Properties

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

$q$-expansion

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

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 \(-0.670318\pi\)
0.509902 0.860233i \(-0.329682\pi\)
\(6\) 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 \(-0.958963\pi\)
0.991701 0.128565i \(-0.0410371\pi\)
\(12\) 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 \(-0.725674\pi\)
0.651057 0.759029i \(-0.274326\pi\)
\(18\) 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 \(0.373704\pi\)
−0.386443 + 0.922313i \(0.626296\pi\)
\(24\) 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 \(-0.913567\pi\)
0.963360 0.268213i \(-0.0864331\pi\)
\(30\) 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 \(-0.898090\pi\)
0.949185 0.314719i \(-0.101910\pi\)
\(42\) 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 \(0.261466\pi\)
−0.681183 + 0.732113i \(0.738534\pi\)
\(48\) 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 \(-0.873543\pi\)
0.922119 0.386907i \(-0.126457\pi\)
\(54\) 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 \(0.198203\pi\)
−0.812323 + 0.583208i \(0.801797\pi\)
\(60\) 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 \(-0.834082\pi\)
0.867199 0.497963i \(-0.165918\pi\)
\(72\) 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 \(-0.821589\pi\)
0.846993 0.531604i \(-0.178411\pi\)
\(90\) 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 \(-0.903647\pi\)
0.954535 0.298100i \(-0.0963530\pi\)
\(102\) 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 \(0.0294918\pi\)
−0.995711 + 0.0925186i \(0.970508\pi\)
\(108\) 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 \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 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 \(-0.871091\pi\)
0.919111 0.394000i \(-0.128909\pi\)
\(132\) 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 \(-0.880571\pi\)
0.930435 0.366457i \(-0.119429\pi\)
\(138\) 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 \(-0.956053\pi\)
0.990484 0.137625i \(-0.0439468\pi\)
\(150\) 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 \(-0.827751\pi\)
0.857125 0.515109i \(-0.172249\pi\)
\(168\) 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 \(0.00791470\pi\)
−0.999691 + 0.0248622i \(0.992085\pi\)
\(174\) 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 \(-0.867991\pi\)
0.915230 0.402932i \(-0.132009\pi\)
\(180\) 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 \(0.241773\pi\)
−0.725145 + 0.688596i \(0.758227\pi\)
\(192\) 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 \(0.137955\pi\)
−0.907544 + 0.419957i \(0.862045\pi\)
\(198\) 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 \(0.0241483\pi\)
−0.997124 + 0.0757914i \(0.975852\pi\)
\(228\) 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 \(-0.803191\pi\)
0.814869 0.579645i \(-0.196809\pi\)
\(234\) 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 \(0.0510736\pi\)
−0.987155 + 0.159765i \(0.948926\pi\)
\(240\) 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 \(-0.911594\pi\)
0.961679 0.274178i \(-0.0884056\pi\)
\(252\) 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 \(-0.838694\pi\)
0.874323 0.485345i \(-0.161306\pi\)
\(258\) 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 \(0.147262\pi\)
−0.894878 + 0.446311i \(0.852738\pi\)
\(264\) 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 \(-0.869102\pi\)
0.916630 0.399736i \(-0.130898\pi\)
\(270\) 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 \(0.307623\pi\)
−0.568243 + 0.822861i \(0.692377\pi\)
\(282\) 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 \(-0.839027\pi\)
0.874829 0.484431i \(-0.160973\pi\)
\(294\) 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 \(-0.854069\pi\)
0.896737 0.442563i \(-0.145931\pi\)
\(312\) 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 \(0.177304\pi\)
−0.848835 + 0.528657i \(0.822696\pi\)
\(318\) 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 \(-0.823787\pi\)
0.850642 0.525745i \(-0.176213\pi\)
\(348\) 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 \(0.0116380\pi\)
−0.999332 + 0.0365538i \(0.988362\pi\)
\(354\) 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 \(-0.861501\pi\)
0.906825 0.421507i \(-0.138499\pi\)
\(360\) 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 \(-0.867520\pi\)
0.914632 0.404287i \(-0.132480\pi\)
\(384\) 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 \(0.0540703\pi\)
−0.985607 + 0.169051i \(0.945930\pi\)
\(390\) 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 \(-0.964565\pi\)
0.993810 0.111092i \(-0.0354347\pi\)
\(402\) 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 \(-0.715136\pi\)
0.625576 0.780163i \(-0.284864\pi\)
\(420\) 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 \(-0.994778\pi\)
0.999865 0.0164062i \(-0.00522249\pi\)
\(432\) 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 \(0.00914668\pi\)
−0.999587 + 0.0287312i \(0.990853\pi\)
\(444\) 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 \(-0.695444\pi\)
0.576147 0.817346i \(-0.304556\pi\)
\(450\) 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 \(-0.821511\pi\)
0.846861 0.531814i \(-0.178489\pi\)
\(462\) 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 \(-0.827505\pi\)
0.856726 0.515771i \(-0.172495\pi\)
\(468\) 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 \(0.0343657\pi\)
−0.994178 + 0.107754i \(0.965634\pi\)
\(480\) 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 \(-0.848579\pi\)
0.888971 0.457963i \(-0.151421\pi\)
\(492\) 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 \(0.184331\pi\)
−0.836959 + 0.547265i \(0.815669\pi\)
\(504\) 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 \(0.00268982\pi\)
−0.999964 + 0.00845022i \(0.997310\pi\)
\(510\) 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 \(-0.971054\pi\)
0.995868 0.0908115i \(-0.0289461\pi\)
\(522\) 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 \(0.0896777\pi\)
−0.960576 + 0.278019i \(0.910322\pi\)
\(558\) 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 \(-0.707717\pi\)
0.607224 0.794531i \(-0.292283\pi\)
\(564\) 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 \(-0.776869\pi\)
0.764207 0.644971i \(-0.223131\pi\)
\(570\) 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 \(-0.875572\pi\)
0.924566 0.381023i \(-0.124428\pi\)
\(588\) 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 \(-0.816912\pi\)
0.839090 0.543992i \(-0.183088\pi\)
\(594\) 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 \(-0.805503\pi\)
0.819057 0.573713i \(-0.194497\pi\)
\(600\) 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 \(0.0524022\pi\)
−0.986480 + 0.163884i \(0.947598\pi\)
\(618\) 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 \(-0.980676\pi\)
0.998158 0.0606722i \(-0.0193244\pi\)
\(642\) 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 \(-0.745072\pi\)
0.696076 0.717968i \(-0.254928\pi\)
\(648\) 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 \(0.162408\pi\)
−0.872637 + 0.488369i \(0.837592\pi\)
\(654\) 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 \(-0.951311\pi\)
0.988324 0.152366i \(-0.0486892\pi\)
\(660\) 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 \(-0.920300\pi\)
0.968817 0.247777i \(-0.0797002\pi\)
\(678\) 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 \(0.0684117\pi\)
−0.976993 + 0.213271i \(0.931588\pi\)
\(684\) 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 \(-0.732240\pi\)
0.666575 0.745438i \(-0.267760\pi\)
\(702\) 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 \(0.223589\pi\)
−0.763277 + 0.646072i \(0.776411\pi\)
\(720\) 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 \(-0.830310\pi\)
0.861237 0.508203i \(-0.169690\pi\)
\(744\) 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 \(0.00899662\pi\)
−0.999601 + 0.0282599i \(0.991003\pi\)
\(762\) 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 \(0.270512\pi\)
−0.660105 + 0.751173i \(0.729488\pi\)
\(774\) 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 \(0.245115\pi\)
−0.717874 + 0.696173i \(0.754885\pi\)
\(798\) 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 \(0.219900\pi\)
−0.770713 + 0.637183i \(0.780100\pi\)
\(810\) 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 \(0.0699215\pi\)
−0.975971 + 0.217903i \(0.930078\pi\)
\(822\) 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 \(-0.962354\pi\)
0.993014 0.117994i \(-0.0376462\pi\)
\(828\) 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 \(0.264334\pi\)
−0.674559 + 0.738221i \(0.735666\pi\)
\(840\) 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 \(-0.671567\pi\)
0.513274 0.858225i \(-0.328433\pi\)
\(858\) 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 \(0.106840\pi\)
−0.944197 + 0.329382i \(0.893160\pi\)
\(864\) 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 \(0.00777093\pi\)
−0.999702 + 0.0244107i \(0.992229\pi\)
\(882\) 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 \(0.0558529\pi\)
−0.984645 + 0.174568i \(0.944147\pi\)
\(888\) 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 \(0.0242361\pi\)
−0.997103 + 0.0760664i \(0.975764\pi\)
\(912\) 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 \(-0.723267\pi\)
0.645298 0.763931i \(-0.276733\pi\)
\(930\) 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 \(-0.741023\pi\)
0.686887 0.726764i \(-0.258977\pi\)
\(942\) 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 \(0.101037\pi\)
−0.950045 + 0.312113i \(0.898963\pi\)
\(948\) 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 \(-0.940851\pi\)
0.982785 0.184753i \(-0.0591485\pi\)
\(954\) 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 \(-0.724269\pi\)
0.647701 0.761895i \(-0.275731\pi\)
\(972\) 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 \(-0.946064\pi\)
0.985679 0.168634i \(-0.0539357\pi\)
\(978\) 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 \(0.169454\pi\)
−0.861614 + 0.507563i \(0.830546\pi\)
\(984\) 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.197.3 yes 4
3.2 odd 2 inner 882.3.b.g.197.2 yes 4
7.2 even 3 882.3.s.h.557.3 8
7.3 odd 6 882.3.s.h.863.1 8
7.4 even 3 882.3.s.h.863.2 8
7.5 odd 6 882.3.s.h.557.4 8
7.6 odd 2 inner 882.3.b.g.197.4 yes 4
21.2 odd 6 882.3.s.h.557.2 8
21.5 even 6 882.3.s.h.557.1 8
21.11 odd 6 882.3.s.h.863.3 8
21.17 even 6 882.3.s.h.863.4 8
21.20 even 2 inner 882.3.b.g.197.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.3.b.g.197.1 4 21.20 even 2 inner
882.3.b.g.197.2 yes 4 3.2 odd 2 inner
882.3.b.g.197.3 yes 4 1.1 even 1 trivial
882.3.b.g.197.4 yes 4 7.6 odd 2 inner
882.3.s.h.557.1 8 21.5 even 6
882.3.s.h.557.2 8 21.2 odd 6
882.3.s.h.557.3 8 7.2 even 3
882.3.s.h.557.4 8 7.5 odd 6
882.3.s.h.863.1 8 7.3 odd 6
882.3.s.h.863.2 8 7.4 even 3
882.3.s.h.863.3 8 21.11 odd 6
882.3.s.h.863.4 8 21.17 even 6