^{67} +33.9411i q^{68} -97.5807i q^{71} -65.0000 q^{73} +77.7817i q^{74} +46.0000 q^{76} +113.000 q^{79} +16.9706i q^{80} -66.0000 q^{82} -29.6985i q^{83} +72.0000 q^{85} -32.5269i q^{86} -36.0000 q^{88} +135.765i q^{89} -33.9411i q^{92} -6.00000 q^{94} -97.5807i q^{95} -104.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + O(q^{10}) \) \( 2 q - 4 q^{4} + 12 q^{10} + 2 q^{13} + 8 q^{16} - 46 q^{19} + 36 q^{22} + 14 q^{25} - 94 q^{31} - 48 q^{34} - 110 q^{37} - 24 q^{40} + 46 q^{43} + 48 q^{46} - 4 q^{52} - 108 q^{55} - 96 q^{58} - 208 q^{61} - 16 q^{64} - 194 q^{67} - 130 q^{73} + 92 q^{76} + 226 q^{79} - 132 q^{82} + 144 q^{85} - 72 q^{88} - 12 q^{94} - 208 q^{97} + O(q^{100}) \)

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
\(5\) 4.24264i 0.848528i 0.905539 + 0.424264i \(0.139467\pi\)
−0.905539 + 0.424264i \(0.860533\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) 6.00000 0.600000
\(11\) 12.7279i 1.15708i 0.815653 + 0.578542i \(0.196378\pi\)
−0.815653 + 0.578542i \(0.803622\pi\)
\(12\) 0 0
\(13\) 1.00000 0.0769231 0.0384615 0.999260i \(-0.487754\pi\)
0.0384615 + 0.999260i \(0.487754\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 16.9706i − 0.998268i −0.866525 0.499134i \(-0.833651\pi\)
0.866525 0.499134i \(-0.166349\pi\)
\(18\) 0 0
\(19\) −23.0000 −1.21053 −0.605263 0.796025i \(-0.706932\pi\)
−0.605263 + 0.796025i \(0.706932\pi\)
\(20\) − 8.48528i − 0.424264i
\(21\) 0 0
\(22\) 18.0000 0.818182
\(23\) 16.9706i 0.737851i 0.929459 + 0.368925i \(0.120274\pi\)
−0.929459 + 0.368925i \(0.879726\pi\)
\(24\) 0 0
\(25\) 7.00000 0.280000
\(26\) − 1.41421i − 0.0543928i
\(27\) 0 0
\(28\) 0 0
\(29\) − 33.9411i − 1.17038i −0.810895 0.585192i \(-0.801019\pi\)
0.810895 0.585192i \(-0.198981\pi\)
\(30\) 0 0
\(31\) −47.0000 −1.51613 −0.758065 0.652180i \(-0.773855\pi\)
−0.758065 + 0.652180i \(0.773855\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) −24.0000 −0.705882
\(35\) 0 0
\(36\) 0 0
\(37\) −55.0000 −1.48649 −0.743243 0.669021i \(-0.766713\pi\)
−0.743243 + 0.669021i \(0.766713\pi\)
\(38\) 32.5269i 0.855971i
\(39\) 0 0
\(40\) −12.0000 −0.300000
\(41\) − 46.6690i − 1.13827i −0.822244 0.569135i \(-0.807278\pi\)
0.822244 0.569135i \(-0.192722\pi\)
\(42\) 0 0
\(43\) 23.0000 0.534884 0.267442 0.963574i \(-0.413822\pi\)
0.267442 + 0.963574i \(0.413822\pi\)
\(44\) − 25.4558i − 0.578542i
\(45\) 0 0
\(46\) 24.0000 0.521739
\(47\) − 4.24264i − 0.0902690i −0.998981 0.0451345i \(-0.985628\pi\)
0.998981 0.0451345i \(-0.0143716\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 9.89949i − 0.197990i
\(51\) 0 0
\(52\) −2.00000 −0.0384615
\(53\) 50.9117i 0.960598i 0.877105 + 0.480299i \(0.159472\pi\)
−0.877105 + 0.480299i \(0.840528\pi\)
\(54\) 0 0
\(55\) −54.0000 −0.981818
\(56\) 0 0
\(57\) 0 0
\(58\) −48.0000 −0.827586
\(59\) 84.8528i 1.43818i 0.694915 + 0.719092i \(0.255442\pi\)
−0.694915 + 0.719092i \(0.744558\pi\)
\(60\) 0 0
\(61\) −104.000 −1.70492 −0.852459 0.522794i \(-0.824890\pi\)
−0.852459 + 0.522794i \(0.824890\pi\)
\(62\) 66.4680i 1.07207i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 4.24264i 0.0652714i
\(66\) 0 0
\(67\) −97.0000 −1.44776 −0.723881 0.689925i \(-0.757643\pi\)
−0.723881 + 0.689925i \(0.757643\pi\)
\(68\) 33.9411i 0.499134i
\(69\) 0 0
\(70\) 0 0
\(71\) − 97.5807i − 1.37438i −0.726479 0.687188i \(-0.758845\pi\)
0.726479 0.687188i \(-0.241155\pi\)
\(72\) 0 0
\(73\) −65.0000 −0.890411 −0.445205 0.895428i \(-0.646869\pi\)
−0.445205 + 0.895428i \(0.646869\pi\)
\(74\) 77.7817i 1.05110i
\(75\) 0 0
\(76\) 46.0000 0.605263
\(77\) 0 0
\(78\) 0 0
\(79\) 113.000 1.43038 0.715190 0.698930i \(-0.246340\pi\)
0.715190 + 0.698930i \(0.246340\pi\)
\(80\) 16.9706i 0.212132i
\(81\) 0 0
\(82\) −66.0000 −0.804878
\(83\) − 29.6985i − 0.357813i −0.983866 0.178907i \(-0.942744\pi\)
0.983866 0.178907i \(-0.0572560\pi\)
\(84\) 0 0
\(85\) 72.0000 0.847059
\(86\) − 32.5269i − 0.378220i
\(87\) 0 0
\(88\) −36.0000 −0.409091
\(89\) 135.765i 1.52544i 0.646727 + 0.762722i \(0.276137\pi\)
−0.646727 + 0.762722i \(0.723863\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 33.9411i − 0.368925i
\(93\) 0 0
\(94\) −6.00000 −0.0638298
\(95\) − 97.5807i − 1.02717i
\(96\) 0 0
\(97\) −104.000 −1.07216 −0.536082 0.844166i \(-0.680096\pi\)
−0.536082 + 0.844166i \(0.680096\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −14.0000 −0.140000
\(101\) − 148.492i − 1.47022i −0.677947 0.735111i \(-0.737130\pi\)
0.677947 0.735111i \(-0.262870\pi\)
\(102\) 0 0
\(103\) −119.000 −1.15534 −0.577670 0.816270i \(-0.696038\pi\)
−0.577670 + 0.816270i \(0.696038\pi\)
\(104\) 2.82843i 0.0271964i
\(105\) 0 0
\(106\) 72.0000 0.679245
\(107\) 118.794i 1.11022i 0.831776 + 0.555112i \(0.187325\pi\)
−0.831776 + 0.555112i \(0.812675\pi\)
\(108\) 0 0
\(109\) −49.0000 −0.449541 −0.224771 0.974412i \(-0.572163\pi\)
−0.224771 + 0.974412i \(0.572163\pi\)
\(110\) 76.3675i 0.694250i
\(111\) 0 0
\(112\) 0 0
\(113\) 97.5807i 0.863546i 0.901982 + 0.431773i \(0.142112\pi\)
−0.901982 + 0.431773i \(0.857888\pi\)
\(114\) 0 0
\(115\) −72.0000 −0.626087
\(116\) 67.8823i 0.585192i
\(117\) 0 0
\(118\) 120.000 1.01695
\(119\) 0 0
\(120\) 0 0
\(121\) −41.0000 −0.338843
\(122\) 147.078i 1.20556i
\(123\) 0 0
\(124\) 94.0000 0.758065
\(125\) 135.765i 1.08612i
\(126\) 0 0
\(127\) 113.000 0.889764 0.444882 0.895589i \(-0.353246\pi\)
0.444882 + 0.895589i \(0.353246\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) 6.00000 0.0461538
\(131\) 21.2132i 0.161933i 0.996717 + 0.0809664i \(0.0258007\pi\)
−0.996717 + 0.0809664i \(0.974199\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 137.179i 1.02372i
\(135\) 0 0
\(136\) 48.0000 0.352941
\(137\) − 67.8823i − 0.495491i −0.968825 0.247745i \(-0.920310\pi\)
0.968825 0.247745i \(-0.0796897\pi\)
\(138\) 0 0
\(139\) 103.000 0.741007 0.370504 0.928831i \(-0.379185\pi\)
0.370504 + 0.928831i \(0.379185\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −138.000 −0.971831
\(143\) 12.7279i 0.0890064i
\(144\) 0 0
\(145\) 144.000 0.993103
\(146\) 91.9239i 0.629616i
\(147\) 0 0
\(148\) 110.000 0.743243
\(149\) 169.706i 1.13896i 0.822004 + 0.569482i \(0.192856\pi\)
−0.822004 + 0.569482i \(0.807144\pi\)
\(150\) 0 0
\(151\) 104.000 0.688742 0.344371 0.938834i \(-0.388092\pi\)
0.344371 + 0.938834i \(0.388092\pi\)
\(152\) − 65.0538i − 0.427986i
\(153\) 0 0
\(154\) 0 0
\(155\) − 199.404i − 1.28648i
\(156\) 0 0
\(157\) −152.000 −0.968153 −0.484076 0.875026i \(-0.660844\pi\)
−0.484076 + 0.875026i \(0.660844\pi\)
\(158\) − 159.806i − 1.01143i
\(159\) 0 0
\(160\) 24.0000 0.150000
\(161\) 0 0
\(162\) 0 0
\(163\) 56.0000 0.343558 0.171779 0.985135i \(-0.445048\pi\)
0.171779 + 0.985135i \(0.445048\pi\)
\(164\) 93.3381i 0.569135i
\(165\) 0 0
\(166\) −42.0000 −0.253012
\(167\) − 4.24264i − 0.0254050i −0.999919 0.0127025i \(-0.995957\pi\)
0.999919 0.0127025i \(-0.00404345\pi\)
\(168\) 0 0
\(169\) −168.000 −0.994083
\(170\) − 101.823i − 0.598961i
\(171\) 0 0
\(172\) −46.0000 −0.267442
\(173\) − 152.735i − 0.882862i −0.897295 0.441431i \(-0.854471\pi\)
0.897295 0.441431i \(-0.145529\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 50.9117i 0.289271i
\(177\) 0 0
\(178\) 192.000 1.07865
\(179\) − 4.24264i − 0.0237019i −0.999930 0.0118510i \(-0.996228\pi\)
0.999930 0.0118510i \(-0.00377236\pi\)
\(180\) 0 0
\(181\) 55.0000 0.303867 0.151934 0.988391i \(-0.451450\pi\)
0.151934 + 0.988391i \(0.451450\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −48.0000 −0.260870
\(185\) − 233.345i − 1.26133i
\(186\) 0 0
\(187\) 216.000 1.15508
\(188\) 8.48528i 0.0451345i
\(189\) 0 0
\(190\) −138.000 −0.726316
\(191\) 63.6396i 0.333192i 0.986025 + 0.166596i \(0.0532775\pi\)
−0.986025 + 0.166596i \(0.946722\pi\)
\(192\) 0 0
\(193\) −151.000 −0.782383 −0.391192 0.920309i \(-0.627937\pi\)
−0.391192 + 0.920309i \(0.627937\pi\)
\(194\) 147.078i 0.758135i
\(195\) 0 0
\(196\) 0 0
\(197\) − 16.9706i − 0.0861450i −0.999072 0.0430725i \(-0.986285\pi\)
0.999072 0.0430725i \(-0.0137146\pi\)
\(198\) 0 0
\(199\) 160.000 0.804020 0.402010 0.915635i \(-0.368312\pi\)
0.402010 + 0.915635i \(0.368312\pi\)
\(200\) 19.7990i 0.0989949i
\(201\) 0 0
\(202\) −210.000 −1.03960
\(203\) 0 0
\(204\) 0 0
\(205\) 198.000 0.965854
\(206\) 168.291i 0.816949i
\(207\) 0 0
\(208\) 4.00000 0.0192308
\(209\) − 292.742i − 1.40068i
\(210\) 0 0
\(211\) −208.000 −0.985782 −0.492891 0.870091i \(-0.664060\pi\)
−0.492891 + 0.870091i \(0.664060\pi\)
\(212\) − 101.823i − 0.480299i
\(213\) 0 0
\(214\) 168.000 0.785047
\(215\) 97.5807i 0.453864i
\(216\) 0 0
\(217\) 0 0
\(218\) 69.2965i 0.317874i
\(219\) 0 0
\(220\) 108.000 0.490909
\(221\) − 16.9706i − 0.0767899i
\(222\) 0 0
\(223\) 40.0000 0.179372 0.0896861 0.995970i \(-0.471414\pi\)
0.0896861 + 0.995970i \(0.471414\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 138.000 0.610619
\(227\) − 224.860i − 0.990572i −0.868730 0.495286i \(-0.835063\pi\)
0.868730 0.495286i \(-0.164937\pi\)
\(228\) 0 0
\(229\) −377.000 −1.64629 −0.823144 0.567833i \(-0.807782\pi\)
−0.823144 + 0.567833i \(0.807782\pi\)
\(230\) 101.823i 0.442710i
\(231\) 0 0
\(232\) 96.0000 0.413793
\(233\) − 343.654i − 1.47491i −0.675397 0.737455i \(-0.736028\pi\)
0.675397 0.737455i \(-0.263972\pi\)
\(234\) 0 0
\(235\) 18.0000 0.0765957
\(236\) − 169.706i − 0.719092i
\(237\) 0 0
\(238\) 0 0
\(239\) 12.7279i 0.0532549i 0.999645 + 0.0266275i \(0.00847678\pi\)
−0.999645 + 0.0266275i \(0.991523\pi\)
\(240\) 0 0
\(241\) 130.000 0.539419 0.269710 0.962942i \(-0.413072\pi\)
0.269710 + 0.962942i \(0.413072\pi\)
\(242\) 57.9828i 0.239598i
\(243\) 0 0
\(244\) 208.000 0.852459
\(245\) 0 0
\(246\) 0 0
\(247\) −23.0000 −0.0931174
\(248\) − 132.936i − 0.536033i
\(249\) 0 0
\(250\) 192.000 0.768000
\(251\) − 50.9117i − 0.202835i −0.994844 0.101418i \(-0.967662\pi\)
0.994844 0.101418i \(-0.0323379\pi\)
\(252\) 0 0
\(253\) −216.000 −0.853755
\(254\) − 159.806i − 0.629158i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 470.933i − 1.83242i −0.400693 0.916212i \(-0.631231\pi\)
0.400693 0.916212i \(-0.368769\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 8.48528i − 0.0326357i
\(261\) 0 0
\(262\) 30.0000 0.114504
\(263\) 101.823i 0.387161i 0.981084 + 0.193581i \(0.0620101\pi\)
−0.981084 + 0.193581i \(0.937990\pi\)
\(264\) 0 0
\(265\) −216.000 −0.815094
\(266\) 0 0
\(267\) 0 0
\(268\) 194.000 0.723881
\(269\) 284.257i 1.05672i 0.849021 + 0.528359i \(0.177192\pi\)
−0.849021 + 0.528359i \(0.822808\pi\)
\(270\) 0 0
\(271\) 520.000 1.91882 0.959410 0.282016i \(-0.0910032\pi\)
0.959410 + 0.282016i \(0.0910032\pi\)
\(272\) − 67.8823i − 0.249567i
\(273\) 0 0
\(274\) −96.0000 −0.350365
\(275\) 89.0955i 0.323983i
\(276\) 0 0
\(277\) 113.000 0.407942 0.203971 0.978977i \(-0.434615\pi\)
0.203971 + 0.978977i \(0.434615\pi\)
\(278\) − 145.664i − 0.523971i
\(279\) 0 0
\(280\) 0 0
\(281\) 458.205i 1.63062i 0.579022 + 0.815312i \(0.303434\pi\)
−0.579022 + 0.815312i \(0.696566\pi\)
\(282\) 0 0
\(283\) −89.0000 −0.314488 −0.157244 0.987560i \(-0.550261\pi\)
−0.157244 + 0.987560i \(0.550261\pi\)
\(284\) 195.161i 0.687188i
\(285\) 0 0
\(286\) 18.0000 0.0629371
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.00346021
\(290\) − 203.647i − 0.702230i
\(291\) 0 0
\(292\) 130.000 0.445205
\(293\) − 67.8823i − 0.231680i −0.993268 0.115840i \(-0.963044\pi\)
0.993268 0.115840i \(-0.0369560\pi\)
\(294\) 0 0
\(295\) −360.000 −1.22034
\(296\) − 155.563i − 0.525552i
\(297\) 0 0
\(298\) 240.000 0.805369
\(299\) 16.9706i 0.0567577i
\(300\) 0 0
\(301\) 0 0
\(302\) − 147.078i − 0.487014i
\(303\) 0 0
\(304\) −92.0000 −0.302632
\(305\) − 441.235i − 1.44667i
\(306\) 0 0
\(307\) −233.000 −0.758958 −0.379479 0.925200i \(-0.623897\pi\)
−0.379479 + 0.925200i \(0.623897\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −282.000 −0.909677
\(311\) − 250.316i − 0.804874i −0.915448 0.402437i \(-0.868163\pi\)
0.915448 0.402437i \(-0.131837\pi\)
\(312\) 0 0
\(313\) 151.000 0.482428 0.241214 0.970472i \(-0.422454\pi\)
0.241214 + 0.970472i \(0.422454\pi\)
\(314\) 214.960i 0.684587i
\(315\) 0 0
\(316\) −226.000 −0.715190
\(317\) 407.294i 1.28484i 0.766354 + 0.642419i \(0.222069\pi\)
−0.766354 + 0.642419i \(0.777931\pi\)
\(318\) 0 0
\(319\) 432.000 1.35423
\(320\) − 33.9411i − 0.106066i
\(321\) 0 0
\(322\) 0 0
\(323\) 390.323i 1.20843i
\(324\) 0 0
\(325\) 7.00000 0.0215385
\(326\) − 79.1960i − 0.242932i
\(327\) 0 0
\(328\) 132.000 0.402439
\(329\) 0 0
\(330\) 0 0
\(331\) −73.0000 −0.220544 −0.110272 0.993901i \(-0.535172\pi\)
−0.110272 + 0.993901i \(0.535172\pi\)
\(332\) 59.3970i 0.178907i
\(333\) 0 0
\(334\) −6.00000 −0.0179641
\(335\) − 411.536i − 1.22847i
\(336\) 0 0
\(337\) 527.000 1.56380 0.781899 0.623405i \(-0.214251\pi\)
0.781899 + 0.623405i \(0.214251\pi\)
\(338\) 237.588i 0.702923i
\(339\) 0 0
\(340\) −144.000 −0.423529
\(341\) − 598.212i − 1.75429i
\(342\) 0 0
\(343\) 0 0
\(344\) 65.0538i 0.189110i
\(345\) 0 0
\(346\) −216.000 −0.624277
\(347\) 424.264i 1.22266i 0.791375 + 0.611332i \(0.209366\pi\)
−0.791375 + 0.611332i \(0.790634\pi\)
\(348\) 0 0
\(349\) 112.000 0.320917 0.160458 0.987043i \(-0.448703\pi\)
0.160458 + 0.987043i \(0.448703\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 72.0000 0.204545
\(353\) − 309.713i − 0.877373i −0.898640 0.438687i \(-0.855444\pi\)
0.898640 0.438687i \(-0.144556\pi\)
\(354\) 0 0
\(355\) 414.000 1.16620
\(356\) − 271.529i − 0.762722i
\(357\) 0 0
\(358\) −6.00000 −0.0167598
\(359\) − 509.117i − 1.41815i −0.705132 0.709076i \(-0.749112\pi\)
0.705132 0.709076i \(-0.250888\pi\)
\(360\) 0 0
\(361\) 168.000 0.465374
\(362\) − 77.7817i − 0.214867i
\(363\) 0 0
\(364\) 0 0
\(365\) − 275.772i − 0.755539i
\(366\) 0 0
\(367\) 319.000 0.869210 0.434605 0.900621i \(-0.356888\pi\)
0.434605 + 0.900621i \(0.356888\pi\)
\(368\) 67.8823i 0.184463i
\(369\) 0 0
\(370\) −330.000 −0.891892
\(371\) 0 0
\(372\) 0 0
\(373\) 209.000 0.560322 0.280161 0.959953i \(-0.409612\pi\)
0.280161 + 0.959953i \(0.409612\pi\)
\(374\) − 305.470i − 0.816765i
\(375\) 0 0
\(376\) 12.0000 0.0319149
\(377\) − 33.9411i − 0.0900295i
\(378\) 0 0
\(379\) −433.000 −1.14248 −0.571240 0.820783i \(-0.693537\pi\)
−0.571240 + 0.820783i \(0.693537\pi\)
\(380\) 195.161i 0.513583i
\(381\) 0 0
\(382\) 90.0000 0.235602
\(383\) 390.323i 1.01912i 0.860435 + 0.509560i \(0.170192\pi\)
−0.860435 + 0.509560i \(0.829808\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 213.546i 0.553229i
\(387\) 0 0
\(388\) 208.000 0.536082
\(389\) − 369.110i − 0.948868i −0.880291 0.474434i \(-0.842653\pi\)
0.880291 0.474434i \(-0.157347\pi\)
\(390\) 0 0
\(391\) 288.000 0.736573
\(392\) 0 0
\(393\) 0 0
\(394\) −24.0000 −0.0609137
\(395\) 479.418i 1.21372i
\(396\) 0 0
\(397\) −113.000 −0.284635 −0.142317 0.989821i \(-0.545455\pi\)
−0.142317 + 0.989821i \(0.545455\pi\)
\(398\) − 226.274i − 0.568528i
\(399\) 0 0
\(400\) 28.0000 0.0700000
\(401\) − 356.382i − 0.888733i −0.895845 0.444366i \(-0.853429\pi\)
0.895845 0.444366i \(-0.146571\pi\)
\(402\) 0 0
\(403\) −47.0000 −0.116625
\(404\) 296.985i 0.735111i
\(405\) 0 0
\(406\) 0 0
\(407\) − 700.036i − 1.71999i
\(408\) 0 0
\(409\) 319.000 0.779951 0.389976 0.920825i \(-0.372483\pi\)
0.389976 + 0.920825i \(0.372483\pi\)
\(410\) − 280.014i − 0.682962i
\(411\) 0 0
\(412\) 238.000 0.577670
\(413\) 0 0
\(414\) 0 0
\(415\) 126.000 0.303614
\(416\) − 5.65685i − 0.0135982i
\(417\) 0 0
\(418\) −414.000 −0.990431
\(419\) 767.918i 1.83274i 0.400333 + 0.916370i \(0.368895\pi\)
−0.400333 + 0.916370i \(0.631105\pi\)
\(420\) 0 0
\(421\) 65.0000 0.154394 0.0771971 0.997016i \(-0.475403\pi\)
0.0771971 + 0.997016i \(0.475403\pi\)
\(422\) 294.156i 0.697053i
\(423\) 0 0
\(424\) −144.000 −0.339623
\(425\) − 118.794i − 0.279515i
\(426\) 0 0
\(427\) 0 0
\(428\) − 237.588i − 0.555112i
\(429\) 0 0
\(430\) 138.000 0.320930
\(431\) 479.418i 1.11234i 0.831069 + 0.556170i \(0.187730\pi\)
−0.831069 + 0.556170i \(0.812270\pi\)
\(432\) 0 0
\(433\) 367.000 0.847575 0.423788 0.905762i \(-0.360700\pi\)
0.423788 + 0.905762i \(0.360700\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 98.0000 0.224771
\(437\) − 390.323i − 0.893188i
\(438\) 0 0
\(439\) −536.000 −1.22096 −0.610478 0.792033i \(-0.709023\pi\)
−0.610478 + 0.792033i \(0.709023\pi\)
\(440\) − 152.735i − 0.347125i
\(441\) 0 0
\(442\) −24.0000 −0.0542986
\(443\) − 84.8528i − 0.191541i −0.995403 0.0957707i \(-0.969468\pi\)
0.995403 0.0957707i \(-0.0305315\pi\)
\(444\) 0 0
\(445\) −576.000 −1.29438
\(446\) − 56.5685i − 0.126835i
\(447\) 0 0
\(448\) 0 0
\(449\) 615.183i 1.37012i 0.728488 + 0.685059i \(0.240224\pi\)
−0.728488 + 0.685059i \(0.759776\pi\)
\(450\) 0 0
\(451\) 594.000 1.31707
\(452\) − 195.161i − 0.431773i
\(453\) 0 0
\(454\) −318.000 −0.700441
\(455\) 0 0
\(456\) 0 0
\(457\) −463.000 −1.01313 −0.506565 0.862202i \(-0.669085\pi\)
−0.506565 + 0.862202i \(0.669085\pi\)
\(458\) 533.159i 1.16410i
\(459\) 0 0
\(460\) 144.000 0.313043
\(461\) 271.529i 0.589000i 0.955651 + 0.294500i \(0.0951531\pi\)
−0.955651 + 0.294500i \(0.904847\pi\)
\(462\) 0 0
\(463\) 47.0000 0.101512 0.0507559 0.998711i \(-0.483837\pi\)
0.0507559 + 0.998711i \(0.483837\pi\)
\(464\) − 135.765i − 0.292596i
\(465\) 0 0
\(466\) −486.000 −1.04292
\(467\) 742.462i 1.58985i 0.606705 + 0.794927i \(0.292491\pi\)
−0.606705 + 0.794927i \(0.707509\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 25.4558i − 0.0541614i
\(471\) 0 0
\(472\) −240.000 −0.508475
\(473\) 292.742i 0.618905i
\(474\) 0 0
\(475\) −161.000 −0.338947
\(476\) 0 0
\(477\) 0 0
\(478\) 18.0000 0.0376569
\(479\) 424.264i 0.885729i 0.896589 + 0.442864i \(0.146038\pi\)
−0.896589 + 0.442864i \(0.853962\pi\)
\(480\) 0 0
\(481\) −55.0000 −0.114345
\(482\) − 183.848i − 0.381427i
\(483\) 0 0
\(484\) 82.0000 0.169421
\(485\) − 441.235i − 0.909762i
\(486\) 0 0
\(487\) 239.000 0.490760 0.245380 0.969427i \(-0.421087\pi\)
0.245380 + 0.969427i \(0.421087\pi\)
\(488\) − 294.156i − 0.602780i
\(489\) 0 0
\(490\) 0 0
\(491\) − 203.647i − 0.414759i −0.978261 0.207380i \(-0.933506\pi\)
0.978261 0.207380i \(-0.0664935\pi\)
\(492\) 0 0
\(493\) −576.000 −1.16836
\(494\) 32.5269i 0.0658440i
\(495\) 0 0
\(496\) −188.000 −0.379032
\(497\) 0 0
\(498\) 0 0
\(499\) −49.0000 −0.0981964 −0.0490982 0.998794i \(-0.515635\pi\)
−0.0490982 + 0.998794i \(0.515635\pi\)
\(500\) − 271.529i − 0.543058i
\(501\) 0 0
\(502\) −72.0000 −0.143426
\(503\) 589.727i 1.17242i 0.810159 + 0.586210i \(0.199381\pi\)
−0.810159 + 0.586210i \(0.800619\pi\)
\(504\) 0 0
\(505\) 630.000 1.24752
\(506\) 305.470i 0.603696i
\(507\) 0 0
\(508\) −226.000 −0.444882
\(509\) − 241.831i − 0.475109i −0.971374 0.237555i \(-0.923654\pi\)
0.971374 0.237555i \(-0.0763458\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) −666.000 −1.29572
\(515\) − 504.874i − 0.980338i
\(516\) 0 0
\(517\) 54.0000 0.104449
\(518\) 0 0
\(519\) 0 0
\(520\) −12.0000 −0.0230769
\(521\) 848.528i 1.62865i 0.580407 + 0.814326i \(0.302893\pi\)
−0.580407 + 0.814326i \(0.697107\pi\)
\(522\) 0 0
\(523\) 391.000 0.747610 0.373805 0.927507i \(-0.378053\pi\)
0.373805 + 0.927507i \(0.378053\pi\)
\(524\) − 42.4264i − 0.0809664i
\(525\) 0 0
\(526\) 144.000 0.273764
\(527\) 797.616i 1.51350i
\(528\) 0 0
\(529\) 241.000 0.455577
\(530\) 305.470i 0.576359i
\(531\) 0 0
\(532\) 0 0
\(533\) − 46.6690i − 0.0875592i
\(534\) 0 0
\(535\) −504.000 −0.942056
\(536\) − 274.357i − 0.511861i
\(537\) 0 0
\(538\) 402.000 0.747212
\(539\) 0 0
\(540\) 0 0
\(541\) −73.0000 −0.134935 −0.0674677 0.997721i \(-0.521492\pi\)
−0.0674677 + 0.997721i \(0.521492\pi\)
\(542\) − 735.391i − 1.35681i
\(543\) 0 0
\(544\) −96.0000 −0.176471
\(545\) − 207.889i − 0.381448i
\(546\) 0 0
\(547\) 32.0000 0.0585009 0.0292505 0.999572i \(-0.490688\pi\)
0.0292505 + 0.999572i \(0.490688\pi\)
\(548\) 135.765i 0.247745i
\(549\) 0 0
\(550\) 126.000 0.229091
\(551\) 780.646i 1.41678i
\(552\) 0 0
\(553\) 0 0
\(554\) − 159.806i − 0.288459i
\(555\) 0 0
\(556\) −206.000 −0.370504
\(557\) 46.6690i 0.0837864i 0.999122 + 0.0418932i \(0.0133389\pi\)
−0.999122 + 0.0418932i \(0.986661\pi\)
\(558\) 0 0
\(559\) 23.0000 0.0411449
\(560\) 0 0
\(561\) 0 0
\(562\) 648.000 1.15302
\(563\) 886.712i 1.57498i 0.616330 + 0.787488i \(0.288619\pi\)
−0.616330 + 0.787488i \(0.711381\pi\)
\(564\) 0 0
\(565\) −414.000 −0.732743
\(566\) 125.865i 0.222376i
\(567\) 0 0
\(568\) 276.000 0.485915
\(569\) 318.198i 0.559223i 0.960113 + 0.279612i \(0.0902057\pi\)
−0.960113 + 0.279612i \(0.909794\pi\)
\(570\) 0 0
\(571\) −535.000 −0.936953 −0.468476 0.883476i \(-0.655197\pi\)
−0.468476 + 0.883476i \(0.655197\pi\)
\(572\) − 25.4558i − 0.0445032i
\(573\) 0 0
\(574\) 0 0
\(575\) 118.794i 0.206598i
\(576\) 0 0
\(577\) −905.000 −1.56846 −0.784229 0.620472i \(-0.786941\pi\)
−0.784229 + 0.620472i \(0.786941\pi\)
\(578\) − 1.41421i − 0.00244674i
\(579\) 0 0
\(580\) −288.000 −0.496552
\(581\) 0 0
\(582\) 0 0
\(583\) −648.000 −1.11149
\(584\) − 183.848i − 0.314808i
\(585\) 0 0
\(586\) −96.0000 −0.163823
\(587\) 576.999i 0.982963i 0.870888 + 0.491481i \(0.163544\pi\)
−0.870888 + 0.491481i \(0.836456\pi\)
\(588\) 0 0
\(589\) 1081.00 1.83531
\(590\) 509.117i 0.862910i
\(591\) 0 0
\(592\) −220.000 −0.371622
\(593\) 555.786i 0.937244i 0.883399 + 0.468622i \(0.155249\pi\)
−0.883399 + 0.468622i \(0.844751\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 339.411i − 0.569482i
\(597\) 0 0
\(598\) 24.0000 0.0401338
\(599\) − 203.647i − 0.339978i −0.985446 0.169989i \(-0.945627\pi\)
0.985446 0.169989i \(-0.0543732\pi\)
\(600\) 0 0
\(601\) −143.000 −0.237937 −0.118968 0.992898i \(-0.537959\pi\)
−0.118968 + 0.992898i \(0.537959\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −208.000 −0.344371
\(605\) − 173.948i − 0.287518i
\(606\) 0 0
\(607\) −599.000 −0.986820 −0.493410 0.869797i \(-0.664250\pi\)
−0.493410 + 0.869797i \(0.664250\pi\)
\(608\) 130.108i 0.213993i
\(609\) 0 0
\(610\) −624.000 −1.02295
\(611\) − 4.24264i − 0.00694377i
\(612\) 0 0
\(613\) −304.000 −0.495922 −0.247961 0.968770i \(-0.579760\pi\)
−0.247961 + 0.968770i \(0.579760\pi\)
\(614\) 329.512i 0.536664i
\(615\) 0 0
\(616\) 0 0
\(617\) 292.742i 0.474461i 0.971453 + 0.237230i \(0.0762396\pi\)
−0.971453 + 0.237230i \(0.923760\pi\)
\(618\) 0 0
\(619\) 343.000 0.554120 0.277060 0.960853i \(-0.410640\pi\)
0.277060 + 0.960853i \(0.410640\pi\)
\(620\) 398.808i 0.643239i
\(621\) 0 0
\(622\) −354.000 −0.569132
\(623\) 0 0
\(624\) 0 0
\(625\) −401.000 −0.641600
\(626\) − 213.546i − 0.341128i
\(627\) 0 0
\(628\) 304.000 0.484076
\(629\) 933.381i 1.48391i
\(630\) 0 0
\(631\) 272.000 0.431062 0.215531 0.976497i \(-0.430852\pi\)
0.215531 + 0.976497i \(0.430852\pi\)
\(632\) 319.612i 0.505716i
\(633\) 0 0
\(634\) 576.000 0.908517
\(635\) 479.418i 0.754990i
\(636\) 0 0
\(637\) 0 0
\(638\) − 610.940i − 0.957587i
\(639\) 0 0
\(640\) −48.0000 −0.0750000
\(641\) 424.264i 0.661878i 0.943652 + 0.330939i \(0.107366\pi\)
−0.943652 + 0.330939i \(0.892634\pi\)
\(642\) 0 0
\(643\) 679.000 1.05599 0.527994 0.849248i \(-0.322944\pi\)
0.527994 + 0.849248i \(0.322944\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 552.000 0.854489
\(647\) 963.079i 1.48853i 0.667884 + 0.744265i \(0.267200\pi\)
−0.667884 + 0.744265i \(0.732800\pi\)
\(648\) 0 0
\(649\) −1080.00 −1.66410
\(650\) − 9.89949i − 0.0152300i
\(651\) 0 0
\(652\) −112.000 −0.171779
\(653\) − 895.197i − 1.37090i −0.728120 0.685450i \(-0.759606\pi\)
0.728120 0.685450i \(-0.240394\pi\)
\(654\) 0 0
\(655\) −90.0000 −0.137405
\(656\) − 186.676i − 0.284567i
\(657\) 0 0
\(658\) 0 0
\(659\) − 16.9706i − 0.0257520i −0.999917 0.0128760i \(-0.995901\pi\)
0.999917 0.0128760i \(-0.00409867\pi\)
\(660\) 0 0
\(661\) 433.000 0.655068 0.327534 0.944839i \(-0.393782\pi\)
0.327534 + 0.944839i \(0.393782\pi\)
\(662\) 103.238i 0.155948i
\(663\) 0 0
\(664\) 84.0000 0.126506
\(665\) 0 0
\(666\) 0 0
\(667\) 576.000 0.863568
\(668\) 8.48528i 0.0127025i
\(669\) 0 0
\(670\) −582.000 −0.868657
\(671\) − 1323.70i − 1.97273i
\(672\) 0 0
\(673\) 737.000 1.09510 0.547548 0.836774i \(-0.315561\pi\)
0.547548 + 0.836774i \(0.315561\pi\)
\(674\) − 745.291i − 1.10577i
\(675\) 0 0
\(676\) 336.000 0.497041
\(677\) 135.765i 0.200538i 0.994960 + 0.100269i \(0.0319704\pi\)
−0.994960 + 0.100269i \(0.968030\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 203.647i 0.299481i
\(681\) 0 0
\(682\) −846.000 −1.24047
\(683\) 1272.79i 1.86353i 0.363060 + 0.931766i \(0.381732\pi\)
−0.363060 + 0.931766i \(0.618268\pi\)
\(684\) 0 0
\(685\) 288.000 0.420438
\(686\) 0 0
\(687\) 0 0
\(688\) 92.0000 0.133721
\(689\) 50.9117i 0.0738921i
\(690\) 0 0
\(691\) 247.000 0.357453 0.178726 0.983899i \(-0.442802\pi\)
0.178726 + 0.983899i \(0.442802\pi\)
\(692\) 305.470i 0.441431i
\(693\) 0 0
\(694\) 600.000 0.864553
\(695\) 436.992i 0.628765i
\(696\) 0 0
\(697\) −792.000 −1.13630
\(698\) − 158.392i − 0.226923i
\(699\) 0 0
\(700\) 0 0
\(701\) 322.441i 0.459972i 0.973194 + 0.229986i \(0.0738681\pi\)
−0.973194 + 0.229986i \(0.926132\pi\)
\(702\) 0 0
\(703\) 1265.00 1.79943
\(704\) − 101.823i − 0.144635i
\(705\) 0 0
\(706\) −438.000 −0.620397
\(707\) 0 0
\(708\) 0 0
\(709\) −1048.00 −1.47814 −0.739069 0.673630i \(-0.764734\pi\)
−0.739069 + 0.673630i \(0.764734\pi\)
\(710\) − 585.484i − 0.824626i
\(711\) 0 0
\(712\) −384.000 −0.539326
\(713\) − 797.616i − 1.11868i
\(714\) 0 0
\(715\) −54.0000 −0.0755245
\(716\) 8.48528i 0.0118510i
\(717\) 0 0
\(718\) −720.000 −1.00279
\(719\) − 250.316i − 0.348144i −0.984733 0.174072i \(-0.944307\pi\)
0.984733 0.174072i \(-0.0556926\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 237.588i − 0.329069i
\(723\) 0 0
\(724\) −110.000 −0.151934
\(725\) − 237.588i − 0.327707i
\(726\) 0 0
\(727\) −641.000 −0.881706 −0.440853 0.897579i \(-0.645324\pi\)
−0.440853 + 0.897579i \(0.645324\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −390.000 −0.534247
\(731\) − 390.323i − 0.533958i
\(732\) 0 0
\(733\) 25.0000 0.0341064 0.0170532 0.999855i \(-0.494572\pi\)
0.0170532 + 0.999855i \(0.494572\pi\)
\(734\) − 451.134i − 0.614624i
\(735\) 0 0
\(736\) 96.0000 0.130435
\(737\) − 1234.61i − 1.67518i
\(738\) 0 0
\(739\) −535.000 −0.723951 −0.361976 0.932188i \(-0.617898\pi\)
−0.361976 + 0.932188i \(0.617898\pi\)
\(740\) 466.690i 0.630663i
\(741\) 0 0
\(742\) 0 0
\(743\) − 937.624i − 1.26194i −0.775806 0.630971i \(-0.782656\pi\)
0.775806 0.630971i \(-0.217344\pi\)
\(744\) 0 0
\(745\) −720.000 −0.966443
\(746\) − 295.571i − 0.396207i
\(747\) 0 0
\(748\) −432.000 −0.577540
\(749\) 0 0
\(750\) 0 0
\(751\) 41.0000 0.0545939 0.0272969 0.999627i \(-0.491310\pi\)
0.0272969 + 0.999627i \(0.491310\pi\)
\(752\) − 16.9706i − 0.0225672i
\(753\) 0 0
\(754\) −48.0000 −0.0636605
\(755\) 441.235i 0.584417i
\(756\) 0 0
\(757\) −1114.00 −1.47160 −0.735799 0.677200i \(-0.763193\pi\)
−0.735799 + 0.677200i \(0.763193\pi\)
\(758\) 612.354i 0.807856i
\(759\) 0 0
\(760\) 276.000 0.363158
\(761\) − 288.500i − 0.379106i −0.981871 0.189553i \(-0.939296\pi\)
0.981871 0.189553i \(-0.0607039\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 127.279i − 0.166596i
\(765\) 0 0
\(766\) 552.000 0.720627
\(767\) 84.8528i 0.110629i
\(768\) 0 0
\(769\) −1127.00 −1.46554 −0.732770 0.680477i \(-0.761773\pi\)
−0.732770 + 0.680477i \(0.761773\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 302.000 0.391192
\(773\) − 216.375i − 0.279915i −0.990157 0.139958i \(-0.955303\pi\)
0.990157 0.139958i \(-0.0446967\pi\)
\(774\) 0 0
\(775\) −329.000 −0.424516
\(776\) − 294.156i − 0.379068i
\(777\) 0 0
\(778\) −522.000 −0.670951
\(779\) 1073.39i 1.37791i
\(780\) 0 0
\(781\) 1242.00 1.59027
\(782\) − 407.294i − 0.520836i
\(783\) 0 0
\(784\) 0 0
\(785\) − 644.881i − 0.821505i
\(786\) 0 0
\(787\) −170.000 −0.216010 −0.108005 0.994150i \(-0.534446\pi\)
−0.108005 + 0.994150i \(0.534446\pi\)
\(788\) 33.9411i 0.0430725i
\(789\) 0 0
\(790\) 678.000 0.858228
\(791\) 0 0
\(792\) 0 0
\(793\) −104.000 −0.131148
\(794\) 159.806i 0.201267i
\(795\) 0 0
\(796\) −320.000 −0.402010
\(797\) − 1170.97i − 1.46922i −0.678489 0.734610i \(-0.737365\pi\)
0.678489 0.734610i \(-0.262635\pi\)
\(798\) 0 0
\(799\) −72.0000 −0.0901126
\(800\) − 39.5980i − 0.0494975i
\(801\) 0 0
\(802\) −504.000 −0.628429
\(803\) − 827.315i − 1.03028i
\(804\) 0 0
\(805\) 0 0
\(806\) 66.4680i 0.0824665i
\(807\) 0 0
\(808\) 420.000 0.519802
\(809\) 335.169i 0.414300i 0.978309 + 0.207150i \(0.0664188\pi\)
−0.978309 + 0.207150i \(0.933581\pi\)
\(810\) 0 0
\(811\) 1114.00 1.37361 0.686806 0.726840i \(-0.259012\pi\)
0.686806 + 0.726840i \(0.259012\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −990.000 −1.21622
\(815\) 237.588i 0.291519i
\(816\) 0 0
\(817\) −529.000 −0.647491
\(818\) − 451.134i − 0.551509i
\(819\) 0 0
\(820\) −396.000 −0.482927
\(821\) 97.5807i 0.118856i 0.998233 + 0.0594280i \(0.0189277\pi\)
−0.998233 + 0.0594280i \(0.981072\pi\)
\(822\) 0 0
\(823\) 1592.00 1.93439 0.967193 0.254042i \(-0.0817601\pi\)
0.967193 + 0.254042i \(0.0817601\pi\)
\(824\) − 336.583i − 0.408474i
\(825\) 0 0
\(826\) 0 0
\(827\) − 1022.48i − 1.23637i −0.786033 0.618184i \(-0.787869\pi\)
0.786033 0.618184i \(-0.212131\pi\)
\(828\) 0 0
\(829\) −383.000 −0.462002 −0.231001 0.972953i \(-0.574200\pi\)
−0.231001 + 0.972953i \(0.574200\pi\)
\(830\) − 178.191i − 0.214688i
\(831\) 0 0
\(832\) −8.00000 −0.00961538
\(833\) 0 0
\(834\) 0 0
\(835\) 18.0000 0.0215569
\(836\) 585.484i 0.700340i
\(837\) 0 0
\(838\) 1086.00 1.29594
\(839\) 576.999i 0.687722i 0.939020 + 0.343861i \(0.111735\pi\)
−0.939020 + 0.343861i \(0.888265\pi\)
\(840\) 0 0
\(841\) −311.000 −0.369798
\(842\) − 91.9239i − 0.109173i
\(843\) 0 0
\(844\) 416.000 0.492891
\(845\) − 712.764i − 0.843507i
\(846\) 0 0
\(847\) 0 0
\(848\) 203.647i 0.240149i
\(849\) 0 0
\(850\) −168.000 −0.197647
\(851\) − 933.381i − 1.09680i
\(852\) 0 0
\(853\) −527.000 −0.617819 −0.308910 0.951091i \(-0.599964\pi\)
−0.308910 + 0.951091i \(0.599964\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −336.000 −0.392523
\(857\) − 1323.70i − 1.54458i −0.635271 0.772289i \(-0.719112\pi\)
0.635271 0.772289i \(-0.280888\pi\)
\(858\) 0 0
\(859\) 1414.00 1.64610 0.823050 0.567969i \(-0.192271\pi\)
0.823050 + 0.567969i \(0.192271\pi\)
\(860\) − 195.161i − 0.226932i
\(861\) 0 0
\(862\) 678.000 0.786543
\(863\) − 1370.37i − 1.58792i −0.607972 0.793959i \(-0.708017\pi\)
0.607972 0.793959i \(-0.291983\pi\)
\(864\) 0 0
\(865\) 648.000 0.749133
\(866\) − 519.016i − 0.599326i
\(867\) 0 0
\(868\) 0 0
\(869\) 1438.26i 1.65507i
\(870\) 0 0
\(871\) −97.0000 −0.111366
\(872\) − 138.593i − 0.158937i
\(873\) 0 0
\(874\) −552.000 −0.631579
\(875\) 0 0
\(876\) 0 0
\(877\) 224.000 0.255416 0.127708 0.991812i \(-0.459238\pi\)
0.127708 + 0.991812i \(0.459238\pi\)
\(878\) 758.018i 0.863347i
\(879\) 0 0
\(880\) −216.000 −0.245455
\(881\) 1612.20i 1.82997i 0.403488 + 0.914985i \(0.367798\pi\)
−0.403488 + 0.914985i \(0.632202\pi\)
\(882\) 0 0
\(883\) 329.000 0.372593 0.186297 0.982494i \(-0.440351\pi\)
0.186297 + 0.982494i \(0.440351\pi\)
\(884\) 33.9411i 0.0383949i
\(885\) 0 0
\(886\) −120.000 −0.135440
\(887\) 55.1543i 0.0621808i 0.999517 + 0.0310904i \(0.00989797\pi\)
−0.999517 + 0.0310904i \(0.990102\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 814.587i 0.915266i
\(891\) 0 0
\(892\) −80.0000 −0.0896861
\(893\) 97.5807i 0.109273i
\(894\) 0 0
\(895\) 18.0000 0.0201117
\(896\) 0 0
\(897\) 0 0
\(898\) 870.000 0.968820
\(899\) 1595.23i 1.77445i
\(900\) 0 0
\(901\) 864.000 0.958935
\(902\) − 840.043i − 0.931311i
\(903\) 0 0
\(904\) −276.000 −0.305310
\(905\) 233.345i 0.257840i
\(906\) 0 0
\(907\) 311.000 0.342889 0.171444 0.985194i \(-0.445157\pi\)
0.171444 + 0.985194i \(0.445157\pi\)
\(908\) 449.720i 0.495286i
\(909\) 0 0
\(910\) 0 0
\(911\) − 356.382i − 0.391198i −0.980684 0.195599i \(-0.937335\pi\)
0.980684 0.195599i \(-0.0626652\pi\)
\(912\) 0 0
\(913\) 378.000 0.414020
\(914\) 654.781i 0.716390i
\(915\) 0 0
\(916\) 754.000 0.823144
\(917\) 0 0
\(918\) 0 0
\(919\) −145.000 −0.157780 −0.0788901 0.996883i \(-0.525138\pi\)
−0.0788901 + 0.996883i \(0.525138\pi\)
\(920\) − 203.647i − 0.221355i
\(921\) 0 0
\(922\) 384.000 0.416486
\(923\) − 97.5807i − 0.105721i
\(924\) 0 0
\(925\) −385.000 −0.416216
\(926\) − 66.4680i − 0.0717797i
\(927\) 0 0
\(928\) −192.000 −0.206897
\(929\) − 309.713i − 0.333383i −0.986009 0.166691i \(-0.946692\pi\)
0.986009 0.166691i \(-0.0533084\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 687.308i 0.737455i
\(933\) 0 0
\(934\) 1050.00 1.12420
\(935\) 916.410i 0.980118i
\(936\) 0 0
\(937\) 487.000 0.519744 0.259872 0.965643i \(-0.416320\pi\)
0.259872 + 0.965643i \(0.416320\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −36.0000 −0.0382979
\(941\) − 1374.62i − 1.46080i −0.683018 0.730401i \(-0.739333\pi\)
0.683018 0.730401i \(-0.260667\pi\)
\(942\) 0 0
\(943\) 792.000 0.839873
\(944\) 339.411i 0.359546i
\(945\) 0 0
\(946\) 414.000 0.437632
\(947\) 190.919i 0.201604i 0.994907 + 0.100802i \(0.0321408\pi\)
−0.994907 + 0.100802i \(0.967859\pi\)
\(948\) 0 0
\(949\) −65.0000 −0.0684932
\(950\) 227.688i 0.239672i
\(951\) 0 0
\(952\) 0 0
\(953\) 1035.20i 1.08626i 0.839649 + 0.543129i \(0.182761\pi\)
−0.839649 + 0.543129i \(0.817239\pi\)
\(954\) 0 0
\(955\) −270.000 −0.282723
\(956\) − 25.4558i − 0.0266275i
\(957\) 0 0
\(958\) 600.000 0.626305
\(959\) 0 0
\(960\) 0 0
\(961\) 1248.00 1.29865
\(962\) 77.7817i 0.0808542i
\(963\) 0 0
\(964\) −260.000 −0.269710
\(965\) − 640.639i − 0.663874i
\(966\) 0 0
\(967\) −1681.00 −1.73837 −0.869183 0.494490i \(-0.835355\pi\)
−0.869183 + 0.494490i \(0.835355\pi\)
\(968\) − 115.966i − 0.119799i
\(969\) 0 0
\(970\) −624.000 −0.643299
\(971\) 1204.91i 1.24090i 0.784248 + 0.620448i \(0.213049\pi\)
−0.784248 + 0.620448i \(0.786951\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 337.997i − 0.347020i
\(975\) 0 0
\(976\) −416.000 −0.426230
\(977\) − 1175.21i − 1.20288i −0.798919 0.601439i \(-0.794594\pi\)
0.798919 0.601439i \(-0.205406\pi\)
\(978\) 0 0
\(979\) −1728.00 −1.76507
\(980\) 0 0
\(981\) 0 0
\(982\) −288.000 −0.293279
\(983\) − 1917.67i − 1.95084i −0.220359 0.975419i \(-0.570723\pi\)
0.220359 0.975419i \(-0.429277\pi\)
\(984\) 0 0
\(985\) 72.0000 0.0730964
\(986\) 814.587i 0.826153i
\(987\) 0 0
\(988\) 46.0000 0.0465587
\(989\) 390.323i 0.394664i
\(990\) 0 0
\(991\) −625.000 −0.630676 −0.315338 0.948979i \(-0.602118\pi\)
−0.315338 + 0.948979i \(0.602118\pi\)
\(992\) 265.872i 0.268016i
\(993\) 0 0
\(994\) 0 0
\(995\) 678.823i 0.682234i
\(996\) 0 0
\(997\) 943.000 0.945838 0.472919 0.881106i \(-0.343200\pi\)
0.472919 + 0.881106i \(0.343200\pi\)
\(998\) 69.2965i 0.0694353i
\(999\) 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.e.197.1 2
3.2 odd 2 inner 882.3.b.e.197.2 2
7.2 even 3 882.3.s.a.557.1 4
7.3 odd 6 126.3.s.a.107.2 yes 4
7.4 even 3 882.3.s.a.863.2 4
7.5 odd 6 126.3.s.a.53.1 4
7.6 odd 2 882.3.b.b.197.1 2
21.2 odd 6 882.3.s.a.557.2 4
21.5 even 6 126.3.s.a.53.2 yes 4
21.11 odd 6 882.3.s.a.863.1 4
21.17 even 6 126.3.s.a.107.1 yes 4
21.20 even 2 882.3.b.b.197.2 2
28.3 even 6 1008.3.dc.b.737.2 4
28.19 even 6 1008.3.dc.b.305.1 4
84.47 odd 6 1008.3.dc.b.305.2 4
84.59 odd 6 1008.3.dc.b.737.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.s.a.53.1 4 7.5 odd 6
126.3.s.a.53.2 yes 4 21.5 even 6
126.3.s.a.107.1 yes 4 21.17 even 6
126.3.s.a.107.2 yes 4 7.3 odd 6
882.3.b.b.197.1 2 7.6 odd 2
882.3.b.b.197.2 2 21.20 even 2
882.3.b.e.197.1 2 1.1 even 1 trivial
882.3.b.e.197.2 2 3.2 odd 2 inner
882.3.s.a.557.1 4 7.2 even 3
882.3.s.a.557.2 4 21.2 odd 6
882.3.s.a.863.1 4 21.11 odd 6
882.3.s.a.863.2 4 7.4 even 3
1008.3.dc.b.305.1 4 28.19 even 6
1008.3.dc.b.305.2 4 84.47 odd 6
1008.3.dc.b.737.1 4 84.59 odd 6
1008.3.dc.b.737.2 4 28.3 even 6