Properties

Label 1143.3.b.a.890.17
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 890.17
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.68

$q$-expansion

\(f(q)\) \(=\) \(q-2.56806i q^{2} -2.59492 q^{4} +0.662170i q^{5} -12.8177 q^{7} -3.60832i q^{8} +O(q^{10})\) \(q-2.56806i q^{2} -2.59492 q^{4} +0.662170i q^{5} -12.8177 q^{7} -3.60832i q^{8} +1.70049 q^{10} -20.5391i q^{11} +4.92172 q^{13} +32.9165i q^{14} -19.6461 q^{16} -3.69398i q^{17} -1.02559 q^{19} -1.71828i q^{20} -52.7457 q^{22} +26.1310i q^{23} +24.5615 q^{25} -12.6393i q^{26} +33.2609 q^{28} -27.6705i q^{29} -29.6107 q^{31} +36.0190i q^{32} -9.48635 q^{34} -8.48748i q^{35} -2.50356 q^{37} +2.63378i q^{38} +2.38932 q^{40} +69.8927i q^{41} -66.7086 q^{43} +53.2974i q^{44} +67.1060 q^{46} -81.6550i q^{47} +115.293 q^{49} -63.0754i q^{50} -12.7715 q^{52} +81.3765i q^{53} +13.6004 q^{55} +46.2502i q^{56} -71.0595 q^{58} +66.1371i q^{59} +5.87636 q^{61} +76.0419i q^{62} +13.9145 q^{64} +3.25902i q^{65} +60.9345 q^{67} +9.58559i q^{68} -21.7963 q^{70} +115.840i q^{71} -104.353 q^{73} +6.42930i q^{74} +2.66133 q^{76} +263.264i q^{77} +15.1002 q^{79} -13.0090i q^{80} +179.489 q^{82} +0.533557i q^{83} +2.44604 q^{85} +171.312i q^{86} -74.1117 q^{88} +76.1149i q^{89} -63.0850 q^{91} -67.8080i q^{92} -209.695 q^{94} -0.679115i q^{95} +89.0106 q^{97} -296.078i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.56806i − 1.28403i −0.766692 0.642015i \(-0.778099\pi\)
0.766692 0.642015i \(-0.221901\pi\)
\(3\) 0 0
\(4\) −2.59492 −0.648731
\(5\) 0.662170i 0.132434i 0.997805 + 0.0662170i \(0.0210930\pi\)
−0.997805 + 0.0662170i \(0.978907\pi\)
\(6\) 0 0
\(7\) −12.8177 −1.83110 −0.915548 0.402210i \(-0.868242\pi\)
−0.915548 + 0.402210i \(0.868242\pi\)
\(8\) − 3.60832i − 0.451040i
\(9\) 0 0
\(10\) 1.70049 0.170049
\(11\) − 20.5391i − 1.86719i −0.358326 0.933597i \(-0.616652\pi\)
0.358326 0.933597i \(-0.383348\pi\)
\(12\) 0 0
\(13\) 4.92172 0.378594 0.189297 0.981920i \(-0.439379\pi\)
0.189297 + 0.981920i \(0.439379\pi\)
\(14\) 32.9165i 2.35118i
\(15\) 0 0
\(16\) −19.6461 −1.22788
\(17\) − 3.69398i − 0.217293i −0.994080 0.108646i \(-0.965348\pi\)
0.994080 0.108646i \(-0.0346516\pi\)
\(18\) 0 0
\(19\) −1.02559 −0.0539784 −0.0269892 0.999636i \(-0.508592\pi\)
−0.0269892 + 0.999636i \(0.508592\pi\)
\(20\) − 1.71828i − 0.0859140i
\(21\) 0 0
\(22\) −52.7457 −2.39753
\(23\) 26.1310i 1.13613i 0.822983 + 0.568066i \(0.192308\pi\)
−0.822983 + 0.568066i \(0.807692\pi\)
\(24\) 0 0
\(25\) 24.5615 0.982461
\(26\) − 12.6393i − 0.486126i
\(27\) 0 0
\(28\) 33.2609 1.18789
\(29\) − 27.6705i − 0.954156i −0.878861 0.477078i \(-0.841696\pi\)
0.878861 0.477078i \(-0.158304\pi\)
\(30\) 0 0
\(31\) −29.6107 −0.955183 −0.477591 0.878582i \(-0.658490\pi\)
−0.477591 + 0.878582i \(0.658490\pi\)
\(32\) 36.0190i 1.12559i
\(33\) 0 0
\(34\) −9.48635 −0.279010
\(35\) − 8.48748i − 0.242499i
\(36\) 0 0
\(37\) −2.50356 −0.0676639 −0.0338319 0.999428i \(-0.510771\pi\)
−0.0338319 + 0.999428i \(0.510771\pi\)
\(38\) 2.63378i 0.0693099i
\(39\) 0 0
\(40\) 2.38932 0.0597331
\(41\) 69.8927i 1.70470i 0.522972 + 0.852350i \(0.324823\pi\)
−0.522972 + 0.852350i \(0.675177\pi\)
\(42\) 0 0
\(43\) −66.7086 −1.55136 −0.775682 0.631124i \(-0.782594\pi\)
−0.775682 + 0.631124i \(0.782594\pi\)
\(44\) 53.2974i 1.21131i
\(45\) 0 0
\(46\) 67.1060 1.45883
\(47\) − 81.6550i − 1.73734i −0.495390 0.868671i \(-0.664975\pi\)
0.495390 0.868671i \(-0.335025\pi\)
\(48\) 0 0
\(49\) 115.293 2.35291
\(50\) − 63.0754i − 1.26151i
\(51\) 0 0
\(52\) −12.7715 −0.245606
\(53\) 81.3765i 1.53541i 0.640806 + 0.767703i \(0.278600\pi\)
−0.640806 + 0.767703i \(0.721400\pi\)
\(54\) 0 0
\(55\) 13.6004 0.247280
\(56\) 46.2502i 0.825897i
\(57\) 0 0
\(58\) −71.0595 −1.22516
\(59\) 66.1371i 1.12097i 0.828165 + 0.560484i \(0.189385\pi\)
−0.828165 + 0.560484i \(0.810615\pi\)
\(60\) 0 0
\(61\) 5.87636 0.0963337 0.0481669 0.998839i \(-0.484662\pi\)
0.0481669 + 0.998839i \(0.484662\pi\)
\(62\) 76.0419i 1.22648i
\(63\) 0 0
\(64\) 13.9145 0.217414
\(65\) 3.25902i 0.0501387i
\(66\) 0 0
\(67\) 60.9345 0.909470 0.454735 0.890627i \(-0.349734\pi\)
0.454735 + 0.890627i \(0.349734\pi\)
\(68\) 9.58559i 0.140965i
\(69\) 0 0
\(70\) −21.7963 −0.311376
\(71\) 115.840i 1.63155i 0.578372 + 0.815773i \(0.303688\pi\)
−0.578372 + 0.815773i \(0.696312\pi\)
\(72\) 0 0
\(73\) −104.353 −1.42950 −0.714750 0.699380i \(-0.753459\pi\)
−0.714750 + 0.699380i \(0.753459\pi\)
\(74\) 6.42930i 0.0868824i
\(75\) 0 0
\(76\) 2.66133 0.0350175
\(77\) 263.264i 3.41901i
\(78\) 0 0
\(79\) 15.1002 0.191142 0.0955710 0.995423i \(-0.469532\pi\)
0.0955710 + 0.995423i \(0.469532\pi\)
\(80\) − 13.0090i − 0.162613i
\(81\) 0 0
\(82\) 179.489 2.18888
\(83\) 0.533557i 0.00642839i 0.999995 + 0.00321420i \(0.00102311\pi\)
−0.999995 + 0.00321420i \(0.998977\pi\)
\(84\) 0 0
\(85\) 2.44604 0.0287770
\(86\) 171.312i 1.99200i
\(87\) 0 0
\(88\) −74.1117 −0.842179
\(89\) 76.1149i 0.855223i 0.903963 + 0.427612i \(0.140645\pi\)
−0.903963 + 0.427612i \(0.859355\pi\)
\(90\) 0 0
\(91\) −63.0850 −0.693242
\(92\) − 67.8080i − 0.737043i
\(93\) 0 0
\(94\) −209.695 −2.23080
\(95\) − 0.679115i − 0.00714858i
\(96\) 0 0
\(97\) 89.0106 0.917635 0.458817 0.888531i \(-0.348273\pi\)
0.458817 + 0.888531i \(0.348273\pi\)
\(98\) − 296.078i − 3.02120i
\(99\) 0 0
\(100\) −63.7353 −0.637353
\(101\) − 50.7640i − 0.502614i −0.967907 0.251307i \(-0.919140\pi\)
0.967907 0.251307i \(-0.0808603\pi\)
\(102\) 0 0
\(103\) −202.537 −1.96638 −0.983191 0.182581i \(-0.941555\pi\)
−0.983191 + 0.182581i \(0.941555\pi\)
\(104\) − 17.7591i − 0.170761i
\(105\) 0 0
\(106\) 208.980 1.97151
\(107\) − 6.15381i − 0.0575122i −0.999586 0.0287561i \(-0.990845\pi\)
0.999586 0.0287561i \(-0.00915461\pi\)
\(108\) 0 0
\(109\) 42.1393 0.386599 0.193300 0.981140i \(-0.438081\pi\)
0.193300 + 0.981140i \(0.438081\pi\)
\(110\) − 34.9266i − 0.317515i
\(111\) 0 0
\(112\) 251.817 2.24836
\(113\) − 48.6558i − 0.430583i −0.976550 0.215291i \(-0.930930\pi\)
0.976550 0.215291i \(-0.0690701\pi\)
\(114\) 0 0
\(115\) −17.3032 −0.150462
\(116\) 71.8028i 0.618990i
\(117\) 0 0
\(118\) 169.844 1.43936
\(119\) 47.3482i 0.397884i
\(120\) 0 0
\(121\) −300.856 −2.48641
\(122\) − 15.0908i − 0.123695i
\(123\) 0 0
\(124\) 76.8374 0.619656
\(125\) 32.8182i 0.262545i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) 108.343i 0.846426i
\(129\) 0 0
\(130\) 8.36935 0.0643796
\(131\) − 98.0583i − 0.748536i −0.927321 0.374268i \(-0.877894\pi\)
0.927321 0.374268i \(-0.122106\pi\)
\(132\) 0 0
\(133\) 13.1457 0.0988397
\(134\) − 156.483i − 1.16779i
\(135\) 0 0
\(136\) −13.3291 −0.0980078
\(137\) 135.480i 0.988904i 0.869205 + 0.494452i \(0.164631\pi\)
−0.869205 + 0.494452i \(0.835369\pi\)
\(138\) 0 0
\(139\) 18.1978 0.130920 0.0654599 0.997855i \(-0.479149\pi\)
0.0654599 + 0.997855i \(0.479149\pi\)
\(140\) 22.0244i 0.157317i
\(141\) 0 0
\(142\) 297.483 2.09495
\(143\) − 101.088i − 0.706908i
\(144\) 0 0
\(145\) 18.3226 0.126363
\(146\) 267.986i 1.83552i
\(147\) 0 0
\(148\) 6.49655 0.0438956
\(149\) − 133.336i − 0.894869i −0.894317 0.447435i \(-0.852338\pi\)
0.894317 0.447435i \(-0.147662\pi\)
\(150\) 0 0
\(151\) 184.374 1.22102 0.610511 0.792008i \(-0.290964\pi\)
0.610511 + 0.792008i \(0.290964\pi\)
\(152\) 3.70066i 0.0243464i
\(153\) 0 0
\(154\) 676.076 4.39011
\(155\) − 19.6073i − 0.126499i
\(156\) 0 0
\(157\) 138.955 0.885065 0.442532 0.896752i \(-0.354080\pi\)
0.442532 + 0.896752i \(0.354080\pi\)
\(158\) − 38.7783i − 0.245432i
\(159\) 0 0
\(160\) −23.8507 −0.149067
\(161\) − 334.939i − 2.08036i
\(162\) 0 0
\(163\) −299.670 −1.83847 −0.919233 0.393715i \(-0.871190\pi\)
−0.919233 + 0.393715i \(0.871190\pi\)
\(164\) − 181.366i − 1.10589i
\(165\) 0 0
\(166\) 1.37020 0.00825424
\(167\) 105.420i 0.631257i 0.948883 + 0.315628i \(0.102215\pi\)
−0.948883 + 0.315628i \(0.897785\pi\)
\(168\) 0 0
\(169\) −144.777 −0.856667
\(170\) − 6.28158i − 0.0369505i
\(171\) 0 0
\(172\) 173.104 1.00642
\(173\) − 110.157i − 0.636744i −0.947966 0.318372i \(-0.896864\pi\)
0.947966 0.318372i \(-0.103136\pi\)
\(174\) 0 0
\(175\) −314.822 −1.79898
\(176\) 403.513i 2.29269i
\(177\) 0 0
\(178\) 195.467 1.09813
\(179\) − 205.760i − 1.14950i −0.818329 0.574750i \(-0.805099\pi\)
0.818329 0.574750i \(-0.194901\pi\)
\(180\) 0 0
\(181\) 43.1373 0.238328 0.119164 0.992875i \(-0.461979\pi\)
0.119164 + 0.992875i \(0.461979\pi\)
\(182\) 162.006i 0.890142i
\(183\) 0 0
\(184\) 94.2891 0.512441
\(185\) − 1.65779i − 0.00896100i
\(186\) 0 0
\(187\) −75.8711 −0.405728
\(188\) 211.889i 1.12707i
\(189\) 0 0
\(190\) −1.74401 −0.00917899
\(191\) − 64.0595i − 0.335390i −0.985839 0.167695i \(-0.946368\pi\)
0.985839 0.167695i \(-0.0536324\pi\)
\(192\) 0 0
\(193\) 191.411 0.991766 0.495883 0.868389i \(-0.334845\pi\)
0.495883 + 0.868389i \(0.334845\pi\)
\(194\) − 228.584i − 1.17827i
\(195\) 0 0
\(196\) −299.175 −1.52640
\(197\) 254.592i 1.29234i 0.763192 + 0.646172i \(0.223631\pi\)
−0.763192 + 0.646172i \(0.776369\pi\)
\(198\) 0 0
\(199\) −49.1976 −0.247224 −0.123612 0.992331i \(-0.539448\pi\)
−0.123612 + 0.992331i \(0.539448\pi\)
\(200\) − 88.6259i − 0.443129i
\(201\) 0 0
\(202\) −130.365 −0.645371
\(203\) 354.671i 1.74715i
\(204\) 0 0
\(205\) −46.2809 −0.225760
\(206\) 520.128i 2.52489i
\(207\) 0 0
\(208\) −96.6925 −0.464868
\(209\) 21.0647i 0.100788i
\(210\) 0 0
\(211\) 164.180 0.778105 0.389053 0.921216i \(-0.372802\pi\)
0.389053 + 0.921216i \(0.372802\pi\)
\(212\) − 211.166i − 0.996065i
\(213\) 0 0
\(214\) −15.8033 −0.0738473
\(215\) − 44.1725i − 0.205453i
\(216\) 0 0
\(217\) 379.540 1.74903
\(218\) − 108.216i − 0.496405i
\(219\) 0 0
\(220\) −35.2920 −0.160418
\(221\) − 18.1807i − 0.0822658i
\(222\) 0 0
\(223\) −173.368 −0.777434 −0.388717 0.921357i \(-0.627082\pi\)
−0.388717 + 0.921357i \(0.627082\pi\)
\(224\) − 461.679i − 2.06107i
\(225\) 0 0
\(226\) −124.951 −0.552881
\(227\) − 394.569i − 1.73819i −0.494645 0.869095i \(-0.664702\pi\)
0.494645 0.869095i \(-0.335298\pi\)
\(228\) 0 0
\(229\) 86.8349 0.379191 0.189596 0.981862i \(-0.439282\pi\)
0.189596 + 0.981862i \(0.439282\pi\)
\(230\) 44.4356i 0.193198i
\(231\) 0 0
\(232\) −99.8441 −0.430362
\(233\) − 344.348i − 1.47789i −0.673767 0.738944i \(-0.735325\pi\)
0.673767 0.738944i \(-0.264675\pi\)
\(234\) 0 0
\(235\) 54.0695 0.230083
\(236\) − 171.621i − 0.727206i
\(237\) 0 0
\(238\) 121.593 0.510895
\(239\) 98.9845i 0.414161i 0.978324 + 0.207081i \(0.0663962\pi\)
−0.978324 + 0.207081i \(0.933604\pi\)
\(240\) 0 0
\(241\) −343.314 −1.42454 −0.712269 0.701906i \(-0.752333\pi\)
−0.712269 + 0.701906i \(0.752333\pi\)
\(242\) 772.615i 3.19262i
\(243\) 0 0
\(244\) −15.2487 −0.0624946
\(245\) 76.3433i 0.311605i
\(246\) 0 0
\(247\) −5.04767 −0.0204359
\(248\) 106.845i 0.430826i
\(249\) 0 0
\(250\) 84.2790 0.337116
\(251\) − 211.652i − 0.843236i −0.906774 0.421618i \(-0.861462\pi\)
0.906774 0.421618i \(-0.138538\pi\)
\(252\) 0 0
\(253\) 536.708 2.12138
\(254\) − 28.9405i − 0.113939i
\(255\) 0 0
\(256\) 333.888 1.30425
\(257\) − 348.091i − 1.35444i −0.735781 0.677219i \(-0.763185\pi\)
0.735781 0.677219i \(-0.236815\pi\)
\(258\) 0 0
\(259\) 32.0898 0.123899
\(260\) − 8.45690i − 0.0325265i
\(261\) 0 0
\(262\) −251.819 −0.961142
\(263\) 290.421i 1.10426i 0.833757 + 0.552131i \(0.186185\pi\)
−0.833757 + 0.552131i \(0.813815\pi\)
\(264\) 0 0
\(265\) −53.8851 −0.203340
\(266\) − 33.7589i − 0.126913i
\(267\) 0 0
\(268\) −158.120 −0.590001
\(269\) − 351.648i − 1.30724i −0.756822 0.653621i \(-0.773249\pi\)
0.756822 0.653621i \(-0.226751\pi\)
\(270\) 0 0
\(271\) −256.884 −0.947912 −0.473956 0.880548i \(-0.657174\pi\)
−0.473956 + 0.880548i \(0.657174\pi\)
\(272\) 72.5722i 0.266809i
\(273\) 0 0
\(274\) 347.920 1.26978
\(275\) − 504.472i − 1.83445i
\(276\) 0 0
\(277\) −162.433 −0.586399 −0.293200 0.956051i \(-0.594720\pi\)
−0.293200 + 0.956051i \(0.594720\pi\)
\(278\) − 46.7331i − 0.168105i
\(279\) 0 0
\(280\) −30.6255 −0.109377
\(281\) − 102.926i − 0.366284i −0.983086 0.183142i \(-0.941373\pi\)
0.983086 0.183142i \(-0.0586268\pi\)
\(282\) 0 0
\(283\) −463.379 −1.63738 −0.818691 0.574235i \(-0.805300\pi\)
−0.818691 + 0.574235i \(0.805300\pi\)
\(284\) − 300.595i − 1.05843i
\(285\) 0 0
\(286\) −259.599 −0.907691
\(287\) − 895.861i − 3.12147i
\(288\) 0 0
\(289\) 275.355 0.952784
\(290\) − 47.0535i − 0.162253i
\(291\) 0 0
\(292\) 270.789 0.927360
\(293\) − 303.314i − 1.03520i −0.855623 0.517600i \(-0.826825\pi\)
0.855623 0.517600i \(-0.173175\pi\)
\(294\) 0 0
\(295\) −43.7940 −0.148454
\(296\) 9.03366i 0.0305191i
\(297\) 0 0
\(298\) −342.413 −1.14904
\(299\) 128.610i 0.430132i
\(300\) 0 0
\(301\) 855.049 2.84069
\(302\) − 473.484i − 1.56783i
\(303\) 0 0
\(304\) 20.1488 0.0662790
\(305\) 3.89115i 0.0127579i
\(306\) 0 0
\(307\) −282.642 −0.920659 −0.460330 0.887748i \(-0.652269\pi\)
−0.460330 + 0.887748i \(0.652269\pi\)
\(308\) − 683.149i − 2.21802i
\(309\) 0 0
\(310\) −50.3527 −0.162428
\(311\) 292.688i 0.941119i 0.882368 + 0.470559i \(0.155948\pi\)
−0.882368 + 0.470559i \(0.844052\pi\)
\(312\) 0 0
\(313\) −189.990 −0.606998 −0.303499 0.952832i \(-0.598155\pi\)
−0.303499 + 0.952832i \(0.598155\pi\)
\(314\) − 356.845i − 1.13645i
\(315\) 0 0
\(316\) −39.1839 −0.124000
\(317\) 427.306i 1.34797i 0.738745 + 0.673985i \(0.235419\pi\)
−0.738745 + 0.673985i \(0.764581\pi\)
\(318\) 0 0
\(319\) −568.328 −1.78159
\(320\) 9.21379i 0.0287931i
\(321\) 0 0
\(322\) −860.142 −2.67125
\(323\) 3.78851i 0.0117291i
\(324\) 0 0
\(325\) 120.885 0.371954
\(326\) 769.570i 2.36064i
\(327\) 0 0
\(328\) 252.195 0.768888
\(329\) 1046.63i 3.18124i
\(330\) 0 0
\(331\) −92.8700 −0.280574 −0.140287 0.990111i \(-0.544803\pi\)
−0.140287 + 0.990111i \(0.544803\pi\)
\(332\) − 1.38454i − 0.00417030i
\(333\) 0 0
\(334\) 270.724 0.810552
\(335\) 40.3490i 0.120445i
\(336\) 0 0
\(337\) −199.485 −0.591944 −0.295972 0.955197i \(-0.595644\pi\)
−0.295972 + 0.955197i \(0.595644\pi\)
\(338\) 371.795i 1.09998i
\(339\) 0 0
\(340\) −6.34729 −0.0186685
\(341\) 608.177i 1.78351i
\(342\) 0 0
\(343\) −849.716 −2.47731
\(344\) 240.706i 0.699727i
\(345\) 0 0
\(346\) −282.889 −0.817597
\(347\) − 12.9677i − 0.0373709i −0.999825 0.0186855i \(-0.994052\pi\)
0.999825 0.0186855i \(-0.00594812\pi\)
\(348\) 0 0
\(349\) −421.158 −1.20676 −0.603378 0.797455i \(-0.706179\pi\)
−0.603378 + 0.797455i \(0.706179\pi\)
\(350\) 808.480i 2.30994i
\(351\) 0 0
\(352\) 739.798 2.10170
\(353\) 668.166i 1.89282i 0.322965 + 0.946411i \(0.395320\pi\)
−0.322965 + 0.946411i \(0.604680\pi\)
\(354\) 0 0
\(355\) −76.7057 −0.216072
\(356\) − 197.512i − 0.554810i
\(357\) 0 0
\(358\) −528.405 −1.47599
\(359\) 368.277i 1.02584i 0.858436 + 0.512920i \(0.171436\pi\)
−0.858436 + 0.512920i \(0.828564\pi\)
\(360\) 0 0
\(361\) −359.948 −0.997086
\(362\) − 110.779i − 0.306020i
\(363\) 0 0
\(364\) 163.701 0.449727
\(365\) − 69.0997i − 0.189314i
\(366\) 0 0
\(367\) −19.2091 −0.0523409 −0.0261705 0.999657i \(-0.508331\pi\)
−0.0261705 + 0.999657i \(0.508331\pi\)
\(368\) − 513.372i − 1.39503i
\(369\) 0 0
\(370\) −4.25729 −0.0115062
\(371\) − 1043.06i − 2.81147i
\(372\) 0 0
\(373\) −79.8145 −0.213980 −0.106990 0.994260i \(-0.534121\pi\)
−0.106990 + 0.994260i \(0.534121\pi\)
\(374\) 194.841i 0.520966i
\(375\) 0 0
\(376\) −294.638 −0.783610
\(377\) − 136.187i − 0.361238i
\(378\) 0 0
\(379\) 116.514 0.307424 0.153712 0.988116i \(-0.450877\pi\)
0.153712 + 0.988116i \(0.450877\pi\)
\(380\) 1.76225i 0.00463751i
\(381\) 0 0
\(382\) −164.508 −0.430650
\(383\) 409.174i 1.06834i 0.845377 + 0.534170i \(0.179376\pi\)
−0.845377 + 0.534170i \(0.820624\pi\)
\(384\) 0 0
\(385\) −174.325 −0.452793
\(386\) − 491.554i − 1.27346i
\(387\) 0 0
\(388\) −230.976 −0.595298
\(389\) − 390.255i − 1.00323i −0.865092 0.501613i \(-0.832740\pi\)
0.865092 0.501613i \(-0.167260\pi\)
\(390\) 0 0
\(391\) 96.5274 0.246873
\(392\) − 416.013i − 1.06126i
\(393\) 0 0
\(394\) 653.806 1.65941
\(395\) 9.99892i 0.0253137i
\(396\) 0 0
\(397\) −564.041 −1.42076 −0.710379 0.703819i \(-0.751476\pi\)
−0.710379 + 0.703819i \(0.751476\pi\)
\(398\) 126.342i 0.317443i
\(399\) 0 0
\(400\) −482.537 −1.20634
\(401\) 383.854i 0.957242i 0.878022 + 0.478621i \(0.158863\pi\)
−0.878022 + 0.478621i \(0.841137\pi\)
\(402\) 0 0
\(403\) −145.735 −0.361626
\(404\) 131.729i 0.326061i
\(405\) 0 0
\(406\) 910.817 2.24339
\(407\) 51.4210i 0.126342i
\(408\) 0 0
\(409\) −134.307 −0.328378 −0.164189 0.986429i \(-0.552501\pi\)
−0.164189 + 0.986429i \(0.552501\pi\)
\(410\) 118.852i 0.289883i
\(411\) 0 0
\(412\) 525.569 1.27565
\(413\) − 847.724i − 2.05260i
\(414\) 0 0
\(415\) −0.353305 −0.000851338 0
\(416\) 177.275i 0.426143i
\(417\) 0 0
\(418\) 54.0955 0.129415
\(419\) − 568.735i − 1.35736i −0.734433 0.678682i \(-0.762552\pi\)
0.734433 0.678682i \(-0.237448\pi\)
\(420\) 0 0
\(421\) 14.6978 0.0349117 0.0174559 0.999848i \(-0.494443\pi\)
0.0174559 + 0.999848i \(0.494443\pi\)
\(422\) − 421.624i − 0.999110i
\(423\) 0 0
\(424\) 293.632 0.692529
\(425\) − 90.7298i − 0.213482i
\(426\) 0 0
\(427\) −75.3212 −0.176396
\(428\) 15.9686i 0.0373099i
\(429\) 0 0
\(430\) −113.437 −0.263808
\(431\) 416.758i 0.966955i 0.875357 + 0.483478i \(0.160627\pi\)
−0.875357 + 0.483478i \(0.839373\pi\)
\(432\) 0 0
\(433\) −714.420 −1.64993 −0.824966 0.565183i \(-0.808806\pi\)
−0.824966 + 0.565183i \(0.808806\pi\)
\(434\) − 974.680i − 2.24581i
\(435\) 0 0
\(436\) −109.348 −0.250799
\(437\) − 26.7997i − 0.0613266i
\(438\) 0 0
\(439\) 205.791 0.468771 0.234386 0.972144i \(-0.424692\pi\)
0.234386 + 0.972144i \(0.424692\pi\)
\(440\) − 49.0746i − 0.111533i
\(441\) 0 0
\(442\) −46.6892 −0.105632
\(443\) 303.789i 0.685755i 0.939380 + 0.342877i \(0.111402\pi\)
−0.939380 + 0.342877i \(0.888598\pi\)
\(444\) 0 0
\(445\) −50.4010 −0.113261
\(446\) 445.219i 0.998248i
\(447\) 0 0
\(448\) −178.352 −0.398107
\(449\) − 81.4465i − 0.181395i −0.995878 0.0906976i \(-0.971090\pi\)
0.995878 0.0906976i \(-0.0289097\pi\)
\(450\) 0 0
\(451\) 1435.53 3.18300
\(452\) 126.258i 0.279332i
\(453\) 0 0
\(454\) −1013.28 −2.23189
\(455\) − 41.7730i − 0.0918088i
\(456\) 0 0
\(457\) 864.984 1.89274 0.946372 0.323079i \(-0.104718\pi\)
0.946372 + 0.323079i \(0.104718\pi\)
\(458\) − 222.997i − 0.486893i
\(459\) 0 0
\(460\) 44.9004 0.0976096
\(461\) 126.563i 0.274540i 0.990534 + 0.137270i \(0.0438328\pi\)
−0.990534 + 0.137270i \(0.956167\pi\)
\(462\) 0 0
\(463\) 27.5640 0.0595334 0.0297667 0.999557i \(-0.490524\pi\)
0.0297667 + 0.999557i \(0.490524\pi\)
\(464\) 543.617i 1.17159i
\(465\) 0 0
\(466\) −884.305 −1.89765
\(467\) 193.863i 0.415124i 0.978222 + 0.207562i \(0.0665529\pi\)
−0.978222 + 0.207562i \(0.933447\pi\)
\(468\) 0 0
\(469\) −781.038 −1.66533
\(470\) − 138.854i − 0.295433i
\(471\) 0 0
\(472\) 238.644 0.505601
\(473\) 1370.14i 2.89670i
\(474\) 0 0
\(475\) −25.1901 −0.0530317
\(476\) − 122.865i − 0.258120i
\(477\) 0 0
\(478\) 254.198 0.531795
\(479\) − 102.074i − 0.213099i −0.994307 0.106549i \(-0.966020\pi\)
0.994307 0.106549i \(-0.0339802\pi\)
\(480\) 0 0
\(481\) −12.3218 −0.0256171
\(482\) 881.650i 1.82915i
\(483\) 0 0
\(484\) 780.697 1.61301
\(485\) 58.9402i 0.121526i
\(486\) 0 0
\(487\) −658.072 −1.35128 −0.675639 0.737233i \(-0.736132\pi\)
−0.675639 + 0.737233i \(0.736132\pi\)
\(488\) − 21.2038i − 0.0434504i
\(489\) 0 0
\(490\) 196.054 0.400110
\(491\) − 99.1405i − 0.201915i −0.994891 0.100958i \(-0.967809\pi\)
0.994891 0.100958i \(-0.0321907\pi\)
\(492\) 0 0
\(493\) −102.214 −0.207331
\(494\) 12.9627i 0.0262403i
\(495\) 0 0
\(496\) 581.733 1.17285
\(497\) − 1484.80i − 2.98752i
\(498\) 0 0
\(499\) 378.592 0.758701 0.379351 0.925253i \(-0.376147\pi\)
0.379351 + 0.925253i \(0.376147\pi\)
\(500\) − 85.1606i − 0.170321i
\(501\) 0 0
\(502\) −543.535 −1.08274
\(503\) 509.788i 1.01349i 0.862095 + 0.506747i \(0.169152\pi\)
−0.862095 + 0.506747i \(0.830848\pi\)
\(504\) 0 0
\(505\) 33.6144 0.0665632
\(506\) − 1378.30i − 2.72391i
\(507\) 0 0
\(508\) −29.2433 −0.0575655
\(509\) − 75.7404i − 0.148802i −0.997228 0.0744012i \(-0.976295\pi\)
0.997228 0.0744012i \(-0.0237045\pi\)
\(510\) 0 0
\(511\) 1337.57 2.61755
\(512\) − 424.074i − 0.828269i
\(513\) 0 0
\(514\) −893.917 −1.73914
\(515\) − 134.114i − 0.260416i
\(516\) 0 0
\(517\) −1677.12 −3.24395
\(518\) − 82.4086i − 0.159090i
\(519\) 0 0
\(520\) 11.7596 0.0226146
\(521\) 16.3142i 0.0313132i 0.999877 + 0.0156566i \(0.00498386\pi\)
−0.999877 + 0.0156566i \(0.995016\pi\)
\(522\) 0 0
\(523\) 430.771 0.823654 0.411827 0.911262i \(-0.364891\pi\)
0.411827 + 0.911262i \(0.364891\pi\)
\(524\) 254.454i 0.485599i
\(525\) 0 0
\(526\) 745.818 1.41790
\(527\) 109.381i 0.207554i
\(528\) 0 0
\(529\) −153.830 −0.290794
\(530\) 138.380i 0.261094i
\(531\) 0 0
\(532\) −34.1120 −0.0641203
\(533\) 343.992i 0.645389i
\(534\) 0 0
\(535\) 4.07487 0.00761657
\(536\) − 219.871i − 0.410207i
\(537\) 0 0
\(538\) −903.052 −1.67854
\(539\) − 2368.01i − 4.39334i
\(540\) 0 0
\(541\) 200.420 0.370461 0.185231 0.982695i \(-0.440697\pi\)
0.185231 + 0.982695i \(0.440697\pi\)
\(542\) 659.694i 1.21715i
\(543\) 0 0
\(544\) 133.053 0.244583
\(545\) 27.9034i 0.0511989i
\(546\) 0 0
\(547\) −842.613 −1.54043 −0.770213 0.637787i \(-0.779850\pi\)
−0.770213 + 0.637787i \(0.779850\pi\)
\(548\) − 351.560i − 0.641532i
\(549\) 0 0
\(550\) −1295.51 −2.35548
\(551\) 28.3786i 0.0515038i
\(552\) 0 0
\(553\) −193.550 −0.349999
\(554\) 417.136i 0.752954i
\(555\) 0 0
\(556\) −47.2220 −0.0849317
\(557\) 730.584i 1.31164i 0.754917 + 0.655821i \(0.227677\pi\)
−0.754917 + 0.655821i \(0.772323\pi\)
\(558\) 0 0
\(559\) −328.321 −0.587337
\(560\) 166.746i 0.297760i
\(561\) 0 0
\(562\) −264.320 −0.470320
\(563\) − 933.078i − 1.65733i −0.559744 0.828666i \(-0.689100\pi\)
0.559744 0.828666i \(-0.310900\pi\)
\(564\) 0 0
\(565\) 32.2184 0.0570238
\(566\) 1189.98i 2.10245i
\(567\) 0 0
\(568\) 417.987 0.735893
\(569\) 327.272i 0.575170i 0.957755 + 0.287585i \(0.0928523\pi\)
−0.957755 + 0.287585i \(0.907148\pi\)
\(570\) 0 0
\(571\) 654.425 1.14610 0.573051 0.819519i \(-0.305760\pi\)
0.573051 + 0.819519i \(0.305760\pi\)
\(572\) 262.315i 0.458593i
\(573\) 0 0
\(574\) −2300.62 −4.00806
\(575\) 641.818i 1.11620i
\(576\) 0 0
\(577\) 220.648 0.382405 0.191203 0.981551i \(-0.438761\pi\)
0.191203 + 0.981551i \(0.438761\pi\)
\(578\) − 707.126i − 1.22340i
\(579\) 0 0
\(580\) −47.5457 −0.0819754
\(581\) − 6.83895i − 0.0117710i
\(582\) 0 0
\(583\) 1671.40 2.86690
\(584\) 376.541i 0.644761i
\(585\) 0 0
\(586\) −778.927 −1.32923
\(587\) − 641.542i − 1.09292i −0.837486 0.546458i \(-0.815976\pi\)
0.837486 0.546458i \(-0.184024\pi\)
\(588\) 0 0
\(589\) 30.3684 0.0515593
\(590\) 112.466i 0.190620i
\(591\) 0 0
\(592\) 49.1852 0.0830831
\(593\) − 798.467i − 1.34649i −0.739421 0.673244i \(-0.764901\pi\)
0.739421 0.673244i \(-0.235099\pi\)
\(594\) 0 0
\(595\) −31.3526 −0.0526934
\(596\) 345.995i 0.580529i
\(597\) 0 0
\(598\) 330.277 0.552302
\(599\) − 486.587i − 0.812332i −0.913799 0.406166i \(-0.866865\pi\)
0.913799 0.406166i \(-0.133135\pi\)
\(600\) 0 0
\(601\) −564.769 −0.939715 −0.469858 0.882742i \(-0.655695\pi\)
−0.469858 + 0.882742i \(0.655695\pi\)
\(602\) − 2195.82i − 3.64753i
\(603\) 0 0
\(604\) −478.437 −0.792114
\(605\) − 199.218i − 0.329286i
\(606\) 0 0
\(607\) −543.288 −0.895038 −0.447519 0.894274i \(-0.647692\pi\)
−0.447519 + 0.894274i \(0.647692\pi\)
\(608\) − 36.9407i − 0.0607577i
\(609\) 0 0
\(610\) 9.99270 0.0163815
\(611\) − 401.883i − 0.657747i
\(612\) 0 0
\(613\) 562.465 0.917561 0.458781 0.888550i \(-0.348286\pi\)
0.458781 + 0.888550i \(0.348286\pi\)
\(614\) 725.842i 1.18215i
\(615\) 0 0
\(616\) 949.940 1.54211
\(617\) 394.884i 0.640006i 0.947416 + 0.320003i \(0.103684\pi\)
−0.947416 + 0.320003i \(0.896316\pi\)
\(618\) 0 0
\(619\) 502.096 0.811141 0.405570 0.914064i \(-0.367073\pi\)
0.405570 + 0.914064i \(0.367073\pi\)
\(620\) 50.8794i 0.0820636i
\(621\) 0 0
\(622\) 751.639 1.20842
\(623\) − 975.615i − 1.56600i
\(624\) 0 0
\(625\) 592.307 0.947691
\(626\) 487.906i 0.779403i
\(627\) 0 0
\(628\) −360.578 −0.574169
\(629\) 9.24811i 0.0147029i
\(630\) 0 0
\(631\) 22.7911 0.0361191 0.0180595 0.999837i \(-0.494251\pi\)
0.0180595 + 0.999837i \(0.494251\pi\)
\(632\) − 54.4864i − 0.0862127i
\(633\) 0 0
\(634\) 1097.35 1.73083
\(635\) 7.46228i 0.0117516i
\(636\) 0 0
\(637\) 567.438 0.890797
\(638\) 1459.50i 2.28762i
\(639\) 0 0
\(640\) −71.7412 −0.112096
\(641\) − 293.293i − 0.457556i −0.973479 0.228778i \(-0.926527\pi\)
0.973479 0.228778i \(-0.0734729\pi\)
\(642\) 0 0
\(643\) −627.634 −0.976103 −0.488051 0.872815i \(-0.662292\pi\)
−0.488051 + 0.872815i \(0.662292\pi\)
\(644\) 869.140i 1.34960i
\(645\) 0 0
\(646\) 9.72911 0.0150605
\(647\) 953.959i 1.47443i 0.675656 + 0.737217i \(0.263860\pi\)
−0.675656 + 0.737217i \(0.736140\pi\)
\(648\) 0 0
\(649\) 1358.40 2.09306
\(650\) − 310.440i − 0.477600i
\(651\) 0 0
\(652\) 777.620 1.19267
\(653\) 1017.63i 1.55839i 0.626783 + 0.779194i \(0.284371\pi\)
−0.626783 + 0.779194i \(0.715629\pi\)
\(654\) 0 0
\(655\) 64.9313 0.0991317
\(656\) − 1373.12i − 2.09317i
\(657\) 0 0
\(658\) 2687.80 4.08480
\(659\) − 961.378i − 1.45884i −0.684064 0.729422i \(-0.739789\pi\)
0.684064 0.729422i \(-0.260211\pi\)
\(660\) 0 0
\(661\) −325.601 −0.492589 −0.246294 0.969195i \(-0.579213\pi\)
−0.246294 + 0.969195i \(0.579213\pi\)
\(662\) 238.496i 0.360265i
\(663\) 0 0
\(664\) 1.92524 0.00289946
\(665\) 8.70468i 0.0130897i
\(666\) 0 0
\(667\) 723.059 1.08405
\(668\) − 273.556i − 0.409516i
\(669\) 0 0
\(670\) 103.619 0.154655
\(671\) − 120.695i − 0.179874i
\(672\) 0 0
\(673\) 467.782 0.695069 0.347535 0.937667i \(-0.387019\pi\)
0.347535 + 0.937667i \(0.387019\pi\)
\(674\) 512.290i 0.760074i
\(675\) 0 0
\(676\) 375.684 0.555746
\(677\) 846.092i 1.24977i 0.780718 + 0.624884i \(0.214854\pi\)
−0.780718 + 0.624884i \(0.785146\pi\)
\(678\) 0 0
\(679\) −1140.91 −1.68028
\(680\) − 8.82611i − 0.0129796i
\(681\) 0 0
\(682\) 1561.83 2.29008
\(683\) − 758.834i − 1.11103i −0.831506 0.555515i \(-0.812521\pi\)
0.831506 0.555515i \(-0.187479\pi\)
\(684\) 0 0
\(685\) −89.7107 −0.130965
\(686\) 2182.12i 3.18093i
\(687\) 0 0
\(688\) 1310.56 1.90489
\(689\) 400.512i 0.581295i
\(690\) 0 0
\(691\) −27.0426 −0.0391354 −0.0195677 0.999809i \(-0.506229\pi\)
−0.0195677 + 0.999809i \(0.506229\pi\)
\(692\) 285.848i 0.413075i
\(693\) 0 0
\(694\) −33.3018 −0.0479854
\(695\) 12.0501i 0.0173382i
\(696\) 0 0
\(697\) 258.182 0.370419
\(698\) 1081.56i 1.54951i
\(699\) 0 0
\(700\) 816.938 1.16705
\(701\) − 1088.82i − 1.55324i −0.629972 0.776618i \(-0.716934\pi\)
0.629972 0.776618i \(-0.283066\pi\)
\(702\) 0 0
\(703\) 2.56763 0.00365239
\(704\) − 285.792i − 0.405955i
\(705\) 0 0
\(706\) 1715.89 2.43044
\(707\) 650.676i 0.920334i
\(708\) 0 0
\(709\) −1141.14 −1.60950 −0.804750 0.593614i \(-0.797701\pi\)
−0.804750 + 0.593614i \(0.797701\pi\)
\(710\) 196.985i 0.277443i
\(711\) 0 0
\(712\) 274.647 0.385740
\(713\) − 773.757i − 1.08521i
\(714\) 0 0
\(715\) 66.9374 0.0936187
\(716\) 533.932i 0.745716i
\(717\) 0 0
\(718\) 945.757 1.31721
\(719\) 358.180i 0.498164i 0.968482 + 0.249082i \(0.0801288\pi\)
−0.968482 + 0.249082i \(0.919871\pi\)
\(720\) 0 0
\(721\) 2596.06 3.60063
\(722\) 924.368i 1.28029i
\(723\) 0 0
\(724\) −111.938 −0.154610
\(725\) − 679.630i − 0.937421i
\(726\) 0 0
\(727\) 500.703 0.688725 0.344362 0.938837i \(-0.388095\pi\)
0.344362 + 0.938837i \(0.388095\pi\)
\(728\) 227.631i 0.312680i
\(729\) 0 0
\(730\) −177.452 −0.243085
\(731\) 246.420i 0.337100i
\(732\) 0 0
\(733\) 174.957 0.238686 0.119343 0.992853i \(-0.461921\pi\)
0.119343 + 0.992853i \(0.461921\pi\)
\(734\) 49.3301i 0.0672073i
\(735\) 0 0
\(736\) −941.212 −1.27882
\(737\) − 1251.54i − 1.69816i
\(738\) 0 0
\(739\) −147.666 −0.199819 −0.0999095 0.994997i \(-0.531855\pi\)
−0.0999095 + 0.994997i \(0.531855\pi\)
\(740\) 4.30182i 0.00581328i
\(741\) 0 0
\(742\) −2678.63 −3.61001
\(743\) 127.109i 0.171075i 0.996335 + 0.0855375i \(0.0272607\pi\)
−0.996335 + 0.0855375i \(0.972739\pi\)
\(744\) 0 0
\(745\) 88.2908 0.118511
\(746\) 204.968i 0.274756i
\(747\) 0 0
\(748\) 196.880 0.263208
\(749\) 78.8774i 0.105310i
\(750\) 0 0
\(751\) −42.3505 −0.0563922 −0.0281961 0.999602i \(-0.508976\pi\)
−0.0281961 + 0.999602i \(0.508976\pi\)
\(752\) 1604.20i 2.13325i
\(753\) 0 0
\(754\) −349.735 −0.463839
\(755\) 122.087i 0.161705i
\(756\) 0 0
\(757\) 192.852 0.254758 0.127379 0.991854i \(-0.459343\pi\)
0.127379 + 0.991854i \(0.459343\pi\)
\(758\) − 299.214i − 0.394742i
\(759\) 0 0
\(760\) −2.45047 −0.00322430
\(761\) − 1239.77i − 1.62914i −0.580068 0.814568i \(-0.696974\pi\)
0.580068 0.814568i \(-0.303026\pi\)
\(762\) 0 0
\(763\) −540.128 −0.707900
\(764\) 166.229i 0.217578i
\(765\) 0 0
\(766\) 1050.78 1.37178
\(767\) 325.508i 0.424392i
\(768\) 0 0
\(769\) −286.471 −0.372524 −0.186262 0.982500i \(-0.559637\pi\)
−0.186262 + 0.982500i \(0.559637\pi\)
\(770\) 447.678i 0.581400i
\(771\) 0 0
\(772\) −496.696 −0.643389
\(773\) − 1184.33i − 1.53212i −0.642767 0.766062i \(-0.722214\pi\)
0.642767 0.766062i \(-0.277786\pi\)
\(774\) 0 0
\(775\) −727.283 −0.938430
\(776\) − 321.179i − 0.413890i
\(777\) 0 0
\(778\) −1002.20 −1.28817
\(779\) − 71.6813i − 0.0920170i
\(780\) 0 0
\(781\) 2379.25 3.04641
\(782\) − 247.888i − 0.316992i
\(783\) 0 0
\(784\) −2265.05 −2.88909
\(785\) 92.0120i 0.117213i
\(786\) 0 0
\(787\) 884.950 1.12446 0.562230 0.826981i \(-0.309944\pi\)
0.562230 + 0.826981i \(0.309944\pi\)
\(788\) − 660.646i − 0.838383i
\(789\) 0 0
\(790\) 25.6778 0.0325036
\(791\) 623.654i 0.788438i
\(792\) 0 0
\(793\) 28.9218 0.0364714
\(794\) 1448.49i 1.82429i
\(795\) 0 0
\(796\) 127.664 0.160382
\(797\) 264.634i 0.332037i 0.986123 + 0.166019i \(0.0530912\pi\)
−0.986123 + 0.166019i \(0.946909\pi\)
\(798\) 0 0
\(799\) −301.632 −0.377512
\(800\) 884.681i 1.10585i
\(801\) 0 0
\(802\) 985.760 1.22913
\(803\) 2143.33i 2.66915i
\(804\) 0 0
\(805\) 221.786 0.275511
\(806\) 374.257i 0.464339i
\(807\) 0 0
\(808\) −183.173 −0.226699
\(809\) − 57.1698i − 0.0706673i −0.999376 0.0353336i \(-0.988751\pi\)
0.999376 0.0353336i \(-0.0112494\pi\)
\(810\) 0 0
\(811\) −1433.72 −1.76784 −0.883918 0.467641i \(-0.845104\pi\)
−0.883918 + 0.467641i \(0.845104\pi\)
\(812\) − 920.345i − 1.13343i
\(813\) 0 0
\(814\) 132.052 0.162226
\(815\) − 198.432i − 0.243475i
\(816\) 0 0
\(817\) 68.4157 0.0837402
\(818\) 344.907i 0.421647i
\(819\) 0 0
\(820\) 120.095 0.146458
\(821\) 582.135i 0.709056i 0.935045 + 0.354528i \(0.115358\pi\)
−0.935045 + 0.354528i \(0.884642\pi\)
\(822\) 0 0
\(823\) −380.700 −0.462576 −0.231288 0.972885i \(-0.574294\pi\)
−0.231288 + 0.972885i \(0.574294\pi\)
\(824\) 730.819i 0.886917i
\(825\) 0 0
\(826\) −2177.00 −2.63560
\(827\) − 66.1038i − 0.0799321i −0.999201 0.0399660i \(-0.987275\pi\)
0.999201 0.0399660i \(-0.0127250\pi\)
\(828\) 0 0
\(829\) −1179.40 −1.42268 −0.711339 0.702849i \(-0.751911\pi\)
−0.711339 + 0.702849i \(0.751911\pi\)
\(830\) 0.907309i 0.00109314i
\(831\) 0 0
\(832\) 68.4834 0.0823118
\(833\) − 425.888i − 0.511271i
\(834\) 0 0
\(835\) −69.8059 −0.0835999
\(836\) − 54.6614i − 0.0653844i
\(837\) 0 0
\(838\) −1460.55 −1.74289
\(839\) 189.991i 0.226449i 0.993569 + 0.113225i \(0.0361179\pi\)
−0.993569 + 0.113225i \(0.963882\pi\)
\(840\) 0 0
\(841\) 75.3427 0.0895870
\(842\) − 37.7449i − 0.0448277i
\(843\) 0 0
\(844\) −426.035 −0.504781
\(845\) − 95.8668i − 0.113452i
\(846\) 0 0
\(847\) 3856.27 4.55286
\(848\) − 1598.73i − 1.88529i
\(849\) 0 0
\(850\) −232.999 −0.274117
\(851\) − 65.4207i − 0.0768750i
\(852\) 0 0
\(853\) −830.684 −0.973838 −0.486919 0.873447i \(-0.661879\pi\)
−0.486919 + 0.873447i \(0.661879\pi\)
\(854\) 193.429i 0.226498i
\(855\) 0 0
\(856\) −22.2049 −0.0259403
\(857\) 157.459i 0.183732i 0.995771 + 0.0918662i \(0.0292832\pi\)
−0.995771 + 0.0918662i \(0.970717\pi\)
\(858\) 0 0
\(859\) 1278.39 1.48823 0.744117 0.668049i \(-0.232870\pi\)
0.744117 + 0.668049i \(0.232870\pi\)
\(860\) 114.624i 0.133284i
\(861\) 0 0
\(862\) 1070.26 1.24160
\(863\) 1164.50i 1.34937i 0.738107 + 0.674683i \(0.235720\pi\)
−0.738107 + 0.674683i \(0.764280\pi\)
\(864\) 0 0
\(865\) 72.9424 0.0843265
\(866\) 1834.67i 2.11856i
\(867\) 0 0
\(868\) −984.876 −1.13465
\(869\) − 310.145i − 0.356899i
\(870\) 0 0
\(871\) 299.903 0.344320
\(872\) − 152.052i − 0.174372i
\(873\) 0 0
\(874\) −68.8232 −0.0787451
\(875\) − 420.652i − 0.480746i
\(876\) 0 0
\(877\) −625.815 −0.713586 −0.356793 0.934183i \(-0.616130\pi\)
−0.356793 + 0.934183i \(0.616130\pi\)
\(878\) − 528.482i − 0.601916i
\(879\) 0 0
\(880\) −267.194 −0.303630
\(881\) 762.102i 0.865042i 0.901624 + 0.432521i \(0.142376\pi\)
−0.901624 + 0.432521i \(0.857624\pi\)
\(882\) 0 0
\(883\) −877.735 −0.994038 −0.497019 0.867740i \(-0.665572\pi\)
−0.497019 + 0.867740i \(0.665572\pi\)
\(884\) 47.1776i 0.0533683i
\(885\) 0 0
\(886\) 780.149 0.880529
\(887\) − 699.683i − 0.788820i −0.918935 0.394410i \(-0.870949\pi\)
0.918935 0.394410i \(-0.129051\pi\)
\(888\) 0 0
\(889\) −144.448 −0.162483
\(890\) 129.433i 0.145430i
\(891\) 0 0
\(892\) 449.876 0.504345
\(893\) 83.7446i 0.0937790i
\(894\) 0 0
\(895\) 136.248 0.152233
\(896\) − 1388.70i − 1.54989i
\(897\) 0 0
\(898\) −209.159 −0.232917
\(899\) 819.342i 0.911393i
\(900\) 0 0
\(901\) 300.603 0.333633
\(902\) − 3686.54i − 4.08707i
\(903\) 0 0
\(904\) −175.566 −0.194210
\(905\) 28.5642i 0.0315627i
\(906\) 0 0
\(907\) −1077.34 −1.18780 −0.593901 0.804538i \(-0.702413\pi\)
−0.593901 + 0.804538i \(0.702413\pi\)
\(908\) 1023.88i 1.12762i
\(909\) 0 0
\(910\) −107.275 −0.117885
\(911\) − 581.178i − 0.637956i −0.947762 0.318978i \(-0.896660\pi\)
0.947762 0.318978i \(-0.103340\pi\)
\(912\) 0 0
\(913\) 10.9588 0.0120031
\(914\) − 2221.33i − 2.43034i
\(915\) 0 0
\(916\) −225.330 −0.245993
\(917\) 1256.88i 1.37064i
\(918\) 0 0
\(919\) 721.090 0.784647 0.392323 0.919827i \(-0.371671\pi\)
0.392323 + 0.919827i \(0.371671\pi\)
\(920\) 62.4354i 0.0678646i
\(921\) 0 0
\(922\) 325.021 0.352517
\(923\) 570.131i 0.617694i
\(924\) 0 0
\(925\) −61.4913 −0.0664771
\(926\) − 70.7858i − 0.0764426i
\(927\) 0 0
\(928\) 996.663 1.07399
\(929\) 553.878i 0.596209i 0.954533 + 0.298105i \(0.0963544\pi\)
−0.954533 + 0.298105i \(0.903646\pi\)
\(930\) 0 0
\(931\) −118.243 −0.127006
\(932\) 893.556i 0.958751i
\(933\) 0 0
\(934\) 497.852 0.533032
\(935\) − 50.2396i − 0.0537322i
\(936\) 0 0
\(937\) −1271.69 −1.35720 −0.678598 0.734510i \(-0.737412\pi\)
−0.678598 + 0.734510i \(0.737412\pi\)
\(938\) 2005.75i 2.13833i
\(939\) 0 0
\(940\) −140.306 −0.149262
\(941\) 318.512i 0.338483i 0.985575 + 0.169241i \(0.0541317\pi\)
−0.985575 + 0.169241i \(0.945868\pi\)
\(942\) 0 0
\(943\) −1826.37 −1.93676
\(944\) − 1299.33i − 1.37641i
\(945\) 0 0
\(946\) 3518.59 3.71944
\(947\) 784.960i 0.828891i 0.910074 + 0.414446i \(0.136025\pi\)
−0.910074 + 0.414446i \(0.863975\pi\)
\(948\) 0 0
\(949\) −513.599 −0.541200
\(950\) 64.6896i 0.0680943i
\(951\) 0 0
\(952\) 170.847 0.179462
\(953\) − 1016.76i − 1.06690i −0.845831 0.533452i \(-0.820895\pi\)
0.845831 0.533452i \(-0.179105\pi\)
\(954\) 0 0
\(955\) 42.4183 0.0444170
\(956\) − 256.857i − 0.268679i
\(957\) 0 0
\(958\) −262.133 −0.273625
\(959\) − 1736.53i − 1.81078i
\(960\) 0 0
\(961\) −84.2084 −0.0876258
\(962\) 31.6432i 0.0328931i
\(963\) 0 0
\(964\) 890.873 0.924142
\(965\) 126.747i 0.131344i
\(966\) 0 0
\(967\) −1823.75 −1.88598 −0.942992 0.332816i \(-0.892001\pi\)
−0.942992 + 0.332816i \(0.892001\pi\)
\(968\) 1085.58i 1.12147i
\(969\) 0 0
\(970\) 151.362 0.156043
\(971\) − 1079.74i − 1.11199i −0.831187 0.555993i \(-0.812338\pi\)
0.831187 0.555993i \(-0.187662\pi\)
\(972\) 0 0
\(973\) −233.254 −0.239727
\(974\) 1689.97i 1.73508i
\(975\) 0 0
\(976\) −115.447 −0.118286
\(977\) 630.788i 0.645637i 0.946461 + 0.322819i \(0.104630\pi\)
−0.946461 + 0.322819i \(0.895370\pi\)
\(978\) 0 0
\(979\) 1563.33 1.59687
\(980\) − 198.105i − 0.202148i
\(981\) 0 0
\(982\) −254.599 −0.259265
\(983\) 1211.78i 1.23274i 0.787458 + 0.616369i \(0.211397\pi\)
−0.787458 + 0.616369i \(0.788603\pi\)
\(984\) 0 0
\(985\) −168.583 −0.171150
\(986\) 262.492i 0.266219i
\(987\) 0 0
\(988\) 13.0983 0.0132574
\(989\) − 1743.16i − 1.76255i
\(990\) 0 0
\(991\) 453.179 0.457295 0.228647 0.973509i \(-0.426570\pi\)
0.228647 + 0.973509i \(0.426570\pi\)
\(992\) − 1066.55i − 1.07515i
\(993\) 0 0
\(994\) −3813.04 −3.83606
\(995\) − 32.5772i − 0.0327409i
\(996\) 0 0
\(997\) −1127.93 −1.13133 −0.565663 0.824637i \(-0.691380\pi\)
−0.565663 + 0.824637i \(0.691380\pi\)
\(998\) − 972.246i − 0.974195i
\(999\) 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.890.17 84
3.2 odd 2 inner 1143.3.b.a.890.68 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.17 84 1.1 even 1 trivial
1143.3.b.a.890.68 yes 84 3.2 odd 2 inner