Properties

Label 882.5.c.c.685.3
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: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.3
Root \(0.765367i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.5.c.c.685.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843 q^{2} +8.00000 q^{4} -27.8477i q^{5} +22.6274 q^{8} +O(q^{10})\) \(q+2.82843 q^{2} +8.00000 q^{4} -27.8477i q^{5} +22.6274 q^{8} -78.7652i q^{10} +139.664 q^{11} +101.931i q^{13} +64.0000 q^{16} -542.887i q^{17} -139.953i q^{19} -222.782i q^{20} +395.029 q^{22} -229.647 q^{23} -150.495 q^{25} +288.303i q^{26} +383.113 q^{29} +397.492i q^{31} +181.019 q^{32} -1535.52i q^{34} -898.912 q^{37} -395.848i q^{38} -630.122i q^{40} -657.862i q^{41} +1155.70 q^{43} +1117.31 q^{44} -649.539 q^{46} +1294.51i q^{47} -425.663 q^{50} +815.445i q^{52} +5168.43 q^{53} -3889.32i q^{55} +1083.61 q^{58} -4155.24i q^{59} +1840.09i q^{61} +1124.28i q^{62} +512.000 q^{64} +2838.53 q^{65} -6315.57 q^{67} -4343.09i q^{68} -1263.28 q^{71} -3995.44i q^{73} -2542.51 q^{74} -1119.63i q^{76} -10614.3 q^{79} -1782.25i q^{80} -1860.71i q^{82} -5750.31i q^{83} -15118.2 q^{85} +3268.82 q^{86} +3160.24 q^{88} -4187.05i q^{89} -1837.17 q^{92} +3661.42i q^{94} -3897.38 q^{95} -3814.28i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 24q^{11} + 256q^{16} + 1648q^{22} - 104q^{23} + 948q^{25} + 1408q^{29} - 3392q^{37} - 2024q^{43} - 192q^{44} - 2304q^{46} - 4384q^{50} + 16680q^{53} + 352q^{58} + 2048q^{64} + 6048q^{65} - 20816q^{67} + 1984q^{71} - 576q^{74} - 29616q^{79} - 29688q^{85} + 18800q^{86} + 13184q^{88} - 832q^{92} - 7240q^{95} + O(q^{100}) \)

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
\(5\) − 27.8477i − 1.11391i −0.830543 0.556954i \(-0.811970\pi\)
0.830543 0.556954i \(-0.188030\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 22.6274 0.353553
\(9\) 0 0
\(10\) − 78.7652i − 0.787652i
\(11\) 139.664 1.15425 0.577124 0.816657i \(-0.304175\pi\)
0.577124 + 0.816657i \(0.304175\pi\)
\(12\) 0 0
\(13\) 101.931i 0.603140i 0.953444 + 0.301570i \(0.0975106\pi\)
−0.953444 + 0.301570i \(0.902489\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 542.887i − 1.87850i −0.343232 0.939251i \(-0.611522\pi\)
0.343232 0.939251i \(-0.388478\pi\)
\(18\) 0 0
\(19\) − 139.953i − 0.387682i −0.981033 0.193841i \(-0.937905\pi\)
0.981033 0.193841i \(-0.0620947\pi\)
\(20\) − 222.782i − 0.556954i
\(21\) 0 0
\(22\) 395.029 0.816177
\(23\) −229.647 −0.434115 −0.217057 0.976159i \(-0.569646\pi\)
−0.217057 + 0.976159i \(0.569646\pi\)
\(24\) 0 0
\(25\) −150.495 −0.240791
\(26\) 288.303i 0.426484i
\(27\) 0 0
\(28\) 0 0
\(29\) 383.113 0.455544 0.227772 0.973714i \(-0.426856\pi\)
0.227772 + 0.973714i \(0.426856\pi\)
\(30\) 0 0
\(31\) 397.492i 0.413623i 0.978381 + 0.206812i \(0.0663087\pi\)
−0.978381 + 0.206812i \(0.933691\pi\)
\(32\) 181.019 0.176777
\(33\) 0 0
\(34\) − 1535.52i − 1.32830i
\(35\) 0 0
\(36\) 0 0
\(37\) −898.912 −0.656619 −0.328310 0.944570i \(-0.606479\pi\)
−0.328310 + 0.944570i \(0.606479\pi\)
\(38\) − 395.848i − 0.274133i
\(39\) 0 0
\(40\) − 630.122i − 0.393826i
\(41\) − 657.862i − 0.391352i −0.980669 0.195676i \(-0.937310\pi\)
0.980669 0.195676i \(-0.0626900\pi\)
\(42\) 0 0
\(43\) 1155.70 0.625041 0.312521 0.949911i \(-0.398827\pi\)
0.312521 + 0.949911i \(0.398827\pi\)
\(44\) 1117.31 0.577124
\(45\) 0 0
\(46\) −649.539 −0.306966
\(47\) 1294.51i 0.586015i 0.956110 + 0.293007i \(0.0946561\pi\)
−0.956110 + 0.293007i \(0.905344\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −425.663 −0.170265
\(51\) 0 0
\(52\) 815.445i 0.301570i
\(53\) 5168.43 1.83996 0.919978 0.391971i \(-0.128207\pi\)
0.919978 + 0.391971i \(0.128207\pi\)
\(54\) 0 0
\(55\) − 3889.32i − 1.28573i
\(56\) 0 0
\(57\) 0 0
\(58\) 1083.61 0.322118
\(59\) − 4155.24i − 1.19369i −0.802356 0.596846i \(-0.796420\pi\)
0.802356 0.596846i \(-0.203580\pi\)
\(60\) 0 0
\(61\) 1840.09i 0.494514i 0.968950 + 0.247257i \(0.0795292\pi\)
−0.968950 + 0.247257i \(0.920471\pi\)
\(62\) 1124.28i 0.292476i
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 2838.53 0.671842
\(66\) 0 0
\(67\) −6315.57 −1.40690 −0.703450 0.710745i \(-0.748358\pi\)
−0.703450 + 0.710745i \(0.748358\pi\)
\(68\) − 4343.09i − 0.939251i
\(69\) 0 0
\(70\) 0 0
\(71\) −1263.28 −0.250601 −0.125301 0.992119i \(-0.539990\pi\)
−0.125301 + 0.992119i \(0.539990\pi\)
\(72\) 0 0
\(73\) − 3995.44i − 0.749753i −0.927075 0.374877i \(-0.877685\pi\)
0.927075 0.374877i \(-0.122315\pi\)
\(74\) −2542.51 −0.464300
\(75\) 0 0
\(76\) − 1119.63i − 0.193841i
\(77\) 0 0
\(78\) 0 0
\(79\) −10614.3 −1.70073 −0.850366 0.526192i \(-0.823619\pi\)
−0.850366 + 0.526192i \(0.823619\pi\)
\(80\) − 1782.25i − 0.278477i
\(81\) 0 0
\(82\) − 1860.71i − 0.276727i
\(83\) − 5750.31i − 0.834709i −0.908744 0.417355i \(-0.862957\pi\)
0.908744 0.417355i \(-0.137043\pi\)
\(84\) 0 0
\(85\) −15118.2 −2.09248
\(86\) 3268.82 0.441971
\(87\) 0 0
\(88\) 3160.24 0.408088
\(89\) − 4187.05i − 0.528601i −0.964440 0.264301i \(-0.914859\pi\)
0.964440 0.264301i \(-0.0851411\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1837.17 −0.217057
\(93\) 0 0
\(94\) 3661.42i 0.414375i
\(95\) −3897.38 −0.431843
\(96\) 0 0
\(97\) − 3814.28i − 0.405386i −0.979242 0.202693i \(-0.935031\pi\)
0.979242 0.202693i \(-0.0649694\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1203.96 −0.120396
\(101\) − 4925.41i − 0.482836i −0.970421 0.241418i \(-0.922387\pi\)
0.970421 0.241418i \(-0.0776126\pi\)
\(102\) 0 0
\(103\) − 18892.6i − 1.78081i −0.455166 0.890407i \(-0.650420\pi\)
0.455166 0.890407i \(-0.349580\pi\)
\(104\) 2306.43i 0.213242i
\(105\) 0 0
\(106\) 14618.5 1.30104
\(107\) 8479.39 0.740623 0.370311 0.928908i \(-0.379251\pi\)
0.370311 + 0.928908i \(0.379251\pi\)
\(108\) 0 0
\(109\) 11678.0 0.982911 0.491455 0.870903i \(-0.336465\pi\)
0.491455 + 0.870903i \(0.336465\pi\)
\(110\) − 11000.7i − 0.909146i
\(111\) 0 0
\(112\) 0 0
\(113\) −19683.8 −1.54153 −0.770763 0.637121i \(-0.780125\pi\)
−0.770763 + 0.637121i \(0.780125\pi\)
\(114\) 0 0
\(115\) 6395.13i 0.483564i
\(116\) 3064.90 0.227772
\(117\) 0 0
\(118\) − 11752.8i − 0.844067i
\(119\) 0 0
\(120\) 0 0
\(121\) 4865.03 0.332288
\(122\) 5204.55i 0.349674i
\(123\) 0 0
\(124\) 3179.94i 0.206812i
\(125\) − 13213.9i − 0.845689i
\(126\) 0 0
\(127\) −27965.5 −1.73386 −0.866931 0.498428i \(-0.833911\pi\)
−0.866931 + 0.498428i \(0.833911\pi\)
\(128\) 1448.15 0.0883883
\(129\) 0 0
\(130\) 8028.58 0.475064
\(131\) − 1169.73i − 0.0681622i −0.999419 0.0340811i \(-0.989150\pi\)
0.999419 0.0340811i \(-0.0108505\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −17863.1 −0.994828
\(135\) 0 0
\(136\) − 12284.1i − 0.664150i
\(137\) 20884.6 1.11272 0.556359 0.830942i \(-0.312198\pi\)
0.556359 + 0.830942i \(0.312198\pi\)
\(138\) 0 0
\(139\) − 3499.87i − 0.181143i −0.995890 0.0905716i \(-0.971131\pi\)
0.995890 0.0905716i \(-0.0288694\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3573.10 −0.177202
\(143\) 14236.0i 0.696173i
\(144\) 0 0
\(145\) − 10668.8i − 0.507434i
\(146\) − 11300.8i − 0.530156i
\(147\) 0 0
\(148\) −7191.29 −0.328310
\(149\) −29533.8 −1.33029 −0.665146 0.746713i \(-0.731631\pi\)
−0.665146 + 0.746713i \(0.731631\pi\)
\(150\) 0 0
\(151\) 25780.8 1.13069 0.565344 0.824856i \(-0.308744\pi\)
0.565344 + 0.824856i \(0.308744\pi\)
\(152\) − 3166.78i − 0.137066i
\(153\) 0 0
\(154\) 0 0
\(155\) 11069.2 0.460738
\(156\) 0 0
\(157\) 25629.4i 1.03977i 0.854235 + 0.519887i \(0.174026\pi\)
−0.854235 + 0.519887i \(0.825974\pi\)
\(158\) −30021.7 −1.20260
\(159\) 0 0
\(160\) − 5040.97i − 0.196913i
\(161\) 0 0
\(162\) 0 0
\(163\) −6182.78 −0.232707 −0.116353 0.993208i \(-0.537120\pi\)
−0.116353 + 0.993208i \(0.537120\pi\)
\(164\) − 5262.90i − 0.195676i
\(165\) 0 0
\(166\) − 16264.3i − 0.590229i
\(167\) − 39270.9i − 1.40811i −0.710144 0.704056i \(-0.751370\pi\)
0.710144 0.704056i \(-0.248630\pi\)
\(168\) 0 0
\(169\) 18171.2 0.636223
\(170\) −42760.6 −1.47961
\(171\) 0 0
\(172\) 9245.61 0.312521
\(173\) 32905.9i 1.09946i 0.835341 + 0.549732i \(0.185270\pi\)
−0.835341 + 0.549732i \(0.814730\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 8938.50 0.288562
\(177\) 0 0
\(178\) − 11842.8i − 0.373777i
\(179\) 4767.88 0.148806 0.0744029 0.997228i \(-0.476295\pi\)
0.0744029 + 0.997228i \(0.476295\pi\)
\(180\) 0 0
\(181\) − 35195.4i − 1.07431i −0.843484 0.537154i \(-0.819499\pi\)
0.843484 0.537154i \(-0.180501\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5196.31 −0.153483
\(185\) 25032.6i 0.731413i
\(186\) 0 0
\(187\) − 75821.7i − 2.16826i
\(188\) 10356.1i 0.293007i
\(189\) 0 0
\(190\) −11023.5 −0.305359
\(191\) −23315.1 −0.639101 −0.319551 0.947569i \(-0.603532\pi\)
−0.319551 + 0.947569i \(0.603532\pi\)
\(192\) 0 0
\(193\) 59183.8 1.58887 0.794434 0.607350i \(-0.207768\pi\)
0.794434 + 0.607350i \(0.207768\pi\)
\(194\) − 10788.4i − 0.286651i
\(195\) 0 0
\(196\) 0 0
\(197\) 60902.2 1.56928 0.784640 0.619952i \(-0.212848\pi\)
0.784640 + 0.619952i \(0.212848\pi\)
\(198\) 0 0
\(199\) − 40116.7i − 1.01302i −0.862233 0.506512i \(-0.830935\pi\)
0.862233 0.506512i \(-0.169065\pi\)
\(200\) −3405.30 −0.0851326
\(201\) 0 0
\(202\) − 13931.2i − 0.341417i
\(203\) 0 0
\(204\) 0 0
\(205\) −18319.9 −0.435930
\(206\) − 53436.5i − 1.25923i
\(207\) 0 0
\(208\) 6523.56i 0.150785i
\(209\) − 19546.4i − 0.447482i
\(210\) 0 0
\(211\) 21929.0 0.492554 0.246277 0.969200i \(-0.420793\pi\)
0.246277 + 0.969200i \(0.420793\pi\)
\(212\) 41347.5 0.919978
\(213\) 0 0
\(214\) 23983.3 0.523699
\(215\) − 32183.6i − 0.696238i
\(216\) 0 0
\(217\) 0 0
\(218\) 33030.3 0.695023
\(219\) 0 0
\(220\) − 31114.6i − 0.642863i
\(221\) 55336.8 1.13300
\(222\) 0 0
\(223\) 64358.0i 1.29418i 0.762416 + 0.647088i \(0.224013\pi\)
−0.762416 + 0.647088i \(0.775987\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −55674.1 −1.09002
\(227\) 43774.8i 0.849517i 0.905307 + 0.424759i \(0.139641\pi\)
−0.905307 + 0.424759i \(0.860359\pi\)
\(228\) 0 0
\(229\) − 28506.4i − 0.543590i −0.962355 0.271795i \(-0.912383\pi\)
0.962355 0.271795i \(-0.0876173\pi\)
\(230\) 18088.2i 0.341931i
\(231\) 0 0
\(232\) 8668.85 0.161059
\(233\) 2194.89 0.0404298 0.0202149 0.999796i \(-0.493565\pi\)
0.0202149 + 0.999796i \(0.493565\pi\)
\(234\) 0 0
\(235\) 36049.0 0.652767
\(236\) − 33241.9i − 0.596846i
\(237\) 0 0
\(238\) 0 0
\(239\) −80942.6 −1.41704 −0.708519 0.705692i \(-0.750636\pi\)
−0.708519 + 0.705692i \(0.750636\pi\)
\(240\) 0 0
\(241\) 43016.6i 0.740631i 0.928906 + 0.370316i \(0.120751\pi\)
−0.928906 + 0.370316i \(0.879249\pi\)
\(242\) 13760.4 0.234963
\(243\) 0 0
\(244\) 14720.7i 0.247257i
\(245\) 0 0
\(246\) 0 0
\(247\) 14265.5 0.233827
\(248\) 8994.21i 0.146238i
\(249\) 0 0
\(250\) − 37374.5i − 0.597992i
\(251\) 31568.5i 0.501080i 0.968106 + 0.250540i \(0.0806082\pi\)
−0.968106 + 0.250540i \(0.919392\pi\)
\(252\) 0 0
\(253\) −32073.4 −0.501076
\(254\) −79098.3 −1.22603
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 68105.9i − 1.03114i −0.856847 0.515571i \(-0.827580\pi\)
0.856847 0.515571i \(-0.172420\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 22708.3 0.335921
\(261\) 0 0
\(262\) − 3308.50i − 0.0481980i
\(263\) 8347.56 0.120684 0.0603418 0.998178i \(-0.480781\pi\)
0.0603418 + 0.998178i \(0.480781\pi\)
\(264\) 0 0
\(265\) − 143929.i − 2.04954i
\(266\) 0 0
\(267\) 0 0
\(268\) −50524.6 −0.703450
\(269\) 91921.0i 1.27031i 0.772384 + 0.635156i \(0.219064\pi\)
−0.772384 + 0.635156i \(0.780936\pi\)
\(270\) 0 0
\(271\) 5134.36i 0.0699114i 0.999389 + 0.0349557i \(0.0111290\pi\)
−0.999389 + 0.0349557i \(0.988871\pi\)
\(272\) − 34744.8i − 0.469625i
\(273\) 0 0
\(274\) 59070.5 0.786810
\(275\) −21018.7 −0.277933
\(276\) 0 0
\(277\) −66254.2 −0.863483 −0.431741 0.901997i \(-0.642101\pi\)
−0.431741 + 0.901997i \(0.642101\pi\)
\(278\) − 9899.12i − 0.128088i
\(279\) 0 0
\(280\) 0 0
\(281\) −38156.4 −0.483230 −0.241615 0.970372i \(-0.577677\pi\)
−0.241615 + 0.970372i \(0.577677\pi\)
\(282\) 0 0
\(283\) 110233.i 1.37638i 0.725528 + 0.688192i \(0.241595\pi\)
−0.725528 + 0.688192i \(0.758405\pi\)
\(284\) −10106.3 −0.125301
\(285\) 0 0
\(286\) 40265.6i 0.492268i
\(287\) 0 0
\(288\) 0 0
\(289\) −211205. −2.52877
\(290\) − 30175.9i − 0.358810i
\(291\) 0 0
\(292\) − 31963.5i − 0.374877i
\(293\) 79011.7i 0.920357i 0.887826 + 0.460179i \(0.152215\pi\)
−0.887826 + 0.460179i \(0.847785\pi\)
\(294\) 0 0
\(295\) −115714. −1.32966
\(296\) −20340.0 −0.232150
\(297\) 0 0
\(298\) −83534.2 −0.940659
\(299\) − 23408.0i − 0.261832i
\(300\) 0 0
\(301\) 0 0
\(302\) 72919.1 0.799517
\(303\) 0 0
\(304\) − 8957.01i − 0.0969206i
\(305\) 51242.2 0.550843
\(306\) 0 0
\(307\) 2801.57i 0.0297252i 0.999890 + 0.0148626i \(0.00473109\pi\)
−0.999890 + 0.0148626i \(0.995269\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 31308.5 0.325791
\(311\) 20659.3i 0.213597i 0.994281 + 0.106799i \(0.0340600\pi\)
−0.994281 + 0.106799i \(0.965940\pi\)
\(312\) 0 0
\(313\) 37376.1i 0.381509i 0.981638 + 0.190755i \(0.0610935\pi\)
−0.981638 + 0.190755i \(0.938907\pi\)
\(314\) 72490.8i 0.735231i
\(315\) 0 0
\(316\) −84914.1 −0.850366
\(317\) 106666. 1.06147 0.530735 0.847538i \(-0.321916\pi\)
0.530735 + 0.847538i \(0.321916\pi\)
\(318\) 0 0
\(319\) 53507.1 0.525811
\(320\) − 14258.0i − 0.139239i
\(321\) 0 0
\(322\) 0 0
\(323\) −75978.8 −0.728262
\(324\) 0 0
\(325\) − 15340.0i − 0.145231i
\(326\) −17487.5 −0.164548
\(327\) 0 0
\(328\) − 14885.7i − 0.138364i
\(329\) 0 0
\(330\) 0 0
\(331\) 108363. 0.989063 0.494532 0.869160i \(-0.335340\pi\)
0.494532 + 0.869160i \(0.335340\pi\)
\(332\) − 46002.5i − 0.417355i
\(333\) 0 0
\(334\) − 111075.i − 0.995686i
\(335\) 175874.i 1.56716i
\(336\) 0 0
\(337\) 186946. 1.64610 0.823050 0.567968i \(-0.192270\pi\)
0.823050 + 0.567968i \(0.192270\pi\)
\(338\) 51395.8 0.449877
\(339\) 0 0
\(340\) −120945. −1.04624
\(341\) 55515.3i 0.477424i
\(342\) 0 0
\(343\) 0 0
\(344\) 26150.5 0.220985
\(345\) 0 0
\(346\) 93071.9i 0.777439i
\(347\) 63227.3 0.525105 0.262552 0.964918i \(-0.415436\pi\)
0.262552 + 0.964918i \(0.415436\pi\)
\(348\) 0 0
\(349\) − 93605.0i − 0.768508i −0.923227 0.384254i \(-0.874459\pi\)
0.923227 0.384254i \(-0.125541\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 25281.9 0.204044
\(353\) − 143251.i − 1.14960i −0.818294 0.574800i \(-0.805080\pi\)
0.818294 0.574800i \(-0.194920\pi\)
\(354\) 0 0
\(355\) 35179.5i 0.279147i
\(356\) − 33496.4i − 0.264301i
\(357\) 0 0
\(358\) 13485.6 0.105222
\(359\) −102340. −0.794065 −0.397032 0.917805i \(-0.629960\pi\)
−0.397032 + 0.917805i \(0.629960\pi\)
\(360\) 0 0
\(361\) 110734. 0.849702
\(362\) − 99547.7i − 0.759651i
\(363\) 0 0
\(364\) 0 0
\(365\) −111264. −0.835156
\(366\) 0 0
\(367\) 15366.4i 0.114088i 0.998372 + 0.0570442i \(0.0181676\pi\)
−0.998372 + 0.0570442i \(0.981832\pi\)
\(368\) −14697.4 −0.108529
\(369\) 0 0
\(370\) 70803.0i 0.517187i
\(371\) 0 0
\(372\) 0 0
\(373\) 97490.8 0.700722 0.350361 0.936615i \(-0.386059\pi\)
0.350361 + 0.936615i \(0.386059\pi\)
\(374\) − 214456.i − 1.53319i
\(375\) 0 0
\(376\) 29291.3i 0.207188i
\(377\) 39050.9i 0.274757i
\(378\) 0 0
\(379\) −66090.7 −0.460110 −0.230055 0.973178i \(-0.573891\pi\)
−0.230055 + 0.973178i \(0.573891\pi\)
\(380\) −31179.0 −0.215921
\(381\) 0 0
\(382\) −65944.9 −0.451913
\(383\) 208901.i 1.42411i 0.702126 + 0.712053i \(0.252234\pi\)
−0.702126 + 0.712053i \(0.747766\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 167397. 1.12350
\(387\) 0 0
\(388\) − 30514.2i − 0.202693i
\(389\) 143999. 0.951614 0.475807 0.879550i \(-0.342156\pi\)
0.475807 + 0.879550i \(0.342156\pi\)
\(390\) 0 0
\(391\) 124672.i 0.815485i
\(392\) 0 0
\(393\) 0 0
\(394\) 172257. 1.10965
\(395\) 295583.i 1.89446i
\(396\) 0 0
\(397\) 284577.i 1.80559i 0.430070 + 0.902796i \(0.358489\pi\)
−0.430070 + 0.902796i \(0.641511\pi\)
\(398\) − 113467.i − 0.716316i
\(399\) 0 0
\(400\) −9631.65 −0.0601978
\(401\) 107846. 0.670680 0.335340 0.942097i \(-0.391149\pi\)
0.335340 + 0.942097i \(0.391149\pi\)
\(402\) 0 0
\(403\) −40516.6 −0.249472
\(404\) − 39403.3i − 0.241418i
\(405\) 0 0
\(406\) 0 0
\(407\) −125546. −0.757901
\(408\) 0 0
\(409\) 84932.2i 0.507722i 0.967241 + 0.253861i \(0.0817005\pi\)
−0.967241 + 0.253861i \(0.918299\pi\)
\(410\) −51816.6 −0.308249
\(411\) 0 0
\(412\) − 151141.i − 0.890407i
\(413\) 0 0
\(414\) 0 0
\(415\) −160133. −0.929789
\(416\) 18451.4i 0.106621i
\(417\) 0 0
\(418\) − 55285.7i − 0.316417i
\(419\) − 331905.i − 1.89054i −0.326292 0.945269i \(-0.605799\pi\)
0.326292 0.945269i \(-0.394201\pi\)
\(420\) 0 0
\(421\) 101490. 0.572609 0.286304 0.958139i \(-0.407573\pi\)
0.286304 + 0.958139i \(0.407573\pi\)
\(422\) 62024.5 0.348288
\(423\) 0 0
\(424\) 116948. 0.650522
\(425\) 81701.5i 0.452327i
\(426\) 0 0
\(427\) 0 0
\(428\) 67835.1 0.370311
\(429\) 0 0
\(430\) − 91029.0i − 0.492315i
\(431\) −229798. −1.23706 −0.618530 0.785761i \(-0.712272\pi\)
−0.618530 + 0.785761i \(0.712272\pi\)
\(432\) 0 0
\(433\) 114264.i 0.609442i 0.952442 + 0.304721i \(0.0985632\pi\)
−0.952442 + 0.304721i \(0.901437\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 93423.7 0.491455
\(437\) 32139.8i 0.168299i
\(438\) 0 0
\(439\) 352339.i 1.82823i 0.405453 + 0.914116i \(0.367114\pi\)
−0.405453 + 0.914116i \(0.632886\pi\)
\(440\) − 88005.3i − 0.454573i
\(441\) 0 0
\(442\) 156516. 0.801151
\(443\) 196316. 1.00034 0.500172 0.865926i \(-0.333270\pi\)
0.500172 + 0.865926i \(0.333270\pi\)
\(444\) 0 0
\(445\) −116600. −0.588813
\(446\) 182032.i 0.915120i
\(447\) 0 0
\(448\) 0 0
\(449\) −77520.3 −0.384523 −0.192262 0.981344i \(-0.561582\pi\)
−0.192262 + 0.981344i \(0.561582\pi\)
\(450\) 0 0
\(451\) − 91879.6i − 0.451717i
\(452\) −157470. −0.770763
\(453\) 0 0
\(454\) 123814.i 0.600699i
\(455\) 0 0
\(456\) 0 0
\(457\) 189872. 0.909137 0.454568 0.890712i \(-0.349794\pi\)
0.454568 + 0.890712i \(0.349794\pi\)
\(458\) − 80628.3i − 0.384376i
\(459\) 0 0
\(460\) 51161.1i 0.241782i
\(461\) − 52599.9i − 0.247505i −0.992313 0.123752i \(-0.960507\pi\)
0.992313 0.123752i \(-0.0394928\pi\)
\(462\) 0 0
\(463\) −244268. −1.13947 −0.569736 0.821828i \(-0.692955\pi\)
−0.569736 + 0.821828i \(0.692955\pi\)
\(464\) 24519.2 0.113886
\(465\) 0 0
\(466\) 6208.09 0.0285882
\(467\) 297615.i 1.36465i 0.731050 + 0.682324i \(0.239031\pi\)
−0.731050 + 0.682324i \(0.760969\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 101962. 0.461576
\(471\) 0 0
\(472\) − 94022.3i − 0.422034i
\(473\) 161410. 0.721452
\(474\) 0 0
\(475\) 21062.2i 0.0933505i
\(476\) 0 0
\(477\) 0 0
\(478\) −228940. −1.00200
\(479\) 46870.8i 0.204283i 0.994770 + 0.102141i \(0.0325694\pi\)
−0.994770 + 0.102141i \(0.967431\pi\)
\(480\) 0 0
\(481\) − 91626.6i − 0.396033i
\(482\) 121669.i 0.523706i
\(483\) 0 0
\(484\) 38920.3 0.166144
\(485\) −106219. −0.451563
\(486\) 0 0
\(487\) 107184. 0.451930 0.225965 0.974135i \(-0.427446\pi\)
0.225965 + 0.974135i \(0.427446\pi\)
\(488\) 41636.4i 0.174837i
\(489\) 0 0
\(490\) 0 0
\(491\) 116367. 0.482687 0.241344 0.970440i \(-0.422412\pi\)
0.241344 + 0.970440i \(0.422412\pi\)
\(492\) 0 0
\(493\) − 207987.i − 0.855740i
\(494\) 40349.0 0.165340
\(495\) 0 0
\(496\) 25439.5i 0.103406i
\(497\) 0 0
\(498\) 0 0
\(499\) 442908. 1.77874 0.889369 0.457190i \(-0.151144\pi\)
0.889369 + 0.457190i \(0.151144\pi\)
\(500\) − 105711.i − 0.422844i
\(501\) 0 0
\(502\) 89289.3i 0.354317i
\(503\) − 68955.1i − 0.272540i −0.990672 0.136270i \(-0.956489\pi\)
0.990672 0.136270i \(-0.0435115\pi\)
\(504\) 0 0
\(505\) −137161. −0.537835
\(506\) −90717.2 −0.354314
\(507\) 0 0
\(508\) −223724. −0.866931
\(509\) − 92773.5i − 0.358087i −0.983841 0.179044i \(-0.942700\pi\)
0.983841 0.179044i \(-0.0573003\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11585.2 0.0441942
\(513\) 0 0
\(514\) − 192632.i − 0.729127i
\(515\) −526117. −1.98366
\(516\) 0 0
\(517\) 180796.i 0.676406i
\(518\) 0 0
\(519\) 0 0
\(520\) 64228.7 0.237532
\(521\) − 142912.i − 0.526495i −0.964728 0.263248i \(-0.915206\pi\)
0.964728 0.263248i \(-0.0847936\pi\)
\(522\) 0 0
\(523\) − 22203.5i − 0.0811742i −0.999176 0.0405871i \(-0.987077\pi\)
0.999176 0.0405871i \(-0.0129228\pi\)
\(524\) − 9357.85i − 0.0340811i
\(525\) 0 0
\(526\) 23610.5 0.0853361
\(527\) 215793. 0.776992
\(528\) 0 0
\(529\) −227103. −0.811544
\(530\) − 407093.i − 1.44924i
\(531\) 0 0
\(532\) 0 0
\(533\) 67056.3 0.236040
\(534\) 0 0
\(535\) − 236132.i − 0.824986i
\(536\) −142905. −0.497414
\(537\) 0 0
\(538\) 259992.i 0.898246i
\(539\) 0 0
\(540\) 0 0
\(541\) 211209. 0.721636 0.360818 0.932636i \(-0.382498\pi\)
0.360818 + 0.932636i \(0.382498\pi\)
\(542\) 14522.2i 0.0494348i
\(543\) 0 0
\(544\) − 98273.0i − 0.332075i
\(545\) − 325204.i − 1.09487i
\(546\) 0 0
\(547\) 254227. 0.849664 0.424832 0.905272i \(-0.360333\pi\)
0.424832 + 0.905272i \(0.360333\pi\)
\(548\) 167077. 0.556359
\(549\) 0 0
\(550\) −59449.8 −0.196528
\(551\) − 53617.9i − 0.176606i
\(552\) 0 0
\(553\) 0 0
\(554\) −187395. −0.610575
\(555\) 0 0
\(556\) − 27998.9i − 0.0905716i
\(557\) −46139.2 −0.148717 −0.0743584 0.997232i \(-0.523691\pi\)
−0.0743584 + 0.997232i \(0.523691\pi\)
\(558\) 0 0
\(559\) 117801.i 0.376987i
\(560\) 0 0
\(561\) 0 0
\(562\) −107922. −0.341695
\(563\) 349629.i 1.10304i 0.834162 + 0.551519i \(0.185952\pi\)
−0.834162 + 0.551519i \(0.814048\pi\)
\(564\) 0 0
\(565\) 548147.i 1.71712i
\(566\) 311787.i 0.973251i
\(567\) 0 0
\(568\) −28584.8 −0.0886010
\(569\) 176196. 0.544217 0.272109 0.962267i \(-0.412279\pi\)
0.272109 + 0.962267i \(0.412279\pi\)
\(570\) 0 0
\(571\) −176527. −0.541426 −0.270713 0.962660i \(-0.587259\pi\)
−0.270713 + 0.962660i \(0.587259\pi\)
\(572\) 113888.i 0.348086i
\(573\) 0 0
\(574\) 0 0
\(575\) 34560.6 0.104531
\(576\) 0 0
\(577\) 226863.i 0.681416i 0.940169 + 0.340708i \(0.110667\pi\)
−0.940169 + 0.340708i \(0.889333\pi\)
\(578\) −597378. −1.78811
\(579\) 0 0
\(580\) − 85350.5i − 0.253717i
\(581\) 0 0
\(582\) 0 0
\(583\) 721844. 2.12376
\(584\) − 90406.4i − 0.265078i
\(585\) 0 0
\(586\) 223479.i 0.650791i
\(587\) 349659.i 1.01477i 0.861719 + 0.507386i \(0.169388\pi\)
−0.861719 + 0.507386i \(0.830612\pi\)
\(588\) 0 0
\(589\) 55630.3 0.160354
\(590\) −327288. −0.940213
\(591\) 0 0
\(592\) −57530.3 −0.164155
\(593\) 137755.i 0.391739i 0.980630 + 0.195870i \(0.0627529\pi\)
−0.980630 + 0.195870i \(0.937247\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −236271. −0.665146
\(597\) 0 0
\(598\) − 66207.9i − 0.185143i
\(599\) −80877.2 −0.225410 −0.112705 0.993629i \(-0.535951\pi\)
−0.112705 + 0.993629i \(0.535951\pi\)
\(600\) 0 0
\(601\) 228798.i 0.633438i 0.948519 + 0.316719i \(0.102581\pi\)
−0.948519 + 0.316719i \(0.897419\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 206246. 0.565344
\(605\) − 135480.i − 0.370139i
\(606\) 0 0
\(607\) − 265951.i − 0.721812i −0.932602 0.360906i \(-0.882468\pi\)
0.932602 0.360906i \(-0.117532\pi\)
\(608\) − 25334.3i − 0.0685332i
\(609\) 0 0
\(610\) 144935. 0.389505
\(611\) −131950. −0.353449
\(612\) 0 0
\(613\) −306522. −0.815718 −0.407859 0.913045i \(-0.633725\pi\)
−0.407859 + 0.913045i \(0.633725\pi\)
\(614\) 7924.04i 0.0210189i
\(615\) 0 0
\(616\) 0 0
\(617\) −491906. −1.29215 −0.646074 0.763275i \(-0.723590\pi\)
−0.646074 + 0.763275i \(0.723590\pi\)
\(618\) 0 0
\(619\) 115230.i 0.300735i 0.988630 + 0.150368i \(0.0480457\pi\)
−0.988630 + 0.150368i \(0.951954\pi\)
\(620\) 88553.9 0.230369
\(621\) 0 0
\(622\) 58433.5i 0.151036i
\(623\) 0 0
\(624\) 0 0
\(625\) −462035. −1.18281
\(626\) 105716.i 0.269768i
\(627\) 0 0
\(628\) 205035.i 0.519887i
\(629\) 488007.i 1.23346i
\(630\) 0 0
\(631\) 72815.9 0.182880 0.0914402 0.995811i \(-0.470853\pi\)
0.0914402 + 0.995811i \(0.470853\pi\)
\(632\) −240173. −0.601299
\(633\) 0 0
\(634\) 301697. 0.750572
\(635\) 778774.i 1.93136i
\(636\) 0 0
\(637\) 0 0
\(638\) 151341. 0.371805
\(639\) 0 0
\(640\) − 40327.8i − 0.0984565i
\(641\) 757794. 1.84431 0.922157 0.386815i \(-0.126425\pi\)
0.922157 + 0.386815i \(0.126425\pi\)
\(642\) 0 0
\(643\) 353082.i 0.853992i 0.904254 + 0.426996i \(0.140428\pi\)
−0.904254 + 0.426996i \(0.859572\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −214901. −0.514959
\(647\) − 214561.i − 0.512557i −0.966603 0.256279i \(-0.917504\pi\)
0.966603 0.256279i \(-0.0824965\pi\)
\(648\) 0 0
\(649\) − 580337.i − 1.37782i
\(650\) − 43388.1i − 0.102694i
\(651\) 0 0
\(652\) −49462.3 −0.116353
\(653\) 497591. 1.16693 0.583467 0.812137i \(-0.301696\pi\)
0.583467 + 0.812137i \(0.301696\pi\)
\(654\) 0 0
\(655\) −32574.3 −0.0759264
\(656\) − 42103.2i − 0.0978379i
\(657\) 0 0
\(658\) 0 0
\(659\) 197464. 0.454692 0.227346 0.973814i \(-0.426995\pi\)
0.227346 + 0.973814i \(0.426995\pi\)
\(660\) 0 0
\(661\) 668423.i 1.52985i 0.644121 + 0.764924i \(0.277223\pi\)
−0.644121 + 0.764924i \(0.722777\pi\)
\(662\) 306496. 0.699373
\(663\) 0 0
\(664\) − 130115.i − 0.295114i
\(665\) 0 0
\(666\) 0 0
\(667\) −87980.6 −0.197759
\(668\) − 314167.i − 0.704056i
\(669\) 0 0
\(670\) 497447.i 1.10815i
\(671\) 256994.i 0.570792i
\(672\) 0 0
\(673\) −404534. −0.893151 −0.446575 0.894746i \(-0.647357\pi\)
−0.446575 + 0.894746i \(0.647357\pi\)
\(674\) 528763. 1.16397
\(675\) 0 0
\(676\) 145369. 0.318111
\(677\) − 211169.i − 0.460737i −0.973103 0.230369i \(-0.926007\pi\)
0.973103 0.230369i \(-0.0739932\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −342085. −0.739803
\(681\) 0 0
\(682\) 157021.i 0.337590i
\(683\) 170109. 0.364657 0.182329 0.983238i \(-0.441637\pi\)
0.182329 + 0.983238i \(0.441637\pi\)
\(684\) 0 0
\(685\) − 581588.i − 1.23946i
\(686\) 0 0
\(687\) 0 0
\(688\) 73964.9 0.156260
\(689\) 526822.i 1.10975i
\(690\) 0 0
\(691\) 168630.i 0.353167i 0.984286 + 0.176583i \(0.0565045\pi\)
−0.984286 + 0.176583i \(0.943496\pi\)
\(692\) 263247.i 0.549732i
\(693\) 0 0
\(694\) 178834. 0.371305
\(695\) −97463.3 −0.201777
\(696\) 0 0
\(697\) −357145. −0.735154
\(698\) − 264755.i − 0.543417i
\(699\) 0 0
\(700\) 0 0
\(701\) 487646. 0.992359 0.496179 0.868220i \(-0.334736\pi\)
0.496179 + 0.868220i \(0.334736\pi\)
\(702\) 0 0
\(703\) 125806.i 0.254560i
\(704\) 71508.0 0.144281
\(705\) 0 0
\(706\) − 405174.i − 0.812890i
\(707\) 0 0
\(708\) 0 0
\(709\) −386929. −0.769731 −0.384865 0.922973i \(-0.625752\pi\)
−0.384865 + 0.922973i \(0.625752\pi\)
\(710\) 99502.6i 0.197387i
\(711\) 0 0
\(712\) − 94742.1i − 0.186889i
\(713\) − 91282.7i − 0.179560i
\(714\) 0 0
\(715\) 396441. 0.775472
\(716\) 38143.1 0.0744029
\(717\) 0 0
\(718\) −289461. −0.561489
\(719\) − 188374.i − 0.364388i −0.983263 0.182194i \(-0.941680\pi\)
0.983263 0.182194i \(-0.0583199\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 313203. 0.600830
\(723\) 0 0
\(724\) − 281563.i − 0.537154i
\(725\) −57656.4 −0.109691
\(726\) 0 0
\(727\) 901538.i 1.70575i 0.522115 + 0.852875i \(0.325143\pi\)
−0.522115 + 0.852875i \(0.674857\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −314701. −0.590545
\(731\) − 627415.i − 1.17414i
\(732\) 0 0
\(733\) 402482.i 0.749098i 0.927207 + 0.374549i \(0.122202\pi\)
−0.927207 + 0.374549i \(0.877798\pi\)
\(734\) 43462.9i 0.0806726i
\(735\) 0 0
\(736\) −41570.5 −0.0767414
\(737\) −882058. −1.62391
\(738\) 0 0
\(739\) −37301.1 −0.0683020 −0.0341510 0.999417i \(-0.510873\pi\)
−0.0341510 + 0.999417i \(0.510873\pi\)
\(740\) 200261.i 0.365707i
\(741\) 0 0
\(742\) 0 0
\(743\) 831221. 1.50570 0.752851 0.658191i \(-0.228678\pi\)
0.752851 + 0.658191i \(0.228678\pi\)
\(744\) 0 0
\(745\) 822449.i 1.48182i
\(746\) 275746. 0.495486
\(747\) 0 0
\(748\) − 606574.i − 1.08413i
\(749\) 0 0
\(750\) 0 0
\(751\) −251098. −0.445209 −0.222605 0.974909i \(-0.571456\pi\)
−0.222605 + 0.974909i \(0.571456\pi\)
\(752\) 82848.4i 0.146504i
\(753\) 0 0
\(754\) 110453.i 0.194282i
\(755\) − 717936.i − 1.25948i
\(756\) 0 0
\(757\) −422399. −0.737108 −0.368554 0.929606i \(-0.620147\pi\)
−0.368554 + 0.929606i \(0.620147\pi\)
\(758\) −186933. −0.325347
\(759\) 0 0
\(760\) −88187.6 −0.152679
\(761\) − 247203.i − 0.426859i −0.976958 0.213430i \(-0.931537\pi\)
0.976958 0.213430i \(-0.0684634\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −186520. −0.319551
\(765\) 0 0
\(766\) 590860.i 1.00699i
\(767\) 423546. 0.719962
\(768\) 0 0
\(769\) − 903767.i − 1.52828i −0.645049 0.764141i \(-0.723163\pi\)
0.645049 0.764141i \(-0.276837\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 473470. 0.794434
\(773\) 244667.i 0.409464i 0.978818 + 0.204732i \(0.0656323\pi\)
−0.978818 + 0.204732i \(0.934368\pi\)
\(774\) 0 0
\(775\) − 59820.4i − 0.0995968i
\(776\) − 86307.3i − 0.143326i
\(777\) 0 0
\(778\) 407291. 0.672893
\(779\) −92070.0 −0.151720
\(780\) 0 0
\(781\) −176435. −0.289256
\(782\) 352626.i 0.576635i
\(783\) 0 0
\(784\) 0 0
\(785\) 713719. 1.15821
\(786\) 0 0
\(787\) − 950431.i − 1.53452i −0.641339 0.767258i \(-0.721621\pi\)
0.641339 0.767258i \(-0.278379\pi\)
\(788\) 487218. 0.784640
\(789\) 0 0
\(790\) 836035.i 1.33958i
\(791\) 0 0
\(792\) 0 0
\(793\) −187561. −0.298261
\(794\) 804907.i 1.27675i
\(795\) 0 0
\(796\) − 320934.i − 0.506512i
\(797\) 560412.i 0.882248i 0.897446 + 0.441124i \(0.145420\pi\)
−0.897446 + 0.441124i \(0.854580\pi\)
\(798\) 0 0
\(799\) 702771. 1.10083
\(800\) −27242.4 −0.0425663
\(801\) 0 0
\(802\) 305035. 0.474242
\(803\) − 558019.i − 0.865401i
\(804\) 0 0
\(805\) 0 0
\(806\) −114598. −0.176404
\(807\) 0 0
\(808\) − 111449.i − 0.170708i
\(809\) 51864.2 0.0792447 0.0396224 0.999215i \(-0.487385\pi\)
0.0396224 + 0.999215i \(0.487385\pi\)
\(810\) 0 0
\(811\) − 978879.i − 1.48829i −0.668019 0.744145i \(-0.732857\pi\)
0.668019 0.744145i \(-0.267143\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −355097. −0.535917
\(815\) 172176.i 0.259214i
\(816\) 0 0
\(817\) − 161744.i − 0.242317i
\(818\) 240224.i 0.359013i
\(819\) 0 0
\(820\) −146560. −0.217965
\(821\) −148798. −0.220755 −0.110377 0.993890i \(-0.535206\pi\)
−0.110377 + 0.993890i \(0.535206\pi\)
\(822\) 0 0
\(823\) 1.33907e6 1.97698 0.988490 0.151287i \(-0.0483416\pi\)
0.988490 + 0.151287i \(0.0483416\pi\)
\(824\) − 427492.i − 0.629613i
\(825\) 0 0
\(826\) 0 0
\(827\) 127284. 0.186108 0.0930538 0.995661i \(-0.470337\pi\)
0.0930538 + 0.995661i \(0.470337\pi\)
\(828\) 0 0
\(829\) − 445598.i − 0.648386i −0.945991 0.324193i \(-0.894907\pi\)
0.945991 0.324193i \(-0.105093\pi\)
\(830\) −452924. −0.657460
\(831\) 0 0
\(832\) 52188.5i 0.0753924i
\(833\) 0 0
\(834\) 0 0
\(835\) −1.09360e6 −1.56851
\(836\) − 156372.i − 0.223741i
\(837\) 0 0
\(838\) − 938768.i − 1.33681i
\(839\) 552232.i 0.784509i 0.919857 + 0.392254i \(0.128305\pi\)
−0.919857 + 0.392254i \(0.871695\pi\)
\(840\) 0 0
\(841\) −560506. −0.792479
\(842\) 287056. 0.404896
\(843\) 0 0
\(844\) 175432. 0.246277
\(845\) − 506025.i − 0.708694i
\(846\) 0 0
\(847\) 0 0
\(848\) 330780. 0.459989
\(849\) 0 0
\(850\) 231087.i 0.319843i
\(851\) 206432. 0.285048
\(852\) 0 0
\(853\) 753308.i 1.03532i 0.855586 + 0.517660i \(0.173197\pi\)
−0.855586 + 0.517660i \(0.826803\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 191867. 0.261850
\(857\) − 1.30117e6i − 1.77163i −0.464037 0.885816i \(-0.653600\pi\)
0.464037 0.885816i \(-0.346400\pi\)
\(858\) 0 0
\(859\) 982372.i 1.33134i 0.746245 + 0.665671i \(0.231855\pi\)
−0.746245 + 0.665671i \(0.768145\pi\)
\(860\) − 257469.i − 0.348119i
\(861\) 0 0
\(862\) −649966. −0.874734
\(863\) 721435. 0.968670 0.484335 0.874883i \(-0.339062\pi\)
0.484335 + 0.874883i \(0.339062\pi\)
\(864\) 0 0
\(865\) 916353. 1.22470
\(866\) 323186.i 0.430940i
\(867\) 0 0
\(868\) 0 0
\(869\) −1.48243e6 −1.96307
\(870\) 0 0
\(871\) − 643750.i − 0.848557i
\(872\) 264242. 0.347511
\(873\) 0 0
\(874\) 90905.2i 0.119005i
\(875\) 0 0
\(876\) 0 0
\(877\) −717008. −0.932234 −0.466117 0.884723i \(-0.654347\pi\)
−0.466117 + 0.884723i \(0.654347\pi\)
\(878\) 996564.i 1.29276i
\(879\) 0 0
\(880\) − 248917.i − 0.321432i
\(881\) − 79128.9i − 0.101949i −0.998700 0.0509745i \(-0.983767\pi\)
0.998700 0.0509745i \(-0.0162327\pi\)
\(882\) 0 0
\(883\) −286591. −0.367571 −0.183786 0.982966i \(-0.558835\pi\)
−0.183786 + 0.982966i \(0.558835\pi\)
\(884\) 442694. 0.566499
\(885\) 0 0
\(886\) 555267. 0.707350
\(887\) 837465.i 1.06444i 0.846607 + 0.532218i \(0.178641\pi\)
−0.846607 + 0.532218i \(0.821359\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −329794. −0.416354
\(891\) 0 0
\(892\) 514864.i 0.647088i
\(893\) 181171. 0.227188
\(894\) 0 0
\(895\) − 132775.i − 0.165756i
\(896\) 0 0
\(897\) 0 0
\(898\) −219261. −0.271899
\(899\) 152284.i 0.188424i
\(900\) 0 0
\(901\) − 2.80588e6i − 3.45636i
\(902\) − 259875.i − 0.319412i
\(903\) 0 0
\(904\) −445393. −0.545012
\(905\) −980112. −1.19668
\(906\) 0 0
\(907\) −846569. −1.02908 −0.514538 0.857467i \(-0.672037\pi\)
−0.514538 + 0.857467i \(0.672037\pi\)
\(908\) 350198.i 0.424759i
\(909\) 0 0
\(910\) 0 0
\(911\) −167017. −0.201244 −0.100622 0.994925i \(-0.532083\pi\)
−0.100622 + 0.994925i \(0.532083\pi\)
\(912\) 0 0
\(913\) − 803112.i − 0.963461i
\(914\) 537040. 0.642857
\(915\) 0 0
\(916\) − 228051.i − 0.271795i
\(917\) 0 0
\(918\) 0 0
\(919\) 1.12203e6 1.32854 0.664268 0.747494i \(-0.268743\pi\)
0.664268 + 0.747494i \(0.268743\pi\)
\(920\) 144705.i 0.170966i
\(921\) 0 0
\(922\) − 148775.i − 0.175012i
\(923\) − 128767.i − 0.151148i
\(924\) 0 0
\(925\) 135281. 0.158108
\(926\) −690893. −0.805729
\(927\) 0 0
\(928\) 69350.8 0.0805296
\(929\) − 1.42663e6i − 1.65302i −0.562921 0.826511i \(-0.690323\pi\)
0.562921 0.826511i \(-0.309677\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 17559.1 0.0202149
\(933\) 0 0
\(934\) 841781.i 0.964952i
\(935\) −2.11146e6 −2.41524
\(936\) 0 0
\(937\) − 265587.i − 0.302502i −0.988495 0.151251i \(-0.951670\pi\)
0.988495 0.151251i \(-0.0483301\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 288392. 0.326383
\(941\) 598591.i 0.676007i 0.941145 + 0.338003i \(0.109751\pi\)
−0.941145 + 0.338003i \(0.890249\pi\)
\(942\) 0 0
\(943\) 151076.i 0.169892i
\(944\) − 265935.i − 0.298423i
\(945\) 0 0
\(946\) 456536. 0.510144
\(947\) −1.15085e6 −1.28328 −0.641639 0.767007i \(-0.721745\pi\)
−0.641639 + 0.767007i \(0.721745\pi\)
\(948\) 0 0
\(949\) 407257. 0.452206
\(950\) 59572.9i 0.0660088i
\(951\) 0 0
\(952\) 0 0
\(953\) −464604. −0.511561 −0.255781 0.966735i \(-0.582332\pi\)
−0.255781 + 0.966735i \(0.582332\pi\)
\(954\) 0 0
\(955\) 649271.i 0.711900i
\(956\) −647541. −0.708519
\(957\) 0 0
\(958\) 132571.i 0.144450i
\(959\) 0 0
\(960\) 0 0
\(961\) 765521. 0.828916
\(962\) − 259159.i − 0.280038i
\(963\) 0 0
\(964\) 344133.i 0.370316i
\(965\) − 1.64813e6i − 1.76985i
\(966\) 0 0
\(967\) 101298. 0.108330 0.0541648 0.998532i \(-0.482750\pi\)
0.0541648 + 0.998532i \(0.482750\pi\)
\(968\) 110083. 0.117482
\(969\) 0 0
\(970\) −300432. −0.319303
\(971\) − 326280.i − 0.346060i −0.984917 0.173030i \(-0.944644\pi\)
0.984917 0.173030i \(-0.0553558\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 303162. 0.319563
\(975\) 0 0
\(976\) 117765.i 0.123628i
\(977\) −741928. −0.777272 −0.388636 0.921391i \(-0.627054\pi\)
−0.388636 + 0.921391i \(0.627054\pi\)
\(978\) 0 0
\(979\) − 584780.i − 0.610137i
\(980\) 0 0
\(981\) 0 0
\(982\) 329135. 0.341311
\(983\) − 584497.i − 0.604888i −0.953167 0.302444i \(-0.902197\pi\)
0.953167 0.302444i \(-0.0978026\pi\)
\(984\) 0 0
\(985\) − 1.69599e6i − 1.74803i
\(986\) − 588276.i − 0.605100i
\(987\) 0 0
\(988\) 114124. 0.116913
\(989\) −265403. −0.271340
\(990\) 0 0
\(991\) 582343. 0.592969 0.296484 0.955038i \(-0.404186\pi\)
0.296484 + 0.955038i \(0.404186\pi\)
\(992\) 71953.7i 0.0731189i
\(993\) 0 0
\(994\) 0 0
\(995\) −1.11716e6 −1.12842
\(996\) 0 0
\(997\) − 1.36934e6i − 1.37759i −0.724954 0.688797i \(-0.758139\pi\)
0.724954 0.688797i \(-0.241861\pi\)
\(998\) 1.25273e6 1.25776
\(999\) 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.c.685.3 4
3.2 odd 2 98.5.b.a.97.1 4
7.6 odd 2 inner 882.5.c.c.685.4 4
12.11 even 2 784.5.c.a.97.4 4
21.2 odd 6 98.5.d.c.31.4 8
21.5 even 6 98.5.d.c.31.3 8
21.11 odd 6 98.5.d.c.19.3 8
21.17 even 6 98.5.d.c.19.4 8
21.20 even 2 98.5.b.a.97.2 yes 4
84.83 odd 2 784.5.c.a.97.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.5.b.a.97.1 4 3.2 odd 2
98.5.b.a.97.2 yes 4 21.20 even 2
98.5.d.c.19.3 8 21.11 odd 6
98.5.d.c.19.4 8 21.17 even 6
98.5.d.c.31.3 8 21.5 even 6
98.5.d.c.31.4 8 21.2 odd 6
784.5.c.a.97.1 4 84.83 odd 2
784.5.c.a.97.4 4 12.11 even 2
882.5.c.c.685.3 4 1.1 even 1 trivial
882.5.c.c.685.4 4 7.6 odd 2 inner