Properties

Label 882.3.b.g.197.2
Level $882$
Weight $3$
Character 882.197
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{37})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 123\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.2
Root \(-2.54138 + 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.3.b.g.197.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +8.60233i q^{5} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} +8.60233i q^{5} +2.82843i q^{8} +12.1655 q^{10} +2.82843i q^{11} -12.1655 q^{13} +4.00000 q^{16} +25.8070i q^{17} +24.3311 q^{19} -17.2047i q^{20} +4.00000 q^{22} -42.4264i q^{23} -49.0000 q^{25} +17.2047i q^{26} +15.5563i q^{29} -24.3311 q^{31} -5.65685i q^{32} +36.4966 q^{34} -6.00000 q^{37} -34.4093i q^{38} -24.3311 q^{40} +25.8070i q^{41} -68.0000 q^{43} -5.65685i q^{44} -60.0000 q^{46} -68.8186i q^{47} +69.2965i q^{50} +24.3311 q^{52} +41.0122i q^{53} -24.3311 q^{55} +22.0000 q^{58} -68.8186i q^{59} -97.3242 q^{61} +34.4093i q^{62} -8.00000 q^{64} -104.652i q^{65} -104.000 q^{67} -51.6140i q^{68} +70.7107i q^{71} -60.8276 q^{73} +8.48528i q^{74} -48.6621 q^{76} -20.0000 q^{79} +34.4093i q^{80} +36.4966 q^{82} -222.000 q^{85} +96.1665i q^{86} -8.00000 q^{88} +94.6256i q^{89} +84.8528i q^{92} -97.3242 q^{94} +209.304i q^{95} +158.152 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} + 16q^{22} - 196q^{25} - 24q^{37} - 272q^{43} - 240q^{46} + 88q^{58} - 32q^{64} - 416q^{67} - 80q^{79} - 888q^{85} - 32q^{88} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 8.60233i 1.72047i 0.509902 + 0.860233i \(0.329682\pi\)
−0.509902 + 0.860233i \(0.670318\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) 12.1655 1.21655
\(11\) 2.82843i 0.257130i 0.991701 + 0.128565i \(0.0410371\pi\)
−0.991701 + 0.128565i \(0.958963\pi\)
\(12\) 0 0
\(13\) −12.1655 −0.935810 −0.467905 0.883779i \(-0.654991\pi\)
−0.467905 + 0.883779i \(0.654991\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 25.8070i 1.51806i 0.651057 + 0.759029i \(0.274326\pi\)
−0.651057 + 0.759029i \(0.725674\pi\)
\(18\) 0 0
\(19\) 24.3311 1.28058 0.640291 0.768133i \(-0.278814\pi\)
0.640291 + 0.768133i \(0.278814\pi\)
\(20\) − 17.2047i − 0.860233i
\(21\) 0 0
\(22\) 4.00000 0.181818
\(23\) − 42.4264i − 1.84463i −0.386443 0.922313i \(-0.626296\pi\)
0.386443 0.922313i \(-0.373704\pi\)
\(24\) 0 0
\(25\) −49.0000 −1.96000
\(26\) 17.2047i 0.661717i
\(27\) 0 0
\(28\) 0 0
\(29\) 15.5563i 0.536426i 0.963360 + 0.268213i \(0.0864331\pi\)
−0.963360 + 0.268213i \(0.913567\pi\)
\(30\) 0 0
\(31\) −24.3311 −0.784873 −0.392436 0.919779i \(-0.628368\pi\)
−0.392436 + 0.919779i \(0.628368\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) 36.4966 1.07343
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.162162 −0.0810811 0.996708i \(-0.525837\pi\)
−0.0810811 + 0.996708i \(0.525837\pi\)
\(38\) − 34.4093i − 0.905508i
\(39\) 0 0
\(40\) −24.3311 −0.608276
\(41\) 25.8070i 0.629438i 0.949185 + 0.314719i \(0.101910\pi\)
−0.949185 + 0.314719i \(0.898090\pi\)
\(42\) 0 0
\(43\) −68.0000 −1.58140 −0.790698 0.612207i \(-0.790282\pi\)
−0.790698 + 0.612207i \(0.790282\pi\)
\(44\) − 5.65685i − 0.128565i
\(45\) 0 0
\(46\) −60.0000 −1.30435
\(47\) − 68.8186i − 1.46423i −0.681183 0.732113i \(-0.738534\pi\)
0.681183 0.732113i \(-0.261466\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 69.2965i 1.38593i
\(51\) 0 0
\(52\) 24.3311 0.467905
\(53\) 41.0122i 0.773815i 0.922119 + 0.386907i \(0.126457\pi\)
−0.922119 + 0.386907i \(0.873543\pi\)
\(54\) 0 0
\(55\) −24.3311 −0.442383
\(56\) 0 0
\(57\) 0 0
\(58\) 22.0000 0.379310
\(59\) − 68.8186i − 1.16642i −0.812323 0.583208i \(-0.801797\pi\)
0.812323 0.583208i \(-0.198203\pi\)
\(60\) 0 0
\(61\) −97.3242 −1.59548 −0.797739 0.603002i \(-0.793971\pi\)
−0.797739 + 0.603002i \(0.793971\pi\)
\(62\) 34.4093i 0.554989i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 104.652i − 1.61003i
\(66\) 0 0
\(67\) −104.000 −1.55224 −0.776119 0.630586i \(-0.782815\pi\)
−0.776119 + 0.630586i \(0.782815\pi\)
\(68\) − 51.6140i − 0.759029i
\(69\) 0 0
\(70\) 0 0
\(71\) 70.7107i 0.995925i 0.867199 + 0.497963i \(0.165918\pi\)
−0.867199 + 0.497963i \(0.834082\pi\)
\(72\) 0 0
\(73\) −60.8276 −0.833255 −0.416628 0.909077i \(-0.636788\pi\)
−0.416628 + 0.909077i \(0.636788\pi\)
\(74\) 8.48528i 0.114666i
\(75\) 0 0
\(76\) −48.6621 −0.640291
\(77\) 0 0
\(78\) 0 0
\(79\) −20.0000 −0.253165 −0.126582 0.991956i \(-0.540401\pi\)
−0.126582 + 0.991956i \(0.540401\pi\)
\(80\) 34.4093i 0.430116i
\(81\) 0 0
\(82\) 36.4966 0.445080
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −222.000 −2.61176
\(86\) 96.1665i 1.11822i
\(87\) 0 0
\(88\) −8.00000 −0.0909091
\(89\) 94.6256i 1.06321i 0.846993 + 0.531604i \(0.178411\pi\)
−0.846993 + 0.531604i \(0.821589\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 84.8528i 0.922313i
\(93\) 0 0
\(94\) −97.3242 −1.03536
\(95\) 209.304i 2.20320i
\(96\) 0 0
\(97\) 158.152 1.63043 0.815216 0.579158i \(-0.196618\pi\)
0.815216 + 0.579158i \(0.196618\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 98.0000 0.980000
\(101\) 60.2163i 0.596201i 0.954535 + 0.298100i \(0.0963530\pi\)
−0.954535 + 0.298100i \(0.903647\pi\)
\(102\) 0 0
\(103\) 170.317 1.65357 0.826783 0.562521i \(-0.190168\pi\)
0.826783 + 0.562521i \(0.190168\pi\)
\(104\) − 34.4093i − 0.330859i
\(105\) 0 0
\(106\) 58.0000 0.547170
\(107\) − 19.7990i − 0.185037i −0.995711 0.0925186i \(-0.970508\pi\)
0.995711 0.0925186i \(-0.0294918\pi\)
\(108\) 0 0
\(109\) −56.0000 −0.513761 −0.256881 0.966443i \(-0.582695\pi\)
−0.256881 + 0.966443i \(0.582695\pi\)
\(110\) 34.4093i 0.312812i
\(111\) 0 0
\(112\) 0 0
\(113\) − 159.806i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(114\) 0 0
\(115\) 364.966 3.17362
\(116\) − 31.1127i − 0.268213i
\(117\) 0 0
\(118\) −97.3242 −0.824781
\(119\) 0 0
\(120\) 0 0
\(121\) 113.000 0.933884
\(122\) 137.637i 1.12817i
\(123\) 0 0
\(124\) 48.6621 0.392436
\(125\) − 206.456i − 1.65165i
\(126\) 0 0
\(127\) −76.0000 −0.598425 −0.299213 0.954186i \(-0.596724\pi\)
−0.299213 + 0.954186i \(0.596724\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) −148.000 −1.13846
\(131\) 103.228i 0.787999i 0.919111 + 0.394000i \(0.128909\pi\)
−0.919111 + 0.394000i \(0.871091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 147.078i 1.09760i
\(135\) 0 0
\(136\) −72.9932 −0.536714
\(137\) 100.409i 0.732914i 0.930435 + 0.366457i \(0.119429\pi\)
−0.930435 + 0.366457i \(0.880571\pi\)
\(138\) 0 0
\(139\) −145.986 −1.05026 −0.525131 0.851022i \(-0.675984\pi\)
−0.525131 + 0.851022i \(0.675984\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 100.000 0.704225
\(143\) − 34.4093i − 0.240624i
\(144\) 0 0
\(145\) −133.821 −0.922902
\(146\) 86.0233i 0.589200i
\(147\) 0 0
\(148\) 12.0000 0.0810811
\(149\) 41.0122i 0.275250i 0.990484 + 0.137625i \(0.0439468\pi\)
−0.990484 + 0.137625i \(0.956053\pi\)
\(150\) 0 0
\(151\) 160.000 1.05960 0.529801 0.848122i \(-0.322266\pi\)
0.529801 + 0.848122i \(0.322266\pi\)
\(152\) 68.8186i 0.452754i
\(153\) 0 0
\(154\) 0 0
\(155\) − 209.304i − 1.35035i
\(156\) 0 0
\(157\) −194.648 −1.23980 −0.619899 0.784681i \(-0.712827\pi\)
−0.619899 + 0.784681i \(0.712827\pi\)
\(158\) 28.2843i 0.179014i
\(159\) 0 0
\(160\) 48.6621 0.304138
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) − 51.6140i − 0.314719i
\(165\) 0 0
\(166\) 0 0
\(167\) 172.047i 1.03022i 0.857125 + 0.515109i \(0.172249\pi\)
−0.857125 + 0.515109i \(0.827751\pi\)
\(168\) 0 0
\(169\) −21.0000 −0.124260
\(170\) 313.955i 1.84680i
\(171\) 0 0
\(172\) 136.000 0.790698
\(173\) − 8.60233i − 0.0497244i −0.999691 0.0248622i \(-0.992085\pi\)
0.999691 0.0248622i \(-0.00791470\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 11.3137i 0.0642824i
\(177\) 0 0
\(178\) 133.821 0.751802
\(179\) 144.250i 0.805865i 0.915230 + 0.402932i \(0.132009\pi\)
−0.915230 + 0.402932i \(0.867991\pi\)
\(180\) 0 0
\(181\) 12.1655 0.0672128 0.0336064 0.999435i \(-0.489301\pi\)
0.0336064 + 0.999435i \(0.489301\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 120.000 0.652174
\(185\) − 51.6140i − 0.278994i
\(186\) 0 0
\(187\) −72.9932 −0.390338
\(188\) 137.637i 0.732113i
\(189\) 0 0
\(190\) 296.000 1.55789
\(191\) − 263.044i − 1.37719i −0.725145 0.688596i \(-0.758227\pi\)
0.725145 0.688596i \(-0.241773\pi\)
\(192\) 0 0
\(193\) 38.0000 0.196891 0.0984456 0.995142i \(-0.468613\pi\)
0.0984456 + 0.995142i \(0.468613\pi\)
\(194\) − 223.660i − 1.15289i
\(195\) 0 0
\(196\) 0 0
\(197\) − 165.463i − 0.839914i −0.907544 0.419957i \(-0.862045\pi\)
0.907544 0.419957i \(-0.137955\pi\)
\(198\) 0 0
\(199\) −243.311 −1.22267 −0.611333 0.791374i \(-0.709366\pi\)
−0.611333 + 0.791374i \(0.709366\pi\)
\(200\) − 138.593i − 0.692965i
\(201\) 0 0
\(202\) 85.1587 0.421578
\(203\) 0 0
\(204\) 0 0
\(205\) −222.000 −1.08293
\(206\) − 240.865i − 1.16925i
\(207\) 0 0
\(208\) −48.6621 −0.233952
\(209\) 68.8186i 0.329276i
\(210\) 0 0
\(211\) 44.0000 0.208531 0.104265 0.994550i \(-0.466751\pi\)
0.104265 + 0.994550i \(0.466751\pi\)
\(212\) − 82.0244i − 0.386907i
\(213\) 0 0
\(214\) −28.0000 −0.130841
\(215\) − 584.958i − 2.72074i
\(216\) 0 0
\(217\) 0 0
\(218\) 79.1960i 0.363284i
\(219\) 0 0
\(220\) 48.6621 0.221191
\(221\) − 313.955i − 1.42061i
\(222\) 0 0
\(223\) 194.648 0.872863 0.436431 0.899738i \(-0.356242\pi\)
0.436431 + 0.899738i \(0.356242\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −226.000 −1.00000
\(227\) − 34.4093i − 0.151583i −0.997124 0.0757914i \(-0.975852\pi\)
0.997124 0.0757914i \(-0.0241483\pi\)
\(228\) 0 0
\(229\) −12.1655 −0.0531246 −0.0265623 0.999647i \(-0.508456\pi\)
−0.0265623 + 0.999647i \(0.508456\pi\)
\(230\) − 516.140i − 2.24408i
\(231\) 0 0
\(232\) −44.0000 −0.189655
\(233\) 270.115i 1.15929i 0.814869 + 0.579645i \(0.196809\pi\)
−0.814869 + 0.579645i \(0.803191\pi\)
\(234\) 0 0
\(235\) 592.000 2.51915
\(236\) 137.637i 0.583208i
\(237\) 0 0
\(238\) 0 0
\(239\) − 76.3675i − 0.319529i −0.987155 0.159765i \(-0.948926\pi\)
0.987155 0.159765i \(-0.0510736\pi\)
\(240\) 0 0
\(241\) −133.821 −0.555273 −0.277636 0.960686i \(-0.589551\pi\)
−0.277636 + 0.960686i \(0.589551\pi\)
\(242\) − 159.806i − 0.660356i
\(243\) 0 0
\(244\) 194.648 0.797739
\(245\) 0 0
\(246\) 0 0
\(247\) −296.000 −1.19838
\(248\) − 68.8186i − 0.277494i
\(249\) 0 0
\(250\) −291.973 −1.16789
\(251\) 137.637i 0.548355i 0.961679 + 0.274178i \(0.0884056\pi\)
−0.961679 + 0.274178i \(0.911594\pi\)
\(252\) 0 0
\(253\) 120.000 0.474308
\(254\) 107.480i 0.423151i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 249.467i 0.970690i 0.874323 + 0.485345i \(0.161306\pi\)
−0.874323 + 0.485345i \(0.838694\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 209.304i 0.805014i
\(261\) 0 0
\(262\) 145.986 0.557200
\(263\) − 234.759i − 0.892621i −0.894878 0.446311i \(-0.852738\pi\)
0.894878 0.446311i \(-0.147262\pi\)
\(264\) 0 0
\(265\) −352.800 −1.33132
\(266\) 0 0
\(267\) 0 0
\(268\) 208.000 0.776119
\(269\) 215.058i 0.799473i 0.916630 + 0.399736i \(0.130898\pi\)
−0.916630 + 0.399736i \(0.869102\pi\)
\(270\) 0 0
\(271\) 316.304 1.16717 0.583586 0.812051i \(-0.301649\pi\)
0.583586 + 0.812051i \(0.301649\pi\)
\(272\) 103.228i 0.379514i
\(273\) 0 0
\(274\) 142.000 0.518248
\(275\) − 138.593i − 0.503974i
\(276\) 0 0
\(277\) 64.0000 0.231047 0.115523 0.993305i \(-0.463145\pi\)
0.115523 + 0.993305i \(0.463145\pi\)
\(278\) 206.456i 0.742647i
\(279\) 0 0
\(280\) 0 0
\(281\) − 462.448i − 1.64572i −0.568243 0.822861i \(-0.692377\pi\)
0.568243 0.822861i \(-0.307623\pi\)
\(282\) 0 0
\(283\) −24.3311 −0.0859754 −0.0429877 0.999076i \(-0.513688\pi\)
−0.0429877 + 0.999076i \(0.513688\pi\)
\(284\) − 141.421i − 0.497963i
\(285\) 0 0
\(286\) −48.6621 −0.170147
\(287\) 0 0
\(288\) 0 0
\(289\) −377.000 −1.30450
\(290\) 189.251i 0.652590i
\(291\) 0 0
\(292\) 121.655 0.416628
\(293\) 283.877i 0.968863i 0.874829 + 0.484431i \(0.160973\pi\)
−0.874829 + 0.484431i \(0.839027\pi\)
\(294\) 0 0
\(295\) 592.000 2.00678
\(296\) − 16.9706i − 0.0573330i
\(297\) 0 0
\(298\) 58.0000 0.194631
\(299\) 516.140i 1.72622i
\(300\) 0 0
\(301\) 0 0
\(302\) − 226.274i − 0.749252i
\(303\) 0 0
\(304\) 97.3242 0.320145
\(305\) − 837.214i − 2.74497i
\(306\) 0 0
\(307\) −316.304 −1.03031 −0.515153 0.857099i \(-0.672265\pi\)
−0.515153 + 0.857099i \(0.672265\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −296.000 −0.954839
\(311\) 275.274i 0.885127i 0.896737 + 0.442563i \(0.145931\pi\)
−0.896737 + 0.442563i \(0.854069\pi\)
\(312\) 0 0
\(313\) 291.973 0.932820 0.466410 0.884569i \(-0.345547\pi\)
0.466410 + 0.884569i \(0.345547\pi\)
\(314\) 275.274i 0.876670i
\(315\) 0 0
\(316\) 40.0000 0.126582
\(317\) − 335.169i − 1.05731i −0.848835 0.528657i \(-0.822696\pi\)
0.848835 0.528657i \(-0.177304\pi\)
\(318\) 0 0
\(319\) −44.0000 −0.137931
\(320\) − 68.8186i − 0.215058i
\(321\) 0 0
\(322\) 0 0
\(323\) 627.911i 1.94400i
\(324\) 0 0
\(325\) 596.111 1.83419
\(326\) 0 0
\(327\) 0 0
\(328\) −72.9932 −0.222540
\(329\) 0 0
\(330\) 0 0
\(331\) −388.000 −1.17221 −0.586103 0.810237i \(-0.699339\pi\)
−0.586103 + 0.810237i \(0.699339\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 243.311 0.728475
\(335\) − 894.642i − 2.67057i
\(336\) 0 0
\(337\) −208.000 −0.617211 −0.308605 0.951190i \(-0.599862\pi\)
−0.308605 + 0.951190i \(0.599862\pi\)
\(338\) 29.6985i 0.0878653i
\(339\) 0 0
\(340\) 444.000 1.30588
\(341\) − 68.8186i − 0.201814i
\(342\) 0 0
\(343\) 0 0
\(344\) − 192.333i − 0.559108i
\(345\) 0 0
\(346\) −12.1655 −0.0351605
\(347\) 364.867i 1.05149i 0.850642 + 0.525745i \(0.176213\pi\)
−0.850642 + 0.525745i \(0.823787\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.0000 0.0454545
\(353\) − 25.8070i − 0.0731076i −0.999332 0.0365538i \(-0.988362\pi\)
0.999332 0.0365538i \(-0.0116380\pi\)
\(354\) 0 0
\(355\) −608.276 −1.71345
\(356\) − 189.251i − 0.531604i
\(357\) 0 0
\(358\) 204.000 0.569832
\(359\) 302.642i 0.843013i 0.906825 + 0.421507i \(0.138499\pi\)
−0.906825 + 0.421507i \(0.861501\pi\)
\(360\) 0 0
\(361\) 231.000 0.639889
\(362\) − 17.2047i − 0.0475267i
\(363\) 0 0
\(364\) 0 0
\(365\) − 523.259i − 1.43359i
\(366\) 0 0
\(367\) −389.297 −1.06075 −0.530377 0.847762i \(-0.677950\pi\)
−0.530377 + 0.847762i \(0.677950\pi\)
\(368\) − 169.706i − 0.461157i
\(369\) 0 0
\(370\) −72.9932 −0.197279
\(371\) 0 0
\(372\) 0 0
\(373\) 706.000 1.89276 0.946381 0.323054i \(-0.104709\pi\)
0.946381 + 0.323054i \(0.104709\pi\)
\(374\) 103.228i 0.276010i
\(375\) 0 0
\(376\) 194.648 0.517682
\(377\) − 189.251i − 0.501992i
\(378\) 0 0
\(379\) 708.000 1.86807 0.934037 0.357176i \(-0.116261\pi\)
0.934037 + 0.357176i \(0.116261\pi\)
\(380\) − 418.607i − 1.10160i
\(381\) 0 0
\(382\) −372.000 −0.973822
\(383\) 309.684i 0.808574i 0.914632 + 0.404287i \(0.132480\pi\)
−0.914632 + 0.404287i \(0.867520\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 53.7401i − 0.139223i
\(387\) 0 0
\(388\) −316.304 −0.815216
\(389\) − 131.522i − 0.338102i −0.985607 0.169051i \(-0.945930\pi\)
0.985607 0.169051i \(-0.0540703\pi\)
\(390\) 0 0
\(391\) 1094.90 2.80025
\(392\) 0 0
\(393\) 0 0
\(394\) −234.000 −0.593909
\(395\) − 172.047i − 0.435561i
\(396\) 0 0
\(397\) 97.3242 0.245149 0.122575 0.992459i \(-0.460885\pi\)
0.122575 + 0.992459i \(0.460885\pi\)
\(398\) 344.093i 0.864555i
\(399\) 0 0
\(400\) −196.000 −0.490000
\(401\) 89.0955i 0.222183i 0.993810 + 0.111092i \(0.0354347\pi\)
−0.993810 + 0.111092i \(0.964565\pi\)
\(402\) 0 0
\(403\) 296.000 0.734491
\(404\) − 120.433i − 0.298100i
\(405\) 0 0
\(406\) 0 0
\(407\) − 16.9706i − 0.0416967i
\(408\) 0 0
\(409\) 547.449 1.33851 0.669253 0.743035i \(-0.266614\pi\)
0.669253 + 0.743035i \(0.266614\pi\)
\(410\) 313.955i 0.765745i
\(411\) 0 0
\(412\) −340.635 −0.826783
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 68.8186i 0.165429i
\(417\) 0 0
\(418\) 97.3242 0.232833
\(419\) 653.777i 1.56033i 0.625576 + 0.780163i \(0.284864\pi\)
−0.625576 + 0.780163i \(0.715136\pi\)
\(420\) 0 0
\(421\) 240.000 0.570071 0.285036 0.958517i \(-0.407995\pi\)
0.285036 + 0.958517i \(0.407995\pi\)
\(422\) − 62.2254i − 0.147454i
\(423\) 0 0
\(424\) −116.000 −0.273585
\(425\) − 1264.54i − 2.97539i
\(426\) 0 0
\(427\) 0 0
\(428\) 39.5980i 0.0925186i
\(429\) 0 0
\(430\) −827.256 −1.92385
\(431\) 14.1421i 0.0328124i 0.999865 + 0.0164062i \(0.00522249\pi\)
−0.999865 + 0.0164062i \(0.994778\pi\)
\(432\) 0 0
\(433\) 389.297 0.899069 0.449534 0.893263i \(-0.351590\pi\)
0.449534 + 0.893263i \(0.351590\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 112.000 0.256881
\(437\) − 1032.28i − 2.36219i
\(438\) 0 0
\(439\) −291.973 −0.665086 −0.332543 0.943088i \(-0.607907\pi\)
−0.332543 + 0.943088i \(0.607907\pi\)
\(440\) − 68.8186i − 0.156406i
\(441\) 0 0
\(442\) −444.000 −1.00452
\(443\) − 25.4558i − 0.0574624i −0.999587 0.0287312i \(-0.990853\pi\)
0.999587 0.0287312i \(-0.00914668\pi\)
\(444\) 0 0
\(445\) −814.000 −1.82921
\(446\) − 275.274i − 0.617207i
\(447\) 0 0
\(448\) 0 0
\(449\) 733.977i 1.63469i 0.576147 + 0.817346i \(0.304556\pi\)
−0.576147 + 0.817346i \(0.695444\pi\)
\(450\) 0 0
\(451\) −72.9932 −0.161847
\(452\) 319.612i 0.707107i
\(453\) 0 0
\(454\) −48.6621 −0.107185
\(455\) 0 0
\(456\) 0 0
\(457\) 272.000 0.595186 0.297593 0.954693i \(-0.403816\pi\)
0.297593 + 0.954693i \(0.403816\pi\)
\(458\) 17.2047i 0.0375647i
\(459\) 0 0
\(460\) −729.932 −1.58681
\(461\) 490.333i 1.06363i 0.846861 + 0.531814i \(0.178489\pi\)
−0.846861 + 0.531814i \(0.821511\pi\)
\(462\) 0 0
\(463\) −436.000 −0.941685 −0.470842 0.882217i \(-0.656050\pi\)
−0.470842 + 0.882217i \(0.656050\pi\)
\(464\) 62.2254i 0.134106i
\(465\) 0 0
\(466\) 382.000 0.819742
\(467\) 481.730i 1.03154i 0.856726 + 0.515771i \(0.172495\pi\)
−0.856726 + 0.515771i \(0.827505\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 837.214i − 1.78131i
\(471\) 0 0
\(472\) 194.648 0.412391
\(473\) − 192.333i − 0.406624i
\(474\) 0 0
\(475\) −1192.22 −2.50994
\(476\) 0 0
\(477\) 0 0
\(478\) −108.000 −0.225941
\(479\) − 103.228i − 0.215507i −0.994178 0.107754i \(-0.965634\pi\)
0.994178 0.107754i \(-0.0343657\pi\)
\(480\) 0 0
\(481\) 72.9932 0.151753
\(482\) 189.251i 0.392637i
\(483\) 0 0
\(484\) −226.000 −0.466942
\(485\) 1360.47i 2.80510i
\(486\) 0 0
\(487\) 232.000 0.476386 0.238193 0.971218i \(-0.423445\pi\)
0.238193 + 0.971218i \(0.423445\pi\)
\(488\) − 275.274i − 0.564087i
\(489\) 0 0
\(490\) 0 0
\(491\) 449.720i 0.915927i 0.888971 + 0.457963i \(0.151421\pi\)
−0.888971 + 0.457963i \(0.848579\pi\)
\(492\) 0 0
\(493\) −401.462 −0.814325
\(494\) 418.607i 0.847383i
\(495\) 0 0
\(496\) −97.3242 −0.196218
\(497\) 0 0
\(498\) 0 0
\(499\) 756.000 1.51503 0.757515 0.652818i \(-0.226413\pi\)
0.757515 + 0.652818i \(0.226413\pi\)
\(500\) 412.912i 0.825823i
\(501\) 0 0
\(502\) 194.648 0.387746
\(503\) − 550.549i − 1.09453i −0.836959 0.547265i \(-0.815669\pi\)
0.836959 0.547265i \(-0.184331\pi\)
\(504\) 0 0
\(505\) −518.000 −1.02574
\(506\) − 169.706i − 0.335387i
\(507\) 0 0
\(508\) 152.000 0.299213
\(509\) − 8.60233i − 0.0169004i −0.999964 0.00845022i \(-0.997310\pi\)
0.999964 0.00845022i \(-0.00268982\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) 352.800 0.686382
\(515\) 1465.13i 2.84490i
\(516\) 0 0
\(517\) 194.648 0.376496
\(518\) 0 0
\(519\) 0 0
\(520\) 296.000 0.569231
\(521\) 94.6256i 0.181623i 0.995868 + 0.0908115i \(0.0289461\pi\)
−0.995868 + 0.0908115i \(0.971054\pi\)
\(522\) 0 0
\(523\) −924.580 −1.76784 −0.883920 0.467639i \(-0.845105\pi\)
−0.883920 + 0.467639i \(0.845105\pi\)
\(524\) − 206.456i − 0.394000i
\(525\) 0 0
\(526\) −332.000 −0.631179
\(527\) − 627.911i − 1.19148i
\(528\) 0 0
\(529\) −1271.00 −2.40265
\(530\) 498.935i 0.941387i
\(531\) 0 0
\(532\) 0 0
\(533\) − 313.955i − 0.589035i
\(534\) 0 0
\(535\) 170.317 0.318350
\(536\) − 294.156i − 0.548799i
\(537\) 0 0
\(538\) 304.138 0.565313
\(539\) 0 0
\(540\) 0 0
\(541\) 592.000 1.09427 0.547135 0.837044i \(-0.315718\pi\)
0.547135 + 0.837044i \(0.315718\pi\)
\(542\) − 447.321i − 0.825315i
\(543\) 0 0
\(544\) 145.986 0.268357
\(545\) − 481.730i − 0.883909i
\(546\) 0 0
\(547\) −416.000 −0.760512 −0.380256 0.924881i \(-0.624164\pi\)
−0.380256 + 0.924881i \(0.624164\pi\)
\(548\) − 200.818i − 0.366457i
\(549\) 0 0
\(550\) −196.000 −0.356364
\(551\) 378.502i 0.686937i
\(552\) 0 0
\(553\) 0 0
\(554\) − 90.5097i − 0.163375i
\(555\) 0 0
\(556\) 291.973 0.525131
\(557\) − 309.713i − 0.556037i −0.960576 0.278019i \(-0.910322\pi\)
0.960576 0.278019i \(-0.0896777\pi\)
\(558\) 0 0
\(559\) 827.256 1.47988
\(560\) 0 0
\(561\) 0 0
\(562\) −654.000 −1.16370
\(563\) 894.642i 1.58906i 0.607224 + 0.794531i \(0.292283\pi\)
−0.607224 + 0.794531i \(0.707717\pi\)
\(564\) 0 0
\(565\) 1374.70 2.43311
\(566\) 34.4093i 0.0607938i
\(567\) 0 0
\(568\) −200.000 −0.352113
\(569\) 733.977i 1.28994i 0.764207 + 0.644971i \(0.223131\pi\)
−0.764207 + 0.644971i \(0.776869\pi\)
\(570\) 0 0
\(571\) 312.000 0.546410 0.273205 0.961956i \(-0.411916\pi\)
0.273205 + 0.961956i \(0.411916\pi\)
\(572\) 68.8186i 0.120312i
\(573\) 0 0
\(574\) 0 0
\(575\) 2078.89i 3.61547i
\(576\) 0 0
\(577\) 194.648 0.337346 0.168673 0.985672i \(-0.446052\pi\)
0.168673 + 0.985672i \(0.446052\pi\)
\(578\) 533.159i 0.922420i
\(579\) 0 0
\(580\) 267.642 0.461451
\(581\) 0 0
\(582\) 0 0
\(583\) −116.000 −0.198971
\(584\) − 172.047i − 0.294600i
\(585\) 0 0
\(586\) 401.462 0.685089
\(587\) 447.321i 0.762046i 0.924566 + 0.381023i \(0.124428\pi\)
−0.924566 + 0.381023i \(0.875572\pi\)
\(588\) 0 0
\(589\) −592.000 −1.00509
\(590\) − 837.214i − 1.41901i
\(591\) 0 0
\(592\) −24.0000 −0.0405405
\(593\) 645.174i 1.08798i 0.839090 + 0.543992i \(0.183088\pi\)
−0.839090 + 0.543992i \(0.816912\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 82.0244i − 0.137625i
\(597\) 0 0
\(598\) 729.932 1.22062
\(599\) 687.308i 1.14743i 0.819057 + 0.573713i \(0.194497\pi\)
−0.819057 + 0.573713i \(0.805503\pi\)
\(600\) 0 0
\(601\) −1070.57 −1.78131 −0.890654 0.454682i \(-0.849753\pi\)
−0.890654 + 0.454682i \(0.849753\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −320.000 −0.529801
\(605\) 972.063i 1.60672i
\(606\) 0 0
\(607\) 729.932 1.20252 0.601262 0.799052i \(-0.294665\pi\)
0.601262 + 0.799052i \(0.294665\pi\)
\(608\) − 137.637i − 0.226377i
\(609\) 0 0
\(610\) −1184.00 −1.94098
\(611\) 837.214i 1.37024i
\(612\) 0 0
\(613\) −514.000 −0.838499 −0.419250 0.907871i \(-0.637707\pi\)
−0.419250 + 0.907871i \(0.637707\pi\)
\(614\) 447.321i 0.728536i
\(615\) 0 0
\(616\) 0 0
\(617\) − 202.233i − 0.327767i −0.986480 0.163884i \(-0.947598\pi\)
0.986480 0.163884i \(-0.0524022\pi\)
\(618\) 0 0
\(619\) −681.269 −1.10060 −0.550298 0.834968i \(-0.685486\pi\)
−0.550298 + 0.834968i \(0.685486\pi\)
\(620\) 418.607i 0.675173i
\(621\) 0 0
\(622\) 389.297 0.625879
\(623\) 0 0
\(624\) 0 0
\(625\) 551.000 0.881600
\(626\) − 412.912i − 0.659603i
\(627\) 0 0
\(628\) 389.297 0.619899
\(629\) − 154.842i − 0.246171i
\(630\) 0 0
\(631\) 636.000 1.00792 0.503962 0.863726i \(-0.331875\pi\)
0.503962 + 0.863726i \(0.331875\pi\)
\(632\) − 56.5685i − 0.0895072i
\(633\) 0 0
\(634\) −474.000 −0.747634
\(635\) − 653.777i − 1.02957i
\(636\) 0 0
\(637\) 0 0
\(638\) 62.2254i 0.0975320i
\(639\) 0 0
\(640\) −97.3242 −0.152069
\(641\) 77.7817i 0.121344i 0.998158 + 0.0606722i \(0.0193244\pi\)
−0.998158 + 0.0606722i \(0.980676\pi\)
\(642\) 0 0
\(643\) −510.952 −0.794638 −0.397319 0.917681i \(-0.630059\pi\)
−0.397319 + 0.917681i \(0.630059\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 888.000 1.37461
\(647\) 929.051i 1.43594i 0.696076 + 0.717968i \(0.254928\pi\)
−0.696076 + 0.717968i \(0.745072\pi\)
\(648\) 0 0
\(649\) 194.648 0.299920
\(650\) − 843.028i − 1.29697i
\(651\) 0 0
\(652\) 0 0
\(653\) − 637.810i − 0.976739i −0.872637 0.488369i \(-0.837592\pi\)
0.872637 0.488369i \(-0.162408\pi\)
\(654\) 0 0
\(655\) −888.000 −1.35573
\(656\) 103.228i 0.157360i
\(657\) 0 0
\(658\) 0 0
\(659\) 200.818i 0.304732i 0.988324 + 0.152366i \(0.0486892\pi\)
−0.988324 + 0.152366i \(0.951311\pi\)
\(660\) 0 0
\(661\) 97.3242 0.147238 0.0736189 0.997286i \(-0.476545\pi\)
0.0736189 + 0.997286i \(0.476545\pi\)
\(662\) 548.715i 0.828874i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 660.000 0.989505
\(668\) − 344.093i − 0.515109i
\(669\) 0 0
\(670\) −1265.21 −1.88838
\(671\) − 275.274i − 0.410245i
\(672\) 0 0
\(673\) −250.000 −0.371471 −0.185736 0.982600i \(-0.559467\pi\)
−0.185736 + 0.982600i \(0.559467\pi\)
\(674\) 294.156i 0.436434i
\(675\) 0 0
\(676\) 42.0000 0.0621302
\(677\) 335.491i 0.495555i 0.968817 + 0.247777i \(0.0797002\pi\)
−0.968817 + 0.247777i \(0.920300\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 627.911i − 0.923398i
\(681\) 0 0
\(682\) −97.3242 −0.142704
\(683\) − 291.328i − 0.426542i −0.976993 0.213271i \(-0.931588\pi\)
0.976993 0.213271i \(-0.0684117\pi\)
\(684\) 0 0
\(685\) −863.752 −1.26095
\(686\) 0 0
\(687\) 0 0
\(688\) −272.000 −0.395349
\(689\) − 498.935i − 0.724143i
\(690\) 0 0
\(691\) −535.283 −0.774650 −0.387325 0.921943i \(-0.626601\pi\)
−0.387325 + 0.921943i \(0.626601\pi\)
\(692\) 17.2047i 0.0248622i
\(693\) 0 0
\(694\) 516.000 0.743516
\(695\) − 1255.82i − 1.80694i
\(696\) 0 0
\(697\) −666.000 −0.955524
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1045.10i 1.49088i 0.666575 + 0.745438i \(0.267760\pi\)
−0.666575 + 0.745438i \(0.732240\pi\)
\(702\) 0 0
\(703\) −145.986 −0.207662
\(704\) − 22.6274i − 0.0321412i
\(705\) 0 0
\(706\) −36.4966 −0.0516949
\(707\) 0 0
\(708\) 0 0
\(709\) −488.000 −0.688293 −0.344147 0.938916i \(-0.611832\pi\)
−0.344147 + 0.938916i \(0.611832\pi\)
\(710\) 860.233i 1.21160i
\(711\) 0 0
\(712\) −267.642 −0.375901
\(713\) 1032.28i 1.44780i
\(714\) 0 0
\(715\) 296.000 0.413986
\(716\) − 288.500i − 0.402932i
\(717\) 0 0
\(718\) 428.000 0.596100
\(719\) − 929.051i − 1.29214i −0.763277 0.646072i \(-0.776411\pi\)
0.763277 0.646072i \(-0.223589\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 326.683i − 0.452470i
\(723\) 0 0
\(724\) −24.3311 −0.0336064
\(725\) − 762.261i − 1.05139i
\(726\) 0 0
\(727\) 900.249 1.23831 0.619153 0.785270i \(-0.287476\pi\)
0.619153 + 0.785270i \(0.287476\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −740.000 −1.01370
\(731\) − 1754.87i − 2.40065i
\(732\) 0 0
\(733\) −1155.72 −1.57671 −0.788353 0.615224i \(-0.789066\pi\)
−0.788353 + 0.615224i \(0.789066\pi\)
\(734\) 550.549i 0.750067i
\(735\) 0 0
\(736\) −240.000 −0.326087
\(737\) − 294.156i − 0.399127i
\(738\) 0 0
\(739\) −920.000 −1.24493 −0.622463 0.782649i \(-0.713868\pi\)
−0.622463 + 0.782649i \(0.713868\pi\)
\(740\) 103.228i 0.139497i
\(741\) 0 0
\(742\) 0 0
\(743\) 755.190i 1.01641i 0.861237 + 0.508203i \(0.169690\pi\)
−0.861237 + 0.508203i \(0.830310\pi\)
\(744\) 0 0
\(745\) −352.800 −0.473557
\(746\) − 998.435i − 1.33838i
\(747\) 0 0
\(748\) 145.986 0.195169
\(749\) 0 0
\(750\) 0 0
\(751\) −120.000 −0.159787 −0.0798935 0.996803i \(-0.525458\pi\)
−0.0798935 + 0.996803i \(0.525458\pi\)
\(752\) − 275.274i − 0.366056i
\(753\) 0 0
\(754\) −267.642 −0.354962
\(755\) 1376.37i 1.82301i
\(756\) 0 0
\(757\) −1016.00 −1.34214 −0.671070 0.741394i \(-0.734165\pi\)
−0.671070 + 0.741394i \(0.734165\pi\)
\(758\) − 1001.26i − 1.32093i
\(759\) 0 0
\(760\) −592.000 −0.778947
\(761\) − 43.0116i − 0.0565199i −0.999601 0.0282599i \(-0.991003\pi\)
0.999601 0.0282599i \(-0.00899662\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 526.087i 0.688596i
\(765\) 0 0
\(766\) 437.959 0.571748
\(767\) 837.214i 1.09154i
\(768\) 0 0
\(769\) 681.269 0.885916 0.442958 0.896542i \(-0.353929\pi\)
0.442958 + 0.896542i \(0.353929\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −76.0000 −0.0984456
\(773\) − 1161.31i − 1.50235i −0.660105 0.751173i \(-0.729488\pi\)
0.660105 0.751173i \(-0.270512\pi\)
\(774\) 0 0
\(775\) 1192.22 1.53835
\(776\) 447.321i 0.576444i
\(777\) 0 0
\(778\) −186.000 −0.239075
\(779\) 627.911i 0.806047i
\(780\) 0 0
\(781\) −200.000 −0.256082
\(782\) − 1548.42i − 1.98007i
\(783\) 0 0
\(784\) 0 0
\(785\) − 1674.43i − 2.13303i
\(786\) 0 0
\(787\) 194.648 0.247330 0.123665 0.992324i \(-0.460535\pi\)
0.123665 + 0.992324i \(0.460535\pi\)
\(788\) 330.926i 0.419957i
\(789\) 0 0
\(790\) −243.311 −0.307988
\(791\) 0 0
\(792\) 0 0
\(793\) 1184.00 1.49306
\(794\) − 137.637i − 0.173347i
\(795\) 0 0
\(796\) 486.621 0.611333
\(797\) − 1109.70i − 1.39235i −0.717874 0.696173i \(-0.754885\pi\)
0.717874 0.696173i \(-0.245115\pi\)
\(798\) 0 0
\(799\) 1776.00 2.22278
\(800\) 277.186i 0.346482i
\(801\) 0 0
\(802\) 126.000 0.157107
\(803\) − 172.047i − 0.214255i
\(804\) 0 0
\(805\) 0 0
\(806\) − 418.607i − 0.519364i
\(807\) 0 0
\(808\) −170.317 −0.210789
\(809\) − 1030.96i − 1.27437i −0.770713 0.637183i \(-0.780100\pi\)
0.770713 0.637183i \(-0.219900\pi\)
\(810\) 0 0
\(811\) 243.311 0.300013 0.150006 0.988685i \(-0.452071\pi\)
0.150006 + 0.988685i \(0.452071\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −24.0000 −0.0294840
\(815\) 0 0
\(816\) 0 0
\(817\) −1654.51 −2.02511
\(818\) − 774.209i − 0.946466i
\(819\) 0 0
\(820\) 444.000 0.541463
\(821\) − 357.796i − 0.435805i −0.975971 0.217903i \(-0.930078\pi\)
0.975971 0.217903i \(-0.0699215\pi\)
\(822\) 0 0
\(823\) 1228.00 1.49210 0.746051 0.665889i \(-0.231947\pi\)
0.746051 + 0.665889i \(0.231947\pi\)
\(824\) 481.730i 0.584624i
\(825\) 0 0
\(826\) 0 0
\(827\) 195.161i 0.235987i 0.993014 + 0.117994i \(0.0376462\pi\)
−0.993014 + 0.117994i \(0.962354\pi\)
\(828\) 0 0
\(829\) −790.759 −0.953871 −0.476936 0.878938i \(-0.658252\pi\)
−0.476936 + 0.878938i \(0.658252\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 97.3242 0.116976
\(833\) 0 0
\(834\) 0 0
\(835\) −1480.00 −1.77246
\(836\) − 137.637i − 0.164638i
\(837\) 0 0
\(838\) 924.580 1.10332
\(839\) − 1238.73i − 1.47644i −0.674559 0.738221i \(-0.735666\pi\)
0.674559 0.738221i \(-0.264334\pi\)
\(840\) 0 0
\(841\) 599.000 0.712247
\(842\) − 339.411i − 0.403101i
\(843\) 0 0
\(844\) −88.0000 −0.104265
\(845\) − 180.649i − 0.213786i
\(846\) 0 0
\(847\) 0 0
\(848\) 164.049i 0.193454i
\(849\) 0 0
\(850\) −1788.33 −2.10392
\(851\) 254.558i 0.299129i
\(852\) 0 0
\(853\) 1557.19 1.82554 0.912771 0.408472i \(-0.133938\pi\)
0.912771 + 0.408472i \(0.133938\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 56.0000 0.0654206
\(857\) 1471.00i 1.71645i 0.513274 + 0.858225i \(0.328433\pi\)
−0.513274 + 0.858225i \(0.671567\pi\)
\(858\) 0 0
\(859\) 510.952 0.594822 0.297411 0.954750i \(-0.403877\pi\)
0.297411 + 0.954750i \(0.403877\pi\)
\(860\) 1169.92i 1.36037i
\(861\) 0 0
\(862\) 20.0000 0.0232019
\(863\) − 568.514i − 0.658765i −0.944197 0.329382i \(-0.893160\pi\)
0.944197 0.329382i \(-0.106840\pi\)
\(864\) 0 0
\(865\) 74.0000 0.0855491
\(866\) − 550.549i − 0.635738i
\(867\) 0 0
\(868\) 0 0
\(869\) − 56.5685i − 0.0650961i
\(870\) 0 0
\(871\) 1265.21 1.45260
\(872\) − 158.392i − 0.181642i
\(873\) 0 0
\(874\) −1459.86 −1.67032
\(875\) 0 0
\(876\) 0 0
\(877\) −882.000 −1.00570 −0.502851 0.864373i \(-0.667715\pi\)
−0.502851 + 0.864373i \(0.667715\pi\)
\(878\) 412.912i 0.470287i
\(879\) 0 0
\(880\) −97.3242 −0.110596
\(881\) − 43.0116i − 0.0488214i −0.999702 0.0244107i \(-0.992229\pi\)
0.999702 0.0244107i \(-0.00777093\pi\)
\(882\) 0 0
\(883\) −140.000 −0.158550 −0.0792752 0.996853i \(-0.525261\pi\)
−0.0792752 + 0.996853i \(0.525261\pi\)
\(884\) 627.911i 0.710306i
\(885\) 0 0
\(886\) −36.0000 −0.0406321
\(887\) − 309.684i − 0.349136i −0.984645 0.174568i \(-0.944147\pi\)
0.984645 0.174568i \(-0.0558529\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1151.17i 1.29345i
\(891\) 0 0
\(892\) −389.297 −0.436431
\(893\) − 1674.43i − 1.87506i
\(894\) 0 0
\(895\) −1240.88 −1.38646
\(896\) 0 0
\(897\) 0 0
\(898\) 1038.00 1.15590
\(899\) − 378.502i − 0.421026i
\(900\) 0 0
\(901\) −1058.40 −1.17470
\(902\) 103.228i 0.114443i
\(903\) 0 0
\(904\) 452.000 0.500000
\(905\) 104.652i 0.115637i
\(906\) 0 0
\(907\) 1228.00 1.35391 0.676957 0.736023i \(-0.263298\pi\)
0.676957 + 0.736023i \(0.263298\pi\)
\(908\) 68.8186i 0.0757914i
\(909\) 0 0
\(910\) 0 0
\(911\) − 138.593i − 0.152133i −0.997103 0.0760664i \(-0.975764\pi\)
0.997103 0.0760664i \(-0.0242361\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 384.666i − 0.420860i
\(915\) 0 0
\(916\) 24.3311 0.0265623
\(917\) 0 0
\(918\) 0 0
\(919\) −1608.00 −1.74973 −0.874864 0.484369i \(-0.839049\pi\)
−0.874864 + 0.484369i \(0.839049\pi\)
\(920\) 1032.28i 1.12204i
\(921\) 0 0
\(922\) 693.435 0.752099
\(923\) − 860.233i − 0.931996i
\(924\) 0 0
\(925\) 294.000 0.317838
\(926\) 616.597i 0.665872i
\(927\) 0 0
\(928\) 88.0000 0.0948276
\(929\) 1419.38i 1.52786i 0.645298 + 0.763931i \(0.276733\pi\)
−0.645298 + 0.763931i \(0.723267\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 540.230i − 0.579645i
\(933\) 0 0
\(934\) 681.269 0.729410
\(935\) − 627.911i − 0.671562i
\(936\) 0 0
\(937\) 291.973 0.311604 0.155802 0.987788i \(-0.450204\pi\)
0.155802 + 0.987788i \(0.450204\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1184.00 −1.25957
\(941\) 1367.77i 1.45353i 0.686887 + 0.726764i \(0.258977\pi\)
−0.686887 + 0.726764i \(0.741023\pi\)
\(942\) 0 0
\(943\) 1094.90 1.16108
\(944\) − 275.274i − 0.291604i
\(945\) 0 0
\(946\) −272.000 −0.287526
\(947\) − 591.141i − 0.624225i −0.950045 0.312113i \(-0.898963\pi\)
0.950045 0.312113i \(-0.101037\pi\)
\(948\) 0 0
\(949\) 740.000 0.779768
\(950\) 1686.06i 1.77480i
\(951\) 0 0
\(952\) 0 0
\(953\) 352.139i 0.369506i 0.982785 + 0.184753i \(0.0591485\pi\)
−0.982785 + 0.184753i \(0.940851\pi\)
\(954\) 0 0
\(955\) 2262.79 2.36941
\(956\) 152.735i 0.159765i
\(957\) 0 0
\(958\) −145.986 −0.152387
\(959\) 0 0
\(960\) 0 0
\(961\) −369.000 −0.383975
\(962\) − 103.228i − 0.107306i
\(963\) 0 0
\(964\) 267.642 0.277636
\(965\) 326.888i 0.338744i
\(966\) 0 0
\(967\) −260.000 −0.268873 −0.134436 0.990922i \(-0.542922\pi\)
−0.134436 + 0.990922i \(0.542922\pi\)
\(968\) 319.612i 0.330178i
\(969\) 0 0
\(970\) 1924.00 1.98351
\(971\) 1479.60i 1.52379i 0.647701 + 0.761895i \(0.275731\pi\)
−0.647701 + 0.761895i \(0.724269\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 328.098i − 0.336856i
\(975\) 0 0
\(976\) −389.297 −0.398870
\(977\) 329.512i 0.337269i 0.985679 + 0.168634i \(0.0539357\pi\)
−0.985679 + 0.168634i \(0.946064\pi\)
\(978\) 0 0
\(979\) −267.642 −0.273383
\(980\) 0 0
\(981\) 0 0
\(982\) 636.000 0.647658
\(983\) − 997.870i − 1.01513i −0.861614 0.507563i \(-0.830546\pi\)
0.861614 0.507563i \(-0.169454\pi\)
\(984\) 0 0
\(985\) 1423.37 1.44504
\(986\) 567.753i 0.575815i
\(987\) 0 0
\(988\) 592.000 0.599190
\(989\) 2885.00i 2.91708i
\(990\) 0 0
\(991\) 1496.00 1.50959 0.754793 0.655963i \(-0.227737\pi\)
0.754793 + 0.655963i \(0.227737\pi\)
\(992\) 137.637i 0.138747i
\(993\) 0 0
\(994\) 0 0
\(995\) − 2093.04i − 2.10355i
\(996\) 0 0
\(997\) 1557.19 1.56187 0.780936 0.624611i \(-0.214742\pi\)
0.780936 + 0.624611i \(0.214742\pi\)
\(998\) − 1069.15i − 1.07129i
\(999\) 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.g.197.2 yes 4
3.2 odd 2 inner 882.3.b.g.197.3 yes 4
7.2 even 3 882.3.s.h.557.2 8
7.3 odd 6 882.3.s.h.863.4 8
7.4 even 3 882.3.s.h.863.3 8
7.5 odd 6 882.3.s.h.557.1 8
7.6 odd 2 inner 882.3.b.g.197.1 4
21.2 odd 6 882.3.s.h.557.3 8
21.5 even 6 882.3.s.h.557.4 8
21.11 odd 6 882.3.s.h.863.2 8
21.17 even 6 882.3.s.h.863.1 8
21.20 even 2 inner 882.3.b.g.197.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.3.b.g.197.1 4 7.6 odd 2 inner
882.3.b.g.197.2 yes 4 1.1 even 1 trivial
882.3.b.g.197.3 yes 4 3.2 odd 2 inner
882.3.b.g.197.4 yes 4 21.20 even 2 inner
882.3.s.h.557.1 8 7.5 odd 6
882.3.s.h.557.2 8 7.2 even 3
882.3.s.h.557.3 8 21.2 odd 6
882.3.s.h.557.4 8 21.5 even 6
882.3.s.h.863.1 8 21.17 even 6
882.3.s.h.863.2 8 21.11 odd 6
882.3.s.h.863.3 8 7.4 even 3
882.3.s.h.863.4 8 7.3 odd 6