Properties

Label 441.6.a.z.1.2
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.22929\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.81507 q^{2} -24.0754 q^{4} +45.9910 q^{5} +157.856 q^{8} +O(q^{10})\) \(q-2.81507 q^{2} -24.0754 q^{4} +45.9910 q^{5} +157.856 q^{8} -129.468 q^{10} +551.781 q^{11} +1094.10 q^{13} +326.035 q^{16} +1180.71 q^{17} -1166.13 q^{19} -1107.25 q^{20} -1553.30 q^{22} -44.3851 q^{23} -1009.82 q^{25} -3079.97 q^{26} -3329.02 q^{29} +8784.01 q^{31} -5969.21 q^{32} -3323.80 q^{34} -2557.12 q^{37} +3282.73 q^{38} +7259.97 q^{40} +12761.3 q^{41} -96.7714 q^{43} -13284.3 q^{44} +124.947 q^{46} -7679.15 q^{47} +2842.73 q^{50} -26340.8 q^{52} +11953.3 q^{53} +25377.0 q^{55} +9371.43 q^{58} -9857.24 q^{59} -38517.9 q^{61} -24727.6 q^{62} +6370.65 q^{64} +50318.7 q^{65} -67548.9 q^{67} -28426.1 q^{68} +61374.6 q^{71} +1850.40 q^{73} +7198.49 q^{74} +28074.9 q^{76} -8.52913 q^{79} +14994.7 q^{80} -35923.9 q^{82} +95039.3 q^{83} +54302.3 q^{85} +272.419 q^{86} +87102.1 q^{88} -53605.6 q^{89} +1068.59 q^{92} +21617.4 q^{94} -53631.4 q^{95} +3110.79 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50} + 102920 q^{53} + 96972 q^{58} - 40318 q^{64} + 63168 q^{65} - 22896 q^{67} + 153824 q^{71} - 17596 q^{74} - 90688 q^{79} + 272656 q^{85} + 161860 q^{86} + 154812 q^{88} + 212200 q^{92} - 108224 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.81507 −0.497639 −0.248820 0.968550i \(-0.580043\pi\)
−0.248820 + 0.968550i \(0.580043\pi\)
\(3\) 0 0
\(4\) −24.0754 −0.752355
\(5\) 45.9910 0.822713 0.411356 0.911475i \(-0.365055\pi\)
0.411356 + 0.911475i \(0.365055\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 157.856 0.872041
\(9\) 0 0
\(10\) −129.468 −0.409414
\(11\) 551.781 1.37494 0.687472 0.726211i \(-0.258720\pi\)
0.687472 + 0.726211i \(0.258720\pi\)
\(12\) 0 0
\(13\) 1094.10 1.79555 0.897776 0.440453i \(-0.145182\pi\)
0.897776 + 0.440453i \(0.145182\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 326.035 0.318393
\(17\) 1180.71 0.990883 0.495442 0.868641i \(-0.335006\pi\)
0.495442 + 0.868641i \(0.335006\pi\)
\(18\) 0 0
\(19\) −1166.13 −0.741074 −0.370537 0.928818i \(-0.620826\pi\)
−0.370537 + 0.928818i \(0.620826\pi\)
\(20\) −1107.25 −0.618972
\(21\) 0 0
\(22\) −1553.30 −0.684226
\(23\) −44.3851 −0.0174951 −0.00874757 0.999962i \(-0.502784\pi\)
−0.00874757 + 0.999962i \(0.502784\pi\)
\(24\) 0 0
\(25\) −1009.82 −0.323143
\(26\) −3079.97 −0.893537
\(27\) 0 0
\(28\) 0 0
\(29\) −3329.02 −0.735057 −0.367529 0.930012i \(-0.619796\pi\)
−0.367529 + 0.930012i \(0.619796\pi\)
\(30\) 0 0
\(31\) 8784.01 1.64168 0.820841 0.571157i \(-0.193505\pi\)
0.820841 + 0.571157i \(0.193505\pi\)
\(32\) −5969.21 −1.03049
\(33\) 0 0
\(34\) −3323.80 −0.493102
\(35\) 0 0
\(36\) 0 0
\(37\) −2557.12 −0.307077 −0.153539 0.988143i \(-0.549067\pi\)
−0.153539 + 0.988143i \(0.549067\pi\)
\(38\) 3282.73 0.368787
\(39\) 0 0
\(40\) 7259.97 0.717439
\(41\) 12761.3 1.18559 0.592794 0.805354i \(-0.298025\pi\)
0.592794 + 0.805354i \(0.298025\pi\)
\(42\) 0 0
\(43\) −96.7714 −0.00798135 −0.00399067 0.999992i \(-0.501270\pi\)
−0.00399067 + 0.999992i \(0.501270\pi\)
\(44\) −13284.3 −1.03445
\(45\) 0 0
\(46\) 124.947 0.00870627
\(47\) −7679.15 −0.507071 −0.253535 0.967326i \(-0.581593\pi\)
−0.253535 + 0.967326i \(0.581593\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2842.73 0.160809
\(51\) 0 0
\(52\) −26340.8 −1.35089
\(53\) 11953.3 0.584520 0.292260 0.956339i \(-0.405593\pi\)
0.292260 + 0.956339i \(0.405593\pi\)
\(54\) 0 0
\(55\) 25377.0 1.13118
\(56\) 0 0
\(57\) 0 0
\(58\) 9371.43 0.365793
\(59\) −9857.24 −0.368659 −0.184330 0.982864i \(-0.559011\pi\)
−0.184330 + 0.982864i \(0.559011\pi\)
\(60\) 0 0
\(61\) −38517.9 −1.32537 −0.662686 0.748897i \(-0.730584\pi\)
−0.662686 + 0.748897i \(0.730584\pi\)
\(62\) −24727.6 −0.816965
\(63\) 0 0
\(64\) 6370.65 0.194417
\(65\) 50318.7 1.47722
\(66\) 0 0
\(67\) −67548.9 −1.83836 −0.919182 0.393833i \(-0.871149\pi\)
−0.919182 + 0.393833i \(0.871149\pi\)
\(68\) −28426.1 −0.745496
\(69\) 0 0
\(70\) 0 0
\(71\) 61374.6 1.44492 0.722458 0.691415i \(-0.243012\pi\)
0.722458 + 0.691415i \(0.243012\pi\)
\(72\) 0 0
\(73\) 1850.40 0.0406404 0.0203202 0.999794i \(-0.493531\pi\)
0.0203202 + 0.999794i \(0.493531\pi\)
\(74\) 7198.49 0.152814
\(75\) 0 0
\(76\) 28074.9 0.557551
\(77\) 0 0
\(78\) 0 0
\(79\) −8.52913 −0.000153758 0 −7.68788e−5 1.00000i \(-0.500024\pi\)
−7.68788e−5 1.00000i \(0.500024\pi\)
\(80\) 14994.7 0.261946
\(81\) 0 0
\(82\) −35923.9 −0.589996
\(83\) 95039.3 1.51429 0.757143 0.653249i \(-0.226595\pi\)
0.757143 + 0.653249i \(0.226595\pi\)
\(84\) 0 0
\(85\) 54302.3 0.815212
\(86\) 272.419 0.00397183
\(87\) 0 0
\(88\) 87102.1 1.19901
\(89\) −53605.6 −0.717357 −0.358678 0.933461i \(-0.616773\pi\)
−0.358678 + 0.933461i \(0.616773\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1068.59 0.0131626
\(93\) 0 0
\(94\) 21617.4 0.252338
\(95\) −53631.4 −0.609691
\(96\) 0 0
\(97\) 3110.79 0.0335693 0.0167846 0.999859i \(-0.494657\pi\)
0.0167846 + 0.999859i \(0.494657\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 24311.9 0.243119
\(101\) −21835.9 −0.212994 −0.106497 0.994313i \(-0.533963\pi\)
−0.106497 + 0.994313i \(0.533963\pi\)
\(102\) 0 0
\(103\) 65341.4 0.606870 0.303435 0.952852i \(-0.401866\pi\)
0.303435 + 0.952852i \(0.401866\pi\)
\(104\) 172710. 1.56579
\(105\) 0 0
\(106\) −33649.5 −0.290880
\(107\) 108957. 0.920020 0.460010 0.887914i \(-0.347846\pi\)
0.460010 + 0.887914i \(0.347846\pi\)
\(108\) 0 0
\(109\) 86728.7 0.699192 0.349596 0.936901i \(-0.386319\pi\)
0.349596 + 0.936901i \(0.386319\pi\)
\(110\) −71438.1 −0.562922
\(111\) 0 0
\(112\) 0 0
\(113\) 101496. 0.747746 0.373873 0.927480i \(-0.378030\pi\)
0.373873 + 0.927480i \(0.378030\pi\)
\(114\) 0 0
\(115\) −2041.32 −0.0143935
\(116\) 80147.3 0.553024
\(117\) 0 0
\(118\) 27748.9 0.183459
\(119\) 0 0
\(120\) 0 0
\(121\) 143411. 0.890470
\(122\) 108431. 0.659557
\(123\) 0 0
\(124\) −211478. −1.23513
\(125\) −190165. −1.08857
\(126\) 0 0
\(127\) −3094.61 −0.0170253 −0.00851267 0.999964i \(-0.502710\pi\)
−0.00851267 + 0.999964i \(0.502710\pi\)
\(128\) 173081. 0.933736
\(129\) 0 0
\(130\) −141651. −0.735124
\(131\) −253431. −1.29027 −0.645136 0.764067i \(-0.723199\pi\)
−0.645136 + 0.764067i \(0.723199\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 190155. 0.914842
\(135\) 0 0
\(136\) 186383. 0.864091
\(137\) 97152.9 0.442236 0.221118 0.975247i \(-0.429029\pi\)
0.221118 + 0.975247i \(0.429029\pi\)
\(138\) 0 0
\(139\) 210308. 0.923249 0.461624 0.887076i \(-0.347267\pi\)
0.461624 + 0.887076i \(0.347267\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −172774. −0.719047
\(143\) 603702. 2.46878
\(144\) 0 0
\(145\) −153105. −0.604741
\(146\) −5209.01 −0.0202243
\(147\) 0 0
\(148\) 61563.7 0.231031
\(149\) −140406. −0.518109 −0.259055 0.965863i \(-0.583411\pi\)
−0.259055 + 0.965863i \(0.583411\pi\)
\(150\) 0 0
\(151\) 119696. 0.427205 0.213603 0.976921i \(-0.431480\pi\)
0.213603 + 0.976921i \(0.431480\pi\)
\(152\) −184080. −0.646247
\(153\) 0 0
\(154\) 0 0
\(155\) 403986. 1.35063
\(156\) 0 0
\(157\) 97616.9 0.316065 0.158032 0.987434i \(-0.449485\pi\)
0.158032 + 0.987434i \(0.449485\pi\)
\(158\) 24.0101 7.65159e−5 0
\(159\) 0 0
\(160\) −274530. −0.847794
\(161\) 0 0
\(162\) 0 0
\(163\) 182678. 0.538539 0.269270 0.963065i \(-0.413218\pi\)
0.269270 + 0.963065i \(0.413218\pi\)
\(164\) −307232. −0.891984
\(165\) 0 0
\(166\) −267542. −0.753568
\(167\) 451674. 1.25324 0.626619 0.779326i \(-0.284438\pi\)
0.626619 + 0.779326i \(0.284438\pi\)
\(168\) 0 0
\(169\) 825757. 2.22400
\(170\) −152865. −0.405682
\(171\) 0 0
\(172\) 2329.81 0.00600481
\(173\) −371647. −0.944095 −0.472047 0.881573i \(-0.656485\pi\)
−0.472047 + 0.881573i \(0.656485\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 179900. 0.437773
\(177\) 0 0
\(178\) 150904. 0.356985
\(179\) −85003.4 −0.198291 −0.0991457 0.995073i \(-0.531611\pi\)
−0.0991457 + 0.995073i \(0.531611\pi\)
\(180\) 0 0
\(181\) −379442. −0.860892 −0.430446 0.902616i \(-0.641644\pi\)
−0.430446 + 0.902616i \(0.641644\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7006.46 −0.0152565
\(185\) −117605. −0.252636
\(186\) 0 0
\(187\) 651496. 1.36241
\(188\) 184878. 0.381497
\(189\) 0 0
\(190\) 150976. 0.303406
\(191\) 922196. 1.82911 0.914555 0.404462i \(-0.132541\pi\)
0.914555 + 0.404462i \(0.132541\pi\)
\(192\) 0 0
\(193\) 505107. 0.976090 0.488045 0.872818i \(-0.337710\pi\)
0.488045 + 0.872818i \(0.337710\pi\)
\(194\) −8757.11 −0.0167054
\(195\) 0 0
\(196\) 0 0
\(197\) −251505. −0.461723 −0.230861 0.972987i \(-0.574154\pi\)
−0.230861 + 0.972987i \(0.574154\pi\)
\(198\) 0 0
\(199\) −208033. −0.372392 −0.186196 0.982513i \(-0.559616\pi\)
−0.186196 + 0.982513i \(0.559616\pi\)
\(200\) −159407. −0.281794
\(201\) 0 0
\(202\) 61469.7 0.105994
\(203\) 0 0
\(204\) 0 0
\(205\) 586904. 0.975399
\(206\) −183941. −0.302002
\(207\) 0 0
\(208\) 356714. 0.571692
\(209\) −643446. −1.01893
\(210\) 0 0
\(211\) −640577. −0.990525 −0.495262 0.868744i \(-0.664928\pi\)
−0.495262 + 0.868744i \(0.664928\pi\)
\(212\) −287781. −0.439767
\(213\) 0 0
\(214\) −306723. −0.457838
\(215\) −4450.62 −0.00656636
\(216\) 0 0
\(217\) 0 0
\(218\) −244148. −0.347945
\(219\) 0 0
\(220\) −610960. −0.851052
\(221\) 1.29182e6 1.77918
\(222\) 0 0
\(223\) 390135. 0.525354 0.262677 0.964884i \(-0.415395\pi\)
0.262677 + 0.964884i \(0.415395\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −285720. −0.372108
\(227\) −291353. −0.375279 −0.187639 0.982238i \(-0.560084\pi\)
−0.187639 + 0.982238i \(0.560084\pi\)
\(228\) 0 0
\(229\) −1.23040e6 −1.55045 −0.775227 0.631682i \(-0.782365\pi\)
−0.775227 + 0.631682i \(0.782365\pi\)
\(230\) 5746.45 0.00716276
\(231\) 0 0
\(232\) −525506. −0.641000
\(233\) −114279. −0.137903 −0.0689517 0.997620i \(-0.521965\pi\)
−0.0689517 + 0.997620i \(0.521965\pi\)
\(234\) 0 0
\(235\) −353172. −0.417174
\(236\) 237317. 0.277363
\(237\) 0 0
\(238\) 0 0
\(239\) 1.14782e6 1.29981 0.649906 0.760014i \(-0.274808\pi\)
0.649906 + 0.760014i \(0.274808\pi\)
\(240\) 0 0
\(241\) 812708. 0.901346 0.450673 0.892689i \(-0.351184\pi\)
0.450673 + 0.892689i \(0.351184\pi\)
\(242\) −403713. −0.443133
\(243\) 0 0
\(244\) 927332. 0.997150
\(245\) 0 0
\(246\) 0 0
\(247\) −1.27586e6 −1.33064
\(248\) 1.38661e6 1.43161
\(249\) 0 0
\(250\) 535328. 0.541714
\(251\) 406772. 0.407537 0.203768 0.979019i \(-0.434681\pi\)
0.203768 + 0.979019i \(0.434681\pi\)
\(252\) 0 0
\(253\) −24490.8 −0.0240548
\(254\) 8711.54 0.00847248
\(255\) 0 0
\(256\) −691096. −0.659081
\(257\) 1.69712e6 1.60281 0.801403 0.598125i \(-0.204087\pi\)
0.801403 + 0.598125i \(0.204087\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.21144e6 −1.11140
\(261\) 0 0
\(262\) 713427. 0.642090
\(263\) −205694. −0.183372 −0.0916859 0.995788i \(-0.529226\pi\)
−0.0916859 + 0.995788i \(0.529226\pi\)
\(264\) 0 0
\(265\) 549746. 0.480892
\(266\) 0 0
\(267\) 0 0
\(268\) 1.62627e6 1.38310
\(269\) −1.73425e6 −1.46127 −0.730635 0.682769i \(-0.760776\pi\)
−0.730635 + 0.682769i \(0.760776\pi\)
\(270\) 0 0
\(271\) 369042. 0.305248 0.152624 0.988284i \(-0.451228\pi\)
0.152624 + 0.988284i \(0.451228\pi\)
\(272\) 384954. 0.315491
\(273\) 0 0
\(274\) −273492. −0.220074
\(275\) −557201. −0.444304
\(276\) 0 0
\(277\) 1.22767e6 0.961351 0.480676 0.876899i \(-0.340391\pi\)
0.480676 + 0.876899i \(0.340391\pi\)
\(278\) −592032. −0.459445
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00671e6 −1.51607 −0.758035 0.652214i \(-0.773841\pi\)
−0.758035 + 0.652214i \(0.773841\pi\)
\(282\) 0 0
\(283\) −1.78581e6 −1.32547 −0.662734 0.748855i \(-0.730604\pi\)
−0.662734 + 0.748855i \(0.730604\pi\)
\(284\) −1.47762e6 −1.08709
\(285\) 0 0
\(286\) −1.69947e6 −1.22856
\(287\) 0 0
\(288\) 0 0
\(289\) −25770.8 −0.0181503
\(290\) 431002. 0.300943
\(291\) 0 0
\(292\) −44549.0 −0.0305760
\(293\) −853248. −0.580639 −0.290319 0.956930i \(-0.593762\pi\)
−0.290319 + 0.956930i \(0.593762\pi\)
\(294\) 0 0
\(295\) −453345. −0.303301
\(296\) −403658. −0.267784
\(297\) 0 0
\(298\) 395254. 0.257831
\(299\) −48561.6 −0.0314134
\(300\) 0 0
\(301\) 0 0
\(302\) −336952. −0.212594
\(303\) 0 0
\(304\) −380198. −0.235953
\(305\) −1.77148e6 −1.09040
\(306\) 0 0
\(307\) −1.96068e6 −1.18730 −0.593652 0.804722i \(-0.702314\pi\)
−0.593652 + 0.804722i \(0.702314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.13725e6 −0.672128
\(311\) −863604. −0.506307 −0.253153 0.967426i \(-0.581468\pi\)
−0.253153 + 0.967426i \(0.581468\pi\)
\(312\) 0 0
\(313\) 1.10047e6 0.634918 0.317459 0.948272i \(-0.397170\pi\)
0.317459 + 0.948272i \(0.397170\pi\)
\(314\) −274799. −0.157286
\(315\) 0 0
\(316\) 205.342 0.000115680 0
\(317\) −1.49591e6 −0.836097 −0.418048 0.908425i \(-0.637286\pi\)
−0.418048 + 0.908425i \(0.637286\pi\)
\(318\) 0 0
\(319\) −1.83689e6 −1.01066
\(320\) 292993. 0.159949
\(321\) 0 0
\(322\) 0 0
\(323\) −1.37686e6 −0.734318
\(324\) 0 0
\(325\) −1.10485e6 −0.580221
\(326\) −514252. −0.267998
\(327\) 0 0
\(328\) 2.01445e6 1.03388
\(329\) 0 0
\(330\) 0 0
\(331\) −2.74015e6 −1.37469 −0.687345 0.726331i \(-0.741224\pi\)
−0.687345 + 0.726331i \(0.741224\pi\)
\(332\) −2.28810e6 −1.13928
\(333\) 0 0
\(334\) −1.27149e6 −0.623661
\(335\) −3.10665e6 −1.51245
\(336\) 0 0
\(337\) −2.31353e6 −1.10968 −0.554842 0.831956i \(-0.687221\pi\)
−0.554842 + 0.831956i \(0.687221\pi\)
\(338\) −2.32457e6 −1.10675
\(339\) 0 0
\(340\) −1.30735e6 −0.613329
\(341\) 4.84685e6 2.25722
\(342\) 0 0
\(343\) 0 0
\(344\) −15276.0 −0.00696006
\(345\) 0 0
\(346\) 1.04621e6 0.469819
\(347\) 3.05926e6 1.36393 0.681966 0.731384i \(-0.261125\pi\)
0.681966 + 0.731384i \(0.261125\pi\)
\(348\) 0 0
\(349\) −210232. −0.0923921 −0.0461961 0.998932i \(-0.514710\pi\)
−0.0461961 + 0.998932i \(0.514710\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.29370e6 −1.41686
\(353\) 3.76790e6 1.60939 0.804697 0.593686i \(-0.202328\pi\)
0.804697 + 0.593686i \(0.202328\pi\)
\(354\) 0 0
\(355\) 2.82268e6 1.18875
\(356\) 1.29057e6 0.539707
\(357\) 0 0
\(358\) 239291. 0.0986776
\(359\) −1.00722e6 −0.412465 −0.206232 0.978503i \(-0.566120\pi\)
−0.206232 + 0.978503i \(0.566120\pi\)
\(360\) 0 0
\(361\) −1.11625e6 −0.450809
\(362\) 1.06816e6 0.428414
\(363\) 0 0
\(364\) 0 0
\(365\) 85101.8 0.0334354
\(366\) 0 0
\(367\) −1.52650e6 −0.591603 −0.295802 0.955249i \(-0.595587\pi\)
−0.295802 + 0.955249i \(0.595587\pi\)
\(368\) −14471.1 −0.00557034
\(369\) 0 0
\(370\) 331066. 0.125722
\(371\) 0 0
\(372\) 0 0
\(373\) 4.86297e6 1.80980 0.904898 0.425629i \(-0.139947\pi\)
0.904898 + 0.425629i \(0.139947\pi\)
\(374\) −1.83401e6 −0.677988
\(375\) 0 0
\(376\) −1.21220e6 −0.442186
\(377\) −3.64227e6 −1.31983
\(378\) 0 0
\(379\) 630878. 0.225604 0.112802 0.993617i \(-0.464017\pi\)
0.112802 + 0.993617i \(0.464017\pi\)
\(380\) 1.29119e6 0.458704
\(381\) 0 0
\(382\) −2.59605e6 −0.910237
\(383\) −565644. −0.197036 −0.0985182 0.995135i \(-0.531410\pi\)
−0.0985182 + 0.995135i \(0.531410\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.42191e6 −0.485741
\(387\) 0 0
\(388\) −74893.5 −0.0252560
\(389\) −592212. −0.198428 −0.0992140 0.995066i \(-0.531633\pi\)
−0.0992140 + 0.995066i \(0.531633\pi\)
\(390\) 0 0
\(391\) −52406.1 −0.0173356
\(392\) 0 0
\(393\) 0 0
\(394\) 708005. 0.229771
\(395\) −392.264 −0.000126498 0
\(396\) 0 0
\(397\) 1.34312e6 0.427698 0.213849 0.976867i \(-0.431400\pi\)
0.213849 + 0.976867i \(0.431400\pi\)
\(398\) 585629. 0.185317
\(399\) 0 0
\(400\) −329238. −0.102887
\(401\) 3.68716e6 1.14507 0.572534 0.819881i \(-0.305960\pi\)
0.572534 + 0.819881i \(0.305960\pi\)
\(402\) 0 0
\(403\) 9.61057e6 2.94772
\(404\) 525707. 0.160247
\(405\) 0 0
\(406\) 0 0
\(407\) −1.41097e6 −0.422214
\(408\) 0 0
\(409\) 1.45630e6 0.430470 0.215235 0.976562i \(-0.430948\pi\)
0.215235 + 0.976562i \(0.430948\pi\)
\(410\) −1.65218e6 −0.485397
\(411\) 0 0
\(412\) −1.57312e6 −0.456582
\(413\) 0 0
\(414\) 0 0
\(415\) 4.37096e6 1.24582
\(416\) −6.53090e6 −1.85029
\(417\) 0 0
\(418\) 1.81135e6 0.507062
\(419\) 2.92192e6 0.813080 0.406540 0.913633i \(-0.366735\pi\)
0.406540 + 0.913633i \(0.366735\pi\)
\(420\) 0 0
\(421\) 2.01999e6 0.555450 0.277725 0.960661i \(-0.410420\pi\)
0.277725 + 0.960661i \(0.410420\pi\)
\(422\) 1.80327e6 0.492924
\(423\) 0 0
\(424\) 1.88691e6 0.509725
\(425\) −1.19231e6 −0.320197
\(426\) 0 0
\(427\) 0 0
\(428\) −2.62319e6 −0.692182
\(429\) 0 0
\(430\) 12528.8 0.00326768
\(431\) 5.43800e6 1.41009 0.705043 0.709164i \(-0.250928\pi\)
0.705043 + 0.709164i \(0.250928\pi\)
\(432\) 0 0
\(433\) −3.77335e6 −0.967179 −0.483590 0.875295i \(-0.660667\pi\)
−0.483590 + 0.875295i \(0.660667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.08802e6 −0.526041
\(437\) 51758.6 0.0129652
\(438\) 0 0
\(439\) 2.35150e6 0.582350 0.291175 0.956670i \(-0.405954\pi\)
0.291175 + 0.956670i \(0.405954\pi\)
\(440\) 4.00591e6 0.986439
\(441\) 0 0
\(442\) −3.63656e6 −0.885391
\(443\) −4.80377e6 −1.16298 −0.581491 0.813553i \(-0.697530\pi\)
−0.581491 + 0.813553i \(0.697530\pi\)
\(444\) 0 0
\(445\) −2.46538e6 −0.590179
\(446\) −1.09826e6 −0.261437
\(447\) 0 0
\(448\) 0 0
\(449\) 2.76805e6 0.647975 0.323987 0.946061i \(-0.394976\pi\)
0.323987 + 0.946061i \(0.394976\pi\)
\(450\) 0 0
\(451\) 7.04142e6 1.63012
\(452\) −2.44356e6 −0.562571
\(453\) 0 0
\(454\) 820179. 0.186754
\(455\) 0 0
\(456\) 0 0
\(457\) 241566. 0.0541061 0.0270530 0.999634i \(-0.491388\pi\)
0.0270530 + 0.999634i \(0.491388\pi\)
\(458\) 3.46368e6 0.771567
\(459\) 0 0
\(460\) 49145.4 0.0108290
\(461\) −990579. −0.217088 −0.108544 0.994092i \(-0.534619\pi\)
−0.108544 + 0.994092i \(0.534619\pi\)
\(462\) 0 0
\(463\) 6.20488e6 1.34518 0.672591 0.740014i \(-0.265181\pi\)
0.672591 + 0.740014i \(0.265181\pi\)
\(464\) −1.08538e6 −0.234037
\(465\) 0 0
\(466\) 321702. 0.0686261
\(467\) 6.88497e6 1.46086 0.730432 0.682985i \(-0.239319\pi\)
0.730432 + 0.682985i \(0.239319\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 994206. 0.207602
\(471\) 0 0
\(472\) −1.55603e6 −0.321486
\(473\) −53396.6 −0.0109739
\(474\) 0 0
\(475\) 1.17758e6 0.239473
\(476\) 0 0
\(477\) 0 0
\(478\) −3.23121e6 −0.646838
\(479\) 5.41288e6 1.07793 0.538963 0.842329i \(-0.318816\pi\)
0.538963 + 0.842329i \(0.318816\pi\)
\(480\) 0 0
\(481\) −2.79774e6 −0.551373
\(482\) −2.28783e6 −0.448545
\(483\) 0 0
\(484\) −3.45268e6 −0.669950
\(485\) 143069. 0.0276179
\(486\) 0 0
\(487\) −3.01043e6 −0.575182 −0.287591 0.957753i \(-0.592854\pi\)
−0.287591 + 0.957753i \(0.592854\pi\)
\(488\) −6.08029e6 −1.15578
\(489\) 0 0
\(490\) 0 0
\(491\) 7.24498e6 1.35623 0.678115 0.734956i \(-0.262797\pi\)
0.678115 + 0.734956i \(0.262797\pi\)
\(492\) 0 0
\(493\) −3.93062e6 −0.728356
\(494\) 3.59163e6 0.662177
\(495\) 0 0
\(496\) 2.86389e6 0.522700
\(497\) 0 0
\(498\) 0 0
\(499\) −5.12788e6 −0.921906 −0.460953 0.887425i \(-0.652492\pi\)
−0.460953 + 0.887425i \(0.652492\pi\)
\(500\) 4.57829e6 0.818989
\(501\) 0 0
\(502\) −1.14509e6 −0.202806
\(503\) −1.05978e7 −1.86766 −0.933830 0.357718i \(-0.883555\pi\)
−0.933830 + 0.357718i \(0.883555\pi\)
\(504\) 0 0
\(505\) −1.00426e6 −0.175233
\(506\) 68943.5 0.0119706
\(507\) 0 0
\(508\) 74503.8 0.0128091
\(509\) −8.78840e6 −1.50354 −0.751770 0.659425i \(-0.770800\pi\)
−0.751770 + 0.659425i \(0.770800\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3.59310e6 −0.605752
\(513\) 0 0
\(514\) −4.77753e6 −0.797619
\(515\) 3.00512e6 0.499280
\(516\) 0 0
\(517\) −4.23721e6 −0.697194
\(518\) 0 0
\(519\) 0 0
\(520\) 7.94312e6 1.28820
\(521\) −162133. −0.0261684 −0.0130842 0.999914i \(-0.504165\pi\)
−0.0130842 + 0.999914i \(0.504165\pi\)
\(522\) 0 0
\(523\) −7.14844e6 −1.14277 −0.571383 0.820684i \(-0.693593\pi\)
−0.571383 + 0.820684i \(0.693593\pi\)
\(524\) 6.10144e6 0.970743
\(525\) 0 0
\(526\) 579044. 0.0912530
\(527\) 1.03714e7 1.62671
\(528\) 0 0
\(529\) −6.43437e6 −0.999694
\(530\) −1.54758e6 −0.239311
\(531\) 0 0
\(532\) 0 0
\(533\) 1.39621e7 2.12879
\(534\) 0 0
\(535\) 5.01106e6 0.756912
\(536\) −1.06630e7 −1.60313
\(537\) 0 0
\(538\) 4.88203e6 0.727185
\(539\) 0 0
\(540\) 0 0
\(541\) −4.83604e6 −0.710390 −0.355195 0.934792i \(-0.615586\pi\)
−0.355195 + 0.934792i \(0.615586\pi\)
\(542\) −1.03888e6 −0.151903
\(543\) 0 0
\(544\) −7.04793e6 −1.02109
\(545\) 3.98874e6 0.575234
\(546\) 0 0
\(547\) 9.98777e6 1.42725 0.713626 0.700527i \(-0.247052\pi\)
0.713626 + 0.700527i \(0.247052\pi\)
\(548\) −2.33899e6 −0.332719
\(549\) 0 0
\(550\) 1.56856e6 0.221103
\(551\) 3.88205e6 0.544732
\(552\) 0 0
\(553\) 0 0
\(554\) −3.45598e6 −0.478406
\(555\) 0 0
\(556\) −5.06324e6 −0.694611
\(557\) −1.74619e6 −0.238481 −0.119241 0.992865i \(-0.538046\pi\)
−0.119241 + 0.992865i \(0.538046\pi\)
\(558\) 0 0
\(559\) −105877. −0.0143309
\(560\) 0 0
\(561\) 0 0
\(562\) 5.64904e6 0.754456
\(563\) 755218. 0.100416 0.0502078 0.998739i \(-0.484012\pi\)
0.0502078 + 0.998739i \(0.484012\pi\)
\(564\) 0 0
\(565\) 4.66792e6 0.615181
\(566\) 5.02719e6 0.659605
\(567\) 0 0
\(568\) 9.68836e6 1.26003
\(569\) 4.39534e6 0.569131 0.284565 0.958657i \(-0.408151\pi\)
0.284565 + 0.958657i \(0.408151\pi\)
\(570\) 0 0
\(571\) 1.16104e7 1.49024 0.745121 0.666930i \(-0.232392\pi\)
0.745121 + 0.666930i \(0.232392\pi\)
\(572\) −1.45344e7 −1.85740
\(573\) 0 0
\(574\) 0 0
\(575\) 44821.1 0.00565344
\(576\) 0 0
\(577\) 1.06643e7 1.33350 0.666748 0.745283i \(-0.267686\pi\)
0.666748 + 0.745283i \(0.267686\pi\)
\(578\) 72546.6 0.00903228
\(579\) 0 0
\(580\) 3.68606e6 0.454980
\(581\) 0 0
\(582\) 0 0
\(583\) 6.59562e6 0.803682
\(584\) 292097. 0.0354401
\(585\) 0 0
\(586\) 2.40195e6 0.288949
\(587\) 1.39482e7 1.67079 0.835396 0.549648i \(-0.185238\pi\)
0.835396 + 0.549648i \(0.185238\pi\)
\(588\) 0 0
\(589\) −1.02433e7 −1.21661
\(590\) 1.27620e6 0.150934
\(591\) 0 0
\(592\) −833711. −0.0977713
\(593\) −1.17933e7 −1.37720 −0.688600 0.725142i \(-0.741774\pi\)
−0.688600 + 0.725142i \(0.741774\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.38033e6 0.389802
\(597\) 0 0
\(598\) 136705. 0.0156326
\(599\) 4.38057e6 0.498843 0.249421 0.968395i \(-0.419760\pi\)
0.249421 + 0.968395i \(0.419760\pi\)
\(600\) 0 0
\(601\) 688570. 0.0777610 0.0388805 0.999244i \(-0.487621\pi\)
0.0388805 + 0.999244i \(0.487621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.88172e6 −0.321410
\(605\) 6.59563e6 0.732601
\(606\) 0 0
\(607\) −9.37319e6 −1.03256 −0.516281 0.856420i \(-0.672684\pi\)
−0.516281 + 0.856420i \(0.672684\pi\)
\(608\) 6.96085e6 0.763666
\(609\) 0 0
\(610\) 4.98684e6 0.542626
\(611\) −8.40174e6 −0.910472
\(612\) 0 0
\(613\) −2.16685e6 −0.232904 −0.116452 0.993196i \(-0.537152\pi\)
−0.116452 + 0.993196i \(0.537152\pi\)
\(614\) 5.51947e6 0.590849
\(615\) 0 0
\(616\) 0 0
\(617\) 5.07951e6 0.537166 0.268583 0.963256i \(-0.413445\pi\)
0.268583 + 0.963256i \(0.413445\pi\)
\(618\) 0 0
\(619\) −2.19034e6 −0.229766 −0.114883 0.993379i \(-0.536649\pi\)
−0.114883 + 0.993379i \(0.536649\pi\)
\(620\) −9.72611e6 −1.01616
\(621\) 0 0
\(622\) 2.43111e6 0.251958
\(623\) 0 0
\(624\) 0 0
\(625\) −5.59018e6 −0.572435
\(626\) −3.09790e6 −0.315960
\(627\) 0 0
\(628\) −2.35016e6 −0.237793
\(629\) −3.01923e6 −0.304278
\(630\) 0 0
\(631\) −7.18693e6 −0.718572 −0.359286 0.933228i \(-0.616980\pi\)
−0.359286 + 0.933228i \(0.616980\pi\)
\(632\) −1346.38 −0.000134083 0
\(633\) 0 0
\(634\) 4.21109e6 0.416075
\(635\) −142324. −0.0140070
\(636\) 0 0
\(637\) 0 0
\(638\) 5.17097e6 0.502945
\(639\) 0 0
\(640\) 7.96017e6 0.768197
\(641\) 1.76500e7 1.69668 0.848340 0.529452i \(-0.177602\pi\)
0.848340 + 0.529452i \(0.177602\pi\)
\(642\) 0 0
\(643\) 898309. 0.0856837 0.0428419 0.999082i \(-0.486359\pi\)
0.0428419 + 0.999082i \(0.486359\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3.87597e6 0.365425
\(647\) 1.38642e6 0.130207 0.0651035 0.997879i \(-0.479262\pi\)
0.0651035 + 0.997879i \(0.479262\pi\)
\(648\) 0 0
\(649\) −5.43904e6 −0.506886
\(650\) 3.11022e6 0.288741
\(651\) 0 0
\(652\) −4.39804e6 −0.405173
\(653\) 1.75425e7 1.60994 0.804968 0.593318i \(-0.202182\pi\)
0.804968 + 0.593318i \(0.202182\pi\)
\(654\) 0 0
\(655\) −1.16556e7 −1.06152
\(656\) 4.16062e6 0.377484
\(657\) 0 0
\(658\) 0 0
\(659\) −9.87522e6 −0.885795 −0.442898 0.896572i \(-0.646050\pi\)
−0.442898 + 0.896572i \(0.646050\pi\)
\(660\) 0 0
\(661\) −8.06792e6 −0.718221 −0.359110 0.933295i \(-0.616920\pi\)
−0.359110 + 0.933295i \(0.616920\pi\)
\(662\) 7.71373e6 0.684100
\(663\) 0 0
\(664\) 1.50025e7 1.32052
\(665\) 0 0
\(666\) 0 0
\(667\) 147759. 0.0128599
\(668\) −1.08742e7 −0.942880
\(669\) 0 0
\(670\) 8.74544e6 0.752652
\(671\) −2.12534e7 −1.82231
\(672\) 0 0
\(673\) −1.12772e7 −0.959762 −0.479881 0.877334i \(-0.659320\pi\)
−0.479881 + 0.877334i \(0.659320\pi\)
\(674\) 6.51274e6 0.552223
\(675\) 0 0
\(676\) −1.98804e7 −1.67324
\(677\) −5.20372e6 −0.436357 −0.218179 0.975909i \(-0.570012\pi\)
−0.218179 + 0.975909i \(0.570012\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 8.57196e6 0.710899
\(681\) 0 0
\(682\) −1.36442e7 −1.12328
\(683\) −6.05915e6 −0.497004 −0.248502 0.968631i \(-0.579938\pi\)
−0.248502 + 0.968631i \(0.579938\pi\)
\(684\) 0 0
\(685\) 4.46816e6 0.363833
\(686\) 0 0
\(687\) 0 0
\(688\) −31550.9 −0.00254121
\(689\) 1.30781e7 1.04954
\(690\) 0 0
\(691\) 7.36498e6 0.586781 0.293391 0.955993i \(-0.405216\pi\)
0.293391 + 0.955993i \(0.405216\pi\)
\(692\) 8.94754e6 0.710295
\(693\) 0 0
\(694\) −8.61204e6 −0.678746
\(695\) 9.67228e6 0.759569
\(696\) 0 0
\(697\) 1.50674e7 1.17478
\(698\) 591818. 0.0459779
\(699\) 0 0
\(700\) 0 0
\(701\) −7.80919e6 −0.600221 −0.300110 0.953904i \(-0.597023\pi\)
−0.300110 + 0.953904i \(0.597023\pi\)
\(702\) 0 0
\(703\) 2.98193e6 0.227567
\(704\) 3.51520e6 0.267312
\(705\) 0 0
\(706\) −1.06069e7 −0.800898
\(707\) 0 0
\(708\) 0 0
\(709\) 1.75650e7 1.31230 0.656150 0.754631i \(-0.272184\pi\)
0.656150 + 0.754631i \(0.272184\pi\)
\(710\) −7.94606e6 −0.591569
\(711\) 0 0
\(712\) −8.46198e6 −0.625564
\(713\) −389879. −0.0287214
\(714\) 0 0
\(715\) 2.77649e7 2.03110
\(716\) 2.04649e6 0.149186
\(717\) 0 0
\(718\) 2.83539e6 0.205259
\(719\) 8.09220e6 0.583773 0.291887 0.956453i \(-0.405717\pi\)
0.291887 + 0.956453i \(0.405717\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.14232e6 0.224341
\(723\) 0 0
\(724\) 9.13520e6 0.647697
\(725\) 3.36172e6 0.237529
\(726\) 0 0
\(727\) 1.51986e7 1.06652 0.533258 0.845952i \(-0.320967\pi\)
0.533258 + 0.845952i \(0.320967\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −239568. −0.0166388
\(731\) −114259. −0.00790858
\(732\) 0 0
\(733\) 5.83402e6 0.401059 0.200530 0.979688i \(-0.435734\pi\)
0.200530 + 0.979688i \(0.435734\pi\)
\(734\) 4.29720e6 0.294405
\(735\) 0 0
\(736\) 264944. 0.0180285
\(737\) −3.72722e7 −2.52765
\(738\) 0 0
\(739\) 6.47719e6 0.436290 0.218145 0.975916i \(-0.429999\pi\)
0.218145 + 0.975916i \(0.429999\pi\)
\(740\) 2.83138e6 0.190072
\(741\) 0 0
\(742\) 0 0
\(743\) 1.50899e7 1.00280 0.501401 0.865215i \(-0.332818\pi\)
0.501401 + 0.865215i \(0.332818\pi\)
\(744\) 0 0
\(745\) −6.45744e6 −0.426255
\(746\) −1.36896e7 −0.900626
\(747\) 0 0
\(748\) −1.56850e7 −1.02502
\(749\) 0 0
\(750\) 0 0
\(751\) −2.13997e6 −0.138455 −0.0692273 0.997601i \(-0.522053\pi\)
−0.0692273 + 0.997601i \(0.522053\pi\)
\(752\) −2.50367e6 −0.161448
\(753\) 0 0
\(754\) 1.02533e7 0.656801
\(755\) 5.50494e6 0.351467
\(756\) 0 0
\(757\) 2.10943e7 1.33791 0.668954 0.743304i \(-0.266742\pi\)
0.668954 + 0.743304i \(0.266742\pi\)
\(758\) −1.77597e6 −0.112270
\(759\) 0 0
\(760\) −8.46605e6 −0.531675
\(761\) −9.79958e6 −0.613403 −0.306701 0.951806i \(-0.599225\pi\)
−0.306701 + 0.951806i \(0.599225\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.22022e7 −1.37614
\(765\) 0 0
\(766\) 1.59233e6 0.0980530
\(767\) −1.07848e7 −0.661947
\(768\) 0 0
\(769\) −3.23493e7 −1.97265 −0.986323 0.164825i \(-0.947294\pi\)
−0.986323 + 0.164825i \(0.947294\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.21606e7 −0.734366
\(773\) −9.19713e6 −0.553609 −0.276805 0.960926i \(-0.589276\pi\)
−0.276805 + 0.960926i \(0.589276\pi\)
\(774\) 0 0
\(775\) −8.87030e6 −0.530499
\(776\) 491058. 0.0292738
\(777\) 0 0
\(778\) 1.66712e6 0.0987456
\(779\) −1.48812e7 −0.878609
\(780\) 0 0
\(781\) 3.38653e7 1.98668
\(782\) 147527. 0.00862690
\(783\) 0 0
\(784\) 0 0
\(785\) 4.48950e6 0.260030
\(786\) 0 0
\(787\) 1.46950e7 0.845731 0.422866 0.906192i \(-0.361024\pi\)
0.422866 + 0.906192i \(0.361024\pi\)
\(788\) 6.05508e6 0.347379
\(789\) 0 0
\(790\) 1104.25 6.29506e−5 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.21423e7 −2.37977
\(794\) −3.78097e6 −0.212839
\(795\) 0 0
\(796\) 5.00847e6 0.280171
\(797\) −2.33344e7 −1.30122 −0.650610 0.759412i \(-0.725487\pi\)
−0.650610 + 0.759412i \(0.725487\pi\)
\(798\) 0 0
\(799\) −9.06688e6 −0.502448
\(800\) 6.02785e6 0.332995
\(801\) 0 0
\(802\) −1.03796e7 −0.569831
\(803\) 1.02101e6 0.0558783
\(804\) 0 0
\(805\) 0 0
\(806\) −2.70545e7 −1.46690
\(807\) 0 0
\(808\) −3.44693e6 −0.185740
\(809\) −1.69301e6 −0.0909468 −0.0454734 0.998966i \(-0.514480\pi\)
−0.0454734 + 0.998966i \(0.514480\pi\)
\(810\) 0 0
\(811\) −2.12400e7 −1.13397 −0.566987 0.823727i \(-0.691891\pi\)
−0.566987 + 0.823727i \(0.691891\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.97199e6 0.210110
\(815\) 8.40155e6 0.443063
\(816\) 0 0
\(817\) 112848. 0.00591477
\(818\) −4.09960e6 −0.214219
\(819\) 0 0
\(820\) −1.41299e7 −0.733847
\(821\) −8.73550e6 −0.452304 −0.226152 0.974092i \(-0.572615\pi\)
−0.226152 + 0.974092i \(0.572615\pi\)
\(822\) 0 0
\(823\) −3.27964e7 −1.68782 −0.843910 0.536485i \(-0.819752\pi\)
−0.843910 + 0.536485i \(0.819752\pi\)
\(824\) 1.03146e7 0.529215
\(825\) 0 0
\(826\) 0 0
\(827\) −1.31248e7 −0.667311 −0.333656 0.942695i \(-0.608282\pi\)
−0.333656 + 0.942695i \(0.608282\pi\)
\(828\) 0 0
\(829\) 2.05402e7 1.03805 0.519026 0.854759i \(-0.326295\pi\)
0.519026 + 0.854759i \(0.326295\pi\)
\(830\) −1.23046e7 −0.619970
\(831\) 0 0
\(832\) 6.97012e6 0.349085
\(833\) 0 0
\(834\) 0 0
\(835\) 2.07729e7 1.03106
\(836\) 1.54912e7 0.766601
\(837\) 0 0
\(838\) −8.22542e6 −0.404621
\(839\) −2.83736e7 −1.39159 −0.695793 0.718243i \(-0.744947\pi\)
−0.695793 + 0.718243i \(0.744947\pi\)
\(840\) 0 0
\(841\) −9.42879e6 −0.459691
\(842\) −5.68643e6 −0.276414
\(843\) 0 0
\(844\) 1.54221e7 0.745226
\(845\) 3.79774e7 1.82972
\(846\) 0 0
\(847\) 0 0
\(848\) 3.89720e6 0.186107
\(849\) 0 0
\(850\) 3.35645e6 0.159343
\(851\) 113498. 0.00537236
\(852\) 0 0
\(853\) 1.02093e7 0.480424 0.240212 0.970720i \(-0.422783\pi\)
0.240212 + 0.970720i \(0.422783\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.71996e7 0.802295
\(857\) 8.33206e6 0.387525 0.193763 0.981048i \(-0.437931\pi\)
0.193763 + 0.981048i \(0.437931\pi\)
\(858\) 0 0
\(859\) 3.12766e7 1.44623 0.723113 0.690729i \(-0.242710\pi\)
0.723113 + 0.690729i \(0.242710\pi\)
\(860\) 107150. 0.00494023
\(861\) 0 0
\(862\) −1.53084e7 −0.701714
\(863\) 3.73573e7 1.70745 0.853726 0.520722i \(-0.174337\pi\)
0.853726 + 0.520722i \(0.174337\pi\)
\(864\) 0 0
\(865\) −1.70924e7 −0.776719
\(866\) 1.06222e7 0.481306
\(867\) 0 0
\(868\) 0 0
\(869\) −4706.21 −0.000211408 0
\(870\) 0 0
\(871\) −7.39051e7 −3.30088
\(872\) 1.36907e7 0.609724
\(873\) 0 0
\(874\) −145704. −0.00645199
\(875\) 0 0
\(876\) 0 0
\(877\) 38996.7 0.00171210 0.000856049 1.00000i \(-0.499728\pi\)
0.000856049 1.00000i \(0.499728\pi\)
\(878\) −6.61965e6 −0.289800
\(879\) 0 0
\(880\) 8.27378e6 0.360162
\(881\) −3.15554e7 −1.36973 −0.684864 0.728671i \(-0.740138\pi\)
−0.684864 + 0.728671i \(0.740138\pi\)
\(882\) 0 0
\(883\) −3.42253e7 −1.47722 −0.738611 0.674132i \(-0.764518\pi\)
−0.738611 + 0.674132i \(0.764518\pi\)
\(884\) −3.11010e7 −1.33858
\(885\) 0 0
\(886\) 1.35230e7 0.578745
\(887\) 2.69886e7 1.15178 0.575892 0.817526i \(-0.304655\pi\)
0.575892 + 0.817526i \(0.304655\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.94022e6 0.293696
\(891\) 0 0
\(892\) −9.39263e6 −0.395253
\(893\) 8.95486e6 0.375777
\(894\) 0 0
\(895\) −3.90940e6 −0.163137
\(896\) 0 0
\(897\) 0 0
\(898\) −7.79226e6 −0.322458
\(899\) −2.92421e7 −1.20673
\(900\) 0 0
\(901\) 1.41135e7 0.579191
\(902\) −1.98221e7 −0.811211
\(903\) 0 0
\(904\) 1.60218e7 0.652065
\(905\) −1.74509e7 −0.708267
\(906\) 0 0
\(907\) 1.92103e7 0.775381 0.387690 0.921790i \(-0.373273\pi\)
0.387690 + 0.921790i \(0.373273\pi\)
\(908\) 7.01442e6 0.282343
\(909\) 0 0
\(910\) 0 0
\(911\) −2.86013e7 −1.14180 −0.570899 0.821020i \(-0.693405\pi\)
−0.570899 + 0.821020i \(0.693405\pi\)
\(912\) 0 0
\(913\) 5.24408e7 2.08206
\(914\) −680027. −0.0269253
\(915\) 0 0
\(916\) 2.96224e7 1.16649
\(917\) 0 0
\(918\) 0 0
\(919\) −4.21754e7 −1.64729 −0.823645 0.567106i \(-0.808063\pi\)
−0.823645 + 0.567106i \(0.808063\pi\)
\(920\) −322235. −0.0125517
\(921\) 0 0
\(922\) 2.78855e6 0.108032
\(923\) 6.71498e7 2.59442
\(924\) 0 0
\(925\) 2.58224e6 0.0992299
\(926\) −1.74672e7 −0.669416
\(927\) 0 0
\(928\) 1.98716e7 0.757466
\(929\) 3.01886e7 1.14763 0.573817 0.818983i \(-0.305462\pi\)
0.573817 + 0.818983i \(0.305462\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.75130e6 0.103752
\(933\) 0 0
\(934\) −1.93817e7 −0.726984
\(935\) 2.99630e7 1.12087
\(936\) 0 0
\(937\) −3.64068e6 −0.135467 −0.0677335 0.997703i \(-0.521577\pi\)
−0.0677335 + 0.997703i \(0.521577\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 8.50275e6 0.313863
\(941\) −1.88601e7 −0.694336 −0.347168 0.937803i \(-0.612857\pi\)
−0.347168 + 0.937803i \(0.612857\pi\)
\(942\) 0 0
\(943\) −566410. −0.0207420
\(944\) −3.21380e6 −0.117379
\(945\) 0 0
\(946\) 150315. 0.00546104
\(947\) 1.82172e7 0.660094 0.330047 0.943965i \(-0.392935\pi\)
0.330047 + 0.943965i \(0.392935\pi\)
\(948\) 0 0
\(949\) 2.02452e6 0.0729720
\(950\) −3.31498e6 −0.119171
\(951\) 0 0
\(952\) 0 0
\(953\) −2.76898e7 −0.987616 −0.493808 0.869571i \(-0.664396\pi\)
−0.493808 + 0.869571i \(0.664396\pi\)
\(954\) 0 0
\(955\) 4.24127e7 1.50483
\(956\) −2.76343e7 −0.977921
\(957\) 0 0
\(958\) −1.52376e7 −0.536419
\(959\) 0 0
\(960\) 0 0
\(961\) 4.85298e7 1.69512
\(962\) 7.87585e6 0.274385
\(963\) 0 0
\(964\) −1.95662e7 −0.678133
\(965\) 2.32304e7 0.803042
\(966\) 0 0
\(967\) −2.44768e7 −0.841761 −0.420881 0.907116i \(-0.638279\pi\)
−0.420881 + 0.907116i \(0.638279\pi\)
\(968\) 2.26383e7 0.776526
\(969\) 0 0
\(970\) −402749. −0.0137437
\(971\) −9.50151e6 −0.323403 −0.161702 0.986840i \(-0.551698\pi\)
−0.161702 + 0.986840i \(0.551698\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 8.47457e6 0.286233
\(975\) 0 0
\(976\) −1.25582e7 −0.421990
\(977\) −4.69012e7 −1.57198 −0.785991 0.618238i \(-0.787847\pi\)
−0.785991 + 0.618238i \(0.787847\pi\)
\(978\) 0 0
\(979\) −2.95785e7 −0.986325
\(980\) 0 0
\(981\) 0 0
\(982\) −2.03951e7 −0.674914
\(983\) −2.35382e7 −0.776945 −0.388473 0.921460i \(-0.626997\pi\)
−0.388473 + 0.921460i \(0.626997\pi\)
\(984\) 0 0
\(985\) −1.15670e7 −0.379865
\(986\) 1.10650e7 0.362458
\(987\) 0 0
\(988\) 3.07167e7 1.00111
\(989\) 4295.21 0.000139635 0
\(990\) 0 0
\(991\) −2.64104e7 −0.854261 −0.427130 0.904190i \(-0.640475\pi\)
−0.427130 + 0.904190i \(0.640475\pi\)
\(992\) −5.24336e7 −1.69173
\(993\) 0 0
\(994\) 0 0
\(995\) −9.56766e6 −0.306371
\(996\) 0 0
\(997\) −1.95164e7 −0.621815 −0.310907 0.950440i \(-0.600633\pi\)
−0.310907 + 0.950440i \(0.600633\pi\)
\(998\) 1.44354e7 0.458777
\(999\) 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 441.6.a.z.1.2 4
3.2 odd 2 49.6.a.g.1.4 yes 4
7.6 odd 2 inner 441.6.a.z.1.1 4
12.11 even 2 784.6.a.bf.1.2 4
21.2 odd 6 49.6.c.h.18.1 8
21.5 even 6 49.6.c.h.18.2 8
21.11 odd 6 49.6.c.h.30.1 8
21.17 even 6 49.6.c.h.30.2 8
21.20 even 2 49.6.a.g.1.3 4
84.83 odd 2 784.6.a.bf.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 4 21.20 even 2
49.6.a.g.1.4 yes 4 3.2 odd 2
49.6.c.h.18.1 8 21.2 odd 6
49.6.c.h.18.2 8 21.5 even 6
49.6.c.h.30.1 8 21.11 odd 6
49.6.c.h.30.2 8 21.17 even 6
441.6.a.z.1.1 4 7.6 odd 2 inner
441.6.a.z.1.2 4 1.1 even 1 trivial
784.6.a.bf.1.2 4 12.11 even 2
784.6.a.bf.1.3 4 84.83 odd 2