Properties

Label 1089.3.b.j.485.1
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.1
Root \(-3.62039i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.j.485.16

$q$-expansion

\(f(q)\) \(=\) \(q-3.62039i q^{2} -9.10725 q^{4} -0.165198i q^{5} -10.2989 q^{7} +18.4903i q^{8} +O(q^{10})\) \(q-3.62039i q^{2} -9.10725 q^{4} -0.165198i q^{5} -10.2989 q^{7} +18.4903i q^{8} -0.598083 q^{10} +20.3554 q^{13} +37.2859i q^{14} +30.5130 q^{16} +23.0147i q^{17} +4.13897 q^{19} +1.50450i q^{20} -13.5095i q^{23} +24.9727 q^{25} -73.6945i q^{26} +93.7943 q^{28} +4.01646i q^{29} -20.5033 q^{31} -36.5080i q^{32} +83.3223 q^{34} +1.70135i q^{35} +43.5939 q^{37} -14.9847i q^{38} +3.05456 q^{40} +52.0734i q^{41} -62.5698 q^{43} -48.9097 q^{46} -38.7052i q^{47} +57.0665 q^{49} -90.4110i q^{50} -185.382 q^{52} -35.3780i q^{53} -190.428i q^{56} +14.5412 q^{58} -19.6883i q^{59} +6.42431 q^{61} +74.2301i q^{62} -10.1214 q^{64} -3.36267i q^{65} -3.98708 q^{67} -209.601i q^{68} +6.15957 q^{70} -54.7712i q^{71} +22.8087 q^{73} -157.827i q^{74} -37.6946 q^{76} +57.1563 q^{79} -5.04069i q^{80} +188.526 q^{82} -56.8060i q^{83} +3.80199 q^{85} +226.527i q^{86} -28.1602i q^{89} -209.637 q^{91} +123.034i q^{92} -140.128 q^{94} -0.683750i q^{95} +82.9323 q^{97} -206.603i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} + 8 q^{7} + O(q^{10}) \) \( 16 q - 32 q^{4} + 8 q^{7} - 24 q^{10} - 4 q^{13} + 28 q^{16} + 20 q^{19} - 44 q^{25} - 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} - 224 q^{40} - 272 q^{43} - 208 q^{46} + 348 q^{49} - 520 q^{52} - 44 q^{58} - 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} + 4 q^{73} - 1052 q^{76} - 216 q^{79} + 348 q^{82} - 416 q^{85} - 168 q^{91} - 1140 q^{94} - 44 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.62039i − 1.81020i −0.425202 0.905098i \(-0.639797\pi\)
0.425202 0.905098i \(-0.360203\pi\)
\(3\) 0 0
\(4\) −9.10725 −2.27681
\(5\) − 0.165198i − 0.0330396i −0.999864 0.0165198i \(-0.994741\pi\)
0.999864 0.0165198i \(-0.00525866\pi\)
\(6\) 0 0
\(7\) −10.2989 −1.47127 −0.735633 0.677381i \(-0.763115\pi\)
−0.735633 + 0.677381i \(0.763115\pi\)
\(8\) 18.4903i 2.31128i
\(9\) 0 0
\(10\) −0.598083 −0.0598083
\(11\) 0 0
\(12\) 0 0
\(13\) 20.3554 1.56580 0.782899 0.622148i \(-0.213740\pi\)
0.782899 + 0.622148i \(0.213740\pi\)
\(14\) 37.2859i 2.66328i
\(15\) 0 0
\(16\) 30.5130 1.90706
\(17\) 23.0147i 1.35381i 0.736072 + 0.676903i \(0.236679\pi\)
−0.736072 + 0.676903i \(0.763321\pi\)
\(18\) 0 0
\(19\) 4.13897 0.217840 0.108920 0.994051i \(-0.465261\pi\)
0.108920 + 0.994051i \(0.465261\pi\)
\(20\) 1.50450i 0.0752251i
\(21\) 0 0
\(22\) 0 0
\(23\) − 13.5095i − 0.587369i −0.955902 0.293685i \(-0.905118\pi\)
0.955902 0.293685i \(-0.0948816\pi\)
\(24\) 0 0
\(25\) 24.9727 0.998908
\(26\) − 73.6945i − 2.83440i
\(27\) 0 0
\(28\) 93.7943 3.34979
\(29\) 4.01646i 0.138499i 0.997599 + 0.0692494i \(0.0220604\pi\)
−0.997599 + 0.0692494i \(0.977940\pi\)
\(30\) 0 0
\(31\) −20.5033 −0.661397 −0.330699 0.943736i \(-0.607284\pi\)
−0.330699 + 0.943736i \(0.607284\pi\)
\(32\) − 36.5080i − 1.14088i
\(33\) 0 0
\(34\) 83.3223 2.45066
\(35\) 1.70135i 0.0486101i
\(36\) 0 0
\(37\) 43.5939 1.17821 0.589107 0.808055i \(-0.299480\pi\)
0.589107 + 0.808055i \(0.299480\pi\)
\(38\) − 14.9847i − 0.394334i
\(39\) 0 0
\(40\) 3.05456 0.0763639
\(41\) 52.0734i 1.27008i 0.772478 + 0.635042i \(0.219017\pi\)
−0.772478 + 0.635042i \(0.780983\pi\)
\(42\) 0 0
\(43\) −62.5698 −1.45511 −0.727556 0.686048i \(-0.759344\pi\)
−0.727556 + 0.686048i \(0.759344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −48.9097 −1.06325
\(47\) − 38.7052i − 0.823515i −0.911293 0.411758i \(-0.864915\pi\)
0.911293 0.411758i \(-0.135085\pi\)
\(48\) 0 0
\(49\) 57.0665 1.16462
\(50\) − 90.4110i − 1.80822i
\(51\) 0 0
\(52\) −185.382 −3.56503
\(53\) − 35.3780i − 0.667510i −0.942660 0.333755i \(-0.891684\pi\)
0.942660 0.333755i \(-0.108316\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 190.428i − 3.40051i
\(57\) 0 0
\(58\) 14.5412 0.250710
\(59\) − 19.6883i − 0.333699i −0.985982 0.166850i \(-0.946641\pi\)
0.985982 0.166850i \(-0.0533594\pi\)
\(60\) 0 0
\(61\) 6.42431 0.105317 0.0526583 0.998613i \(-0.483231\pi\)
0.0526583 + 0.998613i \(0.483231\pi\)
\(62\) 74.2301i 1.19726i
\(63\) 0 0
\(64\) −10.1214 −0.158147
\(65\) − 3.36267i − 0.0517334i
\(66\) 0 0
\(67\) −3.98708 −0.0595087 −0.0297543 0.999557i \(-0.509472\pi\)
−0.0297543 + 0.999557i \(0.509472\pi\)
\(68\) − 209.601i − 3.08236i
\(69\) 0 0
\(70\) 6.15957 0.0879938
\(71\) − 54.7712i − 0.771426i −0.922619 0.385713i \(-0.873956\pi\)
0.922619 0.385713i \(-0.126044\pi\)
\(72\) 0 0
\(73\) 22.8087 0.312448 0.156224 0.987722i \(-0.450068\pi\)
0.156224 + 0.987722i \(0.450068\pi\)
\(74\) − 157.827i − 2.13280i
\(75\) 0 0
\(76\) −37.6946 −0.495981
\(77\) 0 0
\(78\) 0 0
\(79\) 57.1563 0.723497 0.361748 0.932276i \(-0.382180\pi\)
0.361748 + 0.932276i \(0.382180\pi\)
\(80\) − 5.04069i − 0.0630086i
\(81\) 0 0
\(82\) 188.526 2.29910
\(83\) − 56.8060i − 0.684410i −0.939625 0.342205i \(-0.888826\pi\)
0.939625 0.342205i \(-0.111174\pi\)
\(84\) 0 0
\(85\) 3.80199 0.0447293
\(86\) 226.527i 2.63404i
\(87\) 0 0
\(88\) 0 0
\(89\) − 28.1602i − 0.316407i −0.987407 0.158203i \(-0.949430\pi\)
0.987407 0.158203i \(-0.0505701\pi\)
\(90\) 0 0
\(91\) −209.637 −2.30370
\(92\) 123.034i 1.33733i
\(93\) 0 0
\(94\) −140.128 −1.49073
\(95\) − 0.683750i − 0.00719737i
\(96\) 0 0
\(97\) 82.9323 0.854972 0.427486 0.904022i \(-0.359399\pi\)
0.427486 + 0.904022i \(0.359399\pi\)
\(98\) − 206.603i − 2.10819i
\(99\) 0 0
\(100\) −227.433 −2.27433
\(101\) − 153.046i − 1.51531i −0.652658 0.757653i \(-0.726346\pi\)
0.652658 0.757653i \(-0.273654\pi\)
\(102\) 0 0
\(103\) 105.119 1.02057 0.510287 0.860004i \(-0.329539\pi\)
0.510287 + 0.860004i \(0.329539\pi\)
\(104\) 376.376i 3.61900i
\(105\) 0 0
\(106\) −128.082 −1.20832
\(107\) − 112.763i − 1.05386i −0.849910 0.526929i \(-0.823344\pi\)
0.849910 0.526929i \(-0.176656\pi\)
\(108\) 0 0
\(109\) 96.1530 0.882137 0.441069 0.897473i \(-0.354600\pi\)
0.441069 + 0.897473i \(0.354600\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −314.249 −2.80579
\(113\) 104.101i 0.921249i 0.887595 + 0.460625i \(0.152375\pi\)
−0.887595 + 0.460625i \(0.847625\pi\)
\(114\) 0 0
\(115\) −2.23175 −0.0194065
\(116\) − 36.5789i − 0.315336i
\(117\) 0 0
\(118\) −71.2792 −0.604061
\(119\) − 237.025i − 1.99181i
\(120\) 0 0
\(121\) 0 0
\(122\) − 23.2585i − 0.190644i
\(123\) 0 0
\(124\) 186.729 1.50588
\(125\) − 8.25540i − 0.0660432i
\(126\) 0 0
\(127\) 8.05817 0.0634502 0.0317251 0.999497i \(-0.489900\pi\)
0.0317251 + 0.999497i \(0.489900\pi\)
\(128\) − 109.388i − 0.854597i
\(129\) 0 0
\(130\) −12.1742 −0.0936477
\(131\) 240.588i 1.83655i 0.395947 + 0.918273i \(0.370416\pi\)
−0.395947 + 0.918273i \(0.629584\pi\)
\(132\) 0 0
\(133\) −42.6266 −0.320501
\(134\) 14.4348i 0.107722i
\(135\) 0 0
\(136\) −425.548 −3.12903
\(137\) 18.8680i 0.137723i 0.997626 + 0.0688614i \(0.0219366\pi\)
−0.997626 + 0.0688614i \(0.978063\pi\)
\(138\) 0 0
\(139\) 33.4773 0.240844 0.120422 0.992723i \(-0.461575\pi\)
0.120422 + 0.992723i \(0.461575\pi\)
\(140\) − 15.4946i − 0.110676i
\(141\) 0 0
\(142\) −198.293 −1.39643
\(143\) 0 0
\(144\) 0 0
\(145\) 0.663513 0.00457595
\(146\) − 82.5765i − 0.565593i
\(147\) 0 0
\(148\) −397.020 −2.68257
\(149\) 103.421i 0.694098i 0.937847 + 0.347049i \(0.112816\pi\)
−0.937847 + 0.347049i \(0.887184\pi\)
\(150\) 0 0
\(151\) 286.382 1.89657 0.948283 0.317425i \(-0.102818\pi\)
0.948283 + 0.317425i \(0.102818\pi\)
\(152\) 76.5305i 0.503490i
\(153\) 0 0
\(154\) 0 0
\(155\) 3.38711i 0.0218523i
\(156\) 0 0
\(157\) 261.304 1.66436 0.832179 0.554508i \(-0.187093\pi\)
0.832179 + 0.554508i \(0.187093\pi\)
\(158\) − 206.928i − 1.30967i
\(159\) 0 0
\(160\) −6.03106 −0.0376941
\(161\) 139.132i 0.864176i
\(162\) 0 0
\(163\) 192.496 1.18096 0.590480 0.807052i \(-0.298938\pi\)
0.590480 + 0.807052i \(0.298938\pi\)
\(164\) − 474.246i − 2.89174i
\(165\) 0 0
\(166\) −205.660 −1.23892
\(167\) 211.194i 1.26463i 0.774710 + 0.632316i \(0.217896\pi\)
−0.774710 + 0.632316i \(0.782104\pi\)
\(168\) 0 0
\(169\) 245.342 1.45173
\(170\) − 13.7647i − 0.0809688i
\(171\) 0 0
\(172\) 569.839 3.31302
\(173\) 103.773i 0.599844i 0.953964 + 0.299922i \(0.0969607\pi\)
−0.953964 + 0.299922i \(0.903039\pi\)
\(174\) 0 0
\(175\) −257.190 −1.46966
\(176\) 0 0
\(177\) 0 0
\(178\) −101.951 −0.572758
\(179\) − 11.0578i − 0.0617755i −0.999523 0.0308877i \(-0.990167\pi\)
0.999523 0.0308877i \(-0.00983344\pi\)
\(180\) 0 0
\(181\) 124.377 0.687164 0.343582 0.939123i \(-0.388360\pi\)
0.343582 + 0.939123i \(0.388360\pi\)
\(182\) 758.969i 4.17016i
\(183\) 0 0
\(184\) 249.794 1.35758
\(185\) − 7.20163i − 0.0389278i
\(186\) 0 0
\(187\) 0 0
\(188\) 352.498i 1.87499i
\(189\) 0 0
\(190\) −2.47544 −0.0130286
\(191\) − 177.560i − 0.929634i −0.885407 0.464817i \(-0.846120\pi\)
0.885407 0.464817i \(-0.153880\pi\)
\(192\) 0 0
\(193\) −243.331 −1.26078 −0.630391 0.776278i \(-0.717105\pi\)
−0.630391 + 0.776278i \(0.717105\pi\)
\(194\) − 300.248i − 1.54767i
\(195\) 0 0
\(196\) −519.718 −2.65162
\(197\) 215.588i 1.09435i 0.837017 + 0.547177i \(0.184298\pi\)
−0.837017 + 0.547177i \(0.815702\pi\)
\(198\) 0 0
\(199\) −44.0830 −0.221523 −0.110761 0.993847i \(-0.535329\pi\)
−0.110761 + 0.993847i \(0.535329\pi\)
\(200\) 461.752i 2.30876i
\(201\) 0 0
\(202\) −554.086 −2.74300
\(203\) − 41.3650i − 0.203768i
\(204\) 0 0
\(205\) 8.60244 0.0419631
\(206\) − 380.573i − 1.84744i
\(207\) 0 0
\(208\) 621.103 2.98607
\(209\) 0 0
\(210\) 0 0
\(211\) 264.305 1.25263 0.626315 0.779570i \(-0.284562\pi\)
0.626315 + 0.779570i \(0.284562\pi\)
\(212\) 322.197i 1.51980i
\(213\) 0 0
\(214\) −408.245 −1.90769
\(215\) 10.3364i 0.0480764i
\(216\) 0 0
\(217\) 211.161 0.973091
\(218\) − 348.112i − 1.59684i
\(219\) 0 0
\(220\) 0 0
\(221\) 468.473i 2.11979i
\(222\) 0 0
\(223\) 14.5149 0.0650890 0.0325445 0.999470i \(-0.489639\pi\)
0.0325445 + 0.999470i \(0.489639\pi\)
\(224\) 375.991i 1.67853i
\(225\) 0 0
\(226\) 376.887 1.66764
\(227\) 122.517i 0.539722i 0.962899 + 0.269861i \(0.0869778\pi\)
−0.962899 + 0.269861i \(0.913022\pi\)
\(228\) 0 0
\(229\) 3.89370 0.0170030 0.00850152 0.999964i \(-0.497294\pi\)
0.00850152 + 0.999964i \(0.497294\pi\)
\(230\) 8.07980i 0.0351295i
\(231\) 0 0
\(232\) −74.2654 −0.320110
\(233\) − 365.767i − 1.56982i −0.619612 0.784908i \(-0.712710\pi\)
0.619612 0.784908i \(-0.287290\pi\)
\(234\) 0 0
\(235\) −6.39404 −0.0272087
\(236\) 179.306i 0.759770i
\(237\) 0 0
\(238\) −858.125 −3.60557
\(239\) 147.298i 0.616308i 0.951336 + 0.308154i \(0.0997112\pi\)
−0.951336 + 0.308154i \(0.900289\pi\)
\(240\) 0 0
\(241\) −278.601 −1.15602 −0.578010 0.816029i \(-0.696171\pi\)
−0.578010 + 0.816029i \(0.696171\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −58.5078 −0.239786
\(245\) − 9.42728i − 0.0384787i
\(246\) 0 0
\(247\) 84.2502 0.341094
\(248\) − 379.112i − 1.52868i
\(249\) 0 0
\(250\) −29.8878 −0.119551
\(251\) 48.3215i 0.192516i 0.995356 + 0.0962580i \(0.0306874\pi\)
−0.995356 + 0.0962580i \(0.969313\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 29.1737i − 0.114857i
\(255\) 0 0
\(256\) −436.515 −1.70514
\(257\) 199.225i 0.775195i 0.921829 + 0.387597i \(0.126695\pi\)
−0.921829 + 0.387597i \(0.873305\pi\)
\(258\) 0 0
\(259\) −448.967 −1.73346
\(260\) 30.6247i 0.117787i
\(261\) 0 0
\(262\) 871.022 3.32451
\(263\) − 69.3106i − 0.263538i −0.991280 0.131769i \(-0.957934\pi\)
0.991280 0.131769i \(-0.0420658\pi\)
\(264\) 0 0
\(265\) −5.84439 −0.0220543
\(266\) 154.325i 0.580170i
\(267\) 0 0
\(268\) 36.3113 0.135490
\(269\) 184.094i 0.684363i 0.939634 + 0.342181i \(0.111166\pi\)
−0.939634 + 0.342181i \(0.888834\pi\)
\(270\) 0 0
\(271\) 87.7148 0.323671 0.161835 0.986818i \(-0.448259\pi\)
0.161835 + 0.986818i \(0.448259\pi\)
\(272\) 702.248i 2.58179i
\(273\) 0 0
\(274\) 68.3096 0.249305
\(275\) 0 0
\(276\) 0 0
\(277\) 249.233 0.899757 0.449878 0.893090i \(-0.351467\pi\)
0.449878 + 0.893090i \(0.351467\pi\)
\(278\) − 121.201i − 0.435975i
\(279\) 0 0
\(280\) −31.4584 −0.112352
\(281\) 16.8466i 0.0599524i 0.999551 + 0.0299762i \(0.00954314\pi\)
−0.999551 + 0.0299762i \(0.990457\pi\)
\(282\) 0 0
\(283\) −47.6126 −0.168242 −0.0841212 0.996456i \(-0.526808\pi\)
−0.0841212 + 0.996456i \(0.526808\pi\)
\(284\) 498.815i 1.75639i
\(285\) 0 0
\(286\) 0 0
\(287\) − 536.297i − 1.86863i
\(288\) 0 0
\(289\) −240.677 −0.832793
\(290\) − 2.40218i − 0.00828337i
\(291\) 0 0
\(292\) −207.725 −0.711386
\(293\) 425.104i 1.45087i 0.688293 + 0.725433i \(0.258360\pi\)
−0.688293 + 0.725433i \(0.741640\pi\)
\(294\) 0 0
\(295\) −3.25246 −0.0110253
\(296\) 806.062i 2.72318i
\(297\) 0 0
\(298\) 374.423 1.25645
\(299\) − 274.991i − 0.919702i
\(300\) 0 0
\(301\) 644.398 2.14086
\(302\) − 1036.81i − 3.43316i
\(303\) 0 0
\(304\) 126.292 0.415435
\(305\) − 1.06129i − 0.00347962i
\(306\) 0 0
\(307\) 463.469 1.50967 0.754836 0.655914i \(-0.227716\pi\)
0.754836 + 0.655914i \(0.227716\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12.2627 0.0395570
\(311\) − 143.144i − 0.460269i −0.973159 0.230135i \(-0.926083\pi\)
0.973159 0.230135i \(-0.0739166\pi\)
\(312\) 0 0
\(313\) 237.744 0.759565 0.379783 0.925076i \(-0.375999\pi\)
0.379783 + 0.925076i \(0.375999\pi\)
\(314\) − 946.024i − 3.01281i
\(315\) 0 0
\(316\) −520.536 −1.64727
\(317\) − 313.909i − 0.990249i −0.868822 0.495125i \(-0.835122\pi\)
0.868822 0.495125i \(-0.164878\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.67204i 0.00522514i
\(321\) 0 0
\(322\) 503.714 1.56433
\(323\) 95.2571i 0.294914i
\(324\) 0 0
\(325\) 508.329 1.56409
\(326\) − 696.913i − 2.13777i
\(327\) 0 0
\(328\) −962.851 −2.93552
\(329\) 398.620i 1.21161i
\(330\) 0 0
\(331\) −52.4920 −0.158586 −0.0792931 0.996851i \(-0.525266\pi\)
−0.0792931 + 0.996851i \(0.525266\pi\)
\(332\) 517.347i 1.55827i
\(333\) 0 0
\(334\) 764.604 2.28923
\(335\) 0.658659i 0.00196615i
\(336\) 0 0
\(337\) 493.841 1.46540 0.732702 0.680550i \(-0.238259\pi\)
0.732702 + 0.680550i \(0.238259\pi\)
\(338\) − 888.233i − 2.62791i
\(339\) 0 0
\(340\) −34.6257 −0.101840
\(341\) 0 0
\(342\) 0 0
\(343\) −83.0752 −0.242202
\(344\) − 1156.93i − 3.36317i
\(345\) 0 0
\(346\) 375.699 1.08584
\(347\) − 451.711i − 1.30176i −0.759181 0.650880i \(-0.774400\pi\)
0.759181 0.650880i \(-0.225600\pi\)
\(348\) 0 0
\(349\) −177.153 −0.507603 −0.253802 0.967256i \(-0.581681\pi\)
−0.253802 + 0.967256i \(0.581681\pi\)
\(350\) 931.130i 2.66037i
\(351\) 0 0
\(352\) 0 0
\(353\) − 590.460i − 1.67269i −0.548203 0.836346i \(-0.684688\pi\)
0.548203 0.836346i \(-0.315312\pi\)
\(354\) 0 0
\(355\) −9.04811 −0.0254876
\(356\) 256.462i 0.720399i
\(357\) 0 0
\(358\) −40.0336 −0.111826
\(359\) − 592.141i − 1.64942i −0.565558 0.824709i \(-0.691339\pi\)
0.565558 0.824709i \(-0.308661\pi\)
\(360\) 0 0
\(361\) −343.869 −0.952546
\(362\) − 450.292i − 1.24390i
\(363\) 0 0
\(364\) 1909.22 5.24510
\(365\) − 3.76796i − 0.0103232i
\(366\) 0 0
\(367\) 226.762 0.617880 0.308940 0.951082i \(-0.400026\pi\)
0.308940 + 0.951082i \(0.400026\pi\)
\(368\) − 412.215i − 1.12015i
\(369\) 0 0
\(370\) −26.0727 −0.0704669
\(371\) 364.353i 0.982084i
\(372\) 0 0
\(373\) 76.5478 0.205222 0.102611 0.994722i \(-0.467280\pi\)
0.102611 + 0.994722i \(0.467280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 715.669 1.90338
\(377\) 81.7566i 0.216861i
\(378\) 0 0
\(379\) 351.579 0.927649 0.463825 0.885927i \(-0.346477\pi\)
0.463825 + 0.885927i \(0.346477\pi\)
\(380\) 6.22708i 0.0163871i
\(381\) 0 0
\(382\) −642.837 −1.68282
\(383\) − 572.874i − 1.49575i −0.663837 0.747877i \(-0.731073\pi\)
0.663837 0.747877i \(-0.268927\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 880.953i 2.28226i
\(387\) 0 0
\(388\) −755.285 −1.94661
\(389\) 196.178i 0.504313i 0.967687 + 0.252156i \(0.0811397\pi\)
−0.967687 + 0.252156i \(0.918860\pi\)
\(390\) 0 0
\(391\) 310.917 0.795185
\(392\) 1055.17i 2.69177i
\(393\) 0 0
\(394\) 780.512 1.98100
\(395\) − 9.44211i − 0.0239041i
\(396\) 0 0
\(397\) −105.240 −0.265089 −0.132544 0.991177i \(-0.542315\pi\)
−0.132544 + 0.991177i \(0.542315\pi\)
\(398\) 159.598i 0.401000i
\(399\) 0 0
\(400\) 761.992 1.90498
\(401\) − 490.021i − 1.22200i −0.791631 0.610999i \(-0.790768\pi\)
0.791631 0.610999i \(-0.209232\pi\)
\(402\) 0 0
\(403\) −417.353 −1.03562
\(404\) 1393.83i 3.45007i
\(405\) 0 0
\(406\) −149.757 −0.368861
\(407\) 0 0
\(408\) 0 0
\(409\) −533.858 −1.30528 −0.652639 0.757669i \(-0.726338\pi\)
−0.652639 + 0.757669i \(0.726338\pi\)
\(410\) − 31.1442i − 0.0759615i
\(411\) 0 0
\(412\) −957.346 −2.32366
\(413\) 202.767i 0.490960i
\(414\) 0 0
\(415\) −9.38426 −0.0226127
\(416\) − 743.135i − 1.78638i
\(417\) 0 0
\(418\) 0 0
\(419\) 344.904i 0.823160i 0.911374 + 0.411580i \(0.135023\pi\)
−0.911374 + 0.411580i \(0.864977\pi\)
\(420\) 0 0
\(421\) −403.467 −0.958355 −0.479177 0.877718i \(-0.659065\pi\)
−0.479177 + 0.877718i \(0.659065\pi\)
\(422\) − 956.888i − 2.26751i
\(423\) 0 0
\(424\) 654.149 1.54280
\(425\) 574.740i 1.35233i
\(426\) 0 0
\(427\) −66.1631 −0.154949
\(428\) 1026.96i 2.39943i
\(429\) 0 0
\(430\) 37.4219 0.0870277
\(431\) 132.691i 0.307868i 0.988081 + 0.153934i \(0.0491944\pi\)
−0.988081 + 0.153934i \(0.950806\pi\)
\(432\) 0 0
\(433\) −823.887 −1.90274 −0.951370 0.308049i \(-0.900324\pi\)
−0.951370 + 0.308049i \(0.900324\pi\)
\(434\) − 764.485i − 1.76149i
\(435\) 0 0
\(436\) −875.689 −2.00846
\(437\) − 55.9154i − 0.127953i
\(438\) 0 0
\(439\) −109.320 −0.249021 −0.124511 0.992218i \(-0.539736\pi\)
−0.124511 + 0.992218i \(0.539736\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1696.06 3.83723
\(443\) 375.738i 0.848167i 0.905623 + 0.424083i \(0.139404\pi\)
−0.905623 + 0.424083i \(0.860596\pi\)
\(444\) 0 0
\(445\) −4.65201 −0.0104540
\(446\) − 52.5495i − 0.117824i
\(447\) 0 0
\(448\) 104.239 0.232677
\(449\) − 377.057i − 0.839770i −0.907577 0.419885i \(-0.862070\pi\)
0.907577 0.419885i \(-0.137930\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 948.075i − 2.09751i
\(453\) 0 0
\(454\) 443.560 0.977004
\(455\) 34.6317i 0.0761136i
\(456\) 0 0
\(457\) 397.734 0.870316 0.435158 0.900354i \(-0.356693\pi\)
0.435158 + 0.900354i \(0.356693\pi\)
\(458\) − 14.0967i − 0.0307789i
\(459\) 0 0
\(460\) 20.3251 0.0441849
\(461\) − 452.581i − 0.981737i −0.871234 0.490869i \(-0.836680\pi\)
0.871234 0.490869i \(-0.163320\pi\)
\(462\) 0 0
\(463\) −125.818 −0.271745 −0.135873 0.990726i \(-0.543384\pi\)
−0.135873 + 0.990726i \(0.543384\pi\)
\(464\) 122.554i 0.264126i
\(465\) 0 0
\(466\) −1324.22 −2.84168
\(467\) 277.000i 0.593147i 0.955010 + 0.296574i \(0.0958440\pi\)
−0.955010 + 0.296574i \(0.904156\pi\)
\(468\) 0 0
\(469\) 41.0624 0.0875530
\(470\) 23.1489i 0.0492530i
\(471\) 0 0
\(472\) 364.041 0.771273
\(473\) 0 0
\(474\) 0 0
\(475\) 103.361 0.217603
\(476\) 2158.65i 4.53497i
\(477\) 0 0
\(478\) 533.275 1.11564
\(479\) 500.992i 1.04591i 0.852360 + 0.522956i \(0.175171\pi\)
−0.852360 + 0.522956i \(0.824829\pi\)
\(480\) 0 0
\(481\) 887.370 1.84484
\(482\) 1008.65i 2.09263i
\(483\) 0 0
\(484\) 0 0
\(485\) − 13.7003i − 0.0282480i
\(486\) 0 0
\(487\) 690.163 1.41717 0.708586 0.705624i \(-0.249333\pi\)
0.708586 + 0.705624i \(0.249333\pi\)
\(488\) 118.787i 0.243416i
\(489\) 0 0
\(490\) −34.1305 −0.0696540
\(491\) 355.830i 0.724705i 0.932041 + 0.362353i \(0.118026\pi\)
−0.932041 + 0.362353i \(0.881974\pi\)
\(492\) 0 0
\(493\) −92.4378 −0.187501
\(494\) − 305.019i − 0.617447i
\(495\) 0 0
\(496\) −625.618 −1.26133
\(497\) 564.081i 1.13497i
\(498\) 0 0
\(499\) 145.012 0.290604 0.145302 0.989387i \(-0.453585\pi\)
0.145302 + 0.989387i \(0.453585\pi\)
\(500\) 75.1840i 0.150368i
\(501\) 0 0
\(502\) 174.943 0.348492
\(503\) 868.917i 1.72747i 0.503947 + 0.863734i \(0.331881\pi\)
−0.503947 + 0.863734i \(0.668119\pi\)
\(504\) 0 0
\(505\) −25.2829 −0.0500651
\(506\) 0 0
\(507\) 0 0
\(508\) −73.3878 −0.144464
\(509\) − 504.109i − 0.990391i −0.868782 0.495195i \(-0.835096\pi\)
0.868782 0.495195i \(-0.164904\pi\)
\(510\) 0 0
\(511\) −234.904 −0.459694
\(512\) 1142.80i 2.23204i
\(513\) 0 0
\(514\) 721.273 1.40325
\(515\) − 17.3655i − 0.0337194i
\(516\) 0 0
\(517\) 0 0
\(518\) 1625.44i 3.13791i
\(519\) 0 0
\(520\) 62.1767 0.119571
\(521\) − 258.912i − 0.496951i −0.968638 0.248476i \(-0.920070\pi\)
0.968638 0.248476i \(-0.0799296\pi\)
\(522\) 0 0
\(523\) −597.180 −1.14184 −0.570918 0.821007i \(-0.693413\pi\)
−0.570918 + 0.821007i \(0.693413\pi\)
\(524\) − 2191.09i − 4.18147i
\(525\) 0 0
\(526\) −250.932 −0.477056
\(527\) − 471.878i − 0.895404i
\(528\) 0 0
\(529\) 346.493 0.654997
\(530\) 21.1590i 0.0399226i
\(531\) 0 0
\(532\) 388.211 0.729720
\(533\) 1059.97i 1.98870i
\(534\) 0 0
\(535\) −18.6282 −0.0348191
\(536\) − 73.7221i − 0.137541i
\(537\) 0 0
\(538\) 666.491 1.23883
\(539\) 0 0
\(540\) 0 0
\(541\) −241.706 −0.446776 −0.223388 0.974730i \(-0.571712\pi\)
−0.223388 + 0.974730i \(0.571712\pi\)
\(542\) − 317.562i − 0.585908i
\(543\) 0 0
\(544\) 840.222 1.54453
\(545\) − 15.8843i − 0.0291455i
\(546\) 0 0
\(547\) 535.306 0.978622 0.489311 0.872109i \(-0.337248\pi\)
0.489311 + 0.872109i \(0.337248\pi\)
\(548\) − 171.836i − 0.313569i
\(549\) 0 0
\(550\) 0 0
\(551\) 16.6240i 0.0301706i
\(552\) 0 0
\(553\) −588.644 −1.06446
\(554\) − 902.320i − 1.62874i
\(555\) 0 0
\(556\) −304.886 −0.548356
\(557\) − 38.6536i − 0.0693961i −0.999398 0.0346980i \(-0.988953\pi\)
0.999398 0.0346980i \(-0.0110469\pi\)
\(558\) 0 0
\(559\) −1273.63 −2.27841
\(560\) 51.9134i 0.0927024i
\(561\) 0 0
\(562\) 60.9914 0.108526
\(563\) − 356.051i − 0.632418i −0.948690 0.316209i \(-0.897590\pi\)
0.948690 0.316209i \(-0.102410\pi\)
\(564\) 0 0
\(565\) 17.1973 0.0304378
\(566\) 172.376i 0.304552i
\(567\) 0 0
\(568\) 1012.73 1.78298
\(569\) 98.7149i 0.173488i 0.996231 + 0.0867442i \(0.0276463\pi\)
−0.996231 + 0.0867442i \(0.972354\pi\)
\(570\) 0 0
\(571\) 37.7221 0.0660633 0.0330316 0.999454i \(-0.489484\pi\)
0.0330316 + 0.999454i \(0.489484\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1941.61 −3.38259
\(575\) − 337.369i − 0.586728i
\(576\) 0 0
\(577\) −978.346 −1.69557 −0.847787 0.530337i \(-0.822065\pi\)
−0.847787 + 0.530337i \(0.822065\pi\)
\(578\) 871.346i 1.50752i
\(579\) 0 0
\(580\) −6.04277 −0.0104186
\(581\) 585.037i 1.00695i
\(582\) 0 0
\(583\) 0 0
\(584\) 421.739i 0.722156i
\(585\) 0 0
\(586\) 1539.04 2.62635
\(587\) 1115.33i 1.90006i 0.312162 + 0.950029i \(0.398947\pi\)
−0.312162 + 0.950029i \(0.601053\pi\)
\(588\) 0 0
\(589\) −84.8625 −0.144079
\(590\) 11.7752i 0.0199580i
\(591\) 0 0
\(592\) 1330.18 2.24693
\(593\) 1006.79i 1.69779i 0.528563 + 0.848894i \(0.322731\pi\)
−0.528563 + 0.848894i \(0.677269\pi\)
\(594\) 0 0
\(595\) −39.1562 −0.0658087
\(596\) − 941.877i − 1.58033i
\(597\) 0 0
\(598\) −995.576 −1.66484
\(599\) 590.141i 0.985210i 0.870253 + 0.492605i \(0.163955\pi\)
−0.870253 + 0.492605i \(0.836045\pi\)
\(600\) 0 0
\(601\) 750.837 1.24931 0.624656 0.780900i \(-0.285239\pi\)
0.624656 + 0.780900i \(0.285239\pi\)
\(602\) − 2332.97i − 3.87537i
\(603\) 0 0
\(604\) −2608.15 −4.31813
\(605\) 0 0
\(606\) 0 0
\(607\) −176.162 −0.290218 −0.145109 0.989416i \(-0.546353\pi\)
−0.145109 + 0.989416i \(0.546353\pi\)
\(608\) − 151.105i − 0.248529i
\(609\) 0 0
\(610\) −3.84227 −0.00629880
\(611\) − 787.860i − 1.28946i
\(612\) 0 0
\(613\) −908.432 −1.48194 −0.740972 0.671535i \(-0.765635\pi\)
−0.740972 + 0.671535i \(0.765635\pi\)
\(614\) − 1677.94i − 2.73280i
\(615\) 0 0
\(616\) 0 0
\(617\) − 836.440i − 1.35566i −0.735220 0.677828i \(-0.762921\pi\)
0.735220 0.677828i \(-0.237079\pi\)
\(618\) 0 0
\(619\) 960.477 1.55166 0.775830 0.630942i \(-0.217332\pi\)
0.775830 + 0.630942i \(0.217332\pi\)
\(620\) − 30.8473i − 0.0497537i
\(621\) 0 0
\(622\) −518.237 −0.833178
\(623\) 290.018i 0.465518i
\(624\) 0 0
\(625\) 622.954 0.996726
\(626\) − 860.727i − 1.37496i
\(627\) 0 0
\(628\) −2379.76 −3.78943
\(629\) 1003.30i 1.59507i
\(630\) 0 0
\(631\) 230.308 0.364988 0.182494 0.983207i \(-0.441583\pi\)
0.182494 + 0.983207i \(0.441583\pi\)
\(632\) 1056.83i 1.67220i
\(633\) 0 0
\(634\) −1136.47 −1.79255
\(635\) − 1.33120i − 0.00209637i
\(636\) 0 0
\(637\) 1161.61 1.82356
\(638\) 0 0
\(639\) 0 0
\(640\) −18.0708 −0.0282356
\(641\) − 478.911i − 0.747131i −0.927604 0.373565i \(-0.878135\pi\)
0.927604 0.373565i \(-0.121865\pi\)
\(642\) 0 0
\(643\) −716.383 −1.11413 −0.557063 0.830470i \(-0.688072\pi\)
−0.557063 + 0.830470i \(0.688072\pi\)
\(644\) − 1267.11i − 1.96757i
\(645\) 0 0
\(646\) 344.868 0.533852
\(647\) 977.171i 1.51031i 0.655546 + 0.755155i \(0.272438\pi\)
−0.655546 + 0.755155i \(0.727562\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 1840.35i − 2.83131i
\(651\) 0 0
\(652\) −1753.11 −2.68882
\(653\) 892.901i 1.36738i 0.729771 + 0.683692i \(0.239627\pi\)
−0.729771 + 0.683692i \(0.760373\pi\)
\(654\) 0 0
\(655\) 39.7446 0.0606788
\(656\) 1588.92i 2.42213i
\(657\) 0 0
\(658\) 1443.16 2.19325
\(659\) − 350.792i − 0.532309i −0.963930 0.266154i \(-0.914247\pi\)
0.963930 0.266154i \(-0.0857531\pi\)
\(660\) 0 0
\(661\) −285.033 −0.431215 −0.215607 0.976480i \(-0.569173\pi\)
−0.215607 + 0.976480i \(0.569173\pi\)
\(662\) 190.042i 0.287072i
\(663\) 0 0
\(664\) 1050.36 1.58186
\(665\) 7.04184i 0.0105892i
\(666\) 0 0
\(667\) 54.2604 0.0813499
\(668\) − 1923.39i − 2.87933i
\(669\) 0 0
\(670\) 2.38460 0.00355911
\(671\) 0 0
\(672\) 0 0
\(673\) −101.236 −0.150426 −0.0752128 0.997168i \(-0.523964\pi\)
−0.0752128 + 0.997168i \(0.523964\pi\)
\(674\) − 1787.90i − 2.65267i
\(675\) 0 0
\(676\) −2234.39 −3.30531
\(677\) 399.059i 0.589453i 0.955582 + 0.294726i \(0.0952285\pi\)
−0.955582 + 0.294726i \(0.904772\pi\)
\(678\) 0 0
\(679\) −854.108 −1.25789
\(680\) 70.2998i 0.103382i
\(681\) 0 0
\(682\) 0 0
\(683\) 781.537i 1.14427i 0.820159 + 0.572135i \(0.193885\pi\)
−0.820159 + 0.572135i \(0.806115\pi\)
\(684\) 0 0
\(685\) 3.11696 0.00455031
\(686\) 300.765i 0.438433i
\(687\) 0 0
\(688\) −1909.19 −2.77499
\(689\) − 720.133i − 1.04519i
\(690\) 0 0
\(691\) −551.762 −0.798497 −0.399249 0.916843i \(-0.630729\pi\)
−0.399249 + 0.916843i \(0.630729\pi\)
\(692\) − 945.087i − 1.36573i
\(693\) 0 0
\(694\) −1635.37 −2.35644
\(695\) − 5.53039i − 0.00795740i
\(696\) 0 0
\(697\) −1198.46 −1.71945
\(698\) 641.365i 0.918861i
\(699\) 0 0
\(700\) 2342.30 3.34614
\(701\) − 327.564i − 0.467281i −0.972323 0.233640i \(-0.924936\pi\)
0.972323 0.233640i \(-0.0750639\pi\)
\(702\) 0 0
\(703\) 180.434 0.256662
\(704\) 0 0
\(705\) 0 0
\(706\) −2137.70 −3.02790
\(707\) 1576.20i 2.22942i
\(708\) 0 0
\(709\) 1039.76 1.46652 0.733260 0.679949i \(-0.237998\pi\)
0.733260 + 0.679949i \(0.237998\pi\)
\(710\) 32.7577i 0.0461376i
\(711\) 0 0
\(712\) 520.689 0.731305
\(713\) 276.990i 0.388485i
\(714\) 0 0
\(715\) 0 0
\(716\) 100.706i 0.140651i
\(717\) 0 0
\(718\) −2143.78 −2.98577
\(719\) 909.087i 1.26438i 0.774815 + 0.632188i \(0.217843\pi\)
−0.774815 + 0.632188i \(0.782157\pi\)
\(720\) 0 0
\(721\) −1082.61 −1.50153
\(722\) 1244.94i 1.72429i
\(723\) 0 0
\(724\) −1132.73 −1.56454
\(725\) 100.302i 0.138348i
\(726\) 0 0
\(727\) −709.728 −0.976242 −0.488121 0.872776i \(-0.662318\pi\)
−0.488121 + 0.872776i \(0.662318\pi\)
\(728\) − 3876.24i − 5.32451i
\(729\) 0 0
\(730\) −13.6415 −0.0186870
\(731\) − 1440.03i − 1.96994i
\(732\) 0 0
\(733\) −495.869 −0.676493 −0.338246 0.941058i \(-0.609834\pi\)
−0.338246 + 0.941058i \(0.609834\pi\)
\(734\) − 820.967i − 1.11848i
\(735\) 0 0
\(736\) −493.205 −0.670115
\(737\) 0 0
\(738\) 0 0
\(739\) −1099.53 −1.48787 −0.743934 0.668253i \(-0.767042\pi\)
−0.743934 + 0.668253i \(0.767042\pi\)
\(740\) 65.5871i 0.0886312i
\(741\) 0 0
\(742\) 1319.10 1.77777
\(743\) − 723.678i − 0.973994i −0.873404 0.486997i \(-0.838092\pi\)
0.873404 0.486997i \(-0.161908\pi\)
\(744\) 0 0
\(745\) 17.0849 0.0229328
\(746\) − 277.133i − 0.371492i
\(747\) 0 0
\(748\) 0 0
\(749\) 1161.33i 1.55050i
\(750\) 0 0
\(751\) −543.282 −0.723412 −0.361706 0.932292i \(-0.617806\pi\)
−0.361706 + 0.932292i \(0.617806\pi\)
\(752\) − 1181.01i − 1.57049i
\(753\) 0 0
\(754\) 295.991 0.392561
\(755\) − 47.3097i − 0.0626619i
\(756\) 0 0
\(757\) −347.247 −0.458715 −0.229357 0.973342i \(-0.573662\pi\)
−0.229357 + 0.973342i \(0.573662\pi\)
\(758\) − 1272.85i − 1.67923i
\(759\) 0 0
\(760\) 12.6427 0.0166351
\(761\) − 802.222i − 1.05417i −0.849813 0.527084i \(-0.823285\pi\)
0.849813 0.527084i \(-0.176715\pi\)
\(762\) 0 0
\(763\) −990.266 −1.29786
\(764\) 1617.08i 2.11660i
\(765\) 0 0
\(766\) −2074.03 −2.70761
\(767\) − 400.762i − 0.522506i
\(768\) 0 0
\(769\) 1441.35 1.87432 0.937158 0.348905i \(-0.113446\pi\)
0.937158 + 0.348905i \(0.113446\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2216.07 2.87056
\(773\) − 1333.16i − 1.72466i −0.506348 0.862329i \(-0.669005\pi\)
0.506348 0.862329i \(-0.330995\pi\)
\(774\) 0 0
\(775\) −512.023 −0.660675
\(776\) 1533.44i 1.97608i
\(777\) 0 0
\(778\) 710.240 0.912905
\(779\) 215.530i 0.276675i
\(780\) 0 0
\(781\) 0 0
\(782\) − 1125.64i − 1.43944i
\(783\) 0 0
\(784\) 1741.27 2.22100
\(785\) − 43.1670i − 0.0549898i
\(786\) 0 0
\(787\) 442.103 0.561757 0.280878 0.959743i \(-0.409374\pi\)
0.280878 + 0.959743i \(0.409374\pi\)
\(788\) − 1963.41i − 2.49164i
\(789\) 0 0
\(790\) −34.1842 −0.0432711
\(791\) − 1072.12i − 1.35540i
\(792\) 0 0
\(793\) 130.769 0.164905
\(794\) 381.011i 0.479863i
\(795\) 0 0
\(796\) 401.475 0.504365
\(797\) 1298.24i 1.62891i 0.580225 + 0.814456i \(0.302965\pi\)
−0.580225 + 0.814456i \(0.697035\pi\)
\(798\) 0 0
\(799\) 890.790 1.11488
\(800\) − 911.704i − 1.13963i
\(801\) 0 0
\(802\) −1774.07 −2.21206
\(803\) 0 0
\(804\) 0 0
\(805\) 22.9844 0.0285521
\(806\) 1510.98i 1.87467i
\(807\) 0 0
\(808\) 2829.86 3.50230
\(809\) − 913.645i − 1.12935i −0.825313 0.564675i \(-0.809001\pi\)
0.825313 0.564675i \(-0.190999\pi\)
\(810\) 0 0
\(811\) −343.061 −0.423010 −0.211505 0.977377i \(-0.567837\pi\)
−0.211505 + 0.977377i \(0.567837\pi\)
\(812\) 376.721i 0.463942i
\(813\) 0 0
\(814\) 0 0
\(815\) − 31.8001i − 0.0390185i
\(816\) 0 0
\(817\) −258.974 −0.316982
\(818\) 1932.78i 2.36281i
\(819\) 0 0
\(820\) −78.3445 −0.0955421
\(821\) 16.3641i 0.0199319i 0.999950 + 0.00996596i \(0.00317232\pi\)
−0.999950 + 0.00996596i \(0.996828\pi\)
\(822\) 0 0
\(823\) −63.5873 −0.0772628 −0.0386314 0.999254i \(-0.512300\pi\)
−0.0386314 + 0.999254i \(0.512300\pi\)
\(824\) 1943.68i 2.35883i
\(825\) 0 0
\(826\) 734.095 0.888734
\(827\) 1174.88i 1.42066i 0.703871 + 0.710328i \(0.251453\pi\)
−0.703871 + 0.710328i \(0.748547\pi\)
\(828\) 0 0
\(829\) −277.361 −0.334573 −0.167287 0.985908i \(-0.553501\pi\)
−0.167287 + 0.985908i \(0.553501\pi\)
\(830\) 33.9747i 0.0409334i
\(831\) 0 0
\(832\) −206.026 −0.247627
\(833\) 1313.37i 1.57667i
\(834\) 0 0
\(835\) 34.8888 0.0417830
\(836\) 0 0
\(837\) 0 0
\(838\) 1248.69 1.49008
\(839\) 932.487i 1.11143i 0.831374 + 0.555713i \(0.187555\pi\)
−0.831374 + 0.555713i \(0.812445\pi\)
\(840\) 0 0
\(841\) 824.868 0.980818
\(842\) 1460.71i 1.73481i
\(843\) 0 0
\(844\) −2407.09 −2.85200
\(845\) − 40.5300i − 0.0479645i
\(846\) 0 0
\(847\) 0 0
\(848\) − 1079.49i − 1.27298i
\(849\) 0 0
\(850\) 2080.78 2.44798
\(851\) − 588.932i − 0.692047i
\(852\) 0 0
\(853\) 1539.03 1.80425 0.902127 0.431471i \(-0.142005\pi\)
0.902127 + 0.431471i \(0.142005\pi\)
\(854\) 239.536i 0.280488i
\(855\) 0 0
\(856\) 2085.01 2.43576
\(857\) 443.512i 0.517517i 0.965942 + 0.258758i \(0.0833133\pi\)
−0.965942 + 0.258758i \(0.916687\pi\)
\(858\) 0 0
\(859\) 1399.85 1.62962 0.814812 0.579725i \(-0.196840\pi\)
0.814812 + 0.579725i \(0.196840\pi\)
\(860\) − 94.1364i − 0.109461i
\(861\) 0 0
\(862\) 480.395 0.557302
\(863\) − 1447.63i − 1.67744i −0.544566 0.838718i \(-0.683306\pi\)
0.544566 0.838718i \(-0.316694\pi\)
\(864\) 0 0
\(865\) 17.1431 0.0198186
\(866\) 2982.79i 3.44433i
\(867\) 0 0
\(868\) −1923.09 −2.21555
\(869\) 0 0
\(870\) 0 0
\(871\) −81.1586 −0.0931786
\(872\) 1777.89i 2.03887i
\(873\) 0 0
\(874\) −202.436 −0.231620
\(875\) 85.0212i 0.0971671i
\(876\) 0 0
\(877\) 193.706 0.220874 0.110437 0.993883i \(-0.464775\pi\)
0.110437 + 0.993883i \(0.464775\pi\)
\(878\) 395.782i 0.450777i
\(879\) 0 0
\(880\) 0 0
\(881\) − 1111.18i − 1.26127i −0.776079 0.630636i \(-0.782794\pi\)
0.776079 0.630636i \(-0.217206\pi\)
\(882\) 0 0
\(883\) 710.379 0.804507 0.402253 0.915528i \(-0.368227\pi\)
0.402253 + 0.915528i \(0.368227\pi\)
\(884\) − 4266.50i − 4.82636i
\(885\) 0 0
\(886\) 1360.32 1.53535
\(887\) 1134.08i 1.27856i 0.768975 + 0.639279i \(0.220767\pi\)
−0.768975 + 0.639279i \(0.779233\pi\)
\(888\) 0 0
\(889\) −82.9899 −0.0933520
\(890\) 16.8421i 0.0189237i
\(891\) 0 0
\(892\) −132.190 −0.148196
\(893\) − 160.200i − 0.179395i
\(894\) 0 0
\(895\) −1.82673 −0.00204104
\(896\) 1126.58i 1.25734i
\(897\) 0 0
\(898\) −1365.09 −1.52015
\(899\) − 82.3508i − 0.0916027i
\(900\) 0 0
\(901\) 814.215 0.903680
\(902\) 0 0
\(903\) 0 0
\(904\) −1924.86 −2.12927
\(905\) − 20.5468i − 0.0227036i
\(906\) 0 0
\(907\) −987.303 −1.08854 −0.544268 0.838911i \(-0.683193\pi\)
−0.544268 + 0.838911i \(0.683193\pi\)
\(908\) − 1115.79i − 1.22885i
\(909\) 0 0
\(910\) 125.380 0.137781
\(911\) − 1029.67i − 1.13026i −0.825002 0.565130i \(-0.808826\pi\)
0.825002 0.565130i \(-0.191174\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 1439.95i − 1.57544i
\(915\) 0 0
\(916\) −35.4609 −0.0387127
\(917\) − 2477.78i − 2.70205i
\(918\) 0 0
\(919\) 555.392 0.604343 0.302172 0.953254i \(-0.402288\pi\)
0.302172 + 0.953254i \(0.402288\pi\)
\(920\) − 41.2655i − 0.0448538i
\(921\) 0 0
\(922\) −1638.52 −1.77714
\(923\) − 1114.89i − 1.20790i
\(924\) 0 0
\(925\) 1088.66 1.17693
\(926\) 455.511i 0.491913i
\(927\) 0 0
\(928\) 146.633 0.158010
\(929\) 624.021i 0.671713i 0.941913 + 0.335856i \(0.109026\pi\)
−0.941913 + 0.335856i \(0.890974\pi\)
\(930\) 0 0
\(931\) 236.196 0.253701
\(932\) 3331.13i 3.57418i
\(933\) 0 0
\(934\) 1002.85 1.07371
\(935\) 0 0
\(936\) 0 0
\(937\) −270.217 −0.288386 −0.144193 0.989550i \(-0.546059\pi\)
−0.144193 + 0.989550i \(0.546059\pi\)
\(938\) − 148.662i − 0.158488i
\(939\) 0 0
\(940\) 58.2321 0.0619490
\(941\) 1656.43i 1.76029i 0.474703 + 0.880146i \(0.342555\pi\)
−0.474703 + 0.880146i \(0.657445\pi\)
\(942\) 0 0
\(943\) 703.486 0.746008
\(944\) − 600.747i − 0.636385i
\(945\) 0 0
\(946\) 0 0
\(947\) 771.207i 0.814368i 0.913346 + 0.407184i \(0.133489\pi\)
−0.913346 + 0.407184i \(0.866511\pi\)
\(948\) 0 0
\(949\) 464.280 0.489231
\(950\) − 374.208i − 0.393903i
\(951\) 0 0
\(952\) 4382.66 4.60363
\(953\) − 297.525i − 0.312199i −0.987741 0.156099i \(-0.950108\pi\)
0.987741 0.156099i \(-0.0498920\pi\)
\(954\) 0 0
\(955\) −29.3326 −0.0307148
\(956\) − 1341.48i − 1.40322i
\(957\) 0 0
\(958\) 1813.79 1.89331
\(959\) − 194.319i − 0.202627i
\(960\) 0 0
\(961\) −540.614 −0.562553
\(962\) − 3212.63i − 3.33953i
\(963\) 0 0
\(964\) 2537.29 2.63204
\(965\) 40.1978i 0.0416558i
\(966\) 0 0
\(967\) −1078.20 −1.11499 −0.557497 0.830179i \(-0.688238\pi\)
−0.557497 + 0.830179i \(0.688238\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −49.6004 −0.0511344
\(971\) − 687.460i − 0.707992i −0.935247 0.353996i \(-0.884823\pi\)
0.935247 0.353996i \(-0.115177\pi\)
\(972\) 0 0
\(973\) −344.778 −0.354345
\(974\) − 2498.66i − 2.56536i
\(975\) 0 0
\(976\) 196.025 0.200845
\(977\) 1597.66i 1.63527i 0.575738 + 0.817634i \(0.304715\pi\)
−0.575738 + 0.817634i \(0.695285\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 85.8566i 0.0876087i
\(981\) 0 0
\(982\) 1288.25 1.31186
\(983\) 1254.84i 1.27654i 0.769811 + 0.638272i \(0.220350\pi\)
−0.769811 + 0.638272i \(0.779650\pi\)
\(984\) 0 0
\(985\) 35.6147 0.0361571
\(986\) 334.661i 0.339413i
\(987\) 0 0
\(988\) −767.288 −0.776607
\(989\) 845.287i 0.854689i
\(990\) 0 0
\(991\) 973.882 0.982727 0.491363 0.870955i \(-0.336499\pi\)
0.491363 + 0.870955i \(0.336499\pi\)
\(992\) 748.536i 0.754572i
\(993\) 0 0
\(994\) 2042.20 2.05452
\(995\) 7.28243i 0.00731903i
\(996\) 0 0
\(997\) −1601.03 −1.60585 −0.802923 0.596083i \(-0.796723\pi\)
−0.802923 + 0.596083i \(0.796723\pi\)
\(998\) − 524.999i − 0.526051i
\(999\) 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.j.485.1 16
3.2 odd 2 inner 1089.3.b.j.485.16 16
11.7 odd 10 99.3.l.a.71.1 yes 32
11.8 odd 10 99.3.l.a.53.8 yes 32
11.10 odd 2 1089.3.b.i.485.16 16
33.8 even 10 99.3.l.a.53.1 32
33.29 even 10 99.3.l.a.71.8 yes 32
33.32 even 2 1089.3.b.i.485.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.53.1 32 33.8 even 10
99.3.l.a.53.8 yes 32 11.8 odd 10
99.3.l.a.71.1 yes 32 11.7 odd 10
99.3.l.a.71.8 yes 32 33.29 even 10
1089.3.b.i.485.1 16 33.32 even 2
1089.3.b.i.485.16 16 11.10 odd 2
1089.3.b.j.485.1 16 1.1 even 1 trivial
1089.3.b.j.485.16 16 3.2 odd 2 inner