Properties

Label 1143.3.b.a.890.2
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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 890.2
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.83

$q$-expansion

\(f(q)\) \(=\) \(q-3.86719i q^{2} -10.9551 q^{4} -9.44547i q^{5} +11.3204 q^{7} +26.8969i q^{8} +O(q^{10})\) \(q-3.86719i q^{2} -10.9551 q^{4} -9.44547i q^{5} +11.3204 q^{7} +26.8969i q^{8} -36.5274 q^{10} -14.1873i q^{11} +10.8150 q^{13} -43.7780i q^{14} +60.1947 q^{16} +9.53381i q^{17} -24.9634 q^{19} +103.477i q^{20} -54.8648 q^{22} +13.0513i q^{23} -64.2169 q^{25} -41.8237i q^{26} -124.016 q^{28} -18.2561i q^{29} -38.5312 q^{31} -125.197i q^{32} +36.8691 q^{34} -106.926i q^{35} -51.8983 q^{37} +96.5383i q^{38} +254.054 q^{40} +18.2318i q^{41} -31.3978 q^{43} +155.424i q^{44} +50.4719 q^{46} -4.56917i q^{47} +79.1505 q^{49} +248.339i q^{50} -118.480 q^{52} -31.5680i q^{53} -134.005 q^{55} +304.482i q^{56} -70.5998 q^{58} -62.5783i q^{59} +64.5712 q^{61} +149.008i q^{62} -243.380 q^{64} -102.153i q^{65} +74.3116 q^{67} -104.444i q^{68} -413.503 q^{70} +111.127i q^{71} -48.7312 q^{73} +200.701i q^{74} +273.478 q^{76} -160.605i q^{77} -63.2310 q^{79} -568.567i q^{80} +70.5059 q^{82} +51.0401i q^{83} +90.0514 q^{85} +121.421i q^{86} +381.593 q^{88} -19.4944i q^{89} +122.430 q^{91} -142.979i q^{92} -17.6698 q^{94} +235.791i q^{95} +72.1446 q^{97} -306.090i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.86719i − 1.93359i −0.255545 0.966797i \(-0.582255\pi\)
0.255545 0.966797i \(-0.417745\pi\)
\(3\) 0 0
\(4\) −10.9551 −2.73879
\(5\) − 9.44547i − 1.88909i −0.328376 0.944547i \(-0.606501\pi\)
0.328376 0.944547i \(-0.393499\pi\)
\(6\) 0 0
\(7\) 11.3204 1.61719 0.808597 0.588363i \(-0.200227\pi\)
0.808597 + 0.588363i \(0.200227\pi\)
\(8\) 26.8969i 3.36211i
\(9\) 0 0
\(10\) −36.5274 −3.65274
\(11\) − 14.1873i − 1.28975i −0.764288 0.644875i \(-0.776909\pi\)
0.764288 0.644875i \(-0.223091\pi\)
\(12\) 0 0
\(13\) 10.8150 0.831925 0.415962 0.909382i \(-0.363445\pi\)
0.415962 + 0.909382i \(0.363445\pi\)
\(14\) − 43.7780i − 3.12700i
\(15\) 0 0
\(16\) 60.1947 3.76217
\(17\) 9.53381i 0.560813i 0.959881 + 0.280406i \(0.0904692\pi\)
−0.959881 + 0.280406i \(0.909531\pi\)
\(18\) 0 0
\(19\) −24.9634 −1.31387 −0.656933 0.753949i \(-0.728146\pi\)
−0.656933 + 0.753949i \(0.728146\pi\)
\(20\) 103.477i 5.17383i
\(21\) 0 0
\(22\) −54.8648 −2.49385
\(23\) 13.0513i 0.567448i 0.958906 + 0.283724i \(0.0915700\pi\)
−0.958906 + 0.283724i \(0.908430\pi\)
\(24\) 0 0
\(25\) −64.2169 −2.56868
\(26\) − 41.8237i − 1.60860i
\(27\) 0 0
\(28\) −124.016 −4.42915
\(29\) − 18.2561i − 0.629521i −0.949171 0.314761i \(-0.898076\pi\)
0.949171 0.314761i \(-0.101924\pi\)
\(30\) 0 0
\(31\) −38.5312 −1.24294 −0.621472 0.783437i \(-0.713465\pi\)
−0.621472 + 0.783437i \(0.713465\pi\)
\(32\) − 125.197i − 3.91240i
\(33\) 0 0
\(34\) 36.8691 1.08438
\(35\) − 106.926i − 3.05503i
\(36\) 0 0
\(37\) −51.8983 −1.40266 −0.701329 0.712838i \(-0.747410\pi\)
−0.701329 + 0.712838i \(0.747410\pi\)
\(38\) 96.5383i 2.54048i
\(39\) 0 0
\(40\) 254.054 6.35134
\(41\) 18.2318i 0.444679i 0.974969 + 0.222339i \(0.0713693\pi\)
−0.974969 + 0.222339i \(0.928631\pi\)
\(42\) 0 0
\(43\) −31.3978 −0.730180 −0.365090 0.930972i \(-0.618962\pi\)
−0.365090 + 0.930972i \(0.618962\pi\)
\(44\) 155.424i 3.53235i
\(45\) 0 0
\(46\) 50.4719 1.09722
\(47\) − 4.56917i − 0.0972164i −0.998818 0.0486082i \(-0.984521\pi\)
0.998818 0.0486082i \(-0.0154786\pi\)
\(48\) 0 0
\(49\) 79.1505 1.61532
\(50\) 248.339i 4.96678i
\(51\) 0 0
\(52\) −118.480 −2.27846
\(53\) − 31.5680i − 0.595623i −0.954625 0.297812i \(-0.903743\pi\)
0.954625 0.297812i \(-0.0962568\pi\)
\(54\) 0 0
\(55\) −134.005 −2.43646
\(56\) 304.482i 5.43718i
\(57\) 0 0
\(58\) −70.5998 −1.21724
\(59\) − 62.5783i − 1.06065i −0.847795 0.530325i \(-0.822070\pi\)
0.847795 0.530325i \(-0.177930\pi\)
\(60\) 0 0
\(61\) 64.5712 1.05854 0.529272 0.848452i \(-0.322465\pi\)
0.529272 + 0.848452i \(0.322465\pi\)
\(62\) 149.008i 2.40335i
\(63\) 0 0
\(64\) −243.380 −3.80282
\(65\) − 102.153i − 1.57158i
\(66\) 0 0
\(67\) 74.3116 1.10913 0.554564 0.832141i \(-0.312885\pi\)
0.554564 + 0.832141i \(0.312885\pi\)
\(68\) − 104.444i − 1.53595i
\(69\) 0 0
\(70\) −413.503 −5.90719
\(71\) 111.127i 1.56516i 0.622547 + 0.782582i \(0.286098\pi\)
−0.622547 + 0.782582i \(0.713902\pi\)
\(72\) 0 0
\(73\) −48.7312 −0.667550 −0.333775 0.942653i \(-0.608323\pi\)
−0.333775 + 0.942653i \(0.608323\pi\)
\(74\) 200.701i 2.71217i
\(75\) 0 0
\(76\) 273.478 3.59840
\(77\) − 160.605i − 2.08578i
\(78\) 0 0
\(79\) −63.2310 −0.800393 −0.400196 0.916429i \(-0.631058\pi\)
−0.400196 + 0.916429i \(0.631058\pi\)
\(80\) − 568.567i − 7.10709i
\(81\) 0 0
\(82\) 70.5059 0.859829
\(83\) 51.0401i 0.614940i 0.951558 + 0.307470i \(0.0994824\pi\)
−0.951558 + 0.307470i \(0.900518\pi\)
\(84\) 0 0
\(85\) 90.0514 1.05943
\(86\) 121.421i 1.41187i
\(87\) 0 0
\(88\) 381.593 4.33628
\(89\) − 19.4944i − 0.219039i −0.993985 0.109519i \(-0.965069\pi\)
0.993985 0.109519i \(-0.0349312\pi\)
\(90\) 0 0
\(91\) 122.430 1.34538
\(92\) − 142.979i − 1.55412i
\(93\) 0 0
\(94\) −17.6698 −0.187977
\(95\) 235.791i 2.48202i
\(96\) 0 0
\(97\) 72.1446 0.743758 0.371879 0.928281i \(-0.378714\pi\)
0.371879 + 0.928281i \(0.378714\pi\)
\(98\) − 306.090i − 3.12337i
\(99\) 0 0
\(100\) 703.506 7.03506
\(101\) − 130.320i − 1.29030i −0.764055 0.645151i \(-0.776794\pi\)
0.764055 0.645151i \(-0.223206\pi\)
\(102\) 0 0
\(103\) 41.3239 0.401203 0.200602 0.979673i \(-0.435710\pi\)
0.200602 + 0.979673i \(0.435710\pi\)
\(104\) 290.890i 2.79702i
\(105\) 0 0
\(106\) −122.080 −1.15169
\(107\) − 169.058i − 1.57999i −0.613116 0.789993i \(-0.710084\pi\)
0.613116 0.789993i \(-0.289916\pi\)
\(108\) 0 0
\(109\) −67.9965 −0.623821 −0.311910 0.950111i \(-0.600969\pi\)
−0.311910 + 0.950111i \(0.600969\pi\)
\(110\) 518.224i 4.71113i
\(111\) 0 0
\(112\) 681.425 6.08415
\(113\) 44.6199i 0.394866i 0.980316 + 0.197433i \(0.0632605\pi\)
−0.980316 + 0.197433i \(0.936739\pi\)
\(114\) 0 0
\(115\) 123.276 1.07196
\(116\) 199.998i 1.72412i
\(117\) 0 0
\(118\) −242.002 −2.05087
\(119\) 107.926i 0.906943i
\(120\) 0 0
\(121\) −80.2783 −0.663457
\(122\) − 249.709i − 2.04679i
\(123\) 0 0
\(124\) 422.115 3.40416
\(125\) 370.422i 2.96338i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) 440.411i 3.44071i
\(129\) 0 0
\(130\) −395.045 −3.03881
\(131\) − 157.351i − 1.20115i −0.799567 0.600576i \(-0.794938\pi\)
0.799567 0.600576i \(-0.205062\pi\)
\(132\) 0 0
\(133\) −282.595 −2.12477
\(134\) − 287.377i − 2.14460i
\(135\) 0 0
\(136\) −256.430 −1.88551
\(137\) − 112.882i − 0.823958i −0.911193 0.411979i \(-0.864838\pi\)
0.911193 0.411979i \(-0.135162\pi\)
\(138\) 0 0
\(139\) −98.1690 −0.706252 −0.353126 0.935576i \(-0.614881\pi\)
−0.353126 + 0.935576i \(0.614881\pi\)
\(140\) 1171.39i 8.36708i
\(141\) 0 0
\(142\) 429.748 3.02639
\(143\) − 153.435i − 1.07298i
\(144\) 0 0
\(145\) −172.438 −1.18922
\(146\) 188.453i 1.29077i
\(147\) 0 0
\(148\) 568.554 3.84158
\(149\) 198.203i 1.33022i 0.746744 + 0.665112i \(0.231616\pi\)
−0.746744 + 0.665112i \(0.768384\pi\)
\(150\) 0 0
\(151\) 272.334 1.80354 0.901768 0.432219i \(-0.142269\pi\)
0.901768 + 0.432219i \(0.142269\pi\)
\(152\) − 671.438i − 4.41736i
\(153\) 0 0
\(154\) −621.089 −4.03305
\(155\) 363.946i 2.34804i
\(156\) 0 0
\(157\) −15.5821 −0.0992487 −0.0496244 0.998768i \(-0.515802\pi\)
−0.0496244 + 0.998768i \(0.515802\pi\)
\(158\) 244.526i 1.54764i
\(159\) 0 0
\(160\) −1182.54 −7.39088
\(161\) 147.746i 0.917674i
\(162\) 0 0
\(163\) 222.264 1.36358 0.681790 0.731548i \(-0.261202\pi\)
0.681790 + 0.731548i \(0.261202\pi\)
\(164\) − 199.732i − 1.21788i
\(165\) 0 0
\(166\) 197.382 1.18905
\(167\) 0.247721i 0.00148336i 1.00000 0.000741679i \(0.000236084\pi\)
−1.00000 0.000741679i \(0.999764\pi\)
\(168\) 0 0
\(169\) −52.0354 −0.307901
\(170\) − 348.246i − 2.04850i
\(171\) 0 0
\(172\) 343.967 1.99981
\(173\) − 177.674i − 1.02702i −0.858084 0.513509i \(-0.828345\pi\)
0.858084 0.513509i \(-0.171655\pi\)
\(174\) 0 0
\(175\) −726.958 −4.15405
\(176\) − 853.997i − 4.85226i
\(177\) 0 0
\(178\) −75.3887 −0.423532
\(179\) − 132.513i − 0.740298i −0.928972 0.370149i \(-0.879307\pi\)
0.928972 0.370149i \(-0.120693\pi\)
\(180\) 0 0
\(181\) 16.4684 0.0909858 0.0454929 0.998965i \(-0.485514\pi\)
0.0454929 + 0.998965i \(0.485514\pi\)
\(182\) − 473.459i − 2.60143i
\(183\) 0 0
\(184\) −351.039 −1.90782
\(185\) 490.204i 2.64975i
\(186\) 0 0
\(187\) 135.259 0.723308
\(188\) 50.0559i 0.266255i
\(189\) 0 0
\(190\) 911.850 4.79921
\(191\) − 89.1158i − 0.466575i −0.972408 0.233288i \(-0.925052\pi\)
0.972408 0.233288i \(-0.0749483\pi\)
\(192\) 0 0
\(193\) −263.571 −1.36565 −0.682826 0.730581i \(-0.739249\pi\)
−0.682826 + 0.730581i \(0.739249\pi\)
\(194\) − 278.997i − 1.43813i
\(195\) 0 0
\(196\) −867.105 −4.42401
\(197\) − 188.222i − 0.955444i −0.878511 0.477722i \(-0.841463\pi\)
0.878511 0.477722i \(-0.158537\pi\)
\(198\) 0 0
\(199\) 87.7301 0.440855 0.220427 0.975403i \(-0.429255\pi\)
0.220427 + 0.975403i \(0.429255\pi\)
\(200\) − 1727.23i − 8.63617i
\(201\) 0 0
\(202\) −503.974 −2.49492
\(203\) − 206.666i − 1.01806i
\(204\) 0 0
\(205\) 172.208 0.840040
\(206\) − 159.807i − 0.775764i
\(207\) 0 0
\(208\) 651.007 3.12984
\(209\) 354.163i 1.69456i
\(210\) 0 0
\(211\) 42.0224 0.199158 0.0995791 0.995030i \(-0.468250\pi\)
0.0995791 + 0.995030i \(0.468250\pi\)
\(212\) 345.833i 1.63129i
\(213\) 0 0
\(214\) −653.781 −3.05505
\(215\) 296.567i 1.37938i
\(216\) 0 0
\(217\) −436.187 −2.01008
\(218\) 262.955i 1.20622i
\(219\) 0 0
\(220\) 1468.05 6.67295
\(221\) 103.108i 0.466554i
\(222\) 0 0
\(223\) 167.396 0.750656 0.375328 0.926892i \(-0.377530\pi\)
0.375328 + 0.926892i \(0.377530\pi\)
\(224\) − 1417.27i − 6.32710i
\(225\) 0 0
\(226\) 172.553 0.763511
\(227\) − 374.151i − 1.64824i −0.566412 0.824122i \(-0.691669\pi\)
0.566412 0.824122i \(-0.308331\pi\)
\(228\) 0 0
\(229\) −144.506 −0.631030 −0.315515 0.948921i \(-0.602177\pi\)
−0.315515 + 0.948921i \(0.602177\pi\)
\(230\) − 476.731i − 2.07274i
\(231\) 0 0
\(232\) 491.032 2.11652
\(233\) 219.708i 0.942955i 0.881878 + 0.471477i \(0.156279\pi\)
−0.881878 + 0.471477i \(0.843721\pi\)
\(234\) 0 0
\(235\) −43.1580 −0.183651
\(236\) 685.555i 2.90489i
\(237\) 0 0
\(238\) 417.371 1.75366
\(239\) − 434.872i − 1.81955i −0.415103 0.909774i \(-0.636255\pi\)
0.415103 0.909774i \(-0.363745\pi\)
\(240\) 0 0
\(241\) 198.766 0.824756 0.412378 0.911013i \(-0.364698\pi\)
0.412378 + 0.911013i \(0.364698\pi\)
\(242\) 310.451i 1.28286i
\(243\) 0 0
\(244\) −707.387 −2.89913
\(245\) − 747.613i − 3.05148i
\(246\) 0 0
\(247\) −269.980 −1.09304
\(248\) − 1036.37i − 4.17891i
\(249\) 0 0
\(250\) 1432.49 5.72997
\(251\) 343.459i 1.36836i 0.729313 + 0.684180i \(0.239840\pi\)
−0.729313 + 0.684180i \(0.760160\pi\)
\(252\) 0 0
\(253\) 185.162 0.731867
\(254\) − 43.5810i − 0.171579i
\(255\) 0 0
\(256\) 729.632 2.85013
\(257\) 261.700i 1.01829i 0.860681 + 0.509144i \(0.170038\pi\)
−0.860681 + 0.509144i \(0.829962\pi\)
\(258\) 0 0
\(259\) −587.508 −2.26837
\(260\) 1119.10i 4.30423i
\(261\) 0 0
\(262\) −608.506 −2.32254
\(263\) 31.1538i 0.118456i 0.998244 + 0.0592278i \(0.0188638\pi\)
−0.998244 + 0.0592278i \(0.981136\pi\)
\(264\) 0 0
\(265\) −298.175 −1.12519
\(266\) 1092.85i 4.10845i
\(267\) 0 0
\(268\) −814.094 −3.03767
\(269\) 137.914i 0.512693i 0.966585 + 0.256346i \(0.0825188\pi\)
−0.966585 + 0.256346i \(0.917481\pi\)
\(270\) 0 0
\(271\) −194.468 −0.717593 −0.358797 0.933416i \(-0.616813\pi\)
−0.358797 + 0.933416i \(0.616813\pi\)
\(272\) 573.885i 2.10987i
\(273\) 0 0
\(274\) −436.537 −1.59320
\(275\) 911.062i 3.31295i
\(276\) 0 0
\(277\) 205.748 0.742773 0.371386 0.928478i \(-0.378883\pi\)
0.371386 + 0.928478i \(0.378883\pi\)
\(278\) 379.638i 1.36560i
\(279\) 0 0
\(280\) 2875.98 10.2713
\(281\) 94.8530i 0.337555i 0.985654 + 0.168778i \(0.0539819\pi\)
−0.985654 + 0.168778i \(0.946018\pi\)
\(282\) 0 0
\(283\) 193.502 0.683754 0.341877 0.939745i \(-0.388937\pi\)
0.341877 + 0.939745i \(0.388937\pi\)
\(284\) − 1217.41i − 4.28665i
\(285\) 0 0
\(286\) −593.364 −2.07470
\(287\) 206.391i 0.719132i
\(288\) 0 0
\(289\) 198.106 0.685489
\(290\) 666.849i 2.29948i
\(291\) 0 0
\(292\) 533.857 1.82828
\(293\) − 354.078i − 1.20846i −0.796811 0.604229i \(-0.793481\pi\)
0.796811 0.604229i \(-0.206519\pi\)
\(294\) 0 0
\(295\) −591.082 −2.00367
\(296\) − 1395.90i − 4.71589i
\(297\) 0 0
\(298\) 766.490 2.57211
\(299\) 141.150i 0.472074i
\(300\) 0 0
\(301\) −355.434 −1.18084
\(302\) − 1053.17i − 3.48731i
\(303\) 0 0
\(304\) −1502.67 −4.94298
\(305\) − 609.905i − 1.99969i
\(306\) 0 0
\(307\) 151.973 0.495026 0.247513 0.968885i \(-0.420387\pi\)
0.247513 + 0.968885i \(0.420387\pi\)
\(308\) 1759.45i 5.71250i
\(309\) 0 0
\(310\) 1407.45 4.54015
\(311\) − 63.8741i − 0.205383i −0.994713 0.102691i \(-0.967255\pi\)
0.994713 0.102691i \(-0.0327454\pi\)
\(312\) 0 0
\(313\) −37.0453 −0.118356 −0.0591778 0.998247i \(-0.518848\pi\)
−0.0591778 + 0.998247i \(0.518848\pi\)
\(314\) 60.2587i 0.191907i
\(315\) 0 0
\(316\) 692.705 2.19211
\(317\) − 399.581i − 1.26051i −0.776389 0.630254i \(-0.782951\pi\)
0.776389 0.630254i \(-0.217049\pi\)
\(318\) 0 0
\(319\) −259.004 −0.811926
\(320\) 2298.84i 7.18388i
\(321\) 0 0
\(322\) 571.360 1.77441
\(323\) − 237.997i − 0.736832i
\(324\) 0 0
\(325\) −694.507 −2.13695
\(326\) − 859.535i − 2.63661i
\(327\) 0 0
\(328\) −490.379 −1.49506
\(329\) − 51.7246i − 0.157218i
\(330\) 0 0
\(331\) −37.3698 −0.112900 −0.0564498 0.998405i \(-0.517978\pi\)
−0.0564498 + 0.998405i \(0.517978\pi\)
\(332\) − 559.151i − 1.68419i
\(333\) 0 0
\(334\) 0.957984 0.00286821
\(335\) − 701.908i − 2.09525i
\(336\) 0 0
\(337\) −131.462 −0.390094 −0.195047 0.980794i \(-0.562486\pi\)
−0.195047 + 0.980794i \(0.562486\pi\)
\(338\) 201.231i 0.595357i
\(339\) 0 0
\(340\) −986.526 −2.90155
\(341\) 546.653i 1.60309i
\(342\) 0 0
\(343\) 341.314 0.995085
\(344\) − 844.501i − 2.45495i
\(345\) 0 0
\(346\) −687.099 −1.98583
\(347\) − 362.839i − 1.04564i −0.852442 0.522822i \(-0.824879\pi\)
0.852442 0.522822i \(-0.175121\pi\)
\(348\) 0 0
\(349\) −473.272 −1.35608 −0.678040 0.735025i \(-0.737170\pi\)
−0.678040 + 0.735025i \(0.737170\pi\)
\(350\) 2811.28i 8.03224i
\(351\) 0 0
\(352\) −1776.20 −5.04602
\(353\) 264.341i 0.748841i 0.927259 + 0.374420i \(0.122158\pi\)
−0.927259 + 0.374420i \(0.877842\pi\)
\(354\) 0 0
\(355\) 1049.64 2.95674
\(356\) 213.565i 0.599900i
\(357\) 0 0
\(358\) −512.454 −1.43144
\(359\) 54.6728i 0.152292i 0.997097 + 0.0761459i \(0.0242615\pi\)
−0.997097 + 0.0761459i \(0.975739\pi\)
\(360\) 0 0
\(361\) 262.173 0.726242
\(362\) − 63.6865i − 0.175930i
\(363\) 0 0
\(364\) −1341.24 −3.68472
\(365\) 460.289i 1.26107i
\(366\) 0 0
\(367\) −340.491 −0.927769 −0.463885 0.885896i \(-0.653545\pi\)
−0.463885 + 0.885896i \(0.653545\pi\)
\(368\) 785.620i 2.13484i
\(369\) 0 0
\(370\) 1895.71 5.12355
\(371\) − 357.361i − 0.963238i
\(372\) 0 0
\(373\) 168.009 0.450426 0.225213 0.974310i \(-0.427692\pi\)
0.225213 + 0.974310i \(0.427692\pi\)
\(374\) − 523.071i − 1.39859i
\(375\) 0 0
\(376\) 122.896 0.326852
\(377\) − 197.440i − 0.523714i
\(378\) 0 0
\(379\) 29.6118 0.0781315 0.0390657 0.999237i \(-0.487562\pi\)
0.0390657 + 0.999237i \(0.487562\pi\)
\(380\) − 2583.13i − 6.79771i
\(381\) 0 0
\(382\) −344.628 −0.902167
\(383\) − 670.916i − 1.75174i −0.482548 0.875870i \(-0.660289\pi\)
0.482548 0.875870i \(-0.339711\pi\)
\(384\) 0 0
\(385\) −1516.99 −3.94023
\(386\) 1019.28i 2.64062i
\(387\) 0 0
\(388\) −790.354 −2.03700
\(389\) 651.906i 1.67585i 0.545785 + 0.837925i \(0.316232\pi\)
−0.545785 + 0.837925i \(0.683768\pi\)
\(390\) 0 0
\(391\) −124.429 −0.318232
\(392\) 2128.90i 5.43087i
\(393\) 0 0
\(394\) −727.891 −1.84744
\(395\) 597.247i 1.51202i
\(396\) 0 0
\(397\) −434.519 −1.09451 −0.547253 0.836967i \(-0.684326\pi\)
−0.547253 + 0.836967i \(0.684326\pi\)
\(398\) − 339.269i − 0.852434i
\(399\) 0 0
\(400\) −3865.52 −9.66379
\(401\) − 181.485i − 0.452581i −0.974060 0.226290i \(-0.927340\pi\)
0.974060 0.226290i \(-0.0726598\pi\)
\(402\) 0 0
\(403\) −416.716 −1.03403
\(404\) 1427.68i 3.53386i
\(405\) 0 0
\(406\) −799.215 −1.96851
\(407\) 736.295i 1.80908i
\(408\) 0 0
\(409\) 467.124 1.14211 0.571056 0.820911i \(-0.306534\pi\)
0.571056 + 0.820911i \(0.306534\pi\)
\(410\) − 665.962i − 1.62430i
\(411\) 0 0
\(412\) −452.710 −1.09881
\(413\) − 708.409i − 1.71528i
\(414\) 0 0
\(415\) 482.097 1.16168
\(416\) − 1354.00i − 3.25482i
\(417\) 0 0
\(418\) 1369.61 3.27659
\(419\) 457.963i 1.09299i 0.837462 + 0.546495i \(0.184038\pi\)
−0.837462 + 0.546495i \(0.815962\pi\)
\(420\) 0 0
\(421\) −656.507 −1.55940 −0.779699 0.626154i \(-0.784628\pi\)
−0.779699 + 0.626154i \(0.784628\pi\)
\(422\) − 162.508i − 0.385091i
\(423\) 0 0
\(424\) 849.081 2.00255
\(425\) − 612.232i − 1.44055i
\(426\) 0 0
\(427\) 730.968 1.71187
\(428\) 1852.06i 4.32724i
\(429\) 0 0
\(430\) 1146.88 2.66716
\(431\) − 295.176i − 0.684864i −0.939543 0.342432i \(-0.888749\pi\)
0.939543 0.342432i \(-0.111251\pi\)
\(432\) 0 0
\(433\) 359.925 0.831236 0.415618 0.909539i \(-0.363565\pi\)
0.415618 + 0.909539i \(0.363565\pi\)
\(434\) 1686.82i 3.88668i
\(435\) 0 0
\(436\) 744.912 1.70851
\(437\) − 325.806i − 0.745551i
\(438\) 0 0
\(439\) 550.931 1.25497 0.627484 0.778630i \(-0.284085\pi\)
0.627484 + 0.778630i \(0.284085\pi\)
\(440\) − 3604.32i − 8.19165i
\(441\) 0 0
\(442\) 398.740 0.902126
\(443\) 487.185i 1.09974i 0.835250 + 0.549870i \(0.185323\pi\)
−0.835250 + 0.549870i \(0.814677\pi\)
\(444\) 0 0
\(445\) −184.134 −0.413785
\(446\) − 647.353i − 1.45146i
\(447\) 0 0
\(448\) −2755.15 −6.14990
\(449\) − 433.451i − 0.965371i −0.875794 0.482685i \(-0.839662\pi\)
0.875794 0.482685i \(-0.160338\pi\)
\(450\) 0 0
\(451\) 258.660 0.573525
\(452\) − 488.817i − 1.08145i
\(453\) 0 0
\(454\) −1446.91 −3.18704
\(455\) − 1156.41i − 2.54156i
\(456\) 0 0
\(457\) 544.259 1.19094 0.595469 0.803378i \(-0.296966\pi\)
0.595469 + 0.803378i \(0.296966\pi\)
\(458\) 558.831i 1.22016i
\(459\) 0 0
\(460\) −1350.50 −2.93588
\(461\) − 275.026i − 0.596586i −0.954474 0.298293i \(-0.903583\pi\)
0.954474 0.298293i \(-0.0964172\pi\)
\(462\) 0 0
\(463\) 221.762 0.478968 0.239484 0.970900i \(-0.423022\pi\)
0.239484 + 0.970900i \(0.423022\pi\)
\(464\) − 1098.92i − 2.36836i
\(465\) 0 0
\(466\) 849.654 1.82329
\(467\) − 387.031i − 0.828761i −0.910104 0.414380i \(-0.863998\pi\)
0.910104 0.414380i \(-0.136002\pi\)
\(468\) 0 0
\(469\) 841.234 1.79368
\(470\) 166.900i 0.355106i
\(471\) 0 0
\(472\) 1683.16 3.56602
\(473\) 445.448i 0.941751i
\(474\) 0 0
\(475\) 1603.08 3.37490
\(476\) − 1182.35i − 2.48392i
\(477\) 0 0
\(478\) −1681.73 −3.51827
\(479\) 50.1545i 0.104707i 0.998629 + 0.0523533i \(0.0166722\pi\)
−0.998629 + 0.0523533i \(0.983328\pi\)
\(480\) 0 0
\(481\) −561.282 −1.16691
\(482\) − 768.666i − 1.59474i
\(483\) 0 0
\(484\) 879.461 1.81707
\(485\) − 681.439i − 1.40503i
\(486\) 0 0
\(487\) 425.468 0.873650 0.436825 0.899546i \(-0.356103\pi\)
0.436825 + 0.899546i \(0.356103\pi\)
\(488\) 1736.76i 3.55894i
\(489\) 0 0
\(490\) −2891.16 −5.90033
\(491\) 537.662i 1.09503i 0.836794 + 0.547517i \(0.184427\pi\)
−0.836794 + 0.547517i \(0.815573\pi\)
\(492\) 0 0
\(493\) 174.050 0.353043
\(494\) 1044.06i 2.11349i
\(495\) 0 0
\(496\) −2319.37 −4.67616
\(497\) 1257.99i 2.53117i
\(498\) 0 0
\(499\) 215.600 0.432064 0.216032 0.976386i \(-0.430688\pi\)
0.216032 + 0.976386i \(0.430688\pi\)
\(500\) − 4058.03i − 8.11606i
\(501\) 0 0
\(502\) 1328.22 2.64585
\(503\) 231.627i 0.460491i 0.973133 + 0.230245i \(0.0739529\pi\)
−0.973133 + 0.230245i \(0.926047\pi\)
\(504\) 0 0
\(505\) −1230.94 −2.43750
\(506\) − 716.058i − 1.41513i
\(507\) 0 0
\(508\) −123.458 −0.243028
\(509\) − 663.901i − 1.30432i −0.758080 0.652162i \(-0.773862\pi\)
0.758080 0.652162i \(-0.226138\pi\)
\(510\) 0 0
\(511\) −551.654 −1.07956
\(512\) − 1059.98i − 2.07027i
\(513\) 0 0
\(514\) 1012.04 1.96896
\(515\) − 390.324i − 0.757911i
\(516\) 0 0
\(517\) −64.8240 −0.125385
\(518\) 2272.00i 4.38611i
\(519\) 0 0
\(520\) 2747.59 5.28384
\(521\) 629.316i 1.20790i 0.797022 + 0.603950i \(0.206407\pi\)
−0.797022 + 0.603950i \(0.793593\pi\)
\(522\) 0 0
\(523\) 464.271 0.887708 0.443854 0.896099i \(-0.353611\pi\)
0.443854 + 0.896099i \(0.353611\pi\)
\(524\) 1723.80i 3.28970i
\(525\) 0 0
\(526\) 120.478 0.229045
\(527\) − 367.350i − 0.697058i
\(528\) 0 0
\(529\) 358.663 0.678002
\(530\) 1153.10i 2.17566i
\(531\) 0 0
\(532\) 3095.87 5.81931
\(533\) 197.178i 0.369939i
\(534\) 0 0
\(535\) −1596.84 −2.98474
\(536\) 1998.75i 3.72901i
\(537\) 0 0
\(538\) 533.341 0.991340
\(539\) − 1122.93i − 2.08335i
\(540\) 0 0
\(541\) −494.079 −0.913270 −0.456635 0.889654i \(-0.650945\pi\)
−0.456635 + 0.889654i \(0.650945\pi\)
\(542\) 752.044i 1.38753i
\(543\) 0 0
\(544\) 1193.60 2.19412
\(545\) 642.259i 1.17846i
\(546\) 0 0
\(547\) −673.432 −1.23114 −0.615568 0.788084i \(-0.711073\pi\)
−0.615568 + 0.788084i \(0.711073\pi\)
\(548\) 1236.64i 2.25665i
\(549\) 0 0
\(550\) 3523.25 6.40591
\(551\) 455.736i 0.827106i
\(552\) 0 0
\(553\) −715.798 −1.29439
\(554\) − 795.667i − 1.43622i
\(555\) 0 0
\(556\) 1075.46 1.93427
\(557\) − 201.163i − 0.361154i −0.983561 0.180577i \(-0.942203\pi\)
0.983561 0.180577i \(-0.0577965\pi\)
\(558\) 0 0
\(559\) −339.567 −0.607455
\(560\) − 6436.38i − 11.4935i
\(561\) 0 0
\(562\) 366.814 0.652694
\(563\) 848.512i 1.50713i 0.657375 + 0.753563i \(0.271667\pi\)
−0.657375 + 0.753563i \(0.728333\pi\)
\(564\) 0 0
\(565\) 421.456 0.745939
\(566\) − 748.310i − 1.32210i
\(567\) 0 0
\(568\) −2988.96 −5.26225
\(569\) 1066.94i 1.87511i 0.347838 + 0.937555i \(0.386916\pi\)
−0.347838 + 0.937555i \(0.613084\pi\)
\(570\) 0 0
\(571\) −998.217 −1.74819 −0.874095 0.485754i \(-0.838545\pi\)
−0.874095 + 0.485754i \(0.838545\pi\)
\(572\) 1680.91i 2.93865i
\(573\) 0 0
\(574\) 798.152 1.39051
\(575\) − 838.115i − 1.45759i
\(576\) 0 0
\(577\) −433.529 −0.751351 −0.375675 0.926751i \(-0.622589\pi\)
−0.375675 + 0.926751i \(0.622589\pi\)
\(578\) − 766.115i − 1.32546i
\(579\) 0 0
\(580\) 1889.08 3.25703
\(581\) 577.792i 0.994478i
\(582\) 0 0
\(583\) −447.864 −0.768206
\(584\) − 1310.72i − 2.24438i
\(585\) 0 0
\(586\) −1369.29 −2.33667
\(587\) − 643.448i − 1.09616i −0.836425 0.548081i \(-0.815358\pi\)
0.836425 0.548081i \(-0.184642\pi\)
\(588\) 0 0
\(589\) 961.872 1.63306
\(590\) 2285.82i 3.87428i
\(591\) 0 0
\(592\) −3124.00 −5.27703
\(593\) − 398.873i − 0.672635i −0.941749 0.336318i \(-0.890818\pi\)
0.941749 0.336318i \(-0.109182\pi\)
\(594\) 0 0
\(595\) 1019.41 1.71330
\(596\) − 2171.35i − 3.64320i
\(597\) 0 0
\(598\) 545.855 0.912800
\(599\) − 1145.77i − 1.91280i −0.292055 0.956402i \(-0.594339\pi\)
0.292055 0.956402i \(-0.405661\pi\)
\(600\) 0 0
\(601\) 260.499 0.433443 0.216722 0.976233i \(-0.430464\pi\)
0.216722 + 0.976233i \(0.430464\pi\)
\(602\) 1374.53i 2.28327i
\(603\) 0 0
\(604\) −2983.46 −4.93950
\(605\) 758.267i 1.25333i
\(606\) 0 0
\(607\) 938.733 1.54651 0.773256 0.634094i \(-0.218627\pi\)
0.773256 + 0.634094i \(0.218627\pi\)
\(608\) 3125.34i 5.14036i
\(609\) 0 0
\(610\) −2358.62 −3.86659
\(611\) − 49.4157i − 0.0808767i
\(612\) 0 0
\(613\) −608.571 −0.992774 −0.496387 0.868101i \(-0.665340\pi\)
−0.496387 + 0.868101i \(0.665340\pi\)
\(614\) − 587.708i − 0.957179i
\(615\) 0 0
\(616\) 4319.77 7.01261
\(617\) − 5.41005i − 0.00876832i −0.999990 0.00438416i \(-0.998604\pi\)
0.999990 0.00438416i \(-0.00139553\pi\)
\(618\) 0 0
\(619\) 744.538 1.20281 0.601404 0.798945i \(-0.294608\pi\)
0.601404 + 0.798945i \(0.294608\pi\)
\(620\) − 3987.08i − 6.43077i
\(621\) 0 0
\(622\) −247.013 −0.397127
\(623\) − 220.684i − 0.354228i
\(624\) 0 0
\(625\) 1893.39 3.02942
\(626\) 143.261i 0.228852i
\(627\) 0 0
\(628\) 170.704 0.271821
\(629\) − 494.789i − 0.786628i
\(630\) 0 0
\(631\) 566.360 0.897560 0.448780 0.893642i \(-0.351859\pi\)
0.448780 + 0.893642i \(0.351859\pi\)
\(632\) − 1700.72i − 2.69101i
\(633\) 0 0
\(634\) −1545.26 −2.43731
\(635\) − 106.445i − 0.167630i
\(636\) 0 0
\(637\) 856.014 1.34382
\(638\) 1001.62i 1.56993i
\(639\) 0 0
\(640\) 4159.89 6.49983
\(641\) 1034.40i 1.61372i 0.590740 + 0.806862i \(0.298836\pi\)
−0.590740 + 0.806862i \(0.701164\pi\)
\(642\) 0 0
\(643\) 5.47434 0.00851375 0.00425688 0.999991i \(-0.498645\pi\)
0.00425688 + 0.999991i \(0.498645\pi\)
\(644\) − 1618.57i − 2.51331i
\(645\) 0 0
\(646\) −920.379 −1.42473
\(647\) − 639.245i − 0.988015i −0.869458 0.494007i \(-0.835532\pi\)
0.869458 0.494007i \(-0.164468\pi\)
\(648\) 0 0
\(649\) −887.815 −1.36797
\(650\) 2685.79i 4.13198i
\(651\) 0 0
\(652\) −2434.93 −3.73456
\(653\) − 651.138i − 0.997148i −0.866847 0.498574i \(-0.833857\pi\)
0.866847 0.498574i \(-0.166143\pi\)
\(654\) 0 0
\(655\) −1486.25 −2.26909
\(656\) 1097.46i 1.67296i
\(657\) 0 0
\(658\) −200.029 −0.303995
\(659\) − 296.523i − 0.449958i −0.974364 0.224979i \(-0.927769\pi\)
0.974364 0.224979i \(-0.0722314\pi\)
\(660\) 0 0
\(661\) 737.931 1.11639 0.558193 0.829711i \(-0.311495\pi\)
0.558193 + 0.829711i \(0.311495\pi\)
\(662\) 144.516i 0.218302i
\(663\) 0 0
\(664\) −1372.82 −2.06750
\(665\) 2669.24i 4.01390i
\(666\) 0 0
\(667\) 238.266 0.357221
\(668\) − 2.71382i − 0.00406260i
\(669\) 0 0
\(670\) −2714.41 −4.05136
\(671\) − 916.088i − 1.36526i
\(672\) 0 0
\(673\) −475.306 −0.706250 −0.353125 0.935576i \(-0.614881\pi\)
−0.353125 + 0.935576i \(0.614881\pi\)
\(674\) 508.388i 0.754284i
\(675\) 0 0
\(676\) 570.055 0.843277
\(677\) − 881.795i − 1.30250i −0.758862 0.651252i \(-0.774244\pi\)
0.758862 0.651252i \(-0.225756\pi\)
\(678\) 0 0
\(679\) 816.702 1.20280
\(680\) 2422.10i 3.56191i
\(681\) 0 0
\(682\) 2114.01 3.09972
\(683\) − 161.438i − 0.236366i −0.992992 0.118183i \(-0.962293\pi\)
0.992992 0.118183i \(-0.0377070\pi\)
\(684\) 0 0
\(685\) −1066.23 −1.55653
\(686\) − 1319.93i − 1.92409i
\(687\) 0 0
\(688\) −1889.98 −2.74706
\(689\) − 341.409i − 0.495514i
\(690\) 0 0
\(691\) 366.744 0.530744 0.265372 0.964146i \(-0.414505\pi\)
0.265372 + 0.964146i \(0.414505\pi\)
\(692\) 1946.44i 2.81278i
\(693\) 0 0
\(694\) −1403.17 −2.02185
\(695\) 927.252i 1.33418i
\(696\) 0 0
\(697\) −173.819 −0.249382
\(698\) 1830.23i 2.62211i
\(699\) 0 0
\(700\) 7963.94 11.3771
\(701\) − 140.809i − 0.200869i −0.994944 0.100435i \(-0.967977\pi\)
0.994944 0.100435i \(-0.0320233\pi\)
\(702\) 0 0
\(703\) 1295.56 1.84290
\(704\) 3452.90i 4.90469i
\(705\) 0 0
\(706\) 1022.26 1.44795
\(707\) − 1475.27i − 2.08667i
\(708\) 0 0
\(709\) 815.582 1.15033 0.575164 0.818038i \(-0.304938\pi\)
0.575164 + 0.818038i \(0.304938\pi\)
\(710\) − 4059.17i − 5.71714i
\(711\) 0 0
\(712\) 524.340 0.736432
\(713\) − 502.883i − 0.705306i
\(714\) 0 0
\(715\) −1449.27 −2.02695
\(716\) 1451.70i 2.02752i
\(717\) 0 0
\(718\) 211.430 0.294471
\(719\) − 1073.20i − 1.49263i −0.665593 0.746315i \(-0.731821\pi\)
0.665593 0.746315i \(-0.268179\pi\)
\(720\) 0 0
\(721\) 467.802 0.648823
\(722\) − 1013.87i − 1.40426i
\(723\) 0 0
\(724\) −180.414 −0.249191
\(725\) 1172.35i 1.61704i
\(726\) 0 0
\(727\) −511.757 −0.703930 −0.351965 0.936013i \(-0.614486\pi\)
−0.351965 + 0.936013i \(0.614486\pi\)
\(728\) 3292.98i 4.52332i
\(729\) 0 0
\(730\) 1780.02 2.43839
\(731\) − 299.340i − 0.409494i
\(732\) 0 0
\(733\) 630.537 0.860214 0.430107 0.902778i \(-0.358476\pi\)
0.430107 + 0.902778i \(0.358476\pi\)
\(734\) 1316.74i 1.79393i
\(735\) 0 0
\(736\) 1633.98 2.22008
\(737\) − 1054.28i − 1.43050i
\(738\) 0 0
\(739\) 253.573 0.343130 0.171565 0.985173i \(-0.445118\pi\)
0.171565 + 0.985173i \(0.445118\pi\)
\(740\) − 5370.26i − 7.25711i
\(741\) 0 0
\(742\) −1381.98 −1.86251
\(743\) 1027.89i 1.38343i 0.722172 + 0.691714i \(0.243144\pi\)
−0.722172 + 0.691714i \(0.756856\pi\)
\(744\) 0 0
\(745\) 1872.12 2.51292
\(746\) − 649.723i − 0.870942i
\(747\) 0 0
\(748\) −1481.78 −1.98099
\(749\) − 1913.80i − 2.55514i
\(750\) 0 0
\(751\) −1089.29 −1.45045 −0.725224 0.688513i \(-0.758264\pi\)
−0.725224 + 0.688513i \(0.758264\pi\)
\(752\) − 275.040i − 0.365744i
\(753\) 0 0
\(754\) −763.539 −1.01265
\(755\) − 2572.32i − 3.40705i
\(756\) 0 0
\(757\) 1244.71 1.64427 0.822136 0.569291i \(-0.192782\pi\)
0.822136 + 0.569291i \(0.192782\pi\)
\(758\) − 114.514i − 0.151075i
\(759\) 0 0
\(760\) −6342.05 −8.34480
\(761\) − 1341.43i − 1.76272i −0.472446 0.881359i \(-0.656629\pi\)
0.472446 0.881359i \(-0.343371\pi\)
\(762\) 0 0
\(763\) −769.744 −1.00884
\(764\) 976.277i 1.27785i
\(765\) 0 0
\(766\) −2594.56 −3.38715
\(767\) − 676.786i − 0.882380i
\(768\) 0 0
\(769\) 381.351 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(770\) 5866.48i 7.61880i
\(771\) 0 0
\(772\) 2887.46 3.74023
\(773\) − 419.868i − 0.543167i −0.962415 0.271584i \(-0.912453\pi\)
0.962415 0.271584i \(-0.0875474\pi\)
\(774\) 0 0
\(775\) 2474.36 3.19272
\(776\) 1940.46i 2.50060i
\(777\) 0 0
\(778\) 2521.04 3.24042
\(779\) − 455.129i − 0.584248i
\(780\) 0 0
\(781\) 1576.58 2.01867
\(782\) 481.190i 0.615332i
\(783\) 0 0
\(784\) 4764.44 6.07709
\(785\) 147.180i 0.187490i
\(786\) 0 0
\(787\) −1063.32 −1.35110 −0.675551 0.737313i \(-0.736094\pi\)
−0.675551 + 0.737313i \(0.736094\pi\)
\(788\) 2062.00i 2.61676i
\(789\) 0 0
\(790\) 2309.67 2.92363
\(791\) 505.113i 0.638575i
\(792\) 0 0
\(793\) 698.338 0.880628
\(794\) 1680.37i 2.11633i
\(795\) 0 0
\(796\) −961.096 −1.20741
\(797\) 324.161i 0.406727i 0.979103 + 0.203363i \(0.0651873\pi\)
−0.979103 + 0.203363i \(0.934813\pi\)
\(798\) 0 0
\(799\) 43.5616 0.0545202
\(800\) 8039.74i 10.0497i
\(801\) 0 0
\(802\) −701.836 −0.875108
\(803\) 691.362i 0.860973i
\(804\) 0 0
\(805\) 1395.53 1.73357
\(806\) 1611.52i 1.99940i
\(807\) 0 0
\(808\) 3505.21 4.33814
\(809\) − 149.456i − 0.184741i −0.995725 0.0923707i \(-0.970556\pi\)
0.995725 0.0923707i \(-0.0294445\pi\)
\(810\) 0 0
\(811\) 873.810 1.07745 0.538724 0.842483i \(-0.318907\pi\)
0.538724 + 0.842483i \(0.318907\pi\)
\(812\) 2264.05i 2.78824i
\(813\) 0 0
\(814\) 2847.39 3.49803
\(815\) − 2099.38i − 2.57593i
\(816\) 0 0
\(817\) 783.796 0.959359
\(818\) − 1806.46i − 2.20838i
\(819\) 0 0
\(820\) −1886.57 −2.30069
\(821\) 671.265i 0.817619i 0.912620 + 0.408809i \(0.134056\pi\)
−0.912620 + 0.408809i \(0.865944\pi\)
\(822\) 0 0
\(823\) −1205.43 −1.46467 −0.732337 0.680942i \(-0.761571\pi\)
−0.732337 + 0.680942i \(0.761571\pi\)
\(824\) 1111.48i 1.34889i
\(825\) 0 0
\(826\) −2739.55 −3.31665
\(827\) − 380.902i − 0.460583i −0.973122 0.230291i \(-0.926032\pi\)
0.973122 0.230291i \(-0.0739679\pi\)
\(828\) 0 0
\(829\) 1532.03 1.84804 0.924020 0.382343i \(-0.124883\pi\)
0.924020 + 0.382343i \(0.124883\pi\)
\(830\) − 1864.36i − 2.24622i
\(831\) 0 0
\(832\) −2632.16 −3.16366
\(833\) 754.606i 0.905889i
\(834\) 0 0
\(835\) 2.33984 0.00280220
\(836\) − 3879.91i − 4.64104i
\(837\) 0 0
\(838\) 1771.03 2.11340
\(839\) 270.488i 0.322394i 0.986922 + 0.161197i \(0.0515354\pi\)
−0.986922 + 0.161197i \(0.948465\pi\)
\(840\) 0 0
\(841\) 507.714 0.603703
\(842\) 2538.83i 3.01524i
\(843\) 0 0
\(844\) −460.361 −0.545452
\(845\) 491.498i 0.581655i
\(846\) 0 0
\(847\) −908.779 −1.07294
\(848\) − 1900.23i − 2.24083i
\(849\) 0 0
\(850\) −2367.62 −2.78543
\(851\) − 677.342i − 0.795936i
\(852\) 0 0
\(853\) −929.016 −1.08912 −0.544558 0.838723i \(-0.683303\pi\)
−0.544558 + 0.838723i \(0.683303\pi\)
\(854\) − 2826.79i − 3.31006i
\(855\) 0 0
\(856\) 4547.14 5.31208
\(857\) 207.368i 0.241969i 0.992654 + 0.120985i \(0.0386052\pi\)
−0.992654 + 0.120985i \(0.961395\pi\)
\(858\) 0 0
\(859\) −493.994 −0.575080 −0.287540 0.957769i \(-0.592837\pi\)
−0.287540 + 0.957769i \(0.592837\pi\)
\(860\) − 3248.93i − 3.77783i
\(861\) 0 0
\(862\) −1141.50 −1.32425
\(863\) − 294.120i − 0.340811i −0.985374 0.170405i \(-0.945492\pi\)
0.985374 0.170405i \(-0.0545077\pi\)
\(864\) 0 0
\(865\) −1678.21 −1.94013
\(866\) − 1391.90i − 1.60727i
\(867\) 0 0
\(868\) 4778.50 5.50518
\(869\) 897.075i 1.03231i
\(870\) 0 0
\(871\) 803.681 0.922711
\(872\) − 1828.89i − 2.09735i
\(873\) 0 0
\(874\) −1259.95 −1.44159
\(875\) 4193.31i 4.79236i
\(876\) 0 0
\(877\) −1303.57 −1.48640 −0.743198 0.669071i \(-0.766692\pi\)
−0.743198 + 0.669071i \(0.766692\pi\)
\(878\) − 2130.55i − 2.42660i
\(879\) 0 0
\(880\) −8066.41 −9.16637
\(881\) − 1042.89i − 1.18376i −0.806026 0.591880i \(-0.798386\pi\)
0.806026 0.591880i \(-0.201614\pi\)
\(882\) 0 0
\(883\) −895.269 −1.01390 −0.506948 0.861977i \(-0.669226\pi\)
−0.506948 + 0.861977i \(0.669226\pi\)
\(884\) − 1129.57i − 1.27779i
\(885\) 0 0
\(886\) 1884.04 2.12645
\(887\) − 490.009i − 0.552434i −0.961095 0.276217i \(-0.910919\pi\)
0.961095 0.276217i \(-0.0890808\pi\)
\(888\) 0 0
\(889\) 127.574 0.143503
\(890\) 712.082i 0.800092i
\(891\) 0 0
\(892\) −1833.85 −2.05589
\(893\) 114.062i 0.127729i
\(894\) 0 0
\(895\) −1251.65 −1.39849
\(896\) 4985.61i 5.56430i
\(897\) 0 0
\(898\) −1676.24 −1.86663
\(899\) 703.431i 0.782459i
\(900\) 0 0
\(901\) 300.964 0.334033
\(902\) − 1000.29i − 1.10896i
\(903\) 0 0
\(904\) −1200.13 −1.32758
\(905\) − 155.552i − 0.171881i
\(906\) 0 0
\(907\) −1592.09 −1.75534 −0.877668 0.479270i \(-0.840902\pi\)
−0.877668 + 0.479270i \(0.840902\pi\)
\(908\) 4098.88i 4.51419i
\(909\) 0 0
\(910\) −4472.05 −4.91434
\(911\) − 804.999i − 0.883643i −0.897103 0.441821i \(-0.854332\pi\)
0.897103 0.441821i \(-0.145668\pi\)
\(912\) 0 0
\(913\) 724.118 0.793120
\(914\) − 2104.75i − 2.30279i
\(915\) 0 0
\(916\) 1583.08 1.72826
\(917\) − 1781.27i − 1.94250i
\(918\) 0 0
\(919\) 828.169 0.901164 0.450582 0.892735i \(-0.351217\pi\)
0.450582 + 0.892735i \(0.351217\pi\)
\(920\) 3315.73i 3.60406i
\(921\) 0 0
\(922\) −1063.58 −1.15355
\(923\) 1201.84i 1.30210i
\(924\) 0 0
\(925\) 3332.75 3.60297
\(926\) − 857.597i − 0.926131i
\(927\) 0 0
\(928\) −2285.60 −2.46294
\(929\) − 229.494i − 0.247033i −0.992342 0.123517i \(-0.960583\pi\)
0.992342 0.123517i \(-0.0394173\pi\)
\(930\) 0 0
\(931\) −1975.87 −2.12231
\(932\) − 2406.94i − 2.58255i
\(933\) 0 0
\(934\) −1496.72 −1.60249
\(935\) − 1277.58i − 1.36640i
\(936\) 0 0
\(937\) −1193.95 −1.27423 −0.637115 0.770769i \(-0.719872\pi\)
−0.637115 + 0.770769i \(0.719872\pi\)
\(938\) − 3253.21i − 3.46824i
\(939\) 0 0
\(940\) 472.802 0.502981
\(941\) 1658.41i 1.76239i 0.472752 + 0.881196i \(0.343261\pi\)
−0.472752 + 0.881196i \(0.656739\pi\)
\(942\) 0 0
\(943\) −237.949 −0.252332
\(944\) − 3766.88i − 3.99034i
\(945\) 0 0
\(946\) 1722.63 1.82096
\(947\) 106.033i 0.111968i 0.998432 + 0.0559839i \(0.0178295\pi\)
−0.998432 + 0.0559839i \(0.982170\pi\)
\(948\) 0 0
\(949\) −527.028 −0.555351
\(950\) − 6199.39i − 6.52568i
\(951\) 0 0
\(952\) −2902.88 −3.04924
\(953\) − 1530.38i − 1.60586i −0.596073 0.802930i \(-0.703273\pi\)
0.596073 0.802930i \(-0.296727\pi\)
\(954\) 0 0
\(955\) −841.741 −0.881404
\(956\) 4764.09i 4.98336i
\(957\) 0 0
\(958\) 193.957 0.202460
\(959\) − 1277.87i − 1.33250i
\(960\) 0 0
\(961\) 523.656 0.544907
\(962\) 2170.58i 2.25632i
\(963\) 0 0
\(964\) −2177.51 −2.25883
\(965\) 2489.55i 2.57984i
\(966\) 0 0
\(967\) 886.692 0.916951 0.458475 0.888707i \(-0.348396\pi\)
0.458475 + 0.888707i \(0.348396\pi\)
\(968\) − 2159.24i − 2.23062i
\(969\) 0 0
\(970\) −2635.25 −2.71676
\(971\) 819.330i 0.843800i 0.906642 + 0.421900i \(0.138637\pi\)
−0.906642 + 0.421900i \(0.861363\pi\)
\(972\) 0 0
\(973\) −1111.31 −1.14215
\(974\) − 1645.36i − 1.68928i
\(975\) 0 0
\(976\) 3886.84 3.98242
\(977\) 841.090i 0.860890i 0.902617 + 0.430445i \(0.141643\pi\)
−0.902617 + 0.430445i \(0.858357\pi\)
\(978\) 0 0
\(979\) −276.573 −0.282505
\(980\) 8190.22i 8.35736i
\(981\) 0 0
\(982\) 2079.24 2.11735
\(983\) 1409.80i 1.43418i 0.696980 + 0.717091i \(0.254527\pi\)
−0.696980 + 0.717091i \(0.745473\pi\)
\(984\) 0 0
\(985\) −1777.85 −1.80492
\(986\) − 673.086i − 0.682643i
\(987\) 0 0
\(988\) 2957.67 2.99360
\(989\) − 409.782i − 0.414340i
\(990\) 0 0
\(991\) −1664.02 −1.67913 −0.839567 0.543256i \(-0.817191\pi\)
−0.839567 + 0.543256i \(0.817191\pi\)
\(992\) 4823.98i 4.86288i
\(993\) 0 0
\(994\) 4864.90 4.89426
\(995\) − 828.652i − 0.832816i
\(996\) 0 0
\(997\) −193.171 −0.193752 −0.0968761 0.995296i \(-0.530885\pi\)
−0.0968761 + 0.995296i \(0.530885\pi\)
\(998\) − 833.765i − 0.835436i
\(999\) 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.890.2 84
3.2 odd 2 inner 1143.3.b.a.890.83 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.2 84 1.1 even 1 trivial
1143.3.b.a.890.83 yes 84 3.2 odd 2 inner