Properties

Label 1143.3.b.a.890.7
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 890.7
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.78

$q$-expansion

\(f(q)\) \(=\) \(q-3.58466i q^{2} -8.84977 q^{4} -7.20639i q^{5} +4.70987 q^{7} +17.3848i q^{8} +O(q^{10})\) \(q-3.58466i q^{2} -8.84977 q^{4} -7.20639i q^{5} +4.70987 q^{7} +17.3848i q^{8} -25.8324 q^{10} +15.5657i q^{11} -9.78351 q^{13} -16.8833i q^{14} +26.9194 q^{16} +23.5044i q^{17} +25.7089 q^{19} +63.7749i q^{20} +55.7976 q^{22} -5.95651i q^{23} -26.9320 q^{25} +35.0705i q^{26} -41.6813 q^{28} +53.9788i q^{29} +10.8286 q^{31} -26.9577i q^{32} +84.2553 q^{34} -33.9412i q^{35} -62.1323 q^{37} -92.1576i q^{38} +125.281 q^{40} +33.5546i q^{41} +77.7688 q^{43} -137.753i q^{44} -21.3520 q^{46} +59.3622i q^{47} -26.8171 q^{49} +96.5421i q^{50} +86.5819 q^{52} +88.7794i q^{53} +112.172 q^{55} +81.8801i q^{56} +193.496 q^{58} -14.4209i q^{59} -78.6267 q^{61} -38.8169i q^{62} +11.0435 q^{64} +70.5038i q^{65} -18.8207 q^{67} -208.009i q^{68} -121.668 q^{70} -75.8330i q^{71} -45.6238 q^{73} +222.723i q^{74} -227.518 q^{76} +73.3124i q^{77} -56.3446 q^{79} -193.991i q^{80} +120.282 q^{82} +128.210i q^{83} +169.382 q^{85} -278.775i q^{86} -270.606 q^{88} -70.1992i q^{89} -46.0791 q^{91} +52.7137i q^{92} +212.793 q^{94} -185.268i q^{95} -4.59545 q^{97} +96.1301i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.58466i − 1.79233i −0.443722 0.896164i \(-0.646342\pi\)
0.443722 0.896164i \(-0.353658\pi\)
\(3\) 0 0
\(4\) −8.84977 −2.21244
\(5\) − 7.20639i − 1.44128i −0.693311 0.720639i \(-0.743849\pi\)
0.693311 0.720639i \(-0.256151\pi\)
\(6\) 0 0
\(7\) 4.70987 0.672839 0.336420 0.941712i \(-0.390784\pi\)
0.336420 + 0.941712i \(0.390784\pi\)
\(8\) 17.3848i 2.17310i
\(9\) 0 0
\(10\) −25.8324 −2.58324
\(11\) 15.5657i 1.41506i 0.706683 + 0.707531i \(0.250191\pi\)
−0.706683 + 0.707531i \(0.749809\pi\)
\(12\) 0 0
\(13\) −9.78351 −0.752578 −0.376289 0.926502i \(-0.622800\pi\)
−0.376289 + 0.926502i \(0.622800\pi\)
\(14\) − 16.8833i − 1.20595i
\(15\) 0 0
\(16\) 26.9194 1.68246
\(17\) 23.5044i 1.38261i 0.722562 + 0.691306i \(0.242964\pi\)
−0.722562 + 0.691306i \(0.757036\pi\)
\(18\) 0 0
\(19\) 25.7089 1.35310 0.676550 0.736397i \(-0.263474\pi\)
0.676550 + 0.736397i \(0.263474\pi\)
\(20\) 63.7749i 3.18874i
\(21\) 0 0
\(22\) 55.7976 2.53626
\(23\) − 5.95651i − 0.258979i −0.991581 0.129489i \(-0.958666\pi\)
0.991581 0.129489i \(-0.0413338\pi\)
\(24\) 0 0
\(25\) −26.9320 −1.07728
\(26\) 35.0705i 1.34887i
\(27\) 0 0
\(28\) −41.6813 −1.48862
\(29\) 53.9788i 1.86134i 0.365861 + 0.930669i \(0.380774\pi\)
−0.365861 + 0.930669i \(0.619226\pi\)
\(30\) 0 0
\(31\) 10.8286 0.349310 0.174655 0.984630i \(-0.444119\pi\)
0.174655 + 0.984630i \(0.444119\pi\)
\(32\) − 26.9577i − 0.842427i
\(33\) 0 0
\(34\) 84.2553 2.47810
\(35\) − 33.9412i − 0.969748i
\(36\) 0 0
\(37\) −62.1323 −1.67925 −0.839625 0.543166i \(-0.817225\pi\)
−0.839625 + 0.543166i \(0.817225\pi\)
\(38\) − 92.1576i − 2.42520i
\(39\) 0 0
\(40\) 125.281 3.13204
\(41\) 33.5546i 0.818405i 0.912444 + 0.409202i \(0.134193\pi\)
−0.912444 + 0.409202i \(0.865807\pi\)
\(42\) 0 0
\(43\) 77.7688 1.80858 0.904288 0.426922i \(-0.140402\pi\)
0.904288 + 0.426922i \(0.140402\pi\)
\(44\) − 137.753i − 3.13074i
\(45\) 0 0
\(46\) −21.3520 −0.464175
\(47\) 59.3622i 1.26303i 0.775365 + 0.631513i \(0.217566\pi\)
−0.775365 + 0.631513i \(0.782434\pi\)
\(48\) 0 0
\(49\) −26.8171 −0.547287
\(50\) 96.5421i 1.93084i
\(51\) 0 0
\(52\) 86.5819 1.66504
\(53\) 88.7794i 1.67508i 0.546374 + 0.837542i \(0.316008\pi\)
−0.546374 + 0.837542i \(0.683992\pi\)
\(54\) 0 0
\(55\) 112.172 2.03950
\(56\) 81.8801i 1.46214i
\(57\) 0 0
\(58\) 193.496 3.33613
\(59\) − 14.4209i − 0.244421i −0.992504 0.122211i \(-0.961002\pi\)
0.992504 0.122211i \(-0.0389983\pi\)
\(60\) 0 0
\(61\) −78.6267 −1.28896 −0.644481 0.764620i \(-0.722927\pi\)
−0.644481 + 0.764620i \(0.722927\pi\)
\(62\) − 38.8169i − 0.626079i
\(63\) 0 0
\(64\) 11.0435 0.172555
\(65\) 70.5038i 1.08467i
\(66\) 0 0
\(67\) −18.8207 −0.280906 −0.140453 0.990087i \(-0.544856\pi\)
−0.140453 + 0.990087i \(0.544856\pi\)
\(68\) − 208.009i − 3.05895i
\(69\) 0 0
\(70\) −121.668 −1.73811
\(71\) − 75.8330i − 1.06807i −0.845462 0.534035i \(-0.820675\pi\)
0.845462 0.534035i \(-0.179325\pi\)
\(72\) 0 0
\(73\) −45.6238 −0.624984 −0.312492 0.949920i \(-0.601164\pi\)
−0.312492 + 0.949920i \(0.601164\pi\)
\(74\) 222.723i 3.00977i
\(75\) 0 0
\(76\) −227.518 −2.99366
\(77\) 73.3124i 0.952109i
\(78\) 0 0
\(79\) −56.3446 −0.713222 −0.356611 0.934253i \(-0.616068\pi\)
−0.356611 + 0.934253i \(0.616068\pi\)
\(80\) − 193.991i − 2.42489i
\(81\) 0 0
\(82\) 120.282 1.46685
\(83\) 128.210i 1.54470i 0.635199 + 0.772348i \(0.280918\pi\)
−0.635199 + 0.772348i \(0.719082\pi\)
\(84\) 0 0
\(85\) 169.382 1.99273
\(86\) − 278.775i − 3.24156i
\(87\) 0 0
\(88\) −270.606 −3.07506
\(89\) − 70.1992i − 0.788756i −0.918948 0.394378i \(-0.870960\pi\)
0.918948 0.394378i \(-0.129040\pi\)
\(90\) 0 0
\(91\) −46.0791 −0.506364
\(92\) 52.7137i 0.572975i
\(93\) 0 0
\(94\) 212.793 2.26376
\(95\) − 185.268i − 1.95019i
\(96\) 0 0
\(97\) −4.59545 −0.0473758 −0.0236879 0.999719i \(-0.507541\pi\)
−0.0236879 + 0.999719i \(0.507541\pi\)
\(98\) 96.1301i 0.980919i
\(99\) 0 0
\(100\) 238.342 2.38342
\(101\) − 119.961i − 1.18774i −0.804562 0.593868i \(-0.797600\pi\)
0.804562 0.593868i \(-0.202400\pi\)
\(102\) 0 0
\(103\) −158.473 −1.53858 −0.769289 0.638901i \(-0.779389\pi\)
−0.769289 + 0.638901i \(0.779389\pi\)
\(104\) − 170.084i − 1.63542i
\(105\) 0 0
\(106\) 318.244 3.00230
\(107\) − 158.986i − 1.48585i −0.669372 0.742927i \(-0.733437\pi\)
0.669372 0.742927i \(-0.266563\pi\)
\(108\) 0 0
\(109\) 58.4012 0.535790 0.267895 0.963448i \(-0.413672\pi\)
0.267895 + 0.963448i \(0.413672\pi\)
\(110\) − 402.099i − 3.65545i
\(111\) 0 0
\(112\) 126.787 1.13203
\(113\) − 80.9446i − 0.716324i −0.933659 0.358162i \(-0.883404\pi\)
0.933659 0.358162i \(-0.116596\pi\)
\(114\) 0 0
\(115\) −42.9249 −0.373260
\(116\) − 477.700i − 4.11811i
\(117\) 0 0
\(118\) −51.6938 −0.438083
\(119\) 110.703i 0.930276i
\(120\) 0 0
\(121\) −121.290 −1.00240
\(122\) 281.850i 2.31025i
\(123\) 0 0
\(124\) −95.8308 −0.772829
\(125\) 13.9230i 0.111384i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) − 147.418i − 1.15170i
\(129\) 0 0
\(130\) 252.732 1.94409
\(131\) 17.1308i 0.130770i 0.997860 + 0.0653848i \(0.0208275\pi\)
−0.997860 + 0.0653848i \(0.979173\pi\)
\(132\) 0 0
\(133\) 121.086 0.910419
\(134\) 67.4659i 0.503477i
\(135\) 0 0
\(136\) −408.619 −3.00455
\(137\) − 28.8275i − 0.210420i −0.994450 0.105210i \(-0.966449\pi\)
0.994450 0.105210i \(-0.0335515\pi\)
\(138\) 0 0
\(139\) 25.6315 0.184399 0.0921997 0.995741i \(-0.470610\pi\)
0.0921997 + 0.995741i \(0.470610\pi\)
\(140\) 300.372i 2.14551i
\(141\) 0 0
\(142\) −271.836 −1.91433
\(143\) − 152.287i − 1.06494i
\(144\) 0 0
\(145\) 388.992 2.68271
\(146\) 163.546i 1.12018i
\(147\) 0 0
\(148\) 549.856 3.71525
\(149\) 60.3128i 0.404784i 0.979305 + 0.202392i \(0.0648715\pi\)
−0.979305 + 0.202392i \(0.935128\pi\)
\(150\) 0 0
\(151\) 31.2208 0.206760 0.103380 0.994642i \(-0.467034\pi\)
0.103380 + 0.994642i \(0.467034\pi\)
\(152\) 446.943i 2.94042i
\(153\) 0 0
\(154\) 262.800 1.70649
\(155\) − 78.0352i − 0.503453i
\(156\) 0 0
\(157\) −58.7863 −0.374435 −0.187218 0.982318i \(-0.559947\pi\)
−0.187218 + 0.982318i \(0.559947\pi\)
\(158\) 201.976i 1.27833i
\(159\) 0 0
\(160\) −194.267 −1.21417
\(161\) − 28.0544i − 0.174251i
\(162\) 0 0
\(163\) −97.1285 −0.595881 −0.297940 0.954585i \(-0.596300\pi\)
−0.297940 + 0.954585i \(0.596300\pi\)
\(164\) − 296.950i − 1.81067i
\(165\) 0 0
\(166\) 459.588 2.76860
\(167\) 147.196i 0.881415i 0.897651 + 0.440707i \(0.145272\pi\)
−0.897651 + 0.440707i \(0.854728\pi\)
\(168\) 0 0
\(169\) −73.2829 −0.433627
\(170\) − 607.176i − 3.57162i
\(171\) 0 0
\(172\) −688.236 −4.00137
\(173\) − 37.9424i − 0.219320i −0.993969 0.109660i \(-0.965024\pi\)
0.993969 0.109660i \(-0.0349763\pi\)
\(174\) 0 0
\(175\) −126.846 −0.724837
\(176\) 419.018i 2.38079i
\(177\) 0 0
\(178\) −251.640 −1.41371
\(179\) 220.858i 1.23384i 0.787024 + 0.616922i \(0.211621\pi\)
−0.787024 + 0.616922i \(0.788379\pi\)
\(180\) 0 0
\(181\) 136.292 0.752996 0.376498 0.926417i \(-0.377128\pi\)
0.376498 + 0.926417i \(0.377128\pi\)
\(182\) 165.178i 0.907571i
\(183\) 0 0
\(184\) 103.553 0.562785
\(185\) 447.749i 2.42027i
\(186\) 0 0
\(187\) −365.862 −1.95648
\(188\) − 525.342i − 2.79437i
\(189\) 0 0
\(190\) −664.124 −3.49539
\(191\) 202.812i 1.06185i 0.847420 + 0.530923i \(0.178155\pi\)
−0.847420 + 0.530923i \(0.821845\pi\)
\(192\) 0 0
\(193\) 193.960 1.00497 0.502486 0.864585i \(-0.332419\pi\)
0.502486 + 0.864585i \(0.332419\pi\)
\(194\) 16.4731i 0.0849130i
\(195\) 0 0
\(196\) 237.325 1.21084
\(197\) 266.715i 1.35388i 0.736036 + 0.676942i \(0.236695\pi\)
−0.736036 + 0.676942i \(0.763305\pi\)
\(198\) 0 0
\(199\) 160.272 0.805388 0.402694 0.915335i \(-0.368074\pi\)
0.402694 + 0.915335i \(0.368074\pi\)
\(200\) − 468.207i − 2.34104i
\(201\) 0 0
\(202\) −430.020 −2.12881
\(203\) 254.233i 1.25238i
\(204\) 0 0
\(205\) 241.807 1.17955
\(206\) 568.073i 2.75764i
\(207\) 0 0
\(208\) −263.366 −1.26618
\(209\) 400.176i 1.91472i
\(210\) 0 0
\(211\) −210.891 −0.999483 −0.499741 0.866175i \(-0.666572\pi\)
−0.499741 + 0.866175i \(0.666572\pi\)
\(212\) − 785.677i − 3.70603i
\(213\) 0 0
\(214\) −569.912 −2.66314
\(215\) − 560.432i − 2.60666i
\(216\) 0 0
\(217\) 51.0014 0.235030
\(218\) − 209.348i − 0.960313i
\(219\) 0 0
\(220\) −992.699 −4.51227
\(221\) − 229.956i − 1.04052i
\(222\) 0 0
\(223\) 392.407 1.75967 0.879835 0.475279i \(-0.157653\pi\)
0.879835 + 0.475279i \(0.157653\pi\)
\(224\) − 126.967i − 0.566818i
\(225\) 0 0
\(226\) −290.159 −1.28389
\(227\) − 22.0499i − 0.0971360i −0.998820 0.0485680i \(-0.984534\pi\)
0.998820 0.0485680i \(-0.0154658\pi\)
\(228\) 0 0
\(229\) −263.211 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(230\) 153.871i 0.669005i
\(231\) 0 0
\(232\) −938.410 −4.04487
\(233\) 129.088i 0.554025i 0.960866 + 0.277013i \(0.0893444\pi\)
−0.960866 + 0.277013i \(0.910656\pi\)
\(234\) 0 0
\(235\) 427.787 1.82037
\(236\) 127.621i 0.540768i
\(237\) 0 0
\(238\) 396.832 1.66736
\(239\) 289.053i 1.20943i 0.796443 + 0.604713i \(0.206712\pi\)
−0.796443 + 0.604713i \(0.793288\pi\)
\(240\) 0 0
\(241\) −104.175 −0.432260 −0.216130 0.976365i \(-0.569344\pi\)
−0.216130 + 0.976365i \(0.569344\pi\)
\(242\) 434.784i 1.79663i
\(243\) 0 0
\(244\) 695.829 2.85176
\(245\) 193.254i 0.788793i
\(246\) 0 0
\(247\) −251.523 −1.01831
\(248\) 188.253i 0.759085i
\(249\) 0 0
\(250\) 49.9091 0.199636
\(251\) − 440.383i − 1.75451i −0.480023 0.877256i \(-0.659372\pi\)
0.480023 0.877256i \(-0.340628\pi\)
\(252\) 0 0
\(253\) 92.7170 0.366470
\(254\) − 40.3970i − 0.159043i
\(255\) 0 0
\(256\) −484.269 −1.89167
\(257\) − 104.246i − 0.405628i −0.979217 0.202814i \(-0.934991\pi\)
0.979217 0.202814i \(-0.0650087\pi\)
\(258\) 0 0
\(259\) −292.635 −1.12987
\(260\) − 623.942i − 2.39978i
\(261\) 0 0
\(262\) 61.4082 0.234382
\(263\) 106.847i 0.406262i 0.979152 + 0.203131i \(0.0651118\pi\)
−0.979152 + 0.203131i \(0.934888\pi\)
\(264\) 0 0
\(265\) 639.779 2.41426
\(266\) − 434.051i − 1.63177i
\(267\) 0 0
\(268\) 166.559 0.621490
\(269\) 280.299i 1.04200i 0.853556 + 0.521001i \(0.174441\pi\)
−0.853556 + 0.521001i \(0.825559\pi\)
\(270\) 0 0
\(271\) 25.5309 0.0942100 0.0471050 0.998890i \(-0.485000\pi\)
0.0471050 + 0.998890i \(0.485000\pi\)
\(272\) 632.724i 2.32619i
\(273\) 0 0
\(274\) −103.337 −0.377142
\(275\) − 419.215i − 1.52442i
\(276\) 0 0
\(277\) 281.436 1.01602 0.508008 0.861352i \(-0.330382\pi\)
0.508008 + 0.861352i \(0.330382\pi\)
\(278\) − 91.8802i − 0.330504i
\(279\) 0 0
\(280\) 590.060 2.10736
\(281\) 521.602i 1.85623i 0.372289 + 0.928117i \(0.378573\pi\)
−0.372289 + 0.928117i \(0.621427\pi\)
\(282\) 0 0
\(283\) 219.479 0.775543 0.387771 0.921756i \(-0.373245\pi\)
0.387771 + 0.921756i \(0.373245\pi\)
\(284\) 671.105i 2.36305i
\(285\) 0 0
\(286\) −545.897 −1.90873
\(287\) 158.038i 0.550655i
\(288\) 0 0
\(289\) −263.457 −0.911616
\(290\) − 1394.40i − 4.80829i
\(291\) 0 0
\(292\) 403.760 1.38274
\(293\) − 122.921i − 0.419524i −0.977752 0.209762i \(-0.932731\pi\)
0.977752 0.209762i \(-0.0672689\pi\)
\(294\) 0 0
\(295\) −103.922 −0.352279
\(296\) − 1080.16i − 3.64917i
\(297\) 0 0
\(298\) 216.201 0.725506
\(299\) 58.2755i 0.194901i
\(300\) 0 0
\(301\) 366.281 1.21688
\(302\) − 111.916i − 0.370582i
\(303\) 0 0
\(304\) 692.068 2.27654
\(305\) 566.615i 1.85775i
\(306\) 0 0
\(307\) 358.046 1.16627 0.583136 0.812375i \(-0.301825\pi\)
0.583136 + 0.812375i \(0.301825\pi\)
\(308\) − 648.798i − 2.10649i
\(309\) 0 0
\(310\) −279.730 −0.902353
\(311\) − 101.682i − 0.326953i −0.986547 0.163476i \(-0.947729\pi\)
0.986547 0.163476i \(-0.0522707\pi\)
\(312\) 0 0
\(313\) −113.540 −0.362747 −0.181374 0.983414i \(-0.558054\pi\)
−0.181374 + 0.983414i \(0.558054\pi\)
\(314\) 210.729i 0.671111i
\(315\) 0 0
\(316\) 498.637 1.57796
\(317\) − 563.141i − 1.77647i −0.459389 0.888235i \(-0.651932\pi\)
0.459389 0.888235i \(-0.348068\pi\)
\(318\) 0 0
\(319\) −840.217 −2.63391
\(320\) − 79.5838i − 0.248699i
\(321\) 0 0
\(322\) −100.565 −0.312315
\(323\) 604.272i 1.87081i
\(324\) 0 0
\(325\) 263.490 0.810738
\(326\) 348.173i 1.06801i
\(327\) 0 0
\(328\) −583.339 −1.77847
\(329\) 279.589i 0.849814i
\(330\) 0 0
\(331\) 248.907 0.751985 0.375993 0.926623i \(-0.377302\pi\)
0.375993 + 0.926623i \(0.377302\pi\)
\(332\) − 1134.63i − 3.41755i
\(333\) 0 0
\(334\) 527.648 1.57979
\(335\) 135.630i 0.404864i
\(336\) 0 0
\(337\) 81.4574 0.241713 0.120857 0.992670i \(-0.461436\pi\)
0.120857 + 0.992670i \(0.461436\pi\)
\(338\) 262.694i 0.777201i
\(339\) 0 0
\(340\) −1498.99 −4.40880
\(341\) 168.555i 0.494295i
\(342\) 0 0
\(343\) −357.089 −1.04108
\(344\) 1351.99i 3.93021i
\(345\) 0 0
\(346\) −136.011 −0.393094
\(347\) 94.2754i 0.271687i 0.990730 + 0.135844i \(0.0433745\pi\)
−0.990730 + 0.135844i \(0.956626\pi\)
\(348\) 0 0
\(349\) −544.452 −1.56003 −0.780017 0.625759i \(-0.784789\pi\)
−0.780017 + 0.625759i \(0.784789\pi\)
\(350\) 454.701i 1.29915i
\(351\) 0 0
\(352\) 419.614 1.19209
\(353\) − 420.122i − 1.19015i −0.803671 0.595073i \(-0.797123\pi\)
0.803671 0.595073i \(-0.202877\pi\)
\(354\) 0 0
\(355\) −546.482 −1.53939
\(356\) 621.247i 1.74508i
\(357\) 0 0
\(358\) 791.701 2.21145
\(359\) 381.254i 1.06199i 0.847375 + 0.530995i \(0.178182\pi\)
−0.847375 + 0.530995i \(0.821818\pi\)
\(360\) 0 0
\(361\) 299.947 0.830879
\(362\) − 488.561i − 1.34962i
\(363\) 0 0
\(364\) 407.790 1.12030
\(365\) 328.783i 0.900775i
\(366\) 0 0
\(367\) 9.78670 0.0266668 0.0133334 0.999911i \(-0.495756\pi\)
0.0133334 + 0.999911i \(0.495756\pi\)
\(368\) − 160.345i − 0.435721i
\(369\) 0 0
\(370\) 1605.03 4.33791
\(371\) 418.140i 1.12706i
\(372\) 0 0
\(373\) 254.878 0.683319 0.341659 0.939824i \(-0.389011\pi\)
0.341659 + 0.939824i \(0.389011\pi\)
\(374\) 1311.49i 3.50666i
\(375\) 0 0
\(376\) −1032.00 −2.74468
\(377\) − 528.102i − 1.40080i
\(378\) 0 0
\(379\) −99.9256 −0.263656 −0.131828 0.991273i \(-0.542085\pi\)
−0.131828 + 0.991273i \(0.542085\pi\)
\(380\) 1639.58i 4.31469i
\(381\) 0 0
\(382\) 727.013 1.90318
\(383\) − 10.5938i − 0.0276600i −0.999904 0.0138300i \(-0.995598\pi\)
0.999904 0.0138300i \(-0.00440236\pi\)
\(384\) 0 0
\(385\) 528.317 1.37225
\(386\) − 695.279i − 1.80124i
\(387\) 0 0
\(388\) 40.6687 0.104816
\(389\) − 62.8874i − 0.161664i −0.996728 0.0808322i \(-0.974242\pi\)
0.996728 0.0808322i \(-0.0257578\pi\)
\(390\) 0 0
\(391\) 140.004 0.358067
\(392\) − 466.209i − 1.18931i
\(393\) 0 0
\(394\) 956.082 2.42661
\(395\) 406.041i 1.02795i
\(396\) 0 0
\(397\) 461.795 1.16321 0.581605 0.813471i \(-0.302425\pi\)
0.581605 + 0.813471i \(0.302425\pi\)
\(398\) − 574.521i − 1.44352i
\(399\) 0 0
\(400\) −724.994 −1.81248
\(401\) 496.739i 1.23875i 0.785095 + 0.619376i \(0.212614\pi\)
−0.785095 + 0.619376i \(0.787386\pi\)
\(402\) 0 0
\(403\) −105.942 −0.262883
\(404\) 1061.63i 2.62780i
\(405\) 0 0
\(406\) 911.340 2.24468
\(407\) − 967.130i − 2.37624i
\(408\) 0 0
\(409\) 364.666 0.891605 0.445802 0.895131i \(-0.352918\pi\)
0.445802 + 0.895131i \(0.352918\pi\)
\(410\) − 866.797i − 2.11414i
\(411\) 0 0
\(412\) 1402.45 3.40401
\(413\) − 67.9204i − 0.164456i
\(414\) 0 0
\(415\) 923.930 2.22634
\(416\) 263.741i 0.633992i
\(417\) 0 0
\(418\) 1434.50 3.43181
\(419\) 430.068i 1.02642i 0.858264 + 0.513208i \(0.171543\pi\)
−0.858264 + 0.513208i \(0.828457\pi\)
\(420\) 0 0
\(421\) −522.669 −1.24149 −0.620747 0.784011i \(-0.713171\pi\)
−0.620747 + 0.784011i \(0.713171\pi\)
\(422\) 755.971i 1.79140i
\(423\) 0 0
\(424\) −1543.41 −3.64012
\(425\) − 633.021i − 1.48946i
\(426\) 0 0
\(427\) −370.322 −0.867265
\(428\) 1406.99i 3.28737i
\(429\) 0 0
\(430\) −2008.96 −4.67199
\(431\) 110.538i 0.256469i 0.991744 + 0.128235i \(0.0409310\pi\)
−0.991744 + 0.128235i \(0.959069\pi\)
\(432\) 0 0
\(433\) −73.8530 −0.170561 −0.0852806 0.996357i \(-0.527179\pi\)
−0.0852806 + 0.996357i \(0.527179\pi\)
\(434\) − 182.823i − 0.421250i
\(435\) 0 0
\(436\) −516.837 −1.18541
\(437\) − 153.135i − 0.350424i
\(438\) 0 0
\(439\) 330.242 0.752260 0.376130 0.926567i \(-0.377255\pi\)
0.376130 + 0.926567i \(0.377255\pi\)
\(440\) 1950.09i 4.43202i
\(441\) 0 0
\(442\) −824.312 −1.86496
\(443\) 425.440i 0.960360i 0.877170 + 0.480180i \(0.159429\pi\)
−0.877170 + 0.480180i \(0.840571\pi\)
\(444\) 0 0
\(445\) −505.883 −1.13682
\(446\) − 1406.64i − 3.15391i
\(447\) 0 0
\(448\) 52.0135 0.116102
\(449\) − 149.718i − 0.333447i −0.986004 0.166724i \(-0.946681\pi\)
0.986004 0.166724i \(-0.0533187\pi\)
\(450\) 0 0
\(451\) −522.300 −1.15809
\(452\) 716.341i 1.58483i
\(453\) 0 0
\(454\) −79.0412 −0.174100
\(455\) 332.064i 0.729811i
\(456\) 0 0
\(457\) −424.824 −0.929593 −0.464797 0.885418i \(-0.653873\pi\)
−0.464797 + 0.885418i \(0.653873\pi\)
\(458\) 943.520i 2.06009i
\(459\) 0 0
\(460\) 379.876 0.825816
\(461\) − 28.8882i − 0.0626642i −0.999509 0.0313321i \(-0.990025\pi\)
0.999509 0.0313321i \(-0.00997495\pi\)
\(462\) 0 0
\(463\) 514.451 1.11113 0.555563 0.831474i \(-0.312503\pi\)
0.555563 + 0.831474i \(0.312503\pi\)
\(464\) 1453.08i 3.13163i
\(465\) 0 0
\(466\) 462.736 0.992995
\(467\) 424.921i 0.909894i 0.890518 + 0.454947i \(0.150342\pi\)
−0.890518 + 0.454947i \(0.849658\pi\)
\(468\) 0 0
\(469\) −88.6433 −0.189005
\(470\) − 1533.47i − 3.26270i
\(471\) 0 0
\(472\) 250.703 0.531151
\(473\) 1210.52i 2.55925i
\(474\) 0 0
\(475\) −692.393 −1.45767
\(476\) − 979.695i − 2.05818i
\(477\) 0 0
\(478\) 1036.16 2.16769
\(479\) − 400.532i − 0.836183i −0.908405 0.418091i \(-0.862699\pi\)
0.908405 0.418091i \(-0.137301\pi\)
\(480\) 0 0
\(481\) 607.872 1.26377
\(482\) 373.431i 0.774753i
\(483\) 0 0
\(484\) 1073.39 2.21775
\(485\) 33.1166i 0.0682817i
\(486\) 0 0
\(487\) −823.321 −1.69060 −0.845299 0.534293i \(-0.820578\pi\)
−0.845299 + 0.534293i \(0.820578\pi\)
\(488\) − 1366.91i − 2.80104i
\(489\) 0 0
\(490\) 692.751 1.41378
\(491\) − 258.140i − 0.525744i −0.964831 0.262872i \(-0.915330\pi\)
0.964831 0.262872i \(-0.0846698\pi\)
\(492\) 0 0
\(493\) −1268.74 −2.57351
\(494\) 901.625i 1.82515i
\(495\) 0 0
\(496\) 291.500 0.587701
\(497\) − 357.164i − 0.718640i
\(498\) 0 0
\(499\) −49.8261 −0.0998520 −0.0499260 0.998753i \(-0.515899\pi\)
−0.0499260 + 0.998753i \(0.515899\pi\)
\(500\) − 123.215i − 0.246430i
\(501\) 0 0
\(502\) −1578.62 −3.14466
\(503\) − 879.672i − 1.74885i −0.485160 0.874426i \(-0.661238\pi\)
0.485160 0.874426i \(-0.338762\pi\)
\(504\) 0 0
\(505\) −864.488 −1.71186
\(506\) − 332.359i − 0.656836i
\(507\) 0 0
\(508\) −99.7319 −0.196323
\(509\) − 116.561i − 0.229001i −0.993423 0.114500i \(-0.963473\pi\)
0.993423 0.114500i \(-0.0365267\pi\)
\(510\) 0 0
\(511\) −214.882 −0.420513
\(512\) 1146.27i 2.23880i
\(513\) 0 0
\(514\) −373.688 −0.727019
\(515\) 1142.02i 2.21752i
\(516\) 0 0
\(517\) −924.013 −1.78726
\(518\) 1049.00i 2.02509i
\(519\) 0 0
\(520\) −1225.69 −2.35710
\(521\) − 909.374i − 1.74544i −0.488222 0.872719i \(-0.662354\pi\)
0.488222 0.872719i \(-0.337646\pi\)
\(522\) 0 0
\(523\) 942.453 1.80201 0.901007 0.433804i \(-0.142829\pi\)
0.901007 + 0.433804i \(0.142829\pi\)
\(524\) − 151.604i − 0.289320i
\(525\) 0 0
\(526\) 383.010 0.728156
\(527\) 254.520i 0.482960i
\(528\) 0 0
\(529\) 493.520 0.932930
\(530\) − 2293.39i − 4.32715i
\(531\) 0 0
\(532\) −1071.58 −2.01425
\(533\) − 328.282i − 0.615913i
\(534\) 0 0
\(535\) −1145.72 −2.14153
\(536\) − 327.194i − 0.610437i
\(537\) 0 0
\(538\) 1004.77 1.86761
\(539\) − 417.426i − 0.774445i
\(540\) 0 0
\(541\) 697.261 1.28884 0.644419 0.764673i \(-0.277099\pi\)
0.644419 + 0.764673i \(0.277099\pi\)
\(542\) − 91.5196i − 0.168855i
\(543\) 0 0
\(544\) 633.624 1.16475
\(545\) − 420.861i − 0.772223i
\(546\) 0 0
\(547\) 951.659 1.73978 0.869890 0.493246i \(-0.164190\pi\)
0.869890 + 0.493246i \(0.164190\pi\)
\(548\) 255.117i 0.465542i
\(549\) 0 0
\(550\) −1502.74 −2.73226
\(551\) 1387.74i 2.51858i
\(552\) 0 0
\(553\) −265.376 −0.479884
\(554\) − 1008.85i − 1.82103i
\(555\) 0 0
\(556\) −226.833 −0.407973
\(557\) − 840.713i − 1.50936i −0.656093 0.754680i \(-0.727792\pi\)
0.656093 0.754680i \(-0.272208\pi\)
\(558\) 0 0
\(559\) −760.852 −1.36109
\(560\) − 913.676i − 1.63156i
\(561\) 0 0
\(562\) 1869.76 3.32698
\(563\) − 838.241i − 1.48888i −0.667688 0.744442i \(-0.732716\pi\)
0.667688 0.744442i \(-0.267284\pi\)
\(564\) 0 0
\(565\) −583.318 −1.03242
\(566\) − 786.756i − 1.39003i
\(567\) 0 0
\(568\) 1318.34 2.32102
\(569\) 940.180i 1.65234i 0.563423 + 0.826168i \(0.309484\pi\)
−0.563423 + 0.826168i \(0.690516\pi\)
\(570\) 0 0
\(571\) 612.707 1.07304 0.536521 0.843887i \(-0.319738\pi\)
0.536521 + 0.843887i \(0.319738\pi\)
\(572\) 1347.70i 2.35613i
\(573\) 0 0
\(574\) 566.512 0.986954
\(575\) 160.421i 0.278993i
\(576\) 0 0
\(577\) −68.6703 −0.119013 −0.0595064 0.998228i \(-0.518953\pi\)
−0.0595064 + 0.998228i \(0.518953\pi\)
\(578\) 944.404i 1.63392i
\(579\) 0 0
\(580\) −3442.49 −5.93533
\(581\) 603.852i 1.03933i
\(582\) 0 0
\(583\) −1381.91 −2.37035
\(584\) − 793.160i − 1.35815i
\(585\) 0 0
\(586\) −440.628 −0.751925
\(587\) 613.626i 1.04536i 0.852529 + 0.522680i \(0.175068\pi\)
−0.852529 + 0.522680i \(0.824932\pi\)
\(588\) 0 0
\(589\) 278.392 0.472652
\(590\) 372.526i 0.631400i
\(591\) 0 0
\(592\) −1672.56 −2.82527
\(593\) − 10.5482i − 0.0177878i −0.999960 0.00889392i \(-0.997169\pi\)
0.999960 0.00889392i \(-0.00283106\pi\)
\(594\) 0 0
\(595\) 797.767 1.34079
\(596\) − 533.755i − 0.895562i
\(597\) 0 0
\(598\) 208.898 0.349328
\(599\) 12.8106i 0.0213866i 0.999943 + 0.0106933i \(0.00340385\pi\)
−0.999943 + 0.0106933i \(0.996596\pi\)
\(600\) 0 0
\(601\) 1099.50 1.82946 0.914728 0.404071i \(-0.132405\pi\)
0.914728 + 0.404071i \(0.132405\pi\)
\(602\) − 1312.99i − 2.18105i
\(603\) 0 0
\(604\) −276.297 −0.457445
\(605\) 874.064i 1.44473i
\(606\) 0 0
\(607\) 169.097 0.278579 0.139289 0.990252i \(-0.455518\pi\)
0.139289 + 0.990252i \(0.455518\pi\)
\(608\) − 693.052i − 1.13989i
\(609\) 0 0
\(610\) 2031.12 3.32970
\(611\) − 580.771i − 0.950526i
\(612\) 0 0
\(613\) −52.3964 −0.0854754 −0.0427377 0.999086i \(-0.513608\pi\)
−0.0427377 + 0.999086i \(0.513608\pi\)
\(614\) − 1283.47i − 2.09034i
\(615\) 0 0
\(616\) −1274.52 −2.06902
\(617\) 425.530i 0.689677i 0.938662 + 0.344838i \(0.112066\pi\)
−0.938662 + 0.344838i \(0.887934\pi\)
\(618\) 0 0
\(619\) 167.798 0.271079 0.135539 0.990772i \(-0.456723\pi\)
0.135539 + 0.990772i \(0.456723\pi\)
\(620\) 690.594i 1.11386i
\(621\) 0 0
\(622\) −364.496 −0.586007
\(623\) − 330.630i − 0.530706i
\(624\) 0 0
\(625\) −572.966 −0.916746
\(626\) 407.002i 0.650163i
\(627\) 0 0
\(628\) 520.246 0.828416
\(629\) − 1460.38i − 2.32175i
\(630\) 0 0
\(631\) −930.073 −1.47397 −0.736983 0.675911i \(-0.763750\pi\)
−0.736983 + 0.675911i \(0.763750\pi\)
\(632\) − 979.538i − 1.54990i
\(633\) 0 0
\(634\) −2018.67 −3.18402
\(635\) − 81.2119i − 0.127893i
\(636\) 0 0
\(637\) 262.365 0.411876
\(638\) 3011.89i 4.72083i
\(639\) 0 0
\(640\) −1062.35 −1.65992
\(641\) 23.8249i 0.0371683i 0.999827 + 0.0185841i \(0.00591585\pi\)
−0.999827 + 0.0185841i \(0.994084\pi\)
\(642\) 0 0
\(643\) 762.785 1.18629 0.593146 0.805095i \(-0.297886\pi\)
0.593146 + 0.805095i \(0.297886\pi\)
\(644\) 248.275i 0.385520i
\(645\) 0 0
\(646\) 2166.11 3.35311
\(647\) − 1067.19i − 1.64944i −0.565539 0.824722i \(-0.691332\pi\)
0.565539 0.824722i \(-0.308668\pi\)
\(648\) 0 0
\(649\) 224.470 0.345871
\(650\) − 944.521i − 1.45311i
\(651\) 0 0
\(652\) 859.565 1.31835
\(653\) − 437.847i − 0.670516i −0.942126 0.335258i \(-0.891176\pi\)
0.942126 0.335258i \(-0.108824\pi\)
\(654\) 0 0
\(655\) 123.451 0.188475
\(656\) 903.269i 1.37693i
\(657\) 0 0
\(658\) 1002.23 1.52315
\(659\) − 528.030i − 0.801260i −0.916240 0.400630i \(-0.868791\pi\)
0.916240 0.400630i \(-0.131209\pi\)
\(660\) 0 0
\(661\) −1149.03 −1.73831 −0.869157 0.494536i \(-0.835338\pi\)
−0.869157 + 0.494536i \(0.835338\pi\)
\(662\) − 892.247i − 1.34780i
\(663\) 0 0
\(664\) −2228.90 −3.35678
\(665\) − 872.590i − 1.31217i
\(666\) 0 0
\(667\) 321.525 0.482047
\(668\) − 1302.65i − 1.95008i
\(669\) 0 0
\(670\) 486.185 0.725650
\(671\) − 1223.88i − 1.82396i
\(672\) 0 0
\(673\) 932.678 1.38585 0.692926 0.721009i \(-0.256321\pi\)
0.692926 + 0.721009i \(0.256321\pi\)
\(674\) − 291.997i − 0.433230i
\(675\) 0 0
\(676\) 648.537 0.959374
\(677\) − 216.494i − 0.319785i −0.987134 0.159892i \(-0.948885\pi\)
0.987134 0.159892i \(-0.0511147\pi\)
\(678\) 0 0
\(679\) −21.6440 −0.0318763
\(680\) 2944.67i 4.33039i
\(681\) 0 0
\(682\) 604.211 0.885940
\(683\) 230.195i 0.337036i 0.985699 + 0.168518i \(0.0538981\pi\)
−0.985699 + 0.168518i \(0.946102\pi\)
\(684\) 0 0
\(685\) −207.742 −0.303273
\(686\) 1280.04i 1.86595i
\(687\) 0 0
\(688\) 2093.49 3.04286
\(689\) − 868.574i − 1.26063i
\(690\) 0 0
\(691\) −13.7096 −0.0198402 −0.00992011 0.999951i \(-0.503158\pi\)
−0.00992011 + 0.999951i \(0.503158\pi\)
\(692\) 335.782i 0.485234i
\(693\) 0 0
\(694\) 337.945 0.486953
\(695\) − 184.711i − 0.265771i
\(696\) 0 0
\(697\) −788.681 −1.13154
\(698\) 1951.67i 2.79609i
\(699\) 0 0
\(700\) 1122.56 1.60366
\(701\) − 310.382i − 0.442770i −0.975186 0.221385i \(-0.928942\pi\)
0.975186 0.221385i \(-0.0710578\pi\)
\(702\) 0 0
\(703\) −1597.35 −2.27219
\(704\) 171.900i 0.244175i
\(705\) 0 0
\(706\) −1505.99 −2.13313
\(707\) − 565.003i − 0.799155i
\(708\) 0 0
\(709\) −1053.71 −1.48619 −0.743093 0.669188i \(-0.766642\pi\)
−0.743093 + 0.669188i \(0.766642\pi\)
\(710\) 1958.95i 2.75909i
\(711\) 0 0
\(712\) 1220.40 1.71404
\(713\) − 64.5007i − 0.0904638i
\(714\) 0 0
\(715\) −1097.44 −1.53488
\(716\) − 1954.54i − 2.72981i
\(717\) 0 0
\(718\) 1366.67 1.90343
\(719\) 1131.63i 1.57390i 0.617017 + 0.786950i \(0.288341\pi\)
−0.617017 + 0.786950i \(0.711659\pi\)
\(720\) 0 0
\(721\) −746.390 −1.03522
\(722\) − 1075.21i − 1.48921i
\(723\) 0 0
\(724\) −1206.16 −1.66596
\(725\) − 1453.76i − 2.00519i
\(726\) 0 0
\(727\) 70.7147 0.0972692 0.0486346 0.998817i \(-0.484513\pi\)
0.0486346 + 0.998817i \(0.484513\pi\)
\(728\) − 801.075i − 1.10038i
\(729\) 0 0
\(730\) 1178.57 1.61448
\(731\) 1827.91i 2.50056i
\(732\) 0 0
\(733\) 980.186 1.33722 0.668612 0.743611i \(-0.266889\pi\)
0.668612 + 0.743611i \(0.266889\pi\)
\(734\) − 35.0820i − 0.0477956i
\(735\) 0 0
\(736\) −160.574 −0.218171
\(737\) − 292.957i − 0.397500i
\(738\) 0 0
\(739\) −483.337 −0.654042 −0.327021 0.945017i \(-0.606045\pi\)
−0.327021 + 0.945017i \(0.606045\pi\)
\(740\) − 3962.48i − 5.35470i
\(741\) 0 0
\(742\) 1498.89 2.02007
\(743\) − 530.012i − 0.713341i −0.934230 0.356671i \(-0.883912\pi\)
0.934230 0.356671i \(-0.116088\pi\)
\(744\) 0 0
\(745\) 434.638 0.583406
\(746\) − 913.650i − 1.22473i
\(747\) 0 0
\(748\) 3237.79 4.32860
\(749\) − 748.806i − 0.999741i
\(750\) 0 0
\(751\) 263.871 0.351360 0.175680 0.984447i \(-0.443788\pi\)
0.175680 + 0.984447i \(0.443788\pi\)
\(752\) 1597.99i 2.12499i
\(753\) 0 0
\(754\) −1893.07 −2.51070
\(755\) − 224.989i − 0.297999i
\(756\) 0 0
\(757\) −1216.09 −1.60646 −0.803232 0.595667i \(-0.796888\pi\)
−0.803232 + 0.595667i \(0.796888\pi\)
\(758\) 358.199i 0.472558i
\(759\) 0 0
\(760\) 3220.85 4.23796
\(761\) 599.778i 0.788145i 0.919079 + 0.394073i \(0.128934\pi\)
−0.919079 + 0.394073i \(0.871066\pi\)
\(762\) 0 0
\(763\) 275.062 0.360501
\(764\) − 1794.84i − 2.34927i
\(765\) 0 0
\(766\) −37.9750 −0.0495758
\(767\) 141.087i 0.183946i
\(768\) 0 0
\(769\) 378.397 0.492063 0.246032 0.969262i \(-0.420873\pi\)
0.246032 + 0.969262i \(0.420873\pi\)
\(770\) − 1893.84i − 2.45953i
\(771\) 0 0
\(772\) −1716.50 −2.22344
\(773\) − 868.664i − 1.12376i −0.827220 0.561879i \(-0.810079\pi\)
0.827220 0.561879i \(-0.189921\pi\)
\(774\) 0 0
\(775\) −291.637 −0.376305
\(776\) − 79.8909i − 0.102952i
\(777\) 0 0
\(778\) −225.430 −0.289756
\(779\) 862.652i 1.10738i
\(780\) 0 0
\(781\) 1180.39 1.51139
\(782\) − 501.867i − 0.641774i
\(783\) 0 0
\(784\) −721.899 −0.920790
\(785\) 423.637i 0.539665i
\(786\) 0 0
\(787\) 56.4537 0.0717328 0.0358664 0.999357i \(-0.488581\pi\)
0.0358664 + 0.999357i \(0.488581\pi\)
\(788\) − 2360.37i − 2.99539i
\(789\) 0 0
\(790\) 1455.52 1.84243
\(791\) − 381.239i − 0.481971i
\(792\) 0 0
\(793\) 769.245 0.970045
\(794\) − 1655.38i − 2.08486i
\(795\) 0 0
\(796\) −1418.37 −1.78187
\(797\) 457.708i 0.574289i 0.957887 + 0.287145i \(0.0927060\pi\)
−0.957887 + 0.287145i \(0.907294\pi\)
\(798\) 0 0
\(799\) −1395.27 −1.74628
\(800\) 726.025i 0.907531i
\(801\) 0 0
\(802\) 1780.64 2.22025
\(803\) − 710.165i − 0.884390i
\(804\) 0 0
\(805\) −202.171 −0.251144
\(806\) 379.765i 0.471173i
\(807\) 0 0
\(808\) 2085.50 2.58107
\(809\) 1039.81i 1.28530i 0.766159 + 0.642651i \(0.222166\pi\)
−0.766159 + 0.642651i \(0.777834\pi\)
\(810\) 0 0
\(811\) −855.629 −1.05503 −0.527515 0.849546i \(-0.676876\pi\)
−0.527515 + 0.849546i \(0.676876\pi\)
\(812\) − 2249.91i − 2.77082i
\(813\) 0 0
\(814\) −3466.83 −4.25901
\(815\) 699.946i 0.858829i
\(816\) 0 0
\(817\) 1999.35 2.44719
\(818\) − 1307.20i − 1.59805i
\(819\) 0 0
\(820\) −2139.94 −2.60968
\(821\) − 701.585i − 0.854549i −0.904122 0.427275i \(-0.859474\pi\)
0.904122 0.427275i \(-0.140526\pi\)
\(822\) 0 0
\(823\) −507.346 −0.616460 −0.308230 0.951312i \(-0.599737\pi\)
−0.308230 + 0.951312i \(0.599737\pi\)
\(824\) − 2755.03i − 3.34348i
\(825\) 0 0
\(826\) −243.471 −0.294760
\(827\) − 1126.11i − 1.36168i −0.732434 0.680838i \(-0.761616\pi\)
0.732434 0.680838i \(-0.238384\pi\)
\(828\) 0 0
\(829\) −1160.25 −1.39957 −0.699786 0.714352i \(-0.746721\pi\)
−0.699786 + 0.714352i \(0.746721\pi\)
\(830\) − 3311.97i − 3.99033i
\(831\) 0 0
\(832\) −108.044 −0.129861
\(833\) − 630.320i − 0.756686i
\(834\) 0 0
\(835\) 1060.75 1.27036
\(836\) − 3541.47i − 4.23621i
\(837\) 0 0
\(838\) 1541.65 1.83967
\(839\) 607.665i 0.724272i 0.932125 + 0.362136i \(0.117952\pi\)
−0.932125 + 0.362136i \(0.882048\pi\)
\(840\) 0 0
\(841\) −2072.71 −2.46458
\(842\) 1873.59i 2.22517i
\(843\) 0 0
\(844\) 1866.34 2.21130
\(845\) 528.105i 0.624976i
\(846\) 0 0
\(847\) −571.261 −0.674453
\(848\) 2389.89i 2.81826i
\(849\) 0 0
\(850\) −2269.17 −2.66961
\(851\) 370.091i 0.434890i
\(852\) 0 0
\(853\) −273.450 −0.320574 −0.160287 0.987070i \(-0.551242\pi\)
−0.160287 + 0.987070i \(0.551242\pi\)
\(854\) 1327.48i 1.55442i
\(855\) 0 0
\(856\) 2763.94 3.22890
\(857\) 1326.64i 1.54800i 0.633184 + 0.774001i \(0.281747\pi\)
−0.633184 + 0.774001i \(0.718253\pi\)
\(858\) 0 0
\(859\) 251.982 0.293343 0.146672 0.989185i \(-0.453144\pi\)
0.146672 + 0.989185i \(0.453144\pi\)
\(860\) 4959.70i 5.76709i
\(861\) 0 0
\(862\) 396.242 0.459677
\(863\) − 456.453i − 0.528915i −0.964397 0.264457i \(-0.914807\pi\)
0.964397 0.264457i \(-0.0851929\pi\)
\(864\) 0 0
\(865\) −273.428 −0.316101
\(866\) 264.738i 0.305702i
\(867\) 0 0
\(868\) −451.351 −0.519990
\(869\) − 877.041i − 1.00925i
\(870\) 0 0
\(871\) 184.133 0.211404
\(872\) 1015.29i 1.16432i
\(873\) 0 0
\(874\) −548.937 −0.628075
\(875\) 65.5755i 0.0749434i
\(876\) 0 0
\(877\) 537.509 0.612896 0.306448 0.951887i \(-0.400860\pi\)
0.306448 + 0.951887i \(0.400860\pi\)
\(878\) − 1183.81i − 1.34830i
\(879\) 0 0
\(880\) 3019.61 3.43137
\(881\) 1228.12i 1.39401i 0.717067 + 0.697004i \(0.245484\pi\)
−0.717067 + 0.697004i \(0.754516\pi\)
\(882\) 0 0
\(883\) −57.2818 −0.0648718 −0.0324359 0.999474i \(-0.510326\pi\)
−0.0324359 + 0.999474i \(0.510326\pi\)
\(884\) 2035.06i 2.30210i
\(885\) 0 0
\(886\) 1525.06 1.72128
\(887\) 1335.11i 1.50520i 0.658481 + 0.752598i \(0.271199\pi\)
−0.658481 + 0.752598i \(0.728801\pi\)
\(888\) 0 0
\(889\) 53.0776 0.0597048
\(890\) 1813.42i 2.03755i
\(891\) 0 0
\(892\) −3472.71 −3.89317
\(893\) 1526.14i 1.70900i
\(894\) 0 0
\(895\) 1591.59 1.77831
\(896\) − 694.320i − 0.774910i
\(897\) 0 0
\(898\) −536.687 −0.597647
\(899\) 584.516i 0.650184i
\(900\) 0 0
\(901\) −2086.71 −2.31599
\(902\) 1872.27i 2.07568i
\(903\) 0 0
\(904\) 1407.20 1.55664
\(905\) − 982.175i − 1.08528i
\(906\) 0 0
\(907\) −1726.30 −1.90331 −0.951654 0.307173i \(-0.900617\pi\)
−0.951654 + 0.307173i \(0.900617\pi\)
\(908\) 195.136i 0.214908i
\(909\) 0 0
\(910\) 1190.34 1.30806
\(911\) 1710.32i 1.87741i 0.344721 + 0.938705i \(0.387974\pi\)
−0.344721 + 0.938705i \(0.612026\pi\)
\(912\) 0 0
\(913\) −1995.67 −2.18584
\(914\) 1522.85i 1.66614i
\(915\) 0 0
\(916\) 2329.35 2.54296
\(917\) 80.6840i 0.0879870i
\(918\) 0 0
\(919\) −307.462 −0.334561 −0.167280 0.985909i \(-0.553499\pi\)
−0.167280 + 0.985909i \(0.553499\pi\)
\(920\) − 746.240i − 0.811130i
\(921\) 0 0
\(922\) −103.554 −0.112315
\(923\) 741.913i 0.803807i
\(924\) 0 0
\(925\) 1673.35 1.80902
\(926\) − 1844.13i − 1.99150i
\(927\) 0 0
\(928\) 1455.14 1.56804
\(929\) 1510.66i 1.62612i 0.582180 + 0.813060i \(0.302200\pi\)
−0.582180 + 0.813060i \(0.697800\pi\)
\(930\) 0 0
\(931\) −689.438 −0.740535
\(932\) − 1142.40i − 1.22575i
\(933\) 0 0
\(934\) 1523.20 1.63083
\(935\) 2636.54i 2.81983i
\(936\) 0 0
\(937\) −452.031 −0.482423 −0.241212 0.970473i \(-0.577545\pi\)
−0.241212 + 0.970473i \(0.577545\pi\)
\(938\) 317.756i 0.338759i
\(939\) 0 0
\(940\) −3785.82 −4.02747
\(941\) − 706.234i − 0.750515i −0.926921 0.375257i \(-0.877554\pi\)
0.926921 0.375257i \(-0.122446\pi\)
\(942\) 0 0
\(943\) 199.868 0.211949
\(944\) − 388.200i − 0.411229i
\(945\) 0 0
\(946\) 4339.31 4.58701
\(947\) − 663.862i − 0.701016i −0.936560 0.350508i \(-0.886009\pi\)
0.936560 0.350508i \(-0.113991\pi\)
\(948\) 0 0
\(949\) 446.361 0.470349
\(950\) 2481.99i 2.61262i
\(951\) 0 0
\(952\) −1924.54 −2.02158
\(953\) 59.7064i 0.0626510i 0.999509 + 0.0313255i \(0.00997285\pi\)
−0.999509 + 0.0313255i \(0.990027\pi\)
\(954\) 0 0
\(955\) 1461.55 1.53041
\(956\) − 2558.05i − 2.67579i
\(957\) 0 0
\(958\) −1435.77 −1.49871
\(959\) − 135.774i − 0.141579i
\(960\) 0 0
\(961\) −843.741 −0.877982
\(962\) − 2179.01i − 2.26509i
\(963\) 0 0
\(964\) 921.923 0.956351
\(965\) − 1397.75i − 1.44844i
\(966\) 0 0
\(967\) 1441.18 1.49036 0.745182 0.666861i \(-0.232363\pi\)
0.745182 + 0.666861i \(0.232363\pi\)
\(968\) − 2108.60i − 2.17831i
\(969\) 0 0
\(970\) 118.712 0.122383
\(971\) − 942.274i − 0.970416i −0.874399 0.485208i \(-0.838744\pi\)
0.874399 0.485208i \(-0.161256\pi\)
\(972\) 0 0
\(973\) 120.721 0.124071
\(974\) 2951.33i 3.03011i
\(975\) 0 0
\(976\) −2116.58 −2.16863
\(977\) 1642.03i 1.68068i 0.542058 + 0.840341i \(0.317645\pi\)
−0.542058 + 0.840341i \(0.682355\pi\)
\(978\) 0 0
\(979\) 1092.70 1.11614
\(980\) − 1710.26i − 1.74516i
\(981\) 0 0
\(982\) −925.345 −0.942307
\(983\) 704.163i 0.716341i 0.933656 + 0.358170i \(0.116599\pi\)
−0.933656 + 0.358170i \(0.883401\pi\)
\(984\) 0 0
\(985\) 1922.05 1.95132
\(986\) 4548.00i 4.61258i
\(987\) 0 0
\(988\) 2225.92 2.25296
\(989\) − 463.230i − 0.468383i
\(990\) 0 0
\(991\) −1524.19 −1.53803 −0.769015 0.639230i \(-0.779253\pi\)
−0.769015 + 0.639230i \(0.779253\pi\)
\(992\) − 291.914i − 0.294268i
\(993\) 0 0
\(994\) −1280.31 −1.28804
\(995\) − 1154.98i − 1.16079i
\(996\) 0 0
\(997\) −165.044 −0.165540 −0.0827702 0.996569i \(-0.526377\pi\)
−0.0827702 + 0.996569i \(0.526377\pi\)
\(998\) 178.610i 0.178968i
\(999\) 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.890.7 84
3.2 odd 2 inner 1143.3.b.a.890.78 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.7 84 1.1 even 1 trivial
1143.3.b.a.890.78 yes 84 3.2 odd 2 inner