Properties

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

$q$-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 \(-0.783069\pi\)
0.776624 0.629965i \(-0.216931\pi\)
\(3\) 0 0
\(4\) −2.34969 −0.587422
\(5\) 1.17863i 0.235726i 0.993030 + 0.117863i \(0.0376043\pi\)
−0.993030 + 0.117863i \(0.962396\pi\)
\(6\) 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 \(0.00168056\pi\)
−0.999986 + 0.00527960i \(0.998319\pi\)
\(12\) 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 \(-0.961584\pi\)
0.992726 0.120394i \(-0.0384158\pi\)
\(18\) 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 \(-0.972497\pi\)
0.996270 0.0862961i \(-0.0275031\pi\)
\(24\) 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 \(-0.993154\pi\)
0.999769 0.0215070i \(-0.00684642\pi\)
\(30\) 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 \(-0.765978\pi\)
0.741695 0.670737i \(-0.234022\pi\)
\(42\) 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 \(-0.771940\pi\)
0.754127 0.656729i \(-0.228060\pi\)
\(48\) 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 \(-0.917600\pi\)
0.966680 0.255987i \(-0.0824004\pi\)
\(54\) 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 \(0.163592\pi\)
−0.870815 + 0.491611i \(0.836408\pi\)
\(60\) 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 \(0.0719657\pi\)
−0.974551 + 0.224166i \(0.928034\pi\)
\(72\) 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 \(0.191206\pi\)
−0.824945 + 0.565214i \(0.808794\pi\)
\(84\) 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 \(-0.825671\pi\)
0.853740 0.520700i \(-0.174329\pi\)
\(90\) 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 \(0.143625\pi\)
−0.899919 + 0.436057i \(0.856375\pi\)
\(102\) 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 \(-0.919431\pi\)
0.968137 0.250422i \(-0.0805693\pi\)
\(108\) 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 \(0.173456\pi\)
−0.855164 + 0.518358i \(0.826544\pi\)
\(114\) 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 \(-0.697378\pi\)
0.581100 0.813832i \(-0.302622\pi\)
\(132\) 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 \(-0.890230\pi\)
0.941125 0.338058i \(-0.109770\pi\)
\(138\) 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 \(-0.845006\pi\)
0.883775 0.467912i \(-0.154994\pi\)
\(150\) 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 \(0.224754\pi\)
−0.760908 + 0.648860i \(0.775246\pi\)
\(168\) 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 \(-0.859436\pi\)
0.904071 0.427382i \(-0.140564\pi\)
\(174\) 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 \(-0.983380\pi\)
0.998637 0.0521888i \(-0.0166198\pi\)
\(180\) 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 \(0.206404\pi\)
−0.797029 + 0.603941i \(0.793596\pi\)
\(192\) 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 \(-0.962343\pi\)
0.993010 0.118027i \(-0.0376568\pi\)
\(198\) 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 \(0.0742752\pi\)
−0.972899 + 0.231231i \(0.925725\pi\)
\(228\) 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 \(0.0667174\pi\)
−0.978114 + 0.208068i \(0.933283\pi\)
\(234\) 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 \(-0.863389\pi\)
0.909309 0.416122i \(-0.136611\pi\)
\(240\) 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 \(-0.979922\pi\)
0.998011 0.0630363i \(-0.0200784\pi\)
\(252\) 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 \(-0.764972\pi\)
0.739572 0.673077i \(-0.235028\pi\)
\(258\) 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 \(-0.833202\pi\)
0.865819 0.500357i \(-0.166798\pi\)
\(264\) 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 \(0.277665\pi\)
−0.643059 + 0.765816i \(0.722335\pi\)
\(270\) 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 \(-0.605573\pi\)
0.325621 0.945500i \(-0.394427\pi\)
\(282\) 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 \(-0.955723\pi\)
0.990341 0.138651i \(-0.0442766\pi\)
\(294\) 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 \(-0.827209\pi\)
0.856245 0.516569i \(-0.172791\pi\)
\(312\) 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 \(0.177260\pi\)
−0.848909 + 0.528538i \(0.822740\pi\)
\(318\) 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 \(0.0644166\pi\)
−0.979593 + 0.200992i \(0.935583\pi\)
\(348\) 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 \(0.00183989\pi\)
−0.999983 + 0.00578016i \(0.998160\pi\)
\(354\) 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 \(-0.986498\pi\)
0.999101 0.0424044i \(-0.0135018\pi\)
\(360\) 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 \(0.0102272\pi\)
−0.999484 + 0.0321243i \(0.989773\pi\)
\(384\) 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 \(-0.956665\pi\)
0.990747 0.135722i \(-0.0433353\pi\)
\(390\) 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 \(-0.806450\pi\)
0.820760 0.571273i \(-0.193550\pi\)
\(402\) 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 \(-0.652834\pi\)
0.461906 0.886929i \(-0.347166\pi\)
\(420\) 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 \(-0.580509\pi\)
0.250237 0.968185i \(-0.419491\pi\)
\(432\) 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 \(-0.786376\pi\)
0.783126 0.621863i \(-0.213624\pi\)
\(444\) 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 \(-0.736979\pi\)
0.677597 0.735433i \(-0.263021\pi\)
\(450\) 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 \(0.0448730\pi\)
−0.990080 + 0.140506i \(0.955127\pi\)
\(462\) 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 \(-0.895890\pi\)
0.946988 0.321270i \(-0.104110\pi\)
\(468\) 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 \(0.0113739\pi\)
−0.999362 + 0.0357246i \(0.988626\pi\)
\(480\) 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 \(-0.927418\pi\)
0.974115 0.226054i \(-0.0725825\pi\)
\(492\) 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 \(-0.945297\pi\)
0.985269 0.171010i \(-0.0547030\pi\)
\(504\) 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 \(-0.802188\pi\)
0.813038 0.582211i \(-0.197812\pi\)
\(510\) 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 \(0.203719\pi\)
−0.802094 + 0.597197i \(0.796281\pi\)
\(522\) 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 \(0.0265394\pi\)
−0.996526 + 0.0832793i \(0.973461\pi\)
\(558\) 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 \(-0.627316\pi\)
0.389396 0.921071i \(-0.372684\pi\)
\(564\) 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 \(-0.945862\pi\)
0.985571 0.169260i \(-0.0541379\pi\)
\(570\) 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 \(-0.871952\pi\)
0.920173 0.391513i \(-0.128048\pi\)
\(588\) 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 \(0.229329\pi\)
−0.751504 + 0.659729i \(0.770671\pi\)
\(594\) 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 \(0.0432040\pi\)
−0.990803 + 0.135313i \(0.956796\pi\)
\(600\) 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 \(0.0778470\pi\)
−0.970243 + 0.242133i \(0.922153\pi\)
\(618\) 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 \(0.238000\pi\)
−0.733256 + 0.679953i \(0.762000\pi\)
\(642\) 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 \(-0.975416\pi\)
0.997019 0.0771559i \(-0.0245839\pi\)
\(648\) 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 \(0.0794128\pi\)
−0.969040 + 0.246903i \(0.920587\pi\)
\(654\) 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 \(0.291251\pi\)
−0.609796 + 0.792558i \(0.708749\pi\)
\(660\) 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 \(-0.847772\pi\)
0.887807 0.460215i \(-0.152228\pi\)
\(678\) 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 \(-0.975934\pi\)
0.997143 0.0755351i \(-0.0240665\pi\)
\(684\) 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 \(0.216702\pi\)
−0.777076 + 0.629407i \(0.783298\pi\)
\(702\) 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 \(0.0249544\pi\)
−0.996929 + 0.0783163i \(0.975046\pi\)
\(720\) 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 \(0.272271\pi\)
−0.655944 + 0.754810i \(0.727729\pi\)
\(744\) 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 \(0.0655052\pi\)
−0.978900 + 0.204341i \(0.934495\pi\)
\(762\) 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 \(-0.933687\pi\)
0.978378 0.206823i \(-0.0663125\pi\)
\(774\) 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 \(0.0729324\pi\)
−0.973866 + 0.227124i \(0.927068\pi\)
\(798\) 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 \(0.298869\pi\)
−0.590655 + 0.806924i \(0.701131\pi\)
\(810\) 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 \(0.0468051\pi\)
−0.989209 + 0.146513i \(0.953195\pi\)
\(822\) 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 \(0.228012\pi\)
−0.754228 + 0.656613i \(0.771988\pi\)
\(828\) 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 \(-0.848146\pi\)
0.888347 0.459172i \(-0.151854\pi\)
\(840\) 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 \(-0.822935\pi\)
0.849232 0.528020i \(-0.177065\pi\)
\(858\) 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 \(0.178057\pi\)
−0.847583 + 0.530662i \(0.821943\pi\)
\(864\) 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 \(-0.930184\pi\)
0.976043 0.217578i \(-0.0698157\pi\)
\(882\) 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 \(-0.777314\pi\)
0.765107 0.643903i \(-0.222686\pi\)
\(888\) 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 \(0.185549\pi\)
−0.834859 + 0.550464i \(0.814451\pi\)
\(912\) 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 \(0.0613067\pi\)
−0.981510 + 0.191412i \(0.938693\pi\)
\(930\) 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 \(-0.888061\pi\)
0.938800 0.344462i \(-0.111939\pi\)
\(942\) 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 \(0.0480285\pi\)
−0.988638 + 0.150314i \(0.951971\pi\)
\(948\) 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 \(-0.973279\pi\)
0.996479 0.0838468i \(-0.0267206\pi\)
\(954\) 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 \(-0.979195\pi\)
0.997865 0.0653150i \(-0.0208052\pi\)
\(972\) 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 \(0.196604\pi\)
−0.815243 + 0.579119i \(0.803396\pi\)
\(978\) 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 \(0.0194825\pi\)
−0.998127 + 0.0611679i \(0.980517\pi\)
\(984\) 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.890.18 84
3.2 odd 2 inner 1143.3.b.a.890.67 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.18 84 1.1 even 1 trivial
1143.3.b.a.890.67 yes 84 3.2 odd 2 inner