Properties

Label 1143.3.b.a.890.3
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 890.3
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.82

$q$-expansion

\(f(q)\) \(=\) \(q-3.83653i q^{2} -10.7190 q^{4} +7.45843i q^{5} +0.844054 q^{7} +25.7776i q^{8} +O(q^{10})\) \(q-3.83653i q^{2} -10.7190 q^{4} +7.45843i q^{5} +0.844054 q^{7} +25.7776i q^{8} +28.6145 q^{10} +12.8528i q^{11} -4.71633 q^{13} -3.23824i q^{14} +56.0208 q^{16} +5.69167i q^{17} -12.0832 q^{19} -79.9469i q^{20} +49.3100 q^{22} -8.89247i q^{23} -30.6282 q^{25} +18.0944i q^{26} -9.04740 q^{28} -27.8216i q^{29} -26.6004 q^{31} -111.815i q^{32} +21.8363 q^{34} +6.29532i q^{35} -8.01864 q^{37} +46.3577i q^{38} -192.261 q^{40} -74.7809i q^{41} -6.24122 q^{43} -137.769i q^{44} -34.1163 q^{46} -31.6811i q^{47} -48.2876 q^{49} +117.506i q^{50} +50.5543 q^{52} +23.4701i q^{53} -95.8615 q^{55} +21.7577i q^{56} -106.739 q^{58} -82.7889i q^{59} -81.7300 q^{61} +102.053i q^{62} -204.900 q^{64} -35.1764i q^{65} +94.4999 q^{67} -61.0090i q^{68} +24.1522 q^{70} +137.250i q^{71} +104.478 q^{73} +30.7638i q^{74} +129.520 q^{76} +10.8484i q^{77} +44.4068 q^{79} +417.828i q^{80} -286.899 q^{82} -68.1282i q^{83} -42.4509 q^{85} +23.9447i q^{86} -331.314 q^{88} +38.7934i q^{89} -3.98084 q^{91} +95.3183i q^{92} -121.546 q^{94} -90.1219i q^{95} +101.020 q^{97} +185.257i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.83653i − 1.91827i −0.282954 0.959133i \(-0.591314\pi\)
0.282954 0.959133i \(-0.408686\pi\)
\(3\) 0 0
\(4\) −10.7190 −2.67975
\(5\) 7.45843i 1.49169i 0.666121 + 0.745843i \(0.267953\pi\)
−0.666121 + 0.745843i \(0.732047\pi\)
\(6\) 0 0
\(7\) 0.844054 0.120579 0.0602895 0.998181i \(-0.480798\pi\)
0.0602895 + 0.998181i \(0.480798\pi\)
\(8\) 25.7776i 3.22220i
\(9\) 0 0
\(10\) 28.6145 2.86145
\(11\) 12.8528i 1.16843i 0.811598 + 0.584216i \(0.198598\pi\)
−0.811598 + 0.584216i \(0.801402\pi\)
\(12\) 0 0
\(13\) −4.71633 −0.362795 −0.181397 0.983410i \(-0.558062\pi\)
−0.181397 + 0.983410i \(0.558062\pi\)
\(14\) − 3.23824i − 0.231303i
\(15\) 0 0
\(16\) 56.0208 3.50130
\(17\) 5.69167i 0.334804i 0.985889 + 0.167402i \(0.0535378\pi\)
−0.985889 + 0.167402i \(0.946462\pi\)
\(18\) 0 0
\(19\) −12.0832 −0.635959 −0.317979 0.948098i \(-0.603004\pi\)
−0.317979 + 0.948098i \(0.603004\pi\)
\(20\) − 79.9469i − 3.99734i
\(21\) 0 0
\(22\) 49.3100 2.24137
\(23\) − 8.89247i − 0.386629i −0.981137 0.193315i \(-0.938076\pi\)
0.981137 0.193315i \(-0.0619238\pi\)
\(24\) 0 0
\(25\) −30.6282 −1.22513
\(26\) 18.0944i 0.695937i
\(27\) 0 0
\(28\) −9.04740 −0.323122
\(29\) − 27.8216i − 0.959366i −0.877442 0.479683i \(-0.840752\pi\)
0.877442 0.479683i \(-0.159248\pi\)
\(30\) 0 0
\(31\) −26.6004 −0.858078 −0.429039 0.903286i \(-0.641148\pi\)
−0.429039 + 0.903286i \(0.641148\pi\)
\(32\) − 111.815i − 3.49423i
\(33\) 0 0
\(34\) 21.8363 0.642244
\(35\) 6.29532i 0.179866i
\(36\) 0 0
\(37\) −8.01864 −0.216720 −0.108360 0.994112i \(-0.534560\pi\)
−0.108360 + 0.994112i \(0.534560\pi\)
\(38\) 46.3577i 1.21994i
\(39\) 0 0
\(40\) −192.261 −4.80652
\(41\) − 74.7809i − 1.82392i −0.410274 0.911962i \(-0.634567\pi\)
0.410274 0.911962i \(-0.365433\pi\)
\(42\) 0 0
\(43\) −6.24122 −0.145145 −0.0725723 0.997363i \(-0.523121\pi\)
−0.0725723 + 0.997363i \(0.523121\pi\)
\(44\) − 137.769i − 3.13111i
\(45\) 0 0
\(46\) −34.1163 −0.741658
\(47\) − 31.6811i − 0.674066i −0.941493 0.337033i \(-0.890577\pi\)
0.941493 0.337033i \(-0.109423\pi\)
\(48\) 0 0
\(49\) −48.2876 −0.985461
\(50\) 117.506i 2.35012i
\(51\) 0 0
\(52\) 50.5543 0.972198
\(53\) 23.4701i 0.442833i 0.975179 + 0.221416i \(0.0710680\pi\)
−0.975179 + 0.221416i \(0.928932\pi\)
\(54\) 0 0
\(55\) −95.8615 −1.74294
\(56\) 21.7577i 0.388531i
\(57\) 0 0
\(58\) −106.739 −1.84032
\(59\) − 82.7889i − 1.40320i −0.712570 0.701601i \(-0.752469\pi\)
0.712570 0.701601i \(-0.247531\pi\)
\(60\) 0 0
\(61\) −81.7300 −1.33984 −0.669918 0.742435i \(-0.733671\pi\)
−0.669918 + 0.742435i \(0.733671\pi\)
\(62\) 102.053i 1.64602i
\(63\) 0 0
\(64\) −204.900 −3.20156
\(65\) − 35.1764i − 0.541176i
\(66\) 0 0
\(67\) 94.4999 1.41045 0.705223 0.708985i \(-0.250847\pi\)
0.705223 + 0.708985i \(0.250847\pi\)
\(68\) − 61.0090i − 0.897191i
\(69\) 0 0
\(70\) 24.1522 0.345031
\(71\) 137.250i 1.93310i 0.256483 + 0.966549i \(0.417436\pi\)
−0.256483 + 0.966549i \(0.582564\pi\)
\(72\) 0 0
\(73\) 104.478 1.43121 0.715605 0.698505i \(-0.246151\pi\)
0.715605 + 0.698505i \(0.246151\pi\)
\(74\) 30.7638i 0.415727i
\(75\) 0 0
\(76\) 129.520 1.70421
\(77\) 10.8484i 0.140889i
\(78\) 0 0
\(79\) 44.4068 0.562112 0.281056 0.959691i \(-0.409315\pi\)
0.281056 + 0.959691i \(0.409315\pi\)
\(80\) 417.828i 5.22284i
\(81\) 0 0
\(82\) −286.899 −3.49877
\(83\) − 68.1282i − 0.820822i −0.911901 0.410411i \(-0.865385\pi\)
0.911901 0.410411i \(-0.134615\pi\)
\(84\) 0 0
\(85\) −42.4509 −0.499423
\(86\) 23.9447i 0.278426i
\(87\) 0 0
\(88\) −331.314 −3.76493
\(89\) 38.7934i 0.435881i 0.975962 + 0.217941i \(0.0699339\pi\)
−0.975962 + 0.217941i \(0.930066\pi\)
\(90\) 0 0
\(91\) −3.98084 −0.0437454
\(92\) 95.3183i 1.03607i
\(93\) 0 0
\(94\) −121.546 −1.29304
\(95\) − 90.1219i − 0.948651i
\(96\) 0 0
\(97\) 101.020 1.04144 0.520722 0.853726i \(-0.325663\pi\)
0.520722 + 0.853726i \(0.325663\pi\)
\(98\) 185.257i 1.89038i
\(99\) 0 0
\(100\) 328.304 3.28304
\(101\) − 66.5209i − 0.658622i −0.944221 0.329311i \(-0.893183\pi\)
0.944221 0.329311i \(-0.106817\pi\)
\(102\) 0 0
\(103\) −156.271 −1.51719 −0.758595 0.651562i \(-0.774114\pi\)
−0.758595 + 0.651562i \(0.774114\pi\)
\(104\) − 121.576i − 1.16900i
\(105\) 0 0
\(106\) 90.0440 0.849471
\(107\) − 133.785i − 1.25033i −0.780493 0.625164i \(-0.785032\pi\)
0.780493 0.625164i \(-0.214968\pi\)
\(108\) 0 0
\(109\) −134.500 −1.23395 −0.616974 0.786983i \(-0.711642\pi\)
−0.616974 + 0.786983i \(0.711642\pi\)
\(110\) 367.776i 3.34342i
\(111\) 0 0
\(112\) 47.2846 0.422184
\(113\) − 137.454i − 1.21641i −0.793781 0.608203i \(-0.791891\pi\)
0.793781 0.608203i \(-0.208109\pi\)
\(114\) 0 0
\(115\) 66.3239 0.576729
\(116\) 298.220i 2.57086i
\(117\) 0 0
\(118\) −317.622 −2.69171
\(119\) 4.80408i 0.0403704i
\(120\) 0 0
\(121\) −44.1934 −0.365235
\(122\) 313.560i 2.57016i
\(123\) 0 0
\(124\) 285.130 2.29943
\(125\) − 41.9777i − 0.335822i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) 338.843i 2.64721i
\(129\) 0 0
\(130\) −134.956 −1.03812
\(131\) 193.561i 1.47756i 0.673946 + 0.738781i \(0.264598\pi\)
−0.673946 + 0.738781i \(0.735402\pi\)
\(132\) 0 0
\(133\) −10.1989 −0.0766834
\(134\) − 362.552i − 2.70561i
\(135\) 0 0
\(136\) −146.718 −1.07881
\(137\) 117.539i 0.857947i 0.903317 + 0.428973i \(0.141125\pi\)
−0.903317 + 0.428973i \(0.858875\pi\)
\(138\) 0 0
\(139\) −139.824 −1.00593 −0.502964 0.864307i \(-0.667757\pi\)
−0.502964 + 0.864307i \(0.667757\pi\)
\(140\) − 67.4795i − 0.481996i
\(141\) 0 0
\(142\) 526.564 3.70820
\(143\) − 60.6179i − 0.423901i
\(144\) 0 0
\(145\) 207.506 1.43107
\(146\) − 400.835i − 2.74544i
\(147\) 0 0
\(148\) 85.9518 0.580755
\(149\) 6.83029i 0.0458409i 0.999737 + 0.0229204i \(0.00729644\pi\)
−0.999737 + 0.0229204i \(0.992704\pi\)
\(150\) 0 0
\(151\) 142.106 0.941096 0.470548 0.882374i \(-0.344056\pi\)
0.470548 + 0.882374i \(0.344056\pi\)
\(152\) − 311.477i − 2.04919i
\(153\) 0 0
\(154\) 41.6203 0.270262
\(155\) − 198.397i − 1.27998i
\(156\) 0 0
\(157\) 129.637 0.825717 0.412858 0.910795i \(-0.364530\pi\)
0.412858 + 0.910795i \(0.364530\pi\)
\(158\) − 170.368i − 1.07828i
\(159\) 0 0
\(160\) 833.966 5.21229
\(161\) − 7.50572i − 0.0466194i
\(162\) 0 0
\(163\) −28.9706 −0.177734 −0.0888670 0.996044i \(-0.528325\pi\)
−0.0888670 + 0.996044i \(0.528325\pi\)
\(164\) 801.576i 4.88766i
\(165\) 0 0
\(166\) −261.376 −1.57456
\(167\) − 139.833i − 0.837324i −0.908142 0.418662i \(-0.862499\pi\)
0.908142 0.418662i \(-0.137501\pi\)
\(168\) 0 0
\(169\) −146.756 −0.868380
\(170\) 162.864i 0.958026i
\(171\) 0 0
\(172\) 66.8996 0.388951
\(173\) − 152.490i − 0.881443i −0.897644 0.440721i \(-0.854723\pi\)
0.897644 0.440721i \(-0.145277\pi\)
\(174\) 0 0
\(175\) −25.8519 −0.147725
\(176\) 720.022i 4.09104i
\(177\) 0 0
\(178\) 148.832 0.836137
\(179\) − 301.532i − 1.68454i −0.539058 0.842268i \(-0.681220\pi\)
0.539058 0.842268i \(-0.318780\pi\)
\(180\) 0 0
\(181\) −256.150 −1.41519 −0.707597 0.706617i \(-0.750221\pi\)
−0.707597 + 0.706617i \(0.750221\pi\)
\(182\) 15.2726i 0.0839154i
\(183\) 0 0
\(184\) 229.227 1.24580
\(185\) − 59.8065i − 0.323278i
\(186\) 0 0
\(187\) −73.1537 −0.391196
\(188\) 339.589i 1.80633i
\(189\) 0 0
\(190\) −345.756 −1.81977
\(191\) − 186.541i − 0.976656i −0.872660 0.488328i \(-0.837607\pi\)
0.872660 0.488328i \(-0.162393\pi\)
\(192\) 0 0
\(193\) −2.32954 −0.0120701 −0.00603507 0.999982i \(-0.501921\pi\)
−0.00603507 + 0.999982i \(0.501921\pi\)
\(194\) − 387.567i − 1.99777i
\(195\) 0 0
\(196\) 517.594 2.64079
\(197\) − 220.708i − 1.12035i −0.828376 0.560173i \(-0.810735\pi\)
0.828376 0.560173i \(-0.189265\pi\)
\(198\) 0 0
\(199\) −266.198 −1.33768 −0.668840 0.743406i \(-0.733209\pi\)
−0.668840 + 0.743406i \(0.733209\pi\)
\(200\) − 789.523i − 3.94762i
\(201\) 0 0
\(202\) −255.210 −1.26341
\(203\) − 23.4829i − 0.115679i
\(204\) 0 0
\(205\) 557.748 2.72072
\(206\) 599.537i 2.91038i
\(207\) 0 0
\(208\) −264.213 −1.27025
\(209\) − 155.303i − 0.743075i
\(210\) 0 0
\(211\) −196.024 −0.929022 −0.464511 0.885567i \(-0.653770\pi\)
−0.464511 + 0.885567i \(0.653770\pi\)
\(212\) − 251.576i − 1.18668i
\(213\) 0 0
\(214\) −513.271 −2.39846
\(215\) − 46.5497i − 0.216510i
\(216\) 0 0
\(217\) −22.4522 −0.103466
\(218\) 516.015i 2.36704i
\(219\) 0 0
\(220\) 1027.54 4.67063
\(221\) − 26.8438i − 0.121465i
\(222\) 0 0
\(223\) 152.579 0.684211 0.342105 0.939662i \(-0.388860\pi\)
0.342105 + 0.939662i \(0.388860\pi\)
\(224\) − 94.3780i − 0.421331i
\(225\) 0 0
\(226\) −527.347 −2.33339
\(227\) − 171.350i − 0.754846i −0.926041 0.377423i \(-0.876810\pi\)
0.926041 0.377423i \(-0.123190\pi\)
\(228\) 0 0
\(229\) 267.113 1.16643 0.583217 0.812316i \(-0.301794\pi\)
0.583217 + 0.812316i \(0.301794\pi\)
\(230\) − 254.454i − 1.10632i
\(231\) 0 0
\(232\) 717.175 3.09127
\(233\) 321.626i 1.38037i 0.723633 + 0.690185i \(0.242471\pi\)
−0.723633 + 0.690185i \(0.757529\pi\)
\(234\) 0 0
\(235\) 236.291 1.00550
\(236\) 887.413i 3.76023i
\(237\) 0 0
\(238\) 18.4310 0.0774412
\(239\) 387.078i 1.61957i 0.586725 + 0.809786i \(0.300417\pi\)
−0.586725 + 0.809786i \(0.699583\pi\)
\(240\) 0 0
\(241\) −217.992 −0.904533 −0.452266 0.891883i \(-0.649384\pi\)
−0.452266 + 0.891883i \(0.649384\pi\)
\(242\) 169.550i 0.700618i
\(243\) 0 0
\(244\) 876.063 3.59042
\(245\) − 360.150i − 1.47000i
\(246\) 0 0
\(247\) 56.9885 0.230722
\(248\) − 685.696i − 2.76490i
\(249\) 0 0
\(250\) −161.049 −0.644196
\(251\) 341.604i 1.36097i 0.732762 + 0.680485i \(0.238231\pi\)
−0.732762 + 0.680485i \(0.761769\pi\)
\(252\) 0 0
\(253\) 114.293 0.451750
\(254\) 43.2355i 0.170219i
\(255\) 0 0
\(256\) 480.385 1.87651
\(257\) 472.336i 1.83789i 0.394391 + 0.918943i \(0.370955\pi\)
−0.394391 + 0.918943i \(0.629045\pi\)
\(258\) 0 0
\(259\) −6.76816 −0.0261319
\(260\) 377.056i 1.45021i
\(261\) 0 0
\(262\) 742.602 2.83436
\(263\) − 146.715i − 0.557851i −0.960313 0.278926i \(-0.910022\pi\)
0.960313 0.278926i \(-0.0899783\pi\)
\(264\) 0 0
\(265\) −175.050 −0.660568
\(266\) 39.1284i 0.147099i
\(267\) 0 0
\(268\) −1012.94 −3.77964
\(269\) 16.6908i 0.0620474i 0.999519 + 0.0310237i \(0.00987674\pi\)
−0.999519 + 0.0310237i \(0.990123\pi\)
\(270\) 0 0
\(271\) −120.702 −0.445396 −0.222698 0.974888i \(-0.571486\pi\)
−0.222698 + 0.974888i \(0.571486\pi\)
\(272\) 318.852i 1.17225i
\(273\) 0 0
\(274\) 450.941 1.64577
\(275\) − 393.657i − 1.43148i
\(276\) 0 0
\(277\) −488.270 −1.76271 −0.881355 0.472456i \(-0.843368\pi\)
−0.881355 + 0.472456i \(0.843368\pi\)
\(278\) 536.440i 1.92964i
\(279\) 0 0
\(280\) −162.278 −0.579566
\(281\) 329.738i 1.17345i 0.809788 + 0.586723i \(0.199582\pi\)
−0.809788 + 0.586723i \(0.800418\pi\)
\(282\) 0 0
\(283\) −452.880 −1.60028 −0.800141 0.599812i \(-0.795242\pi\)
−0.800141 + 0.599812i \(0.795242\pi\)
\(284\) − 1471.18i − 5.18021i
\(285\) 0 0
\(286\) −232.562 −0.813155
\(287\) − 63.1191i − 0.219927i
\(288\) 0 0
\(289\) 256.605 0.887906
\(290\) − 796.102i − 2.74518i
\(291\) 0 0
\(292\) −1119.90 −3.83528
\(293\) 95.4461i 0.325755i 0.986646 + 0.162877i \(0.0520775\pi\)
−0.986646 + 0.162877i \(0.947923\pi\)
\(294\) 0 0
\(295\) 617.475 2.09314
\(296\) − 206.702i − 0.698316i
\(297\) 0 0
\(298\) 26.2046 0.0879351
\(299\) 41.9398i 0.140267i
\(300\) 0 0
\(301\) −5.26793 −0.0175014
\(302\) − 545.193i − 1.80527i
\(303\) 0 0
\(304\) −676.912 −2.22668
\(305\) − 609.578i − 1.99862i
\(306\) 0 0
\(307\) −274.354 −0.893663 −0.446831 0.894618i \(-0.647448\pi\)
−0.446831 + 0.894618i \(0.647448\pi\)
\(308\) − 116.284i − 0.377546i
\(309\) 0 0
\(310\) −761.158 −2.45535
\(311\) − 244.443i − 0.785989i −0.919541 0.392995i \(-0.871439\pi\)
0.919541 0.392995i \(-0.128561\pi\)
\(312\) 0 0
\(313\) −200.326 −0.640021 −0.320010 0.947414i \(-0.603686\pi\)
−0.320010 + 0.947414i \(0.603686\pi\)
\(314\) − 497.359i − 1.58394i
\(315\) 0 0
\(316\) −475.997 −1.50632
\(317\) 490.938i 1.54870i 0.632758 + 0.774350i \(0.281923\pi\)
−0.632758 + 0.774350i \(0.718077\pi\)
\(318\) 0 0
\(319\) 357.584 1.12095
\(320\) − 1528.23i − 4.77572i
\(321\) 0 0
\(322\) −28.7960 −0.0894284
\(323\) − 68.7737i − 0.212922i
\(324\) 0 0
\(325\) 144.453 0.444470
\(326\) 111.147i 0.340941i
\(327\) 0 0
\(328\) 1927.68 5.87706
\(329\) − 26.7405i − 0.0812783i
\(330\) 0 0
\(331\) 63.0543 0.190496 0.0952482 0.995454i \(-0.469636\pi\)
0.0952482 + 0.995454i \(0.469636\pi\)
\(332\) 730.266i 2.19960i
\(333\) 0 0
\(334\) −536.474 −1.60621
\(335\) 704.821i 2.10394i
\(336\) 0 0
\(337\) −64.9287 −0.192667 −0.0963333 0.995349i \(-0.530711\pi\)
−0.0963333 + 0.995349i \(0.530711\pi\)
\(338\) 563.035i 1.66578i
\(339\) 0 0
\(340\) 455.031 1.33833
\(341\) − 341.889i − 1.00261i
\(342\) 0 0
\(343\) −82.1159 −0.239405
\(344\) − 160.884i − 0.467686i
\(345\) 0 0
\(346\) −585.032 −1.69084
\(347\) − 157.767i − 0.454659i −0.973818 0.227330i \(-0.927000\pi\)
0.973818 0.227330i \(-0.0729995\pi\)
\(348\) 0 0
\(349\) 348.778 0.999365 0.499682 0.866209i \(-0.333450\pi\)
0.499682 + 0.866209i \(0.333450\pi\)
\(350\) 99.1816i 0.283376i
\(351\) 0 0
\(352\) 1437.13 4.08277
\(353\) 361.738i 1.02475i 0.858761 + 0.512377i \(0.171235\pi\)
−0.858761 + 0.512377i \(0.828765\pi\)
\(354\) 0 0
\(355\) −1023.67 −2.88358
\(356\) − 415.827i − 1.16805i
\(357\) 0 0
\(358\) −1156.84 −3.23139
\(359\) − 440.805i − 1.22787i −0.789357 0.613934i \(-0.789586\pi\)
0.789357 0.613934i \(-0.210414\pi\)
\(360\) 0 0
\(361\) −214.996 −0.595556
\(362\) 982.728i 2.71472i
\(363\) 0 0
\(364\) 42.6705 0.117227
\(365\) 779.245i 2.13492i
\(366\) 0 0
\(367\) −21.4041 −0.0583219 −0.0291609 0.999575i \(-0.509284\pi\)
−0.0291609 + 0.999575i \(0.509284\pi\)
\(368\) − 498.163i − 1.35371i
\(369\) 0 0
\(370\) −229.450 −0.620134
\(371\) 19.8101i 0.0533964i
\(372\) 0 0
\(373\) −573.207 −1.53675 −0.768374 0.640001i \(-0.778934\pi\)
−0.768374 + 0.640001i \(0.778934\pi\)
\(374\) 280.657i 0.750419i
\(375\) 0 0
\(376\) 816.664 2.17198
\(377\) 131.216i 0.348053i
\(378\) 0 0
\(379\) −469.913 −1.23988 −0.619938 0.784651i \(-0.712842\pi\)
−0.619938 + 0.784651i \(0.712842\pi\)
\(380\) 966.016i 2.54215i
\(381\) 0 0
\(382\) −715.672 −1.87349
\(383\) − 116.747i − 0.304821i −0.988317 0.152411i \(-0.951296\pi\)
0.988317 0.152411i \(-0.0487036\pi\)
\(384\) 0 0
\(385\) −80.9122 −0.210162
\(386\) 8.93735i 0.0231538i
\(387\) 0 0
\(388\) −1082.83 −2.79081
\(389\) − 357.982i − 0.920261i −0.887851 0.460130i \(-0.847803\pi\)
0.887851 0.460130i \(-0.152197\pi\)
\(390\) 0 0
\(391\) 50.6130 0.129445
\(392\) − 1244.74i − 3.17536i
\(393\) 0 0
\(394\) −846.755 −2.14912
\(395\) 331.206i 0.838495i
\(396\) 0 0
\(397\) 327.680 0.825389 0.412695 0.910869i \(-0.364588\pi\)
0.412695 + 0.910869i \(0.364588\pi\)
\(398\) 1021.28i 2.56603i
\(399\) 0 0
\(400\) −1715.82 −4.28955
\(401\) 326.702i 0.814717i 0.913268 + 0.407359i \(0.133550\pi\)
−0.913268 + 0.407359i \(0.866450\pi\)
\(402\) 0 0
\(403\) 125.456 0.311306
\(404\) 713.037i 1.76494i
\(405\) 0 0
\(406\) −90.0931 −0.221904
\(407\) − 103.062i − 0.253223i
\(408\) 0 0
\(409\) 431.498 1.05501 0.527504 0.849553i \(-0.323128\pi\)
0.527504 + 0.849553i \(0.323128\pi\)
\(410\) − 2139.82i − 5.21907i
\(411\) 0 0
\(412\) 1675.06 4.06569
\(413\) − 69.8783i − 0.169197i
\(414\) 0 0
\(415\) 508.130 1.22441
\(416\) 527.357i 1.26769i
\(417\) 0 0
\(418\) −595.824 −1.42542
\(419\) − 440.104i − 1.05037i −0.850989 0.525184i \(-0.823996\pi\)
0.850989 0.525184i \(-0.176004\pi\)
\(420\) 0 0
\(421\) −192.973 −0.458369 −0.229184 0.973383i \(-0.573606\pi\)
−0.229184 + 0.973383i \(0.573606\pi\)
\(422\) 752.051i 1.78211i
\(423\) 0 0
\(424\) −605.005 −1.42690
\(425\) − 174.326i − 0.410178i
\(426\) 0 0
\(427\) −68.9845 −0.161556
\(428\) 1434.04i 3.35057i
\(429\) 0 0
\(430\) −178.590 −0.415325
\(431\) − 186.224i − 0.432074i −0.976385 0.216037i \(-0.930687\pi\)
0.976385 0.216037i \(-0.0693132\pi\)
\(432\) 0 0
\(433\) 835.214 1.92890 0.964450 0.264266i \(-0.0851296\pi\)
0.964450 + 0.264266i \(0.0851296\pi\)
\(434\) 86.1385i 0.198476i
\(435\) 0 0
\(436\) 1441.71 3.30667
\(437\) 107.450i 0.245880i
\(438\) 0 0
\(439\) 65.1154 0.148327 0.0741633 0.997246i \(-0.476371\pi\)
0.0741633 + 0.997246i \(0.476371\pi\)
\(440\) − 2471.08i − 5.61610i
\(441\) 0 0
\(442\) −102.987 −0.233003
\(443\) − 146.097i − 0.329790i −0.986311 0.164895i \(-0.947271\pi\)
0.986311 0.164895i \(-0.0527286\pi\)
\(444\) 0 0
\(445\) −289.338 −0.650198
\(446\) − 585.375i − 1.31250i
\(447\) 0 0
\(448\) −172.946 −0.386041
\(449\) − 415.030i − 0.924342i −0.886791 0.462171i \(-0.847071\pi\)
0.886791 0.462171i \(-0.152929\pi\)
\(450\) 0 0
\(451\) 961.141 2.13113
\(452\) 1473.37i 3.25966i
\(453\) 0 0
\(454\) −657.391 −1.44800
\(455\) − 29.6908i − 0.0652545i
\(456\) 0 0
\(457\) −600.629 −1.31429 −0.657143 0.753766i \(-0.728235\pi\)
−0.657143 + 0.753766i \(0.728235\pi\)
\(458\) − 1024.79i − 2.23753i
\(459\) 0 0
\(460\) −710.925 −1.54549
\(461\) − 623.725i − 1.35298i −0.736451 0.676491i \(-0.763500\pi\)
0.736451 0.676491i \(-0.236500\pi\)
\(462\) 0 0
\(463\) −242.413 −0.523570 −0.261785 0.965126i \(-0.584311\pi\)
−0.261785 + 0.965126i \(0.584311\pi\)
\(464\) − 1558.59i − 3.35903i
\(465\) 0 0
\(466\) 1233.93 2.64792
\(467\) 394.124i 0.843948i 0.906608 + 0.421974i \(0.138663\pi\)
−0.906608 + 0.421974i \(0.861337\pi\)
\(468\) 0 0
\(469\) 79.7630 0.170070
\(470\) − 906.540i − 1.92881i
\(471\) 0 0
\(472\) 2134.10 4.52140
\(473\) − 80.2169i − 0.169592i
\(474\) 0 0
\(475\) 370.088 0.779132
\(476\) − 51.4948i − 0.108182i
\(477\) 0 0
\(478\) 1485.04 3.10677
\(479\) − 560.196i − 1.16951i −0.811210 0.584755i \(-0.801191\pi\)
0.811210 0.584755i \(-0.198809\pi\)
\(480\) 0 0
\(481\) 37.8186 0.0786249
\(482\) 836.335i 1.73514i
\(483\) 0 0
\(484\) 473.709 0.978738
\(485\) 753.451i 1.55351i
\(486\) 0 0
\(487\) 473.772 0.972837 0.486418 0.873726i \(-0.338303\pi\)
0.486418 + 0.873726i \(0.338303\pi\)
\(488\) − 2106.81i − 4.31723i
\(489\) 0 0
\(490\) −1381.73 −2.81985
\(491\) 599.215i 1.22040i 0.792249 + 0.610198i \(0.208910\pi\)
−0.792249 + 0.610198i \(0.791090\pi\)
\(492\) 0 0
\(493\) 158.351 0.321200
\(494\) − 218.638i − 0.442587i
\(495\) 0 0
\(496\) −1490.18 −3.00439
\(497\) 115.846i 0.233091i
\(498\) 0 0
\(499\) −799.002 −1.60121 −0.800603 0.599195i \(-0.795487\pi\)
−0.800603 + 0.599195i \(0.795487\pi\)
\(500\) 449.959i 0.899918i
\(501\) 0 0
\(502\) 1310.57 2.61071
\(503\) − 613.592i − 1.21986i −0.792454 0.609932i \(-0.791197\pi\)
0.792454 0.609932i \(-0.208803\pi\)
\(504\) 0 0
\(505\) 496.141 0.982458
\(506\) − 438.488i − 0.866577i
\(507\) 0 0
\(508\) 120.797 0.237789
\(509\) − 178.490i − 0.350667i −0.984509 0.175334i \(-0.943900\pi\)
0.984509 0.175334i \(-0.0561004\pi\)
\(510\) 0 0
\(511\) 88.1854 0.172574
\(512\) − 487.642i − 0.952425i
\(513\) 0 0
\(514\) 1812.13 3.52555
\(515\) − 1165.53i − 2.26317i
\(516\) 0 0
\(517\) 407.190 0.787601
\(518\) 25.9663i 0.0501280i
\(519\) 0 0
\(520\) 906.765 1.74378
\(521\) − 824.524i − 1.58258i −0.611441 0.791290i \(-0.709410\pi\)
0.611441 0.791290i \(-0.290590\pi\)
\(522\) 0 0
\(523\) −35.4452 −0.0677729 −0.0338864 0.999426i \(-0.510788\pi\)
−0.0338864 + 0.999426i \(0.510788\pi\)
\(524\) − 2074.77i − 3.95949i
\(525\) 0 0
\(526\) −562.877 −1.07011
\(527\) − 151.401i − 0.287288i
\(528\) 0 0
\(529\) 449.924 0.850518
\(530\) 671.587i 1.26714i
\(531\) 0 0
\(532\) 109.322 0.205492
\(533\) 352.691i 0.661710i
\(534\) 0 0
\(535\) 997.828 1.86510
\(536\) 2435.98i 4.54475i
\(537\) 0 0
\(538\) 64.0347 0.119024
\(539\) − 620.629i − 1.15144i
\(540\) 0 0
\(541\) 277.739 0.513381 0.256691 0.966494i \(-0.417368\pi\)
0.256691 + 0.966494i \(0.417368\pi\)
\(542\) 463.078i 0.854388i
\(543\) 0 0
\(544\) 636.415 1.16988
\(545\) − 1003.16i − 1.84066i
\(546\) 0 0
\(547\) −746.686 −1.36506 −0.682529 0.730859i \(-0.739120\pi\)
−0.682529 + 0.730859i \(0.739120\pi\)
\(548\) − 1259.90i − 2.29908i
\(549\) 0 0
\(550\) −1510.28 −2.74596
\(551\) 336.175i 0.610117i
\(552\) 0 0
\(553\) 37.4818 0.0677790
\(554\) 1873.27i 3.38135i
\(555\) 0 0
\(556\) 1498.77 2.69563
\(557\) 702.305i 1.26087i 0.776241 + 0.630436i \(0.217124\pi\)
−0.776241 + 0.630436i \(0.782876\pi\)
\(558\) 0 0
\(559\) 29.4357 0.0526577
\(560\) 352.669i 0.629766i
\(561\) 0 0
\(562\) 1265.05 2.25098
\(563\) − 403.982i − 0.717553i −0.933424 0.358776i \(-0.883194\pi\)
0.933424 0.358776i \(-0.116806\pi\)
\(564\) 0 0
\(565\) 1025.19 1.81450
\(566\) 1737.49i 3.06977i
\(567\) 0 0
\(568\) −3537.98 −6.22884
\(569\) 597.989i 1.05095i 0.850810 + 0.525474i \(0.176112\pi\)
−0.850810 + 0.525474i \(0.823888\pi\)
\(570\) 0 0
\(571\) 498.679 0.873343 0.436672 0.899621i \(-0.356157\pi\)
0.436672 + 0.899621i \(0.356157\pi\)
\(572\) 649.762i 1.13595i
\(573\) 0 0
\(574\) −242.159 −0.421879
\(575\) 272.361i 0.473671i
\(576\) 0 0
\(577\) 704.908 1.22168 0.610839 0.791755i \(-0.290832\pi\)
0.610839 + 0.791755i \(0.290832\pi\)
\(578\) − 984.473i − 1.70324i
\(579\) 0 0
\(580\) −2224.25 −3.83492
\(581\) − 57.5039i − 0.0989740i
\(582\) 0 0
\(583\) −301.656 −0.517420
\(584\) 2693.21i 4.61165i
\(585\) 0 0
\(586\) 366.182 0.624884
\(587\) − 685.573i − 1.16793i −0.811780 0.583964i \(-0.801501\pi\)
0.811780 0.583964i \(-0.198499\pi\)
\(588\) 0 0
\(589\) 321.419 0.545702
\(590\) − 2368.97i − 4.01520i
\(591\) 0 0
\(592\) −449.211 −0.758802
\(593\) 265.529i 0.447773i 0.974615 + 0.223886i \(0.0718745\pi\)
−0.974615 + 0.223886i \(0.928126\pi\)
\(594\) 0 0
\(595\) −35.8309 −0.0602200
\(596\) − 73.2139i − 0.122842i
\(597\) 0 0
\(598\) 160.904 0.269069
\(599\) 667.031i 1.11357i 0.830655 + 0.556787i \(0.187966\pi\)
−0.830655 + 0.556787i \(0.812034\pi\)
\(600\) 0 0
\(601\) 136.316 0.226816 0.113408 0.993549i \(-0.463823\pi\)
0.113408 + 0.993549i \(0.463823\pi\)
\(602\) 20.2106i 0.0335724i
\(603\) 0 0
\(604\) −1523.23 −2.52190
\(605\) − 329.614i − 0.544816i
\(606\) 0 0
\(607\) −883.875 −1.45614 −0.728068 0.685504i \(-0.759582\pi\)
−0.728068 + 0.685504i \(0.759582\pi\)
\(608\) 1351.09i 2.22218i
\(609\) 0 0
\(610\) −2338.67 −3.83388
\(611\) 149.419i 0.244548i
\(612\) 0 0
\(613\) −343.773 −0.560805 −0.280403 0.959883i \(-0.590468\pi\)
−0.280403 + 0.959883i \(0.590468\pi\)
\(614\) 1052.57i 1.71428i
\(615\) 0 0
\(616\) −279.647 −0.453972
\(617\) 556.940i 0.902657i 0.892358 + 0.451329i \(0.149050\pi\)
−0.892358 + 0.451329i \(0.850950\pi\)
\(618\) 0 0
\(619\) 600.216 0.969654 0.484827 0.874610i \(-0.338882\pi\)
0.484827 + 0.874610i \(0.338882\pi\)
\(620\) 2126.62i 3.43003i
\(621\) 0 0
\(622\) −937.812 −1.50774
\(623\) 32.7437i 0.0525582i
\(624\) 0 0
\(625\) −452.617 −0.724188
\(626\) 768.559i 1.22773i
\(627\) 0 0
\(628\) −1389.58 −2.21271
\(629\) − 45.6395i − 0.0725588i
\(630\) 0 0
\(631\) −293.725 −0.465491 −0.232746 0.972538i \(-0.574771\pi\)
−0.232746 + 0.972538i \(0.574771\pi\)
\(632\) 1144.70i 1.81124i
\(633\) 0 0
\(634\) 1883.50 2.97082
\(635\) − 84.0523i − 0.132366i
\(636\) 0 0
\(637\) 227.740 0.357520
\(638\) − 1371.88i − 2.15029i
\(639\) 0 0
\(640\) −2527.24 −3.94881
\(641\) 520.355i 0.811786i 0.913921 + 0.405893i \(0.133039\pi\)
−0.913921 + 0.405893i \(0.866961\pi\)
\(642\) 0 0
\(643\) 786.529 1.22322 0.611609 0.791160i \(-0.290523\pi\)
0.611609 + 0.791160i \(0.290523\pi\)
\(644\) 80.4538i 0.124928i
\(645\) 0 0
\(646\) −263.853 −0.408441
\(647\) − 628.680i − 0.971685i −0.874046 0.485842i \(-0.838513\pi\)
0.874046 0.485842i \(-0.161487\pi\)
\(648\) 0 0
\(649\) 1064.07 1.63955
\(650\) − 554.198i − 0.852613i
\(651\) 0 0
\(652\) 310.536 0.476282
\(653\) 703.238i 1.07693i 0.842646 + 0.538467i \(0.180996\pi\)
−0.842646 + 0.538467i \(0.819004\pi\)
\(654\) 0 0
\(655\) −1443.66 −2.20406
\(656\) − 4189.29i − 6.38611i
\(657\) 0 0
\(658\) −102.591 −0.155913
\(659\) − 930.433i − 1.41189i −0.708268 0.705943i \(-0.750523\pi\)
0.708268 0.705943i \(-0.249477\pi\)
\(660\) 0 0
\(661\) −590.748 −0.893718 −0.446859 0.894604i \(-0.647458\pi\)
−0.446859 + 0.894604i \(0.647458\pi\)
\(662\) − 241.910i − 0.365423i
\(663\) 0 0
\(664\) 1756.18 2.64486
\(665\) − 76.0677i − 0.114388i
\(666\) 0 0
\(667\) −247.403 −0.370919
\(668\) 1498.87i 2.24382i
\(669\) 0 0
\(670\) 2704.07 4.03593
\(671\) − 1050.46i − 1.56551i
\(672\) 0 0
\(673\) −272.854 −0.405429 −0.202714 0.979238i \(-0.564976\pi\)
−0.202714 + 0.979238i \(0.564976\pi\)
\(674\) 249.101i 0.369586i
\(675\) 0 0
\(676\) 1573.08 2.32704
\(677\) 1232.99i 1.82126i 0.413224 + 0.910629i \(0.364403\pi\)
−0.413224 + 0.910629i \(0.635597\pi\)
\(678\) 0 0
\(679\) 85.2663 0.125576
\(680\) − 1094.29i − 1.60924i
\(681\) 0 0
\(682\) −1311.67 −1.92327
\(683\) 1079.42i 1.58041i 0.612840 + 0.790207i \(0.290027\pi\)
−0.612840 + 0.790207i \(0.709973\pi\)
\(684\) 0 0
\(685\) −876.655 −1.27979
\(686\) 315.041i 0.459243i
\(687\) 0 0
\(688\) −349.638 −0.508195
\(689\) − 110.693i − 0.160657i
\(690\) 0 0
\(691\) 521.237 0.754323 0.377161 0.926148i \(-0.376900\pi\)
0.377161 + 0.926148i \(0.376900\pi\)
\(692\) 1634.54i 2.36204i
\(693\) 0 0
\(694\) −605.278 −0.872158
\(695\) − 1042.87i − 1.50053i
\(696\) 0 0
\(697\) 425.628 0.610657
\(698\) − 1338.10i − 1.91705i
\(699\) 0 0
\(700\) 277.106 0.395866
\(701\) − 240.666i − 0.343319i −0.985156 0.171659i \(-0.945087\pi\)
0.985156 0.171659i \(-0.0549128\pi\)
\(702\) 0 0
\(703\) 96.8910 0.137825
\(704\) − 2633.53i − 3.74080i
\(705\) 0 0
\(706\) 1387.82 1.96575
\(707\) − 56.1472i − 0.0794161i
\(708\) 0 0
\(709\) 205.759 0.290210 0.145105 0.989416i \(-0.453648\pi\)
0.145105 + 0.989416i \(0.453648\pi\)
\(710\) 3927.34i 5.53147i
\(711\) 0 0
\(712\) −1000.00 −1.40450
\(713\) 236.543i 0.331758i
\(714\) 0 0
\(715\) 452.114 0.632328
\(716\) 3232.12i 4.51413i
\(717\) 0 0
\(718\) −1691.16 −2.35538
\(719\) − 450.051i − 0.625941i −0.949763 0.312970i \(-0.898676\pi\)
0.949763 0.312970i \(-0.101324\pi\)
\(720\) 0 0
\(721\) −131.901 −0.182941
\(722\) 824.839i 1.14244i
\(723\) 0 0
\(724\) 2745.67 3.79236
\(725\) 852.127i 1.17535i
\(726\) 0 0
\(727\) 16.4297 0.0225993 0.0112996 0.999936i \(-0.496403\pi\)
0.0112996 + 0.999936i \(0.496403\pi\)
\(728\) − 102.617i − 0.140957i
\(729\) 0 0
\(730\) 2989.60 4.09534
\(731\) − 35.5230i − 0.0485950i
\(732\) 0 0
\(733\) −556.901 −0.759756 −0.379878 0.925037i \(-0.624034\pi\)
−0.379878 + 0.925037i \(0.624034\pi\)
\(734\) 82.1177i 0.111877i
\(735\) 0 0
\(736\) −994.313 −1.35097
\(737\) 1214.58i 1.64801i
\(738\) 0 0
\(739\) −850.227 −1.15051 −0.575255 0.817974i \(-0.695097\pi\)
−0.575255 + 0.817974i \(0.695097\pi\)
\(740\) 641.065i 0.866305i
\(741\) 0 0
\(742\) 76.0019 0.102428
\(743\) − 63.6337i − 0.0856443i −0.999083 0.0428221i \(-0.986365\pi\)
0.999083 0.0428221i \(-0.0136349\pi\)
\(744\) 0 0
\(745\) −50.9433 −0.0683802
\(746\) 2199.13i 2.94789i
\(747\) 0 0
\(748\) 784.134 1.04831
\(749\) − 112.922i − 0.150764i
\(750\) 0 0
\(751\) 1198.70 1.59614 0.798072 0.602562i \(-0.205853\pi\)
0.798072 + 0.602562i \(0.205853\pi\)
\(752\) − 1774.80i − 2.36011i
\(753\) 0 0
\(754\) 503.414 0.667658
\(755\) 1059.88i 1.40382i
\(756\) 0 0
\(757\) −426.213 −0.563028 −0.281514 0.959557i \(-0.590837\pi\)
−0.281514 + 0.959557i \(0.590837\pi\)
\(758\) 1802.84i 2.37841i
\(759\) 0 0
\(760\) 2323.13 3.05675
\(761\) 885.119i 1.16310i 0.813511 + 0.581550i \(0.197553\pi\)
−0.813511 + 0.581550i \(0.802447\pi\)
\(762\) 0 0
\(763\) −113.526 −0.148788
\(764\) 1999.53i 2.61719i
\(765\) 0 0
\(766\) −447.902 −0.584729
\(767\) 390.460i 0.509074i
\(768\) 0 0
\(769\) 489.713 0.636818 0.318409 0.947953i \(-0.396851\pi\)
0.318409 + 0.947953i \(0.396851\pi\)
\(770\) 310.422i 0.403146i
\(771\) 0 0
\(772\) 24.9703 0.0323449
\(773\) − 478.239i − 0.618679i −0.950952 0.309340i \(-0.899892\pi\)
0.950952 0.309340i \(-0.100108\pi\)
\(774\) 0 0
\(775\) 814.723 1.05126
\(776\) 2604.06i 3.35575i
\(777\) 0 0
\(778\) −1373.41 −1.76531
\(779\) 903.594i 1.15994i
\(780\) 0 0
\(781\) −1764.04 −2.25869
\(782\) − 194.179i − 0.248310i
\(783\) 0 0
\(784\) −2705.11 −3.45039
\(785\) 966.893i 1.23171i
\(786\) 0 0
\(787\) −636.491 −0.808757 −0.404378 0.914592i \(-0.632512\pi\)
−0.404378 + 0.914592i \(0.632512\pi\)
\(788\) 2365.77i 3.00225i
\(789\) 0 0
\(790\) 1270.68 1.60846
\(791\) − 116.019i − 0.146673i
\(792\) 0 0
\(793\) 385.466 0.486085
\(794\) − 1257.15i − 1.58332i
\(795\) 0 0
\(796\) 2853.38 3.58465
\(797\) − 382.445i − 0.479855i −0.970791 0.239928i \(-0.922876\pi\)
0.970791 0.239928i \(-0.0771237\pi\)
\(798\) 0 0
\(799\) 180.318 0.225680
\(800\) 3424.70i 4.28088i
\(801\) 0 0
\(802\) 1253.40 1.56285
\(803\) 1342.84i 1.67227i
\(804\) 0 0
\(805\) 55.9809 0.0695415
\(806\) − 481.317i − 0.597168i
\(807\) 0 0
\(808\) 1714.75 2.12222
\(809\) − 1031.19i − 1.27465i −0.770597 0.637323i \(-0.780042\pi\)
0.770597 0.637323i \(-0.219958\pi\)
\(810\) 0 0
\(811\) 407.070 0.501936 0.250968 0.967995i \(-0.419251\pi\)
0.250968 + 0.967995i \(0.419251\pi\)
\(812\) 251.713i 0.309992i
\(813\) 0 0
\(814\) −395.400 −0.485749
\(815\) − 216.076i − 0.265123i
\(816\) 0 0
\(817\) 75.4140 0.0923061
\(818\) − 1655.46i − 2.02379i
\(819\) 0 0
\(820\) −5978.50 −7.29085
\(821\) − 446.485i − 0.543830i −0.962321 0.271915i \(-0.912343\pi\)
0.962321 0.271915i \(-0.0876570\pi\)
\(822\) 0 0
\(823\) 706.495 0.858438 0.429219 0.903200i \(-0.358789\pi\)
0.429219 + 0.903200i \(0.358789\pi\)
\(824\) − 4028.29i − 4.88870i
\(825\) 0 0
\(826\) −268.090 −0.324565
\(827\) − 1052.40i − 1.27255i −0.771461 0.636276i \(-0.780474\pi\)
0.771461 0.636276i \(-0.219526\pi\)
\(828\) 0 0
\(829\) −1352.99 −1.63208 −0.816040 0.577995i \(-0.803835\pi\)
−0.816040 + 0.577995i \(0.803835\pi\)
\(830\) − 1949.46i − 2.34874i
\(831\) 0 0
\(832\) 966.374 1.16151
\(833\) − 274.837i − 0.329936i
\(834\) 0 0
\(835\) 1042.94 1.24902
\(836\) 1664.69i 1.99125i
\(837\) 0 0
\(838\) −1688.47 −2.01489
\(839\) 1124.03i 1.33973i 0.742485 + 0.669863i \(0.233647\pi\)
−0.742485 + 0.669863i \(0.766353\pi\)
\(840\) 0 0
\(841\) 66.9580 0.0796172
\(842\) 740.348i 0.879274i
\(843\) 0 0
\(844\) 2101.18 2.48954
\(845\) − 1094.57i − 1.29535i
\(846\) 0 0
\(847\) −37.3016 −0.0440397
\(848\) 1314.82i 1.55049i
\(849\) 0 0
\(850\) −668.807 −0.786831
\(851\) 71.3055i 0.0837903i
\(852\) 0 0
\(853\) −224.933 −0.263697 −0.131848 0.991270i \(-0.542091\pi\)
−0.131848 + 0.991270i \(0.542091\pi\)
\(854\) 264.661i 0.309908i
\(855\) 0 0
\(856\) 3448.67 4.02882
\(857\) − 971.825i − 1.13398i −0.823723 0.566992i \(-0.808107\pi\)
0.823723 0.566992i \(-0.191893\pi\)
\(858\) 0 0
\(859\) 871.022 1.01399 0.506997 0.861948i \(-0.330755\pi\)
0.506997 + 0.861948i \(0.330755\pi\)
\(860\) 498.966i 0.580193i
\(861\) 0 0
\(862\) −714.454 −0.828833
\(863\) 1554.76i 1.80157i 0.434262 + 0.900786i \(0.357009\pi\)
−0.434262 + 0.900786i \(0.642991\pi\)
\(864\) 0 0
\(865\) 1137.33 1.31484
\(866\) − 3204.33i − 3.70014i
\(867\) 0 0
\(868\) 240.665 0.277263
\(869\) 570.751i 0.656790i
\(870\) 0 0
\(871\) −445.693 −0.511702
\(872\) − 3467.10i − 3.97603i
\(873\) 0 0
\(874\) 412.234 0.471664
\(875\) − 35.4315i − 0.0404931i
\(876\) 0 0
\(877\) −1171.39 −1.33568 −0.667839 0.744306i \(-0.732780\pi\)
−0.667839 + 0.744306i \(0.732780\pi\)
\(878\) − 249.817i − 0.284530i
\(879\) 0 0
\(880\) −5370.24 −6.10254
\(881\) − 212.080i − 0.240727i −0.992730 0.120363i \(-0.961594\pi\)
0.992730 0.120363i \(-0.0384060\pi\)
\(882\) 0 0
\(883\) −534.932 −0.605812 −0.302906 0.953020i \(-0.597957\pi\)
−0.302906 + 0.953020i \(0.597957\pi\)
\(884\) 287.738i 0.325496i
\(885\) 0 0
\(886\) −560.507 −0.632626
\(887\) 649.299i 0.732017i 0.930611 + 0.366009i \(0.119276\pi\)
−0.930611 + 0.366009i \(0.880724\pi\)
\(888\) 0 0
\(889\) −9.51200 −0.0106997
\(890\) 1110.06i 1.24725i
\(891\) 0 0
\(892\) −1635.49 −1.83351
\(893\) 382.810i 0.428678i
\(894\) 0 0
\(895\) 2248.96 2.51280
\(896\) 286.002i 0.319199i
\(897\) 0 0
\(898\) −1592.28 −1.77314
\(899\) 740.066i 0.823210i
\(900\) 0 0
\(901\) −133.584 −0.148262
\(902\) − 3687.45i − 4.08808i
\(903\) 0 0
\(904\) 3543.24 3.91951
\(905\) − 1910.48i − 2.11102i
\(906\) 0 0
\(907\) 960.314 1.05878 0.529390 0.848378i \(-0.322421\pi\)
0.529390 + 0.848378i \(0.322421\pi\)
\(908\) 1836.70i 2.02280i
\(909\) 0 0
\(910\) −113.910 −0.125176
\(911\) 19.1928i 0.0210679i 0.999945 + 0.0105339i \(0.00335312\pi\)
−0.999945 + 0.0105339i \(0.996647\pi\)
\(912\) 0 0
\(913\) 875.636 0.959075
\(914\) 2304.33i 2.52115i
\(915\) 0 0
\(916\) −2863.19 −3.12575
\(917\) 163.376i 0.178163i
\(918\) 0 0
\(919\) −222.712 −0.242341 −0.121171 0.992632i \(-0.538665\pi\)
−0.121171 + 0.992632i \(0.538665\pi\)
\(920\) 1709.67i 1.85834i
\(921\) 0 0
\(922\) −2392.94 −2.59538
\(923\) − 647.316i − 0.701317i
\(924\) 0 0
\(925\) 245.597 0.265510
\(926\) 930.025i 1.00435i
\(927\) 0 0
\(928\) −3110.88 −3.35224
\(929\) 1217.46i 1.31051i 0.755408 + 0.655254i \(0.227438\pi\)
−0.755408 + 0.655254i \(0.772562\pi\)
\(930\) 0 0
\(931\) 583.469 0.626713
\(932\) − 3447.51i − 3.69904i
\(933\) 0 0
\(934\) 1512.07 1.61892
\(935\) − 545.612i − 0.583542i
\(936\) 0 0
\(937\) −676.327 −0.721800 −0.360900 0.932605i \(-0.617530\pi\)
−0.360900 + 0.932605i \(0.617530\pi\)
\(938\) − 306.013i − 0.326240i
\(939\) 0 0
\(940\) −2532.81 −2.69447
\(941\) 367.884i 0.390951i 0.980709 + 0.195475i \(0.0626249\pi\)
−0.980709 + 0.195475i \(0.937375\pi\)
\(942\) 0 0
\(943\) −664.987 −0.705182
\(944\) − 4637.90i − 4.91303i
\(945\) 0 0
\(946\) −307.755 −0.325322
\(947\) − 1836.82i − 1.93962i −0.243862 0.969810i \(-0.578415\pi\)
0.243862 0.969810i \(-0.421585\pi\)
\(948\) 0 0
\(949\) −492.755 −0.519236
\(950\) − 1419.85i − 1.49458i
\(951\) 0 0
\(952\) −123.838 −0.130082
\(953\) 340.367i 0.357154i 0.983926 + 0.178577i \(0.0571493\pi\)
−0.983926 + 0.178577i \(0.942851\pi\)
\(954\) 0 0
\(955\) 1391.31 1.45686
\(956\) − 4149.09i − 4.34005i
\(957\) 0 0
\(958\) −2149.21 −2.24343
\(959\) 99.2090i 0.103450i
\(960\) 0 0
\(961\) −253.418 −0.263703
\(962\) − 145.092i − 0.150823i
\(963\) 0 0
\(964\) 2336.66 2.42392
\(965\) − 17.3747i − 0.0180049i
\(966\) 0 0
\(967\) 242.367 0.250638 0.125319 0.992116i \(-0.460005\pi\)
0.125319 + 0.992116i \(0.460005\pi\)
\(968\) − 1139.20i − 1.17686i
\(969\) 0 0
\(970\) 2890.64 2.98004
\(971\) 210.675i 0.216967i 0.994098 + 0.108483i \(0.0345995\pi\)
−0.994098 + 0.108483i \(0.965401\pi\)
\(972\) 0 0
\(973\) −118.019 −0.121294
\(974\) − 1817.64i − 1.86616i
\(975\) 0 0
\(976\) −4578.58 −4.69117
\(977\) 45.8883i 0.0469686i 0.999724 + 0.0234843i \(0.00747596\pi\)
−0.999724 + 0.0234843i \(0.992524\pi\)
\(978\) 0 0
\(979\) −498.603 −0.509298
\(980\) 3860.44i 3.93923i
\(981\) 0 0
\(982\) 2298.91 2.34105
\(983\) 146.995i 0.149538i 0.997201 + 0.0747688i \(0.0238219\pi\)
−0.997201 + 0.0747688i \(0.976178\pi\)
\(984\) 0 0
\(985\) 1646.14 1.67121
\(986\) − 607.521i − 0.616147i
\(987\) 0 0
\(988\) −610.859 −0.618278
\(989\) 55.4999i 0.0561172i
\(990\) 0 0
\(991\) 1099.39 1.10938 0.554689 0.832058i \(-0.312837\pi\)
0.554689 + 0.832058i \(0.312837\pi\)
\(992\) 2974.33i 2.99832i
\(993\) 0 0
\(994\) 444.448 0.447131
\(995\) − 1985.42i − 1.99540i
\(996\) 0 0
\(997\) −1142.73 −1.14617 −0.573085 0.819496i \(-0.694254\pi\)
−0.573085 + 0.819496i \(0.694254\pi\)
\(998\) 3065.40i 3.07154i
\(999\) 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.890.3 84
3.2 odd 2 inner 1143.3.b.a.890.82 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.3 84 1.1 even 1 trivial
1143.3.b.a.890.82 yes 84 3.2 odd 2 inner