Properties

Label 1089.3.b.g.485.2
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.65306824704.6
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.2
Root \(-1.75726 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.g.485.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.48514i q^{2} -2.17590 q^{4} +4.68911i q^{5} +3.30128 q^{7} -4.53313i q^{8} +O(q^{10})\) \(q-2.48514i q^{2} -2.17590 q^{4} +4.68911i q^{5} +3.30128 q^{7} -4.53313i q^{8} +11.6531 q^{10} -1.00848 q^{13} -8.20412i q^{14} -19.9691 q^{16} +6.45594i q^{17} -25.1291 q^{19} -10.2030i q^{20} -24.4236i q^{23} +3.01224 q^{25} +2.50622i q^{26} -7.18325 q^{28} -50.8695i q^{29} +48.4055 q^{31} +31.4933i q^{32} +16.0439 q^{34} +15.4800i q^{35} +65.8199 q^{37} +62.4493i q^{38} +21.2564 q^{40} +6.41379i q^{41} +48.5582 q^{43} -60.6959 q^{46} -56.8721i q^{47} -38.1016 q^{49} -7.48582i q^{50} +2.19436 q^{52} -67.3272i q^{53} -14.9651i q^{56} -126.418 q^{58} -0.307118i q^{59} +86.8383 q^{61} -120.294i q^{62} -1.61108 q^{64} -4.72888i q^{65} +29.6749 q^{67} -14.0475i q^{68} +38.4700 q^{70} -40.9307i q^{71} +61.0447 q^{73} -163.571i q^{74} +54.6785 q^{76} -85.4268 q^{79} -93.6371i q^{80} +15.9391 q^{82} -64.7194i q^{83} -30.2726 q^{85} -120.674i q^{86} +64.1065i q^{89} -3.32928 q^{91} +53.1433i q^{92} -141.335 q^{94} -117.833i q^{95} -86.8082 q^{97} +94.6876i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 16 q^{7} + 48 q^{10} + 8 q^{13} + 104 q^{16} - 40 q^{19} - 112 q^{25} + 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} - 432 q^{40} + 104 q^{43} - 24 q^{46} - 96 q^{49} - 280 q^{52} - 432 q^{58} + 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} - 448 q^{73} + 344 q^{76} - 448 q^{79} + 504 q^{82} - 48 q^{85} - 544 q^{91} - 360 q^{94} - 152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.48514i − 1.24257i −0.783585 0.621284i \(-0.786611\pi\)
0.783585 0.621284i \(-0.213389\pi\)
\(3\) 0 0
\(4\) −2.17590 −0.543976
\(5\) 4.68911i 0.937822i 0.883245 + 0.468911i \(0.155354\pi\)
−0.883245 + 0.468911i \(0.844646\pi\)
\(6\) 0 0
\(7\) 3.30128 0.471611 0.235805 0.971800i \(-0.424227\pi\)
0.235805 + 0.971800i \(0.424227\pi\)
\(8\) − 4.53313i − 0.566641i
\(9\) 0 0
\(10\) 11.6531 1.16531
\(11\) 0 0
\(12\) 0 0
\(13\) −1.00848 −0.0775756 −0.0387878 0.999247i \(-0.512350\pi\)
−0.0387878 + 0.999247i \(0.512350\pi\)
\(14\) − 8.20412i − 0.586009i
\(15\) 0 0
\(16\) −19.9691 −1.24807
\(17\) 6.45594i 0.379761i 0.981807 + 0.189881i \(0.0608101\pi\)
−0.981807 + 0.189881i \(0.939190\pi\)
\(18\) 0 0
\(19\) −25.1291 −1.32259 −0.661293 0.750128i \(-0.729992\pi\)
−0.661293 + 0.750128i \(0.729992\pi\)
\(20\) − 10.2030i − 0.510152i
\(21\) 0 0
\(22\) 0 0
\(23\) − 24.4236i − 1.06189i −0.847405 0.530947i \(-0.821836\pi\)
0.847405 0.530947i \(-0.178164\pi\)
\(24\) 0 0
\(25\) 3.01224 0.120489
\(26\) 2.50622i 0.0963929i
\(27\) 0 0
\(28\) −7.18325 −0.256545
\(29\) − 50.8695i − 1.75412i −0.480379 0.877061i \(-0.659501\pi\)
0.480379 0.877061i \(-0.340499\pi\)
\(30\) 0 0
\(31\) 48.4055 1.56147 0.780734 0.624864i \(-0.214846\pi\)
0.780734 + 0.624864i \(0.214846\pi\)
\(32\) 31.4933i 0.984166i
\(33\) 0 0
\(34\) 16.0439 0.471879
\(35\) 15.4800i 0.442287i
\(36\) 0 0
\(37\) 65.8199 1.77892 0.889458 0.457017i \(-0.151082\pi\)
0.889458 + 0.457017i \(0.151082\pi\)
\(38\) 62.4493i 1.64340i
\(39\) 0 0
\(40\) 21.2564 0.531409
\(41\) 6.41379i 0.156434i 0.996936 + 0.0782169i \(0.0249227\pi\)
−0.996936 + 0.0782169i \(0.975077\pi\)
\(42\) 0 0
\(43\) 48.5582 1.12926 0.564631 0.825344i \(-0.309019\pi\)
0.564631 + 0.825344i \(0.309019\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −60.6959 −1.31948
\(47\) − 56.8721i − 1.21004i −0.796209 0.605022i \(-0.793164\pi\)
0.796209 0.605022i \(-0.206836\pi\)
\(48\) 0 0
\(49\) −38.1016 −0.777583
\(50\) − 7.48582i − 0.149716i
\(51\) 0 0
\(52\) 2.19436 0.0421992
\(53\) − 67.3272i − 1.27033i −0.772379 0.635163i \(-0.780933\pi\)
0.772379 0.635163i \(-0.219067\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 14.9651i − 0.267234i
\(57\) 0 0
\(58\) −126.418 −2.17962
\(59\) − 0.307118i − 0.00520539i −0.999997 0.00260270i \(-0.999172\pi\)
0.999997 0.00260270i \(-0.000828465\pi\)
\(60\) 0 0
\(61\) 86.8383 1.42358 0.711790 0.702393i \(-0.247885\pi\)
0.711790 + 0.702393i \(0.247885\pi\)
\(62\) − 120.294i − 1.94023i
\(63\) 0 0
\(64\) −1.61108 −0.0251731
\(65\) − 4.72888i − 0.0727521i
\(66\) 0 0
\(67\) 29.6749 0.442909 0.221455 0.975171i \(-0.428920\pi\)
0.221455 + 0.975171i \(0.428920\pi\)
\(68\) − 14.0475i − 0.206581i
\(69\) 0 0
\(70\) 38.4700 0.549572
\(71\) − 40.9307i − 0.576489i −0.957557 0.288244i \(-0.906928\pi\)
0.957557 0.288244i \(-0.0930716\pi\)
\(72\) 0 0
\(73\) 61.0447 0.836229 0.418115 0.908394i \(-0.362691\pi\)
0.418115 + 0.908394i \(0.362691\pi\)
\(74\) − 163.571i − 2.21042i
\(75\) 0 0
\(76\) 54.6785 0.719454
\(77\) 0 0
\(78\) 0 0
\(79\) −85.4268 −1.08135 −0.540676 0.841231i \(-0.681831\pi\)
−0.540676 + 0.841231i \(0.681831\pi\)
\(80\) − 93.6371i − 1.17046i
\(81\) 0 0
\(82\) 15.9391 0.194380
\(83\) − 64.7194i − 0.779751i −0.920867 0.389876i \(-0.872518\pi\)
0.920867 0.389876i \(-0.127482\pi\)
\(84\) 0 0
\(85\) −30.2726 −0.356149
\(86\) − 120.674i − 1.40318i
\(87\) 0 0
\(88\) 0 0
\(89\) 64.1065i 0.720297i 0.932895 + 0.360149i \(0.117274\pi\)
−0.932895 + 0.360149i \(0.882726\pi\)
\(90\) 0 0
\(91\) −3.32928 −0.0365855
\(92\) 53.1433i 0.577644i
\(93\) 0 0
\(94\) −141.335 −1.50356
\(95\) − 117.833i − 1.24035i
\(96\) 0 0
\(97\) −86.8082 −0.894930 −0.447465 0.894302i \(-0.647673\pi\)
−0.447465 + 0.894302i \(0.647673\pi\)
\(98\) 94.6876i 0.966200i
\(99\) 0 0
\(100\) −6.55433 −0.0655433
\(101\) − 114.951i − 1.13813i −0.822292 0.569065i \(-0.807305\pi\)
0.822292 0.569065i \(-0.192695\pi\)
\(102\) 0 0
\(103\) 28.6367 0.278026 0.139013 0.990291i \(-0.455607\pi\)
0.139013 + 0.990291i \(0.455607\pi\)
\(104\) 4.57158i 0.0439575i
\(105\) 0 0
\(106\) −167.317 −1.57847
\(107\) 119.479i 1.11662i 0.829631 + 0.558312i \(0.188551\pi\)
−0.829631 + 0.558312i \(0.811449\pi\)
\(108\) 0 0
\(109\) 76.7577 0.704199 0.352100 0.935962i \(-0.385468\pi\)
0.352100 + 0.935962i \(0.385468\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −65.9234 −0.588601
\(113\) − 84.4614i − 0.747446i −0.927540 0.373723i \(-0.878081\pi\)
0.927540 0.373723i \(-0.121919\pi\)
\(114\) 0 0
\(115\) 114.525 0.995867
\(116\) 110.687i 0.954199i
\(117\) 0 0
\(118\) −0.763231 −0.00646806
\(119\) 21.3129i 0.179100i
\(120\) 0 0
\(121\) 0 0
\(122\) − 215.805i − 1.76889i
\(123\) 0 0
\(124\) −105.326 −0.849400
\(125\) 131.352i 1.05082i
\(126\) 0 0
\(127\) 99.8661 0.786347 0.393174 0.919464i \(-0.371377\pi\)
0.393174 + 0.919464i \(0.371377\pi\)
\(128\) 129.977i 1.01545i
\(129\) 0 0
\(130\) −11.7519 −0.0903994
\(131\) 165.140i 1.26061i 0.776347 + 0.630305i \(0.217070\pi\)
−0.776347 + 0.630305i \(0.782930\pi\)
\(132\) 0 0
\(133\) −82.9582 −0.623746
\(134\) − 73.7462i − 0.550345i
\(135\) 0 0
\(136\) 29.2656 0.215189
\(137\) − 23.5961i − 0.172234i −0.996285 0.0861170i \(-0.972554\pi\)
0.996285 0.0861170i \(-0.0274459\pi\)
\(138\) 0 0
\(139\) −272.764 −1.96233 −0.981167 0.193162i \(-0.938126\pi\)
−0.981167 + 0.193162i \(0.938126\pi\)
\(140\) − 33.6831i − 0.240593i
\(141\) 0 0
\(142\) −101.718 −0.716327
\(143\) 0 0
\(144\) 0 0
\(145\) 238.533 1.64505
\(146\) − 151.704i − 1.03907i
\(147\) 0 0
\(148\) −143.218 −0.967687
\(149\) − 34.7567i − 0.233266i −0.993175 0.116633i \(-0.962790\pi\)
0.993175 0.116633i \(-0.0372102\pi\)
\(150\) 0 0
\(151\) −123.990 −0.821124 −0.410562 0.911833i \(-0.634667\pi\)
−0.410562 + 0.911833i \(0.634667\pi\)
\(152\) 113.914i 0.749432i
\(153\) 0 0
\(154\) 0 0
\(155\) 226.979i 1.46438i
\(156\) 0 0
\(157\) 157.296 1.00188 0.500942 0.865481i \(-0.332987\pi\)
0.500942 + 0.865481i \(0.332987\pi\)
\(158\) 212.297i 1.34365i
\(159\) 0 0
\(160\) −147.676 −0.922973
\(161\) − 80.6289i − 0.500800i
\(162\) 0 0
\(163\) −300.686 −1.84470 −0.922350 0.386356i \(-0.873734\pi\)
−0.922350 + 0.386356i \(0.873734\pi\)
\(164\) − 13.9558i − 0.0850962i
\(165\) 0 0
\(166\) −160.836 −0.968894
\(167\) − 90.7043i − 0.543139i −0.962419 0.271570i \(-0.912457\pi\)
0.962419 0.271570i \(-0.0875427\pi\)
\(168\) 0 0
\(169\) −167.983 −0.993982
\(170\) 75.2316i 0.442539i
\(171\) 0 0
\(172\) −105.658 −0.614291
\(173\) − 194.822i − 1.12614i −0.826410 0.563069i \(-0.809620\pi\)
0.826410 0.563069i \(-0.190380\pi\)
\(174\) 0 0
\(175\) 9.94422 0.0568241
\(176\) 0 0
\(177\) 0 0
\(178\) 159.313 0.895018
\(179\) 27.8164i 0.155399i 0.996977 + 0.0776994i \(0.0247574\pi\)
−0.996977 + 0.0776994i \(0.975243\pi\)
\(180\) 0 0
\(181\) 194.094 1.07234 0.536171 0.844109i \(-0.319870\pi\)
0.536171 + 0.844109i \(0.319870\pi\)
\(182\) 8.27371i 0.0454599i
\(183\) 0 0
\(184\) −110.715 −0.601713
\(185\) 308.637i 1.66831i
\(186\) 0 0
\(187\) 0 0
\(188\) 123.748i 0.658235i
\(189\) 0 0
\(190\) −292.832 −1.54122
\(191\) − 297.244i − 1.55625i −0.628108 0.778126i \(-0.716171\pi\)
0.628108 0.778126i \(-0.283829\pi\)
\(192\) 0 0
\(193\) 197.890 1.02533 0.512667 0.858587i \(-0.328657\pi\)
0.512667 + 0.858587i \(0.328657\pi\)
\(194\) 215.730i 1.11201i
\(195\) 0 0
\(196\) 82.9053 0.422986
\(197\) 249.339i 1.26568i 0.774282 + 0.632841i \(0.218111\pi\)
−0.774282 + 0.632841i \(0.781889\pi\)
\(198\) 0 0
\(199\) 101.803 0.511571 0.255786 0.966733i \(-0.417666\pi\)
0.255786 + 0.966733i \(0.417666\pi\)
\(200\) − 13.6549i − 0.0682743i
\(201\) 0 0
\(202\) −285.669 −1.41420
\(203\) − 167.934i − 0.827263i
\(204\) 0 0
\(205\) −30.0750 −0.146707
\(206\) − 71.1661i − 0.345466i
\(207\) 0 0
\(208\) 20.1384 0.0968194
\(209\) 0 0
\(210\) 0 0
\(211\) 37.9452 0.179835 0.0899175 0.995949i \(-0.471340\pi\)
0.0899175 + 0.995949i \(0.471340\pi\)
\(212\) 146.497i 0.691026i
\(213\) 0 0
\(214\) 296.921 1.38748
\(215\) 227.695i 1.05905i
\(216\) 0 0
\(217\) 159.800 0.736405
\(218\) − 190.753i − 0.875016i
\(219\) 0 0
\(220\) 0 0
\(221\) − 6.51071i − 0.0294602i
\(222\) 0 0
\(223\) 119.504 0.535893 0.267946 0.963434i \(-0.413655\pi\)
0.267946 + 0.963434i \(0.413655\pi\)
\(224\) 103.968i 0.464143i
\(225\) 0 0
\(226\) −209.898 −0.928753
\(227\) 233.484i 1.02856i 0.857621 + 0.514282i \(0.171941\pi\)
−0.857621 + 0.514282i \(0.828059\pi\)
\(228\) 0 0
\(229\) −170.303 −0.743683 −0.371841 0.928296i \(-0.621274\pi\)
−0.371841 + 0.928296i \(0.621274\pi\)
\(230\) − 284.610i − 1.23743i
\(231\) 0 0
\(232\) −230.598 −0.993958
\(233\) 149.406i 0.641228i 0.947210 + 0.320614i \(0.103889\pi\)
−0.947210 + 0.320614i \(0.896111\pi\)
\(234\) 0 0
\(235\) 266.680 1.13481
\(236\) 0.668259i 0.00283161i
\(237\) 0 0
\(238\) 52.9653 0.222543
\(239\) 267.637i 1.11982i 0.828554 + 0.559909i \(0.189164\pi\)
−0.828554 + 0.559909i \(0.810836\pi\)
\(240\) 0 0
\(241\) −202.338 −0.839578 −0.419789 0.907622i \(-0.637896\pi\)
−0.419789 + 0.907622i \(0.637896\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −188.952 −0.774392
\(245\) − 178.663i − 0.729235i
\(246\) 0 0
\(247\) 25.3423 0.102600
\(248\) − 219.428i − 0.884792i
\(249\) 0 0
\(250\) 326.429 1.30572
\(251\) 169.400i 0.674901i 0.941343 + 0.337450i \(0.109565\pi\)
−0.941343 + 0.337450i \(0.890435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 248.181i − 0.977090i
\(255\) 0 0
\(256\) 316.566 1.23659
\(257\) 208.654i 0.811885i 0.913899 + 0.405942i \(0.133057\pi\)
−0.913899 + 0.405942i \(0.866943\pi\)
\(258\) 0 0
\(259\) 217.290 0.838956
\(260\) 10.2896i 0.0395754i
\(261\) 0 0
\(262\) 410.395 1.56639
\(263\) − 117.552i − 0.446967i −0.974708 0.223483i \(-0.928257\pi\)
0.974708 0.223483i \(-0.0717429\pi\)
\(264\) 0 0
\(265\) 315.705 1.19134
\(266\) 206.162i 0.775047i
\(267\) 0 0
\(268\) −64.5697 −0.240932
\(269\) − 467.823i − 1.73912i −0.493830 0.869559i \(-0.664403\pi\)
0.493830 0.869559i \(-0.335597\pi\)
\(270\) 0 0
\(271\) 116.293 0.429126 0.214563 0.976710i \(-0.431167\pi\)
0.214563 + 0.976710i \(0.431167\pi\)
\(272\) − 128.919i − 0.473967i
\(273\) 0 0
\(274\) −58.6394 −0.214013
\(275\) 0 0
\(276\) 0 0
\(277\) −290.493 −1.04871 −0.524356 0.851499i \(-0.675694\pi\)
−0.524356 + 0.851499i \(0.675694\pi\)
\(278\) 677.857i 2.43833i
\(279\) 0 0
\(280\) 70.1731 0.250618
\(281\) − 26.7812i − 0.0953066i −0.998864 0.0476533i \(-0.984826\pi\)
0.998864 0.0476533i \(-0.0151743\pi\)
\(282\) 0 0
\(283\) −23.7154 −0.0838001 −0.0419001 0.999122i \(-0.513341\pi\)
−0.0419001 + 0.999122i \(0.513341\pi\)
\(284\) 89.0612i 0.313596i
\(285\) 0 0
\(286\) 0 0
\(287\) 21.1737i 0.0737759i
\(288\) 0 0
\(289\) 247.321 0.855781
\(290\) − 592.787i − 2.04409i
\(291\) 0 0
\(292\) −132.827 −0.454888
\(293\) − 379.831i − 1.29635i −0.761491 0.648175i \(-0.775532\pi\)
0.761491 0.648175i \(-0.224468\pi\)
\(294\) 0 0
\(295\) 1.44011 0.00488173
\(296\) − 298.370i − 1.00801i
\(297\) 0 0
\(298\) −86.3750 −0.289849
\(299\) 24.6307i 0.0823770i
\(300\) 0 0
\(301\) 160.304 0.532572
\(302\) 308.131i 1.02030i
\(303\) 0 0
\(304\) 501.805 1.65067
\(305\) 407.195i 1.33506i
\(306\) 0 0
\(307\) −41.2092 −0.134232 −0.0671160 0.997745i \(-0.521380\pi\)
−0.0671160 + 0.997745i \(0.521380\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 564.073 1.81959
\(311\) 520.132i 1.67245i 0.548387 + 0.836224i \(0.315242\pi\)
−0.548387 + 0.836224i \(0.684758\pi\)
\(312\) 0 0
\(313\) −558.820 −1.78537 −0.892684 0.450683i \(-0.851181\pi\)
−0.892684 + 0.450683i \(0.851181\pi\)
\(314\) − 390.902i − 1.24491i
\(315\) 0 0
\(316\) 185.880 0.588229
\(317\) − 434.973i − 1.37215i −0.727529 0.686077i \(-0.759331\pi\)
0.727529 0.686077i \(-0.240669\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 7.55451i − 0.0236079i
\(321\) 0 0
\(322\) −200.374 −0.622279
\(323\) − 162.232i − 0.502267i
\(324\) 0 0
\(325\) −3.03779 −0.00934704
\(326\) 747.246i 2.29216i
\(327\) 0 0
\(328\) 29.0745 0.0886419
\(329\) − 187.750i − 0.570670i
\(330\) 0 0
\(331\) 185.292 0.559795 0.279897 0.960030i \(-0.409700\pi\)
0.279897 + 0.960030i \(0.409700\pi\)
\(332\) 140.823i 0.424166i
\(333\) 0 0
\(334\) −225.412 −0.674888
\(335\) 139.149i 0.415370i
\(336\) 0 0
\(337\) 366.174 1.08657 0.543285 0.839549i \(-0.317180\pi\)
0.543285 + 0.839549i \(0.317180\pi\)
\(338\) 417.461i 1.23509i
\(339\) 0 0
\(340\) 65.8703 0.193736
\(341\) 0 0
\(342\) 0 0
\(343\) −287.546 −0.838327
\(344\) − 220.121i − 0.639886i
\(345\) 0 0
\(346\) −484.159 −1.39930
\(347\) 232.794i 0.670876i 0.942062 + 0.335438i \(0.108884\pi\)
−0.942062 + 0.335438i \(0.891116\pi\)
\(348\) 0 0
\(349\) 190.183 0.544938 0.272469 0.962165i \(-0.412160\pi\)
0.272469 + 0.962165i \(0.412160\pi\)
\(350\) − 24.7127i − 0.0706078i
\(351\) 0 0
\(352\) 0 0
\(353\) 650.001i 1.84136i 0.390316 + 0.920681i \(0.372366\pi\)
−0.390316 + 0.920681i \(0.627634\pi\)
\(354\) 0 0
\(355\) 191.929 0.540644
\(356\) − 139.489i − 0.391824i
\(357\) 0 0
\(358\) 69.1275 0.193094
\(359\) − 318.134i − 0.886167i −0.896480 0.443083i \(-0.853885\pi\)
0.896480 0.443083i \(-0.146115\pi\)
\(360\) 0 0
\(361\) 270.473 0.749233
\(362\) − 482.350i − 1.33246i
\(363\) 0 0
\(364\) 7.24418 0.0199016
\(365\) 286.246i 0.784234i
\(366\) 0 0
\(367\) −10.0024 −0.0272546 −0.0136273 0.999907i \(-0.504338\pi\)
−0.0136273 + 0.999907i \(0.504338\pi\)
\(368\) 487.715i 1.32531i
\(369\) 0 0
\(370\) 767.004 2.07299
\(371\) − 222.266i − 0.599099i
\(372\) 0 0
\(373\) −373.434 −1.00116 −0.500582 0.865689i \(-0.666881\pi\)
−0.500582 + 0.865689i \(0.666881\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −257.809 −0.685661
\(377\) 51.3010i 0.136077i
\(378\) 0 0
\(379\) −273.614 −0.721937 −0.360969 0.932578i \(-0.617554\pi\)
−0.360969 + 0.932578i \(0.617554\pi\)
\(380\) 256.394i 0.674720i
\(381\) 0 0
\(382\) −738.692 −1.93375
\(383\) 129.063i 0.336978i 0.985703 + 0.168489i \(0.0538888\pi\)
−0.985703 + 0.168489i \(0.946111\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 491.783i − 1.27405i
\(387\) 0 0
\(388\) 188.886 0.486820
\(389\) − 311.684i − 0.801244i −0.916243 0.400622i \(-0.868794\pi\)
0.916243 0.400622i \(-0.131206\pi\)
\(390\) 0 0
\(391\) 157.677 0.403266
\(392\) 172.719i 0.440611i
\(393\) 0 0
\(394\) 619.642 1.57270
\(395\) − 400.576i − 1.01412i
\(396\) 0 0
\(397\) 449.573 1.13243 0.566213 0.824259i \(-0.308408\pi\)
0.566213 + 0.824259i \(0.308408\pi\)
\(398\) − 252.994i − 0.635662i
\(399\) 0 0
\(400\) −60.1515 −0.150379
\(401\) 348.583i 0.869284i 0.900603 + 0.434642i \(0.143125\pi\)
−0.900603 + 0.434642i \(0.856875\pi\)
\(402\) 0 0
\(403\) −48.8161 −0.121132
\(404\) 250.122i 0.619115i
\(405\) 0 0
\(406\) −417.340 −1.02793
\(407\) 0 0
\(408\) 0 0
\(409\) 532.578 1.30215 0.651073 0.759015i \(-0.274319\pi\)
0.651073 + 0.759015i \(0.274319\pi\)
\(410\) 74.7404i 0.182294i
\(411\) 0 0
\(412\) −62.3106 −0.151239
\(413\) − 1.01388i − 0.00245492i
\(414\) 0 0
\(415\) 303.476 0.731268
\(416\) − 31.7604i − 0.0763472i
\(417\) 0 0
\(418\) 0 0
\(419\) − 267.800i − 0.639141i −0.947563 0.319570i \(-0.896461\pi\)
0.947563 0.319570i \(-0.103539\pi\)
\(420\) 0 0
\(421\) 371.455 0.882315 0.441158 0.897430i \(-0.354568\pi\)
0.441158 + 0.897430i \(0.354568\pi\)
\(422\) − 94.2990i − 0.223457i
\(423\) 0 0
\(424\) −305.203 −0.719819
\(425\) 19.4468i 0.0457572i
\(426\) 0 0
\(427\) 286.677 0.671375
\(428\) − 259.974i − 0.607416i
\(429\) 0 0
\(430\) 565.853 1.31594
\(431\) 285.915i 0.663376i 0.943389 + 0.331688i \(0.107618\pi\)
−0.943389 + 0.331688i \(0.892382\pi\)
\(432\) 0 0
\(433\) 228.990 0.528845 0.264423 0.964407i \(-0.414819\pi\)
0.264423 + 0.964407i \(0.414819\pi\)
\(434\) − 397.124i − 0.915033i
\(435\) 0 0
\(436\) −167.017 −0.383067
\(437\) 613.743i 1.40445i
\(438\) 0 0
\(439\) 815.871 1.85848 0.929238 0.369481i \(-0.120464\pi\)
0.929238 + 0.369481i \(0.120464\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −16.1800 −0.0366063
\(443\) 434.432i 0.980660i 0.871537 + 0.490330i \(0.163124\pi\)
−0.871537 + 0.490330i \(0.836876\pi\)
\(444\) 0 0
\(445\) −300.602 −0.675511
\(446\) − 296.984i − 0.665883i
\(447\) 0 0
\(448\) −5.31860 −0.0118719
\(449\) − 73.1022i − 0.162811i −0.996681 0.0814056i \(-0.974059\pi\)
0.996681 0.0814056i \(-0.0259409\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 183.780i 0.406592i
\(453\) 0 0
\(454\) 580.239 1.27806
\(455\) − 15.6114i − 0.0343107i
\(456\) 0 0
\(457\) 49.1411 0.107530 0.0537649 0.998554i \(-0.482878\pi\)
0.0537649 + 0.998554i \(0.482878\pi\)
\(458\) 423.227i 0.924077i
\(459\) 0 0
\(460\) −249.195 −0.541728
\(461\) 657.645i 1.42656i 0.700878 + 0.713281i \(0.252792\pi\)
−0.700878 + 0.713281i \(0.747208\pi\)
\(462\) 0 0
\(463\) −195.703 −0.422685 −0.211343 0.977412i \(-0.567784\pi\)
−0.211343 + 0.977412i \(0.567784\pi\)
\(464\) 1015.82i 2.18926i
\(465\) 0 0
\(466\) 371.295 0.796770
\(467\) 13.5937i 0.0291086i 0.999894 + 0.0145543i \(0.00463295\pi\)
−0.999894 + 0.0145543i \(0.995367\pi\)
\(468\) 0 0
\(469\) 97.9651 0.208881
\(470\) − 662.735i − 1.41007i
\(471\) 0 0
\(472\) −1.39221 −0.00294959
\(473\) 0 0
\(474\) 0 0
\(475\) −75.6949 −0.159358
\(476\) − 46.3747i − 0.0974258i
\(477\) 0 0
\(478\) 665.113 1.39145
\(479\) 369.614i 0.771637i 0.922575 + 0.385818i \(0.126081\pi\)
−0.922575 + 0.385818i \(0.873919\pi\)
\(480\) 0 0
\(481\) −66.3782 −0.138000
\(482\) 502.838i 1.04323i
\(483\) 0 0
\(484\) 0 0
\(485\) − 407.053i − 0.839285i
\(486\) 0 0
\(487\) −485.893 −0.997727 −0.498863 0.866681i \(-0.666249\pi\)
−0.498863 + 0.866681i \(0.666249\pi\)
\(488\) − 393.650i − 0.806659i
\(489\) 0 0
\(490\) −444.001 −0.906124
\(491\) − 538.550i − 1.09684i −0.836202 0.548421i \(-0.815229\pi\)
0.836202 0.548421i \(-0.184771\pi\)
\(492\) 0 0
\(493\) 328.411 0.666148
\(494\) − 62.9790i − 0.127488i
\(495\) 0 0
\(496\) −966.612 −1.94881
\(497\) − 135.124i − 0.271878i
\(498\) 0 0
\(499\) −949.252 −1.90231 −0.951154 0.308716i \(-0.900101\pi\)
−0.951154 + 0.308716i \(0.900101\pi\)
\(500\) − 285.810i − 0.571620i
\(501\) 0 0
\(502\) 420.982 0.838610
\(503\) 207.815i 0.413151i 0.978431 + 0.206575i \(0.0662319\pi\)
−0.978431 + 0.206575i \(0.933768\pi\)
\(504\) 0 0
\(505\) 539.019 1.06736
\(506\) 0 0
\(507\) 0 0
\(508\) −217.299 −0.427754
\(509\) 473.501i 0.930258i 0.885243 + 0.465129i \(0.153992\pi\)
−0.885243 + 0.465129i \(0.846008\pi\)
\(510\) 0 0
\(511\) 201.526 0.394375
\(512\) − 266.802i − 0.521098i
\(513\) 0 0
\(514\) 518.535 1.00882
\(515\) 134.281i 0.260739i
\(516\) 0 0
\(517\) 0 0
\(518\) − 539.994i − 1.04246i
\(519\) 0 0
\(520\) −21.4367 −0.0412243
\(521\) 756.599i 1.45221i 0.687586 + 0.726103i \(0.258670\pi\)
−0.687586 + 0.726103i \(0.741330\pi\)
\(522\) 0 0
\(523\) 782.107 1.49543 0.747713 0.664022i \(-0.231152\pi\)
0.747713 + 0.664022i \(0.231152\pi\)
\(524\) − 359.329i − 0.685741i
\(525\) 0 0
\(526\) −292.134 −0.555387
\(527\) 312.503i 0.592985i
\(528\) 0 0
\(529\) −67.5098 −0.127618
\(530\) − 784.570i − 1.48032i
\(531\) 0 0
\(532\) 180.509 0.339302
\(533\) − 6.46819i − 0.0121354i
\(534\) 0 0
\(535\) −560.249 −1.04719
\(536\) − 134.520i − 0.250971i
\(537\) 0 0
\(538\) −1162.60 −2.16097
\(539\) 0 0
\(540\) 0 0
\(541\) 55.2149 0.102061 0.0510304 0.998697i \(-0.483749\pi\)
0.0510304 + 0.998697i \(0.483749\pi\)
\(542\) − 289.005i − 0.533219i
\(543\) 0 0
\(544\) −203.319 −0.373748
\(545\) 359.926i 0.660414i
\(546\) 0 0
\(547\) 358.392 0.655195 0.327597 0.944817i \(-0.393761\pi\)
0.327597 + 0.944817i \(0.393761\pi\)
\(548\) 51.3427i 0.0936911i
\(549\) 0 0
\(550\) 0 0
\(551\) 1278.31i 2.31998i
\(552\) 0 0
\(553\) −282.017 −0.509977
\(554\) 721.916i 1.30310i
\(555\) 0 0
\(556\) 593.509 1.06746
\(557\) 644.969i 1.15793i 0.815351 + 0.578967i \(0.196544\pi\)
−0.815351 + 0.578967i \(0.803456\pi\)
\(558\) 0 0
\(559\) −48.9701 −0.0876031
\(560\) − 309.122i − 0.552004i
\(561\) 0 0
\(562\) −66.5548 −0.118425
\(563\) − 730.584i − 1.29766i −0.760932 0.648831i \(-0.775258\pi\)
0.760932 0.648831i \(-0.224742\pi\)
\(564\) 0 0
\(565\) 396.049 0.700972
\(566\) 58.9361i 0.104127i
\(567\) 0 0
\(568\) −185.544 −0.326662
\(569\) − 1037.85i − 1.82399i −0.410200 0.911996i \(-0.634541\pi\)
0.410200 0.911996i \(-0.365459\pi\)
\(570\) 0 0
\(571\) 161.245 0.282391 0.141196 0.989982i \(-0.454905\pi\)
0.141196 + 0.989982i \(0.454905\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 52.6195 0.0916716
\(575\) − 73.5695i − 0.127947i
\(576\) 0 0
\(577\) 988.690 1.71350 0.856750 0.515732i \(-0.172480\pi\)
0.856750 + 0.515732i \(0.172480\pi\)
\(578\) − 614.626i − 1.06337i
\(579\) 0 0
\(580\) −519.024 −0.894869
\(581\) − 213.656i − 0.367739i
\(582\) 0 0
\(583\) 0 0
\(584\) − 276.724i − 0.473842i
\(585\) 0 0
\(586\) −943.931 −1.61080
\(587\) − 517.799i − 0.882110i −0.897480 0.441055i \(-0.854604\pi\)
0.897480 0.441055i \(-0.145396\pi\)
\(588\) 0 0
\(589\) −1216.39 −2.06517
\(590\) − 3.57887i − 0.00606589i
\(591\) 0 0
\(592\) −1314.36 −2.22020
\(593\) 396.829i 0.669188i 0.942362 + 0.334594i \(0.108599\pi\)
−0.942362 + 0.334594i \(0.891401\pi\)
\(594\) 0 0
\(595\) −99.9383 −0.167964
\(596\) 75.6271i 0.126891i
\(597\) 0 0
\(598\) 61.2107 0.102359
\(599\) − 865.942i − 1.44565i −0.691034 0.722823i \(-0.742844\pi\)
0.691034 0.722823i \(-0.257156\pi\)
\(600\) 0 0
\(601\) −412.367 −0.686134 −0.343067 0.939311i \(-0.611466\pi\)
−0.343067 + 0.939311i \(0.611466\pi\)
\(602\) − 398.378i − 0.661757i
\(603\) 0 0
\(604\) 269.789 0.446671
\(605\) 0 0
\(606\) 0 0
\(607\) 248.291 0.409045 0.204523 0.978862i \(-0.434436\pi\)
0.204523 + 0.978862i \(0.434436\pi\)
\(608\) − 791.399i − 1.30164i
\(609\) 0 0
\(610\) 1011.93 1.65891
\(611\) 57.3545i 0.0938699i
\(612\) 0 0
\(613\) −59.7914 −0.0975390 −0.0487695 0.998810i \(-0.515530\pi\)
−0.0487695 + 0.998810i \(0.515530\pi\)
\(614\) 102.410i 0.166792i
\(615\) 0 0
\(616\) 0 0
\(617\) 615.297i 0.997240i 0.866821 + 0.498620i \(0.166160\pi\)
−0.866821 + 0.498620i \(0.833840\pi\)
\(618\) 0 0
\(619\) −308.597 −0.498541 −0.249270 0.968434i \(-0.580191\pi\)
−0.249270 + 0.968434i \(0.580191\pi\)
\(620\) − 493.883i − 0.796586i
\(621\) 0 0
\(622\) 1292.60 2.07813
\(623\) 211.633i 0.339700i
\(624\) 0 0
\(625\) −540.621 −0.864993
\(626\) 1388.74i 2.21844i
\(627\) 0 0
\(628\) −342.260 −0.545001
\(629\) 424.930i 0.675564i
\(630\) 0 0
\(631\) −473.277 −0.750042 −0.375021 0.927016i \(-0.622365\pi\)
−0.375021 + 0.927016i \(0.622365\pi\)
\(632\) 387.251i 0.612739i
\(633\) 0 0
\(634\) −1080.97 −1.70500
\(635\) 468.283i 0.737454i
\(636\) 0 0
\(637\) 38.4248 0.0603214
\(638\) 0 0
\(639\) 0 0
\(640\) −609.476 −0.952307
\(641\) − 280.129i − 0.437019i −0.975835 0.218509i \(-0.929881\pi\)
0.975835 0.218509i \(-0.0701194\pi\)
\(642\) 0 0
\(643\) −1239.37 −1.92748 −0.963740 0.266843i \(-0.914019\pi\)
−0.963740 + 0.266843i \(0.914019\pi\)
\(644\) 175.441i 0.272423i
\(645\) 0 0
\(646\) −403.169 −0.624101
\(647\) − 19.9618i − 0.0308529i −0.999881 0.0154264i \(-0.995089\pi\)
0.999881 0.0154264i \(-0.00491059\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 7.54931i 0.0116143i
\(651\) 0 0
\(652\) 654.263 1.00347
\(653\) 420.634i 0.644156i 0.946713 + 0.322078i \(0.104381\pi\)
−0.946713 + 0.322078i \(0.895619\pi\)
\(654\) 0 0
\(655\) −774.360 −1.18223
\(656\) − 128.077i − 0.195240i
\(657\) 0 0
\(658\) −466.585 −0.709097
\(659\) − 380.713i − 0.577714i −0.957372 0.288857i \(-0.906725\pi\)
0.957372 0.288857i \(-0.0932751\pi\)
\(660\) 0 0
\(661\) 339.630 0.513812 0.256906 0.966436i \(-0.417297\pi\)
0.256906 + 0.966436i \(0.417297\pi\)
\(662\) − 460.476i − 0.695583i
\(663\) 0 0
\(664\) −293.381 −0.441839
\(665\) − 389.000i − 0.584963i
\(666\) 0 0
\(667\) −1242.41 −1.86269
\(668\) 197.364i 0.295455i
\(669\) 0 0
\(670\) 345.804 0.516126
\(671\) 0 0
\(672\) 0 0
\(673\) 219.920 0.326775 0.163388 0.986562i \(-0.447758\pi\)
0.163388 + 0.986562i \(0.447758\pi\)
\(674\) − 909.992i − 1.35014i
\(675\) 0 0
\(676\) 365.515 0.540702
\(677\) 336.004i 0.496313i 0.968720 + 0.248157i \(0.0798248\pi\)
−0.968720 + 0.248157i \(0.920175\pi\)
\(678\) 0 0
\(679\) −286.578 −0.422058
\(680\) 137.230i 0.201809i
\(681\) 0 0
\(682\) 0 0
\(683\) − 231.915i − 0.339553i −0.985483 0.169777i \(-0.945695\pi\)
0.985483 0.169777i \(-0.0543046\pi\)
\(684\) 0 0
\(685\) 110.645 0.161525
\(686\) 714.592i 1.04168i
\(687\) 0 0
\(688\) −969.662 −1.40939
\(689\) 67.8983i 0.0985462i
\(690\) 0 0
\(691\) 404.654 0.585606 0.292803 0.956173i \(-0.405412\pi\)
0.292803 + 0.956173i \(0.405412\pi\)
\(692\) 423.914i 0.612592i
\(693\) 0 0
\(694\) 578.525 0.833609
\(695\) − 1279.02i − 1.84032i
\(696\) 0 0
\(697\) −41.4071 −0.0594075
\(698\) − 472.632i − 0.677123i
\(699\) 0 0
\(700\) −21.6377 −0.0309109
\(701\) − 144.543i − 0.206196i −0.994671 0.103098i \(-0.967125\pi\)
0.994671 0.103098i \(-0.0328755\pi\)
\(702\) 0 0
\(703\) −1654.00 −2.35277
\(704\) 0 0
\(705\) 0 0
\(706\) 1615.34 2.28802
\(707\) − 379.485i − 0.536755i
\(708\) 0 0
\(709\) 1033.34 1.45746 0.728730 0.684802i \(-0.240111\pi\)
0.728730 + 0.684802i \(0.240111\pi\)
\(710\) − 476.969i − 0.671787i
\(711\) 0 0
\(712\) 290.603 0.408150
\(713\) − 1182.23i − 1.65811i
\(714\) 0 0
\(715\) 0 0
\(716\) − 60.5258i − 0.0845332i
\(717\) 0 0
\(718\) −790.606 −1.10112
\(719\) 1286.83i 1.78974i 0.446324 + 0.894872i \(0.352733\pi\)
−0.446324 + 0.894872i \(0.647267\pi\)
\(720\) 0 0
\(721\) 94.5376 0.131120
\(722\) − 672.162i − 0.930973i
\(723\) 0 0
\(724\) −422.329 −0.583328
\(725\) − 153.231i − 0.211353i
\(726\) 0 0
\(727\) −835.384 −1.14908 −0.574542 0.818475i \(-0.694820\pi\)
−0.574542 + 0.818475i \(0.694820\pi\)
\(728\) 15.0921i 0.0207308i
\(729\) 0 0
\(730\) 711.359 0.974465
\(731\) 313.489i 0.428850i
\(732\) 0 0
\(733\) −1300.03 −1.77358 −0.886790 0.462173i \(-0.847070\pi\)
−0.886790 + 0.462173i \(0.847070\pi\)
\(734\) 24.8574i 0.0338657i
\(735\) 0 0
\(736\) 769.178 1.04508
\(737\) 0 0
\(738\) 0 0
\(739\) −993.426 −1.34428 −0.672142 0.740422i \(-0.734626\pi\)
−0.672142 + 0.740422i \(0.734626\pi\)
\(740\) − 671.563i − 0.907518i
\(741\) 0 0
\(742\) −552.361 −0.744421
\(743\) 762.477i 1.02621i 0.858325 + 0.513107i \(0.171506\pi\)
−0.858325 + 0.513107i \(0.828494\pi\)
\(744\) 0 0
\(745\) 162.978 0.218762
\(746\) 928.035i 1.24401i
\(747\) 0 0
\(748\) 0 0
\(749\) 394.432i 0.526612i
\(750\) 0 0
\(751\) −647.654 −0.862389 −0.431194 0.902259i \(-0.641908\pi\)
−0.431194 + 0.902259i \(0.641908\pi\)
\(752\) 1135.68i 1.51022i
\(753\) 0 0
\(754\) 127.490 0.169085
\(755\) − 581.401i − 0.770068i
\(756\) 0 0
\(757\) −43.6130 −0.0576130 −0.0288065 0.999585i \(-0.509171\pi\)
−0.0288065 + 0.999585i \(0.509171\pi\)
\(758\) 679.969i 0.897057i
\(759\) 0 0
\(760\) −534.154 −0.702834
\(761\) 130.629i 0.171655i 0.996310 + 0.0858274i \(0.0273534\pi\)
−0.996310 + 0.0858274i \(0.972647\pi\)
\(762\) 0 0
\(763\) 253.398 0.332108
\(764\) 646.774i 0.846563i
\(765\) 0 0
\(766\) 320.739 0.418719
\(767\) 0.309723i 0 0.000403811i
\(768\) 0 0
\(769\) 1261.92 1.64099 0.820495 0.571654i \(-0.193698\pi\)
0.820495 + 0.571654i \(0.193698\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −430.588 −0.557757
\(773\) − 436.493i − 0.564674i −0.959315 0.282337i \(-0.908890\pi\)
0.959315 0.282337i \(-0.0911097\pi\)
\(774\) 0 0
\(775\) 145.809 0.188140
\(776\) 393.513i 0.507104i
\(777\) 0 0
\(778\) −774.577 −0.995600
\(779\) − 161.173i − 0.206897i
\(780\) 0 0
\(781\) 0 0
\(782\) − 391.849i − 0.501086i
\(783\) 0 0
\(784\) 760.853 0.970475
\(785\) 737.578i 0.939590i
\(786\) 0 0
\(787\) −38.8895 −0.0494148 −0.0247074 0.999695i \(-0.507865\pi\)
−0.0247074 + 0.999695i \(0.507865\pi\)
\(788\) − 542.538i − 0.688500i
\(789\) 0 0
\(790\) −995.485 −1.26011
\(791\) − 278.830i − 0.352504i
\(792\) 0 0
\(793\) −87.5749 −0.110435
\(794\) − 1117.25i − 1.40712i
\(795\) 0 0
\(796\) −221.513 −0.278282
\(797\) 864.722i 1.08497i 0.840065 + 0.542486i \(0.182517\pi\)
−0.840065 + 0.542486i \(0.817483\pi\)
\(798\) 0 0
\(799\) 367.163 0.459528
\(800\) 94.8653i 0.118582i
\(801\) 0 0
\(802\) 866.276 1.08014
\(803\) 0 0
\(804\) 0 0
\(805\) 378.078 0.469662
\(806\) 121.315i 0.150514i
\(807\) 0 0
\(808\) −521.089 −0.644912
\(809\) − 267.814i − 0.331043i −0.986206 0.165521i \(-0.947069\pi\)
0.986206 0.165521i \(-0.0529307\pi\)
\(810\) 0 0
\(811\) −62.9645 −0.0776381 −0.0388190 0.999246i \(-0.512360\pi\)
−0.0388190 + 0.999246i \(0.512360\pi\)
\(812\) 365.409i 0.450011i
\(813\) 0 0
\(814\) 0 0
\(815\) − 1409.95i − 1.73000i
\(816\) 0 0
\(817\) −1220.23 −1.49354
\(818\) − 1323.53i − 1.61800i
\(819\) 0 0
\(820\) 65.4402 0.0798051
\(821\) 205.138i 0.249863i 0.992165 + 0.124932i \(0.0398712\pi\)
−0.992165 + 0.124932i \(0.960129\pi\)
\(822\) 0 0
\(823\) 92.1457 0.111963 0.0559816 0.998432i \(-0.482171\pi\)
0.0559816 + 0.998432i \(0.482171\pi\)
\(824\) − 129.814i − 0.157541i
\(825\) 0 0
\(826\) −2.51963 −0.00305041
\(827\) 152.560i 0.184475i 0.995737 + 0.0922373i \(0.0294018\pi\)
−0.995737 + 0.0922373i \(0.970598\pi\)
\(828\) 0 0
\(829\) −527.763 −0.636626 −0.318313 0.947986i \(-0.603116\pi\)
−0.318313 + 0.947986i \(0.603116\pi\)
\(830\) − 754.180i − 0.908651i
\(831\) 0 0
\(832\) 1.62474 0.00195281
\(833\) − 245.982i − 0.295296i
\(834\) 0 0
\(835\) 425.322 0.509368
\(836\) 0 0
\(837\) 0 0
\(838\) −665.519 −0.794176
\(839\) − 548.065i − 0.653237i −0.945156 0.326618i \(-0.894091\pi\)
0.945156 0.326618i \(-0.105909\pi\)
\(840\) 0 0
\(841\) −1746.71 −2.07694
\(842\) − 923.116i − 1.09634i
\(843\) 0 0
\(844\) −82.5650 −0.0978259
\(845\) − 787.691i − 0.932178i
\(846\) 0 0
\(847\) 0 0
\(848\) 1344.46i 1.58545i
\(849\) 0 0
\(850\) 48.3280 0.0568565
\(851\) − 1607.56i − 1.88902i
\(852\) 0 0
\(853\) 88.0263 0.103196 0.0515980 0.998668i \(-0.483569\pi\)
0.0515980 + 0.998668i \(0.483569\pi\)
\(854\) − 712.432i − 0.834230i
\(855\) 0 0
\(856\) 541.613 0.632725
\(857\) 1341.27i 1.56508i 0.622601 + 0.782539i \(0.286076\pi\)
−0.622601 + 0.782539i \(0.713924\pi\)
\(858\) 0 0
\(859\) 1230.21 1.43214 0.716069 0.698030i \(-0.245940\pi\)
0.716069 + 0.698030i \(0.245940\pi\)
\(860\) − 495.442i − 0.576095i
\(861\) 0 0
\(862\) 710.538 0.824290
\(863\) 218.670i 0.253383i 0.991942 + 0.126692i \(0.0404358\pi\)
−0.991942 + 0.126692i \(0.959564\pi\)
\(864\) 0 0
\(865\) 913.542 1.05612
\(866\) − 569.071i − 0.657126i
\(867\) 0 0
\(868\) −347.709 −0.400586
\(869\) 0 0
\(870\) 0 0
\(871\) −29.9266 −0.0343589
\(872\) − 347.953i − 0.399029i
\(873\) 0 0
\(874\) 1525.23 1.74512
\(875\) 433.631i 0.495578i
\(876\) 0 0
\(877\) −547.663 −0.624474 −0.312237 0.950004i \(-0.601078\pi\)
−0.312237 + 0.950004i \(0.601078\pi\)
\(878\) − 2027.55i − 2.30928i
\(879\) 0 0
\(880\) 0 0
\(881\) − 235.271i − 0.267050i −0.991045 0.133525i \(-0.957370\pi\)
0.991045 0.133525i \(-0.0426297\pi\)
\(882\) 0 0
\(883\) −246.902 −0.279617 −0.139808 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(884\) 14.1667i 0.0160256i
\(885\) 0 0
\(886\) 1079.62 1.21854
\(887\) − 667.037i − 0.752014i −0.926617 0.376007i \(-0.877297\pi\)
0.926617 0.376007i \(-0.122703\pi\)
\(888\) 0 0
\(889\) 329.686 0.370850
\(890\) 747.038i 0.839368i
\(891\) 0 0
\(892\) −260.029 −0.291513
\(893\) 1429.15i 1.60039i
\(894\) 0 0
\(895\) −130.434 −0.145736
\(896\) 429.090i 0.478895i
\(897\) 0 0
\(898\) −181.669 −0.202304
\(899\) − 2462.36i − 2.73900i
\(900\) 0 0
\(901\) 434.661 0.482420
\(902\) 0 0
\(903\) 0 0
\(904\) −382.875 −0.423534
\(905\) 910.128i 1.00567i
\(906\) 0 0
\(907\) −115.024 −0.126819 −0.0634093 0.997988i \(-0.520197\pi\)
−0.0634093 + 0.997988i \(0.520197\pi\)
\(908\) − 508.038i − 0.559513i
\(909\) 0 0
\(910\) −38.7963 −0.0426333
\(911\) 545.185i 0.598447i 0.954183 + 0.299223i \(0.0967276\pi\)
−0.954183 + 0.299223i \(0.903272\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 122.122i − 0.133613i
\(915\) 0 0
\(916\) 370.564 0.404545
\(917\) 545.173i 0.594518i
\(918\) 0 0
\(919\) 116.321 0.126573 0.0632866 0.997995i \(-0.479842\pi\)
0.0632866 + 0.997995i \(0.479842\pi\)
\(920\) − 519.156i − 0.564300i
\(921\) 0 0
\(922\) 1634.34 1.77260
\(923\) 41.2779i 0.0447214i
\(924\) 0 0
\(925\) 198.265 0.214341
\(926\) 486.349i 0.525215i
\(927\) 0 0
\(928\) 1602.05 1.72635
\(929\) 25.3182i 0.0272531i 0.999907 + 0.0136266i \(0.00433760\pi\)
−0.999907 + 0.0136266i \(0.995662\pi\)
\(930\) 0 0
\(931\) 957.459 1.02842
\(932\) − 325.093i − 0.348812i
\(933\) 0 0
\(934\) 33.7823 0.0361695
\(935\) 0 0
\(936\) 0 0
\(937\) −1360.70 −1.45219 −0.726093 0.687597i \(-0.758666\pi\)
−0.726093 + 0.687597i \(0.758666\pi\)
\(938\) − 243.457i − 0.259549i
\(939\) 0 0
\(940\) −580.269 −0.617307
\(941\) − 908.864i − 0.965850i −0.875662 0.482925i \(-0.839574\pi\)
0.875662 0.482925i \(-0.160426\pi\)
\(942\) 0 0
\(943\) 156.647 0.166116
\(944\) 6.13286i 0.00649668i
\(945\) 0 0
\(946\) 0 0
\(947\) − 1593.80i − 1.68300i −0.540256 0.841501i \(-0.681673\pi\)
0.540256 0.841501i \(-0.318327\pi\)
\(948\) 0 0
\(949\) −61.5625 −0.0648709
\(950\) 188.112i 0.198013i
\(951\) 0 0
\(952\) 96.6140 0.101485
\(953\) 163.175i 0.171222i 0.996329 + 0.0856110i \(0.0272842\pi\)
−0.996329 + 0.0856110i \(0.972716\pi\)
\(954\) 0 0
\(955\) 1393.81 1.45949
\(956\) − 582.351i − 0.609154i
\(957\) 0 0
\(958\) 918.541 0.958811
\(959\) − 77.8971i − 0.0812274i
\(960\) 0 0
\(961\) 1382.09 1.43818
\(962\) 164.959i 0.171475i
\(963\) 0 0
\(964\) 440.269 0.456710
\(965\) 927.926i 0.961582i
\(966\) 0 0
\(967\) −751.225 −0.776862 −0.388431 0.921478i \(-0.626983\pi\)
−0.388431 + 0.921478i \(0.626983\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −1011.58 −1.04287
\(971\) 134.041i 0.138044i 0.997615 + 0.0690221i \(0.0219879\pi\)
−0.997615 + 0.0690221i \(0.978012\pi\)
\(972\) 0 0
\(973\) −900.470 −0.925458
\(974\) 1207.51i 1.23974i
\(975\) 0 0
\(976\) −1734.08 −1.77672
\(977\) 1081.18i 1.10664i 0.832970 + 0.553319i \(0.186639\pi\)
−0.832970 + 0.553319i \(0.813361\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 388.752i 0.396686i
\(981\) 0 0
\(982\) −1338.37 −1.36290
\(983\) − 1114.11i − 1.13337i −0.823933 0.566687i \(-0.808225\pi\)
0.823933 0.566687i \(-0.191775\pi\)
\(984\) 0 0
\(985\) −1169.18 −1.18698
\(986\) − 816.146i − 0.827734i
\(987\) 0 0
\(988\) −55.1423 −0.0558121
\(989\) − 1185.96i − 1.19916i
\(990\) 0 0
\(991\) −1199.41 −1.21031 −0.605153 0.796109i \(-0.706888\pi\)
−0.605153 + 0.796109i \(0.706888\pi\)
\(992\) 1524.45i 1.53674i
\(993\) 0 0
\(994\) −335.800 −0.337827
\(995\) 477.364i 0.479763i
\(996\) 0 0
\(997\) 995.763 0.998760 0.499380 0.866383i \(-0.333561\pi\)
0.499380 + 0.866383i \(0.333561\pi\)
\(998\) 2359.02i 2.36375i
\(999\) 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 1089.3.b.g.485.2 8
3.2 odd 2 inner 1089.3.b.g.485.7 8
11.10 odd 2 99.3.b.a.89.7 yes 8
33.32 even 2 99.3.b.a.89.2 8
44.43 even 2 1584.3.i.b.881.5 8
132.131 odd 2 1584.3.i.b.881.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.b.a.89.2 8 33.32 even 2
99.3.b.a.89.7 yes 8 11.10 odd 2
1089.3.b.g.485.2 8 1.1 even 1 trivial
1089.3.b.g.485.7 8 3.2 odd 2 inner
1584.3.i.b.881.4 8 132.131 odd 2
1584.3.i.b.881.5 8 44.43 even 2