Properties

Label 1143.3.b.a.890.16
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 890.16
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.69

$q$-expansion

\(f(q)\) \(=\) \(q-2.71611i q^{2} -3.37726 q^{4} -6.95811i q^{5} +4.31599 q^{7} -1.69142i q^{8} +O(q^{10})\) \(q-2.71611i q^{2} -3.37726 q^{4} -6.95811i q^{5} +4.31599 q^{7} -1.69142i q^{8} -18.8990 q^{10} +10.6740i q^{11} +12.5086 q^{13} -11.7227i q^{14} -18.1031 q^{16} -32.6178i q^{17} -15.7237 q^{19} +23.4994i q^{20} +28.9918 q^{22} +11.1858i q^{23} -23.4154 q^{25} -33.9747i q^{26} -14.5762 q^{28} -42.0647i q^{29} -12.3795 q^{31} +42.4045i q^{32} -88.5935 q^{34} -30.0312i q^{35} -7.23275 q^{37} +42.7075i q^{38} -11.7691 q^{40} +3.87379i q^{41} -16.1935 q^{43} -36.0489i q^{44} +30.3820 q^{46} -2.07073i q^{47} -30.3722 q^{49} +63.5987i q^{50} -42.2448 q^{52} -3.76342i q^{53} +74.2709 q^{55} -7.30016i q^{56} -114.252 q^{58} -64.3455i q^{59} -52.2457 q^{61} +33.6241i q^{62} +42.7627 q^{64} -87.0362i q^{65} +112.184 q^{67} +110.159i q^{68} -81.5680 q^{70} -56.2639i q^{71} -25.1935 q^{73} +19.6450i q^{74} +53.1032 q^{76} +46.0689i q^{77} +58.5144 q^{79} +125.964i q^{80} +10.5216 q^{82} +90.6579i q^{83} -226.958 q^{85} +43.9834i q^{86} +18.0542 q^{88} +76.9304i q^{89} +53.9869 q^{91} -37.7775i q^{92} -5.62433 q^{94} +109.408i q^{95} -41.6816 q^{97} +82.4944i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.71611i − 1.35806i −0.734112 0.679028i \(-0.762402\pi\)
0.734112 0.679028i \(-0.237598\pi\)
\(3\) 0 0
\(4\) −3.37726 −0.844316
\(5\) − 6.95811i − 1.39162i −0.718224 0.695811i \(-0.755045\pi\)
0.718224 0.695811i \(-0.244955\pi\)
\(6\) 0 0
\(7\) 4.31599 0.616570 0.308285 0.951294i \(-0.400245\pi\)
0.308285 + 0.951294i \(0.400245\pi\)
\(8\) − 1.69142i − 0.211428i
\(9\) 0 0
\(10\) −18.8990 −1.88990
\(11\) 10.6740i 0.970364i 0.874413 + 0.485182i \(0.161247\pi\)
−0.874413 + 0.485182i \(0.838753\pi\)
\(12\) 0 0
\(13\) 12.5086 0.962199 0.481099 0.876666i \(-0.340238\pi\)
0.481099 + 0.876666i \(0.340238\pi\)
\(14\) − 11.7227i − 0.837337i
\(15\) 0 0
\(16\) −18.1031 −1.13145
\(17\) − 32.6178i − 1.91869i −0.282232 0.959346i \(-0.591075\pi\)
0.282232 0.959346i \(-0.408925\pi\)
\(18\) 0 0
\(19\) −15.7237 −0.827566 −0.413783 0.910376i \(-0.635793\pi\)
−0.413783 + 0.910376i \(0.635793\pi\)
\(20\) 23.4994i 1.17497i
\(21\) 0 0
\(22\) 28.9918 1.31781
\(23\) 11.1858i 0.486341i 0.969984 + 0.243170i \(0.0781874\pi\)
−0.969984 + 0.243170i \(0.921813\pi\)
\(24\) 0 0
\(25\) −23.4154 −0.936615
\(26\) − 33.9747i − 1.30672i
\(27\) 0 0
\(28\) −14.5762 −0.520580
\(29\) − 42.0647i − 1.45051i −0.688482 0.725254i \(-0.741722\pi\)
0.688482 0.725254i \(-0.258278\pi\)
\(30\) 0 0
\(31\) −12.3795 −0.399339 −0.199669 0.979863i \(-0.563987\pi\)
−0.199669 + 0.979863i \(0.563987\pi\)
\(32\) 42.4045i 1.32514i
\(33\) 0 0
\(34\) −88.5935 −2.60569
\(35\) − 30.0312i − 0.858033i
\(36\) 0 0
\(37\) −7.23275 −0.195480 −0.0977399 0.995212i \(-0.531161\pi\)
−0.0977399 + 0.995212i \(0.531161\pi\)
\(38\) 42.7075i 1.12388i
\(39\) 0 0
\(40\) −11.7691 −0.294228
\(41\) 3.87379i 0.0944826i 0.998884 + 0.0472413i \(0.0150430\pi\)
−0.998884 + 0.0472413i \(0.984957\pi\)
\(42\) 0 0
\(43\) −16.1935 −0.376593 −0.188297 0.982112i \(-0.560297\pi\)
−0.188297 + 0.982112i \(0.560297\pi\)
\(44\) − 36.0489i − 0.819293i
\(45\) 0 0
\(46\) 30.3820 0.660478
\(47\) − 2.07073i − 0.0440580i −0.999757 0.0220290i \(-0.992987\pi\)
0.999757 0.0220290i \(-0.00701262\pi\)
\(48\) 0 0
\(49\) −30.3722 −0.619841
\(50\) 63.5987i 1.27197i
\(51\) 0 0
\(52\) −42.2448 −0.812400
\(53\) − 3.76342i − 0.0710079i −0.999370 0.0355039i \(-0.988696\pi\)
0.999370 0.0355039i \(-0.0113036\pi\)
\(54\) 0 0
\(55\) 74.2709 1.35038
\(56\) − 7.30016i − 0.130360i
\(57\) 0 0
\(58\) −114.252 −1.96987
\(59\) − 64.3455i − 1.09060i −0.838240 0.545301i \(-0.816415\pi\)
0.838240 0.545301i \(-0.183585\pi\)
\(60\) 0 0
\(61\) −52.2457 −0.856486 −0.428243 0.903664i \(-0.640867\pi\)
−0.428243 + 0.903664i \(0.640867\pi\)
\(62\) 33.6241i 0.542325i
\(63\) 0 0
\(64\) 42.7627 0.668167
\(65\) − 87.0362i − 1.33902i
\(66\) 0 0
\(67\) 112.184 1.67438 0.837192 0.546909i \(-0.184196\pi\)
0.837192 + 0.546909i \(0.184196\pi\)
\(68\) 110.159i 1.61998i
\(69\) 0 0
\(70\) −81.5680 −1.16526
\(71\) − 56.2639i − 0.792449i −0.918154 0.396224i \(-0.870320\pi\)
0.918154 0.396224i \(-0.129680\pi\)
\(72\) 0 0
\(73\) −25.1935 −0.345117 −0.172558 0.984999i \(-0.555203\pi\)
−0.172558 + 0.984999i \(0.555203\pi\)
\(74\) 19.6450i 0.265472i
\(75\) 0 0
\(76\) 53.1032 0.698727
\(77\) 46.0689i 0.598297i
\(78\) 0 0
\(79\) 58.5144 0.740688 0.370344 0.928895i \(-0.379240\pi\)
0.370344 + 0.928895i \(0.379240\pi\)
\(80\) 125.964i 1.57455i
\(81\) 0 0
\(82\) 10.5216 0.128313
\(83\) 90.6579i 1.09226i 0.837699 + 0.546132i \(0.183900\pi\)
−0.837699 + 0.546132i \(0.816100\pi\)
\(84\) 0 0
\(85\) −226.958 −2.67010
\(86\) 43.9834i 0.511435i
\(87\) 0 0
\(88\) 18.0542 0.205162
\(89\) 76.9304i 0.864387i 0.901781 + 0.432193i \(0.142260\pi\)
−0.901781 + 0.432193i \(0.857740\pi\)
\(90\) 0 0
\(91\) 53.9869 0.593263
\(92\) − 37.7775i − 0.410625i
\(93\) 0 0
\(94\) −5.62433 −0.0598333
\(95\) 109.408i 1.15166i
\(96\) 0 0
\(97\) −41.6816 −0.429707 −0.214854 0.976646i \(-0.568927\pi\)
−0.214854 + 0.976646i \(0.568927\pi\)
\(98\) 82.4944i 0.841779i
\(99\) 0 0
\(100\) 79.0798 0.790798
\(101\) 18.4549i 0.182722i 0.995818 + 0.0913608i \(0.0291216\pi\)
−0.995818 + 0.0913608i \(0.970878\pi\)
\(102\) 0 0
\(103\) 4.37878 0.0425124 0.0212562 0.999774i \(-0.493233\pi\)
0.0212562 + 0.999774i \(0.493233\pi\)
\(104\) − 21.1573i − 0.203436i
\(105\) 0 0
\(106\) −10.2219 −0.0964326
\(107\) 120.961i 1.13048i 0.824927 + 0.565240i \(0.191216\pi\)
−0.824927 + 0.565240i \(0.808784\pi\)
\(108\) 0 0
\(109\) 22.7798 0.208989 0.104494 0.994525i \(-0.466678\pi\)
0.104494 + 0.994525i \(0.466678\pi\)
\(110\) − 201.728i − 1.83389i
\(111\) 0 0
\(112\) −78.1330 −0.697616
\(113\) − 22.4867i − 0.198997i −0.995038 0.0994985i \(-0.968276\pi\)
0.995038 0.0994985i \(-0.0317239\pi\)
\(114\) 0 0
\(115\) 77.8323 0.676803
\(116\) 142.064i 1.22469i
\(117\) 0 0
\(118\) −174.770 −1.48110
\(119\) − 140.778i − 1.18301i
\(120\) 0 0
\(121\) 7.06569 0.0583942
\(122\) 141.905i 1.16316i
\(123\) 0 0
\(124\) 41.8088 0.337168
\(125\) − 11.0261i − 0.0882086i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) 53.4696i 0.417731i
\(129\) 0 0
\(130\) −236.400 −1.81846
\(131\) 91.7796i 0.700608i 0.936636 + 0.350304i \(0.113922\pi\)
−0.936636 + 0.350304i \(0.886078\pi\)
\(132\) 0 0
\(133\) −67.8636 −0.510252
\(134\) − 304.703i − 2.27391i
\(135\) 0 0
\(136\) −55.1704 −0.405665
\(137\) 89.3843i 0.652440i 0.945294 + 0.326220i \(0.105775\pi\)
−0.945294 + 0.326220i \(0.894225\pi\)
\(138\) 0 0
\(139\) −149.686 −1.07687 −0.538437 0.842666i \(-0.680985\pi\)
−0.538437 + 0.842666i \(0.680985\pi\)
\(140\) 101.423i 0.724451i
\(141\) 0 0
\(142\) −152.819 −1.07619
\(143\) 133.517i 0.933683i
\(144\) 0 0
\(145\) −292.691 −2.01856
\(146\) 68.4284i 0.468688i
\(147\) 0 0
\(148\) 24.4269 0.165047
\(149\) − 124.068i − 0.832674i −0.909210 0.416337i \(-0.863314\pi\)
0.909210 0.416337i \(-0.136686\pi\)
\(150\) 0 0
\(151\) 216.165 1.43156 0.715779 0.698327i \(-0.246072\pi\)
0.715779 + 0.698327i \(0.246072\pi\)
\(152\) 26.5955i 0.174970i
\(153\) 0 0
\(154\) 125.128 0.812521
\(155\) 86.1380i 0.555729i
\(156\) 0 0
\(157\) −229.144 −1.45952 −0.729758 0.683706i \(-0.760367\pi\)
−0.729758 + 0.683706i \(0.760367\pi\)
\(158\) − 158.932i − 1.00590i
\(159\) 0 0
\(160\) 295.055 1.84410
\(161\) 48.2780i 0.299863i
\(162\) 0 0
\(163\) −123.348 −0.756736 −0.378368 0.925655i \(-0.623515\pi\)
−0.378368 + 0.925655i \(0.623515\pi\)
\(164\) − 13.0828i − 0.0797732i
\(165\) 0 0
\(166\) 246.237 1.48336
\(167\) 63.7232i 0.381576i 0.981631 + 0.190788i \(0.0611043\pi\)
−0.981631 + 0.190788i \(0.938896\pi\)
\(168\) 0 0
\(169\) −12.5353 −0.0741735
\(170\) 616.444i 3.62614i
\(171\) 0 0
\(172\) 54.6898 0.317964
\(173\) 35.2926i 0.204004i 0.994784 + 0.102002i \(0.0325248\pi\)
−0.994784 + 0.102002i \(0.967475\pi\)
\(174\) 0 0
\(175\) −101.060 −0.577489
\(176\) − 193.233i − 1.09791i
\(177\) 0 0
\(178\) 208.952 1.17389
\(179\) 241.836i 1.35104i 0.737342 + 0.675519i \(0.236080\pi\)
−0.737342 + 0.675519i \(0.763920\pi\)
\(180\) 0 0
\(181\) 288.741 1.59526 0.797628 0.603150i \(-0.206088\pi\)
0.797628 + 0.603150i \(0.206088\pi\)
\(182\) − 146.635i − 0.805684i
\(183\) 0 0
\(184\) 18.9200 0.102826
\(185\) 50.3263i 0.272034i
\(186\) 0 0
\(187\) 348.162 1.86183
\(188\) 6.99339i 0.0371989i
\(189\) 0 0
\(190\) 297.163 1.56402
\(191\) − 91.5192i − 0.479158i −0.970877 0.239579i \(-0.922991\pi\)
0.970877 0.239579i \(-0.0770095\pi\)
\(192\) 0 0
\(193\) 226.506 1.17361 0.586803 0.809730i \(-0.300386\pi\)
0.586803 + 0.809730i \(0.300386\pi\)
\(194\) 113.212i 0.583566i
\(195\) 0 0
\(196\) 102.575 0.523342
\(197\) − 272.020i − 1.38081i −0.723423 0.690405i \(-0.757432\pi\)
0.723423 0.690405i \(-0.242568\pi\)
\(198\) 0 0
\(199\) −152.048 −0.764059 −0.382029 0.924150i \(-0.624775\pi\)
−0.382029 + 0.924150i \(0.624775\pi\)
\(200\) 39.6053i 0.198026i
\(201\) 0 0
\(202\) 50.1255 0.248146
\(203\) − 181.551i − 0.894340i
\(204\) 0 0
\(205\) 26.9543 0.131484
\(206\) − 11.8933i − 0.0577343i
\(207\) 0 0
\(208\) −226.445 −1.08868
\(209\) − 167.835i − 0.803040i
\(210\) 0 0
\(211\) 210.301 0.996685 0.498342 0.866980i \(-0.333942\pi\)
0.498342 + 0.866980i \(0.333942\pi\)
\(212\) 12.7100i 0.0599531i
\(213\) 0 0
\(214\) 328.544 1.53525
\(215\) 112.676i 0.524076i
\(216\) 0 0
\(217\) −53.4298 −0.246220
\(218\) − 61.8725i − 0.283819i
\(219\) 0 0
\(220\) −250.832 −1.14015
\(221\) − 408.002i − 1.84616i
\(222\) 0 0
\(223\) 198.451 0.889913 0.444956 0.895552i \(-0.353219\pi\)
0.444956 + 0.895552i \(0.353219\pi\)
\(224\) 183.017i 0.817042i
\(225\) 0 0
\(226\) −61.0763 −0.270249
\(227\) − 161.325i − 0.710685i −0.934736 0.355342i \(-0.884364\pi\)
0.934736 0.355342i \(-0.115636\pi\)
\(228\) 0 0
\(229\) 394.501 1.72271 0.861356 0.508002i \(-0.169616\pi\)
0.861356 + 0.508002i \(0.169616\pi\)
\(230\) − 211.401i − 0.919136i
\(231\) 0 0
\(232\) −71.1492 −0.306678
\(233\) − 385.938i − 1.65639i −0.560442 0.828194i \(-0.689369\pi\)
0.560442 0.828194i \(-0.310631\pi\)
\(234\) 0 0
\(235\) −14.4084 −0.0613122
\(236\) 217.312i 0.920812i
\(237\) 0 0
\(238\) −382.369 −1.60659
\(239\) − 393.051i − 1.64456i −0.569081 0.822281i \(-0.692701\pi\)
0.569081 0.822281i \(-0.307299\pi\)
\(240\) 0 0
\(241\) 80.5872 0.334387 0.167193 0.985924i \(-0.446530\pi\)
0.167193 + 0.985924i \(0.446530\pi\)
\(242\) − 19.1912i − 0.0793025i
\(243\) 0 0
\(244\) 176.447 0.723145
\(245\) 211.333i 0.862585i
\(246\) 0 0
\(247\) −196.682 −0.796283
\(248\) 20.9390i 0.0844314i
\(249\) 0 0
\(250\) −29.9481 −0.119792
\(251\) − 156.298i − 0.622699i −0.950295 0.311350i \(-0.899219\pi\)
0.950295 0.311350i \(-0.100781\pi\)
\(252\) 0 0
\(253\) −119.398 −0.471927
\(254\) 30.6090i 0.120508i
\(255\) 0 0
\(256\) 316.280 1.23547
\(257\) 334.933i 1.30324i 0.758546 + 0.651620i \(0.225910\pi\)
−0.758546 + 0.651620i \(0.774090\pi\)
\(258\) 0 0
\(259\) −31.2165 −0.120527
\(260\) 293.944i 1.13055i
\(261\) 0 0
\(262\) 249.284 0.951465
\(263\) − 403.738i − 1.53513i −0.640973 0.767563i \(-0.721469\pi\)
0.640973 0.767563i \(-0.278531\pi\)
\(264\) 0 0
\(265\) −26.1863 −0.0988162
\(266\) 184.325i 0.692951i
\(267\) 0 0
\(268\) −378.874 −1.41371
\(269\) − 96.1414i − 0.357403i −0.983903 0.178701i \(-0.942810\pi\)
0.983903 0.178701i \(-0.0571896\pi\)
\(270\) 0 0
\(271\) 189.580 0.699556 0.349778 0.936833i \(-0.386257\pi\)
0.349778 + 0.936833i \(0.386257\pi\)
\(272\) 590.484i 2.17090i
\(273\) 0 0
\(274\) 242.778 0.886050
\(275\) − 249.936i − 0.908857i
\(276\) 0 0
\(277\) 171.642 0.619646 0.309823 0.950794i \(-0.399730\pi\)
0.309823 + 0.950794i \(0.399730\pi\)
\(278\) 406.563i 1.46246i
\(279\) 0 0
\(280\) −50.7954 −0.181412
\(281\) − 45.9693i − 0.163592i −0.996649 0.0817959i \(-0.973934\pi\)
0.996649 0.0817959i \(-0.0260656\pi\)
\(282\) 0 0
\(283\) −130.208 −0.460098 −0.230049 0.973179i \(-0.573889\pi\)
−0.230049 + 0.973179i \(0.573889\pi\)
\(284\) 190.018i 0.669077i
\(285\) 0 0
\(286\) 362.646 1.26799
\(287\) 16.7192i 0.0582552i
\(288\) 0 0
\(289\) −774.919 −2.68138
\(290\) 794.982i 2.74132i
\(291\) 0 0
\(292\) 85.0851 0.291387
\(293\) − 500.515i − 1.70824i −0.520075 0.854121i \(-0.674096\pi\)
0.520075 0.854121i \(-0.325904\pi\)
\(294\) 0 0
\(295\) −447.723 −1.51771
\(296\) 12.2336i 0.0413299i
\(297\) 0 0
\(298\) −336.984 −1.13082
\(299\) 139.919i 0.467956i
\(300\) 0 0
\(301\) −69.8911 −0.232196
\(302\) − 587.129i − 1.94414i
\(303\) 0 0
\(304\) 284.649 0.936347
\(305\) 363.531i 1.19191i
\(306\) 0 0
\(307\) 65.9196 0.214722 0.107361 0.994220i \(-0.465760\pi\)
0.107361 + 0.994220i \(0.465760\pi\)
\(308\) − 155.587i − 0.505152i
\(309\) 0 0
\(310\) 233.961 0.754711
\(311\) 309.588i 0.995459i 0.867332 + 0.497730i \(0.165833\pi\)
−0.867332 + 0.497730i \(0.834167\pi\)
\(312\) 0 0
\(313\) 147.814 0.472248 0.236124 0.971723i \(-0.424123\pi\)
0.236124 + 0.971723i \(0.424123\pi\)
\(314\) 622.381i 1.98210i
\(315\) 0 0
\(316\) −197.618 −0.625375
\(317\) 148.170i 0.467413i 0.972307 + 0.233706i \(0.0750854\pi\)
−0.972307 + 0.233706i \(0.924915\pi\)
\(318\) 0 0
\(319\) 448.999 1.40752
\(320\) − 297.548i − 0.929837i
\(321\) 0 0
\(322\) 131.128 0.407231
\(323\) 512.874i 1.58784i
\(324\) 0 0
\(325\) −292.893 −0.901209
\(326\) 335.027i 1.02769i
\(327\) 0 0
\(328\) 6.55221 0.0199763
\(329\) − 8.93724i − 0.0271649i
\(330\) 0 0
\(331\) −286.761 −0.866348 −0.433174 0.901310i \(-0.642606\pi\)
−0.433174 + 0.901310i \(0.642606\pi\)
\(332\) − 306.176i − 0.922216i
\(333\) 0 0
\(334\) 173.079 0.518201
\(335\) − 780.587i − 2.33011i
\(336\) 0 0
\(337\) −480.605 −1.42613 −0.713063 0.701100i \(-0.752693\pi\)
−0.713063 + 0.701100i \(0.752693\pi\)
\(338\) 34.0473i 0.100732i
\(339\) 0 0
\(340\) 766.498 2.25440
\(341\) − 132.139i − 0.387504i
\(342\) 0 0
\(343\) −342.570 −0.998746
\(344\) 27.3901i 0.0796223i
\(345\) 0 0
\(346\) 95.8588 0.277049
\(347\) − 421.056i − 1.21342i −0.794924 0.606709i \(-0.792489\pi\)
0.794924 0.606709i \(-0.207511\pi\)
\(348\) 0 0
\(349\) −192.726 −0.552224 −0.276112 0.961125i \(-0.589046\pi\)
−0.276112 + 0.961125i \(0.589046\pi\)
\(350\) 274.492i 0.784262i
\(351\) 0 0
\(352\) −452.625 −1.28587
\(353\) − 687.481i − 1.94754i −0.227541 0.973769i \(-0.573069\pi\)
0.227541 0.973769i \(-0.426931\pi\)
\(354\) 0 0
\(355\) −391.490 −1.10279
\(356\) − 259.814i − 0.729815i
\(357\) 0 0
\(358\) 656.853 1.83479
\(359\) − 620.707i − 1.72899i −0.502643 0.864494i \(-0.667639\pi\)
0.502643 0.864494i \(-0.332361\pi\)
\(360\) 0 0
\(361\) −113.764 −0.315135
\(362\) − 784.254i − 2.16645i
\(363\) 0 0
\(364\) −182.328 −0.500901
\(365\) 175.299i 0.480272i
\(366\) 0 0
\(367\) 189.875 0.517371 0.258686 0.965962i \(-0.416711\pi\)
0.258686 + 0.965962i \(0.416711\pi\)
\(368\) − 202.499i − 0.550268i
\(369\) 0 0
\(370\) 136.692 0.369438
\(371\) − 16.2429i − 0.0437813i
\(372\) 0 0
\(373\) 439.303 1.17776 0.588878 0.808222i \(-0.299570\pi\)
0.588878 + 0.808222i \(0.299570\pi\)
\(374\) − 945.647i − 2.52847i
\(375\) 0 0
\(376\) −3.50247 −0.00931509
\(377\) − 526.170i − 1.39568i
\(378\) 0 0
\(379\) −514.039 −1.35630 −0.678152 0.734922i \(-0.737219\pi\)
−0.678152 + 0.734922i \(0.737219\pi\)
\(380\) − 369.498i − 0.972364i
\(381\) 0 0
\(382\) −248.576 −0.650724
\(383\) 400.504i 1.04570i 0.852424 + 0.522851i \(0.175132\pi\)
−0.852424 + 0.522851i \(0.824868\pi\)
\(384\) 0 0
\(385\) 320.553 0.832604
\(386\) − 615.215i − 1.59382i
\(387\) 0 0
\(388\) 140.770 0.362808
\(389\) − 714.324i − 1.83631i −0.396222 0.918155i \(-0.629679\pi\)
0.396222 0.918155i \(-0.370321\pi\)
\(390\) 0 0
\(391\) 364.857 0.933138
\(392\) 51.3723i 0.131052i
\(393\) 0 0
\(394\) −738.836 −1.87522
\(395\) − 407.150i − 1.03076i
\(396\) 0 0
\(397\) 45.5606 0.114762 0.0573811 0.998352i \(-0.481725\pi\)
0.0573811 + 0.998352i \(0.481725\pi\)
\(398\) 412.979i 1.03763i
\(399\) 0 0
\(400\) 423.892 1.05973
\(401\) 378.312i 0.943423i 0.881753 + 0.471711i \(0.156364\pi\)
−0.881753 + 0.471711i \(0.843636\pi\)
\(402\) 0 0
\(403\) −154.850 −0.384243
\(404\) − 62.3270i − 0.154275i
\(405\) 0 0
\(406\) −493.113 −1.21456
\(407\) − 77.2024i − 0.189686i
\(408\) 0 0
\(409\) −226.210 −0.553080 −0.276540 0.961002i \(-0.589188\pi\)
−0.276540 + 0.961002i \(0.589188\pi\)
\(410\) − 73.2108i − 0.178563i
\(411\) 0 0
\(412\) −14.7883 −0.0358939
\(413\) − 277.715i − 0.672432i
\(414\) 0 0
\(415\) 630.808 1.52002
\(416\) 530.420i 1.27505i
\(417\) 0 0
\(418\) −455.860 −1.09057
\(419\) 36.9462i 0.0881771i 0.999028 + 0.0440885i \(0.0140384\pi\)
−0.999028 + 0.0440885i \(0.985962\pi\)
\(420\) 0 0
\(421\) 162.731 0.386535 0.193268 0.981146i \(-0.438091\pi\)
0.193268 + 0.981146i \(0.438091\pi\)
\(422\) − 571.200i − 1.35355i
\(423\) 0 0
\(424\) −6.36553 −0.0150130
\(425\) 763.757i 1.79708i
\(426\) 0 0
\(427\) −225.492 −0.528084
\(428\) − 408.518i − 0.954482i
\(429\) 0 0
\(430\) 306.042 0.711725
\(431\) 680.810i 1.57960i 0.613361 + 0.789802i \(0.289817\pi\)
−0.613361 + 0.789802i \(0.710183\pi\)
\(432\) 0 0
\(433\) −43.8082 −0.101174 −0.0505869 0.998720i \(-0.516109\pi\)
−0.0505869 + 0.998720i \(0.516109\pi\)
\(434\) 145.121i 0.334381i
\(435\) 0 0
\(436\) −76.9334 −0.176453
\(437\) − 175.883i − 0.402479i
\(438\) 0 0
\(439\) −208.320 −0.474532 −0.237266 0.971445i \(-0.576251\pi\)
−0.237266 + 0.971445i \(0.576251\pi\)
\(440\) − 125.624i − 0.285508i
\(441\) 0 0
\(442\) −1108.18 −2.50719
\(443\) − 63.6117i − 0.143593i −0.997419 0.0717965i \(-0.977127\pi\)
0.997419 0.0717965i \(-0.0228732\pi\)
\(444\) 0 0
\(445\) 535.291 1.20290
\(446\) − 539.014i − 1.20855i
\(447\) 0 0
\(448\) 184.563 0.411972
\(449\) − 803.599i − 1.78975i −0.446314 0.894876i \(-0.647264\pi\)
0.446314 0.894876i \(-0.352736\pi\)
\(450\) 0 0
\(451\) −41.3488 −0.0916825
\(452\) 75.9434i 0.168016i
\(453\) 0 0
\(454\) −438.178 −0.965150
\(455\) − 375.647i − 0.825598i
\(456\) 0 0
\(457\) 576.381 1.26123 0.630613 0.776097i \(-0.282803\pi\)
0.630613 + 0.776097i \(0.282803\pi\)
\(458\) − 1071.51i − 2.33954i
\(459\) 0 0
\(460\) −262.860 −0.571435
\(461\) − 309.110i − 0.670521i −0.942126 0.335260i \(-0.891176\pi\)
0.942126 0.335260i \(-0.108824\pi\)
\(462\) 0 0
\(463\) 699.328 1.51043 0.755214 0.655479i \(-0.227533\pi\)
0.755214 + 0.655479i \(0.227533\pi\)
\(464\) 761.504i 1.64117i
\(465\) 0 0
\(466\) −1048.25 −2.24947
\(467\) 785.231i 1.68144i 0.541472 + 0.840719i \(0.317867\pi\)
−0.541472 + 0.840719i \(0.682133\pi\)
\(468\) 0 0
\(469\) 484.184 1.03237
\(470\) 39.1347i 0.0832653i
\(471\) 0 0
\(472\) −108.835 −0.230584
\(473\) − 172.850i − 0.365433i
\(474\) 0 0
\(475\) 368.177 0.775110
\(476\) 475.444i 0.998833i
\(477\) 0 0
\(478\) −1067.57 −2.23341
\(479\) − 404.198i − 0.843836i −0.906634 0.421918i \(-0.861357\pi\)
0.906634 0.421918i \(-0.138643\pi\)
\(480\) 0 0
\(481\) −90.4715 −0.188090
\(482\) − 218.884i − 0.454116i
\(483\) 0 0
\(484\) −23.8627 −0.0493031
\(485\) 290.025i 0.597990i
\(486\) 0 0
\(487\) −301.492 −0.619081 −0.309541 0.950886i \(-0.600175\pi\)
−0.309541 + 0.950886i \(0.600175\pi\)
\(488\) 88.3695i 0.181085i
\(489\) 0 0
\(490\) 574.005 1.17144
\(491\) − 825.810i − 1.68189i −0.541118 0.840947i \(-0.681999\pi\)
0.541118 0.840947i \(-0.318001\pi\)
\(492\) 0 0
\(493\) −1372.06 −2.78308
\(494\) 534.210i 1.08140i
\(495\) 0 0
\(496\) 224.108 0.451831
\(497\) − 242.834i − 0.488600i
\(498\) 0 0
\(499\) 814.889 1.63304 0.816522 0.577315i \(-0.195899\pi\)
0.816522 + 0.577315i \(0.195899\pi\)
\(500\) 37.2380i 0.0744760i
\(501\) 0 0
\(502\) −424.522 −0.845661
\(503\) 727.680i 1.44668i 0.690493 + 0.723340i \(0.257394\pi\)
−0.690493 + 0.723340i \(0.742606\pi\)
\(504\) 0 0
\(505\) 128.411 0.254280
\(506\) 324.297i 0.640904i
\(507\) 0 0
\(508\) 38.0598 0.0749209
\(509\) − 81.3689i − 0.159860i −0.996800 0.0799302i \(-0.974530\pi\)
0.996800 0.0799302i \(-0.0254697\pi\)
\(510\) 0 0
\(511\) −108.735 −0.212789
\(512\) − 645.174i − 1.26011i
\(513\) 0 0
\(514\) 909.714 1.76987
\(515\) − 30.4681i − 0.0591613i
\(516\) 0 0
\(517\) 22.1029 0.0427523
\(518\) 84.7875i 0.163682i
\(519\) 0 0
\(520\) −147.215 −0.283106
\(521\) 980.158i 1.88130i 0.339376 + 0.940651i \(0.389784\pi\)
−0.339376 + 0.940651i \(0.610216\pi\)
\(522\) 0 0
\(523\) 7.12508 0.0136235 0.00681174 0.999977i \(-0.497832\pi\)
0.00681174 + 0.999977i \(0.497832\pi\)
\(524\) − 309.964i − 0.591534i
\(525\) 0 0
\(526\) −1096.60 −2.08479
\(527\) 403.792i 0.766209i
\(528\) 0 0
\(529\) 403.877 0.763473
\(530\) 71.1249i 0.134198i
\(531\) 0 0
\(532\) 229.193 0.430814
\(533\) 48.4556i 0.0909111i
\(534\) 0 0
\(535\) 841.663 1.57320
\(536\) − 189.750i − 0.354011i
\(537\) 0 0
\(538\) −261.131 −0.485373
\(539\) − 324.193i − 0.601472i
\(540\) 0 0
\(541\) 880.813 1.62812 0.814060 0.580780i \(-0.197252\pi\)
0.814060 + 0.580780i \(0.197252\pi\)
\(542\) − 514.919i − 0.950036i
\(543\) 0 0
\(544\) 1383.14 2.54254
\(545\) − 158.504i − 0.290834i
\(546\) 0 0
\(547\) 1050.90 1.92121 0.960607 0.277910i \(-0.0896417\pi\)
0.960607 + 0.277910i \(0.0896417\pi\)
\(548\) − 301.874i − 0.550865i
\(549\) 0 0
\(550\) −678.853 −1.23428
\(551\) 661.415i 1.20039i
\(552\) 0 0
\(553\) 252.548 0.456686
\(554\) − 466.199i − 0.841514i
\(555\) 0 0
\(556\) 505.527 0.909222
\(557\) 1054.84i 1.89379i 0.321544 + 0.946895i \(0.395798\pi\)
−0.321544 + 0.946895i \(0.604202\pi\)
\(558\) 0 0
\(559\) −202.558 −0.362358
\(560\) 543.658i 0.970819i
\(561\) 0 0
\(562\) −124.858 −0.222167
\(563\) − 125.889i − 0.223603i −0.993731 0.111802i \(-0.964338\pi\)
0.993731 0.111802i \(-0.0356621\pi\)
\(564\) 0 0
\(565\) −156.465 −0.276929
\(566\) 353.659i 0.624839i
\(567\) 0 0
\(568\) −95.1660 −0.167546
\(569\) − 68.3275i − 0.120083i −0.998196 0.0600417i \(-0.980877\pi\)
0.998196 0.0600417i \(-0.0191234\pi\)
\(570\) 0 0
\(571\) −517.216 −0.905807 −0.452904 0.891559i \(-0.649612\pi\)
−0.452904 + 0.891559i \(0.649612\pi\)
\(572\) − 450.921i − 0.788323i
\(573\) 0 0
\(574\) 45.4113 0.0791138
\(575\) − 261.920i − 0.455514i
\(576\) 0 0
\(577\) 724.772 1.25610 0.628052 0.778171i \(-0.283852\pi\)
0.628052 + 0.778171i \(0.283852\pi\)
\(578\) 2104.77i 3.64147i
\(579\) 0 0
\(580\) 988.495 1.70430
\(581\) 391.279i 0.673458i
\(582\) 0 0
\(583\) 40.1707 0.0689035
\(584\) 42.6129i 0.0729673i
\(585\) 0 0
\(586\) −1359.45 −2.31989
\(587\) 96.7186i 0.164768i 0.996601 + 0.0823838i \(0.0262533\pi\)
−0.996601 + 0.0823838i \(0.973747\pi\)
\(588\) 0 0
\(589\) 194.652 0.330479
\(590\) 1216.07i 2.06113i
\(591\) 0 0
\(592\) 130.936 0.221175
\(593\) 644.219i 1.08637i 0.839612 + 0.543186i \(0.182782\pi\)
−0.839612 + 0.543186i \(0.817218\pi\)
\(594\) 0 0
\(595\) −979.550 −1.64630
\(596\) 419.012i 0.703040i
\(597\) 0 0
\(598\) 380.036 0.635511
\(599\) − 756.993i − 1.26376i −0.775066 0.631881i \(-0.782283\pi\)
0.775066 0.631881i \(-0.217717\pi\)
\(600\) 0 0
\(601\) −898.672 −1.49529 −0.747647 0.664096i \(-0.768817\pi\)
−0.747647 + 0.664096i \(0.768817\pi\)
\(602\) 189.832i 0.315335i
\(603\) 0 0
\(604\) −730.047 −1.20869
\(605\) − 49.1639i − 0.0812627i
\(606\) 0 0
\(607\) 1046.26 1.72366 0.861832 0.507193i \(-0.169317\pi\)
0.861832 + 0.507193i \(0.169317\pi\)
\(608\) − 666.757i − 1.09664i
\(609\) 0 0
\(610\) 987.392 1.61867
\(611\) − 25.9019i − 0.0423926i
\(612\) 0 0
\(613\) −328.003 −0.535078 −0.267539 0.963547i \(-0.586210\pi\)
−0.267539 + 0.963547i \(0.586210\pi\)
\(614\) − 179.045i − 0.291604i
\(615\) 0 0
\(616\) 77.9220 0.126497
\(617\) − 638.293i − 1.03451i −0.855831 0.517255i \(-0.826954\pi\)
0.855831 0.517255i \(-0.173046\pi\)
\(618\) 0 0
\(619\) 713.100 1.15202 0.576010 0.817443i \(-0.304609\pi\)
0.576010 + 0.817443i \(0.304609\pi\)
\(620\) − 290.911i − 0.469211i
\(621\) 0 0
\(622\) 840.875 1.35189
\(623\) 332.031i 0.532955i
\(624\) 0 0
\(625\) −662.105 −1.05937
\(626\) − 401.478i − 0.641339i
\(627\) 0 0
\(628\) 773.879 1.23229
\(629\) 235.916i 0.375066i
\(630\) 0 0
\(631\) 376.709 0.597003 0.298502 0.954409i \(-0.403513\pi\)
0.298502 + 0.954409i \(0.403513\pi\)
\(632\) − 98.9725i − 0.156602i
\(633\) 0 0
\(634\) 402.446 0.634772
\(635\) 78.4140i 0.123487i
\(636\) 0 0
\(637\) −379.914 −0.596411
\(638\) − 1219.53i − 1.91149i
\(639\) 0 0
\(640\) 372.048 0.581324
\(641\) 7.88021i 0.0122936i 0.999981 + 0.00614681i \(0.00195660\pi\)
−0.999981 + 0.00614681i \(0.998043\pi\)
\(642\) 0 0
\(643\) 115.955 0.180335 0.0901674 0.995927i \(-0.471260\pi\)
0.0901674 + 0.995927i \(0.471260\pi\)
\(644\) − 163.047i − 0.253179i
\(645\) 0 0
\(646\) 1393.02 2.15638
\(647\) 509.586i 0.787613i 0.919193 + 0.393806i \(0.128842\pi\)
−0.919193 + 0.393806i \(0.871158\pi\)
\(648\) 0 0
\(649\) 686.824 1.05828
\(650\) 795.530i 1.22389i
\(651\) 0 0
\(652\) 416.579 0.638924
\(653\) − 430.316i − 0.658983i −0.944159 0.329491i \(-0.893123\pi\)
0.944159 0.329491i \(-0.106877\pi\)
\(654\) 0 0
\(655\) 638.613 0.974982
\(656\) − 70.1277i − 0.106902i
\(657\) 0 0
\(658\) −24.2745 −0.0368914
\(659\) − 497.970i − 0.755646i −0.925878 0.377823i \(-0.876673\pi\)
0.925878 0.377823i \(-0.123327\pi\)
\(660\) 0 0
\(661\) −55.0164 −0.0832321 −0.0416161 0.999134i \(-0.513251\pi\)
−0.0416161 + 0.999134i \(0.513251\pi\)
\(662\) 778.875i 1.17655i
\(663\) 0 0
\(664\) 153.341 0.230935
\(665\) 472.202i 0.710079i
\(666\) 0 0
\(667\) 470.529 0.705441
\(668\) − 215.210i − 0.322171i
\(669\) 0 0
\(670\) −2120.16 −3.16442
\(671\) − 557.670i − 0.831103i
\(672\) 0 0
\(673\) 819.232 1.21728 0.608642 0.793445i \(-0.291715\pi\)
0.608642 + 0.793445i \(0.291715\pi\)
\(674\) 1305.38i 1.93676i
\(675\) 0 0
\(676\) 42.3351 0.0626258
\(677\) − 876.071i − 1.29405i −0.762469 0.647025i \(-0.776013\pi\)
0.762469 0.647025i \(-0.223987\pi\)
\(678\) 0 0
\(679\) −179.897 −0.264945
\(680\) 383.882i 0.564533i
\(681\) 0 0
\(682\) −358.904 −0.526252
\(683\) 1239.06i 1.81414i 0.420978 + 0.907071i \(0.361687\pi\)
−0.420978 + 0.907071i \(0.638313\pi\)
\(684\) 0 0
\(685\) 621.946 0.907950
\(686\) 930.458i 1.35635i
\(687\) 0 0
\(688\) 293.154 0.426095
\(689\) − 47.0750i − 0.0683237i
\(690\) 0 0
\(691\) −737.495 −1.06729 −0.533643 0.845710i \(-0.679178\pi\)
−0.533643 + 0.845710i \(0.679178\pi\)
\(692\) − 119.193i − 0.172244i
\(693\) 0 0
\(694\) −1143.63 −1.64789
\(695\) 1041.53i 1.49860i
\(696\) 0 0
\(697\) 126.354 0.181283
\(698\) 523.466i 0.749951i
\(699\) 0 0
\(700\) 341.308 0.487583
\(701\) 86.5794i 0.123508i 0.998091 + 0.0617542i \(0.0196695\pi\)
−0.998091 + 0.0617542i \(0.980330\pi\)
\(702\) 0 0
\(703\) 113.726 0.161772
\(704\) 456.449i 0.648365i
\(705\) 0 0
\(706\) −1867.27 −2.64486
\(707\) 79.6511i 0.112661i
\(708\) 0 0
\(709\) −879.191 −1.24004 −0.620022 0.784585i \(-0.712876\pi\)
−0.620022 + 0.784585i \(0.712876\pi\)
\(710\) 1063.33i 1.49765i
\(711\) 0 0
\(712\) 130.122 0.182755
\(713\) − 138.475i − 0.194215i
\(714\) 0 0
\(715\) 929.024 1.29933
\(716\) − 816.743i − 1.14070i
\(717\) 0 0
\(718\) −1685.91 −2.34806
\(719\) 553.678i 0.770066i 0.922903 + 0.385033i \(0.125810\pi\)
−0.922903 + 0.385033i \(0.874190\pi\)
\(720\) 0 0
\(721\) 18.8988 0.0262119
\(722\) 308.995i 0.427971i
\(723\) 0 0
\(724\) −975.155 −1.34690
\(725\) 984.961i 1.35857i
\(726\) 0 0
\(727\) 8.52118 0.0117210 0.00586051 0.999983i \(-0.498135\pi\)
0.00586051 + 0.999983i \(0.498135\pi\)
\(728\) − 91.3147i − 0.125432i
\(729\) 0 0
\(730\) 476.133 0.652237
\(731\) 528.196i 0.722567i
\(732\) 0 0
\(733\) −1240.93 −1.69294 −0.846472 0.532433i \(-0.821278\pi\)
−0.846472 + 0.532433i \(0.821278\pi\)
\(734\) − 515.722i − 0.702619i
\(735\) 0 0
\(736\) −474.329 −0.644469
\(737\) 1197.45i 1.62476i
\(738\) 0 0
\(739\) −236.646 −0.320225 −0.160113 0.987099i \(-0.551186\pi\)
−0.160113 + 0.987099i \(0.551186\pi\)
\(740\) − 169.965i − 0.229683i
\(741\) 0 0
\(742\) −44.1175 −0.0594575
\(743\) − 962.986i − 1.29608i −0.761607 0.648039i \(-0.775589\pi\)
0.761607 0.648039i \(-0.224411\pi\)
\(744\) 0 0
\(745\) −863.282 −1.15877
\(746\) − 1193.20i − 1.59946i
\(747\) 0 0
\(748\) −1175.84 −1.57197
\(749\) 522.068i 0.697020i
\(750\) 0 0
\(751\) 192.651 0.256526 0.128263 0.991740i \(-0.459060\pi\)
0.128263 + 0.991740i \(0.459060\pi\)
\(752\) 37.4867i 0.0498493i
\(753\) 0 0
\(754\) −1429.14 −1.89541
\(755\) − 1504.10i − 1.99219i
\(756\) 0 0
\(757\) −473.144 −0.625026 −0.312513 0.949914i \(-0.601171\pi\)
−0.312513 + 0.949914i \(0.601171\pi\)
\(758\) 1396.19i 1.84194i
\(759\) 0 0
\(760\) 185.055 0.243493
\(761\) − 306.374i − 0.402593i −0.979530 0.201297i \(-0.935484\pi\)
0.979530 0.201297i \(-0.0645155\pi\)
\(762\) 0 0
\(763\) 98.3174 0.128856
\(764\) 309.085i 0.404561i
\(765\) 0 0
\(766\) 1087.81 1.42012
\(767\) − 804.871i − 1.04938i
\(768\) 0 0
\(769\) 1329.71 1.72914 0.864572 0.502508i \(-0.167589\pi\)
0.864572 + 0.502508i \(0.167589\pi\)
\(770\) − 870.657i − 1.13072i
\(771\) 0 0
\(772\) −764.970 −0.990894
\(773\) 1473.99i 1.90685i 0.301634 + 0.953424i \(0.402468\pi\)
−0.301634 + 0.953424i \(0.597532\pi\)
\(774\) 0 0
\(775\) 289.871 0.374027
\(776\) 70.5012i 0.0908520i
\(777\) 0 0
\(778\) −1940.18 −2.49381
\(779\) − 60.9105i − 0.0781906i
\(780\) 0 0
\(781\) 600.561 0.768964
\(782\) − 990.992i − 1.26725i
\(783\) 0 0
\(784\) 549.833 0.701317
\(785\) 1594.41i 2.03110i
\(786\) 0 0
\(787\) 542.649 0.689516 0.344758 0.938692i \(-0.387961\pi\)
0.344758 + 0.938692i \(0.387961\pi\)
\(788\) 918.682i 1.16584i
\(789\) 0 0
\(790\) −1105.86 −1.39983
\(791\) − 97.0523i − 0.122696i
\(792\) 0 0
\(793\) −653.519 −0.824110
\(794\) − 123.748i − 0.155853i
\(795\) 0 0
\(796\) 513.505 0.645107
\(797\) 1010.34i 1.26768i 0.773462 + 0.633842i \(0.218523\pi\)
−0.773462 + 0.633842i \(0.781477\pi\)
\(798\) 0 0
\(799\) −67.5425 −0.0845338
\(800\) − 992.916i − 1.24115i
\(801\) 0 0
\(802\) 1027.54 1.28122
\(803\) − 268.916i − 0.334889i
\(804\) 0 0
\(805\) 335.924 0.417296
\(806\) 420.590i 0.521824i
\(807\) 0 0
\(808\) 31.2150 0.0386324
\(809\) − 476.543i − 0.589051i −0.955644 0.294526i \(-0.904838\pi\)
0.955644 0.294526i \(-0.0951616\pi\)
\(810\) 0 0
\(811\) −1459.91 −1.80014 −0.900069 0.435748i \(-0.856484\pi\)
−0.900069 + 0.435748i \(0.856484\pi\)
\(812\) 613.145i 0.755105i
\(813\) 0 0
\(814\) −209.690 −0.257605
\(815\) 858.269i 1.05309i
\(816\) 0 0
\(817\) 254.623 0.311656
\(818\) 614.411i 0.751113i
\(819\) 0 0
\(820\) −91.0316 −0.111014
\(821\) 990.138i 1.20602i 0.797735 + 0.603008i \(0.206031\pi\)
−0.797735 + 0.603008i \(0.793969\pi\)
\(822\) 0 0
\(823\) −108.153 −0.131413 −0.0657065 0.997839i \(-0.520930\pi\)
−0.0657065 + 0.997839i \(0.520930\pi\)
\(824\) − 7.40637i − 0.00898831i
\(825\) 0 0
\(826\) −754.304 −0.913201
\(827\) − 755.525i − 0.913573i −0.889576 0.456787i \(-0.849000\pi\)
0.889576 0.456787i \(-0.151000\pi\)
\(828\) 0 0
\(829\) 603.366 0.727824 0.363912 0.931433i \(-0.381441\pi\)
0.363912 + 0.931433i \(0.381441\pi\)
\(830\) − 1713.35i − 2.06427i
\(831\) 0 0
\(832\) 534.901 0.642910
\(833\) 990.674i 1.18928i
\(834\) 0 0
\(835\) 443.393 0.531010
\(836\) 566.824i 0.678019i
\(837\) 0 0
\(838\) 100.350 0.119749
\(839\) − 81.2325i − 0.0968206i −0.998828 0.0484103i \(-0.984585\pi\)
0.998828 0.0484103i \(-0.0154155\pi\)
\(840\) 0 0
\(841\) −928.440 −1.10397
\(842\) − 441.996i − 0.524936i
\(843\) 0 0
\(844\) −710.240 −0.841517
\(845\) 87.2222i 0.103222i
\(846\) 0 0
\(847\) 30.4955 0.0360041
\(848\) 68.1297i 0.0803416i
\(849\) 0 0
\(850\) 2074.45 2.44053
\(851\) − 80.9044i − 0.0950698i
\(852\) 0 0
\(853\) 532.064 0.623756 0.311878 0.950122i \(-0.399042\pi\)
0.311878 + 0.950122i \(0.399042\pi\)
\(854\) 612.461i 0.717167i
\(855\) 0 0
\(856\) 204.597 0.239015
\(857\) − 98.9583i − 0.115471i −0.998332 0.0577353i \(-0.981612\pi\)
0.998332 0.0577353i \(-0.0183879\pi\)
\(858\) 0 0
\(859\) −124.582 −0.145032 −0.0725158 0.997367i \(-0.523103\pi\)
−0.0725158 + 0.997367i \(0.523103\pi\)
\(860\) − 380.538i − 0.442486i
\(861\) 0 0
\(862\) 1849.16 2.14519
\(863\) − 1163.67i − 1.34840i −0.738547 0.674202i \(-0.764488\pi\)
0.738547 0.674202i \(-0.235512\pi\)
\(864\) 0 0
\(865\) 245.570 0.283896
\(866\) 118.988i 0.137400i
\(867\) 0 0
\(868\) 180.447 0.207888
\(869\) 624.583i 0.718737i
\(870\) 0 0
\(871\) 1403.26 1.61109
\(872\) − 38.5303i − 0.0441861i
\(873\) 0 0
\(874\) −477.719 −0.546589
\(875\) − 47.5885i − 0.0543868i
\(876\) 0 0
\(877\) 93.2868 0.106370 0.0531852 0.998585i \(-0.483063\pi\)
0.0531852 + 0.998585i \(0.483063\pi\)
\(878\) 565.819i 0.644441i
\(879\) 0 0
\(880\) −1344.54 −1.52788
\(881\) 674.774i 0.765918i 0.923765 + 0.382959i \(0.125095\pi\)
−0.923765 + 0.382959i \(0.874905\pi\)
\(882\) 0 0
\(883\) −595.635 −0.674558 −0.337279 0.941405i \(-0.609507\pi\)
−0.337279 + 0.941405i \(0.609507\pi\)
\(884\) 1377.93i 1.55875i
\(885\) 0 0
\(886\) −172.776 −0.195007
\(887\) 432.077i 0.487121i 0.969886 + 0.243561i \(0.0783155\pi\)
−0.969886 + 0.243561i \(0.921684\pi\)
\(888\) 0 0
\(889\) −48.6387 −0.0547118
\(890\) − 1453.91i − 1.63361i
\(891\) 0 0
\(892\) −670.220 −0.751368
\(893\) 32.5596i 0.0364609i
\(894\) 0 0
\(895\) 1682.72 1.88014
\(896\) 230.774i 0.257561i
\(897\) 0 0
\(898\) −2182.66 −2.43058
\(899\) 520.740i 0.579244i
\(900\) 0 0
\(901\) −122.754 −0.136242
\(902\) 112.308i 0.124510i
\(903\) 0 0
\(904\) −38.0345 −0.0420735
\(905\) − 2009.10i − 2.21999i
\(906\) 0 0
\(907\) 1369.79 1.51025 0.755123 0.655583i \(-0.227577\pi\)
0.755123 + 0.655583i \(0.227577\pi\)
\(908\) 544.839i 0.600043i
\(909\) 0 0
\(910\) −1020.30 −1.12121
\(911\) − 759.290i − 0.833469i −0.909028 0.416734i \(-0.863175\pi\)
0.909028 0.416734i \(-0.136825\pi\)
\(912\) 0 0
\(913\) −967.683 −1.05989
\(914\) − 1565.51i − 1.71282i
\(915\) 0 0
\(916\) −1332.33 −1.45451
\(917\) 396.120i 0.431974i
\(918\) 0 0
\(919\) 560.626 0.610039 0.305019 0.952346i \(-0.401337\pi\)
0.305019 + 0.952346i \(0.401337\pi\)
\(920\) − 131.647i − 0.143095i
\(921\) 0 0
\(922\) −839.578 −0.910605
\(923\) − 703.781i − 0.762493i
\(924\) 0 0
\(925\) 169.358 0.183089
\(926\) − 1899.45i − 2.05125i
\(927\) 0 0
\(928\) 1783.73 1.92213
\(929\) 91.4794i 0.0984708i 0.998787 + 0.0492354i \(0.0156785\pi\)
−0.998787 + 0.0492354i \(0.984322\pi\)
\(930\) 0 0
\(931\) 477.565 0.512959
\(932\) 1303.42i 1.39851i
\(933\) 0 0
\(934\) 2132.78 2.28349
\(935\) − 2422.55i − 2.59097i
\(936\) 0 0
\(937\) −858.127 −0.915824 −0.457912 0.888998i \(-0.651402\pi\)
−0.457912 + 0.888998i \(0.651402\pi\)
\(938\) − 1315.10i − 1.40202i
\(939\) 0 0
\(940\) 48.6608 0.0517668
\(941\) 475.385i 0.505191i 0.967572 + 0.252596i \(0.0812842\pi\)
−0.967572 + 0.252596i \(0.918716\pi\)
\(942\) 0 0
\(943\) −43.3315 −0.0459507
\(944\) 1164.86i 1.23396i
\(945\) 0 0
\(946\) −469.479 −0.496278
\(947\) − 955.194i − 1.00865i −0.863513 0.504326i \(-0.831741\pi\)
0.863513 0.504326i \(-0.168259\pi\)
\(948\) 0 0
\(949\) −315.135 −0.332071
\(950\) − 1000.01i − 1.05264i
\(951\) 0 0
\(952\) −238.115 −0.250121
\(953\) 1128.75i 1.18441i 0.805786 + 0.592206i \(0.201743\pi\)
−0.805786 + 0.592206i \(0.798257\pi\)
\(954\) 0 0
\(955\) −636.801 −0.666808
\(956\) 1327.44i 1.38853i
\(957\) 0 0
\(958\) −1097.85 −1.14598
\(959\) 385.782i 0.402275i
\(960\) 0 0
\(961\) −807.748 −0.840528
\(962\) 245.731i 0.255437i
\(963\) 0 0
\(964\) −272.164 −0.282328
\(965\) − 1576.05i − 1.63322i
\(966\) 0 0
\(967\) −873.591 −0.903404 −0.451702 0.892169i \(-0.649183\pi\)
−0.451702 + 0.892169i \(0.649183\pi\)
\(968\) − 11.9511i − 0.0123462i
\(969\) 0 0
\(970\) 787.741 0.812104
\(971\) 1506.38i 1.55137i 0.631120 + 0.775685i \(0.282596\pi\)
−0.631120 + 0.775685i \(0.717404\pi\)
\(972\) 0 0
\(973\) −646.041 −0.663969
\(974\) 818.887i 0.840747i
\(975\) 0 0
\(976\) 945.811 0.969068
\(977\) − 648.296i − 0.663558i −0.943357 0.331779i \(-0.892351\pi\)
0.943357 0.331779i \(-0.107649\pi\)
\(978\) 0 0
\(979\) −821.155 −0.838769
\(980\) − 713.729i − 0.728294i
\(981\) 0 0
\(982\) −2242.99 −2.28411
\(983\) − 1659.56i − 1.68826i −0.536140 0.844129i \(-0.680118\pi\)
0.536140 0.844129i \(-0.319882\pi\)
\(984\) 0 0
\(985\) −1892.74 −1.92157
\(986\) 3726.66i 3.77958i
\(987\) 0 0
\(988\) 664.246 0.672314
\(989\) − 181.138i − 0.183153i
\(990\) 0 0
\(991\) 106.575 0.107542 0.0537712 0.998553i \(-0.482876\pi\)
0.0537712 + 0.998553i \(0.482876\pi\)
\(992\) − 524.946i − 0.529180i
\(993\) 0 0
\(994\) −659.565 −0.663546
\(995\) 1057.97i 1.06328i
\(996\) 0 0
\(997\) 888.924 0.891598 0.445799 0.895133i \(-0.352920\pi\)
0.445799 + 0.895133i \(0.352920\pi\)
\(998\) − 2213.33i − 2.21776i
\(999\) 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.890.16 84
3.2 odd 2 inner 1143.3.b.a.890.69 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.16 84 1.1 even 1 trivial
1143.3.b.a.890.69 yes 84 3.2 odd 2 inner