Properties

Label 1143.3.b.a.890.14
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 890.14
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.71

$q$-expansion

\(f(q)\) \(=\) \(q-2.96858i q^{2} -4.81247 q^{4} +3.90126i q^{5} +11.9179 q^{7} +2.41189i q^{8} +O(q^{10})\) \(q-2.96858i q^{2} -4.81247 q^{4} +3.90126i q^{5} +11.9179 q^{7} +2.41189i q^{8} +11.5812 q^{10} +2.00981i q^{11} +10.4679 q^{13} -35.3793i q^{14} -12.0900 q^{16} +15.0361i q^{17} -12.2139 q^{19} -18.7747i q^{20} +5.96627 q^{22} +16.9967i q^{23} +9.78019 q^{25} -31.0748i q^{26} -57.3547 q^{28} +13.3323i q^{29} +9.79569 q^{31} +45.5377i q^{32} +44.6358 q^{34} +46.4949i q^{35} +23.1196 q^{37} +36.2579i q^{38} -9.40942 q^{40} +38.8506i q^{41} -8.44282 q^{43} -9.67214i q^{44} +50.4561 q^{46} +61.5275i q^{47} +93.0369 q^{49} -29.0333i q^{50} -50.3764 q^{52} -61.3578i q^{53} -7.84077 q^{55} +28.7448i q^{56} +39.5779 q^{58} +38.6575i q^{59} +49.1446 q^{61} -29.0793i q^{62} +86.8224 q^{64} +40.8379i q^{65} +8.24716 q^{67} -72.3606i q^{68} +138.024 q^{70} +20.5967i q^{71} -24.6261 q^{73} -68.6325i q^{74} +58.7790 q^{76} +23.9527i q^{77} +22.7212 q^{79} -47.1662i q^{80} +115.331 q^{82} -71.4711i q^{83} -58.6595 q^{85} +25.0632i q^{86} -4.84744 q^{88} -11.0072i q^{89} +124.755 q^{91} -81.7962i q^{92} +182.650 q^{94} -47.6495i q^{95} -93.8083 q^{97} -276.188i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.96858i − 1.48429i −0.670239 0.742145i \(-0.733808\pi\)
0.670239 0.742145i \(-0.266192\pi\)
\(3\) 0 0
\(4\) −4.81247 −1.20312
\(5\) 3.90126i 0.780251i 0.920762 + 0.390126i \(0.127568\pi\)
−0.920762 + 0.390126i \(0.872432\pi\)
\(6\) 0 0
\(7\) 11.9179 1.70256 0.851280 0.524711i \(-0.175827\pi\)
0.851280 + 0.524711i \(0.175827\pi\)
\(8\) 2.41189i 0.301487i
\(9\) 0 0
\(10\) 11.5812 1.15812
\(11\) 2.00981i 0.182710i 0.995818 + 0.0913548i \(0.0291197\pi\)
−0.995818 + 0.0913548i \(0.970880\pi\)
\(12\) 0 0
\(13\) 10.4679 0.805222 0.402611 0.915371i \(-0.368103\pi\)
0.402611 + 0.915371i \(0.368103\pi\)
\(14\) − 35.3793i − 2.52709i
\(15\) 0 0
\(16\) −12.0900 −0.755625
\(17\) 15.0361i 0.884474i 0.896898 + 0.442237i \(0.145815\pi\)
−0.896898 + 0.442237i \(0.854185\pi\)
\(18\) 0 0
\(19\) −12.2139 −0.642836 −0.321418 0.946937i \(-0.604159\pi\)
−0.321418 + 0.946937i \(0.604159\pi\)
\(20\) − 18.7747i − 0.938735i
\(21\) 0 0
\(22\) 5.96627 0.271194
\(23\) 16.9967i 0.738987i 0.929233 + 0.369494i \(0.120469\pi\)
−0.929233 + 0.369494i \(0.879531\pi\)
\(24\) 0 0
\(25\) 9.78019 0.391208
\(26\) − 31.0748i − 1.19518i
\(27\) 0 0
\(28\) −57.3547 −2.04838
\(29\) 13.3323i 0.459734i 0.973222 + 0.229867i \(0.0738291\pi\)
−0.973222 + 0.229867i \(0.926171\pi\)
\(30\) 0 0
\(31\) 9.79569 0.315990 0.157995 0.987440i \(-0.449497\pi\)
0.157995 + 0.987440i \(0.449497\pi\)
\(32\) 45.5377i 1.42305i
\(33\) 0 0
\(34\) 44.6358 1.31282
\(35\) 46.4949i 1.32843i
\(36\) 0 0
\(37\) 23.1196 0.624855 0.312428 0.949942i \(-0.398858\pi\)
0.312428 + 0.949942i \(0.398858\pi\)
\(38\) 36.2579i 0.954155i
\(39\) 0 0
\(40\) −9.40942 −0.235235
\(41\) 38.8506i 0.947575i 0.880639 + 0.473787i \(0.157113\pi\)
−0.880639 + 0.473787i \(0.842887\pi\)
\(42\) 0 0
\(43\) −8.44282 −0.196345 −0.0981724 0.995169i \(-0.531300\pi\)
−0.0981724 + 0.995169i \(0.531300\pi\)
\(44\) − 9.67214i − 0.219821i
\(45\) 0 0
\(46\) 50.4561 1.09687
\(47\) 61.5275i 1.30910i 0.756020 + 0.654548i \(0.227141\pi\)
−0.756020 + 0.654548i \(0.772859\pi\)
\(48\) 0 0
\(49\) 93.0369 1.89871
\(50\) − 29.0333i − 0.580666i
\(51\) 0 0
\(52\) −50.3764 −0.968777
\(53\) − 61.3578i − 1.15769i −0.815436 0.578847i \(-0.803503\pi\)
0.815436 0.578847i \(-0.196497\pi\)
\(54\) 0 0
\(55\) −7.84077 −0.142559
\(56\) 28.7448i 0.513299i
\(57\) 0 0
\(58\) 39.5779 0.682378
\(59\) 38.6575i 0.655213i 0.944814 + 0.327606i \(0.106242\pi\)
−0.944814 + 0.327606i \(0.893758\pi\)
\(60\) 0 0
\(61\) 49.1446 0.805650 0.402825 0.915277i \(-0.368028\pi\)
0.402825 + 0.915277i \(0.368028\pi\)
\(62\) − 29.0793i − 0.469021i
\(63\) 0 0
\(64\) 86.8224 1.35660
\(65\) 40.8379i 0.628275i
\(66\) 0 0
\(67\) 8.24716 0.123092 0.0615460 0.998104i \(-0.480397\pi\)
0.0615460 + 0.998104i \(0.480397\pi\)
\(68\) − 72.3606i − 1.06413i
\(69\) 0 0
\(70\) 138.024 1.97177
\(71\) 20.5967i 0.290095i 0.989425 + 0.145047i \(0.0463334\pi\)
−0.989425 + 0.145047i \(0.953667\pi\)
\(72\) 0 0
\(73\) −24.6261 −0.337344 −0.168672 0.985672i \(-0.553948\pi\)
−0.168672 + 0.985672i \(0.553948\pi\)
\(74\) − 68.6325i − 0.927466i
\(75\) 0 0
\(76\) 58.7790 0.773408
\(77\) 23.9527i 0.311074i
\(78\) 0 0
\(79\) 22.7212 0.287610 0.143805 0.989606i \(-0.454066\pi\)
0.143805 + 0.989606i \(0.454066\pi\)
\(80\) − 47.1662i − 0.589577i
\(81\) 0 0
\(82\) 115.331 1.40648
\(83\) − 71.4711i − 0.861097i −0.902567 0.430549i \(-0.858320\pi\)
0.902567 0.430549i \(-0.141680\pi\)
\(84\) 0 0
\(85\) −58.6595 −0.690112
\(86\) 25.0632i 0.291433i
\(87\) 0 0
\(88\) −4.84744 −0.0550845
\(89\) − 11.0072i − 0.123676i −0.998086 0.0618381i \(-0.980304\pi\)
0.998086 0.0618381i \(-0.0196962\pi\)
\(90\) 0 0
\(91\) 124.755 1.37094
\(92\) − 81.7962i − 0.889089i
\(93\) 0 0
\(94\) 182.650 1.94308
\(95\) − 47.6495i − 0.501574i
\(96\) 0 0
\(97\) −93.8083 −0.967096 −0.483548 0.875318i \(-0.660652\pi\)
−0.483548 + 0.875318i \(0.660652\pi\)
\(98\) − 276.188i − 2.81824i
\(99\) 0 0
\(100\) −47.0669 −0.470669
\(101\) − 80.1451i − 0.793516i −0.917923 0.396758i \(-0.870135\pi\)
0.917923 0.396758i \(-0.129865\pi\)
\(102\) 0 0
\(103\) 106.830 1.03718 0.518591 0.855022i \(-0.326457\pi\)
0.518591 + 0.855022i \(0.326457\pi\)
\(104\) 25.2474i 0.242764i
\(105\) 0 0
\(106\) −182.146 −1.71836
\(107\) − 184.771i − 1.72683i −0.504494 0.863415i \(-0.668321\pi\)
0.504494 0.863415i \(-0.331679\pi\)
\(108\) 0 0
\(109\) 16.2143 0.148755 0.0743775 0.997230i \(-0.476303\pi\)
0.0743775 + 0.997230i \(0.476303\pi\)
\(110\) 23.2760i 0.211600i
\(111\) 0 0
\(112\) −144.088 −1.28650
\(113\) − 83.0951i − 0.735354i −0.929953 0.367677i \(-0.880153\pi\)
0.929953 0.367677i \(-0.119847\pi\)
\(114\) 0 0
\(115\) −66.3085 −0.576596
\(116\) − 64.1612i − 0.553114i
\(117\) 0 0
\(118\) 114.758 0.972526
\(119\) 179.199i 1.50587i
\(120\) 0 0
\(121\) 116.961 0.966617
\(122\) − 145.890i − 1.19582i
\(123\) 0 0
\(124\) −47.1415 −0.380173
\(125\) 135.686i 1.08549i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) − 75.5885i − 0.590535i
\(129\) 0 0
\(130\) 121.231 0.932543
\(131\) 46.6377i 0.356013i 0.984029 + 0.178006i \(0.0569648\pi\)
−0.984029 + 0.178006i \(0.943035\pi\)
\(132\) 0 0
\(133\) −145.564 −1.09447
\(134\) − 24.4824i − 0.182704i
\(135\) 0 0
\(136\) −36.2654 −0.266657
\(137\) − 2.14478i − 0.0156553i −0.999969 0.00782765i \(-0.997508\pi\)
0.999969 0.00782765i \(-0.00249164\pi\)
\(138\) 0 0
\(139\) 68.4845 0.492694 0.246347 0.969182i \(-0.420770\pi\)
0.246347 + 0.969182i \(0.420770\pi\)
\(140\) − 223.755i − 1.59825i
\(141\) 0 0
\(142\) 61.1430 0.430585
\(143\) 21.0384i 0.147122i
\(144\) 0 0
\(145\) −52.0126 −0.358708
\(146\) 73.1047i 0.500717i
\(147\) 0 0
\(148\) −111.263 −0.751775
\(149\) 127.883i 0.858274i 0.903239 + 0.429137i \(0.141182\pi\)
−0.903239 + 0.429137i \(0.858818\pi\)
\(150\) 0 0
\(151\) 42.5770 0.281967 0.140983 0.990012i \(-0.454974\pi\)
0.140983 + 0.990012i \(0.454974\pi\)
\(152\) − 29.4586i − 0.193806i
\(153\) 0 0
\(154\) 71.1056 0.461725
\(155\) 38.2155i 0.246552i
\(156\) 0 0
\(157\) −185.690 −1.18274 −0.591370 0.806400i \(-0.701413\pi\)
−0.591370 + 0.806400i \(0.701413\pi\)
\(158\) − 67.4496i − 0.426896i
\(159\) 0 0
\(160\) −177.654 −1.11034
\(161\) 202.565i 1.25817i
\(162\) 0 0
\(163\) 218.551 1.34080 0.670402 0.741998i \(-0.266122\pi\)
0.670402 + 0.741998i \(0.266122\pi\)
\(164\) − 186.967i − 1.14004i
\(165\) 0 0
\(166\) −212.168 −1.27812
\(167\) 58.1575i 0.348248i 0.984724 + 0.174124i \(0.0557094\pi\)
−0.984724 + 0.174124i \(0.944291\pi\)
\(168\) 0 0
\(169\) −59.4234 −0.351618
\(170\) 174.136i 1.02433i
\(171\) 0 0
\(172\) 40.6309 0.236226
\(173\) − 287.885i − 1.66407i −0.554720 0.832037i \(-0.687175\pi\)
0.554720 0.832037i \(-0.312825\pi\)
\(174\) 0 0
\(175\) 116.560 0.666055
\(176\) − 24.2985i − 0.138060i
\(177\) 0 0
\(178\) −32.6757 −0.183571
\(179\) 25.5291i 0.142620i 0.997454 + 0.0713102i \(0.0227180\pi\)
−0.997454 + 0.0713102i \(0.977282\pi\)
\(180\) 0 0
\(181\) −134.987 −0.745785 −0.372892 0.927875i \(-0.621634\pi\)
−0.372892 + 0.927875i \(0.621634\pi\)
\(182\) − 370.347i − 2.03487i
\(183\) 0 0
\(184\) −40.9942 −0.222795
\(185\) 90.1957i 0.487544i
\(186\) 0 0
\(187\) −30.2196 −0.161602
\(188\) − 296.100i − 1.57500i
\(189\) 0 0
\(190\) −141.451 −0.744481
\(191\) − 188.564i − 0.987244i −0.869677 0.493622i \(-0.835673\pi\)
0.869677 0.493622i \(-0.164327\pi\)
\(192\) 0 0
\(193\) 181.955 0.942773 0.471387 0.881927i \(-0.343754\pi\)
0.471387 + 0.881927i \(0.343754\pi\)
\(194\) 278.478i 1.43545i
\(195\) 0 0
\(196\) −447.738 −2.28438
\(197\) 77.3544i 0.392662i 0.980538 + 0.196331i \(0.0629027\pi\)
−0.980538 + 0.196331i \(0.937097\pi\)
\(198\) 0 0
\(199\) −176.017 −0.884509 −0.442255 0.896890i \(-0.645821\pi\)
−0.442255 + 0.896890i \(0.645821\pi\)
\(200\) 23.5888i 0.117944i
\(201\) 0 0
\(202\) −237.917 −1.17781
\(203\) 158.893i 0.782724i
\(204\) 0 0
\(205\) −151.566 −0.739347
\(206\) − 317.133i − 1.53948i
\(207\) 0 0
\(208\) −126.557 −0.608445
\(209\) − 24.5475i − 0.117452i
\(210\) 0 0
\(211\) 243.707 1.15501 0.577504 0.816388i \(-0.304027\pi\)
0.577504 + 0.816388i \(0.304027\pi\)
\(212\) 295.283i 1.39284i
\(213\) 0 0
\(214\) −548.507 −2.56312
\(215\) − 32.9376i − 0.153198i
\(216\) 0 0
\(217\) 116.744 0.537992
\(218\) − 48.1335i − 0.220796i
\(219\) 0 0
\(220\) 37.7335 0.171516
\(221\) 157.396i 0.712198i
\(222\) 0 0
\(223\) −351.699 −1.57713 −0.788564 0.614953i \(-0.789175\pi\)
−0.788564 + 0.614953i \(0.789175\pi\)
\(224\) 542.715i 2.42283i
\(225\) 0 0
\(226\) −246.674 −1.09148
\(227\) − 199.659i − 0.879556i −0.898107 0.439778i \(-0.855057\pi\)
0.898107 0.439778i \(-0.144943\pi\)
\(228\) 0 0
\(229\) 239.891 1.04756 0.523781 0.851853i \(-0.324521\pi\)
0.523781 + 0.851853i \(0.324521\pi\)
\(230\) 196.842i 0.855836i
\(231\) 0 0
\(232\) −32.1560 −0.138604
\(233\) 130.240i 0.558970i 0.960150 + 0.279485i \(0.0901637\pi\)
−0.960150 + 0.279485i \(0.909836\pi\)
\(234\) 0 0
\(235\) −240.035 −1.02142
\(236\) − 186.038i − 0.788298i
\(237\) 0 0
\(238\) 531.966 2.23515
\(239\) 424.397i 1.77572i 0.460113 + 0.887861i \(0.347809\pi\)
−0.460113 + 0.887861i \(0.652191\pi\)
\(240\) 0 0
\(241\) −330.531 −1.37150 −0.685749 0.727838i \(-0.740525\pi\)
−0.685749 + 0.727838i \(0.740525\pi\)
\(242\) − 347.207i − 1.43474i
\(243\) 0 0
\(244\) −236.507 −0.969292
\(245\) 362.961i 1.48147i
\(246\) 0 0
\(247\) −127.854 −0.517626
\(248\) 23.6262i 0.0952668i
\(249\) 0 0
\(250\) 402.796 1.61119
\(251\) 186.099i 0.741429i 0.928747 + 0.370714i \(0.120887\pi\)
−0.928747 + 0.370714i \(0.879113\pi\)
\(252\) 0 0
\(253\) −34.1601 −0.135020
\(254\) − 33.4542i − 0.131709i
\(255\) 0 0
\(256\) 122.899 0.480074
\(257\) − 81.5366i − 0.317263i −0.987338 0.158632i \(-0.949292\pi\)
0.987338 0.158632i \(-0.0507082\pi\)
\(258\) 0 0
\(259\) 275.538 1.06385
\(260\) − 196.531i − 0.755890i
\(261\) 0 0
\(262\) 138.448 0.528427
\(263\) − 480.944i − 1.82868i −0.404943 0.914342i \(-0.632709\pi\)
0.404943 0.914342i \(-0.367291\pi\)
\(264\) 0 0
\(265\) 239.373 0.903293
\(266\) 432.119i 1.62451i
\(267\) 0 0
\(268\) −39.6893 −0.148094
\(269\) − 454.168i − 1.68836i −0.536063 0.844178i \(-0.680089\pi\)
0.536063 0.844178i \(-0.319911\pi\)
\(270\) 0 0
\(271\) 197.293 0.728018 0.364009 0.931395i \(-0.381408\pi\)
0.364009 + 0.931395i \(0.381408\pi\)
\(272\) − 181.786i − 0.668330i
\(273\) 0 0
\(274\) −6.36694 −0.0232370
\(275\) 19.6563i 0.0714774i
\(276\) 0 0
\(277\) 357.402 1.29026 0.645130 0.764073i \(-0.276803\pi\)
0.645130 + 0.764073i \(0.276803\pi\)
\(278\) − 203.302i − 0.731302i
\(279\) 0 0
\(280\) −112.141 −0.400503
\(281\) − 236.862i − 0.842925i −0.906846 0.421462i \(-0.861517\pi\)
0.906846 0.421462i \(-0.138483\pi\)
\(282\) 0 0
\(283\) 276.813 0.978138 0.489069 0.872245i \(-0.337337\pi\)
0.489069 + 0.872245i \(0.337337\pi\)
\(284\) − 99.1211i − 0.349018i
\(285\) 0 0
\(286\) 62.4542 0.218371
\(287\) 463.018i 1.61330i
\(288\) 0 0
\(289\) 62.9170 0.217706
\(290\) 154.404i 0.532427i
\(291\) 0 0
\(292\) 118.513 0.405865
\(293\) − 26.4288i − 0.0902007i −0.998982 0.0451004i \(-0.985639\pi\)
0.998982 0.0451004i \(-0.0143608\pi\)
\(294\) 0 0
\(295\) −150.813 −0.511231
\(296\) 55.7621i 0.188385i
\(297\) 0 0
\(298\) 379.631 1.27393
\(299\) 177.919i 0.595048i
\(300\) 0 0
\(301\) −100.621 −0.334289
\(302\) − 126.393i − 0.418520i
\(303\) 0 0
\(304\) 147.666 0.485743
\(305\) 191.726i 0.628609i
\(306\) 0 0
\(307\) −187.955 −0.612232 −0.306116 0.951994i \(-0.599030\pi\)
−0.306116 + 0.951994i \(0.599030\pi\)
\(308\) − 115.272i − 0.374259i
\(309\) 0 0
\(310\) 113.446 0.365954
\(311\) 168.017i 0.540248i 0.962825 + 0.270124i \(0.0870648\pi\)
−0.962825 + 0.270124i \(0.912935\pi\)
\(312\) 0 0
\(313\) −498.185 −1.59164 −0.795822 0.605530i \(-0.792961\pi\)
−0.795822 + 0.605530i \(0.792961\pi\)
\(314\) 551.237i 1.75553i
\(315\) 0 0
\(316\) −109.345 −0.346028
\(317\) 170.371i 0.537447i 0.963217 + 0.268724i \(0.0866019\pi\)
−0.963217 + 0.268724i \(0.913398\pi\)
\(318\) 0 0
\(319\) −26.7953 −0.0839978
\(320\) 338.716i 1.05849i
\(321\) 0 0
\(322\) 601.332 1.86749
\(323\) − 183.649i − 0.568572i
\(324\) 0 0
\(325\) 102.378 0.315009
\(326\) − 648.787i − 1.99014i
\(327\) 0 0
\(328\) −93.7034 −0.285681
\(329\) 733.281i 2.22882i
\(330\) 0 0
\(331\) −91.5223 −0.276502 −0.138251 0.990397i \(-0.544148\pi\)
−0.138251 + 0.990397i \(0.544148\pi\)
\(332\) 343.953i 1.03600i
\(333\) 0 0
\(334\) 172.645 0.516902
\(335\) 32.1743i 0.0960427i
\(336\) 0 0
\(337\) 391.711 1.16235 0.581174 0.813779i \(-0.302594\pi\)
0.581174 + 0.813779i \(0.302594\pi\)
\(338\) 176.403i 0.521903i
\(339\) 0 0
\(340\) 282.297 0.830286
\(341\) 19.6874i 0.0577344i
\(342\) 0 0
\(343\) 524.829 1.53011
\(344\) − 20.3632i − 0.0591953i
\(345\) 0 0
\(346\) −854.609 −2.46997
\(347\) − 266.022i − 0.766633i −0.923617 0.383316i \(-0.874782\pi\)
0.923617 0.383316i \(-0.125218\pi\)
\(348\) 0 0
\(349\) −140.116 −0.401479 −0.200739 0.979645i \(-0.564334\pi\)
−0.200739 + 0.979645i \(0.564334\pi\)
\(350\) − 346.017i − 0.988619i
\(351\) 0 0
\(352\) −91.5219 −0.260006
\(353\) − 43.8632i − 0.124258i −0.998068 0.0621292i \(-0.980211\pi\)
0.998068 0.0621292i \(-0.0197891\pi\)
\(354\) 0 0
\(355\) −80.3531 −0.226347
\(356\) 52.9717i 0.148797i
\(357\) 0 0
\(358\) 75.7851 0.211690
\(359\) − 383.231i − 1.06750i −0.845644 0.533748i \(-0.820783\pi\)
0.845644 0.533748i \(-0.179217\pi\)
\(360\) 0 0
\(361\) −211.821 −0.586762
\(362\) 400.720i 1.10696i
\(363\) 0 0
\(364\) −600.382 −1.64940
\(365\) − 96.0729i − 0.263213i
\(366\) 0 0
\(367\) 83.5467 0.227648 0.113824 0.993501i \(-0.463690\pi\)
0.113824 + 0.993501i \(0.463690\pi\)
\(368\) − 205.490i − 0.558397i
\(369\) 0 0
\(370\) 267.753 0.723657
\(371\) − 731.258i − 1.97105i
\(372\) 0 0
\(373\) −254.080 −0.681181 −0.340590 0.940212i \(-0.610627\pi\)
−0.340590 + 0.940212i \(0.610627\pi\)
\(374\) 89.7092i 0.239864i
\(375\) 0 0
\(376\) −148.398 −0.394675
\(377\) 139.561i 0.370187i
\(378\) 0 0
\(379\) 628.995 1.65962 0.829809 0.558047i \(-0.188449\pi\)
0.829809 + 0.558047i \(0.188449\pi\)
\(380\) 229.312i 0.603453i
\(381\) 0 0
\(382\) −559.766 −1.46536
\(383\) 367.947i 0.960698i 0.877077 + 0.480349i \(0.159490\pi\)
−0.877077 + 0.480349i \(0.840510\pi\)
\(384\) 0 0
\(385\) −93.4457 −0.242716
\(386\) − 540.149i − 1.39935i
\(387\) 0 0
\(388\) 451.450 1.16353
\(389\) 530.476i 1.36369i 0.731496 + 0.681845i \(0.238822\pi\)
−0.731496 + 0.681845i \(0.761178\pi\)
\(390\) 0 0
\(391\) −255.563 −0.653615
\(392\) 224.395i 0.572437i
\(393\) 0 0
\(394\) 229.633 0.582824
\(395\) 88.6411i 0.224408i
\(396\) 0 0
\(397\) −606.103 −1.52671 −0.763354 0.645981i \(-0.776449\pi\)
−0.763354 + 0.645981i \(0.776449\pi\)
\(398\) 522.522i 1.31287i
\(399\) 0 0
\(400\) −118.242 −0.295606
\(401\) − 437.995i − 1.09226i −0.837702 0.546128i \(-0.816101\pi\)
0.837702 0.546128i \(-0.183899\pi\)
\(402\) 0 0
\(403\) 102.540 0.254442
\(404\) 385.696i 0.954693i
\(405\) 0 0
\(406\) 471.687 1.16179
\(407\) 46.4660i 0.114167i
\(408\) 0 0
\(409\) −470.257 −1.14977 −0.574886 0.818233i \(-0.694954\pi\)
−0.574886 + 0.818233i \(0.694954\pi\)
\(410\) 449.936i 1.09741i
\(411\) 0 0
\(412\) −514.115 −1.24785
\(413\) 460.718i 1.11554i
\(414\) 0 0
\(415\) 278.827 0.671872
\(416\) 476.683i 1.14587i
\(417\) 0 0
\(418\) −72.8714 −0.174333
\(419\) − 148.317i − 0.353979i −0.984213 0.176990i \(-0.943364\pi\)
0.984213 0.176990i \(-0.0566359\pi\)
\(420\) 0 0
\(421\) 329.588 0.782868 0.391434 0.920206i \(-0.371979\pi\)
0.391434 + 0.920206i \(0.371979\pi\)
\(422\) − 723.463i − 1.71437i
\(423\) 0 0
\(424\) 147.989 0.349030
\(425\) 147.056i 0.346013i
\(426\) 0 0
\(427\) 585.702 1.37167
\(428\) 889.205i 2.07758i
\(429\) 0 0
\(430\) −97.7780 −0.227391
\(431\) 739.306i 1.71533i 0.514211 + 0.857663i \(0.328085\pi\)
−0.514211 + 0.857663i \(0.671915\pi\)
\(432\) 0 0
\(433\) −210.577 −0.486320 −0.243160 0.969986i \(-0.578184\pi\)
−0.243160 + 0.969986i \(0.578184\pi\)
\(434\) − 346.565i − 0.798537i
\(435\) 0 0
\(436\) −78.0309 −0.178970
\(437\) − 207.596i − 0.475047i
\(438\) 0 0
\(439\) −795.985 −1.81318 −0.906589 0.422016i \(-0.861323\pi\)
−0.906589 + 0.422016i \(0.861323\pi\)
\(440\) − 18.9111i − 0.0429798i
\(441\) 0 0
\(442\) 467.242 1.05711
\(443\) 186.660i 0.421355i 0.977556 + 0.210678i \(0.0675671\pi\)
−0.977556 + 0.210678i \(0.932433\pi\)
\(444\) 0 0
\(445\) 42.9418 0.0964985
\(446\) 1044.05i 2.34091i
\(447\) 0 0
\(448\) 1034.74 2.30969
\(449\) − 414.327i − 0.922776i −0.887198 0.461388i \(-0.847352\pi\)
0.887198 0.461388i \(-0.152648\pi\)
\(450\) 0 0
\(451\) −78.0821 −0.173131
\(452\) 399.893i 0.884718i
\(453\) 0 0
\(454\) −592.704 −1.30552
\(455\) 486.703i 1.06968i
\(456\) 0 0
\(457\) −672.941 −1.47252 −0.736259 0.676699i \(-0.763410\pi\)
−0.736259 + 0.676699i \(0.763410\pi\)
\(458\) − 712.137i − 1.55488i
\(459\) 0 0
\(460\) 319.108 0.693713
\(461\) 876.511i 1.90133i 0.310225 + 0.950663i \(0.399596\pi\)
−0.310225 + 0.950663i \(0.600404\pi\)
\(462\) 0 0
\(463\) 89.6318 0.193589 0.0967946 0.995304i \(-0.469141\pi\)
0.0967946 + 0.995304i \(0.469141\pi\)
\(464\) − 161.187i − 0.347386i
\(465\) 0 0
\(466\) 386.628 0.829673
\(467\) 95.3908i 0.204263i 0.994771 + 0.102131i \(0.0325662\pi\)
−0.994771 + 0.102131i \(0.967434\pi\)
\(468\) 0 0
\(469\) 98.2891 0.209572
\(470\) 712.563i 1.51609i
\(471\) 0 0
\(472\) −93.2379 −0.197538
\(473\) − 16.9684i − 0.0358741i
\(474\) 0 0
\(475\) −119.454 −0.251482
\(476\) − 862.388i − 1.81174i
\(477\) 0 0
\(478\) 1259.86 2.63569
\(479\) 748.279i 1.56217i 0.624425 + 0.781085i \(0.285333\pi\)
−0.624425 + 0.781085i \(0.714667\pi\)
\(480\) 0 0
\(481\) 242.014 0.503147
\(482\) 981.208i 2.03570i
\(483\) 0 0
\(484\) −562.870 −1.16295
\(485\) − 365.970i − 0.754578i
\(486\) 0 0
\(487\) −674.647 −1.38531 −0.692656 0.721268i \(-0.743559\pi\)
−0.692656 + 0.721268i \(0.743559\pi\)
\(488\) 118.532i 0.242893i
\(489\) 0 0
\(490\) 1077.48 2.19894
\(491\) − 319.707i − 0.651135i −0.945519 0.325568i \(-0.894445\pi\)
0.945519 0.325568i \(-0.105555\pi\)
\(492\) 0 0
\(493\) −200.465 −0.406622
\(494\) 379.544i 0.768307i
\(495\) 0 0
\(496\) −118.430 −0.238770
\(497\) 245.470i 0.493903i
\(498\) 0 0
\(499\) −912.107 −1.82787 −0.913935 0.405862i \(-0.866972\pi\)
−0.913935 + 0.405862i \(0.866972\pi\)
\(500\) − 652.988i − 1.30598i
\(501\) 0 0
\(502\) 552.449 1.10050
\(503\) 442.736i 0.880191i 0.897951 + 0.440096i \(0.145055\pi\)
−0.897951 + 0.440096i \(0.854945\pi\)
\(504\) 0 0
\(505\) 312.667 0.619142
\(506\) 101.407i 0.200409i
\(507\) 0 0
\(508\) −54.2338 −0.106759
\(509\) 240.686i 0.472860i 0.971649 + 0.236430i \(0.0759774\pi\)
−0.971649 + 0.236430i \(0.924023\pi\)
\(510\) 0 0
\(511\) −293.493 −0.574349
\(512\) − 667.190i − 1.30310i
\(513\) 0 0
\(514\) −242.048 −0.470911
\(515\) 416.770i 0.809263i
\(516\) 0 0
\(517\) −123.658 −0.239185
\(518\) − 817.957i − 1.57907i
\(519\) 0 0
\(520\) −98.4967 −0.189417
\(521\) − 416.011i − 0.798486i −0.916845 0.399243i \(-0.869273\pi\)
0.916845 0.399243i \(-0.130727\pi\)
\(522\) 0 0
\(523\) 195.597 0.373991 0.186995 0.982361i \(-0.440125\pi\)
0.186995 + 0.982361i \(0.440125\pi\)
\(524\) − 224.443i − 0.428326i
\(525\) 0 0
\(526\) −1427.72 −2.71430
\(527\) 147.289i 0.279485i
\(528\) 0 0
\(529\) 240.112 0.453898
\(530\) − 710.597i − 1.34075i
\(531\) 0 0
\(532\) 700.524 1.31677
\(533\) 406.683i 0.763008i
\(534\) 0 0
\(535\) 720.839 1.34736
\(536\) 19.8913i 0.0371106i
\(537\) 0 0
\(538\) −1348.23 −2.50601
\(539\) 186.986i 0.346913i
\(540\) 0 0
\(541\) 203.303 0.375792 0.187896 0.982189i \(-0.439833\pi\)
0.187896 + 0.982189i \(0.439833\pi\)
\(542\) − 585.680i − 1.08059i
\(543\) 0 0
\(544\) −684.707 −1.25865
\(545\) 63.2562i 0.116066i
\(546\) 0 0
\(547\) −1006.26 −1.83960 −0.919801 0.392385i \(-0.871650\pi\)
−0.919801 + 0.392385i \(0.871650\pi\)
\(548\) 10.3217i 0.0188352i
\(549\) 0 0
\(550\) 58.3513 0.106093
\(551\) − 162.839i − 0.295533i
\(552\) 0 0
\(553\) 270.789 0.489673
\(554\) − 1060.98i − 1.91512i
\(555\) 0 0
\(556\) −329.580 −0.592770
\(557\) 882.622i 1.58460i 0.610132 + 0.792300i \(0.291116\pi\)
−0.610132 + 0.792300i \(0.708884\pi\)
\(558\) 0 0
\(559\) −88.3785 −0.158101
\(560\) − 562.123i − 1.00379i
\(561\) 0 0
\(562\) −703.143 −1.25114
\(563\) 15.7127i 0.0279089i 0.999903 + 0.0139545i \(0.00444199\pi\)
−0.999903 + 0.0139545i \(0.995558\pi\)
\(564\) 0 0
\(565\) 324.175 0.573761
\(566\) − 821.742i − 1.45184i
\(567\) 0 0
\(568\) −49.6771 −0.0874596
\(569\) − 1032.00i − 1.81371i −0.421447 0.906853i \(-0.638478\pi\)
0.421447 0.906853i \(-0.361522\pi\)
\(570\) 0 0
\(571\) −241.590 −0.423100 −0.211550 0.977367i \(-0.567851\pi\)
−0.211550 + 0.977367i \(0.567851\pi\)
\(572\) − 101.247i − 0.177005i
\(573\) 0 0
\(574\) 1374.51 2.39461
\(575\) 166.231i 0.289097i
\(576\) 0 0
\(577\) −661.808 −1.14698 −0.573490 0.819212i \(-0.694411\pi\)
−0.573490 + 0.819212i \(0.694411\pi\)
\(578\) − 186.774i − 0.323139i
\(579\) 0 0
\(580\) 250.309 0.431568
\(581\) − 851.787i − 1.46607i
\(582\) 0 0
\(583\) 123.317 0.211522
\(584\) − 59.3956i − 0.101705i
\(585\) 0 0
\(586\) −78.4561 −0.133884
\(587\) 315.909i 0.538175i 0.963116 + 0.269088i \(0.0867221\pi\)
−0.963116 + 0.269088i \(0.913278\pi\)
\(588\) 0 0
\(589\) −119.643 −0.203130
\(590\) 447.701i 0.758815i
\(591\) 0 0
\(592\) −279.516 −0.472156
\(593\) − 269.585i − 0.454612i −0.973823 0.227306i \(-0.927008\pi\)
0.973823 0.227306i \(-0.0729918\pi\)
\(594\) 0 0
\(595\) −699.100 −1.17496
\(596\) − 615.433i − 1.03261i
\(597\) 0 0
\(598\) 528.168 0.883225
\(599\) 136.849i 0.228462i 0.993454 + 0.114231i \(0.0364404\pi\)
−0.993454 + 0.114231i \(0.963560\pi\)
\(600\) 0 0
\(601\) 306.108 0.509332 0.254666 0.967029i \(-0.418034\pi\)
0.254666 + 0.967029i \(0.418034\pi\)
\(602\) 298.701i 0.496182i
\(603\) 0 0
\(604\) −204.900 −0.339239
\(605\) 456.294i 0.754204i
\(606\) 0 0
\(607\) −1101.53 −1.81470 −0.907352 0.420372i \(-0.861900\pi\)
−0.907352 + 0.420372i \(0.861900\pi\)
\(608\) − 556.192i − 0.914790i
\(609\) 0 0
\(610\) 569.154 0.933039
\(611\) 644.063i 1.05411i
\(612\) 0 0
\(613\) −273.907 −0.446831 −0.223415 0.974723i \(-0.571721\pi\)
−0.223415 + 0.974723i \(0.571721\pi\)
\(614\) 557.961i 0.908731i
\(615\) 0 0
\(616\) −57.7714 −0.0937847
\(617\) 307.717i 0.498731i 0.968409 + 0.249366i \(0.0802221\pi\)
−0.968409 + 0.249366i \(0.919778\pi\)
\(618\) 0 0
\(619\) 132.268 0.213680 0.106840 0.994276i \(-0.465927\pi\)
0.106840 + 0.994276i \(0.465927\pi\)
\(620\) − 183.911i − 0.296631i
\(621\) 0 0
\(622\) 498.773 0.801886
\(623\) − 131.183i − 0.210566i
\(624\) 0 0
\(625\) −284.843 −0.455749
\(626\) 1478.90i 2.36246i
\(627\) 0 0
\(628\) 893.629 1.42298
\(629\) 347.628i 0.552668i
\(630\) 0 0
\(631\) 1019.72 1.61604 0.808018 0.589157i \(-0.200540\pi\)
0.808018 + 0.589157i \(0.200540\pi\)
\(632\) 54.8010i 0.0867105i
\(633\) 0 0
\(634\) 505.760 0.797728
\(635\) 43.9649i 0.0692361i
\(636\) 0 0
\(637\) 973.900 1.52888
\(638\) 79.5440i 0.124677i
\(639\) 0 0
\(640\) 294.890 0.460766
\(641\) − 176.132i − 0.274777i −0.990517 0.137388i \(-0.956129\pi\)
0.990517 0.137388i \(-0.0438708\pi\)
\(642\) 0 0
\(643\) 1087.93 1.69196 0.845978 0.533217i \(-0.179017\pi\)
0.845978 + 0.533217i \(0.179017\pi\)
\(644\) − 974.841i − 1.51373i
\(645\) 0 0
\(646\) −545.176 −0.843926
\(647\) − 169.613i − 0.262153i −0.991372 0.131076i \(-0.958157\pi\)
0.991372 0.131076i \(-0.0418433\pi\)
\(648\) 0 0
\(649\) −77.6942 −0.119714
\(650\) − 303.917i − 0.467565i
\(651\) 0 0
\(652\) −1051.77 −1.61315
\(653\) − 581.712i − 0.890830i −0.895324 0.445415i \(-0.853056\pi\)
0.895324 0.445415i \(-0.146944\pi\)
\(654\) 0 0
\(655\) −181.946 −0.277780
\(656\) − 469.703i − 0.716011i
\(657\) 0 0
\(658\) 2176.80 3.30821
\(659\) 918.820i 1.39426i 0.716942 + 0.697132i \(0.245541\pi\)
−0.716942 + 0.697132i \(0.754459\pi\)
\(660\) 0 0
\(661\) −250.125 −0.378403 −0.189202 0.981938i \(-0.560590\pi\)
−0.189202 + 0.981938i \(0.560590\pi\)
\(662\) 271.691i 0.410410i
\(663\) 0 0
\(664\) 172.381 0.259609
\(665\) − 567.883i − 0.853960i
\(666\) 0 0
\(667\) −226.605 −0.339737
\(668\) − 279.881i − 0.418984i
\(669\) 0 0
\(670\) 95.5120 0.142555
\(671\) 98.7712i 0.147200i
\(672\) 0 0
\(673\) 876.265 1.30203 0.651014 0.759065i \(-0.274344\pi\)
0.651014 + 0.759065i \(0.274344\pi\)
\(674\) − 1162.83i − 1.72526i
\(675\) 0 0
\(676\) 285.974 0.423038
\(677\) − 814.452i − 1.20303i −0.798861 0.601515i \(-0.794564\pi\)
0.798861 0.601515i \(-0.205436\pi\)
\(678\) 0 0
\(679\) −1118.00 −1.64654
\(680\) − 141.481i − 0.208060i
\(681\) 0 0
\(682\) 58.4437 0.0856946
\(683\) − 1331.38i − 1.94931i −0.223722 0.974653i \(-0.571821\pi\)
0.223722 0.974653i \(-0.428179\pi\)
\(684\) 0 0
\(685\) 8.36733 0.0122151
\(686\) − 1558.00i − 2.27113i
\(687\) 0 0
\(688\) 102.074 0.148363
\(689\) − 642.287i − 0.932201i
\(690\) 0 0
\(691\) 698.547 1.01092 0.505461 0.862850i \(-0.331323\pi\)
0.505461 + 0.862850i \(0.331323\pi\)
\(692\) 1385.44i 2.00208i
\(693\) 0 0
\(694\) −789.707 −1.13791
\(695\) 267.176i 0.384426i
\(696\) 0 0
\(697\) −584.159 −0.838105
\(698\) 415.946i 0.595911i
\(699\) 0 0
\(700\) −560.940 −0.801343
\(701\) − 1006.22i − 1.43540i −0.696351 0.717701i \(-0.745194\pi\)
0.696351 0.717701i \(-0.254806\pi\)
\(702\) 0 0
\(703\) −282.381 −0.401679
\(704\) 174.496i 0.247864i
\(705\) 0 0
\(706\) −130.211 −0.184435
\(707\) − 955.163i − 1.35101i
\(708\) 0 0
\(709\) −995.815 −1.40453 −0.702267 0.711913i \(-0.747829\pi\)
−0.702267 + 0.711913i \(0.747829\pi\)
\(710\) 238.535i 0.335964i
\(711\) 0 0
\(712\) 26.5481 0.0372867
\(713\) 166.494i 0.233513i
\(714\) 0 0
\(715\) −82.0763 −0.114792
\(716\) − 122.858i − 0.171589i
\(717\) 0 0
\(718\) −1137.65 −1.58447
\(719\) − 140.868i − 0.195922i −0.995190 0.0979608i \(-0.968768\pi\)
0.995190 0.0979608i \(-0.0312320\pi\)
\(720\) 0 0
\(721\) 1273.19 1.76587
\(722\) 628.808i 0.870925i
\(723\) 0 0
\(724\) 649.621 0.897267
\(725\) 130.392i 0.179851i
\(726\) 0 0
\(727\) −719.188 −0.989254 −0.494627 0.869105i \(-0.664695\pi\)
−0.494627 + 0.869105i \(0.664695\pi\)
\(728\) 300.897i 0.413320i
\(729\) 0 0
\(730\) −285.200 −0.390685
\(731\) − 126.947i − 0.173662i
\(732\) 0 0
\(733\) 1368.27 1.86668 0.933338 0.359000i \(-0.116882\pi\)
0.933338 + 0.359000i \(0.116882\pi\)
\(734\) − 248.015i − 0.337895i
\(735\) 0 0
\(736\) −773.991 −1.05162
\(737\) 16.5752i 0.0224901i
\(738\) 0 0
\(739\) −97.8408 −0.132396 −0.0661981 0.997806i \(-0.521087\pi\)
−0.0661981 + 0.997806i \(0.521087\pi\)
\(740\) − 434.064i − 0.586573i
\(741\) 0 0
\(742\) −2170.80 −2.92560
\(743\) 393.158i 0.529149i 0.964365 + 0.264575i \(0.0852315\pi\)
−0.964365 + 0.264575i \(0.914768\pi\)
\(744\) 0 0
\(745\) −498.904 −0.669670
\(746\) 754.258i 1.01107i
\(747\) 0 0
\(748\) 145.431 0.194426
\(749\) − 2202.09i − 2.94003i
\(750\) 0 0
\(751\) −1155.22 −1.53824 −0.769122 0.639102i \(-0.779306\pi\)
−0.769122 + 0.639102i \(0.779306\pi\)
\(752\) − 743.868i − 0.989186i
\(753\) 0 0
\(754\) 414.297 0.549466
\(755\) 166.104i 0.220005i
\(756\) 0 0
\(757\) −1031.03 −1.36200 −0.680998 0.732285i \(-0.738454\pi\)
−0.680998 + 0.732285i \(0.738454\pi\)
\(758\) − 1867.22i − 2.46336i
\(759\) 0 0
\(760\) 114.926 0.151218
\(761\) − 587.059i − 0.771431i −0.922618 0.385715i \(-0.873955\pi\)
0.922618 0.385715i \(-0.126045\pi\)
\(762\) 0 0
\(763\) 193.241 0.253265
\(764\) 907.457i 1.18777i
\(765\) 0 0
\(766\) 1092.28 1.42596
\(767\) 404.663i 0.527592i
\(768\) 0 0
\(769\) −803.230 −1.04451 −0.522256 0.852789i \(-0.674910\pi\)
−0.522256 + 0.852789i \(0.674910\pi\)
\(770\) 277.401i 0.360261i
\(771\) 0 0
\(772\) −875.655 −1.13427
\(773\) − 653.000i − 0.844761i −0.906419 0.422381i \(-0.861195\pi\)
0.906419 0.422381i \(-0.138805\pi\)
\(774\) 0 0
\(775\) 95.8037 0.123618
\(776\) − 226.256i − 0.291567i
\(777\) 0 0
\(778\) 1574.76 2.02411
\(779\) − 474.516i − 0.609135i
\(780\) 0 0
\(781\) −41.3954 −0.0530031
\(782\) 758.661i 0.970154i
\(783\) 0 0
\(784\) −1124.82 −1.43471
\(785\) − 724.425i − 0.922835i
\(786\) 0 0
\(787\) −299.587 −0.380669 −0.190335 0.981719i \(-0.560957\pi\)
−0.190335 + 0.981719i \(0.560957\pi\)
\(788\) − 372.266i − 0.472419i
\(789\) 0 0
\(790\) 263.138 0.333086
\(791\) − 990.321i − 1.25199i
\(792\) 0 0
\(793\) 514.440 0.648727
\(794\) 1799.27i 2.26608i
\(795\) 0 0
\(796\) 847.079 1.06417
\(797\) − 698.510i − 0.876424i −0.898872 0.438212i \(-0.855612\pi\)
0.898872 0.438212i \(-0.144388\pi\)
\(798\) 0 0
\(799\) −925.132 −1.15786
\(800\) 445.367i 0.556709i
\(801\) 0 0
\(802\) −1300.22 −1.62122
\(803\) − 49.4938i − 0.0616361i
\(804\) 0 0
\(805\) −790.260 −0.981689
\(806\) − 304.399i − 0.377666i
\(807\) 0 0
\(808\) 193.301 0.239234
\(809\) − 803.469i − 0.993163i −0.867990 0.496582i \(-0.834588\pi\)
0.867990 0.496582i \(-0.165412\pi\)
\(810\) 0 0
\(811\) −1104.38 −1.36175 −0.680874 0.732400i \(-0.738400\pi\)
−0.680874 + 0.732400i \(0.738400\pi\)
\(812\) − 764.668i − 0.941710i
\(813\) 0 0
\(814\) 137.938 0.169457
\(815\) 852.624i 1.04616i
\(816\) 0 0
\(817\) 103.120 0.126217
\(818\) 1396.00i 1.70660i
\(819\) 0 0
\(820\) 729.408 0.889521
\(821\) − 1038.73i − 1.26520i −0.774479 0.632600i \(-0.781988\pi\)
0.774479 0.632600i \(-0.218012\pi\)
\(822\) 0 0
\(823\) −273.226 −0.331988 −0.165994 0.986127i \(-0.553083\pi\)
−0.165994 + 0.986127i \(0.553083\pi\)
\(824\) 257.662i 0.312697i
\(825\) 0 0
\(826\) 1367.68 1.65578
\(827\) − 723.298i − 0.874604i −0.899315 0.437302i \(-0.855934\pi\)
0.899315 0.437302i \(-0.144066\pi\)
\(828\) 0 0
\(829\) −314.714 −0.379631 −0.189815 0.981820i \(-0.560789\pi\)
−0.189815 + 0.981820i \(0.560789\pi\)
\(830\) − 827.720i − 0.997254i
\(831\) 0 0
\(832\) 908.846 1.09236
\(833\) 1398.91i 1.67936i
\(834\) 0 0
\(835\) −226.887 −0.271721
\(836\) 118.134i 0.141309i
\(837\) 0 0
\(838\) −440.292 −0.525408
\(839\) 418.313i 0.498585i 0.968428 + 0.249293i \(0.0801981\pi\)
−0.968428 + 0.249293i \(0.919802\pi\)
\(840\) 0 0
\(841\) 663.250 0.788645
\(842\) − 978.407i − 1.16200i
\(843\) 0 0
\(844\) −1172.83 −1.38961
\(845\) − 231.826i − 0.274350i
\(846\) 0 0
\(847\) 1393.93 1.64572
\(848\) 741.816i 0.874783i
\(849\) 0 0
\(850\) 436.546 0.513584
\(851\) 392.958i 0.461760i
\(852\) 0 0
\(853\) 1438.86 1.68682 0.843412 0.537267i \(-0.180543\pi\)
0.843412 + 0.537267i \(0.180543\pi\)
\(854\) − 1738.70i − 2.03595i
\(855\) 0 0
\(856\) 445.648 0.520616
\(857\) − 1244.60i − 1.45228i −0.687549 0.726138i \(-0.741313\pi\)
0.687549 0.726138i \(-0.258687\pi\)
\(858\) 0 0
\(859\) 121.302 0.141214 0.0706068 0.997504i \(-0.477506\pi\)
0.0706068 + 0.997504i \(0.477506\pi\)
\(860\) 158.511i 0.184316i
\(861\) 0 0
\(862\) 2194.69 2.54604
\(863\) − 802.161i − 0.929503i −0.885441 0.464752i \(-0.846144\pi\)
0.885441 0.464752i \(-0.153856\pi\)
\(864\) 0 0
\(865\) 1123.11 1.29840
\(866\) 625.114i 0.721841i
\(867\) 0 0
\(868\) −561.829 −0.647268
\(869\) 45.6651i 0.0525491i
\(870\) 0 0
\(871\) 86.3303 0.0991163
\(872\) 39.1072i 0.0448477i
\(873\) 0 0
\(874\) −616.265 −0.705108
\(875\) 1617.10i 1.84812i
\(876\) 0 0
\(877\) 181.374 0.206812 0.103406 0.994639i \(-0.467026\pi\)
0.103406 + 0.994639i \(0.467026\pi\)
\(878\) 2362.95i 2.69128i
\(879\) 0 0
\(880\) 94.7949 0.107721
\(881\) − 750.315i − 0.851663i −0.904802 0.425832i \(-0.859982\pi\)
0.904802 0.425832i \(-0.140018\pi\)
\(882\) 0 0
\(883\) 1709.63 1.93616 0.968082 0.250635i \(-0.0806393\pi\)
0.968082 + 0.250635i \(0.0806393\pi\)
\(884\) − 757.463i − 0.856858i
\(885\) 0 0
\(886\) 554.117 0.625414
\(887\) 876.953i 0.988673i 0.869271 + 0.494336i \(0.164589\pi\)
−0.869271 + 0.494336i \(0.835411\pi\)
\(888\) 0 0
\(889\) 134.308 0.151078
\(890\) − 127.476i − 0.143232i
\(891\) 0 0
\(892\) 1692.54 1.89747
\(893\) − 751.490i − 0.841535i
\(894\) 0 0
\(895\) −99.5954 −0.111280
\(896\) − 900.858i − 1.00542i
\(897\) 0 0
\(898\) −1229.96 −1.36967
\(899\) 130.599i 0.145271i
\(900\) 0 0
\(901\) 922.580 1.02395
\(902\) 231.793i 0.256977i
\(903\) 0 0
\(904\) 200.416 0.221700
\(905\) − 526.619i − 0.581900i
\(906\) 0 0
\(907\) −604.074 −0.666013 −0.333007 0.942924i \(-0.608063\pi\)
−0.333007 + 0.942924i \(0.608063\pi\)
\(908\) 960.854i 1.05821i
\(909\) 0 0
\(910\) 1444.82 1.58771
\(911\) − 360.889i − 0.396146i −0.980187 0.198073i \(-0.936532\pi\)
0.980187 0.198073i \(-0.0634683\pi\)
\(912\) 0 0
\(913\) 143.643 0.157331
\(914\) 1997.68i 2.18565i
\(915\) 0 0
\(916\) −1154.47 −1.26034
\(917\) 555.825i 0.606134i
\(918\) 0 0
\(919\) −267.944 −0.291561 −0.145780 0.989317i \(-0.546569\pi\)
−0.145780 + 0.989317i \(0.546569\pi\)
\(920\) − 159.929i − 0.173836i
\(921\) 0 0
\(922\) 2602.00 2.82212
\(923\) 215.604i 0.233590i
\(924\) 0 0
\(925\) 226.114 0.244448
\(926\) − 266.079i − 0.287343i
\(927\) 0 0
\(928\) −607.121 −0.654225
\(929\) 219.959i 0.236770i 0.992968 + 0.118385i \(0.0377717\pi\)
−0.992968 + 0.118385i \(0.962228\pi\)
\(930\) 0 0
\(931\) −1136.34 −1.22056
\(932\) − 626.776i − 0.672507i
\(933\) 0 0
\(934\) 283.175 0.303185
\(935\) − 117.894i − 0.126090i
\(936\) 0 0
\(937\) −252.272 −0.269234 −0.134617 0.990898i \(-0.542980\pi\)
−0.134617 + 0.990898i \(0.542980\pi\)
\(938\) − 291.779i − 0.311065i
\(939\) 0 0
\(940\) 1155.16 1.22889
\(941\) 600.523i 0.638176i 0.947725 + 0.319088i \(0.103377\pi\)
−0.947725 + 0.319088i \(0.896623\pi\)
\(942\) 0 0
\(943\) −660.331 −0.700245
\(944\) − 467.369i − 0.495095i
\(945\) 0 0
\(946\) −50.3722 −0.0532476
\(947\) 38.7868i 0.0409575i 0.999790 + 0.0204788i \(0.00651905\pi\)
−0.999790 + 0.0204788i \(0.993481\pi\)
\(948\) 0 0
\(949\) −257.784 −0.271637
\(950\) 354.609i 0.373273i
\(951\) 0 0
\(952\) −432.208 −0.454000
\(953\) 101.007i 0.105989i 0.998595 + 0.0529944i \(0.0168765\pi\)
−0.998595 + 0.0529944i \(0.983123\pi\)
\(954\) 0 0
\(955\) 735.635 0.770299
\(956\) − 2042.40i − 2.13640i
\(957\) 0 0
\(958\) 2221.33 2.31871
\(959\) − 25.5613i − 0.0266541i
\(960\) 0 0
\(961\) −865.044 −0.900150
\(962\) − 718.437i − 0.746816i
\(963\) 0 0
\(964\) 1590.67 1.65007
\(965\) 709.854i 0.735600i
\(966\) 0 0
\(967\) −768.260 −0.794478 −0.397239 0.917715i \(-0.630032\pi\)
−0.397239 + 0.917715i \(0.630032\pi\)
\(968\) 282.097i 0.291422i
\(969\) 0 0
\(970\) −1086.41 −1.12001
\(971\) − 688.506i − 0.709069i −0.935043 0.354535i \(-0.884639\pi\)
0.935043 0.354535i \(-0.115361\pi\)
\(972\) 0 0
\(973\) 816.193 0.838842
\(974\) 2002.74i 2.05620i
\(975\) 0 0
\(976\) −594.158 −0.608769
\(977\) − 926.097i − 0.947898i −0.880552 0.473949i \(-0.842828\pi\)
0.880552 0.473949i \(-0.157172\pi\)
\(978\) 0 0
\(979\) 22.1223 0.0225968
\(980\) − 1746.74i − 1.78239i
\(981\) 0 0
\(982\) −949.077 −0.966474
\(983\) 1142.89i 1.16266i 0.813669 + 0.581328i \(0.197467\pi\)
−0.813669 + 0.581328i \(0.802533\pi\)
\(984\) 0 0
\(985\) −301.779 −0.306375
\(986\) 595.096i 0.603546i
\(987\) 0 0
\(988\) 615.292 0.622765
\(989\) − 143.500i − 0.145096i
\(990\) 0 0
\(991\) 634.172 0.639931 0.319966 0.947429i \(-0.396329\pi\)
0.319966 + 0.947429i \(0.396329\pi\)
\(992\) 446.073i 0.449671i
\(993\) 0 0
\(994\) 728.698 0.733096
\(995\) − 686.689i − 0.690140i
\(996\) 0 0
\(997\) 1132.63 1.13604 0.568020 0.823015i \(-0.307710\pi\)
0.568020 + 0.823015i \(0.307710\pi\)
\(998\) 2707.66i 2.71309i
\(999\) 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.890.14 84
3.2 odd 2 inner 1143.3.b.a.890.71 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.14 84 1.1 even 1 trivial
1143.3.b.a.890.71 yes 84 3.2 odd 2 inner