Properties

Label 882.5.c.a.685.2
Level $882$
Weight $5$
Character 882.685
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.2
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.5.c.a.685.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843 q^{2} +8.00000 q^{4} +14.1536i q^{5} -22.6274 q^{8} +O(q^{10})\) \(q-2.82843 q^{2} +8.00000 q^{4} +14.1536i q^{5} -22.6274 q^{8} -40.0324i q^{10} -64.0294 q^{11} +228.919i q^{13} +64.0000 q^{16} -225.455i q^{17} -294.737i q^{19} +113.229i q^{20} +181.103 q^{22} -709.499 q^{23} +424.676 q^{25} -647.481i q^{26} -740.397 q^{29} +666.713i q^{31} -181.019 q^{32} +637.683i q^{34} -833.765 q^{37} +833.642i q^{38} -320.259i q^{40} -2817.60i q^{41} +3066.41 q^{43} -512.235 q^{44} +2006.77 q^{46} +613.726i q^{47} -1201.17 q^{50} +1831.35i q^{52} +1152.60 q^{53} -906.246i q^{55} +2094.16 q^{58} -3492.59i q^{59} +2272.21i q^{61} -1885.75i q^{62} +512.000 q^{64} -3240.03 q^{65} -8674.62 q^{67} -1803.64i q^{68} +353.591 q^{71} -4069.95i q^{73} +2358.24 q^{74} -2357.90i q^{76} +6472.83 q^{79} +905.829i q^{80} +7969.38i q^{82} -8225.83i q^{83} +3191.00 q^{85} -8673.11 q^{86} +1448.82 q^{88} +15538.3i q^{89} -5675.99 q^{92} -1735.88i q^{94} +4171.58 q^{95} -1558.61i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 324q^{11} + 256q^{16} - 192q^{22} + 624q^{23} + 952q^{25} - 2724q^{29} - 2792q^{37} - 632q^{43} - 2592q^{44} + 9792q^{46} - 2112q^{50} - 2076q^{53} + 672q^{58} + 2048q^{64} - 1488q^{65} - 29200q^{67} + 9696q^{71} + 1536q^{74} - 7948q^{79} + 1224q^{85} - 36480q^{86} - 1536q^{88} + 4992q^{92} + 6504q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82843 −0.707107
\(3\) 0 0
\(4\) 8.00000 0.500000
\(5\) 14.1536i 0.566143i 0.959099 + 0.283072i \(0.0913534\pi\)
−0.959099 + 0.283072i \(0.908647\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −22.6274 −0.353553
\(9\) 0 0
\(10\) − 40.0324i − 0.400324i
\(11\) −64.0294 −0.529169 −0.264584 0.964363i \(-0.585235\pi\)
−0.264584 + 0.964363i \(0.585235\pi\)
\(12\) 0 0
\(13\) 228.919i 1.35455i 0.735729 + 0.677276i \(0.236839\pi\)
−0.735729 + 0.677276i \(0.763161\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 225.455i − 0.780121i −0.920789 0.390061i \(-0.872454\pi\)
0.920789 0.390061i \(-0.127546\pi\)
\(18\) 0 0
\(19\) − 294.737i − 0.816446i −0.912882 0.408223i \(-0.866149\pi\)
0.912882 0.408223i \(-0.133851\pi\)
\(20\) 113.229i 0.283072i
\(21\) 0 0
\(22\) 181.103 0.374179
\(23\) −709.499 −1.34121 −0.670604 0.741816i \(-0.733965\pi\)
−0.670604 + 0.741816i \(0.733965\pi\)
\(24\) 0 0
\(25\) 424.676 0.679482
\(26\) − 647.481i − 0.957812i
\(27\) 0 0
\(28\) 0 0
\(29\) −740.397 −0.880377 −0.440188 0.897905i \(-0.645088\pi\)
−0.440188 + 0.897905i \(0.645088\pi\)
\(30\) 0 0
\(31\) 666.713i 0.693770i 0.937908 + 0.346885i \(0.112761\pi\)
−0.937908 + 0.346885i \(0.887239\pi\)
\(32\) −181.019 −0.176777
\(33\) 0 0
\(34\) 637.683i 0.551629i
\(35\) 0 0
\(36\) 0 0
\(37\) −833.765 −0.609032 −0.304516 0.952507i \(-0.598495\pi\)
−0.304516 + 0.952507i \(0.598495\pi\)
\(38\) 833.642i 0.577315i
\(39\) 0 0
\(40\) − 320.259i − 0.200162i
\(41\) − 2817.60i − 1.67615i −0.545558 0.838073i \(-0.683682\pi\)
0.545558 0.838073i \(-0.316318\pi\)
\(42\) 0 0
\(43\) 3066.41 1.65841 0.829207 0.558942i \(-0.188793\pi\)
0.829207 + 0.558942i \(0.188793\pi\)
\(44\) −512.235 −0.264584
\(45\) 0 0
\(46\) 2006.77 0.948377
\(47\) 613.726i 0.277830i 0.990304 + 0.138915i \(0.0443614\pi\)
−0.990304 + 0.138915i \(0.955639\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1201.17 −0.480466
\(51\) 0 0
\(52\) 1831.35i 0.677276i
\(53\) 1152.60 0.410324 0.205162 0.978728i \(-0.434228\pi\)
0.205162 + 0.978728i \(0.434228\pi\)
\(54\) 0 0
\(55\) − 906.246i − 0.299585i
\(56\) 0 0
\(57\) 0 0
\(58\) 2094.16 0.622520
\(59\) − 3492.59i − 1.00333i −0.865062 0.501665i \(-0.832721\pi\)
0.865062 0.501665i \(-0.167279\pi\)
\(60\) 0 0
\(61\) 2272.21i 0.610645i 0.952249 + 0.305323i \(0.0987643\pi\)
−0.952249 + 0.305323i \(0.901236\pi\)
\(62\) − 1885.75i − 0.490569i
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) −3240.03 −0.766870
\(66\) 0 0
\(67\) −8674.62 −1.93242 −0.966208 0.257764i \(-0.917014\pi\)
−0.966208 + 0.257764i \(0.917014\pi\)
\(68\) − 1803.64i − 0.390061i
\(69\) 0 0
\(70\) 0 0
\(71\) 353.591 0.0701431 0.0350715 0.999385i \(-0.488834\pi\)
0.0350715 + 0.999385i \(0.488834\pi\)
\(72\) 0 0
\(73\) − 4069.95i − 0.763736i −0.924217 0.381868i \(-0.875281\pi\)
0.924217 0.381868i \(-0.124719\pi\)
\(74\) 2358.24 0.430650
\(75\) 0 0
\(76\) − 2357.90i − 0.408223i
\(77\) 0 0
\(78\) 0 0
\(79\) 6472.83 1.03715 0.518573 0.855033i \(-0.326464\pi\)
0.518573 + 0.855033i \(0.326464\pi\)
\(80\) 905.829i 0.141536i
\(81\) 0 0
\(82\) 7969.38i 1.18521i
\(83\) − 8225.83i − 1.19405i −0.802222 0.597026i \(-0.796349\pi\)
0.802222 0.597026i \(-0.203651\pi\)
\(84\) 0 0
\(85\) 3191.00 0.441660
\(86\) −8673.11 −1.17268
\(87\) 0 0
\(88\) 1448.82 0.187089
\(89\) 15538.3i 1.96166i 0.194878 + 0.980828i \(0.437569\pi\)
−0.194878 + 0.980828i \(0.562431\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5675.99 −0.670604
\(93\) 0 0
\(94\) − 1735.88i − 0.196455i
\(95\) 4171.58 0.462225
\(96\) 0 0
\(97\) − 1558.61i − 0.165651i −0.996564 0.0828254i \(-0.973606\pi\)
0.996564 0.0828254i \(-0.0263944\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3397.41 0.339741
\(101\) 15735.3i 1.54253i 0.636516 + 0.771264i \(0.280375\pi\)
−0.636516 + 0.771264i \(0.719625\pi\)
\(102\) 0 0
\(103\) 33.7287i 0.00317926i 0.999999 + 0.00158963i \(0.000505995\pi\)
−0.999999 + 0.00158963i \(0.999494\pi\)
\(104\) − 5179.85i − 0.478906i
\(105\) 0 0
\(106\) −3260.05 −0.290143
\(107\) 5446.46 0.475715 0.237858 0.971300i \(-0.423555\pi\)
0.237858 + 0.971300i \(0.423555\pi\)
\(108\) 0 0
\(109\) 16697.8 1.40542 0.702709 0.711477i \(-0.251974\pi\)
0.702709 + 0.711477i \(0.251974\pi\)
\(110\) 2563.25i 0.211839i
\(111\) 0 0
\(112\) 0 0
\(113\) −9455.64 −0.740515 −0.370258 0.928929i \(-0.620731\pi\)
−0.370258 + 0.928929i \(0.620731\pi\)
\(114\) 0 0
\(115\) − 10041.9i − 0.759315i
\(116\) −5923.18 −0.440188
\(117\) 0 0
\(118\) 9878.54i 0.709461i
\(119\) 0 0
\(120\) 0 0
\(121\) −10541.2 −0.719980
\(122\) − 6426.79i − 0.431792i
\(123\) 0 0
\(124\) 5333.70i 0.346885i
\(125\) 14856.7i 0.950827i
\(126\) 0 0
\(127\) −2380.07 −0.147564 −0.0737822 0.997274i \(-0.523507\pi\)
−0.0737822 + 0.997274i \(0.523507\pi\)
\(128\) −1448.15 −0.0883883
\(129\) 0 0
\(130\) 9164.17 0.542259
\(131\) 4339.79i 0.252887i 0.991974 + 0.126443i \(0.0403562\pi\)
−0.991974 + 0.126443i \(0.959644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 24535.5 1.36642
\(135\) 0 0
\(136\) 5101.46i 0.275814i
\(137\) 30759.2 1.63883 0.819414 0.573202i \(-0.194299\pi\)
0.819414 + 0.573202i \(0.194299\pi\)
\(138\) 0 0
\(139\) − 27186.6i − 1.40710i −0.710644 0.703551i \(-0.751596\pi\)
0.710644 0.703551i \(-0.248404\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1000.11 −0.0495987
\(143\) − 14657.6i − 0.716786i
\(144\) 0 0
\(145\) − 10479.3i − 0.498419i
\(146\) 11511.6i 0.540043i
\(147\) 0 0
\(148\) −6670.12 −0.304516
\(149\) 5826.92 0.262462 0.131231 0.991352i \(-0.458107\pi\)
0.131231 + 0.991352i \(0.458107\pi\)
\(150\) 0 0
\(151\) 25186.8 1.10463 0.552317 0.833634i \(-0.313744\pi\)
0.552317 + 0.833634i \(0.313744\pi\)
\(152\) 6669.14i 0.288657i
\(153\) 0 0
\(154\) 0 0
\(155\) −9436.37 −0.392773
\(156\) 0 0
\(157\) − 23896.0i − 0.969453i −0.874666 0.484726i \(-0.838919\pi\)
0.874666 0.484726i \(-0.161081\pi\)
\(158\) −18307.9 −0.733373
\(159\) 0 0
\(160\) − 2562.07i − 0.100081i
\(161\) 0 0
\(162\) 0 0
\(163\) −42585.9 −1.60284 −0.801422 0.598100i \(-0.795923\pi\)
−0.801422 + 0.598100i \(0.795923\pi\)
\(164\) − 22540.8i − 0.838073i
\(165\) 0 0
\(166\) 23266.2i 0.844323i
\(167\) − 26356.3i − 0.945043i −0.881319 0.472521i \(-0.843344\pi\)
0.881319 0.472521i \(-0.156656\pi\)
\(168\) 0 0
\(169\) −23843.0 −0.834809
\(170\) −9025.50 −0.312301
\(171\) 0 0
\(172\) 24531.3 0.829207
\(173\) − 28019.2i − 0.936188i −0.883679 0.468094i \(-0.844941\pi\)
0.883679 0.468094i \(-0.155059\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4097.88 −0.132292
\(177\) 0 0
\(178\) − 43948.9i − 1.38710i
\(179\) −25398.5 −0.792687 −0.396343 0.918102i \(-0.629721\pi\)
−0.396343 + 0.918102i \(0.629721\pi\)
\(180\) 0 0
\(181\) − 44097.2i − 1.34603i −0.739630 0.673014i \(-0.764999\pi\)
0.739630 0.673014i \(-0.235001\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 16054.1 0.474188
\(185\) − 11800.8i − 0.344799i
\(186\) 0 0
\(187\) 14435.8i 0.412816i
\(188\) 4909.81i 0.138915i
\(189\) 0 0
\(190\) −11799.0 −0.326843
\(191\) 64559.1 1.76966 0.884832 0.465911i \(-0.154273\pi\)
0.884832 + 0.465911i \(0.154273\pi\)
\(192\) 0 0
\(193\) 36674.6 0.984579 0.492290 0.870432i \(-0.336160\pi\)
0.492290 + 0.870432i \(0.336160\pi\)
\(194\) 4408.41i 0.117133i
\(195\) 0 0
\(196\) 0 0
\(197\) 73147.0 1.88480 0.942398 0.334494i \(-0.108566\pi\)
0.942398 + 0.334494i \(0.108566\pi\)
\(198\) 0 0
\(199\) 1401.03i 0.0353786i 0.999844 + 0.0176893i \(0.00563097\pi\)
−0.999844 + 0.0176893i \(0.994369\pi\)
\(200\) −9609.33 −0.240233
\(201\) 0 0
\(202\) − 44506.2i − 1.09073i
\(203\) 0 0
\(204\) 0 0
\(205\) 39879.2 0.948939
\(206\) − 95.3993i − 0.00224807i
\(207\) 0 0
\(208\) 14650.8i 0.338638i
\(209\) 18871.8i 0.432038i
\(210\) 0 0
\(211\) 58231.7 1.30796 0.653980 0.756512i \(-0.273098\pi\)
0.653980 + 0.756512i \(0.273098\pi\)
\(212\) 9220.80 0.205162
\(213\) 0 0
\(214\) −15404.9 −0.336381
\(215\) 43400.6i 0.938900i
\(216\) 0 0
\(217\) 0 0
\(218\) −47228.4 −0.993781
\(219\) 0 0
\(220\) − 7249.97i − 0.149793i
\(221\) 51611.0 1.05671
\(222\) 0 0
\(223\) − 61050.9i − 1.22767i −0.789434 0.613836i \(-0.789626\pi\)
0.789434 0.613836i \(-0.210374\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 26744.6 0.523623
\(227\) 24993.7i 0.485041i 0.970146 + 0.242520i \(0.0779742\pi\)
−0.970146 + 0.242520i \(0.922026\pi\)
\(228\) 0 0
\(229\) − 18928.9i − 0.360956i −0.983579 0.180478i \(-0.942236\pi\)
0.983579 0.180478i \(-0.0577644\pi\)
\(230\) 28402.9i 0.536917i
\(231\) 0 0
\(232\) 16753.3 0.311260
\(233\) −6684.74 −0.123132 −0.0615662 0.998103i \(-0.519610\pi\)
−0.0615662 + 0.998103i \(0.519610\pi\)
\(234\) 0 0
\(235\) −8686.42 −0.157291
\(236\) − 27940.7i − 0.501665i
\(237\) 0 0
\(238\) 0 0
\(239\) 96461.0 1.68871 0.844356 0.535782i \(-0.179983\pi\)
0.844356 + 0.535782i \(0.179983\pi\)
\(240\) 0 0
\(241\) − 54700.2i − 0.941792i −0.882189 0.470896i \(-0.843931\pi\)
0.882189 0.470896i \(-0.156069\pi\)
\(242\) 29815.1 0.509103
\(243\) 0 0
\(244\) 18177.7i 0.305323i
\(245\) 0 0
\(246\) 0 0
\(247\) 67471.0 1.10592
\(248\) − 15086.0i − 0.245285i
\(249\) 0 0
\(250\) − 42021.0i − 0.672336i
\(251\) − 108137.i − 1.71643i −0.513286 0.858217i \(-0.671572\pi\)
0.513286 0.858217i \(-0.328428\pi\)
\(252\) 0 0
\(253\) 45428.8 0.709725
\(254\) 6731.84 0.104344
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 35886.1i − 0.543325i −0.962393 0.271663i \(-0.912427\pi\)
0.962393 0.271663i \(-0.0875735\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −25920.2 −0.383435
\(261\) 0 0
\(262\) − 12274.8i − 0.178818i
\(263\) 65431.6 0.945967 0.472984 0.881071i \(-0.343177\pi\)
0.472984 + 0.881071i \(0.343177\pi\)
\(264\) 0 0
\(265\) 16313.4i 0.232302i
\(266\) 0 0
\(267\) 0 0
\(268\) −69396.9 −0.966208
\(269\) − 69211.3i − 0.956472i −0.878231 0.478236i \(-0.841276\pi\)
0.878231 0.478236i \(-0.158724\pi\)
\(270\) 0 0
\(271\) 6550.55i 0.0891947i 0.999005 + 0.0445973i \(0.0142005\pi\)
−0.999005 + 0.0445973i \(0.985800\pi\)
\(272\) − 14429.1i − 0.195030i
\(273\) 0 0
\(274\) −87000.1 −1.15883
\(275\) −27191.8 −0.359561
\(276\) 0 0
\(277\) 79103.2 1.03094 0.515471 0.856907i \(-0.327617\pi\)
0.515471 + 0.856907i \(0.327617\pi\)
\(278\) 76895.4i 0.994972i
\(279\) 0 0
\(280\) 0 0
\(281\) −34363.1 −0.435191 −0.217596 0.976039i \(-0.569821\pi\)
−0.217596 + 0.976039i \(0.569821\pi\)
\(282\) 0 0
\(283\) − 91912.3i − 1.14763i −0.818986 0.573813i \(-0.805463\pi\)
0.818986 0.573813i \(-0.194537\pi\)
\(284\) 2828.73 0.0350715
\(285\) 0 0
\(286\) 41457.8i 0.506844i
\(287\) 0 0
\(288\) 0 0
\(289\) 32691.0 0.391411
\(290\) 29639.8i 0.352436i
\(291\) 0 0
\(292\) − 32559.6i − 0.381868i
\(293\) 23218.2i 0.270453i 0.990815 + 0.135227i \(0.0431763\pi\)
−0.990815 + 0.135227i \(0.956824\pi\)
\(294\) 0 0
\(295\) 49432.7 0.568028
\(296\) 18865.9 0.215325
\(297\) 0 0
\(298\) −16481.0 −0.185589
\(299\) − 162418.i − 1.81673i
\(300\) 0 0
\(301\) 0 0
\(302\) −71239.0 −0.781095
\(303\) 0 0
\(304\) − 18863.2i − 0.204112i
\(305\) −32159.9 −0.345713
\(306\) 0 0
\(307\) 66385.9i 0.704367i 0.935931 + 0.352183i \(0.114561\pi\)
−0.935931 + 0.352183i \(0.885439\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 26690.1 0.277733
\(311\) − 95700.5i − 0.989449i −0.869050 0.494724i \(-0.835269\pi\)
0.869050 0.494724i \(-0.164731\pi\)
\(312\) 0 0
\(313\) 47904.0i 0.488971i 0.969653 + 0.244486i \(0.0786191\pi\)
−0.969653 + 0.244486i \(0.921381\pi\)
\(314\) 67588.2i 0.685507i
\(315\) 0 0
\(316\) 51782.6 0.518573
\(317\) 34128.2 0.339621 0.169811 0.985477i \(-0.445684\pi\)
0.169811 + 0.985477i \(0.445684\pi\)
\(318\) 0 0
\(319\) 47407.2 0.465868
\(320\) 7246.63i 0.0707679i
\(321\) 0 0
\(322\) 0 0
\(323\) −66450.0 −0.636927
\(324\) 0 0
\(325\) 97216.5i 0.920393i
\(326\) 120451. 1.13338
\(327\) 0 0
\(328\) 63755.1i 0.592607i
\(329\) 0 0
\(330\) 0 0
\(331\) 133285. 1.21654 0.608270 0.793730i \(-0.291864\pi\)
0.608270 + 0.793730i \(0.291864\pi\)
\(332\) − 65806.6i − 0.597026i
\(333\) 0 0
\(334\) 74546.9i 0.668246i
\(335\) − 122777.i − 1.09402i
\(336\) 0 0
\(337\) −49734.4 −0.437922 −0.218961 0.975734i \(-0.570267\pi\)
−0.218961 + 0.975734i \(0.570267\pi\)
\(338\) 67438.1 0.590299
\(339\) 0 0
\(340\) 25528.0 0.220830
\(341\) − 42689.2i − 0.367121i
\(342\) 0 0
\(343\) 0 0
\(344\) −69384.9 −0.586338
\(345\) 0 0
\(346\) 79250.2i 0.661985i
\(347\) −18769.5 −0.155882 −0.0779408 0.996958i \(-0.524835\pi\)
−0.0779408 + 0.996958i \(0.524835\pi\)
\(348\) 0 0
\(349\) 4574.17i 0.0375545i 0.999824 + 0.0187772i \(0.00597733\pi\)
−0.999824 + 0.0187772i \(0.994023\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11590.6 0.0935447
\(353\) 60893.0i 0.488673i 0.969691 + 0.244336i \(0.0785701\pi\)
−0.969691 + 0.244336i \(0.921430\pi\)
\(354\) 0 0
\(355\) 5004.58i 0.0397110i
\(356\) 124306.i 0.980828i
\(357\) 0 0
\(358\) 71837.7 0.560514
\(359\) −232118. −1.80103 −0.900513 0.434830i \(-0.856808\pi\)
−0.900513 + 0.434830i \(0.856808\pi\)
\(360\) 0 0
\(361\) 43451.1 0.333416
\(362\) 124726.i 0.951786i
\(363\) 0 0
\(364\) 0 0
\(365\) 57604.4 0.432384
\(366\) 0 0
\(367\) 144718.i 1.07446i 0.843436 + 0.537229i \(0.180529\pi\)
−0.843436 + 0.537229i \(0.819471\pi\)
\(368\) −45407.9 −0.335302
\(369\) 0 0
\(370\) 33377.6i 0.243810i
\(371\) 0 0
\(372\) 0 0
\(373\) −148883. −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(374\) − 40830.5i − 0.291905i
\(375\) 0 0
\(376\) − 13887.0i − 0.0982276i
\(377\) − 169491.i − 1.19252i
\(378\) 0 0
\(379\) 140667. 0.979299 0.489649 0.871919i \(-0.337125\pi\)
0.489649 + 0.871919i \(0.337125\pi\)
\(380\) 33372.7 0.231113
\(381\) 0 0
\(382\) −182601. −1.25134
\(383\) − 29322.3i − 0.199895i −0.994993 0.0999473i \(-0.968133\pi\)
0.994993 0.0999473i \(-0.0318674\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −103731. −0.696202
\(387\) 0 0
\(388\) − 12468.9i − 0.0828254i
\(389\) −132420. −0.875092 −0.437546 0.899196i \(-0.644152\pi\)
−0.437546 + 0.899196i \(0.644152\pi\)
\(390\) 0 0
\(391\) 159960.i 1.04630i
\(392\) 0 0
\(393\) 0 0
\(394\) −206891. −1.33275
\(395\) 91613.7i 0.587173i
\(396\) 0 0
\(397\) 34227.6i 0.217168i 0.994087 + 0.108584i \(0.0346316\pi\)
−0.994087 + 0.108584i \(0.965368\pi\)
\(398\) − 3962.70i − 0.0250164i
\(399\) 0 0
\(400\) 27179.3 0.169870
\(401\) 179081. 1.11368 0.556841 0.830619i \(-0.312013\pi\)
0.556841 + 0.830619i \(0.312013\pi\)
\(402\) 0 0
\(403\) −152623. −0.939747
\(404\) 125883.i 0.771264i
\(405\) 0 0
\(406\) 0 0
\(407\) 53385.5 0.322281
\(408\) 0 0
\(409\) − 49766.8i − 0.297504i −0.988875 0.148752i \(-0.952474\pi\)
0.988875 0.148752i \(-0.0475257\pi\)
\(410\) −112795. −0.671001
\(411\) 0 0
\(412\) 269.830i 0.00158963i
\(413\) 0 0
\(414\) 0 0
\(415\) 116425. 0.676005
\(416\) − 41438.8i − 0.239453i
\(417\) 0 0
\(418\) − 53377.6i − 0.305497i
\(419\) − 43951.2i − 0.250347i −0.992135 0.125174i \(-0.960051\pi\)
0.992135 0.125174i \(-0.0399488\pi\)
\(420\) 0 0
\(421\) −218257. −1.23141 −0.615707 0.787975i \(-0.711129\pi\)
−0.615707 + 0.787975i \(0.711129\pi\)
\(422\) −164704. −0.924867
\(423\) 0 0
\(424\) −26080.4 −0.145071
\(425\) − 95745.4i − 0.530078i
\(426\) 0 0
\(427\) 0 0
\(428\) 43571.7 0.237858
\(429\) 0 0
\(430\) − 122756.i − 0.663902i
\(431\) 76739.6 0.413109 0.206555 0.978435i \(-0.433775\pi\)
0.206555 + 0.978435i \(0.433775\pi\)
\(432\) 0 0
\(433\) − 216713.i − 1.15587i −0.816083 0.577935i \(-0.803859\pi\)
0.816083 0.577935i \(-0.196141\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 133582. 0.702709
\(437\) 209116.i 1.09502i
\(438\) 0 0
\(439\) 23762.1i 0.123298i 0.998098 + 0.0616489i \(0.0196359\pi\)
−0.998098 + 0.0616489i \(0.980364\pi\)
\(440\) 20506.0i 0.105919i
\(441\) 0 0
\(442\) −145978. −0.747210
\(443\) −174865. −0.891035 −0.445517 0.895273i \(-0.646980\pi\)
−0.445517 + 0.895273i \(0.646980\pi\)
\(444\) 0 0
\(445\) −219922. −1.11058
\(446\) 172678.i 0.868095i
\(447\) 0 0
\(448\) 0 0
\(449\) 195154. 0.968023 0.484011 0.875062i \(-0.339179\pi\)
0.484011 + 0.875062i \(0.339179\pi\)
\(450\) 0 0
\(451\) 180409.i 0.886965i
\(452\) −75645.1 −0.370258
\(453\) 0 0
\(454\) − 70692.8i − 0.342976i
\(455\) 0 0
\(456\) 0 0
\(457\) 58472.2 0.279974 0.139987 0.990153i \(-0.455294\pi\)
0.139987 + 0.990153i \(0.455294\pi\)
\(458\) 53539.0i 0.255234i
\(459\) 0 0
\(460\) − 80335.6i − 0.379658i
\(461\) 307243.i 1.44571i 0.691002 + 0.722853i \(0.257169\pi\)
−0.691002 + 0.722853i \(0.742831\pi\)
\(462\) 0 0
\(463\) −17772.2 −0.0829049 −0.0414524 0.999140i \(-0.513199\pi\)
−0.0414524 + 0.999140i \(0.513199\pi\)
\(464\) −47385.4 −0.220094
\(465\) 0 0
\(466\) 18907.3 0.0870678
\(467\) − 64965.1i − 0.297883i −0.988846 0.148942i \(-0.952413\pi\)
0.988846 0.148942i \(-0.0475867\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 24568.9 0.111222
\(471\) 0 0
\(472\) 79028.3i 0.354731i
\(473\) −196340. −0.877581
\(474\) 0 0
\(475\) − 125168.i − 0.554760i
\(476\) 0 0
\(477\) 0 0
\(478\) −272833. −1.19410
\(479\) 210937.i 0.919351i 0.888087 + 0.459675i \(0.152034\pi\)
−0.888087 + 0.459675i \(0.847966\pi\)
\(480\) 0 0
\(481\) − 190865.i − 0.824965i
\(482\) 154716.i 0.665948i
\(483\) 0 0
\(484\) −84329.8 −0.359990
\(485\) 22059.9 0.0937821
\(486\) 0 0
\(487\) −228888. −0.965084 −0.482542 0.875873i \(-0.660286\pi\)
−0.482542 + 0.875873i \(0.660286\pi\)
\(488\) − 51414.3i − 0.215896i
\(489\) 0 0
\(490\) 0 0
\(491\) −140350. −0.582169 −0.291085 0.956697i \(-0.594016\pi\)
−0.291085 + 0.956697i \(0.594016\pi\)
\(492\) 0 0
\(493\) 166926.i 0.686801i
\(494\) −190837. −0.782002
\(495\) 0 0
\(496\) 42669.6i 0.173442i
\(497\) 0 0
\(498\) 0 0
\(499\) −345327. −1.38685 −0.693424 0.720529i \(-0.743899\pi\)
−0.693424 + 0.720529i \(0.743899\pi\)
\(500\) 118853.i 0.475414i
\(501\) 0 0
\(502\) 305858.i 1.21370i
\(503\) − 58979.0i − 0.233110i −0.993184 0.116555i \(-0.962815\pi\)
0.993184 0.116555i \(-0.0371851\pi\)
\(504\) 0 0
\(505\) −222711. −0.873291
\(506\) −128492. −0.501852
\(507\) 0 0
\(508\) −19040.5 −0.0737822
\(509\) − 23848.7i − 0.0920511i −0.998940 0.0460255i \(-0.985344\pi\)
0.998940 0.0460255i \(-0.0146556\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11585.2 −0.0441942
\(513\) 0 0
\(514\) 101501.i 0.384189i
\(515\) −477.382 −0.00179991
\(516\) 0 0
\(517\) − 39296.5i − 0.147019i
\(518\) 0 0
\(519\) 0 0
\(520\) 73313.4 0.271129
\(521\) 481708.i 1.77463i 0.461163 + 0.887315i \(0.347432\pi\)
−0.461163 + 0.887315i \(0.652568\pi\)
\(522\) 0 0
\(523\) − 461410.i − 1.68688i −0.537226 0.843438i \(-0.680528\pi\)
0.537226 0.843438i \(-0.319472\pi\)
\(524\) 34718.3i 0.126443i
\(525\) 0 0
\(526\) −185068. −0.668900
\(527\) 150314. 0.541225
\(528\) 0 0
\(529\) 223547. 0.798837
\(530\) − 46141.3i − 0.164262i
\(531\) 0 0
\(532\) 0 0
\(533\) 645003. 2.27043
\(534\) 0 0
\(535\) 77086.9i 0.269323i
\(536\) 196284. 0.683212
\(537\) 0 0
\(538\) 195759.i 0.676328i
\(539\) 0 0
\(540\) 0 0
\(541\) −508196. −1.73635 −0.868174 0.496260i \(-0.834706\pi\)
−0.868174 + 0.496260i \(0.834706\pi\)
\(542\) − 18527.7i − 0.0630702i
\(543\) 0 0
\(544\) 40811.7i 0.137907i
\(545\) 236333.i 0.795668i
\(546\) 0 0
\(547\) 40170.8 0.134257 0.0671283 0.997744i \(-0.478616\pi\)
0.0671283 + 0.997744i \(0.478616\pi\)
\(548\) 246073. 0.819414
\(549\) 0 0
\(550\) 76910.0 0.254248
\(551\) 218222.i 0.718780i
\(552\) 0 0
\(553\) 0 0
\(554\) −223738. −0.728987
\(555\) 0 0
\(556\) − 217493.i − 0.703551i
\(557\) 78460.4 0.252895 0.126447 0.991973i \(-0.459642\pi\)
0.126447 + 0.991973i \(0.459642\pi\)
\(558\) 0 0
\(559\) 701959.i 2.24641i
\(560\) 0 0
\(561\) 0 0
\(562\) 97193.7 0.307727
\(563\) − 410625.i − 1.29547i −0.761864 0.647737i \(-0.775716\pi\)
0.761864 0.647737i \(-0.224284\pi\)
\(564\) 0 0
\(565\) − 133831.i − 0.419238i
\(566\) 259967.i 0.811495i
\(567\) 0 0
\(568\) −8000.86 −0.0247993
\(569\) 148809. 0.459625 0.229812 0.973235i \(-0.426189\pi\)
0.229812 + 0.973235i \(0.426189\pi\)
\(570\) 0 0
\(571\) 72292.6 0.221729 0.110864 0.993836i \(-0.464638\pi\)
0.110864 + 0.993836i \(0.464638\pi\)
\(572\) − 117261.i − 0.358393i
\(573\) 0 0
\(574\) 0 0
\(575\) −301307. −0.911326
\(576\) 0 0
\(577\) 557335.i 1.67404i 0.547176 + 0.837018i \(0.315703\pi\)
−0.547176 + 0.837018i \(0.684297\pi\)
\(578\) −92464.2 −0.276769
\(579\) 0 0
\(580\) − 83834.1i − 0.249210i
\(581\) 0 0
\(582\) 0 0
\(583\) −73800.4 −0.217131
\(584\) 92092.4i 0.270021i
\(585\) 0 0
\(586\) − 65670.9i − 0.191239i
\(587\) − 308119.i − 0.894217i −0.894480 0.447109i \(-0.852454\pi\)
0.894480 0.447109i \(-0.147546\pi\)
\(588\) 0 0
\(589\) 196505. 0.566426
\(590\) −139817. −0.401657
\(591\) 0 0
\(592\) −53360.9 −0.152258
\(593\) 231870.i 0.659378i 0.944090 + 0.329689i \(0.106944\pi\)
−0.944090 + 0.329689i \(0.893056\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 46615.3 0.131231
\(597\) 0 0
\(598\) 459387.i 1.28462i
\(599\) 334083. 0.931110 0.465555 0.885019i \(-0.345855\pi\)
0.465555 + 0.885019i \(0.345855\pi\)
\(600\) 0 0
\(601\) − 645072.i − 1.78591i −0.450147 0.892955i \(-0.648628\pi\)
0.450147 0.892955i \(-0.351372\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 201494. 0.552317
\(605\) − 149196.i − 0.407612i
\(606\) 0 0
\(607\) 139306.i 0.378088i 0.981969 + 0.189044i \(0.0605388\pi\)
−0.981969 + 0.189044i \(0.939461\pi\)
\(608\) 53353.1i 0.144329i
\(609\) 0 0
\(610\) 90962.0 0.244456
\(611\) −140494. −0.376335
\(612\) 0 0
\(613\) −407331. −1.08399 −0.541997 0.840381i \(-0.682331\pi\)
−0.541997 + 0.840381i \(0.682331\pi\)
\(614\) − 187768.i − 0.498063i
\(615\) 0 0
\(616\) 0 0
\(617\) −276504. −0.726325 −0.363163 0.931726i \(-0.618303\pi\)
−0.363163 + 0.931726i \(0.618303\pi\)
\(618\) 0 0
\(619\) − 223901.i − 0.584352i −0.956365 0.292176i \(-0.905621\pi\)
0.956365 0.292176i \(-0.0943793\pi\)
\(620\) −75491.0 −0.196387
\(621\) 0 0
\(622\) 270682.i 0.699646i
\(623\) 0 0
\(624\) 0 0
\(625\) 55147.5 0.141178
\(626\) − 135493.i − 0.345755i
\(627\) 0 0
\(628\) − 191168.i − 0.484726i
\(629\) 187976.i 0.475119i
\(630\) 0 0
\(631\) 299528. 0.752278 0.376139 0.926563i \(-0.377251\pi\)
0.376139 + 0.926563i \(0.377251\pi\)
\(632\) −146463. −0.366686
\(633\) 0 0
\(634\) −96529.1 −0.240148
\(635\) − 33686.4i − 0.0835425i
\(636\) 0 0
\(637\) 0 0
\(638\) −134088. −0.329418
\(639\) 0 0
\(640\) − 20496.6i − 0.0500405i
\(641\) −577884. −1.40645 −0.703226 0.710967i \(-0.748258\pi\)
−0.703226 + 0.710967i \(0.748258\pi\)
\(642\) 0 0
\(643\) 135320.i 0.327295i 0.986519 + 0.163647i \(0.0523259\pi\)
−0.986519 + 0.163647i \(0.947674\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 187949. 0.450375
\(647\) 218100.i 0.521011i 0.965472 + 0.260506i \(0.0838892\pi\)
−0.965472 + 0.260506i \(0.916111\pi\)
\(648\) 0 0
\(649\) 223629.i 0.530931i
\(650\) − 274970.i − 0.650816i
\(651\) 0 0
\(652\) −340688. −0.801422
\(653\) −144254. −0.338301 −0.169150 0.985590i \(-0.554102\pi\)
−0.169150 + 0.985590i \(0.554102\pi\)
\(654\) 0 0
\(655\) −61423.6 −0.143170
\(656\) − 180327.i − 0.419037i
\(657\) 0 0
\(658\) 0 0
\(659\) 159392. 0.367024 0.183512 0.983017i \(-0.441253\pi\)
0.183512 + 0.983017i \(0.441253\pi\)
\(660\) 0 0
\(661\) − 352962.i − 0.807839i −0.914795 0.403919i \(-0.867648\pi\)
0.914795 0.403919i \(-0.132352\pi\)
\(662\) −376988. −0.860223
\(663\) 0 0
\(664\) 186129.i 0.422161i
\(665\) 0 0
\(666\) 0 0
\(667\) 525311. 1.18077
\(668\) − 210850.i − 0.472521i
\(669\) 0 0
\(670\) 347265.i 0.773592i
\(671\) − 145488.i − 0.323135i
\(672\) 0 0
\(673\) 504858. 1.11465 0.557326 0.830294i \(-0.311827\pi\)
0.557326 + 0.830294i \(0.311827\pi\)
\(674\) 140670. 0.309658
\(675\) 0 0
\(676\) −190744. −0.417404
\(677\) 195706.i 0.426999i 0.976943 + 0.213500i \(0.0684862\pi\)
−0.976943 + 0.213500i \(0.931514\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −72204.0 −0.156150
\(681\) 0 0
\(682\) 120743.i 0.259594i
\(683\) 234441. 0.502566 0.251283 0.967914i \(-0.419148\pi\)
0.251283 + 0.967914i \(0.419148\pi\)
\(684\) 0 0
\(685\) 435352.i 0.927812i
\(686\) 0 0
\(687\) 0 0
\(688\) 196250. 0.414603
\(689\) 263852.i 0.555805i
\(690\) 0 0
\(691\) − 89309.2i − 0.187042i −0.995617 0.0935212i \(-0.970188\pi\)
0.995617 0.0935212i \(-0.0298123\pi\)
\(692\) − 224153.i − 0.468094i
\(693\) 0 0
\(694\) 53088.3 0.110225
\(695\) 384788. 0.796622
\(696\) 0 0
\(697\) −635243. −1.30760
\(698\) − 12937.7i − 0.0265550i
\(699\) 0 0
\(700\) 0 0
\(701\) −122213. −0.248704 −0.124352 0.992238i \(-0.539685\pi\)
−0.124352 + 0.992238i \(0.539685\pi\)
\(702\) 0 0
\(703\) 245741.i 0.497242i
\(704\) −32783.1 −0.0661461
\(705\) 0 0
\(706\) − 172231.i − 0.345544i
\(707\) 0 0
\(708\) 0 0
\(709\) 84019.8 0.167143 0.0835717 0.996502i \(-0.473367\pi\)
0.0835717 + 0.996502i \(0.473367\pi\)
\(710\) − 14155.1i − 0.0280799i
\(711\) 0 0
\(712\) − 351591.i − 0.693550i
\(713\) − 473032.i − 0.930489i
\(714\) 0 0
\(715\) 207457. 0.405804
\(716\) −203188. −0.396343
\(717\) 0 0
\(718\) 656529. 1.27352
\(719\) − 759119.i − 1.46843i −0.678919 0.734213i \(-0.737551\pi\)
0.678919 0.734213i \(-0.262449\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −122898. −0.235760
\(723\) 0 0
\(724\) − 352778.i − 0.673014i
\(725\) −314429. −0.598200
\(726\) 0 0
\(727\) − 92384.1i − 0.174795i −0.996174 0.0873974i \(-0.972145\pi\)
0.996174 0.0873974i \(-0.0278550\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −162930. −0.305742
\(731\) − 691337.i − 1.29376i
\(732\) 0 0
\(733\) 3352.68i 0.00623999i 0.999995 + 0.00312000i \(0.000993127\pi\)
−0.999995 + 0.00312000i \(0.999007\pi\)
\(734\) − 409324.i − 0.759757i
\(735\) 0 0
\(736\) 128433. 0.237094
\(737\) 555431. 1.02257
\(738\) 0 0
\(739\) −27185.6 −0.0497795 −0.0248897 0.999690i \(-0.507923\pi\)
−0.0248897 + 0.999690i \(0.507923\pi\)
\(740\) − 94406.0i − 0.172400i
\(741\) 0 0
\(742\) 0 0
\(743\) −773801. −1.40169 −0.700845 0.713314i \(-0.747193\pi\)
−0.700845 + 0.713314i \(0.747193\pi\)
\(744\) 0 0
\(745\) 82471.7i 0.148591i
\(746\) 421105. 0.756681
\(747\) 0 0
\(748\) 115486.i 0.206408i
\(749\) 0 0
\(750\) 0 0
\(751\) 197844. 0.350787 0.175394 0.984498i \(-0.443880\pi\)
0.175394 + 0.984498i \(0.443880\pi\)
\(752\) 39278.5i 0.0694574i
\(753\) 0 0
\(754\) 479393.i 0.843236i
\(755\) 356483.i 0.625381i
\(756\) 0 0
\(757\) −770706. −1.34492 −0.672461 0.740132i \(-0.734763\pi\)
−0.672461 + 0.740132i \(0.734763\pi\)
\(758\) −397868. −0.692469
\(759\) 0 0
\(760\) −94392.2 −0.163421
\(761\) 152140.i 0.262708i 0.991336 + 0.131354i \(0.0419325\pi\)
−0.991336 + 0.131354i \(0.958068\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 516473. 0.884832
\(765\) 0 0
\(766\) 82936.1i 0.141347i
\(767\) 799521. 1.35906
\(768\) 0 0
\(769\) 961897.i 1.62658i 0.581857 + 0.813291i \(0.302326\pi\)
−0.581857 + 0.813291i \(0.697674\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 293397. 0.492290
\(773\) − 151069.i − 0.252823i −0.991978 0.126411i \(-0.959654\pi\)
0.991978 0.126411i \(-0.0403459\pi\)
\(774\) 0 0
\(775\) 283137.i 0.471404i
\(776\) 35267.3i 0.0585664i
\(777\) 0 0
\(778\) 374540. 0.618783
\(779\) −830452. −1.36848
\(780\) 0 0
\(781\) −22640.3 −0.0371175
\(782\) − 452435.i − 0.739849i
\(783\) 0 0
\(784\) 0 0
\(785\) 338215. 0.548849
\(786\) 0 0
\(787\) 198891.i 0.321118i 0.987026 + 0.160559i \(0.0513297\pi\)
−0.987026 + 0.160559i \(0.948670\pi\)
\(788\) 585176. 0.942398
\(789\) 0 0
\(790\) − 259123.i − 0.415194i
\(791\) 0 0
\(792\) 0 0
\(793\) −520153. −0.827150
\(794\) − 96810.3i − 0.153561i
\(795\) 0 0
\(796\) 11208.2i 0.0176893i
\(797\) 990756.i 1.55973i 0.625947 + 0.779866i \(0.284713\pi\)
−0.625947 + 0.779866i \(0.715287\pi\)
\(798\) 0 0
\(799\) 138368. 0.216741
\(800\) −76874.6 −0.120117
\(801\) 0 0
\(802\) −506518. −0.787492
\(803\) 260597.i 0.404145i
\(804\) 0 0
\(805\) 0 0
\(806\) 431684. 0.664501
\(807\) 0 0
\(808\) − 356050.i − 0.545366i
\(809\) 92288.6 0.141010 0.0705052 0.997511i \(-0.477539\pi\)
0.0705052 + 0.997511i \(0.477539\pi\)
\(810\) 0 0
\(811\) − 617125.i − 0.938277i −0.883125 0.469139i \(-0.844564\pi\)
0.883125 0.469139i \(-0.155436\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −150997. −0.227887
\(815\) − 602743.i − 0.907439i
\(816\) 0 0
\(817\) − 903784.i − 1.35401i
\(818\) 140762.i 0.210367i
\(819\) 0 0
\(820\) 319033. 0.474469
\(821\) −307066. −0.455559 −0.227780 0.973713i \(-0.573147\pi\)
−0.227780 + 0.973713i \(0.573147\pi\)
\(822\) 0 0
\(823\) 846858. 1.25029 0.625146 0.780508i \(-0.285040\pi\)
0.625146 + 0.780508i \(0.285040\pi\)
\(824\) − 763.194i − 0.00112404i
\(825\) 0 0
\(826\) 0 0
\(827\) −294059. −0.429956 −0.214978 0.976619i \(-0.568968\pi\)
−0.214978 + 0.976619i \(0.568968\pi\)
\(828\) 0 0
\(829\) − 1.03438e6i − 1.50513i −0.658520 0.752563i \(-0.728817\pi\)
0.658520 0.752563i \(-0.271183\pi\)
\(830\) −329299. −0.478007
\(831\) 0 0
\(832\) 117207.i 0.169319i
\(833\) 0 0
\(834\) 0 0
\(835\) 373036. 0.535030
\(836\) 150975.i 0.216019i
\(837\) 0 0
\(838\) 124313.i 0.177022i
\(839\) − 307258.i − 0.436495i −0.975893 0.218248i \(-0.929966\pi\)
0.975893 0.218248i \(-0.0700340\pi\)
\(840\) 0 0
\(841\) −159093. −0.224937
\(842\) 617324. 0.870741
\(843\) 0 0
\(844\) 465853. 0.653980
\(845\) − 337463.i − 0.472621i
\(846\) 0 0
\(847\) 0 0
\(848\) 73766.4 0.102581
\(849\) 0 0
\(850\) 270809.i 0.374822i
\(851\) 591555. 0.816838
\(852\) 0 0
\(853\) 70737.4i 0.0972190i 0.998818 + 0.0486095i \(0.0154790\pi\)
−0.998818 + 0.0486095i \(0.984521\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −123239. −0.168191
\(857\) 329863.i 0.449130i 0.974459 + 0.224565i \(0.0720961\pi\)
−0.974459 + 0.224565i \(0.927904\pi\)
\(858\) 0 0
\(859\) − 69962.5i − 0.0948154i −0.998876 0.0474077i \(-0.984904\pi\)
0.998876 0.0474077i \(-0.0150960\pi\)
\(860\) 347205.i 0.469450i
\(861\) 0 0
\(862\) −217052. −0.292113
\(863\) 772223. 1.03686 0.518431 0.855119i \(-0.326516\pi\)
0.518431 + 0.855119i \(0.326516\pi\)
\(864\) 0 0
\(865\) 396572. 0.530017
\(866\) 612957.i 0.817324i
\(867\) 0 0
\(868\) 0 0
\(869\) −414451. −0.548825
\(870\) 0 0
\(871\) − 1.98579e6i − 2.61756i
\(872\) −377827. −0.496890
\(873\) 0 0
\(874\) − 591468.i − 0.774299i
\(875\) 0 0
\(876\) 0 0
\(877\) 242520. 0.315317 0.157659 0.987494i \(-0.449605\pi\)
0.157659 + 0.987494i \(0.449605\pi\)
\(878\) − 67209.3i − 0.0871847i
\(879\) 0 0
\(880\) − 57999.7i − 0.0748963i
\(881\) − 646528.i − 0.832982i −0.909140 0.416491i \(-0.863260\pi\)
0.909140 0.416491i \(-0.136740\pi\)
\(882\) 0 0
\(883\) −461877. −0.592387 −0.296193 0.955128i \(-0.595717\pi\)
−0.296193 + 0.955128i \(0.595717\pi\)
\(884\) 412888. 0.528357
\(885\) 0 0
\(886\) 494592. 0.630057
\(887\) 215226.i 0.273557i 0.990602 + 0.136778i \(0.0436748\pi\)
−0.990602 + 0.136778i \(0.956325\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 622034. 0.785297
\(891\) 0 0
\(892\) − 488407.i − 0.613836i
\(893\) 180888. 0.226833
\(894\) 0 0
\(895\) − 359479.i − 0.448774i
\(896\) 0 0
\(897\) 0 0
\(898\) −551980. −0.684496
\(899\) − 493632.i − 0.610779i
\(900\) 0 0
\(901\) − 259860.i − 0.320103i
\(902\) − 510275.i − 0.627179i
\(903\) 0 0
\(904\) 213957. 0.261812
\(905\) 624134. 0.762045
\(906\) 0 0
\(907\) 1.09827e6 1.33504 0.667519 0.744592i \(-0.267356\pi\)
0.667519 + 0.744592i \(0.267356\pi\)
\(908\) 199949.i 0.242520i
\(909\) 0 0
\(910\) 0 0
\(911\) 1.32312e6 1.59427 0.797133 0.603803i \(-0.206349\pi\)
0.797133 + 0.603803i \(0.206349\pi\)
\(912\) 0 0
\(913\) 526695.i 0.631855i
\(914\) −165384. −0.197971
\(915\) 0 0
\(916\) − 151431.i − 0.180478i
\(917\) 0 0
\(918\) 0 0
\(919\) 516528. 0.611594 0.305797 0.952097i \(-0.401077\pi\)
0.305797 + 0.952097i \(0.401077\pi\)
\(920\) 227223.i 0.268459i
\(921\) 0 0
\(922\) − 869014.i − 1.02227i
\(923\) 80943.8i 0.0950124i
\(924\) 0 0
\(925\) −354080. −0.413826
\(926\) 50267.5 0.0586226
\(927\) 0 0
\(928\) 134026. 0.155630
\(929\) − 1.23939e6i − 1.43608i −0.696003 0.718039i \(-0.745040\pi\)
0.696003 0.718039i \(-0.254960\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −53477.9 −0.0615662
\(933\) 0 0
\(934\) 183749.i 0.210635i
\(935\) −204318. −0.233713
\(936\) 0 0
\(937\) − 1.15260e6i − 1.31281i −0.754411 0.656403i \(-0.772077\pi\)
0.754411 0.656403i \(-0.227923\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −69491.3 −0.0786457
\(941\) 107507.i 0.121411i 0.998156 + 0.0607057i \(0.0193351\pi\)
−0.998156 + 0.0607057i \(0.980665\pi\)
\(942\) 0 0
\(943\) 1.99909e6i 2.24806i
\(944\) − 223526.i − 0.250832i
\(945\) 0 0
\(946\) 555334. 0.620543
\(947\) 412614. 0.460092 0.230046 0.973180i \(-0.426112\pi\)
0.230046 + 0.973180i \(0.426112\pi\)
\(948\) 0 0
\(949\) 931689. 1.03452
\(950\) 354028.i 0.392275i
\(951\) 0 0
\(952\) 0 0
\(953\) −1.43052e6 −1.57510 −0.787550 0.616251i \(-0.788651\pi\)
−0.787550 + 0.616251i \(0.788651\pi\)
\(954\) 0 0
\(955\) 913742.i 1.00188i
\(956\) 771688. 0.844356
\(957\) 0 0
\(958\) − 596619.i − 0.650079i
\(959\) 0 0
\(960\) 0 0
\(961\) 479015. 0.518683
\(962\) 539847.i 0.583338i
\(963\) 0 0
\(964\) − 437602.i − 0.470896i
\(965\) 519077.i 0.557413i
\(966\) 0 0
\(967\) −344533. −0.368449 −0.184225 0.982884i \(-0.558977\pi\)
−0.184225 + 0.982884i \(0.558977\pi\)
\(968\) 238521. 0.254551
\(969\) 0 0
\(970\) −62394.8 −0.0663139
\(971\) 1.45316e6i 1.54126i 0.637283 + 0.770630i \(0.280058\pi\)
−0.637283 + 0.770630i \(0.719942\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 647393. 0.682417
\(975\) 0 0
\(976\) 145422.i 0.152661i
\(977\) −1.54571e6 −1.61934 −0.809669 0.586887i \(-0.800353\pi\)
−0.809669 + 0.586887i \(0.800353\pi\)
\(978\) 0 0
\(979\) − 994907.i − 1.03805i
\(980\) 0 0
\(981\) 0 0
\(982\) 396970. 0.411656
\(983\) 134268.i 0.138952i 0.997584 + 0.0694761i \(0.0221328\pi\)
−0.997584 + 0.0694761i \(0.977867\pi\)
\(984\) 0 0
\(985\) 1.03529e6i 1.06706i
\(986\) − 472139.i − 0.485641i
\(987\) 0 0
\(988\) 539768. 0.552959
\(989\) −2.17561e6 −2.22428
\(990\) 0 0
\(991\) −791191. −0.805627 −0.402814 0.915282i \(-0.631968\pi\)
−0.402814 + 0.915282i \(0.631968\pi\)
\(992\) − 120688.i − 0.122642i
\(993\) 0 0
\(994\) 0 0
\(995\) −19829.5 −0.0200293
\(996\) 0 0
\(997\) − 672757.i − 0.676811i −0.941000 0.338406i \(-0.890112\pi\)
0.941000 0.338406i \(-0.109888\pi\)
\(998\) 976732. 0.980650
\(999\) 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 882.5.c.a.685.2 4
3.2 odd 2 294.5.c.a.97.4 4
7.2 even 3 126.5.n.b.73.2 4
7.3 odd 6 126.5.n.b.19.2 4
7.6 odd 2 inner 882.5.c.a.685.1 4
21.2 odd 6 42.5.g.a.31.1 yes 4
21.5 even 6 294.5.g.c.31.1 4
21.11 odd 6 294.5.g.c.19.1 4
21.17 even 6 42.5.g.a.19.1 4
21.20 even 2 294.5.c.a.97.3 4
84.23 even 6 336.5.bh.d.241.2 4
84.59 odd 6 336.5.bh.d.145.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.5.g.a.19.1 4 21.17 even 6
42.5.g.a.31.1 yes 4 21.2 odd 6
126.5.n.b.19.2 4 7.3 odd 6
126.5.n.b.73.2 4 7.2 even 3
294.5.c.a.97.3 4 21.20 even 2
294.5.c.a.97.4 4 3.2 odd 2
294.5.g.c.19.1 4 21.11 odd 6
294.5.g.c.31.1 4 21.5 even 6
336.5.bh.d.145.2 4 84.59 odd 6
336.5.bh.d.241.2 4 84.23 even 6
882.5.c.a.685.1 4 7.6 odd 2 inner
882.5.c.a.685.2 4 1.1 even 1 trivial