Properties

Label 1143.3.b.a.890.67
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.67
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.18

$q$-expansion

\(f(q)\) \(=\) \(q+2.51986i q^{2} -2.34969 q^{4} -1.17863i q^{5} +3.92062 q^{7} +4.15855i q^{8} +O(q^{10})\) \(q+2.51986i q^{2} -2.34969 q^{4} -1.17863i q^{5} +3.92062 q^{7} +4.15855i q^{8} +2.96998 q^{10} -0.116151i q^{11} -19.8712 q^{13} +9.87941i q^{14} -19.8777 q^{16} +4.09340i q^{17} +0.981142 q^{19} +2.76941i q^{20} +0.292685 q^{22} +3.96962i q^{23} +23.6108 q^{25} -50.0726i q^{26} -9.21224 q^{28} +1.24741i q^{29} -1.50131 q^{31} -33.4548i q^{32} -10.3148 q^{34} -4.62096i q^{35} -62.1179 q^{37} +2.47234i q^{38} +4.90139 q^{40} +55.0004i q^{41} -48.4928 q^{43} +0.272919i q^{44} -10.0029 q^{46} +61.7325i q^{47} -33.6287 q^{49} +59.4960i q^{50} +46.6911 q^{52} +27.1346i q^{53} -0.136899 q^{55} +16.3041i q^{56} -3.14329 q^{58} -58.0100i q^{59} +7.12498 q^{61} -3.78310i q^{62} +4.79062 q^{64} +23.4207i q^{65} -103.496 q^{67} -9.61821i q^{68} +11.6442 q^{70} -31.8315i q^{71} -47.2798 q^{73} -156.528i q^{74} -2.30538 q^{76} -0.455385i q^{77} -40.9794 q^{79} +23.4284i q^{80} -138.593 q^{82} -93.8255i q^{83} +4.82460 q^{85} -122.195i q^{86} +0.483021 q^{88} +92.6845i q^{89} -77.9074 q^{91} -9.32737i q^{92} -155.557 q^{94} -1.15640i q^{95} +156.948 q^{97} -84.7396i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.51986i 1.25993i 0.776624 + 0.629965i \(0.216931\pi\)
−0.776624 + 0.629965i \(0.783069\pi\)
\(3\) 0 0
\(4\) −2.34969 −0.587422
\(5\) − 1.17863i − 0.235726i −0.993030 0.117863i \(-0.962396\pi\)
0.993030 0.117863i \(-0.0376043\pi\)
\(6\) 0 0
\(7\) 3.92062 0.560089 0.280044 0.959987i \(-0.409651\pi\)
0.280044 + 0.959987i \(0.409651\pi\)
\(8\) 4.15855i 0.519819i
\(9\) 0 0
\(10\) 2.96998 0.296998
\(11\) − 0.116151i − 0.0105592i −0.999986 0.00527960i \(-0.998319\pi\)
0.999986 0.00527960i \(-0.00168056\pi\)
\(12\) 0 0
\(13\) −19.8712 −1.52855 −0.764276 0.644889i \(-0.776904\pi\)
−0.764276 + 0.644889i \(0.776904\pi\)
\(14\) 9.87941i 0.705672i
\(15\) 0 0
\(16\) −19.8777 −1.24236
\(17\) 4.09340i 0.240788i 0.992726 + 0.120394i \(0.0384158\pi\)
−0.992726 + 0.120394i \(0.961584\pi\)
\(18\) 0 0
\(19\) 0.981142 0.0516391 0.0258195 0.999667i \(-0.491780\pi\)
0.0258195 + 0.999667i \(0.491780\pi\)
\(20\) 2.76941i 0.138471i
\(21\) 0 0
\(22\) 0.292685 0.0133038
\(23\) 3.96962i 0.172592i 0.996270 + 0.0862961i \(0.0275031\pi\)
−0.996270 + 0.0862961i \(0.972497\pi\)
\(24\) 0 0
\(25\) 23.6108 0.944433
\(26\) − 50.0726i − 1.92587i
\(27\) 0 0
\(28\) −9.21224 −0.329009
\(29\) 1.24741i 0.0430140i 0.999769 + 0.0215070i \(0.00684642\pi\)
−0.999769 + 0.0215070i \(0.993154\pi\)
\(30\) 0 0
\(31\) −1.50131 −0.0484294 −0.0242147 0.999707i \(-0.507709\pi\)
−0.0242147 + 0.999707i \(0.507709\pi\)
\(32\) − 33.4548i − 1.04546i
\(33\) 0 0
\(34\) −10.3148 −0.303376
\(35\) − 4.62096i − 0.132027i
\(36\) 0 0
\(37\) −62.1179 −1.67886 −0.839430 0.543467i \(-0.817111\pi\)
−0.839430 + 0.543467i \(0.817111\pi\)
\(38\) 2.47234i 0.0650616i
\(39\) 0 0
\(40\) 4.90139 0.122535
\(41\) 55.0004i 1.34147i 0.741695 + 0.670737i \(0.234022\pi\)
−0.741695 + 0.670737i \(0.765978\pi\)
\(42\) 0 0
\(43\) −48.4928 −1.12774 −0.563870 0.825863i \(-0.690688\pi\)
−0.563870 + 0.825863i \(0.690688\pi\)
\(44\) 0.272919i 0.00620271i
\(45\) 0 0
\(46\) −10.0029 −0.217454
\(47\) 61.7325i 1.31346i 0.754127 + 0.656729i \(0.228060\pi\)
−0.754127 + 0.656729i \(0.771940\pi\)
\(48\) 0 0
\(49\) −33.6287 −0.686301
\(50\) 59.4960i 1.18992i
\(51\) 0 0
\(52\) 46.6911 0.897906
\(53\) 27.1346i 0.511974i 0.966680 + 0.255987i \(0.0824004\pi\)
−0.966680 + 0.255987i \(0.917600\pi\)
\(54\) 0 0
\(55\) −0.136899 −0.00248907
\(56\) 16.3041i 0.291145i
\(57\) 0 0
\(58\) −3.14329 −0.0541946
\(59\) − 58.0100i − 0.983221i −0.870815 0.491611i \(-0.836408\pi\)
0.870815 0.491611i \(-0.163592\pi\)
\(60\) 0 0
\(61\) 7.12498 0.116803 0.0584015 0.998293i \(-0.481400\pi\)
0.0584015 + 0.998293i \(0.481400\pi\)
\(62\) − 3.78310i − 0.0610177i
\(63\) 0 0
\(64\) 4.79062 0.0748535
\(65\) 23.4207i 0.360319i
\(66\) 0 0
\(67\) −103.496 −1.54472 −0.772358 0.635188i \(-0.780923\pi\)
−0.772358 + 0.635188i \(0.780923\pi\)
\(68\) − 9.61821i − 0.141444i
\(69\) 0 0
\(70\) 11.6442 0.166345
\(71\) − 31.8315i − 0.448331i −0.974551 0.224166i \(-0.928034\pi\)
0.974551 0.224166i \(-0.0719657\pi\)
\(72\) 0 0
\(73\) −47.2798 −0.647668 −0.323834 0.946114i \(-0.604972\pi\)
−0.323834 + 0.946114i \(0.604972\pi\)
\(74\) − 156.528i − 2.11525i
\(75\) 0 0
\(76\) −2.30538 −0.0303339
\(77\) − 0.455385i − 0.00591409i
\(78\) 0 0
\(79\) −40.9794 −0.518726 −0.259363 0.965780i \(-0.583513\pi\)
−0.259363 + 0.965780i \(0.583513\pi\)
\(80\) 23.4284i 0.292856i
\(81\) 0 0
\(82\) −138.593 −1.69016
\(83\) − 93.8255i − 1.13043i −0.824945 0.565214i \(-0.808794\pi\)
0.824945 0.565214i \(-0.191206\pi\)
\(84\) 0 0
\(85\) 4.82460 0.0567600
\(86\) − 122.195i − 1.42087i
\(87\) 0 0
\(88\) 0.483021 0.00548887
\(89\) 92.6845i 1.04140i 0.853740 + 0.520700i \(0.174329\pi\)
−0.853740 + 0.520700i \(0.825671\pi\)
\(90\) 0 0
\(91\) −77.9074 −0.856125
\(92\) − 9.32737i − 0.101385i
\(93\) 0 0
\(94\) −155.557 −1.65486
\(95\) − 1.15640i − 0.0121727i
\(96\) 0 0
\(97\) 156.948 1.61802 0.809008 0.587797i \(-0.200005\pi\)
0.809008 + 0.587797i \(0.200005\pi\)
\(98\) − 84.7396i − 0.864690i
\(99\) 0 0
\(100\) −55.4781 −0.554781
\(101\) − 88.0834i − 0.872113i −0.899919 0.436057i \(-0.856375\pi\)
0.899919 0.436057i \(-0.143625\pi\)
\(102\) 0 0
\(103\) −24.6377 −0.239201 −0.119600 0.992822i \(-0.538161\pi\)
−0.119600 + 0.992822i \(0.538161\pi\)
\(104\) − 82.6353i − 0.794570i
\(105\) 0 0
\(106\) −68.3754 −0.645051
\(107\) 53.5903i 0.500844i 0.968137 + 0.250422i \(0.0805693\pi\)
−0.968137 + 0.250422i \(0.919431\pi\)
\(108\) 0 0
\(109\) −110.122 −1.01029 −0.505147 0.863033i \(-0.668562\pi\)
−0.505147 + 0.863033i \(0.668562\pi\)
\(110\) − 0.344966i − 0.00313606i
\(111\) 0 0
\(112\) −77.9330 −0.695830
\(113\) − 117.149i − 1.03672i −0.855164 0.518358i \(-0.826544\pi\)
0.855164 0.518358i \(-0.173456\pi\)
\(114\) 0 0
\(115\) 4.67871 0.0406844
\(116\) − 2.93102i − 0.0252674i
\(117\) 0 0
\(118\) 146.177 1.23879
\(119\) 16.0487i 0.134863i
\(120\) 0 0
\(121\) 120.987 0.999889
\(122\) 17.9539i 0.147163i
\(123\) 0 0
\(124\) 3.52762 0.0284485
\(125\) − 57.2941i − 0.458353i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) − 121.748i − 0.951154i
\(129\) 0 0
\(130\) −59.0170 −0.453977
\(131\) 213.224i 1.62766i 0.581100 + 0.813832i \(0.302622\pi\)
−0.581100 + 0.813832i \(0.697378\pi\)
\(132\) 0 0
\(133\) 3.84669 0.0289225
\(134\) − 260.795i − 1.94623i
\(135\) 0 0
\(136\) −17.0226 −0.125166
\(137\) 92.6280i 0.676117i 0.941125 + 0.338058i \(0.109770\pi\)
−0.941125 + 0.338058i \(0.890230\pi\)
\(138\) 0 0
\(139\) −46.4344 −0.334060 −0.167030 0.985952i \(-0.553418\pi\)
−0.167030 + 0.985952i \(0.553418\pi\)
\(140\) 10.8578i 0.0775558i
\(141\) 0 0
\(142\) 80.2109 0.564866
\(143\) 2.30806i 0.0161403i
\(144\) 0 0
\(145\) 1.47023 0.0101395
\(146\) − 119.138i − 0.816016i
\(147\) 0 0
\(148\) 145.958 0.986200
\(149\) 139.438i 0.935824i 0.883775 + 0.467912i \(0.154994\pi\)
−0.883775 + 0.467912i \(0.845006\pi\)
\(150\) 0 0
\(151\) 281.925 1.86705 0.933526 0.358510i \(-0.116715\pi\)
0.933526 + 0.358510i \(0.116715\pi\)
\(152\) 4.08013i 0.0268430i
\(153\) 0 0
\(154\) 1.14751 0.00745134
\(155\) 1.76949i 0.0114161i
\(156\) 0 0
\(157\) −89.4645 −0.569837 −0.284919 0.958552i \(-0.591967\pi\)
−0.284919 + 0.958552i \(0.591967\pi\)
\(158\) − 103.262i − 0.653559i
\(159\) 0 0
\(160\) −39.4308 −0.246443
\(161\) 15.5634i 0.0966669i
\(162\) 0 0
\(163\) −89.8376 −0.551151 −0.275575 0.961279i \(-0.588868\pi\)
−0.275575 + 0.961279i \(0.588868\pi\)
\(164\) − 129.234i − 0.788012i
\(165\) 0 0
\(166\) 236.427 1.42426
\(167\) − 216.719i − 1.29772i −0.760908 0.648860i \(-0.775246\pi\)
0.760908 0.648860i \(-0.224754\pi\)
\(168\) 0 0
\(169\) 225.864 1.33647
\(170\) 12.1573i 0.0715135i
\(171\) 0 0
\(172\) 113.943 0.662460
\(173\) 147.874i 0.854763i 0.904071 + 0.427382i \(0.140564\pi\)
−0.904071 + 0.427382i \(0.859436\pi\)
\(174\) 0 0
\(175\) 92.5691 0.528967
\(176\) 2.30882i 0.0131183i
\(177\) 0 0
\(178\) −233.552 −1.31209
\(179\) 18.6836i 0.104378i 0.998637 + 0.0521888i \(0.0166198\pi\)
−0.998637 + 0.0521888i \(0.983380\pi\)
\(180\) 0 0
\(181\) −5.48027 −0.0302777 −0.0151389 0.999885i \(-0.504819\pi\)
−0.0151389 + 0.999885i \(0.504819\pi\)
\(182\) − 196.316i − 1.07866i
\(183\) 0 0
\(184\) −16.5079 −0.0897167
\(185\) 73.2139i 0.395751i
\(186\) 0 0
\(187\) 0.475453 0.00254253
\(188\) − 145.052i − 0.771554i
\(189\) 0 0
\(190\) 2.91397 0.0153367
\(191\) − 230.705i − 1.20788i −0.797029 0.603941i \(-0.793596\pi\)
0.797029 0.603941i \(-0.206404\pi\)
\(192\) 0 0
\(193\) −232.264 −1.20344 −0.601721 0.798706i \(-0.705518\pi\)
−0.601721 + 0.798706i \(0.705518\pi\)
\(194\) 395.486i 2.03859i
\(195\) 0 0
\(196\) 79.0171 0.403148
\(197\) 46.5025i 0.236053i 0.993010 + 0.118027i \(0.0376568\pi\)
−0.993010 + 0.118027i \(0.962343\pi\)
\(198\) 0 0
\(199\) 230.998 1.16079 0.580397 0.814334i \(-0.302897\pi\)
0.580397 + 0.814334i \(0.302897\pi\)
\(200\) 98.1868i 0.490934i
\(201\) 0 0
\(202\) 221.958 1.09880
\(203\) 4.89061i 0.0240917i
\(204\) 0 0
\(205\) 64.8251 0.316220
\(206\) − 62.0835i − 0.301376i
\(207\) 0 0
\(208\) 394.994 1.89901
\(209\) − 0.113961i 0 0.000545267i
\(210\) 0 0
\(211\) −132.024 −0.625704 −0.312852 0.949802i \(-0.601284\pi\)
−0.312852 + 0.949802i \(0.601284\pi\)
\(212\) − 63.7579i − 0.300745i
\(213\) 0 0
\(214\) −135.040 −0.631028
\(215\) 57.1550i 0.265837i
\(216\) 0 0
\(217\) −5.88608 −0.0271248
\(218\) − 277.492i − 1.27290i
\(219\) 0 0
\(220\) 0.321670 0.00146214
\(221\) − 81.3407i − 0.368057i
\(222\) 0 0
\(223\) −268.297 −1.20312 −0.601562 0.798826i \(-0.705455\pi\)
−0.601562 + 0.798826i \(0.705455\pi\)
\(224\) − 131.164i − 0.585553i
\(225\) 0 0
\(226\) 295.199 1.30619
\(227\) − 104.979i − 0.462461i −0.972899 0.231231i \(-0.925725\pi\)
0.972899 0.231231i \(-0.0742752\pi\)
\(228\) 0 0
\(229\) −146.937 −0.641647 −0.320823 0.947139i \(-0.603960\pi\)
−0.320823 + 0.947139i \(0.603960\pi\)
\(230\) 11.7897i 0.0512595i
\(231\) 0 0
\(232\) −5.18740 −0.0223595
\(233\) − 96.9595i − 0.416135i −0.978114 0.208068i \(-0.933283\pi\)
0.978114 0.208068i \(-0.0667174\pi\)
\(234\) 0 0
\(235\) 72.7597 0.309616
\(236\) 136.306i 0.577566i
\(237\) 0 0
\(238\) −40.4404 −0.169918
\(239\) 198.906i 0.832244i 0.909309 + 0.416122i \(0.136611\pi\)
−0.909309 + 0.416122i \(0.863389\pi\)
\(240\) 0 0
\(241\) 264.284 1.09661 0.548307 0.836277i \(-0.315273\pi\)
0.548307 + 0.836277i \(0.315273\pi\)
\(242\) 304.869i 1.25979i
\(243\) 0 0
\(244\) −16.7415 −0.0686126
\(245\) 39.6358i 0.161779i
\(246\) 0 0
\(247\) −19.4965 −0.0789330
\(248\) − 6.24328i − 0.0251745i
\(249\) 0 0
\(250\) 144.373 0.577492
\(251\) 31.6442i 0.126073i 0.998011 + 0.0630363i \(0.0200784\pi\)
−0.998011 + 0.0630363i \(0.979922\pi\)
\(252\) 0 0
\(253\) 0.461076 0.00182244
\(254\) − 28.3974i − 0.111801i
\(255\) 0 0
\(256\) 325.949 1.27324
\(257\) 345.962i 1.34615i 0.739572 + 0.673077i \(0.235028\pi\)
−0.739572 + 0.673077i \(0.764972\pi\)
\(258\) 0 0
\(259\) −243.541 −0.940311
\(260\) − 55.0315i − 0.211659i
\(261\) 0 0
\(262\) −537.294 −2.05074
\(263\) 263.188i 1.00071i 0.865819 + 0.500357i \(0.166798\pi\)
−0.865819 + 0.500357i \(0.833202\pi\)
\(264\) 0 0
\(265\) 31.9816 0.120685
\(266\) 9.69311i 0.0364403i
\(267\) 0 0
\(268\) 243.183 0.907400
\(269\) − 412.009i − 1.53163i −0.643059 0.765816i \(-0.722335\pi\)
0.643059 0.765816i \(-0.277665\pi\)
\(270\) 0 0
\(271\) −339.138 −1.25143 −0.625716 0.780051i \(-0.715193\pi\)
−0.625716 + 0.780051i \(0.715193\pi\)
\(272\) − 81.3674i − 0.299145i
\(273\) 0 0
\(274\) −233.410 −0.851860
\(275\) − 2.74243i − 0.00997246i
\(276\) 0 0
\(277\) 345.943 1.24889 0.624447 0.781068i \(-0.285325\pi\)
0.624447 + 0.781068i \(0.285325\pi\)
\(278\) − 117.008i − 0.420892i
\(279\) 0 0
\(280\) 19.2165 0.0686303
\(281\) 531.371i 1.89100i 0.325621 + 0.945500i \(0.394427\pi\)
−0.325621 + 0.945500i \(0.605573\pi\)
\(282\) 0 0
\(283\) 168.014 0.593689 0.296844 0.954926i \(-0.404066\pi\)
0.296844 + 0.954926i \(0.404066\pi\)
\(284\) 74.7942i 0.263360i
\(285\) 0 0
\(286\) −5.81599 −0.0203356
\(287\) 215.636i 0.751344i
\(288\) 0 0
\(289\) 272.244 0.942021
\(290\) 3.70477i 0.0127751i
\(291\) 0 0
\(292\) 111.093 0.380455
\(293\) 81.2495i 0.277302i 0.990341 + 0.138651i \(0.0442766\pi\)
−0.990341 + 0.138651i \(0.955723\pi\)
\(294\) 0 0
\(295\) −68.3723 −0.231770
\(296\) − 258.320i − 0.872703i
\(297\) 0 0
\(298\) −351.363 −1.17907
\(299\) − 78.8810i − 0.263816i
\(300\) 0 0
\(301\) −190.122 −0.631635
\(302\) 710.411i 2.35235i
\(303\) 0 0
\(304\) −19.5029 −0.0641542
\(305\) − 8.39770i − 0.0275334i
\(306\) 0 0
\(307\) 65.7639 0.214215 0.107107 0.994247i \(-0.465841\pi\)
0.107107 + 0.994247i \(0.465841\pi\)
\(308\) 1.07001i 0.00347407i
\(309\) 0 0
\(310\) −4.45886 −0.0143834
\(311\) 321.306i 1.03314i 0.856245 + 0.516569i \(0.172791\pi\)
−0.856245 + 0.516569i \(0.827209\pi\)
\(312\) 0 0
\(313\) −175.188 −0.559705 −0.279852 0.960043i \(-0.590286\pi\)
−0.279852 + 0.960043i \(0.590286\pi\)
\(314\) − 225.438i − 0.717955i
\(315\) 0 0
\(316\) 96.2888 0.304711
\(317\) − 335.093i − 1.05708i −0.848909 0.528538i \(-0.822740\pi\)
0.848909 0.528538i \(-0.177260\pi\)
\(318\) 0 0
\(319\) 0.144888 0.000454194 0
\(320\) − 5.64636i − 0.0176449i
\(321\) 0 0
\(322\) −39.2175 −0.121794
\(323\) 4.01621i 0.0124341i
\(324\) 0 0
\(325\) −469.175 −1.44362
\(326\) − 226.378i − 0.694411i
\(327\) 0 0
\(328\) −228.722 −0.697323
\(329\) 242.030i 0.735653i
\(330\) 0 0
\(331\) 133.280 0.402659 0.201330 0.979524i \(-0.435474\pi\)
0.201330 + 0.979524i \(0.435474\pi\)
\(332\) 220.461i 0.664038i
\(333\) 0 0
\(334\) 546.102 1.63504
\(335\) 121.983i 0.364129i
\(336\) 0 0
\(337\) 303.240 0.899822 0.449911 0.893073i \(-0.351456\pi\)
0.449911 + 0.893073i \(0.351456\pi\)
\(338\) 569.145i 1.68386i
\(339\) 0 0
\(340\) −11.3363 −0.0333421
\(341\) 0.174379i 0 0.000511376i
\(342\) 0 0
\(343\) −323.956 −0.944478
\(344\) − 201.660i − 0.586221i
\(345\) 0 0
\(346\) −372.622 −1.07694
\(347\) − 139.489i − 0.401985i −0.979593 0.200992i \(-0.935583\pi\)
0.979593 0.200992i \(-0.0644166\pi\)
\(348\) 0 0
\(349\) 412.489 1.18192 0.590958 0.806702i \(-0.298750\pi\)
0.590958 + 0.806702i \(0.298750\pi\)
\(350\) 233.261i 0.666461i
\(351\) 0 0
\(352\) −3.88582 −0.0110393
\(353\) − 4.08080i − 0.0115603i −0.999983 0.00578016i \(-0.998160\pi\)
0.999983 0.00578016i \(-0.00183989\pi\)
\(354\) 0 0
\(355\) −37.5175 −0.105683
\(356\) − 217.780i − 0.611741i
\(357\) 0 0
\(358\) −47.0800 −0.131508
\(359\) 30.4463i 0.0848087i 0.999101 + 0.0424044i \(0.0135018\pi\)
−0.999101 + 0.0424044i \(0.986498\pi\)
\(360\) 0 0
\(361\) −360.037 −0.997333
\(362\) − 13.8095i − 0.0381478i
\(363\) 0 0
\(364\) 183.058 0.502907
\(365\) 55.7253i 0.152672i
\(366\) 0 0
\(367\) 599.427 1.63332 0.816658 0.577122i \(-0.195824\pi\)
0.816658 + 0.577122i \(0.195824\pi\)
\(368\) − 78.9070i − 0.214421i
\(369\) 0 0
\(370\) −184.489 −0.498618
\(371\) 106.385i 0.286751i
\(372\) 0 0
\(373\) 96.8953 0.259773 0.129886 0.991529i \(-0.458539\pi\)
0.129886 + 0.991529i \(0.458539\pi\)
\(374\) 1.19807i 0.00320341i
\(375\) 0 0
\(376\) −256.718 −0.682760
\(377\) − 24.7874i − 0.0657492i
\(378\) 0 0
\(379\) 678.905 1.79131 0.895653 0.444753i \(-0.146709\pi\)
0.895653 + 0.444753i \(0.146709\pi\)
\(380\) 2.71719i 0.00715049i
\(381\) 0 0
\(382\) 581.345 1.52185
\(383\) − 24.6072i − 0.0642486i −0.999484 0.0321243i \(-0.989773\pi\)
0.999484 0.0321243i \(-0.0102272\pi\)
\(384\) 0 0
\(385\) −0.536730 −0.00139410
\(386\) − 585.273i − 1.51625i
\(387\) 0 0
\(388\) −368.778 −0.950459
\(389\) 105.592i 0.271444i 0.990747 + 0.135722i \(0.0433353\pi\)
−0.990747 + 0.135722i \(0.956665\pi\)
\(390\) 0 0
\(391\) −16.2492 −0.0415582
\(392\) − 139.847i − 0.356752i
\(393\) 0 0
\(394\) −117.180 −0.297410
\(395\) 48.2995i 0.122277i
\(396\) 0 0
\(397\) −656.200 −1.65290 −0.826449 0.563012i \(-0.809643\pi\)
−0.826449 + 0.563012i \(0.809643\pi\)
\(398\) 582.083i 1.46252i
\(399\) 0 0
\(400\) −469.330 −1.17332
\(401\) 458.161i 1.14255i 0.820760 + 0.571273i \(0.193550\pi\)
−0.820760 + 0.571273i \(0.806450\pi\)
\(402\) 0 0
\(403\) 29.8329 0.0740269
\(404\) 206.969i 0.512299i
\(405\) 0 0
\(406\) −12.3236 −0.0303538
\(407\) 7.21506i 0.0177274i
\(408\) 0 0
\(409\) 303.807 0.742803 0.371402 0.928472i \(-0.378877\pi\)
0.371402 + 0.928472i \(0.378877\pi\)
\(410\) 163.350i 0.398415i
\(411\) 0 0
\(412\) 57.8909 0.140512
\(413\) − 227.435i − 0.550691i
\(414\) 0 0
\(415\) −110.585 −0.266471
\(416\) 664.787i 1.59805i
\(417\) 0 0
\(418\) 0.287165 0.000686998 0
\(419\) 743.246i 1.77386i 0.461906 + 0.886929i \(0.347166\pi\)
−0.461906 + 0.886929i \(0.652834\pi\)
\(420\) 0 0
\(421\) −487.274 −1.15742 −0.578710 0.815533i \(-0.696444\pi\)
−0.578710 + 0.815533i \(0.696444\pi\)
\(422\) − 332.681i − 0.788343i
\(423\) 0 0
\(424\) −112.841 −0.266134
\(425\) 96.6486i 0.227408i
\(426\) 0 0
\(427\) 27.9343 0.0654200
\(428\) − 125.921i − 0.294207i
\(429\) 0 0
\(430\) −144.023 −0.334936
\(431\) 834.575i 1.93637i 0.250237 + 0.968185i \(0.419491\pi\)
−0.250237 + 0.968185i \(0.580509\pi\)
\(432\) 0 0
\(433\) −454.402 −1.04943 −0.524713 0.851279i \(-0.675828\pi\)
−0.524713 + 0.851279i \(0.675828\pi\)
\(434\) − 14.8321i − 0.0341753i
\(435\) 0 0
\(436\) 258.753 0.593469
\(437\) 3.89476i 0.00891250i
\(438\) 0 0
\(439\) −69.1547 −0.157528 −0.0787639 0.996893i \(-0.525097\pi\)
−0.0787639 + 0.996893i \(0.525097\pi\)
\(440\) − 0.569302i − 0.00129387i
\(441\) 0 0
\(442\) 204.967 0.463726
\(443\) 550.971i 1.24373i 0.783126 + 0.621863i \(0.213624\pi\)
−0.783126 + 0.621863i \(0.786376\pi\)
\(444\) 0 0
\(445\) 109.241 0.245485
\(446\) − 676.070i − 1.51585i
\(447\) 0 0
\(448\) 18.7822 0.0419246
\(449\) 660.419i 1.47087i 0.677597 + 0.735433i \(0.263021\pi\)
−0.677597 + 0.735433i \(0.736979\pi\)
\(450\) 0 0
\(451\) 6.38837 0.0141649
\(452\) 275.263i 0.608990i
\(453\) 0 0
\(454\) 264.532 0.582669
\(455\) 91.8239i 0.201811i
\(456\) 0 0
\(457\) 332.594 0.727777 0.363888 0.931443i \(-0.381449\pi\)
0.363888 + 0.931443i \(0.381449\pi\)
\(458\) − 370.261i − 0.808429i
\(459\) 0 0
\(460\) −10.9935 −0.0238989
\(461\) − 129.547i − 0.281013i −0.990080 0.140506i \(-0.955127\pi\)
0.990080 0.140506i \(-0.0448730\pi\)
\(462\) 0 0
\(463\) 26.7590 0.0577949 0.0288975 0.999582i \(-0.490800\pi\)
0.0288975 + 0.999582i \(0.490800\pi\)
\(464\) − 24.7956i − 0.0534388i
\(465\) 0 0
\(466\) 244.324 0.524301
\(467\) 300.066i 0.642540i 0.946988 + 0.321270i \(0.104110\pi\)
−0.946988 + 0.321270i \(0.895890\pi\)
\(468\) 0 0
\(469\) −405.768 −0.865178
\(470\) 183.344i 0.390094i
\(471\) 0 0
\(472\) 241.238 0.511097
\(473\) 5.63250i 0.0119080i
\(474\) 0 0
\(475\) 23.1656 0.0487697
\(476\) − 37.7094i − 0.0792214i
\(477\) 0 0
\(478\) −501.216 −1.04857
\(479\) − 34.2241i − 0.0714491i −0.999362 0.0357246i \(-0.988626\pi\)
0.999362 0.0357246i \(-0.0113739\pi\)
\(480\) 0 0
\(481\) 1234.36 2.56623
\(482\) 665.958i 1.38166i
\(483\) 0 0
\(484\) −284.281 −0.587357
\(485\) − 184.983i − 0.381408i
\(486\) 0 0
\(487\) 241.028 0.494924 0.247462 0.968898i \(-0.420404\pi\)
0.247462 + 0.968898i \(0.420404\pi\)
\(488\) 29.6296i 0.0607163i
\(489\) 0 0
\(490\) −99.8766 −0.203830
\(491\) 221.985i 0.452107i 0.974115 + 0.226054i \(0.0725825\pi\)
−0.974115 + 0.226054i \(0.927418\pi\)
\(492\) 0 0
\(493\) −5.10613 −0.0103573
\(494\) − 49.1283i − 0.0994501i
\(495\) 0 0
\(496\) 29.8427 0.0601667
\(497\) − 124.799i − 0.251105i
\(498\) 0 0
\(499\) 618.851 1.24018 0.620091 0.784530i \(-0.287096\pi\)
0.620091 + 0.784530i \(0.287096\pi\)
\(500\) 134.623i 0.269247i
\(501\) 0 0
\(502\) −79.7389 −0.158843
\(503\) 172.036i 0.342020i 0.985269 + 0.171010i \(0.0547030\pi\)
−0.985269 + 0.171010i \(0.945297\pi\)
\(504\) 0 0
\(505\) −103.818 −0.205580
\(506\) 1.16185i 0.00229614i
\(507\) 0 0
\(508\) 26.4797 0.0521253
\(509\) 592.690i 1.16442i 0.813038 + 0.582211i \(0.197812\pi\)
−0.813038 + 0.582211i \(0.802188\pi\)
\(510\) 0 0
\(511\) −185.366 −0.362751
\(512\) 334.356i 0.653039i
\(513\) 0 0
\(514\) −871.774 −1.69606
\(515\) 29.0387i 0.0563858i
\(516\) 0 0
\(517\) 7.17030 0.0138691
\(518\) − 613.688i − 1.18473i
\(519\) 0 0
\(520\) −97.3963 −0.187301
\(521\) − 622.280i − 1.19439i −0.802094 0.597197i \(-0.796281\pi\)
0.802094 0.597197i \(-0.203719\pi\)
\(522\) 0 0
\(523\) −357.150 −0.682887 −0.341443 0.939902i \(-0.610916\pi\)
−0.341443 + 0.939902i \(0.610916\pi\)
\(524\) − 501.010i − 0.956126i
\(525\) 0 0
\(526\) −663.195 −1.26083
\(527\) − 6.14547i − 0.0116612i
\(528\) 0 0
\(529\) 513.242 0.970212
\(530\) 80.5892i 0.152055i
\(531\) 0 0
\(532\) −9.03852 −0.0169897
\(533\) − 1092.92i − 2.05051i
\(534\) 0 0
\(535\) 63.1630 0.118062
\(536\) − 430.393i − 0.802972i
\(537\) 0 0
\(538\) 1038.21 1.92975
\(539\) 3.90602i 0.00724678i
\(540\) 0 0
\(541\) −277.130 −0.512255 −0.256128 0.966643i \(-0.582447\pi\)
−0.256128 + 0.966643i \(0.582447\pi\)
\(542\) − 854.580i − 1.57672i
\(543\) 0 0
\(544\) 136.944 0.251735
\(545\) 129.793i 0.238152i
\(546\) 0 0
\(547\) 629.328 1.15051 0.575254 0.817975i \(-0.304903\pi\)
0.575254 + 0.817975i \(0.304903\pi\)
\(548\) − 217.647i − 0.397166i
\(549\) 0 0
\(550\) 6.91053 0.0125646
\(551\) 1.22388i 0.00222120i
\(552\) 0 0
\(553\) −160.665 −0.290533
\(554\) 871.729i 1.57352i
\(555\) 0 0
\(556\) 109.106 0.196234
\(557\) − 92.7731i − 0.166559i −0.996526 0.0832793i \(-0.973461\pi\)
0.996526 0.0832793i \(-0.0265394\pi\)
\(558\) 0 0
\(559\) 963.610 1.72381
\(560\) 91.8541i 0.164025i
\(561\) 0 0
\(562\) −1338.98 −2.38253
\(563\) 1037.13i 1.84214i 0.389396 + 0.921071i \(0.372684\pi\)
−0.389396 + 0.921071i \(0.627316\pi\)
\(564\) 0 0
\(565\) −138.075 −0.244380
\(566\) 423.371i 0.748006i
\(567\) 0 0
\(568\) 132.373 0.233051
\(569\) 192.618i 0.338521i 0.985571 + 0.169260i \(0.0541379\pi\)
−0.985571 + 0.169260i \(0.945862\pi\)
\(570\) 0 0
\(571\) 447.153 0.783105 0.391552 0.920156i \(-0.371938\pi\)
0.391552 + 0.920156i \(0.371938\pi\)
\(572\) − 5.42323i − 0.00948117i
\(573\) 0 0
\(574\) −543.372 −0.946641
\(575\) 93.7260i 0.163002i
\(576\) 0 0
\(577\) −345.436 −0.598675 −0.299338 0.954147i \(-0.596766\pi\)
−0.299338 + 0.954147i \(0.596766\pi\)
\(578\) 686.017i 1.18688i
\(579\) 0 0
\(580\) −3.45458 −0.00595617
\(581\) − 367.854i − 0.633140i
\(582\) 0 0
\(583\) 3.15172 0.00540603
\(584\) − 196.615i − 0.336670i
\(585\) 0 0
\(586\) −204.737 −0.349381
\(587\) 459.636i 0.783026i 0.920173 + 0.391513i \(0.128048\pi\)
−0.920173 + 0.391513i \(0.871952\pi\)
\(588\) 0 0
\(589\) −1.47300 −0.00250085
\(590\) − 172.289i − 0.292014i
\(591\) 0 0
\(592\) 1234.76 2.08575
\(593\) − 782.439i − 1.31946i −0.751504 0.659729i \(-0.770671\pi\)
0.751504 0.659729i \(-0.229329\pi\)
\(594\) 0 0
\(595\) 18.9154 0.0317906
\(596\) − 327.635i − 0.549724i
\(597\) 0 0
\(598\) 198.769 0.332390
\(599\) − 162.105i − 0.270626i −0.990803 0.135313i \(-0.956796\pi\)
0.990803 0.135313i \(-0.0432040\pi\)
\(600\) 0 0
\(601\) −410.144 −0.682436 −0.341218 0.939984i \(-0.610839\pi\)
−0.341218 + 0.939984i \(0.610839\pi\)
\(602\) − 479.081i − 0.795815i
\(603\) 0 0
\(604\) −662.436 −1.09675
\(605\) − 142.598i − 0.235699i
\(606\) 0 0
\(607\) −564.201 −0.929490 −0.464745 0.885444i \(-0.653854\pi\)
−0.464745 + 0.885444i \(0.653854\pi\)
\(608\) − 32.8240i − 0.0539868i
\(609\) 0 0
\(610\) 21.1610 0.0346902
\(611\) − 1226.70i − 2.00769i
\(612\) 0 0
\(613\) 842.376 1.37419 0.687093 0.726569i \(-0.258886\pi\)
0.687093 + 0.726569i \(0.258886\pi\)
\(614\) 165.716i 0.269896i
\(615\) 0 0
\(616\) 1.89374 0.00307425
\(617\) − 298.792i − 0.484266i −0.970243 0.242133i \(-0.922153\pi\)
0.970243 0.242133i \(-0.0778470\pi\)
\(618\) 0 0
\(619\) 402.914 0.650910 0.325455 0.945557i \(-0.394482\pi\)
0.325455 + 0.945557i \(0.394482\pi\)
\(620\) − 4.15775i − 0.00670605i
\(621\) 0 0
\(622\) −809.646 −1.30168
\(623\) 363.381i 0.583276i
\(624\) 0 0
\(625\) 522.742 0.836388
\(626\) − 441.448i − 0.705189i
\(627\) 0 0
\(628\) 210.214 0.334735
\(629\) − 254.273i − 0.404250i
\(630\) 0 0
\(631\) 617.724 0.978960 0.489480 0.872015i \(-0.337187\pi\)
0.489480 + 0.872015i \(0.337187\pi\)
\(632\) − 170.415i − 0.269644i
\(633\) 0 0
\(634\) 844.388 1.33184
\(635\) 13.2825i 0.0209173i
\(636\) 0 0
\(637\) 668.243 1.04905
\(638\) 0.365097i 0 0.000572252i
\(639\) 0 0
\(640\) −143.495 −0.224211
\(641\) − 871.699i − 1.35991i −0.733256 0.679953i \(-0.762000\pi\)
0.733256 0.679953i \(-0.238000\pi\)
\(642\) 0 0
\(643\) −915.020 −1.42305 −0.711524 0.702661i \(-0.751995\pi\)
−0.711524 + 0.702661i \(0.751995\pi\)
\(644\) − 36.5691i − 0.0567843i
\(645\) 0 0
\(646\) −10.1203 −0.0156661
\(647\) 99.8398i 0.154312i 0.997019 + 0.0771559i \(0.0245839\pi\)
−0.997019 + 0.0771559i \(0.975416\pi\)
\(648\) 0 0
\(649\) −6.73794 −0.0103820
\(650\) − 1182.26i − 1.81885i
\(651\) 0 0
\(652\) 211.090 0.323758
\(653\) − 322.455i − 0.493805i −0.969040 0.246903i \(-0.920587\pi\)
0.969040 0.246903i \(-0.0794128\pi\)
\(654\) 0 0
\(655\) 251.312 0.383682
\(656\) − 1093.28i − 1.66659i
\(657\) 0 0
\(658\) −609.881 −0.926871
\(659\) − 1044.59i − 1.58512i −0.609796 0.792558i \(-0.708749\pi\)
0.609796 0.792558i \(-0.291251\pi\)
\(660\) 0 0
\(661\) −557.410 −0.843282 −0.421641 0.906763i \(-0.638546\pi\)
−0.421641 + 0.906763i \(0.638546\pi\)
\(662\) 335.848i 0.507323i
\(663\) 0 0
\(664\) 390.178 0.587617
\(665\) − 4.53382i − 0.00681777i
\(666\) 0 0
\(667\) −4.95173 −0.00742388
\(668\) 509.223i 0.762310i
\(669\) 0 0
\(670\) −307.381 −0.458777
\(671\) − 0.827575i − 0.00123335i
\(672\) 0 0
\(673\) −1035.16 −1.53812 −0.769062 0.639174i \(-0.779276\pi\)
−0.769062 + 0.639174i \(0.779276\pi\)
\(674\) 764.122i 1.13371i
\(675\) 0 0
\(676\) −530.710 −0.785074
\(677\) 623.131i 0.920430i 0.887807 + 0.460215i \(0.152228\pi\)
−0.887807 + 0.460215i \(0.847772\pi\)
\(678\) 0 0
\(679\) 615.332 0.906233
\(680\) 20.0633i 0.0295049i
\(681\) 0 0
\(682\) −0.439411 −0.000644298 0
\(683\) 103.181i 0.151070i 0.997143 + 0.0755351i \(0.0240665\pi\)
−0.997143 + 0.0755351i \(0.975934\pi\)
\(684\) 0 0
\(685\) 109.174 0.159378
\(686\) − 816.323i − 1.18998i
\(687\) 0 0
\(688\) 963.927 1.40106
\(689\) − 539.197i − 0.782579i
\(690\) 0 0
\(691\) 326.699 0.472791 0.236396 0.971657i \(-0.424034\pi\)
0.236396 + 0.971657i \(0.424034\pi\)
\(692\) − 347.458i − 0.502107i
\(693\) 0 0
\(694\) 351.492 0.506472
\(695\) 54.7289i 0.0787466i
\(696\) 0 0
\(697\) −225.139 −0.323011
\(698\) 1039.41i 1.48913i
\(699\) 0 0
\(700\) −217.509 −0.310727
\(701\) − 882.429i − 1.25881i −0.777076 0.629407i \(-0.783298\pi\)
0.777076 0.629407i \(-0.216702\pi\)
\(702\) 0 0
\(703\) −60.9465 −0.0866948
\(704\) − 0.556436i 0 0.000790393i
\(705\) 0 0
\(706\) 10.2830 0.0145652
\(707\) − 345.342i − 0.488461i
\(708\) 0 0
\(709\) 29.3380 0.0413794 0.0206897 0.999786i \(-0.493414\pi\)
0.0206897 + 0.999786i \(0.493414\pi\)
\(710\) − 94.5389i − 0.133153i
\(711\) 0 0
\(712\) −385.433 −0.541339
\(713\) − 5.95964i − 0.00835854i
\(714\) 0 0
\(715\) 2.72035 0.00380468
\(716\) − 43.9006i − 0.0613138i
\(717\) 0 0
\(718\) −76.7205 −0.106853
\(719\) − 112.619i − 0.156633i −0.996929 0.0783163i \(-0.975046\pi\)
0.996929 0.0783163i \(-0.0249544\pi\)
\(720\) 0 0
\(721\) −96.5950 −0.133974
\(722\) − 907.243i − 1.25657i
\(723\) 0 0
\(724\) 12.8769 0.0177858
\(725\) 29.4523i 0.0406239i
\(726\) 0 0
\(727\) 60.0874 0.0826511 0.0413256 0.999146i \(-0.486842\pi\)
0.0413256 + 0.999146i \(0.486842\pi\)
\(728\) − 323.982i − 0.445030i
\(729\) 0 0
\(730\) −140.420 −0.192356
\(731\) − 198.500i − 0.271546i
\(732\) 0 0
\(733\) 497.573 0.678817 0.339409 0.940639i \(-0.389773\pi\)
0.339409 + 0.940639i \(0.389773\pi\)
\(734\) 1510.47i 2.05786i
\(735\) 0 0
\(736\) 132.803 0.180439
\(737\) 12.0212i 0.0163110i
\(738\) 0 0
\(739\) 216.118 0.292447 0.146223 0.989252i \(-0.453288\pi\)
0.146223 + 0.989252i \(0.453288\pi\)
\(740\) − 172.030i − 0.232473i
\(741\) 0 0
\(742\) −268.074 −0.361286
\(743\) − 1121.65i − 1.50962i −0.655944 0.754810i \(-0.727729\pi\)
0.655944 0.754810i \(-0.272271\pi\)
\(744\) 0 0
\(745\) 164.345 0.220598
\(746\) 244.162i 0.327296i
\(747\) 0 0
\(748\) −1.11717 −0.00149354
\(749\) 210.107i 0.280517i
\(750\) 0 0
\(751\) 235.599 0.313714 0.156857 0.987621i \(-0.449864\pi\)
0.156857 + 0.987621i \(0.449864\pi\)
\(752\) − 1227.10i − 1.63178i
\(753\) 0 0
\(754\) 62.4609 0.0828393
\(755\) − 332.285i − 0.440112i
\(756\) 0 0
\(757\) −2.57579 −0.00340263 −0.00170132 0.999999i \(-0.500542\pi\)
−0.00170132 + 0.999999i \(0.500542\pi\)
\(758\) 1710.75i 2.25692i
\(759\) 0 0
\(760\) 4.80896 0.00632758
\(761\) − 311.007i − 0.408682i −0.978900 0.204341i \(-0.934495\pi\)
0.978900 0.204341i \(-0.0655052\pi\)
\(762\) 0 0
\(763\) −431.747 −0.565854
\(764\) 542.086i 0.709537i
\(765\) 0 0
\(766\) 62.0067 0.0809486
\(767\) 1152.73i 1.50291i
\(768\) 0 0
\(769\) −794.978 −1.03378 −0.516891 0.856051i \(-0.672911\pi\)
−0.516891 + 0.856051i \(0.672911\pi\)
\(770\) − 1.35248i − 0.00175647i
\(771\) 0 0
\(772\) 545.749 0.706929
\(773\) 319.749i 0.413647i 0.978378 + 0.206823i \(0.0663125\pi\)
−0.978378 + 0.206823i \(0.933687\pi\)
\(774\) 0 0
\(775\) −35.4472 −0.0457384
\(776\) 652.675i 0.841075i
\(777\) 0 0
\(778\) −266.076 −0.342000
\(779\) 53.9633i 0.0692725i
\(780\) 0 0
\(781\) −3.69727 −0.00473402
\(782\) − 40.9458i − 0.0523603i
\(783\) 0 0
\(784\) 668.462 0.852631
\(785\) 105.445i 0.134325i
\(786\) 0 0
\(787\) 548.874 0.697426 0.348713 0.937230i \(-0.386619\pi\)
0.348713 + 0.937230i \(0.386619\pi\)
\(788\) − 109.266i − 0.138663i
\(789\) 0 0
\(790\) −121.708 −0.154061
\(791\) − 459.296i − 0.580653i
\(792\) 0 0
\(793\) −141.582 −0.178539
\(794\) − 1653.53i − 2.08253i
\(795\) 0 0
\(796\) −542.774 −0.681876
\(797\) − 362.036i − 0.454249i −0.973866 0.227124i \(-0.927068\pi\)
0.973866 0.227124i \(-0.0729324\pi\)
\(798\) 0 0
\(799\) −252.696 −0.316265
\(800\) − 789.897i − 0.987371i
\(801\) 0 0
\(802\) −1154.50 −1.43953
\(803\) 5.49160i 0.00683885i
\(804\) 0 0
\(805\) 18.3434 0.0227869
\(806\) 75.1746i 0.0932687i
\(807\) 0 0
\(808\) 366.299 0.453341
\(809\) − 1305.60i − 1.61385i −0.590655 0.806924i \(-0.701131\pi\)
0.590655 0.806924i \(-0.298869\pi\)
\(810\) 0 0
\(811\) 6.45607 0.00796063 0.00398032 0.999992i \(-0.498733\pi\)
0.00398032 + 0.999992i \(0.498733\pi\)
\(812\) − 11.4914i − 0.0141520i
\(813\) 0 0
\(814\) −18.1809 −0.0223353
\(815\) 105.885i 0.129920i
\(816\) 0 0
\(817\) −47.5784 −0.0582355
\(818\) 765.550i 0.935880i
\(819\) 0 0
\(820\) −152.319 −0.185755
\(821\) − 240.575i − 0.293027i −0.989209 0.146513i \(-0.953195\pi\)
0.989209 0.146513i \(-0.0468051\pi\)
\(822\) 0 0
\(823\) −226.340 −0.275018 −0.137509 0.990501i \(-0.543910\pi\)
−0.137509 + 0.990501i \(0.543910\pi\)
\(824\) − 102.457i − 0.124341i
\(825\) 0 0
\(826\) 573.105 0.693832
\(827\) − 1086.04i − 1.31323i −0.754228 0.656613i \(-0.771988\pi\)
0.754228 0.656613i \(-0.228012\pi\)
\(828\) 0 0
\(829\) 458.391 0.552945 0.276473 0.961022i \(-0.410835\pi\)
0.276473 + 0.961022i \(0.410835\pi\)
\(830\) − 278.660i − 0.335734i
\(831\) 0 0
\(832\) −95.1953 −0.114417
\(833\) − 137.656i − 0.165253i
\(834\) 0 0
\(835\) −255.432 −0.305906
\(836\) 0.267773i 0 0.000320302i
\(837\) 0 0
\(838\) −1872.88 −2.23494
\(839\) 770.491i 0.918344i 0.888347 + 0.459172i \(0.151854\pi\)
−0.888347 + 0.459172i \(0.848146\pi\)
\(840\) 0 0
\(841\) 839.444 0.998150
\(842\) − 1227.86i − 1.45827i
\(843\) 0 0
\(844\) 310.214 0.367553
\(845\) − 266.210i − 0.315041i
\(846\) 0 0
\(847\) 474.342 0.560026
\(848\) − 539.374i − 0.636055i
\(849\) 0 0
\(850\) −243.541 −0.286519
\(851\) − 246.584i − 0.289758i
\(852\) 0 0
\(853\) −142.876 −0.167498 −0.0837489 0.996487i \(-0.526689\pi\)
−0.0837489 + 0.996487i \(0.526689\pi\)
\(854\) 70.3906i 0.0824246i
\(855\) 0 0
\(856\) −222.858 −0.260348
\(857\) 905.026i 1.05604i 0.849232 + 0.528020i \(0.177065\pi\)
−0.849232 + 0.528020i \(0.822935\pi\)
\(858\) 0 0
\(859\) 143.642 0.167220 0.0836101 0.996499i \(-0.473355\pi\)
0.0836101 + 0.996499i \(0.473355\pi\)
\(860\) − 134.297i − 0.156159i
\(861\) 0 0
\(862\) −2103.01 −2.43969
\(863\) − 915.923i − 1.06132i −0.847583 0.530662i \(-0.821943\pi\)
0.847583 0.530662i \(-0.178057\pi\)
\(864\) 0 0
\(865\) 174.289 0.201490
\(866\) − 1145.03i − 1.32220i
\(867\) 0 0
\(868\) 13.8305 0.0159337
\(869\) 4.75980i 0.00547733i
\(870\) 0 0
\(871\) 2056.59 2.36118
\(872\) − 457.948i − 0.525170i
\(873\) 0 0
\(874\) −9.81425 −0.0112291
\(875\) − 224.629i − 0.256718i
\(876\) 0 0
\(877\) −172.726 −0.196951 −0.0984753 0.995139i \(-0.531397\pi\)
−0.0984753 + 0.995139i \(0.531397\pi\)
\(878\) − 174.260i − 0.198474i
\(879\) 0 0
\(880\) 2.72124 0.00309232
\(881\) 383.373i 0.435156i 0.976043 + 0.217578i \(0.0698157\pi\)
−0.976043 + 0.217578i \(0.930184\pi\)
\(882\) 0 0
\(883\) −1187.34 −1.34467 −0.672334 0.740248i \(-0.734708\pi\)
−0.672334 + 0.740248i \(0.734708\pi\)
\(884\) 191.125i 0.216205i
\(885\) 0 0
\(886\) −1388.37 −1.56701
\(887\) 1142.28i 1.28781i 0.765107 + 0.643903i \(0.222686\pi\)
−0.765107 + 0.643903i \(0.777314\pi\)
\(888\) 0 0
\(889\) −44.1832 −0.0496998
\(890\) 275.271i 0.309293i
\(891\) 0 0
\(892\) 630.414 0.706742
\(893\) 60.5684i 0.0678257i
\(894\) 0 0
\(895\) 22.0210 0.0246045
\(896\) − 477.327i − 0.532731i
\(897\) 0 0
\(898\) −1664.16 −1.85319
\(899\) − 1.87275i − 0.00208314i
\(900\) 0 0
\(901\) −111.073 −0.123277
\(902\) 16.0978i 0.0178468i
\(903\) 0 0
\(904\) 487.169 0.538904
\(905\) 6.45920i 0.00713724i
\(906\) 0 0
\(907\) −412.022 −0.454269 −0.227135 0.973863i \(-0.572936\pi\)
−0.227135 + 0.973863i \(0.572936\pi\)
\(908\) 246.667i 0.271660i
\(909\) 0 0
\(910\) −231.383 −0.254267
\(911\) − 1002.94i − 1.10093i −0.834859 0.550464i \(-0.814451\pi\)
0.834859 0.550464i \(-0.185549\pi\)
\(912\) 0 0
\(913\) −10.8979 −0.0119364
\(914\) 838.090i 0.916947i
\(915\) 0 0
\(916\) 345.256 0.376918
\(917\) 835.970i 0.911636i
\(918\) 0 0
\(919\) −1339.72 −1.45781 −0.728903 0.684616i \(-0.759970\pi\)
−0.728903 + 0.684616i \(0.759970\pi\)
\(920\) 19.4566i 0.0211485i
\(921\) 0 0
\(922\) 326.440 0.354056
\(923\) 632.530i 0.685298i
\(924\) 0 0
\(925\) −1466.65 −1.58557
\(926\) 67.4290i 0.0728175i
\(927\) 0 0
\(928\) 41.7318 0.0449696
\(929\) − 355.644i − 0.382824i −0.981510 0.191412i \(-0.938693\pi\)
0.981510 0.191412i \(-0.0613067\pi\)
\(930\) 0 0
\(931\) −32.9946 −0.0354399
\(932\) 227.825i 0.244447i
\(933\) 0 0
\(934\) −756.125 −0.809556
\(935\) − 0.560383i 0 0.000599340i
\(936\) 0 0
\(937\) 548.887 0.585792 0.292896 0.956144i \(-0.405381\pi\)
0.292896 + 0.956144i \(0.405381\pi\)
\(938\) − 1022.48i − 1.09006i
\(939\) 0 0
\(940\) −170.963 −0.181875
\(941\) 648.277i 0.688924i 0.938800 + 0.344462i \(0.111939\pi\)
−0.938800 + 0.344462i \(0.888061\pi\)
\(942\) 0 0
\(943\) −218.331 −0.231528
\(944\) 1153.11i 1.22151i
\(945\) 0 0
\(946\) −14.1931 −0.0150033
\(947\) − 284.695i − 0.300628i −0.988638 0.150314i \(-0.951971\pi\)
0.988638 0.150314i \(-0.0480285\pi\)
\(948\) 0 0
\(949\) 939.504 0.989994
\(950\) 58.3740i 0.0614463i
\(951\) 0 0
\(952\) −66.7392 −0.0701042
\(953\) 159.812i 0.167694i 0.996479 + 0.0838468i \(0.0267206\pi\)
−0.996479 + 0.0838468i \(0.973279\pi\)
\(954\) 0 0
\(955\) −271.916 −0.284729
\(956\) − 467.368i − 0.488879i
\(957\) 0 0
\(958\) 86.2400 0.0900208
\(959\) 363.159i 0.378686i
\(960\) 0 0
\(961\) −958.746 −0.997655
\(962\) 3110.40i 3.23326i
\(963\) 0 0
\(964\) −620.985 −0.644176
\(965\) 273.753i 0.283682i
\(966\) 0 0
\(967\) 487.083 0.503705 0.251852 0.967766i \(-0.418960\pi\)
0.251852 + 0.967766i \(0.418960\pi\)
\(968\) 503.128i 0.519761i
\(969\) 0 0
\(970\) 466.131 0.480547
\(971\) 126.842i 0.130630i 0.997865 + 0.0653150i \(0.0208052\pi\)
−0.997865 + 0.0653150i \(0.979195\pi\)
\(972\) 0 0
\(973\) −182.052 −0.187103
\(974\) 607.356i 0.623569i
\(975\) 0 0
\(976\) −141.628 −0.145111
\(977\) − 1131.60i − 1.15824i −0.815243 0.579119i \(-0.803396\pi\)
0.815243 0.579119i \(-0.196604\pi\)
\(978\) 0 0
\(979\) 10.7654 0.0109963
\(980\) − 93.1318i − 0.0950324i
\(981\) 0 0
\(982\) −559.370 −0.569623
\(983\) − 120.256i − 0.122336i −0.998127 0.0611679i \(-0.980517\pi\)
0.998127 0.0611679i \(-0.0194825\pi\)
\(984\) 0 0
\(985\) 54.8091 0.0556438
\(986\) − 12.8667i − 0.0130494i
\(987\) 0 0
\(988\) 45.8106 0.0463670
\(989\) − 192.498i − 0.194639i
\(990\) 0 0
\(991\) 1347.36 1.35959 0.679796 0.733401i \(-0.262068\pi\)
0.679796 + 0.733401i \(0.262068\pi\)
\(992\) 50.2262i 0.0506312i
\(993\) 0 0
\(994\) 314.477 0.316375
\(995\) − 272.261i − 0.273629i
\(996\) 0 0
\(997\) −275.154 −0.275982 −0.137991 0.990433i \(-0.544065\pi\)
−0.137991 + 0.990433i \(0.544065\pi\)
\(998\) 1559.42i 1.56254i
\(999\) 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 1143.3.b.a.890.67 yes 84
3.2 odd 2 inner 1143.3.b.a.890.18 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.18 84 3.2 odd 2 inner
1143.3.b.a.890.67 yes 84 1.1 even 1 trivial