Properties

Label 1143.3.b.a.890.9
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 890.9
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.76

$q$-expansion

\(f(q)\) \(=\) \(q-3.22737i q^{2} -6.41592 q^{4} +1.45400i q^{5} +2.15873 q^{7} +7.79707i q^{8} +O(q^{10})\) \(q-3.22737i q^{2} -6.41592 q^{4} +1.45400i q^{5} +2.15873 q^{7} +7.79707i q^{8} +4.69259 q^{10} -6.04547i q^{11} +15.2005 q^{13} -6.96703i q^{14} -0.499646 q^{16} +16.9441i q^{17} -5.63400 q^{19} -9.32873i q^{20} -19.5110 q^{22} +12.2914i q^{23} +22.8859 q^{25} -49.0577i q^{26} -13.8503 q^{28} +20.9473i q^{29} +44.7052 q^{31} +32.8008i q^{32} +54.6849 q^{34} +3.13879i q^{35} +50.2331 q^{37} +18.1830i q^{38} -11.3369 q^{40} -45.3146i q^{41} +63.6233 q^{43} +38.7873i q^{44} +39.6689 q^{46} -57.8837i q^{47} -44.3399 q^{49} -73.8613i q^{50} -97.5252 q^{52} -16.7003i q^{53} +8.79009 q^{55} +16.8318i q^{56} +67.6048 q^{58} -111.728i q^{59} -102.642 q^{61} -144.280i q^{62} +103.862 q^{64} +22.1015i q^{65} -69.5730 q^{67} -108.712i q^{68} +10.1300 q^{70} +87.3147i q^{71} -14.5578 q^{73} -162.121i q^{74} +36.1473 q^{76} -13.0506i q^{77} +146.673 q^{79} -0.726483i q^{80} -146.247 q^{82} +59.3483i q^{83} -24.6367 q^{85} -205.336i q^{86} +47.1370 q^{88} -10.9134i q^{89} +32.8138 q^{91} -78.8606i q^{92} -186.812 q^{94} -8.19182i q^{95} +21.3026 q^{97} +143.101i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.22737i − 1.61369i −0.590766 0.806843i \(-0.701175\pi\)
0.590766 0.806843i \(-0.298825\pi\)
\(3\) 0 0
\(4\) −6.41592 −1.60398
\(5\) 1.45400i 0.290799i 0.989373 + 0.145400i \(0.0464468\pi\)
−0.989373 + 0.145400i \(0.953553\pi\)
\(6\) 0 0
\(7\) 2.15873 0.308390 0.154195 0.988040i \(-0.450722\pi\)
0.154195 + 0.988040i \(0.450722\pi\)
\(8\) 7.79707i 0.974634i
\(9\) 0 0
\(10\) 4.69259 0.469259
\(11\) − 6.04547i − 0.549588i −0.961503 0.274794i \(-0.911390\pi\)
0.961503 0.274794i \(-0.0886097\pi\)
\(12\) 0 0
\(13\) 15.2005 1.16927 0.584635 0.811297i \(-0.301238\pi\)
0.584635 + 0.811297i \(0.301238\pi\)
\(14\) − 6.96703i − 0.497645i
\(15\) 0 0
\(16\) −0.499646 −0.0312279
\(17\) 16.9441i 0.996713i 0.866972 + 0.498356i \(0.166063\pi\)
−0.866972 + 0.498356i \(0.833937\pi\)
\(18\) 0 0
\(19\) −5.63400 −0.296527 −0.148263 0.988948i \(-0.547368\pi\)
−0.148263 + 0.988948i \(0.547368\pi\)
\(20\) − 9.32873i − 0.466436i
\(21\) 0 0
\(22\) −19.5110 −0.886862
\(23\) 12.2914i 0.534408i 0.963640 + 0.267204i \(0.0860998\pi\)
−0.963640 + 0.267204i \(0.913900\pi\)
\(24\) 0 0
\(25\) 22.8859 0.915436
\(26\) − 49.0577i − 1.88683i
\(27\) 0 0
\(28\) −13.8503 −0.494652
\(29\) 20.9473i 0.722321i 0.932504 + 0.361161i \(0.117619\pi\)
−0.932504 + 0.361161i \(0.882381\pi\)
\(30\) 0 0
\(31\) 44.7052 1.44210 0.721052 0.692881i \(-0.243659\pi\)
0.721052 + 0.692881i \(0.243659\pi\)
\(32\) 32.8008i 1.02503i
\(33\) 0 0
\(34\) 54.6849 1.60838
\(35\) 3.13879i 0.0896797i
\(36\) 0 0
\(37\) 50.2331 1.35765 0.678826 0.734299i \(-0.262489\pi\)
0.678826 + 0.734299i \(0.262489\pi\)
\(38\) 18.1830i 0.478501i
\(39\) 0 0
\(40\) −11.3369 −0.283423
\(41\) − 45.3146i − 1.10523i −0.833435 0.552617i \(-0.813629\pi\)
0.833435 0.552617i \(-0.186371\pi\)
\(42\) 0 0
\(43\) 63.6233 1.47961 0.739806 0.672820i \(-0.234917\pi\)
0.739806 + 0.672820i \(0.234917\pi\)
\(44\) 38.7873i 0.881529i
\(45\) 0 0
\(46\) 39.6689 0.862367
\(47\) − 57.8837i − 1.23157i −0.787915 0.615784i \(-0.788839\pi\)
0.787915 0.615784i \(-0.211161\pi\)
\(48\) 0 0
\(49\) −44.3399 −0.904895
\(50\) − 73.8613i − 1.47723i
\(51\) 0 0
\(52\) −97.5252 −1.87548
\(53\) − 16.7003i − 0.315099i −0.987511 0.157550i \(-0.949641\pi\)
0.987511 0.157550i \(-0.0503594\pi\)
\(54\) 0 0
\(55\) 8.79009 0.159820
\(56\) 16.8318i 0.300568i
\(57\) 0 0
\(58\) 67.6048 1.16560
\(59\) − 111.728i − 1.89370i −0.321682 0.946848i \(-0.604248\pi\)
0.321682 0.946848i \(-0.395752\pi\)
\(60\) 0 0
\(61\) −102.642 −1.68265 −0.841325 0.540529i \(-0.818224\pi\)
−0.841325 + 0.540529i \(0.818224\pi\)
\(62\) − 144.280i − 2.32710i
\(63\) 0 0
\(64\) 103.862 1.62284
\(65\) 22.1015i 0.340023i
\(66\) 0 0
\(67\) −69.5730 −1.03840 −0.519201 0.854652i \(-0.673770\pi\)
−0.519201 + 0.854652i \(0.673770\pi\)
\(68\) − 108.712i − 1.59871i
\(69\) 0 0
\(70\) 10.1300 0.144715
\(71\) 87.3147i 1.22979i 0.788611 + 0.614893i \(0.210801\pi\)
−0.788611 + 0.614893i \(0.789199\pi\)
\(72\) 0 0
\(73\) −14.5578 −0.199422 −0.0997108 0.995016i \(-0.531792\pi\)
−0.0997108 + 0.995016i \(0.531792\pi\)
\(74\) − 162.121i − 2.19082i
\(75\) 0 0
\(76\) 36.1473 0.475623
\(77\) − 13.0506i − 0.169488i
\(78\) 0 0
\(79\) 146.673 1.85662 0.928310 0.371807i \(-0.121262\pi\)
0.928310 + 0.371807i \(0.121262\pi\)
\(80\) − 0.726483i − 0.00908104i
\(81\) 0 0
\(82\) −146.247 −1.78350
\(83\) 59.3483i 0.715040i 0.933905 + 0.357520i \(0.116378\pi\)
−0.933905 + 0.357520i \(0.883622\pi\)
\(84\) 0 0
\(85\) −24.6367 −0.289843
\(86\) − 205.336i − 2.38763i
\(87\) 0 0
\(88\) 47.1370 0.535647
\(89\) − 10.9134i − 0.122623i −0.998119 0.0613114i \(-0.980472\pi\)
0.998119 0.0613114i \(-0.0195283\pi\)
\(90\) 0 0
\(91\) 32.8138 0.360591
\(92\) − 78.8606i − 0.857180i
\(93\) 0 0
\(94\) −186.812 −1.98736
\(95\) − 8.19182i − 0.0862297i
\(96\) 0 0
\(97\) 21.3026 0.219614 0.109807 0.993953i \(-0.464977\pi\)
0.109807 + 0.993953i \(0.464977\pi\)
\(98\) 143.101i 1.46022i
\(99\) 0 0
\(100\) −146.834 −1.46834
\(101\) − 191.133i − 1.89240i −0.323576 0.946202i \(-0.604885\pi\)
0.323576 0.946202i \(-0.395115\pi\)
\(102\) 0 0
\(103\) 27.2875 0.264928 0.132464 0.991188i \(-0.457711\pi\)
0.132464 + 0.991188i \(0.457711\pi\)
\(104\) 118.519i 1.13961i
\(105\) 0 0
\(106\) −53.8980 −0.508471
\(107\) 77.1669i 0.721186i 0.932723 + 0.360593i \(0.117426\pi\)
−0.932723 + 0.360593i \(0.882574\pi\)
\(108\) 0 0
\(109\) −1.49702 −0.0137341 −0.00686707 0.999976i \(-0.502186\pi\)
−0.00686707 + 0.999976i \(0.502186\pi\)
\(110\) − 28.3689i − 0.257899i
\(111\) 0 0
\(112\) −1.07860 −0.00963037
\(113\) − 131.194i − 1.16101i −0.814256 0.580506i \(-0.802855\pi\)
0.814256 0.580506i \(-0.197145\pi\)
\(114\) 0 0
\(115\) −17.8716 −0.155406
\(116\) − 134.396i − 1.15859i
\(117\) 0 0
\(118\) −360.588 −3.05583
\(119\) 36.5778i 0.307376i
\(120\) 0 0
\(121\) 84.4523 0.697953
\(122\) 331.263i 2.71527i
\(123\) 0 0
\(124\) −286.825 −2.31311
\(125\) 69.6259i 0.557007i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) − 203.997i − 1.59373i
\(129\) 0 0
\(130\) 71.3297 0.548690
\(131\) − 143.034i − 1.09186i −0.837830 0.545931i \(-0.816176\pi\)
0.837830 0.545931i \(-0.183824\pi\)
\(132\) 0 0
\(133\) −12.1623 −0.0914459
\(134\) 224.538i 1.67565i
\(135\) 0 0
\(136\) −132.114 −0.971430
\(137\) − 4.85061i − 0.0354059i −0.999843 0.0177030i \(-0.994365\pi\)
0.999843 0.0177030i \(-0.00563532\pi\)
\(138\) 0 0
\(139\) 115.522 0.831097 0.415548 0.909571i \(-0.363590\pi\)
0.415548 + 0.909571i \(0.363590\pi\)
\(140\) − 20.1382i − 0.143844i
\(141\) 0 0
\(142\) 281.797 1.98449
\(143\) − 91.8942i − 0.642617i
\(144\) 0 0
\(145\) −30.4573 −0.210051
\(146\) 46.9833i 0.321804i
\(147\) 0 0
\(148\) −322.292 −2.17765
\(149\) − 101.662i − 0.682295i −0.940010 0.341148i \(-0.889184\pi\)
0.940010 0.341148i \(-0.110816\pi\)
\(150\) 0 0
\(151\) 147.676 0.977990 0.488995 0.872287i \(-0.337364\pi\)
0.488995 + 0.872287i \(0.337364\pi\)
\(152\) − 43.9287i − 0.289005i
\(153\) 0 0
\(154\) −42.1190 −0.273500
\(155\) 65.0012i 0.419363i
\(156\) 0 0
\(157\) −31.8716 −0.203004 −0.101502 0.994835i \(-0.532365\pi\)
−0.101502 + 0.994835i \(0.532365\pi\)
\(158\) − 473.368i − 2.99600i
\(159\) 0 0
\(160\) −47.6923 −0.298077
\(161\) 26.5338i 0.164806i
\(162\) 0 0
\(163\) −8.77669 −0.0538447 −0.0269224 0.999638i \(-0.508571\pi\)
−0.0269224 + 0.999638i \(0.508571\pi\)
\(164\) 290.735i 1.77277i
\(165\) 0 0
\(166\) 191.539 1.15385
\(167\) 117.208i 0.701842i 0.936405 + 0.350921i \(0.114131\pi\)
−0.936405 + 0.350921i \(0.885869\pi\)
\(168\) 0 0
\(169\) 62.0553 0.367191
\(170\) 79.5117i 0.467716i
\(171\) 0 0
\(172\) −408.202 −2.37327
\(173\) 206.099i 1.19132i 0.803235 + 0.595662i \(0.203110\pi\)
−0.803235 + 0.595662i \(0.796890\pi\)
\(174\) 0 0
\(175\) 49.4045 0.282311
\(176\) 3.02059i 0.0171625i
\(177\) 0 0
\(178\) −35.2217 −0.197875
\(179\) 114.251i 0.638271i 0.947709 + 0.319136i \(0.103393\pi\)
−0.947709 + 0.319136i \(0.896607\pi\)
\(180\) 0 0
\(181\) 320.232 1.76923 0.884617 0.466317i \(-0.154420\pi\)
0.884617 + 0.466317i \(0.154420\pi\)
\(182\) − 105.902i − 0.581881i
\(183\) 0 0
\(184\) −95.8368 −0.520852
\(185\) 73.0388i 0.394804i
\(186\) 0 0
\(187\) 102.435 0.547781
\(188\) 371.377i 1.97541i
\(189\) 0 0
\(190\) −26.4381 −0.139148
\(191\) − 104.041i − 0.544719i −0.962196 0.272360i \(-0.912196\pi\)
0.962196 0.272360i \(-0.0878040\pi\)
\(192\) 0 0
\(193\) −333.803 −1.72955 −0.864774 0.502162i \(-0.832538\pi\)
−0.864774 + 0.502162i \(0.832538\pi\)
\(194\) − 68.7514i − 0.354389i
\(195\) 0 0
\(196\) 284.481 1.45143
\(197\) 111.309i 0.565020i 0.959264 + 0.282510i \(0.0911670\pi\)
−0.959264 + 0.282510i \(0.908833\pi\)
\(198\) 0 0
\(199\) 179.209 0.900549 0.450274 0.892890i \(-0.351326\pi\)
0.450274 + 0.892890i \(0.351326\pi\)
\(200\) 178.443i 0.892215i
\(201\) 0 0
\(202\) −616.856 −3.05374
\(203\) 45.2196i 0.222757i
\(204\) 0 0
\(205\) 65.8873 0.321401
\(206\) − 88.0670i − 0.427510i
\(207\) 0 0
\(208\) −7.59487 −0.0365138
\(209\) 34.0602i 0.162968i
\(210\) 0 0
\(211\) −211.720 −1.00341 −0.501706 0.865038i \(-0.667294\pi\)
−0.501706 + 0.865038i \(0.667294\pi\)
\(212\) 107.148i 0.505413i
\(213\) 0 0
\(214\) 249.046 1.16377
\(215\) 92.5081i 0.430270i
\(216\) 0 0
\(217\) 96.5066 0.444731
\(218\) 4.83144i 0.0221626i
\(219\) 0 0
\(220\) −56.3965 −0.256348
\(221\) 257.559i 1.16543i
\(222\) 0 0
\(223\) 252.821 1.13373 0.566864 0.823812i \(-0.308157\pi\)
0.566864 + 0.823812i \(0.308157\pi\)
\(224\) 70.8082i 0.316108i
\(225\) 0 0
\(226\) −423.413 −1.87351
\(227\) 160.992i 0.709217i 0.935015 + 0.354609i \(0.115386\pi\)
−0.935015 + 0.354609i \(0.884614\pi\)
\(228\) 0 0
\(229\) 16.8026 0.0733739 0.0366870 0.999327i \(-0.488320\pi\)
0.0366870 + 0.999327i \(0.488320\pi\)
\(230\) 57.6784i 0.250776i
\(231\) 0 0
\(232\) −163.328 −0.703999
\(233\) 56.2312i 0.241336i 0.992693 + 0.120668i \(0.0385036\pi\)
−0.992693 + 0.120668i \(0.961496\pi\)
\(234\) 0 0
\(235\) 84.1627 0.358139
\(236\) 716.838i 3.03745i
\(237\) 0 0
\(238\) 118.050 0.496009
\(239\) − 365.589i − 1.52966i −0.644232 0.764830i \(-0.722823\pi\)
0.644232 0.764830i \(-0.277177\pi\)
\(240\) 0 0
\(241\) −314.533 −1.30512 −0.652558 0.757739i \(-0.726304\pi\)
−0.652558 + 0.757739i \(0.726304\pi\)
\(242\) − 272.559i − 1.12628i
\(243\) 0 0
\(244\) 658.541 2.69894
\(245\) − 64.4700i − 0.263143i
\(246\) 0 0
\(247\) −85.6397 −0.346719
\(248\) 348.570i 1.40552i
\(249\) 0 0
\(250\) 224.709 0.898835
\(251\) − 413.825i − 1.64871i −0.566076 0.824353i \(-0.691539\pi\)
0.566076 0.824353i \(-0.308461\pi\)
\(252\) 0 0
\(253\) 74.3072 0.293704
\(254\) 36.3706i 0.143191i
\(255\) 0 0
\(256\) −242.928 −0.948936
\(257\) 129.894i 0.505426i 0.967541 + 0.252713i \(0.0813228\pi\)
−0.967541 + 0.252713i \(0.918677\pi\)
\(258\) 0 0
\(259\) 108.440 0.418687
\(260\) − 141.801i − 0.545390i
\(261\) 0 0
\(262\) −461.623 −1.76192
\(263\) 395.319i 1.50311i 0.659668 + 0.751557i \(0.270697\pi\)
−0.659668 + 0.751557i \(0.729303\pi\)
\(264\) 0 0
\(265\) 24.2821 0.0916307
\(266\) 39.2523i 0.147565i
\(267\) 0 0
\(268\) 446.375 1.66558
\(269\) 424.438i 1.57784i 0.614497 + 0.788919i \(0.289359\pi\)
−0.614497 + 0.788919i \(0.710641\pi\)
\(270\) 0 0
\(271\) −296.500 −1.09410 −0.547048 0.837101i \(-0.684248\pi\)
−0.547048 + 0.837101i \(0.684248\pi\)
\(272\) − 8.46606i − 0.0311252i
\(273\) 0 0
\(274\) −15.6547 −0.0571340
\(275\) − 138.356i − 0.503113i
\(276\) 0 0
\(277\) −111.921 −0.404045 −0.202023 0.979381i \(-0.564751\pi\)
−0.202023 + 0.979381i \(0.564751\pi\)
\(278\) − 372.834i − 1.34113i
\(279\) 0 0
\(280\) −24.4734 −0.0874049
\(281\) − 202.966i − 0.722298i −0.932508 0.361149i \(-0.882385\pi\)
0.932508 0.361149i \(-0.117615\pi\)
\(282\) 0 0
\(283\) 20.8527 0.0736843 0.0368422 0.999321i \(-0.488270\pi\)
0.0368422 + 0.999321i \(0.488270\pi\)
\(284\) − 560.204i − 1.97255i
\(285\) 0 0
\(286\) −296.577 −1.03698
\(287\) − 97.8221i − 0.340844i
\(288\) 0 0
\(289\) 1.89702 0.00656409
\(290\) 98.2971i 0.338955i
\(291\) 0 0
\(292\) 93.4015 0.319868
\(293\) − 506.171i − 1.72755i −0.503881 0.863773i \(-0.668095\pi\)
0.503881 0.863773i \(-0.331905\pi\)
\(294\) 0 0
\(295\) 162.452 0.550685
\(296\) 391.671i 1.32321i
\(297\) 0 0
\(298\) −328.101 −1.10101
\(299\) 186.835i 0.624867i
\(300\) 0 0
\(301\) 137.346 0.456298
\(302\) − 476.607i − 1.57817i
\(303\) 0 0
\(304\) 2.81501 0.00925989
\(305\) − 149.241i − 0.489314i
\(306\) 0 0
\(307\) 446.721 1.45512 0.727559 0.686045i \(-0.240655\pi\)
0.727559 + 0.686045i \(0.240655\pi\)
\(308\) 83.7313i 0.271855i
\(309\) 0 0
\(310\) 209.783 0.676720
\(311\) 523.253i 1.68249i 0.540657 + 0.841243i \(0.318176\pi\)
−0.540657 + 0.841243i \(0.681824\pi\)
\(312\) 0 0
\(313\) 354.860 1.13374 0.566869 0.823808i \(-0.308155\pi\)
0.566869 + 0.823808i \(0.308155\pi\)
\(314\) 102.861i 0.327584i
\(315\) 0 0
\(316\) −941.042 −2.97798
\(317\) 387.303i 1.22178i 0.791717 + 0.610888i \(0.209187\pi\)
−0.791717 + 0.610888i \(0.790813\pi\)
\(318\) 0 0
\(319\) 126.636 0.396979
\(320\) 151.015i 0.471921i
\(321\) 0 0
\(322\) 85.6344 0.265945
\(323\) − 95.4632i − 0.295552i
\(324\) 0 0
\(325\) 347.877 1.07039
\(326\) 28.3256i 0.0868884i
\(327\) 0 0
\(328\) 353.321 1.07720
\(329\) − 124.955i − 0.379804i
\(330\) 0 0
\(331\) −266.449 −0.804981 −0.402490 0.915424i \(-0.631855\pi\)
−0.402490 + 0.915424i \(0.631855\pi\)
\(332\) − 380.774i − 1.14691i
\(333\) 0 0
\(334\) 378.272 1.13255
\(335\) − 101.159i − 0.301967i
\(336\) 0 0
\(337\) 555.631 1.64876 0.824379 0.566039i \(-0.191525\pi\)
0.824379 + 0.566039i \(0.191525\pi\)
\(338\) − 200.275i − 0.592531i
\(339\) 0 0
\(340\) 158.067 0.464903
\(341\) − 270.264i − 0.792563i
\(342\) 0 0
\(343\) −201.496 −0.587451
\(344\) 496.076i 1.44208i
\(345\) 0 0
\(346\) 665.158 1.92242
\(347\) − 51.5775i − 0.148638i −0.997235 0.0743192i \(-0.976322\pi\)
0.997235 0.0743192i \(-0.0236784\pi\)
\(348\) 0 0
\(349\) 593.799 1.70143 0.850716 0.525626i \(-0.176169\pi\)
0.850716 + 0.525626i \(0.176169\pi\)
\(350\) − 159.447i − 0.455562i
\(351\) 0 0
\(352\) 198.296 0.563342
\(353\) − 461.155i − 1.30639i −0.757190 0.653195i \(-0.773428\pi\)
0.757190 0.653195i \(-0.226572\pi\)
\(354\) 0 0
\(355\) −126.955 −0.357621
\(356\) 70.0197i 0.196685i
\(357\) 0 0
\(358\) 368.729 1.02997
\(359\) − 231.114i − 0.643771i −0.946779 0.321885i \(-0.895683\pi\)
0.946779 0.321885i \(-0.104317\pi\)
\(360\) 0 0
\(361\) −329.258 −0.912072
\(362\) − 1033.51i − 2.85499i
\(363\) 0 0
\(364\) −210.531 −0.578381
\(365\) − 21.1670i − 0.0579916i
\(366\) 0 0
\(367\) −26.4050 −0.0719482 −0.0359741 0.999353i \(-0.511453\pi\)
−0.0359741 + 0.999353i \(0.511453\pi\)
\(368\) − 6.14134i − 0.0166884i
\(369\) 0 0
\(370\) 235.723 0.637090
\(371\) − 36.0514i − 0.0971736i
\(372\) 0 0
\(373\) −529.266 −1.41894 −0.709472 0.704734i \(-0.751066\pi\)
−0.709472 + 0.704734i \(0.751066\pi\)
\(374\) − 330.596i − 0.883947i
\(375\) 0 0
\(376\) 451.323 1.20033
\(377\) 318.410i 0.844588i
\(378\) 0 0
\(379\) −361.693 −0.954336 −0.477168 0.878812i \(-0.658337\pi\)
−0.477168 + 0.878812i \(0.658337\pi\)
\(380\) 52.5581i 0.138311i
\(381\) 0 0
\(382\) −335.780 −0.879005
\(383\) − 93.1447i − 0.243198i −0.992579 0.121599i \(-0.961198\pi\)
0.992579 0.121599i \(-0.0388021\pi\)
\(384\) 0 0
\(385\) 18.9755 0.0492869
\(386\) 1077.30i 2.79095i
\(387\) 0 0
\(388\) −136.676 −0.352257
\(389\) 205.427i 0.528091i 0.964510 + 0.264045i \(0.0850569\pi\)
−0.964510 + 0.264045i \(0.914943\pi\)
\(390\) 0 0
\(391\) −208.267 −0.532651
\(392\) − 345.721i − 0.881942i
\(393\) 0 0
\(394\) 359.235 0.911764
\(395\) 213.262i 0.539904i
\(396\) 0 0
\(397\) 178.575 0.449812 0.224906 0.974380i \(-0.427793\pi\)
0.224906 + 0.974380i \(0.427793\pi\)
\(398\) − 578.375i − 1.45320i
\(399\) 0 0
\(400\) −11.4348 −0.0285871
\(401\) − 80.5359i − 0.200838i −0.994945 0.100419i \(-0.967982\pi\)
0.994945 0.100419i \(-0.0320183\pi\)
\(402\) 0 0
\(403\) 679.542 1.68621
\(404\) 1226.29i 3.03538i
\(405\) 0 0
\(406\) 145.941 0.359459
\(407\) − 303.683i − 0.746149i
\(408\) 0 0
\(409\) −50.7083 −0.123981 −0.0619906 0.998077i \(-0.519745\pi\)
−0.0619906 + 0.998077i \(0.519745\pi\)
\(410\) − 212.643i − 0.518641i
\(411\) 0 0
\(412\) −175.075 −0.424939
\(413\) − 241.191i − 0.583997i
\(414\) 0 0
\(415\) −86.2923 −0.207933
\(416\) 498.589i 1.19853i
\(417\) 0 0
\(418\) 109.925 0.262978
\(419\) 421.505i 1.00598i 0.864292 + 0.502990i \(0.167767\pi\)
−0.864292 + 0.502990i \(0.832233\pi\)
\(420\) 0 0
\(421\) 126.531 0.300548 0.150274 0.988644i \(-0.451984\pi\)
0.150274 + 0.988644i \(0.451984\pi\)
\(422\) 683.298i 1.61919i
\(423\) 0 0
\(424\) 130.213 0.307107
\(425\) 387.781i 0.912426i
\(426\) 0 0
\(427\) −221.576 −0.518913
\(428\) − 495.097i − 1.15677i
\(429\) 0 0
\(430\) 298.558 0.694321
\(431\) 69.7232i 0.161771i 0.996723 + 0.0808854i \(0.0257748\pi\)
−0.996723 + 0.0808854i \(0.974225\pi\)
\(432\) 0 0
\(433\) 433.061 1.00014 0.500070 0.865985i \(-0.333308\pi\)
0.500070 + 0.865985i \(0.333308\pi\)
\(434\) − 311.462i − 0.717656i
\(435\) 0 0
\(436\) 9.60477 0.0220293
\(437\) − 69.2497i − 0.158466i
\(438\) 0 0
\(439\) 730.260 1.66346 0.831731 0.555179i \(-0.187350\pi\)
0.831731 + 0.555179i \(0.187350\pi\)
\(440\) 68.5370i 0.155766i
\(441\) 0 0
\(442\) 831.238 1.88063
\(443\) 113.431i 0.256053i 0.991771 + 0.128026i \(0.0408642\pi\)
−0.991771 + 0.128026i \(0.959136\pi\)
\(444\) 0 0
\(445\) 15.8681 0.0356586
\(446\) − 815.948i − 1.82948i
\(447\) 0 0
\(448\) 224.210 0.500468
\(449\) 372.379i 0.829352i 0.909969 + 0.414676i \(0.136105\pi\)
−0.909969 + 0.414676i \(0.863895\pi\)
\(450\) 0 0
\(451\) −273.948 −0.607424
\(452\) 841.733i 1.86224i
\(453\) 0 0
\(454\) 519.582 1.14445
\(455\) 47.7112i 0.104860i
\(456\) 0 0
\(457\) −525.639 −1.15020 −0.575098 0.818085i \(-0.695036\pi\)
−0.575098 + 0.818085i \(0.695036\pi\)
\(458\) − 54.2283i − 0.118402i
\(459\) 0 0
\(460\) 114.663 0.249267
\(461\) 86.0937i 0.186754i 0.995631 + 0.0933771i \(0.0297662\pi\)
−0.995631 + 0.0933771i \(0.970234\pi\)
\(462\) 0 0
\(463\) 395.423 0.854046 0.427023 0.904241i \(-0.359562\pi\)
0.427023 + 0.904241i \(0.359562\pi\)
\(464\) − 10.4662i − 0.0225566i
\(465\) 0 0
\(466\) 181.479 0.389440
\(467\) 398.158i 0.852587i 0.904585 + 0.426294i \(0.140181\pi\)
−0.904585 + 0.426294i \(0.859819\pi\)
\(468\) 0 0
\(469\) −150.189 −0.320233
\(470\) − 271.624i − 0.577924i
\(471\) 0 0
\(472\) 871.151 1.84566
\(473\) − 384.633i − 0.813177i
\(474\) 0 0
\(475\) −128.939 −0.271451
\(476\) − 234.680i − 0.493026i
\(477\) 0 0
\(478\) −1179.89 −2.46839
\(479\) 298.231i 0.622611i 0.950310 + 0.311305i \(0.100766\pi\)
−0.950310 + 0.311305i \(0.899234\pi\)
\(480\) 0 0
\(481\) 763.568 1.58746
\(482\) 1015.11i 2.10605i
\(483\) 0 0
\(484\) −541.839 −1.11950
\(485\) 30.9739i 0.0638637i
\(486\) 0 0
\(487\) 4.07802 0.00837376 0.00418688 0.999991i \(-0.498667\pi\)
0.00418688 + 0.999991i \(0.498667\pi\)
\(488\) − 800.304i − 1.63997i
\(489\) 0 0
\(490\) −208.069 −0.424630
\(491\) − 515.872i − 1.05065i −0.850900 0.525327i \(-0.823943\pi\)
0.850900 0.525327i \(-0.176057\pi\)
\(492\) 0 0
\(493\) −354.934 −0.719947
\(494\) 276.391i 0.559496i
\(495\) 0 0
\(496\) −22.3368 −0.0450338
\(497\) 188.489i 0.379254i
\(498\) 0 0
\(499\) −26.3742 −0.0528541 −0.0264270 0.999651i \(-0.508413\pi\)
−0.0264270 + 0.999651i \(0.508413\pi\)
\(500\) − 446.714i − 0.893429i
\(501\) 0 0
\(502\) −1335.57 −2.66049
\(503\) 11.6037i 0.0230690i 0.999933 + 0.0115345i \(0.00367163\pi\)
−0.999933 + 0.0115345i \(0.996328\pi\)
\(504\) 0 0
\(505\) 277.907 0.550310
\(506\) − 239.817i − 0.473946i
\(507\) 0 0
\(508\) 72.3038 0.142330
\(509\) 623.478i 1.22491i 0.790506 + 0.612454i \(0.209818\pi\)
−0.790506 + 0.612454i \(0.790182\pi\)
\(510\) 0 0
\(511\) −31.4263 −0.0614997
\(512\) − 31.9719i − 0.0624451i
\(513\) 0 0
\(514\) 419.218 0.815598
\(515\) 39.6760i 0.0770408i
\(516\) 0 0
\(517\) −349.934 −0.676855
\(518\) − 349.975i − 0.675628i
\(519\) 0 0
\(520\) −172.327 −0.331398
\(521\) 126.020i 0.241881i 0.992660 + 0.120941i \(0.0385911\pi\)
−0.992660 + 0.120941i \(0.961409\pi\)
\(522\) 0 0
\(523\) 272.380 0.520802 0.260401 0.965500i \(-0.416145\pi\)
0.260401 + 0.965500i \(0.416145\pi\)
\(524\) 917.694i 1.75132i
\(525\) 0 0
\(526\) 1275.84 2.42555
\(527\) 757.490i 1.43736i
\(528\) 0 0
\(529\) 377.922 0.714408
\(530\) − 78.3674i − 0.147863i
\(531\) 0 0
\(532\) 78.0324 0.146677
\(533\) − 688.805i − 1.29232i
\(534\) 0 0
\(535\) −112.200 −0.209721
\(536\) − 542.465i − 1.01206i
\(537\) 0 0
\(538\) 1369.82 2.54613
\(539\) 268.055i 0.497320i
\(540\) 0 0
\(541\) −893.640 −1.65183 −0.825915 0.563795i \(-0.809341\pi\)
−0.825915 + 0.563795i \(0.809341\pi\)
\(542\) 956.915i 1.76553i
\(543\) 0 0
\(544\) −555.781 −1.02166
\(545\) − 2.17666i − 0.00399388i
\(546\) 0 0
\(547\) 271.399 0.496158 0.248079 0.968740i \(-0.420201\pi\)
0.248079 + 0.968740i \(0.420201\pi\)
\(548\) 31.1211i 0.0567904i
\(549\) 0 0
\(550\) −446.526 −0.811866
\(551\) − 118.017i − 0.214187i
\(552\) 0 0
\(553\) 316.628 0.572564
\(554\) 361.209i 0.652002i
\(555\) 0 0
\(556\) −741.183 −1.33306
\(557\) 356.461i 0.639967i 0.947423 + 0.319983i \(0.103677\pi\)
−0.947423 + 0.319983i \(0.896323\pi\)
\(558\) 0 0
\(559\) 967.106 1.73007
\(560\) − 1.56828i − 0.00280051i
\(561\) 0 0
\(562\) −655.045 −1.16556
\(563\) 255.522i 0.453858i 0.973911 + 0.226929i \(0.0728686\pi\)
−0.973911 + 0.226929i \(0.927131\pi\)
\(564\) 0 0
\(565\) 190.756 0.337622
\(566\) − 67.2993i − 0.118903i
\(567\) 0 0
\(568\) −680.799 −1.19859
\(569\) 182.702i 0.321093i 0.987028 + 0.160547i \(0.0513257\pi\)
−0.987028 + 0.160547i \(0.948674\pi\)
\(570\) 0 0
\(571\) 159.637 0.279574 0.139787 0.990182i \(-0.455358\pi\)
0.139787 + 0.990182i \(0.455358\pi\)
\(572\) 589.586i 1.03074i
\(573\) 0 0
\(574\) −315.708 −0.550014
\(575\) 281.299i 0.489216i
\(576\) 0 0
\(577\) −920.566 −1.59543 −0.797717 0.603031i \(-0.793959\pi\)
−0.797717 + 0.603031i \(0.793959\pi\)
\(578\) − 6.12239i − 0.0105924i
\(579\) 0 0
\(580\) 195.412 0.336917
\(581\) 128.117i 0.220511i
\(582\) 0 0
\(583\) −100.961 −0.173175
\(584\) − 113.508i − 0.194363i
\(585\) 0 0
\(586\) −1633.60 −2.78772
\(587\) 790.574i 1.34680i 0.739276 + 0.673402i \(0.235168\pi\)
−0.739276 + 0.673402i \(0.764832\pi\)
\(588\) 0 0
\(589\) −251.869 −0.427622
\(590\) − 524.293i − 0.888633i
\(591\) 0 0
\(592\) −25.0988 −0.0423966
\(593\) − 42.5943i − 0.0718285i −0.999355 0.0359142i \(-0.988566\pi\)
0.999355 0.0359142i \(-0.0114343\pi\)
\(594\) 0 0
\(595\) −53.1840 −0.0893849
\(596\) 652.255i 1.09439i
\(597\) 0 0
\(598\) 602.987 1.00834
\(599\) 532.138i 0.888378i 0.895933 + 0.444189i \(0.146508\pi\)
−0.895933 + 0.444189i \(0.853492\pi\)
\(600\) 0 0
\(601\) −20.6373 −0.0343382 −0.0171691 0.999853i \(-0.505465\pi\)
−0.0171691 + 0.999853i \(0.505465\pi\)
\(602\) − 443.265i − 0.736321i
\(603\) 0 0
\(604\) −947.480 −1.56868
\(605\) 122.793i 0.202964i
\(606\) 0 0
\(607\) 611.889 1.00805 0.504027 0.863688i \(-0.331851\pi\)
0.504027 + 0.863688i \(0.331851\pi\)
\(608\) − 184.800i − 0.303947i
\(609\) 0 0
\(610\) −481.655 −0.789598
\(611\) − 879.861i − 1.44003i
\(612\) 0 0
\(613\) 939.218 1.53217 0.766083 0.642742i \(-0.222203\pi\)
0.766083 + 0.642742i \(0.222203\pi\)
\(614\) − 1441.74i − 2.34810i
\(615\) 0 0
\(616\) 101.756 0.165188
\(617\) − 1182.83i − 1.91707i −0.284978 0.958534i \(-0.591986\pi\)
0.284978 0.958534i \(-0.408014\pi\)
\(618\) 0 0
\(619\) −535.727 −0.865472 −0.432736 0.901521i \(-0.642452\pi\)
−0.432736 + 0.901521i \(0.642452\pi\)
\(620\) − 417.043i − 0.672650i
\(621\) 0 0
\(622\) 1688.73 2.71500
\(623\) − 23.5592i − 0.0378157i
\(624\) 0 0
\(625\) 470.911 0.753458
\(626\) − 1145.26i − 1.82950i
\(627\) 0 0
\(628\) 204.485 0.325614
\(629\) 851.155i 1.35319i
\(630\) 0 0
\(631\) −517.306 −0.819819 −0.409910 0.912126i \(-0.634440\pi\)
−0.409910 + 0.912126i \(0.634440\pi\)
\(632\) 1143.62i 1.80952i
\(633\) 0 0
\(634\) 1249.97 1.97156
\(635\) − 16.3857i − 0.0258043i
\(636\) 0 0
\(637\) −673.988 −1.05807
\(638\) − 408.703i − 0.640600i
\(639\) 0 0
\(640\) 296.611 0.463455
\(641\) 289.081i 0.450984i 0.974245 + 0.225492i \(0.0723989\pi\)
−0.974245 + 0.225492i \(0.927601\pi\)
\(642\) 0 0
\(643\) −311.244 −0.484050 −0.242025 0.970270i \(-0.577812\pi\)
−0.242025 + 0.970270i \(0.577812\pi\)
\(644\) − 170.239i − 0.264346i
\(645\) 0 0
\(646\) −308.095 −0.476928
\(647\) 764.050i 1.18091i 0.807070 + 0.590456i \(0.201052\pi\)
−0.807070 + 0.590456i \(0.798948\pi\)
\(648\) 0 0
\(649\) −675.449 −1.04075
\(650\) − 1122.73i − 1.72727i
\(651\) 0 0
\(652\) 56.3105 0.0863659
\(653\) 442.765i 0.678047i 0.940778 + 0.339023i \(0.110097\pi\)
−0.940778 + 0.339023i \(0.889903\pi\)
\(654\) 0 0
\(655\) 207.971 0.317513
\(656\) 22.6413i 0.0345141i
\(657\) 0 0
\(658\) −403.277 −0.612883
\(659\) − 67.8358i − 0.102937i −0.998675 0.0514687i \(-0.983610\pi\)
0.998675 0.0514687i \(-0.0163903\pi\)
\(660\) 0 0
\(661\) 776.372 1.17454 0.587271 0.809391i \(-0.300202\pi\)
0.587271 + 0.809391i \(0.300202\pi\)
\(662\) 859.929i 1.29899i
\(663\) 0 0
\(664\) −462.743 −0.696902
\(665\) − 17.6840i − 0.0265924i
\(666\) 0 0
\(667\) −257.472 −0.386014
\(668\) − 751.995i − 1.12574i
\(669\) 0 0
\(670\) −326.477 −0.487279
\(671\) 620.517i 0.924765i
\(672\) 0 0
\(673\) −1207.59 −1.79434 −0.897172 0.441681i \(-0.854382\pi\)
−0.897172 + 0.441681i \(0.854382\pi\)
\(674\) − 1793.23i − 2.66058i
\(675\) 0 0
\(676\) −398.142 −0.588967
\(677\) − 920.516i − 1.35970i −0.733352 0.679849i \(-0.762045\pi\)
0.733352 0.679849i \(-0.237955\pi\)
\(678\) 0 0
\(679\) 45.9866 0.0677270
\(680\) − 192.094i − 0.282491i
\(681\) 0 0
\(682\) −872.242 −1.27895
\(683\) − 382.840i − 0.560528i −0.959923 0.280264i \(-0.909578\pi\)
0.959923 0.280264i \(-0.0904220\pi\)
\(684\) 0 0
\(685\) 7.05277 0.0102960
\(686\) 650.302i 0.947961i
\(687\) 0 0
\(688\) −31.7891 −0.0462051
\(689\) − 253.852i − 0.368436i
\(690\) 0 0
\(691\) −678.861 −0.982433 −0.491217 0.871037i \(-0.663448\pi\)
−0.491217 + 0.871037i \(0.663448\pi\)
\(692\) − 1322.31i − 1.91086i
\(693\) 0 0
\(694\) −166.460 −0.239856
\(695\) 167.969i 0.241682i
\(696\) 0 0
\(697\) 767.816 1.10160
\(698\) − 1916.41i − 2.74557i
\(699\) 0 0
\(700\) −316.975 −0.452822
\(701\) 684.318i 0.976202i 0.872787 + 0.488101i \(0.162310\pi\)
−0.872787 + 0.488101i \(0.837690\pi\)
\(702\) 0 0
\(703\) −283.014 −0.402580
\(704\) − 627.894i − 0.891894i
\(705\) 0 0
\(706\) −1488.32 −2.10810
\(707\) − 412.605i − 0.583599i
\(708\) 0 0
\(709\) −1302.62 −1.83727 −0.918633 0.395112i \(-0.870706\pi\)
−0.918633 + 0.395112i \(0.870706\pi\)
\(710\) 409.732i 0.577087i
\(711\) 0 0
\(712\) 85.0928 0.119512
\(713\) 549.489i 0.770672i
\(714\) 0 0
\(715\) 133.614 0.186873
\(716\) − 733.023i − 1.02377i
\(717\) 0 0
\(718\) −745.890 −1.03884
\(719\) − 733.683i − 1.02042i −0.860050 0.510210i \(-0.829568\pi\)
0.860050 0.510210i \(-0.170432\pi\)
\(720\) 0 0
\(721\) 58.9065 0.0817011
\(722\) 1062.64i 1.47180i
\(723\) 0 0
\(724\) −2054.58 −2.83782
\(725\) 479.398i 0.661239i
\(726\) 0 0
\(727\) −509.080 −0.700247 −0.350123 0.936704i \(-0.613860\pi\)
−0.350123 + 0.936704i \(0.613860\pi\)
\(728\) 255.852i 0.351445i
\(729\) 0 0
\(730\) −68.3136 −0.0935803
\(731\) 1078.04i 1.47475i
\(732\) 0 0
\(733\) −1281.50 −1.74830 −0.874149 0.485658i \(-0.838580\pi\)
−0.874149 + 0.485658i \(0.838580\pi\)
\(734\) 85.2186i 0.116102i
\(735\) 0 0
\(736\) −403.168 −0.547782
\(737\) 420.601i 0.570694i
\(738\) 0 0
\(739\) −651.069 −0.881013 −0.440507 0.897749i \(-0.645201\pi\)
−0.440507 + 0.897749i \(0.645201\pi\)
\(740\) − 468.611i − 0.633258i
\(741\) 0 0
\(742\) −116.351 −0.156808
\(743\) 1290.81i 1.73730i 0.495429 + 0.868649i \(0.335011\pi\)
−0.495429 + 0.868649i \(0.664989\pi\)
\(744\) 0 0
\(745\) 147.816 0.198411
\(746\) 1708.14i 2.28973i
\(747\) 0 0
\(748\) −657.216 −0.878631
\(749\) 166.583i 0.222407i
\(750\) 0 0
\(751\) −1270.00 −1.69107 −0.845536 0.533918i \(-0.820719\pi\)
−0.845536 + 0.533918i \(0.820719\pi\)
\(752\) 28.9213i 0.0384592i
\(753\) 0 0
\(754\) 1027.63 1.36290
\(755\) 214.721i 0.284399i
\(756\) 0 0
\(757\) −835.003 −1.10304 −0.551521 0.834161i \(-0.685952\pi\)
−0.551521 + 0.834161i \(0.685952\pi\)
\(758\) 1167.32i 1.54000i
\(759\) 0 0
\(760\) 63.8722 0.0840424
\(761\) − 907.753i − 1.19284i −0.802672 0.596421i \(-0.796589\pi\)
0.802672 0.596421i \(-0.203411\pi\)
\(762\) 0 0
\(763\) −3.23167 −0.00423548
\(764\) 667.521i 0.873719i
\(765\) 0 0
\(766\) −300.613 −0.392445
\(767\) − 1698.32i − 2.21424i
\(768\) 0 0
\(769\) −396.941 −0.516178 −0.258089 0.966121i \(-0.583093\pi\)
−0.258089 + 0.966121i \(0.583093\pi\)
\(770\) − 61.2408i − 0.0795335i
\(771\) 0 0
\(772\) 2141.65 2.77416
\(773\) − 1381.35i − 1.78700i −0.449059 0.893502i \(-0.648241\pi\)
0.449059 0.893502i \(-0.351759\pi\)
\(774\) 0 0
\(775\) 1023.12 1.32015
\(776\) 166.098i 0.214044i
\(777\) 0 0
\(778\) 662.990 0.852172
\(779\) 255.303i 0.327731i
\(780\) 0 0
\(781\) 527.859 0.675875
\(782\) 672.154i 0.859532i
\(783\) 0 0
\(784\) 22.1542 0.0282580
\(785\) − 46.3412i − 0.0590333i
\(786\) 0 0
\(787\) −113.599 −0.144344 −0.0721722 0.997392i \(-0.522993\pi\)
−0.0721722 + 0.997392i \(0.522993\pi\)
\(788\) − 714.149i − 0.906281i
\(789\) 0 0
\(790\) 688.276 0.871235
\(791\) − 283.214i − 0.358045i
\(792\) 0 0
\(793\) −1560.20 −1.96747
\(794\) − 576.329i − 0.725855i
\(795\) 0 0
\(796\) −1149.79 −1.44446
\(797\) − 81.8545i − 0.102703i −0.998681 0.0513516i \(-0.983647\pi\)
0.998681 0.0513516i \(-0.0163529\pi\)
\(798\) 0 0
\(799\) 980.788 1.22752
\(800\) 750.676i 0.938345i
\(801\) 0 0
\(802\) −259.919 −0.324089
\(803\) 88.0086i 0.109600i
\(804\) 0 0
\(805\) −38.5801 −0.0479256
\(806\) − 2193.13i − 2.72101i
\(807\) 0 0
\(808\) 1490.28 1.84440
\(809\) 330.667i 0.408735i 0.978894 + 0.204368i \(0.0655138\pi\)
−0.978894 + 0.204368i \(0.934486\pi\)
\(810\) 0 0
\(811\) −1106.38 −1.36422 −0.682110 0.731249i \(-0.738938\pi\)
−0.682110 + 0.731249i \(0.738938\pi\)
\(812\) − 290.126i − 0.357298i
\(813\) 0 0
\(814\) −980.097 −1.20405
\(815\) − 12.7613i − 0.0156580i
\(816\) 0 0
\(817\) −358.454 −0.438744
\(818\) 163.655i 0.200067i
\(819\) 0 0
\(820\) −422.728 −0.515521
\(821\) − 1212.58i − 1.47696i −0.674276 0.738479i \(-0.735544\pi\)
0.674276 0.738479i \(-0.264456\pi\)
\(822\) 0 0
\(823\) 384.766 0.467517 0.233759 0.972295i \(-0.424897\pi\)
0.233759 + 0.972295i \(0.424897\pi\)
\(824\) 212.763i 0.258207i
\(825\) 0 0
\(826\) −778.412 −0.942388
\(827\) 836.135i 1.01105i 0.862813 + 0.505523i \(0.168700\pi\)
−0.862813 + 0.505523i \(0.831300\pi\)
\(828\) 0 0
\(829\) −1008.06 −1.21599 −0.607995 0.793941i \(-0.708026\pi\)
−0.607995 + 0.793941i \(0.708026\pi\)
\(830\) 278.497i 0.335539i
\(831\) 0 0
\(832\) 1578.75 1.89754
\(833\) − 751.300i − 0.901921i
\(834\) 0 0
\(835\) −170.419 −0.204095
\(836\) − 218.528i − 0.261397i
\(837\) 0 0
\(838\) 1360.35 1.62333
\(839\) 788.289i 0.939558i 0.882784 + 0.469779i \(0.155666\pi\)
−0.882784 + 0.469779i \(0.844334\pi\)
\(840\) 0 0
\(841\) 402.210 0.478252
\(842\) − 408.361i − 0.484990i
\(843\) 0 0
\(844\) 1358.38 1.60945
\(845\) 90.2281i 0.106779i
\(846\) 0 0
\(847\) 182.310 0.215242
\(848\) 8.34422i 0.00983988i
\(849\) 0 0
\(850\) 1251.51 1.47237
\(851\) 617.434i 0.725540i
\(852\) 0 0
\(853\) −355.835 −0.417157 −0.208579 0.978006i \(-0.566884\pi\)
−0.208579 + 0.978006i \(0.566884\pi\)
\(854\) 715.107i 0.837362i
\(855\) 0 0
\(856\) −601.676 −0.702893
\(857\) 160.752i 0.187576i 0.995592 + 0.0937878i \(0.0298975\pi\)
−0.995592 + 0.0937878i \(0.970102\pi\)
\(858\) 0 0
\(859\) 378.660 0.440815 0.220408 0.975408i \(-0.429261\pi\)
0.220408 + 0.975408i \(0.429261\pi\)
\(860\) − 593.525i − 0.690145i
\(861\) 0 0
\(862\) 225.023 0.261047
\(863\) − 116.099i − 0.134529i −0.997735 0.0672646i \(-0.978573\pi\)
0.997735 0.0672646i \(-0.0214272\pi\)
\(864\) 0 0
\(865\) −299.667 −0.346436
\(866\) − 1397.65i − 1.61391i
\(867\) 0 0
\(868\) −619.179 −0.713339
\(869\) − 886.707i − 1.02038i
\(870\) 0 0
\(871\) −1057.54 −1.21417
\(872\) − 11.6724i − 0.0133858i
\(873\) 0 0
\(874\) −223.495 −0.255715
\(875\) 150.304i 0.171776i
\(876\) 0 0
\(877\) −21.0296 −0.0239790 −0.0119895 0.999928i \(-0.503816\pi\)
−0.0119895 + 0.999928i \(0.503816\pi\)
\(878\) − 2356.82i − 2.68430i
\(879\) 0 0
\(880\) −4.39193 −0.00499083
\(881\) − 176.808i − 0.200690i −0.994953 0.100345i \(-0.968005\pi\)
0.994953 0.100345i \(-0.0319947\pi\)
\(882\) 0 0
\(883\) 1552.93 1.75870 0.879349 0.476177i \(-0.157978\pi\)
0.879349 + 0.476177i \(0.157978\pi\)
\(884\) − 1652.48i − 1.86932i
\(885\) 0 0
\(886\) 366.085 0.413189
\(887\) 1436.68i 1.61971i 0.586631 + 0.809854i \(0.300454\pi\)
−0.586631 + 0.809854i \(0.699546\pi\)
\(888\) 0 0
\(889\) −24.3277 −0.0273652
\(890\) − 51.2122i − 0.0575418i
\(891\) 0 0
\(892\) −1622.08 −1.81848
\(893\) 326.117i 0.365193i
\(894\) 0 0
\(895\) −166.120 −0.185609
\(896\) − 440.376i − 0.491491i
\(897\) 0 0
\(898\) 1201.80 1.33831
\(899\) 936.454i 1.04166i
\(900\) 0 0
\(901\) 282.971 0.314064
\(902\) 884.132i 0.980191i
\(903\) 0 0
\(904\) 1022.93 1.13156
\(905\) 465.616i 0.514492i
\(906\) 0 0
\(907\) −1361.80 −1.50144 −0.750718 0.660623i \(-0.770292\pi\)
−0.750718 + 0.660623i \(0.770292\pi\)
\(908\) − 1032.91i − 1.13757i
\(909\) 0 0
\(910\) 153.982 0.169211
\(911\) − 1434.07i − 1.57417i −0.616844 0.787085i \(-0.711589\pi\)
0.616844 0.787085i \(-0.288411\pi\)
\(912\) 0 0
\(913\) 358.789 0.392978
\(914\) 1696.43i 1.85605i
\(915\) 0 0
\(916\) −107.804 −0.117690
\(917\) − 308.772i − 0.336720i
\(918\) 0 0
\(919\) 752.268 0.818573 0.409286 0.912406i \(-0.365778\pi\)
0.409286 + 0.912406i \(0.365778\pi\)
\(920\) − 139.346i − 0.151463i
\(921\) 0 0
\(922\) 277.856 0.301363
\(923\) 1327.23i 1.43795i
\(924\) 0 0
\(925\) 1149.63 1.24284
\(926\) − 1276.18i − 1.37816i
\(927\) 0 0
\(928\) −687.089 −0.740398
\(929\) − 74.4733i − 0.0801650i −0.999196 0.0400825i \(-0.987238\pi\)
0.999196 0.0400825i \(-0.0127621\pi\)
\(930\) 0 0
\(931\) 249.811 0.268326
\(932\) − 360.775i − 0.387098i
\(933\) 0 0
\(934\) 1285.00 1.37581
\(935\) 148.940i 0.159294i
\(936\) 0 0
\(937\) 363.649 0.388099 0.194050 0.980992i \(-0.437838\pi\)
0.194050 + 0.980992i \(0.437838\pi\)
\(938\) 484.717i 0.516756i
\(939\) 0 0
\(940\) −539.981 −0.574448
\(941\) − 201.166i − 0.213779i −0.994271 0.106890i \(-0.965911\pi\)
0.994271 0.106890i \(-0.0340892\pi\)
\(942\) 0 0
\(943\) 556.979 0.590646
\(944\) 55.8244i 0.0591361i
\(945\) 0 0
\(946\) −1241.35 −1.31221
\(947\) − 1001.85i − 1.05792i −0.848647 0.528959i \(-0.822582\pi\)
0.848647 0.528959i \(-0.177418\pi\)
\(948\) 0 0
\(949\) −221.285 −0.233177
\(950\) 416.135i 0.438037i
\(951\) 0 0
\(952\) −285.200 −0.299580
\(953\) − 1431.96i − 1.50258i −0.659971 0.751291i \(-0.729432\pi\)
0.659971 0.751291i \(-0.270568\pi\)
\(954\) 0 0
\(955\) 151.276 0.158404
\(956\) 2345.59i 2.45355i
\(957\) 0 0
\(958\) 962.500 1.00470
\(959\) − 10.4712i − 0.0109188i
\(960\) 0 0
\(961\) 1037.56 1.07966
\(962\) − 2464.32i − 2.56166i
\(963\) 0 0
\(964\) 2018.02 2.09338
\(965\) − 485.348i − 0.502951i
\(966\) 0 0
\(967\) 812.242 0.839961 0.419980 0.907533i \(-0.362037\pi\)
0.419980 + 0.907533i \(0.362037\pi\)
\(968\) 658.480i 0.680248i
\(969\) 0 0
\(970\) 99.9643 0.103056
\(971\) 1589.54i 1.63701i 0.574496 + 0.818507i \(0.305198\pi\)
−0.574496 + 0.818507i \(0.694802\pi\)
\(972\) 0 0
\(973\) 249.382 0.256302
\(974\) − 13.1613i − 0.0135126i
\(975\) 0 0
\(976\) 51.2845 0.0525456
\(977\) − 1147.24i − 1.17425i −0.809495 0.587126i \(-0.800259\pi\)
0.809495 0.587126i \(-0.199741\pi\)
\(978\) 0 0
\(979\) −65.9768 −0.0673920
\(980\) 413.635i 0.422076i
\(981\) 0 0
\(982\) −1664.91 −1.69543
\(983\) − 1293.14i − 1.31551i −0.753234 0.657753i \(-0.771507\pi\)
0.753234 0.657753i \(-0.228493\pi\)
\(984\) 0 0
\(985\) −161.843 −0.164307
\(986\) 1145.50i 1.16177i
\(987\) 0 0
\(988\) 549.458 0.556131
\(989\) 782.019i 0.790717i
\(990\) 0 0
\(991\) −605.355 −0.610852 −0.305426 0.952216i \(-0.598799\pi\)
−0.305426 + 0.952216i \(0.598799\pi\)
\(992\) 1466.37i 1.47819i
\(993\) 0 0
\(994\) 608.324 0.611996
\(995\) 260.570i 0.261879i
\(996\) 0 0
\(997\) 927.609 0.930400 0.465200 0.885206i \(-0.345982\pi\)
0.465200 + 0.885206i \(0.345982\pi\)
\(998\) 85.1193i 0.0852898i
\(999\) 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.890.9 84
3.2 odd 2 inner 1143.3.b.a.890.76 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.9 84 1.1 even 1 trivial
1143.3.b.a.890.76 yes 84 3.2 odd 2 inner