Properties

Label 1143.3.b.a.890.12
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 890.12
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.73

$q$-expansion

\(f(q)\) \(=\) \(q-3.03383i q^{2} -5.20412 q^{4} +9.69466i q^{5} -0.465658 q^{7} +3.65310i q^{8} +O(q^{10})\) \(q-3.03383i q^{2} -5.20412 q^{4} +9.69466i q^{5} -0.465658 q^{7} +3.65310i q^{8} +29.4119 q^{10} -0.825606i q^{11} +10.5786 q^{13} +1.41273i q^{14} -9.73361 q^{16} +10.2606i q^{17} +25.2136 q^{19} -50.4522i q^{20} -2.50475 q^{22} -0.704676i q^{23} -68.9864 q^{25} -32.0937i q^{26} +2.42334 q^{28} +8.67552i q^{29} -57.9325 q^{31} +44.1425i q^{32} +31.1288 q^{34} -4.51440i q^{35} -13.5115 q^{37} -76.4937i q^{38} -35.4155 q^{40} +10.5730i q^{41} -34.4833 q^{43} +4.29656i q^{44} -2.13787 q^{46} +62.5318i q^{47} -48.7832 q^{49} +209.293i q^{50} -55.0523 q^{52} -91.4802i q^{53} +8.00397 q^{55} -1.70109i q^{56} +26.3200 q^{58} +17.6740i q^{59} +67.4200 q^{61} +175.757i q^{62} +94.9864 q^{64} +102.556i q^{65} -30.5164 q^{67} -53.3972i q^{68} -13.6959 q^{70} +133.968i q^{71} +117.109 q^{73} +40.9917i q^{74} -131.215 q^{76} +0.384450i q^{77} -93.5817 q^{79} -94.3640i q^{80} +32.0767 q^{82} +104.813i q^{83} -99.4726 q^{85} +104.616i q^{86} +3.01602 q^{88} -5.45973i q^{89} -4.92601 q^{91} +3.66722i q^{92} +189.711 q^{94} +244.437i q^{95} -46.1082 q^{97} +148.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.03383i − 1.51691i −0.651723 0.758457i \(-0.725953\pi\)
0.651723 0.758457i \(-0.274047\pi\)
\(3\) 0 0
\(4\) −5.20412 −1.30103
\(5\) 9.69466i 1.93893i 0.245227 + 0.969466i \(0.421138\pi\)
−0.245227 + 0.969466i \(0.578862\pi\)
\(6\) 0 0
\(7\) −0.465658 −0.0665226 −0.0332613 0.999447i \(-0.510589\pi\)
−0.0332613 + 0.999447i \(0.510589\pi\)
\(8\) 3.65310i 0.456637i
\(9\) 0 0
\(10\) 29.4119 2.94119
\(11\) − 0.825606i − 0.0750551i −0.999296 0.0375276i \(-0.988052\pi\)
0.999296 0.0375276i \(-0.0119482\pi\)
\(12\) 0 0
\(13\) 10.5786 0.813739 0.406869 0.913486i \(-0.366620\pi\)
0.406869 + 0.913486i \(0.366620\pi\)
\(14\) 1.41273i 0.100909i
\(15\) 0 0
\(16\) −9.73361 −0.608350
\(17\) 10.2606i 0.603562i 0.953377 + 0.301781i \(0.0975812\pi\)
−0.953377 + 0.301781i \(0.902419\pi\)
\(18\) 0 0
\(19\) 25.2136 1.32703 0.663515 0.748163i \(-0.269064\pi\)
0.663515 + 0.748163i \(0.269064\pi\)
\(20\) − 50.4522i − 2.52261i
\(21\) 0 0
\(22\) −2.50475 −0.113852
\(23\) − 0.704676i − 0.0306381i −0.999883 0.0153190i \(-0.995124\pi\)
0.999883 0.0153190i \(-0.00487639\pi\)
\(24\) 0 0
\(25\) −68.9864 −2.75946
\(26\) − 32.0937i − 1.23437i
\(27\) 0 0
\(28\) 2.42334 0.0865479
\(29\) 8.67552i 0.299156i 0.988750 + 0.149578i \(0.0477915\pi\)
−0.988750 + 0.149578i \(0.952209\pi\)
\(30\) 0 0
\(31\) −57.9325 −1.86879 −0.934394 0.356240i \(-0.884059\pi\)
−0.934394 + 0.356240i \(0.884059\pi\)
\(32\) 44.1425i 1.37945i
\(33\) 0 0
\(34\) 31.1288 0.915552
\(35\) − 4.51440i − 0.128983i
\(36\) 0 0
\(37\) −13.5115 −0.365177 −0.182588 0.983189i \(-0.558448\pi\)
−0.182588 + 0.983189i \(0.558448\pi\)
\(38\) − 76.4937i − 2.01299i
\(39\) 0 0
\(40\) −35.4155 −0.885388
\(41\) 10.5730i 0.257878i 0.991653 + 0.128939i \(0.0411572\pi\)
−0.991653 + 0.128939i \(0.958843\pi\)
\(42\) 0 0
\(43\) −34.4833 −0.801937 −0.400969 0.916092i \(-0.631326\pi\)
−0.400969 + 0.916092i \(0.631326\pi\)
\(44\) 4.29656i 0.0976490i
\(45\) 0 0
\(46\) −2.13787 −0.0464754
\(47\) 62.5318i 1.33046i 0.746637 + 0.665232i \(0.231667\pi\)
−0.746637 + 0.665232i \(0.768333\pi\)
\(48\) 0 0
\(49\) −48.7832 −0.995575
\(50\) 209.293i 4.18586i
\(51\) 0 0
\(52\) −55.0523 −1.05870
\(53\) − 91.4802i − 1.72604i −0.505168 0.863021i \(-0.668570\pi\)
0.505168 0.863021i \(-0.331430\pi\)
\(54\) 0 0
\(55\) 8.00397 0.145527
\(56\) − 1.70109i − 0.0303767i
\(57\) 0 0
\(58\) 26.3200 0.453794
\(59\) 17.6740i 0.299559i 0.988719 + 0.149779i \(0.0478564\pi\)
−0.988719 + 0.149779i \(0.952144\pi\)
\(60\) 0 0
\(61\) 67.4200 1.10525 0.552623 0.833431i \(-0.313627\pi\)
0.552623 + 0.833431i \(0.313627\pi\)
\(62\) 175.757i 2.83479i
\(63\) 0 0
\(64\) 94.9864 1.48416
\(65\) 102.556i 1.57778i
\(66\) 0 0
\(67\) −30.5164 −0.455469 −0.227734 0.973723i \(-0.573132\pi\)
−0.227734 + 0.973723i \(0.573132\pi\)
\(68\) − 53.3972i − 0.785253i
\(69\) 0 0
\(70\) −13.6959 −0.195656
\(71\) 133.968i 1.88687i 0.331553 + 0.943437i \(0.392427\pi\)
−0.331553 + 0.943437i \(0.607573\pi\)
\(72\) 0 0
\(73\) 117.109 1.60423 0.802117 0.597167i \(-0.203707\pi\)
0.802117 + 0.597167i \(0.203707\pi\)
\(74\) 40.9917i 0.553942i
\(75\) 0 0
\(76\) −131.215 −1.72651
\(77\) 0.384450i 0.00499286i
\(78\) 0 0
\(79\) −93.5817 −1.18458 −0.592289 0.805726i \(-0.701776\pi\)
−0.592289 + 0.805726i \(0.701776\pi\)
\(80\) − 94.3640i − 1.17955i
\(81\) 0 0
\(82\) 32.0767 0.391179
\(83\) 104.813i 1.26281i 0.775454 + 0.631404i \(0.217521\pi\)
−0.775454 + 0.631404i \(0.782479\pi\)
\(84\) 0 0
\(85\) −99.4726 −1.17027
\(86\) 104.616i 1.21647i
\(87\) 0 0
\(88\) 3.01602 0.0342730
\(89\) − 5.45973i − 0.0613453i −0.999529 0.0306727i \(-0.990235\pi\)
0.999529 0.0306727i \(-0.00976494\pi\)
\(90\) 0 0
\(91\) −4.92601 −0.0541320
\(92\) 3.66722i 0.0398611i
\(93\) 0 0
\(94\) 189.711 2.01820
\(95\) 244.437i 2.57302i
\(96\) 0 0
\(97\) −46.1082 −0.475342 −0.237671 0.971346i \(-0.576384\pi\)
−0.237671 + 0.971346i \(0.576384\pi\)
\(98\) 148.000i 1.51020i
\(99\) 0 0
\(100\) 359.013 3.59013
\(101\) 162.423i 1.60815i 0.594529 + 0.804074i \(0.297339\pi\)
−0.594529 + 0.804074i \(0.702661\pi\)
\(102\) 0 0
\(103\) −38.9286 −0.377947 −0.188974 0.981982i \(-0.560516\pi\)
−0.188974 + 0.981982i \(0.560516\pi\)
\(104\) 38.6447i 0.371584i
\(105\) 0 0
\(106\) −277.535 −2.61826
\(107\) 4.84185i 0.0452509i 0.999744 + 0.0226255i \(0.00720252\pi\)
−0.999744 + 0.0226255i \(0.992797\pi\)
\(108\) 0 0
\(109\) −93.9791 −0.862194 −0.431097 0.902306i \(-0.641873\pi\)
−0.431097 + 0.902306i \(0.641873\pi\)
\(110\) − 24.2827i − 0.220752i
\(111\) 0 0
\(112\) 4.53253 0.0404690
\(113\) 80.6426i 0.713651i 0.934171 + 0.356826i \(0.116141\pi\)
−0.934171 + 0.356826i \(0.883859\pi\)
\(114\) 0 0
\(115\) 6.83159 0.0594052
\(116\) − 45.1484i − 0.389211i
\(117\) 0 0
\(118\) 53.6198 0.454405
\(119\) − 4.77791i − 0.0401505i
\(120\) 0 0
\(121\) 120.318 0.994367
\(122\) − 204.541i − 1.67656i
\(123\) 0 0
\(124\) 301.488 2.43135
\(125\) − 426.433i − 3.41146i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) − 111.602i − 0.871894i
\(129\) 0 0
\(130\) 311.137 2.39336
\(131\) − 34.3037i − 0.261861i −0.991392 0.130930i \(-0.958204\pi\)
0.991392 0.130930i \(-0.0417964\pi\)
\(132\) 0 0
\(133\) −11.7409 −0.0882775
\(134\) 92.5816i 0.690907i
\(135\) 0 0
\(136\) −37.4828 −0.275609
\(137\) − 49.9509i − 0.364605i −0.983242 0.182303i \(-0.941645\pi\)
0.983242 0.182303i \(-0.0583551\pi\)
\(138\) 0 0
\(139\) −124.555 −0.896081 −0.448040 0.894013i \(-0.647878\pi\)
−0.448040 + 0.894013i \(0.647878\pi\)
\(140\) 23.4935i 0.167810i
\(141\) 0 0
\(142\) 406.436 2.86223
\(143\) − 8.73377i − 0.0610753i
\(144\) 0 0
\(145\) −84.1062 −0.580042
\(146\) − 355.289i − 2.43349i
\(147\) 0 0
\(148\) 70.3157 0.475106
\(149\) − 88.0919i − 0.591221i −0.955309 0.295610i \(-0.904477\pi\)
0.955309 0.295610i \(-0.0955230\pi\)
\(150\) 0 0
\(151\) −186.337 −1.23402 −0.617010 0.786955i \(-0.711656\pi\)
−0.617010 + 0.786955i \(0.711656\pi\)
\(152\) 92.1077i 0.605972i
\(153\) 0 0
\(154\) 1.16636 0.00757375
\(155\) − 561.635i − 3.62345i
\(156\) 0 0
\(157\) 183.153 1.16658 0.583289 0.812265i \(-0.301766\pi\)
0.583289 + 0.812265i \(0.301766\pi\)
\(158\) 283.911i 1.79690i
\(159\) 0 0
\(160\) −427.946 −2.67466
\(161\) 0.328138i 0.00203813i
\(162\) 0 0
\(163\) −199.823 −1.22591 −0.612954 0.790118i \(-0.710019\pi\)
−0.612954 + 0.790118i \(0.710019\pi\)
\(164\) − 55.0232i − 0.335507i
\(165\) 0 0
\(166\) 317.985 1.91557
\(167\) − 5.57997i − 0.0334130i −0.999860 0.0167065i \(-0.994682\pi\)
0.999860 0.0167065i \(-0.00531809\pi\)
\(168\) 0 0
\(169\) −57.0931 −0.337829
\(170\) 301.783i 1.77519i
\(171\) 0 0
\(172\) 179.455 1.04334
\(173\) 106.519i 0.615717i 0.951432 + 0.307858i \(0.0996123\pi\)
−0.951432 + 0.307858i \(0.900388\pi\)
\(174\) 0 0
\(175\) 32.1241 0.183566
\(176\) 8.03613i 0.0456598i
\(177\) 0 0
\(178\) −16.5639 −0.0930556
\(179\) 308.791i 1.72509i 0.505982 + 0.862544i \(0.331130\pi\)
−0.505982 + 0.862544i \(0.668870\pi\)
\(180\) 0 0
\(181\) 157.986 0.872849 0.436424 0.899741i \(-0.356245\pi\)
0.436424 + 0.899741i \(0.356245\pi\)
\(182\) 14.9447i 0.0821137i
\(183\) 0 0
\(184\) 2.57425 0.0139905
\(185\) − 130.990i − 0.708053i
\(186\) 0 0
\(187\) 8.47118 0.0453004
\(188\) − 325.423i − 1.73097i
\(189\) 0 0
\(190\) 741.580 3.90305
\(191\) − 151.699i − 0.794233i −0.917768 0.397117i \(-0.870011\pi\)
0.917768 0.397117i \(-0.129989\pi\)
\(192\) 0 0
\(193\) −101.412 −0.525449 −0.262724 0.964871i \(-0.584621\pi\)
−0.262724 + 0.964871i \(0.584621\pi\)
\(194\) 139.884i 0.721054i
\(195\) 0 0
\(196\) 253.873 1.29527
\(197\) − 60.6071i − 0.307650i −0.988098 0.153825i \(-0.950841\pi\)
0.988098 0.153825i \(-0.0491592\pi\)
\(198\) 0 0
\(199\) −93.6473 −0.470590 −0.235295 0.971924i \(-0.575606\pi\)
−0.235295 + 0.971924i \(0.575606\pi\)
\(200\) − 252.014i − 1.26007i
\(201\) 0 0
\(202\) 492.764 2.43942
\(203\) − 4.03982i − 0.0199006i
\(204\) 0 0
\(205\) −102.502 −0.500008
\(206\) 118.103i 0.573314i
\(207\) 0 0
\(208\) −102.968 −0.495038
\(209\) − 20.8165i − 0.0996005i
\(210\) 0 0
\(211\) −193.953 −0.919208 −0.459604 0.888124i \(-0.652009\pi\)
−0.459604 + 0.888124i \(0.652009\pi\)
\(212\) 476.074i 2.24563i
\(213\) 0 0
\(214\) 14.6893 0.0686418
\(215\) − 334.304i − 1.55490i
\(216\) 0 0
\(217\) 26.9767 0.124317
\(218\) 285.117i 1.30787i
\(219\) 0 0
\(220\) −41.6536 −0.189335
\(221\) 108.542i 0.491142i
\(222\) 0 0
\(223\) −96.5930 −0.433152 −0.216576 0.976266i \(-0.569489\pi\)
−0.216576 + 0.976266i \(0.569489\pi\)
\(224\) − 20.5553i − 0.0917648i
\(225\) 0 0
\(226\) 244.656 1.08255
\(227\) − 255.360i − 1.12493i −0.826820 0.562467i \(-0.809852\pi\)
0.826820 0.562467i \(-0.190148\pi\)
\(228\) 0 0
\(229\) −153.818 −0.671694 −0.335847 0.941917i \(-0.609022\pi\)
−0.335847 + 0.941917i \(0.609022\pi\)
\(230\) − 20.7259i − 0.0901126i
\(231\) 0 0
\(232\) −31.6925 −0.136606
\(233\) 345.735i 1.48384i 0.670488 + 0.741920i \(0.266085\pi\)
−0.670488 + 0.741920i \(0.733915\pi\)
\(234\) 0 0
\(235\) −606.224 −2.57968
\(236\) − 91.9775i − 0.389735i
\(237\) 0 0
\(238\) −14.4954 −0.0609049
\(239\) − 17.6299i − 0.0737652i −0.999320 0.0368826i \(-0.988257\pi\)
0.999320 0.0368826i \(-0.0117428\pi\)
\(240\) 0 0
\(241\) 312.932 1.29847 0.649236 0.760587i \(-0.275089\pi\)
0.649236 + 0.760587i \(0.275089\pi\)
\(242\) − 365.025i − 1.50837i
\(243\) 0 0
\(244\) −350.862 −1.43796
\(245\) − 472.936i − 1.93035i
\(246\) 0 0
\(247\) 266.725 1.07986
\(248\) − 211.633i − 0.853359i
\(249\) 0 0
\(250\) −1293.72 −5.17490
\(251\) − 230.937i − 0.920066i −0.887902 0.460033i \(-0.847837\pi\)
0.887902 0.460033i \(-0.152163\pi\)
\(252\) 0 0
\(253\) −0.581785 −0.00229955
\(254\) − 34.1895i − 0.134604i
\(255\) 0 0
\(256\) 41.3626 0.161573
\(257\) 258.980i 1.00770i 0.863790 + 0.503852i \(0.168084\pi\)
−0.863790 + 0.503852i \(0.831916\pi\)
\(258\) 0 0
\(259\) 6.29176 0.0242925
\(260\) − 533.714i − 2.05274i
\(261\) 0 0
\(262\) −104.072 −0.397220
\(263\) 188.471i 0.716620i 0.933603 + 0.358310i \(0.116647\pi\)
−0.933603 + 0.358310i \(0.883353\pi\)
\(264\) 0 0
\(265\) 886.869 3.34668
\(266\) 35.6199i 0.133909i
\(267\) 0 0
\(268\) 158.811 0.592579
\(269\) − 66.7539i − 0.248156i −0.992272 0.124078i \(-0.960403\pi\)
0.992272 0.124078i \(-0.0395973\pi\)
\(270\) 0 0
\(271\) 16.5305 0.0609980 0.0304990 0.999535i \(-0.490290\pi\)
0.0304990 + 0.999535i \(0.490290\pi\)
\(272\) − 99.8722i − 0.367177i
\(273\) 0 0
\(274\) −151.543 −0.553075
\(275\) 56.9556i 0.207111i
\(276\) 0 0
\(277\) −304.859 −1.10058 −0.550288 0.834975i \(-0.685482\pi\)
−0.550288 + 0.834975i \(0.685482\pi\)
\(278\) 377.879i 1.35928i
\(279\) 0 0
\(280\) 16.4915 0.0588983
\(281\) 114.652i 0.408014i 0.978969 + 0.204007i \(0.0653965\pi\)
−0.978969 + 0.204007i \(0.934604\pi\)
\(282\) 0 0
\(283\) 213.495 0.754400 0.377200 0.926132i \(-0.376887\pi\)
0.377200 + 0.926132i \(0.376887\pi\)
\(284\) − 697.186i − 2.45488i
\(285\) 0 0
\(286\) −26.4968 −0.0926460
\(287\) − 4.92341i − 0.0171547i
\(288\) 0 0
\(289\) 183.721 0.635713
\(290\) 255.164i 0.879875i
\(291\) 0 0
\(292\) −609.450 −2.08716
\(293\) − 499.490i − 1.70475i −0.522935 0.852373i \(-0.675163\pi\)
0.522935 0.852373i \(-0.324837\pi\)
\(294\) 0 0
\(295\) −171.343 −0.580824
\(296\) − 49.3590i − 0.166753i
\(297\) 0 0
\(298\) −267.256 −0.896832
\(299\) − 7.45449i − 0.0249314i
\(300\) 0 0
\(301\) 16.0574 0.0533470
\(302\) 565.315i 1.87190i
\(303\) 0 0
\(304\) −245.419 −0.807300
\(305\) 653.614i 2.14300i
\(306\) 0 0
\(307\) 252.944 0.823922 0.411961 0.911202i \(-0.364844\pi\)
0.411961 + 0.911202i \(0.364844\pi\)
\(308\) − 2.00073i − 0.00649586i
\(309\) 0 0
\(310\) −1703.91 −5.49647
\(311\) 365.311i 1.17463i 0.809357 + 0.587317i \(0.199816\pi\)
−0.809357 + 0.587317i \(0.800184\pi\)
\(312\) 0 0
\(313\) 70.6417 0.225692 0.112846 0.993612i \(-0.464003\pi\)
0.112846 + 0.993612i \(0.464003\pi\)
\(314\) − 555.654i − 1.76960i
\(315\) 0 0
\(316\) 487.010 1.54117
\(317\) − 219.350i − 0.691955i −0.938243 0.345978i \(-0.887547\pi\)
0.938243 0.345978i \(-0.112453\pi\)
\(318\) 0 0
\(319\) 7.16256 0.0224532
\(320\) 920.860i 2.87769i
\(321\) 0 0
\(322\) 0.995515 0.00309166
\(323\) 258.705i 0.800945i
\(324\) 0 0
\(325\) −729.780 −2.24548
\(326\) 606.229i 1.85960i
\(327\) 0 0
\(328\) −38.6242 −0.117757
\(329\) − 29.1184i − 0.0885059i
\(330\) 0 0
\(331\) 198.861 0.600789 0.300394 0.953815i \(-0.402882\pi\)
0.300394 + 0.953815i \(0.402882\pi\)
\(332\) − 545.460i − 1.64295i
\(333\) 0 0
\(334\) −16.9287 −0.0506847
\(335\) − 295.846i − 0.883123i
\(336\) 0 0
\(337\) 469.626 1.39355 0.696774 0.717291i \(-0.254618\pi\)
0.696774 + 0.717291i \(0.254618\pi\)
\(338\) 173.211i 0.512458i
\(339\) 0 0
\(340\) 517.667 1.52255
\(341\) 47.8294i 0.140262i
\(342\) 0 0
\(343\) 45.5335 0.132751
\(344\) − 125.971i − 0.366195i
\(345\) 0 0
\(346\) 323.160 0.933990
\(347\) 298.679i 0.860745i 0.902651 + 0.430373i \(0.141618\pi\)
−0.902651 + 0.430373i \(0.858382\pi\)
\(348\) 0 0
\(349\) −412.332 −1.18147 −0.590733 0.806867i \(-0.701161\pi\)
−0.590733 + 0.806867i \(0.701161\pi\)
\(350\) − 97.4589i − 0.278454i
\(351\) 0 0
\(352\) 36.4443 0.103535
\(353\) − 363.649i − 1.03017i −0.857140 0.515084i \(-0.827761\pi\)
0.857140 0.515084i \(-0.172239\pi\)
\(354\) 0 0
\(355\) −1298.77 −3.65852
\(356\) 28.4131i 0.0798121i
\(357\) 0 0
\(358\) 936.818 2.61681
\(359\) 415.792i 1.15819i 0.815259 + 0.579097i \(0.196595\pi\)
−0.815259 + 0.579097i \(0.803405\pi\)
\(360\) 0 0
\(361\) 274.725 0.761010
\(362\) − 479.302i − 1.32404i
\(363\) 0 0
\(364\) 25.6356 0.0704274
\(365\) 1135.33i 3.11050i
\(366\) 0 0
\(367\) −53.1936 −0.144942 −0.0724708 0.997371i \(-0.523088\pi\)
−0.0724708 + 0.997371i \(0.523088\pi\)
\(368\) 6.85904i 0.0186387i
\(369\) 0 0
\(370\) −397.401 −1.07406
\(371\) 42.5985i 0.114821i
\(372\) 0 0
\(373\) 394.389 1.05734 0.528671 0.848827i \(-0.322691\pi\)
0.528671 + 0.848827i \(0.322691\pi\)
\(374\) − 25.7001i − 0.0687169i
\(375\) 0 0
\(376\) −228.435 −0.607539
\(377\) 91.7749i 0.243435i
\(378\) 0 0
\(379\) 357.810 0.944091 0.472045 0.881574i \(-0.343516\pi\)
0.472045 + 0.881574i \(0.343516\pi\)
\(380\) − 1272.08i − 3.34758i
\(381\) 0 0
\(382\) −460.228 −1.20478
\(383\) 500.587i 1.30702i 0.756919 + 0.653508i \(0.226704\pi\)
−0.756919 + 0.653508i \(0.773296\pi\)
\(384\) 0 0
\(385\) −3.72711 −0.00968082
\(386\) 307.666i 0.797061i
\(387\) 0 0
\(388\) 239.953 0.618435
\(389\) − 44.6053i − 0.114667i −0.998355 0.0573333i \(-0.981740\pi\)
0.998355 0.0573333i \(-0.0182598\pi\)
\(390\) 0 0
\(391\) 7.23037 0.0184920
\(392\) − 178.210i − 0.454617i
\(393\) 0 0
\(394\) −183.872 −0.466679
\(395\) − 907.242i − 2.29682i
\(396\) 0 0
\(397\) 213.697 0.538279 0.269139 0.963101i \(-0.413261\pi\)
0.269139 + 0.963101i \(0.413261\pi\)
\(398\) 284.110i 0.713844i
\(399\) 0 0
\(400\) 671.486 1.67872
\(401\) − 282.656i − 0.704877i −0.935835 0.352439i \(-0.885353\pi\)
0.935835 0.352439i \(-0.114647\pi\)
\(402\) 0 0
\(403\) −612.845 −1.52071
\(404\) − 845.269i − 2.09225i
\(405\) 0 0
\(406\) −12.2561 −0.0301875
\(407\) 11.1552i 0.0274084i
\(408\) 0 0
\(409\) 415.230 1.01523 0.507616 0.861584i \(-0.330527\pi\)
0.507616 + 0.861584i \(0.330527\pi\)
\(410\) 310.973i 0.758470i
\(411\) 0 0
\(412\) 202.589 0.491721
\(413\) − 8.23003i − 0.0199274i
\(414\) 0 0
\(415\) −1016.13 −2.44850
\(416\) 466.966i 1.12251i
\(417\) 0 0
\(418\) −63.1537 −0.151085
\(419\) − 402.985i − 0.961778i −0.876782 0.480889i \(-0.840314\pi\)
0.876782 0.480889i \(-0.159686\pi\)
\(420\) 0 0
\(421\) 410.489 0.975033 0.487517 0.873114i \(-0.337903\pi\)
0.487517 + 0.873114i \(0.337903\pi\)
\(422\) 588.420i 1.39436i
\(423\) 0 0
\(424\) 334.186 0.788175
\(425\) − 707.839i − 1.66550i
\(426\) 0 0
\(427\) −31.3947 −0.0735238
\(428\) − 25.1976i − 0.0588728i
\(429\) 0 0
\(430\) −1014.22 −2.35865
\(431\) − 481.326i − 1.11677i −0.829583 0.558383i \(-0.811422\pi\)
0.829583 0.558383i \(-0.188578\pi\)
\(432\) 0 0
\(433\) −716.958 −1.65579 −0.827896 0.560881i \(-0.810462\pi\)
−0.827896 + 0.560881i \(0.810462\pi\)
\(434\) − 81.8428i − 0.188578i
\(435\) 0 0
\(436\) 489.079 1.12174
\(437\) − 17.7674i − 0.0406577i
\(438\) 0 0
\(439\) 440.326 1.00302 0.501510 0.865152i \(-0.332778\pi\)
0.501510 + 0.865152i \(0.332778\pi\)
\(440\) 29.2393i 0.0664529i
\(441\) 0 0
\(442\) 329.299 0.745021
\(443\) 146.953i 0.331723i 0.986149 + 0.165862i \(0.0530405\pi\)
−0.986149 + 0.165862i \(0.946959\pi\)
\(444\) 0 0
\(445\) 52.9302 0.118944
\(446\) 293.047i 0.657055i
\(447\) 0 0
\(448\) −44.2312 −0.0987303
\(449\) − 249.942i − 0.556663i −0.960485 0.278332i \(-0.910219\pi\)
0.960485 0.278332i \(-0.0897815\pi\)
\(450\) 0 0
\(451\) 8.72915 0.0193551
\(452\) − 419.674i − 0.928482i
\(453\) 0 0
\(454\) −774.718 −1.70643
\(455\) − 47.7560i − 0.104958i
\(456\) 0 0
\(457\) 105.026 0.229815 0.114908 0.993376i \(-0.463343\pi\)
0.114908 + 0.993376i \(0.463343\pi\)
\(458\) 466.657i 1.01890i
\(459\) 0 0
\(460\) −35.5524 −0.0772879
\(461\) − 125.250i − 0.271692i −0.990730 0.135846i \(-0.956625\pi\)
0.990730 0.135846i \(-0.0433753\pi\)
\(462\) 0 0
\(463\) 505.017 1.09075 0.545375 0.838192i \(-0.316387\pi\)
0.545375 + 0.838192i \(0.316387\pi\)
\(464\) − 84.4441i − 0.181992i
\(465\) 0 0
\(466\) 1048.90 2.25086
\(467\) 78.9893i 0.169142i 0.996417 + 0.0845710i \(0.0269520\pi\)
−0.996417 + 0.0845710i \(0.973048\pi\)
\(468\) 0 0
\(469\) 14.2102 0.0302990
\(470\) 1839.18i 3.91315i
\(471\) 0 0
\(472\) −64.5648 −0.136790
\(473\) 28.4696i 0.0601895i
\(474\) 0 0
\(475\) −1739.39 −3.66188
\(476\) 24.8648i 0.0522370i
\(477\) 0 0
\(478\) −53.4861 −0.111896
\(479\) 695.723i 1.45245i 0.687458 + 0.726225i \(0.258727\pi\)
−0.687458 + 0.726225i \(0.741273\pi\)
\(480\) 0 0
\(481\) −142.933 −0.297159
\(482\) − 949.382i − 1.96967i
\(483\) 0 0
\(484\) −626.151 −1.29370
\(485\) − 447.003i − 0.921656i
\(486\) 0 0
\(487\) −539.311 −1.10741 −0.553707 0.832711i \(-0.686787\pi\)
−0.553707 + 0.832711i \(0.686787\pi\)
\(488\) 246.292i 0.504697i
\(489\) 0 0
\(490\) −1434.81 −2.92818
\(491\) − 669.576i − 1.36370i −0.731493 0.681849i \(-0.761176\pi\)
0.731493 0.681849i \(-0.238824\pi\)
\(492\) 0 0
\(493\) −89.0156 −0.180559
\(494\) − 809.197i − 1.63805i
\(495\) 0 0
\(496\) 563.892 1.13688
\(497\) − 62.3833i − 0.125520i
\(498\) 0 0
\(499\) 409.869 0.821382 0.410691 0.911775i \(-0.365288\pi\)
0.410691 + 0.911775i \(0.365288\pi\)
\(500\) 2219.21i 4.43842i
\(501\) 0 0
\(502\) −700.622 −1.39566
\(503\) − 626.217i − 1.24496i −0.782634 0.622482i \(-0.786125\pi\)
0.782634 0.622482i \(-0.213875\pi\)
\(504\) 0 0
\(505\) −1574.64 −3.11809
\(506\) 1.76504i 0.00348822i
\(507\) 0 0
\(508\) −58.6475 −0.115448
\(509\) − 957.198i − 1.88055i −0.340421 0.940273i \(-0.610570\pi\)
0.340421 0.940273i \(-0.389430\pi\)
\(510\) 0 0
\(511\) −54.5328 −0.106718
\(512\) − 571.897i − 1.11699i
\(513\) 0 0
\(514\) 785.701 1.52860
\(515\) − 377.399i − 0.732814i
\(516\) 0 0
\(517\) 51.6266 0.0998581
\(518\) − 19.0881i − 0.0368497i
\(519\) 0 0
\(520\) −374.647 −0.720475
\(521\) − 614.917i − 1.18026i −0.807307 0.590132i \(-0.799076\pi\)
0.807307 0.590132i \(-0.200924\pi\)
\(522\) 0 0
\(523\) 1029.64 1.96871 0.984356 0.176194i \(-0.0563785\pi\)
0.984356 + 0.176194i \(0.0563785\pi\)
\(524\) 178.521i 0.340688i
\(525\) 0 0
\(526\) 571.789 1.08705
\(527\) − 594.419i − 1.12793i
\(528\) 0 0
\(529\) 528.503 0.999061
\(530\) − 2690.61i − 5.07662i
\(531\) 0 0
\(532\) 61.1011 0.114852
\(533\) 111.848i 0.209846i
\(534\) 0 0
\(535\) −46.9401 −0.0877384
\(536\) − 111.479i − 0.207984i
\(537\) 0 0
\(538\) −202.520 −0.376431
\(539\) 40.2757i 0.0747230i
\(540\) 0 0
\(541\) 674.507 1.24678 0.623390 0.781911i \(-0.285755\pi\)
0.623390 + 0.781911i \(0.285755\pi\)
\(542\) − 50.1506i − 0.0925288i
\(543\) 0 0
\(544\) −452.927 −0.832586
\(545\) − 911.095i − 1.67173i
\(546\) 0 0
\(547\) 378.731 0.692378 0.346189 0.938165i \(-0.387476\pi\)
0.346189 + 0.938165i \(0.387476\pi\)
\(548\) 259.951i 0.474363i
\(549\) 0 0
\(550\) 172.794 0.314170
\(551\) 218.741i 0.396989i
\(552\) 0 0
\(553\) 43.5771 0.0788012
\(554\) 924.891i 1.66948i
\(555\) 0 0
\(556\) 648.201 1.16583
\(557\) 326.784i 0.586686i 0.956007 + 0.293343i \(0.0947678\pi\)
−0.956007 + 0.293343i \(0.905232\pi\)
\(558\) 0 0
\(559\) −364.785 −0.652568
\(560\) 43.9414i 0.0784667i
\(561\) 0 0
\(562\) 347.834 0.618922
\(563\) 164.132i 0.291531i 0.989319 + 0.145766i \(0.0465645\pi\)
−0.989319 + 0.145766i \(0.953435\pi\)
\(564\) 0 0
\(565\) −781.802 −1.38372
\(566\) − 647.708i − 1.14436i
\(567\) 0 0
\(568\) −489.398 −0.861617
\(569\) − 450.365i − 0.791502i −0.918358 0.395751i \(-0.870484\pi\)
0.918358 0.395751i \(-0.129516\pi\)
\(570\) 0 0
\(571\) −614.662 −1.07647 −0.538233 0.842796i \(-0.680908\pi\)
−0.538233 + 0.842796i \(0.680908\pi\)
\(572\) 45.4516i 0.0794608i
\(573\) 0 0
\(574\) −14.9368 −0.0260223
\(575\) 48.6130i 0.0845444i
\(576\) 0 0
\(577\) 498.675 0.864254 0.432127 0.901813i \(-0.357763\pi\)
0.432127 + 0.901813i \(0.357763\pi\)
\(578\) − 557.378i − 0.964322i
\(579\) 0 0
\(580\) 437.699 0.754653
\(581\) − 48.8071i − 0.0840053i
\(582\) 0 0
\(583\) −75.5266 −0.129548
\(584\) 427.811i 0.732553i
\(585\) 0 0
\(586\) −1515.37 −2.58595
\(587\) 773.400i 1.31755i 0.752342 + 0.658773i \(0.228924\pi\)
−0.752342 + 0.658773i \(0.771076\pi\)
\(588\) 0 0
\(589\) −1460.68 −2.47994
\(590\) 519.826i 0.881061i
\(591\) 0 0
\(592\) 131.516 0.222156
\(593\) 1037.15i 1.74899i 0.485032 + 0.874496i \(0.338808\pi\)
−0.485032 + 0.874496i \(0.661192\pi\)
\(594\) 0 0
\(595\) 46.3202 0.0778491
\(596\) 458.441i 0.769196i
\(597\) 0 0
\(598\) −22.6157 −0.0378188
\(599\) 266.824i 0.445448i 0.974882 + 0.222724i \(0.0714949\pi\)
−0.974882 + 0.222724i \(0.928505\pi\)
\(600\) 0 0
\(601\) −603.952 −1.00491 −0.502456 0.864603i \(-0.667570\pi\)
−0.502456 + 0.864603i \(0.667570\pi\)
\(602\) − 48.7155i − 0.0809228i
\(603\) 0 0
\(604\) 969.721 1.60550
\(605\) 1166.45i 1.92801i
\(606\) 0 0
\(607\) −270.545 −0.445708 −0.222854 0.974852i \(-0.571537\pi\)
−0.222854 + 0.974852i \(0.571537\pi\)
\(608\) 1112.99i 1.83058i
\(609\) 0 0
\(610\) 1982.95 3.25074
\(611\) 661.499i 1.08265i
\(612\) 0 0
\(613\) −290.634 −0.474117 −0.237059 0.971495i \(-0.576183\pi\)
−0.237059 + 0.971495i \(0.576183\pi\)
\(614\) − 767.389i − 1.24982i
\(615\) 0 0
\(616\) −1.40443 −0.00227993
\(617\) − 1097.41i − 1.77862i −0.457302 0.889311i \(-0.651184\pi\)
0.457302 0.889311i \(-0.348816\pi\)
\(618\) 0 0
\(619\) −540.438 −0.873083 −0.436542 0.899684i \(-0.643797\pi\)
−0.436542 + 0.899684i \(0.643797\pi\)
\(620\) 2922.82i 4.71422i
\(621\) 0 0
\(622\) 1108.29 1.78182
\(623\) 2.54237i 0.00408085i
\(624\) 0 0
\(625\) 2409.46 3.85514
\(626\) − 214.315i − 0.342356i
\(627\) 0 0
\(628\) −953.149 −1.51775
\(629\) − 138.636i − 0.220407i
\(630\) 0 0
\(631\) 1145.02 1.81462 0.907310 0.420463i \(-0.138132\pi\)
0.907310 + 0.420463i \(0.138132\pi\)
\(632\) − 341.863i − 0.540923i
\(633\) 0 0
\(634\) −665.470 −1.04964
\(635\) 109.253i 0.172052i
\(636\) 0 0
\(637\) −516.058 −0.810138
\(638\) − 21.7300i − 0.0340595i
\(639\) 0 0
\(640\) 1081.95 1.69054
\(641\) − 904.089i − 1.41044i −0.708991 0.705218i \(-0.750849\pi\)
0.708991 0.705218i \(-0.249151\pi\)
\(642\) 0 0
\(643\) 101.162 0.157328 0.0786642 0.996901i \(-0.474934\pi\)
0.0786642 + 0.996901i \(0.474934\pi\)
\(644\) − 1.70767i − 0.00265166i
\(645\) 0 0
\(646\) 784.868 1.21497
\(647\) 799.354i 1.23548i 0.786384 + 0.617738i \(0.211951\pi\)
−0.786384 + 0.617738i \(0.788049\pi\)
\(648\) 0 0
\(649\) 14.5918 0.0224834
\(650\) 2214.03i 3.40620i
\(651\) 0 0
\(652\) 1039.90 1.59494
\(653\) 405.770i 0.621394i 0.950509 + 0.310697i \(0.100562\pi\)
−0.950509 + 0.310697i \(0.899438\pi\)
\(654\) 0 0
\(655\) 332.563 0.507730
\(656\) − 102.914i − 0.156880i
\(657\) 0 0
\(658\) −88.3403 −0.134256
\(659\) 87.7483i 0.133154i 0.997781 + 0.0665768i \(0.0212077\pi\)
−0.997781 + 0.0665768i \(0.978792\pi\)
\(660\) 0 0
\(661\) −252.832 −0.382499 −0.191250 0.981541i \(-0.561254\pi\)
−0.191250 + 0.981541i \(0.561254\pi\)
\(662\) − 603.310i − 0.911345i
\(663\) 0 0
\(664\) −382.892 −0.576645
\(665\) − 113.824i − 0.171164i
\(666\) 0 0
\(667\) 6.11343 0.00916556
\(668\) 29.0388i 0.0434713i
\(669\) 0 0
\(670\) −897.547 −1.33962
\(671\) − 55.6624i − 0.0829544i
\(672\) 0 0
\(673\) −20.5389 −0.0305184 −0.0152592 0.999884i \(-0.504857\pi\)
−0.0152592 + 0.999884i \(0.504857\pi\)
\(674\) − 1424.76i − 2.11389i
\(675\) 0 0
\(676\) 297.119 0.439526
\(677\) 201.518i 0.297663i 0.988863 + 0.148832i \(0.0475512\pi\)
−0.988863 + 0.148832i \(0.952449\pi\)
\(678\) 0 0
\(679\) 21.4707 0.0316210
\(680\) − 363.383i − 0.534387i
\(681\) 0 0
\(682\) 145.106 0.212766
\(683\) 638.072i 0.934219i 0.884199 + 0.467110i \(0.154705\pi\)
−0.884199 + 0.467110i \(0.845295\pi\)
\(684\) 0 0
\(685\) 484.257 0.706945
\(686\) − 138.141i − 0.201372i
\(687\) 0 0
\(688\) 335.647 0.487859
\(689\) − 967.733i − 1.40455i
\(690\) 0 0
\(691\) 1306.30 1.89045 0.945226 0.326417i \(-0.105841\pi\)
0.945226 + 0.326417i \(0.105841\pi\)
\(692\) − 554.338i − 0.801066i
\(693\) 0 0
\(694\) 906.140 1.30568
\(695\) − 1207.52i − 1.73744i
\(696\) 0 0
\(697\) −108.485 −0.155646
\(698\) 1250.94i 1.79218i
\(699\) 0 0
\(700\) −167.178 −0.238825
\(701\) 1065.94i 1.52059i 0.649576 + 0.760297i \(0.274947\pi\)
−0.649576 + 0.760297i \(0.725053\pi\)
\(702\) 0 0
\(703\) −340.674 −0.484601
\(704\) − 78.4214i − 0.111394i
\(705\) 0 0
\(706\) −1103.25 −1.56268
\(707\) − 75.6336i − 0.106978i
\(708\) 0 0
\(709\) 472.012 0.665744 0.332872 0.942972i \(-0.391982\pi\)
0.332872 + 0.942972i \(0.391982\pi\)
\(710\) 3940.26i 5.54966i
\(711\) 0 0
\(712\) 19.9449 0.0280126
\(713\) 40.8236i 0.0572561i
\(714\) 0 0
\(715\) 84.6709 0.118421
\(716\) − 1606.98i − 2.24439i
\(717\) 0 0
\(718\) 1261.44 1.75688
\(719\) − 209.385i − 0.291217i −0.989342 0.145609i \(-0.953486\pi\)
0.989342 0.145609i \(-0.0465140\pi\)
\(720\) 0 0
\(721\) 18.1274 0.0251420
\(722\) − 833.468i − 1.15439i
\(723\) 0 0
\(724\) −822.176 −1.13560
\(725\) − 598.492i − 0.825507i
\(726\) 0 0
\(727\) 973.645 1.33926 0.669632 0.742693i \(-0.266452\pi\)
0.669632 + 0.742693i \(0.266452\pi\)
\(728\) − 17.9952i − 0.0247187i
\(729\) 0 0
\(730\) 3444.41 4.71836
\(731\) − 353.818i − 0.484019i
\(732\) 0 0
\(733\) 666.657 0.909491 0.454746 0.890621i \(-0.349730\pi\)
0.454746 + 0.890621i \(0.349730\pi\)
\(734\) 161.380i 0.219864i
\(735\) 0 0
\(736\) 31.1062 0.0422638
\(737\) 25.1945i 0.0341853i
\(738\) 0 0
\(739\) 1070.77 1.44894 0.724472 0.689304i \(-0.242084\pi\)
0.724472 + 0.689304i \(0.242084\pi\)
\(740\) 681.687i 0.921198i
\(741\) 0 0
\(742\) 129.237 0.174173
\(743\) − 486.119i − 0.654265i −0.944979 0.327132i \(-0.893918\pi\)
0.944979 0.327132i \(-0.106082\pi\)
\(744\) 0 0
\(745\) 854.021 1.14634
\(746\) − 1196.51i − 1.60390i
\(747\) 0 0
\(748\) −44.0851 −0.0589372
\(749\) − 2.25465i − 0.00301021i
\(750\) 0 0
\(751\) −945.864 −1.25947 −0.629737 0.776809i \(-0.716837\pi\)
−0.629737 + 0.776809i \(0.716837\pi\)
\(752\) − 608.660i − 0.809388i
\(753\) 0 0
\(754\) 278.429 0.369270
\(755\) − 1806.47i − 2.39268i
\(756\) 0 0
\(757\) −1232.83 −1.62858 −0.814290 0.580458i \(-0.802873\pi\)
−0.814290 + 0.580458i \(0.802873\pi\)
\(758\) − 1085.54i − 1.43210i
\(759\) 0 0
\(760\) −892.953 −1.17494
\(761\) 1036.56i 1.36210i 0.732235 + 0.681052i \(0.238477\pi\)
−0.732235 + 0.681052i \(0.761523\pi\)
\(762\) 0 0
\(763\) 43.7621 0.0573554
\(764\) 789.458i 1.03332i
\(765\) 0 0
\(766\) 1518.70 1.98263
\(767\) 186.966i 0.243763i
\(768\) 0 0
\(769\) 255.317 0.332012 0.166006 0.986125i \(-0.446913\pi\)
0.166006 + 0.986125i \(0.446913\pi\)
\(770\) 11.3074i 0.0146850i
\(771\) 0 0
\(772\) 527.758 0.683625
\(773\) − 355.381i − 0.459743i −0.973221 0.229871i \(-0.926169\pi\)
0.973221 0.229871i \(-0.0738305\pi\)
\(774\) 0 0
\(775\) 3996.55 5.15684
\(776\) − 168.438i − 0.217059i
\(777\) 0 0
\(778\) −135.325 −0.173939
\(779\) 266.583i 0.342212i
\(780\) 0 0
\(781\) 110.605 0.141620
\(782\) − 21.9357i − 0.0280508i
\(783\) 0 0
\(784\) 474.836 0.605658
\(785\) 1775.60i 2.26191i
\(786\) 0 0
\(787\) 149.041 0.189378 0.0946892 0.995507i \(-0.469814\pi\)
0.0946892 + 0.995507i \(0.469814\pi\)
\(788\) 315.407i 0.400262i
\(789\) 0 0
\(790\) −2752.42 −3.48407
\(791\) − 37.5519i − 0.0474739i
\(792\) 0 0
\(793\) 713.210 0.899382
\(794\) − 648.319i − 0.816523i
\(795\) 0 0
\(796\) 487.352 0.612251
\(797\) 1254.91i 1.57454i 0.616605 + 0.787272i \(0.288507\pi\)
−0.616605 + 0.787272i \(0.711493\pi\)
\(798\) 0 0
\(799\) −641.611 −0.803017
\(800\) − 3045.23i − 3.80654i
\(801\) 0 0
\(802\) −857.530 −1.06924
\(803\) − 96.6860i − 0.120406i
\(804\) 0 0
\(805\) −3.18119 −0.00395178
\(806\) 1859.27i 2.30678i
\(807\) 0 0
\(808\) −593.347 −0.734341
\(809\) − 495.041i − 0.611917i −0.952045 0.305958i \(-0.901023\pi\)
0.952045 0.305958i \(-0.0989769\pi\)
\(810\) 0 0
\(811\) 378.712 0.466969 0.233485 0.972360i \(-0.424987\pi\)
0.233485 + 0.972360i \(0.424987\pi\)
\(812\) 21.0237i 0.0258913i
\(813\) 0 0
\(814\) 33.8430 0.0415762
\(815\) − 1937.22i − 2.37695i
\(816\) 0 0
\(817\) −869.448 −1.06420
\(818\) − 1259.74i − 1.54002i
\(819\) 0 0
\(820\) 533.431 0.650526
\(821\) − 26.1980i − 0.0319099i −0.999873 0.0159549i \(-0.994921\pi\)
0.999873 0.0159549i \(-0.00507883\pi\)
\(822\) 0 0
\(823\) −1424.46 −1.73081 −0.865405 0.501074i \(-0.832939\pi\)
−0.865405 + 0.501074i \(0.832939\pi\)
\(824\) − 142.210i − 0.172585i
\(825\) 0 0
\(826\) −24.9685 −0.0302282
\(827\) − 691.197i − 0.835789i −0.908496 0.417894i \(-0.862768\pi\)
0.908496 0.417894i \(-0.137232\pi\)
\(828\) 0 0
\(829\) 954.895 1.15186 0.575932 0.817498i \(-0.304639\pi\)
0.575932 + 0.817498i \(0.304639\pi\)
\(830\) 3082.76i 3.71416i
\(831\) 0 0
\(832\) 1004.82 1.20772
\(833\) − 500.542i − 0.600891i
\(834\) 0 0
\(835\) 54.0959 0.0647855
\(836\) 108.332i 0.129583i
\(837\) 0 0
\(838\) −1222.59 −1.45893
\(839\) 197.809i 0.235767i 0.993027 + 0.117884i \(0.0376110\pi\)
−0.993027 + 0.117884i \(0.962389\pi\)
\(840\) 0 0
\(841\) 765.735 0.910506
\(842\) − 1245.35i − 1.47904i
\(843\) 0 0
\(844\) 1009.35 1.19592
\(845\) − 553.498i − 0.655027i
\(846\) 0 0
\(847\) −56.0272 −0.0661478
\(848\) 890.432i 1.05004i
\(849\) 0 0
\(850\) −2147.46 −2.52643
\(851\) 9.52126i 0.0111883i
\(852\) 0 0
\(853\) 221.837 0.260066 0.130033 0.991510i \(-0.458492\pi\)
0.130033 + 0.991510i \(0.458492\pi\)
\(854\) 95.2461i 0.111529i
\(855\) 0 0
\(856\) −17.6878 −0.0206633
\(857\) 1660.75i 1.93787i 0.247325 + 0.968933i \(0.420448\pi\)
−0.247325 + 0.968933i \(0.579552\pi\)
\(858\) 0 0
\(859\) −638.952 −0.743832 −0.371916 0.928266i \(-0.621299\pi\)
−0.371916 + 0.928266i \(0.621299\pi\)
\(860\) 1739.76i 2.02297i
\(861\) 0 0
\(862\) −1460.26 −1.69404
\(863\) − 419.217i − 0.485767i −0.970055 0.242884i \(-0.921907\pi\)
0.970055 0.242884i \(-0.0780933\pi\)
\(864\) 0 0
\(865\) −1032.66 −1.19383
\(866\) 2175.13i 2.51170i
\(867\) 0 0
\(868\) −140.390 −0.161740
\(869\) 77.2616i 0.0889087i
\(870\) 0 0
\(871\) −322.821 −0.370633
\(872\) − 343.315i − 0.393710i
\(873\) 0 0
\(874\) −53.9033 −0.0616742
\(875\) 198.572i 0.226939i
\(876\) 0 0
\(877\) 1354.65 1.54465 0.772323 0.635230i \(-0.219095\pi\)
0.772323 + 0.635230i \(0.219095\pi\)
\(878\) − 1335.87i − 1.52150i
\(879\) 0 0
\(880\) −77.9075 −0.0885313
\(881\) 563.515i 0.639631i 0.947480 + 0.319816i \(0.103621\pi\)
−0.947480 + 0.319816i \(0.896379\pi\)
\(882\) 0 0
\(883\) −1272.91 −1.44157 −0.720785 0.693159i \(-0.756219\pi\)
−0.720785 + 0.693159i \(0.756219\pi\)
\(884\) − 564.868i − 0.638991i
\(885\) 0 0
\(886\) 445.832 0.503196
\(887\) 716.899i 0.808229i 0.914708 + 0.404114i \(0.132420\pi\)
−0.914708 + 0.404114i \(0.867580\pi\)
\(888\) 0 0
\(889\) −5.24770 −0.00590293
\(890\) − 160.581i − 0.180428i
\(891\) 0 0
\(892\) 502.682 0.563544
\(893\) 1576.65i 1.76557i
\(894\) 0 0
\(895\) −2993.62 −3.34483
\(896\) 51.9686i 0.0580007i
\(897\) 0 0
\(898\) −758.281 −0.844411
\(899\) − 502.594i − 0.559059i
\(900\) 0 0
\(901\) 938.638 1.04177
\(902\) − 26.4827i − 0.0293600i
\(903\) 0 0
\(904\) −294.595 −0.325880
\(905\) 1531.62i 1.69239i
\(906\) 0 0
\(907\) 1550.93 1.70996 0.854981 0.518660i \(-0.173569\pi\)
0.854981 + 0.518660i \(0.173569\pi\)
\(908\) 1328.92i 1.46357i
\(909\) 0 0
\(910\) −144.884 −0.159213
\(911\) − 472.805i − 0.518996i −0.965744 0.259498i \(-0.916443\pi\)
0.965744 0.259498i \(-0.0835571\pi\)
\(912\) 0 0
\(913\) 86.5343 0.0947802
\(914\) − 318.630i − 0.348610i
\(915\) 0 0
\(916\) 800.487 0.873894
\(917\) 15.9738i 0.0174196i
\(918\) 0 0
\(919\) −1256.65 −1.36741 −0.683704 0.729759i \(-0.739632\pi\)
−0.683704 + 0.729759i \(0.739632\pi\)
\(920\) 24.9565i 0.0271266i
\(921\) 0 0
\(922\) −379.988 −0.412134
\(923\) 1417.19i 1.53542i
\(924\) 0 0
\(925\) 932.113 1.00769
\(926\) − 1532.14i − 1.65457i
\(927\) 0 0
\(928\) −382.959 −0.412671
\(929\) − 1347.86i − 1.45087i −0.688291 0.725435i \(-0.741639\pi\)
0.688291 0.725435i \(-0.258361\pi\)
\(930\) 0 0
\(931\) −1230.00 −1.32116
\(932\) − 1799.25i − 1.93052i
\(933\) 0 0
\(934\) 239.640 0.256574
\(935\) 82.1252i 0.0878344i
\(936\) 0 0
\(937\) 89.2394 0.0952395 0.0476198 0.998866i \(-0.484836\pi\)
0.0476198 + 0.998866i \(0.484836\pi\)
\(938\) − 43.1114i − 0.0459609i
\(939\) 0 0
\(940\) 3154.86 3.35624
\(941\) 1728.75i 1.83715i 0.395252 + 0.918573i \(0.370657\pi\)
−0.395252 + 0.918573i \(0.629343\pi\)
\(942\) 0 0
\(943\) 7.45055 0.00790090
\(944\) − 172.032i − 0.182237i
\(945\) 0 0
\(946\) 86.3720 0.0913024
\(947\) − 1500.82i − 1.58482i −0.609992 0.792408i \(-0.708827\pi\)
0.609992 0.792408i \(-0.291173\pi\)
\(948\) 0 0
\(949\) 1238.85 1.30543
\(950\) 5277.02i 5.55476i
\(951\) 0 0
\(952\) 17.4542 0.0183342
\(953\) 847.145i 0.888924i 0.895798 + 0.444462i \(0.146605\pi\)
−0.895798 + 0.444462i \(0.853395\pi\)
\(954\) 0 0
\(955\) 1470.67 1.53996
\(956\) 91.7480i 0.0959708i
\(957\) 0 0
\(958\) 2110.71 2.20324
\(959\) 23.2601i 0.0242545i
\(960\) 0 0
\(961\) 2395.17 2.49237
\(962\) 433.635i 0.450764i
\(963\) 0 0
\(964\) −1628.53 −1.68935
\(965\) − 983.151i − 1.01881i
\(966\) 0 0
\(967\) −1781.12 −1.84191 −0.920953 0.389673i \(-0.872588\pi\)
−0.920953 + 0.389673i \(0.872588\pi\)
\(968\) 439.535i 0.454065i
\(969\) 0 0
\(970\) −1356.13 −1.39807
\(971\) 821.559i 0.846096i 0.906107 + 0.423048i \(0.139040\pi\)
−0.906107 + 0.423048i \(0.860960\pi\)
\(972\) 0 0
\(973\) 58.0002 0.0596096
\(974\) 1636.18i 1.67985i
\(975\) 0 0
\(976\) −656.240 −0.672377
\(977\) 195.226i 0.199822i 0.994996 + 0.0999111i \(0.0318558\pi\)
−0.994996 + 0.0999111i \(0.968144\pi\)
\(978\) 0 0
\(979\) −4.50759 −0.00460428
\(980\) 2461.22i 2.51145i
\(981\) 0 0
\(982\) −2031.38 −2.06861
\(983\) 288.267i 0.293253i 0.989192 + 0.146626i \(0.0468415\pi\)
−0.989192 + 0.146626i \(0.953158\pi\)
\(984\) 0 0
\(985\) 587.565 0.596513
\(986\) 270.058i 0.273893i
\(987\) 0 0
\(988\) −1388.07 −1.40493
\(989\) 24.2996i 0.0245698i
\(990\) 0 0
\(991\) 575.624 0.580852 0.290426 0.956898i \(-0.406203\pi\)
0.290426 + 0.956898i \(0.406203\pi\)
\(992\) − 2557.28i − 2.57791i
\(993\) 0 0
\(994\) −189.260 −0.190403
\(995\) − 907.879i − 0.912441i
\(996\) 0 0
\(997\) 181.307 0.181853 0.0909265 0.995858i \(-0.471017\pi\)
0.0909265 + 0.995858i \(0.471017\pi\)
\(998\) − 1243.47i − 1.24597i
\(999\) 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.890.12 84
3.2 odd 2 inner 1143.3.b.a.890.73 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.12 84 1.1 even 1 trivial
1143.3.b.a.890.73 yes 84 3.2 odd 2 inner