Properties

Label 531.5.c.d.235.19
Level $531$
Weight $5$
Character 531.235
Analytic conductor $54.889$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 235.19
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.22

$q$-expansion

\(f(q)\) \(=\) \(q-0.850217i q^{2} +15.2771 q^{4} +11.2738 q^{5} -26.4494 q^{7} -26.5923i q^{8} +O(q^{10})\) \(q-0.850217i q^{2} +15.2771 q^{4} +11.2738 q^{5} -26.4494 q^{7} -26.5923i q^{8} -9.58520i q^{10} -19.9957i q^{11} -190.208i q^{13} +22.4877i q^{14} +221.825 q^{16} +159.452 q^{17} -294.035 q^{19} +172.232 q^{20} -17.0007 q^{22} +165.303i q^{23} -497.901 q^{25} -161.718 q^{26} -404.071 q^{28} -513.249 q^{29} -1627.89i q^{31} -614.077i q^{32} -135.569i q^{34} -298.186 q^{35} -1773.75i q^{37} +249.994i q^{38} -299.797i q^{40} +2060.39 q^{41} -2180.85i q^{43} -305.478i q^{44} +140.543 q^{46} +4253.57i q^{47} -1701.43 q^{49} +423.323i q^{50} -2905.83i q^{52} -246.901 q^{53} -225.429i q^{55} +703.351i q^{56} +436.373i q^{58} +(1203.62 - 3266.29i) q^{59} +1494.54i q^{61} -1384.06 q^{62} +3027.10 q^{64} -2144.37i q^{65} +3881.60i q^{67} +2435.97 q^{68} +253.523i q^{70} -1862.97 q^{71} -619.232i q^{73} -1508.07 q^{74} -4492.02 q^{76} +528.875i q^{77} +2372.28 q^{79} +2500.82 q^{80} -1751.78i q^{82} -4603.55i q^{83} +1797.63 q^{85} -1854.19 q^{86} -531.733 q^{88} -8914.96i q^{89} +5030.88i q^{91} +2525.36i q^{92} +3616.46 q^{94} -3314.91 q^{95} +1374.52i q^{97} +1446.58i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.850217i 0.212554i −0.994337 0.106277i \(-0.966107\pi\)
0.994337 0.106277i \(-0.0338930\pi\)
\(3\) 0 0
\(4\) 15.2771 0.954821
\(5\) 11.2738 0.450953 0.225477 0.974249i \(-0.427606\pi\)
0.225477 + 0.974249i \(0.427606\pi\)
\(6\) 0 0
\(7\) −26.4494 −0.539783 −0.269892 0.962891i \(-0.586988\pi\)
−0.269892 + 0.962891i \(0.586988\pi\)
\(8\) 26.5923i 0.415505i
\(9\) 0 0
\(10\) 9.58520i 0.0958520i
\(11\) 19.9957i 0.165254i −0.996581 0.0826270i \(-0.973669\pi\)
0.996581 0.0826270i \(-0.0263310\pi\)
\(12\) 0 0
\(13\) 190.208i 1.12549i −0.826630 0.562745i \(-0.809745\pi\)
0.826630 0.562745i \(-0.190255\pi\)
\(14\) 22.4877i 0.114733i
\(15\) 0 0
\(16\) 221.825 0.866503
\(17\) 159.452 0.551736 0.275868 0.961195i \(-0.411035\pi\)
0.275868 + 0.961195i \(0.411035\pi\)
\(18\) 0 0
\(19\) −294.035 −0.814503 −0.407251 0.913316i \(-0.633513\pi\)
−0.407251 + 0.913316i \(0.633513\pi\)
\(20\) 172.232 0.430579
\(21\) 0 0
\(22\) −17.0007 −0.0351254
\(23\) 165.303i 0.312482i 0.987719 + 0.156241i \(0.0499376\pi\)
−0.987719 + 0.156241i \(0.950062\pi\)
\(24\) 0 0
\(25\) −497.901 −0.796641
\(26\) −161.718 −0.239228
\(27\) 0 0
\(28\) −404.071 −0.515396
\(29\) −513.249 −0.610284 −0.305142 0.952307i \(-0.598704\pi\)
−0.305142 + 0.952307i \(0.598704\pi\)
\(30\) 0 0
\(31\) 1627.89i 1.69396i −0.531626 0.846979i \(-0.678419\pi\)
0.531626 0.846979i \(-0.321581\pi\)
\(32\) 614.077i 0.599684i
\(33\) 0 0
\(34\) 135.569i 0.117274i
\(35\) −298.186 −0.243417
\(36\) 0 0
\(37\) 1773.75i 1.29565i −0.761787 0.647827i \(-0.775678\pi\)
0.761787 0.647827i \(-0.224322\pi\)
\(38\) 249.994i 0.173126i
\(39\) 0 0
\(40\) 299.797i 0.187373i
\(41\) 2060.39 1.22570 0.612848 0.790201i \(-0.290024\pi\)
0.612848 + 0.790201i \(0.290024\pi\)
\(42\) 0 0
\(43\) 2180.85i 1.17947i −0.807596 0.589736i \(-0.799232\pi\)
0.807596 0.589736i \(-0.200768\pi\)
\(44\) 305.478i 0.157788i
\(45\) 0 0
\(46\) 140.543 0.0664193
\(47\) 4253.57i 1.92557i 0.270277 + 0.962783i \(0.412885\pi\)
−0.270277 + 0.962783i \(0.587115\pi\)
\(48\) 0 0
\(49\) −1701.43 −0.708634
\(50\) 423.323i 0.169329i
\(51\) 0 0
\(52\) 2905.83i 1.07464i
\(53\) −246.901 −0.0878965 −0.0439482 0.999034i \(-0.513994\pi\)
−0.0439482 + 0.999034i \(0.513994\pi\)
\(54\) 0 0
\(55\) 225.429i 0.0745219i
\(56\) 703.351i 0.224283i
\(57\) 0 0
\(58\) 436.373i 0.129718i
\(59\) 1203.62 3266.29i 0.345769 0.938320i
\(60\) 0 0
\(61\) 1494.54i 0.401650i 0.979627 + 0.200825i \(0.0643623\pi\)
−0.979627 + 0.200825i \(0.935638\pi\)
\(62\) −1384.06 −0.360058
\(63\) 0 0
\(64\) 3027.10 0.739038
\(65\) 2144.37i 0.507543i
\(66\) 0 0
\(67\) 3881.60i 0.864691i 0.901708 + 0.432346i \(0.142314\pi\)
−0.901708 + 0.432346i \(0.857686\pi\)
\(68\) 2435.97 0.526809
\(69\) 0 0
\(70\) 253.523i 0.0517393i
\(71\) −1862.97 −0.369564 −0.184782 0.982780i \(-0.559158\pi\)
−0.184782 + 0.982780i \(0.559158\pi\)
\(72\) 0 0
\(73\) 619.232i 0.116200i −0.998311 0.0581002i \(-0.981496\pi\)
0.998311 0.0581002i \(-0.0185043\pi\)
\(74\) −1508.07 −0.275397
\(75\) 0 0
\(76\) −4492.02 −0.777704
\(77\) 528.875i 0.0892014i
\(78\) 0 0
\(79\) 2372.28 0.380112 0.190056 0.981773i \(-0.439133\pi\)
0.190056 + 0.981773i \(0.439133\pi\)
\(80\) 2500.82 0.390752
\(81\) 0 0
\(82\) 1751.78i 0.260527i
\(83\) 4603.55i 0.668246i −0.942529 0.334123i \(-0.891560\pi\)
0.942529 0.334123i \(-0.108440\pi\)
\(84\) 0 0
\(85\) 1797.63 0.248807
\(86\) −1854.19 −0.250702
\(87\) 0 0
\(88\) −531.733 −0.0686639
\(89\) 8914.96i 1.12548i −0.826633 0.562742i \(-0.809747\pi\)
0.826633 0.562742i \(-0.190253\pi\)
\(90\) 0 0
\(91\) 5030.88i 0.607521i
\(92\) 2525.36i 0.298364i
\(93\) 0 0
\(94\) 3616.46 0.409287
\(95\) −3314.91 −0.367303
\(96\) 0 0
\(97\) 1374.52i 0.146085i 0.997329 + 0.0730427i \(0.0232709\pi\)
−0.997329 + 0.0730427i \(0.976729\pi\)
\(98\) 1446.58i 0.150623i
\(99\) 0 0
\(100\) −7606.50 −0.760650
\(101\) 3317.53i 0.325216i −0.986691 0.162608i \(-0.948009\pi\)
0.986691 0.162608i \(-0.0519905\pi\)
\(102\) 0 0
\(103\) 13068.4i 1.23182i −0.787817 0.615910i \(-0.788789\pi\)
0.787817 0.615910i \(-0.211211\pi\)
\(104\) −5058.07 −0.467647
\(105\) 0 0
\(106\) 209.919i 0.0186828i
\(107\) 12221.9 1.06751 0.533754 0.845640i \(-0.320781\pi\)
0.533754 + 0.845640i \(0.320781\pi\)
\(108\) 0 0
\(109\) 5683.83i 0.478397i −0.970971 0.239198i \(-0.923115\pi\)
0.970971 0.239198i \(-0.0768846\pi\)
\(110\) −191.663 −0.0158399
\(111\) 0 0
\(112\) −5867.13 −0.467724
\(113\) 16579.1i 1.29839i −0.760624 0.649193i \(-0.775107\pi\)
0.760624 0.649193i \(-0.224893\pi\)
\(114\) 0 0
\(115\) 1863.60i 0.140915i
\(116\) −7840.97 −0.582712
\(117\) 0 0
\(118\) −2777.05 1023.34i −0.199444 0.0734946i
\(119\) −4217.40 −0.297818
\(120\) 0 0
\(121\) 14241.2 0.972691
\(122\) 1270.68 0.0853724
\(123\) 0 0
\(124\) 24869.5i 1.61743i
\(125\) −12659.4 −0.810201
\(126\) 0 0
\(127\) −10677.2 −0.661986 −0.330993 0.943633i \(-0.607384\pi\)
−0.330993 + 0.943633i \(0.607384\pi\)
\(128\) 12398.9i 0.756770i
\(129\) 0 0
\(130\) −1823.18 −0.107880
\(131\) 13777.4i 0.802832i −0.915896 0.401416i \(-0.868518\pi\)
0.915896 0.401416i \(-0.131482\pi\)
\(132\) 0 0
\(133\) 7777.06 0.439655
\(134\) 3300.20 0.183794
\(135\) 0 0
\(136\) 4240.19i 0.229249i
\(137\) −17518.6 −0.933377 −0.466689 0.884422i \(-0.654553\pi\)
−0.466689 + 0.884422i \(0.654553\pi\)
\(138\) 0 0
\(139\) −34138.1 −1.76689 −0.883445 0.468535i \(-0.844782\pi\)
−0.883445 + 0.468535i \(0.844782\pi\)
\(140\) −4555.43 −0.232420
\(141\) 0 0
\(142\) 1583.93i 0.0785524i
\(143\) −3803.35 −0.185992
\(144\) 0 0
\(145\) −5786.28 −0.275210
\(146\) −526.481 −0.0246989
\(147\) 0 0
\(148\) 27097.8i 1.23712i
\(149\) 21713.6i 0.978045i −0.872271 0.489023i \(-0.837354\pi\)
0.872271 0.489023i \(-0.162646\pi\)
\(150\) 0 0
\(151\) 23115.7i 1.01380i 0.862004 + 0.506901i \(0.169209\pi\)
−0.862004 + 0.506901i \(0.830791\pi\)
\(152\) 7819.09i 0.338430i
\(153\) 0 0
\(154\) 449.658 0.0189601
\(155\) 18352.6i 0.763896i
\(156\) 0 0
\(157\) 27512.2i 1.11616i −0.829787 0.558080i \(-0.811538\pi\)
0.829787 0.558080i \(-0.188462\pi\)
\(158\) 2016.95i 0.0807944i
\(159\) 0 0
\(160\) 6922.99i 0.270429i
\(161\) 4372.16i 0.168673i
\(162\) 0 0
\(163\) 28943.8 1.08938 0.544691 0.838637i \(-0.316647\pi\)
0.544691 + 0.838637i \(0.316647\pi\)
\(164\) 31476.9 1.17032
\(165\) 0 0
\(166\) −3914.01 −0.142039
\(167\) −15906.8 −0.570362 −0.285181 0.958474i \(-0.592054\pi\)
−0.285181 + 0.958474i \(0.592054\pi\)
\(168\) 0 0
\(169\) −7618.03 −0.266728
\(170\) 1528.38i 0.0528850i
\(171\) 0 0
\(172\) 33317.1i 1.12619i
\(173\) 31592.5i 1.05558i 0.849375 + 0.527790i \(0.176979\pi\)
−0.849375 + 0.527790i \(0.823021\pi\)
\(174\) 0 0
\(175\) 13169.2 0.430014
\(176\) 4435.55i 0.143193i
\(177\) 0 0
\(178\) −7579.64 −0.239226
\(179\) 61910.6i 1.93223i 0.258113 + 0.966115i \(0.416899\pi\)
−0.258113 + 0.966115i \(0.583101\pi\)
\(180\) 0 0
\(181\) 4663.31 0.142343 0.0711717 0.997464i \(-0.477326\pi\)
0.0711717 + 0.997464i \(0.477326\pi\)
\(182\) 4277.34 0.129131
\(183\) 0 0
\(184\) 4395.79 0.129838
\(185\) 19997.0i 0.584280i
\(186\) 0 0
\(187\) 3188.36i 0.0911767i
\(188\) 64982.4i 1.83857i
\(189\) 0 0
\(190\) 2818.39i 0.0780717i
\(191\) 28272.6i 0.774995i 0.921871 + 0.387497i \(0.126660\pi\)
−0.921871 + 0.387497i \(0.873340\pi\)
\(192\) 0 0
\(193\) −20835.7 −0.559362 −0.279681 0.960093i \(-0.590229\pi\)
−0.279681 + 0.960093i \(0.590229\pi\)
\(194\) 1168.64 0.0310511
\(195\) 0 0
\(196\) −25993.0 −0.676618
\(197\) 51531.0 1.32781 0.663905 0.747817i \(-0.268898\pi\)
0.663905 + 0.747817i \(0.268898\pi\)
\(198\) 0 0
\(199\) −32606.5 −0.823374 −0.411687 0.911325i \(-0.635060\pi\)
−0.411687 + 0.911325i \(0.635060\pi\)
\(200\) 13240.3i 0.331009i
\(201\) 0 0
\(202\) −2820.62 −0.0691259
\(203\) 13575.1 0.329421
\(204\) 0 0
\(205\) 23228.5 0.552731
\(206\) −11110.9 −0.261828
\(207\) 0 0
\(208\) 42192.8i 0.975241i
\(209\) 5879.46i 0.134600i
\(210\) 0 0
\(211\) 16107.1i 0.361787i −0.983503 0.180894i \(-0.942101\pi\)
0.983503 0.180894i \(-0.0578989\pi\)
\(212\) −3771.94 −0.0839254
\(213\) 0 0
\(214\) 10391.3i 0.226903i
\(215\) 24586.5i 0.531887i
\(216\) 0 0
\(217\) 43056.8i 0.914371i
\(218\) −4832.49 −0.101685
\(219\) 0 0
\(220\) 3443.90i 0.0711550i
\(221\) 30329.0i 0.620974i
\(222\) 0 0
\(223\) 15954.1 0.320821 0.160411 0.987050i \(-0.448718\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(224\) 16241.9i 0.323700i
\(225\) 0 0
\(226\) −14095.8 −0.275977
\(227\) 49130.8i 0.953459i 0.879050 + 0.476730i \(0.158178\pi\)
−0.879050 + 0.476730i \(0.841822\pi\)
\(228\) 0 0
\(229\) 32670.4i 0.622993i 0.950247 + 0.311496i \(0.100830\pi\)
−0.950247 + 0.311496i \(0.899170\pi\)
\(230\) 1584.46 0.0299520
\(231\) 0 0
\(232\) 13648.5i 0.253576i
\(233\) 14015.4i 0.258164i −0.991634 0.129082i \(-0.958797\pi\)
0.991634 0.129082i \(-0.0412030\pi\)
\(234\) 0 0
\(235\) 47954.1i 0.868340i
\(236\) 18387.9 49899.6i 0.330147 0.895927i
\(237\) 0 0
\(238\) 3585.70i 0.0633025i
\(239\) −100221. −1.75454 −0.877271 0.479996i \(-0.840638\pi\)
−0.877271 + 0.479996i \(0.840638\pi\)
\(240\) 0 0
\(241\) 29390.7 0.506030 0.253015 0.967462i \(-0.418578\pi\)
0.253015 + 0.967462i \(0.418578\pi\)
\(242\) 12108.1i 0.206750i
\(243\) 0 0
\(244\) 22832.3i 0.383504i
\(245\) −19181.6 −0.319561
\(246\) 0 0
\(247\) 55927.8i 0.916715i
\(248\) −43289.5 −0.703849
\(249\) 0 0
\(250\) 10763.2i 0.172212i
\(251\) 100346. 1.59277 0.796385 0.604790i \(-0.206743\pi\)
0.796385 + 0.604790i \(0.206743\pi\)
\(252\) 0 0
\(253\) 3305.36 0.0516389
\(254\) 9077.91i 0.140708i
\(255\) 0 0
\(256\) 37891.8 0.578184
\(257\) −35228.4 −0.533367 −0.266684 0.963784i \(-0.585928\pi\)
−0.266684 + 0.963784i \(0.585928\pi\)
\(258\) 0 0
\(259\) 46914.6i 0.699373i
\(260\) 32759.8i 0.484613i
\(261\) 0 0
\(262\) −11713.8 −0.170645
\(263\) 118656. 1.71545 0.857725 0.514109i \(-0.171877\pi\)
0.857725 + 0.514109i \(0.171877\pi\)
\(264\) 0 0
\(265\) −2783.52 −0.0396372
\(266\) 6612.18i 0.0934505i
\(267\) 0 0
\(268\) 59299.7i 0.825625i
\(269\) 36762.0i 0.508035i −0.967200 0.254018i \(-0.918248\pi\)
0.967200 0.254018i \(-0.0817522\pi\)
\(270\) 0 0
\(271\) 91224.9 1.24215 0.621076 0.783750i \(-0.286696\pi\)
0.621076 + 0.783750i \(0.286696\pi\)
\(272\) 35370.4 0.478081
\(273\) 0 0
\(274\) 14894.6i 0.198393i
\(275\) 9955.90i 0.131648i
\(276\) 0 0
\(277\) 81627.4 1.06384 0.531920 0.846794i \(-0.321471\pi\)
0.531920 + 0.846794i \(0.321471\pi\)
\(278\) 29024.8i 0.375560i
\(279\) 0 0
\(280\) 7929.46i 0.101141i
\(281\) −58538.2 −0.741355 −0.370678 0.928762i \(-0.620875\pi\)
−0.370678 + 0.928762i \(0.620875\pi\)
\(282\) 0 0
\(283\) 45800.4i 0.571869i −0.958249 0.285935i \(-0.907696\pi\)
0.958249 0.285935i \(-0.0923040\pi\)
\(284\) −28460.9 −0.352867
\(285\) 0 0
\(286\) 3233.67i 0.0395333i
\(287\) −54496.1 −0.661610
\(288\) 0 0
\(289\) −58096.1 −0.695587
\(290\) 4919.59i 0.0584969i
\(291\) 0 0
\(292\) 9460.09i 0.110951i
\(293\) 85610.9 0.997226 0.498613 0.866825i \(-0.333843\pi\)
0.498613 + 0.866825i \(0.333843\pi\)
\(294\) 0 0
\(295\) 13569.4 36823.6i 0.155925 0.423138i
\(296\) −47168.2 −0.538351
\(297\) 0 0
\(298\) −18461.2 −0.207888
\(299\) 31441.9 0.351695
\(300\) 0 0
\(301\) 57682.0i 0.636660i
\(302\) 19653.3 0.215488
\(303\) 0 0
\(304\) −65224.4 −0.705769
\(305\) 16849.2i 0.181125i
\(306\) 0 0
\(307\) 121584. 1.29002 0.645012 0.764172i \(-0.276852\pi\)
0.645012 + 0.764172i \(0.276852\pi\)
\(308\) 8079.70i 0.0851714i
\(309\) 0 0
\(310\) −15603.7 −0.162369
\(311\) 17980.2 0.185898 0.0929488 0.995671i \(-0.470371\pi\)
0.0929488 + 0.995671i \(0.470371\pi\)
\(312\) 0 0
\(313\) 65025.1i 0.663731i 0.943327 + 0.331865i \(0.107678\pi\)
−0.943327 + 0.331865i \(0.892322\pi\)
\(314\) −23391.3 −0.237244
\(315\) 0 0
\(316\) 36241.6 0.362939
\(317\) 6089.75 0.0606011 0.0303006 0.999541i \(-0.490354\pi\)
0.0303006 + 0.999541i \(0.490354\pi\)
\(318\) 0 0
\(319\) 10262.8i 0.100852i
\(320\) 34127.0 0.333272
\(321\) 0 0
\(322\) −3717.28 −0.0358521
\(323\) −46884.5 −0.449391
\(324\) 0 0
\(325\) 94704.6i 0.896612i
\(326\) 24608.5i 0.231553i
\(327\) 0 0
\(328\) 54790.7i 0.509283i
\(329\) 112504.i 1.03939i
\(330\) 0 0
\(331\) −159934. −1.45977 −0.729884 0.683571i \(-0.760426\pi\)
−0.729884 + 0.683571i \(0.760426\pi\)
\(332\) 70329.0i 0.638055i
\(333\) 0 0
\(334\) 13524.3i 0.121233i
\(335\) 43760.5i 0.389935i
\(336\) 0 0
\(337\) 120034.i 1.05693i 0.848956 + 0.528463i \(0.177231\pi\)
−0.848956 + 0.528463i \(0.822769\pi\)
\(338\) 6476.98i 0.0566942i
\(339\) 0 0
\(340\) 27462.7 0.237566
\(341\) −32550.9 −0.279933
\(342\) 0 0
\(343\) 108507. 0.922292
\(344\) −57993.8 −0.490077
\(345\) 0 0
\(346\) 26860.4 0.224368
\(347\) 184056.i 1.52859i 0.644869 + 0.764293i \(0.276912\pi\)
−0.644869 + 0.764293i \(0.723088\pi\)
\(348\) 0 0
\(349\) 179218.i 1.47140i 0.677305 + 0.735702i \(0.263148\pi\)
−0.677305 + 0.735702i \(0.736852\pi\)
\(350\) 11196.6i 0.0914012i
\(351\) 0 0
\(352\) −12278.9 −0.0991002
\(353\) 124737.i 1.00103i 0.865728 + 0.500514i \(0.166856\pi\)
−0.865728 + 0.500514i \(0.833144\pi\)
\(354\) 0 0
\(355\) −21002.8 −0.166656
\(356\) 136195.i 1.07464i
\(357\) 0 0
\(358\) 52637.4 0.410703
\(359\) 77449.2 0.600936 0.300468 0.953792i \(-0.402857\pi\)
0.300468 + 0.953792i \(0.402857\pi\)
\(360\) 0 0
\(361\) −43864.2 −0.336586
\(362\) 3964.83i 0.0302557i
\(363\) 0 0
\(364\) 76857.4i 0.580074i
\(365\) 6981.12i 0.0524009i
\(366\) 0 0
\(367\) 42555.8i 0.315956i −0.987443 0.157978i \(-0.949502\pi\)
0.987443 0.157978i \(-0.0504975\pi\)
\(368\) 36668.3i 0.270767i
\(369\) 0 0
\(370\) −17001.8 −0.124191
\(371\) 6530.38 0.0474451
\(372\) 0 0
\(373\) 86818.1 0.624012 0.312006 0.950080i \(-0.398999\pi\)
0.312006 + 0.950080i \(0.398999\pi\)
\(374\) −2710.79 −0.0193800
\(375\) 0 0
\(376\) 113112. 0.800082
\(377\) 97624.0i 0.686869i
\(378\) 0 0
\(379\) 45077.2 0.313819 0.156909 0.987613i \(-0.449847\pi\)
0.156909 + 0.987613i \(0.449847\pi\)
\(380\) −50642.2 −0.350708
\(381\) 0 0
\(382\) 24037.8 0.164728
\(383\) 275505. 1.87815 0.939077 0.343707i \(-0.111683\pi\)
0.939077 + 0.343707i \(0.111683\pi\)
\(384\) 0 0
\(385\) 5962.45i 0.0402257i
\(386\) 17714.8i 0.118895i
\(387\) 0 0
\(388\) 20998.7i 0.139485i
\(389\) −104159. −0.688333 −0.344167 0.938909i \(-0.611839\pi\)
−0.344167 + 0.938909i \(0.611839\pi\)
\(390\) 0 0
\(391\) 26357.9i 0.172408i
\(392\) 45245.0i 0.294441i
\(393\) 0 0
\(394\) 43812.5i 0.282232i
\(395\) 26744.7 0.171413
\(396\) 0 0
\(397\) 30109.0i 0.191036i 0.995428 + 0.0955181i \(0.0304508\pi\)
−0.995428 + 0.0955181i \(0.969549\pi\)
\(398\) 27722.5i 0.175012i
\(399\) 0 0
\(400\) −110447. −0.690292
\(401\) 80257.6i 0.499111i −0.968360 0.249556i \(-0.919715\pi\)
0.968360 0.249556i \(-0.0802846\pi\)
\(402\) 0 0
\(403\) −309638. −1.90653
\(404\) 50682.3i 0.310523i
\(405\) 0 0
\(406\) 11541.8i 0.0700199i
\(407\) −35467.5 −0.214112
\(408\) 0 0
\(409\) 99079.5i 0.592294i −0.955142 0.296147i \(-0.904298\pi\)
0.955142 0.296147i \(-0.0957018\pi\)
\(410\) 19749.3i 0.117485i
\(411\) 0 0
\(412\) 199647.i 1.17617i
\(413\) −31835.0 + 86391.4i −0.186640 + 0.506489i
\(414\) 0 0
\(415\) 51899.6i 0.301348i
\(416\) −116802. −0.674939
\(417\) 0 0
\(418\) 4998.81 0.0286098
\(419\) 311022.i 1.77159i 0.464078 + 0.885795i \(0.346386\pi\)
−0.464078 + 0.885795i \(0.653614\pi\)
\(420\) 0 0
\(421\) 89938.3i 0.507435i −0.967278 0.253718i \(-0.918347\pi\)
0.967278 0.253718i \(-0.0816534\pi\)
\(422\) −13694.5 −0.0768993
\(423\) 0 0
\(424\) 6565.68i 0.0365214i
\(425\) −79391.2 −0.439536
\(426\) 0 0
\(427\) 39529.7i 0.216804i
\(428\) 186715. 1.01928
\(429\) 0 0
\(430\) −20903.8 −0.113055
\(431\) 73388.3i 0.395068i 0.980296 + 0.197534i \(0.0632934\pi\)
−0.980296 + 0.197534i \(0.936707\pi\)
\(432\) 0 0
\(433\) 68075.2 0.363089 0.181545 0.983383i \(-0.441890\pi\)
0.181545 + 0.983383i \(0.441890\pi\)
\(434\) 36607.6 0.194353
\(435\) 0 0
\(436\) 86832.7i 0.456783i
\(437\) 48604.9i 0.254517i
\(438\) 0 0
\(439\) 137778. 0.714909 0.357454 0.933931i \(-0.383645\pi\)
0.357454 + 0.933931i \(0.383645\pi\)
\(440\) −5994.67 −0.0309642
\(441\) 0 0
\(442\) −25786.2 −0.131991
\(443\) 19727.4i 0.100522i 0.998736 + 0.0502611i \(0.0160054\pi\)
−0.998736 + 0.0502611i \(0.983995\pi\)
\(444\) 0 0
\(445\) 100506.i 0.507540i
\(446\) 13564.5i 0.0681919i
\(447\) 0 0
\(448\) −80064.9 −0.398920
\(449\) 279194. 1.38489 0.692443 0.721473i \(-0.256534\pi\)
0.692443 + 0.721473i \(0.256534\pi\)
\(450\) 0 0
\(451\) 41199.1i 0.202551i
\(452\) 253281.i 1.23973i
\(453\) 0 0
\(454\) 41771.8 0.202662
\(455\) 56717.3i 0.273964i
\(456\) 0 0
\(457\) 193755.i 0.927728i 0.885906 + 0.463864i \(0.153537\pi\)
−0.885906 + 0.463864i \(0.846463\pi\)
\(458\) 27776.9 0.132420
\(459\) 0 0
\(460\) 28470.4i 0.134548i
\(461\) −40358.7 −0.189905 −0.0949524 0.995482i \(-0.530270\pi\)
−0.0949524 + 0.995482i \(0.530270\pi\)
\(462\) 0 0
\(463\) 219514.i 1.02400i 0.858985 + 0.512000i \(0.171095\pi\)
−0.858985 + 0.512000i \(0.828905\pi\)
\(464\) −113851. −0.528813
\(465\) 0 0
\(466\) −11916.2 −0.0548737
\(467\) 321723.i 1.47519i 0.675243 + 0.737596i \(0.264039\pi\)
−0.675243 + 0.737596i \(0.735961\pi\)
\(468\) 0 0
\(469\) 102666.i 0.466746i
\(470\) 40771.3 0.184569
\(471\) 0 0
\(472\) −86858.3 32007.1i −0.389877 0.143669i
\(473\) −43607.6 −0.194913
\(474\) 0 0
\(475\) 146400. 0.648866
\(476\) −64429.8 −0.284363
\(477\) 0 0
\(478\) 85209.7i 0.372935i
\(479\) −42962.3 −0.187248 −0.0936238 0.995608i \(-0.529845\pi\)
−0.0936238 + 0.995608i \(0.529845\pi\)
\(480\) 0 0
\(481\) −337381. −1.45825
\(482\) 24988.5i 0.107559i
\(483\) 0 0
\(484\) 217564. 0.928746
\(485\) 15496.1i 0.0658777i
\(486\) 0 0
\(487\) −268645. −1.13271 −0.566357 0.824160i \(-0.691648\pi\)
−0.566357 + 0.824160i \(0.691648\pi\)
\(488\) 39743.3 0.166888
\(489\) 0 0
\(490\) 16308.5i 0.0679239i
\(491\) −331744. −1.37607 −0.688034 0.725678i \(-0.741526\pi\)
−0.688034 + 0.725678i \(0.741526\pi\)
\(492\) 0 0
\(493\) −81838.5 −0.336716
\(494\) 47550.8 0.194852
\(495\) 0 0
\(496\) 361107.i 1.46782i
\(497\) 49274.5 0.199485
\(498\) 0 0
\(499\) 112576. 0.452111 0.226055 0.974114i \(-0.427417\pi\)
0.226055 + 0.974114i \(0.427417\pi\)
\(500\) −193399. −0.773597
\(501\) 0 0
\(502\) 85315.9i 0.338550i
\(503\) 350747.i 1.38630i 0.720791 + 0.693152i \(0.243779\pi\)
−0.720791 + 0.693152i \(0.756221\pi\)
\(504\) 0 0
\(505\) 37401.2i 0.146657i
\(506\) 2810.27i 0.0109761i
\(507\) 0 0
\(508\) −163117. −0.632078
\(509\) 95019.3i 0.366755i −0.983043 0.183378i \(-0.941297\pi\)
0.983043 0.183378i \(-0.0587031\pi\)
\(510\) 0 0
\(511\) 16378.3i 0.0627231i
\(512\) 230599.i 0.879665i
\(513\) 0 0
\(514\) 29951.7i 0.113369i
\(515\) 147331.i 0.555493i
\(516\) 0 0
\(517\) 85053.4 0.318208
\(518\) 39887.6 0.148655
\(519\) 0 0
\(520\) −57023.8 −0.210887
\(521\) 131693. 0.485161 0.242581 0.970131i \(-0.422006\pi\)
0.242581 + 0.970131i \(0.422006\pi\)
\(522\) 0 0
\(523\) 432940. 1.58280 0.791398 0.611302i \(-0.209354\pi\)
0.791398 + 0.611302i \(0.209354\pi\)
\(524\) 210479.i 0.766561i
\(525\) 0 0
\(526\) 100883.i 0.364626i
\(527\) 259571.i 0.934618i
\(528\) 0 0
\(529\) 252516. 0.902355
\(530\) 2366.60i 0.00842505i
\(531\) 0 0
\(532\) 118811. 0.419792
\(533\) 391903.i 1.37951i
\(534\) 0 0
\(535\) 137788. 0.481396
\(536\) 103221. 0.359284
\(537\) 0 0
\(538\) −31255.6 −0.107985
\(539\) 34021.4i 0.117105i
\(540\) 0 0
\(541\) 369539.i 1.26260i −0.775539 0.631300i \(-0.782522\pi\)
0.775539 0.631300i \(-0.217478\pi\)
\(542\) 77560.9i 0.264024i
\(543\) 0 0
\(544\) 97915.6i 0.330867i
\(545\) 64078.6i 0.215735i
\(546\) 0 0
\(547\) −145339. −0.485745 −0.242873 0.970058i \(-0.578090\pi\)
−0.242873 + 0.970058i \(0.578090\pi\)
\(548\) −267633. −0.891208
\(549\) 0 0
\(550\) 8464.67 0.0279824
\(551\) 150913. 0.497078
\(552\) 0 0
\(553\) −62745.4 −0.205178
\(554\) 69401.0i 0.226124i
\(555\) 0 0
\(556\) −521532. −1.68706
\(557\) −120254. −0.387604 −0.193802 0.981041i \(-0.562082\pi\)
−0.193802 + 0.981041i \(0.562082\pi\)
\(558\) 0 0
\(559\) −414814. −1.32749
\(560\) −66145.0 −0.210922
\(561\) 0 0
\(562\) 49770.1i 0.157578i
\(563\) 388859.i 1.22680i 0.789771 + 0.613402i \(0.210200\pi\)
−0.789771 + 0.613402i \(0.789800\pi\)
\(564\) 0 0
\(565\) 186910.i 0.585511i
\(566\) −38940.3 −0.121553
\(567\) 0 0
\(568\) 49540.8i 0.153556i
\(569\) 441305.i 1.36306i −0.731792 0.681528i \(-0.761316\pi\)
0.731792 0.681528i \(-0.238684\pi\)
\(570\) 0 0
\(571\) 18999.3i 0.0582726i 0.999575 + 0.0291363i \(0.00927569\pi\)
−0.999575 + 0.0291363i \(0.990724\pi\)
\(572\) −58104.2 −0.177589
\(573\) 0 0
\(574\) 46333.5i 0.140628i
\(575\) 82304.5i 0.248936i
\(576\) 0 0
\(577\) 70278.9 0.211093 0.105546 0.994414i \(-0.466341\pi\)
0.105546 + 0.994414i \(0.466341\pi\)
\(578\) 49394.3i 0.147850i
\(579\) 0 0
\(580\) −88397.8 −0.262776
\(581\) 121761.i 0.360708i
\(582\) 0 0
\(583\) 4936.97i 0.0145252i
\(584\) −16466.8 −0.0482819
\(585\) 0 0
\(586\) 72787.8i 0.211965i
\(587\) 325818.i 0.945581i −0.881175 0.472790i \(-0.843247\pi\)
0.881175 0.472790i \(-0.156753\pi\)
\(588\) 0 0
\(589\) 478658.i 1.37973i
\(590\) −31308.0 11536.9i −0.0899398 0.0331426i
\(591\) 0 0
\(592\) 393462.i 1.12269i
\(593\) −607064. −1.72634 −0.863168 0.504917i \(-0.831523\pi\)
−0.863168 + 0.504917i \(0.831523\pi\)
\(594\) 0 0
\(595\) −47546.3 −0.134302
\(596\) 331721.i 0.933858i
\(597\) 0 0
\(598\) 26732.4i 0.0747543i
\(599\) −23980.8 −0.0668360 −0.0334180 0.999441i \(-0.510639\pi\)
−0.0334180 + 0.999441i \(0.510639\pi\)
\(600\) 0 0
\(601\) 226771.i 0.627825i −0.949452 0.313913i \(-0.898360\pi\)
0.949452 0.313913i \(-0.101640\pi\)
\(602\) 49042.2 0.135325
\(603\) 0 0
\(604\) 353141.i 0.967999i
\(605\) 160553. 0.438638
\(606\) 0 0
\(607\) −172250. −0.467499 −0.233749 0.972297i \(-0.575100\pi\)
−0.233749 + 0.972297i \(0.575100\pi\)
\(608\) 180560.i 0.488444i
\(609\) 0 0
\(610\) 14325.5 0.0384989
\(611\) 809063. 2.16721
\(612\) 0 0
\(613\) 283915.i 0.755559i −0.925896 0.377779i \(-0.876688\pi\)
0.925896 0.377779i \(-0.123312\pi\)
\(614\) 103372.i 0.274200i
\(615\) 0 0
\(616\) 14064.0 0.0370637
\(617\) 117535. 0.308743 0.154372 0.988013i \(-0.450665\pi\)
0.154372 + 0.988013i \(0.450665\pi\)
\(618\) 0 0
\(619\) 232785. 0.607537 0.303769 0.952746i \(-0.401755\pi\)
0.303769 + 0.952746i \(0.401755\pi\)
\(620\) 280375.i 0.729384i
\(621\) 0 0
\(622\) 15287.1i 0.0395133i
\(623\) 235795.i 0.607517i
\(624\) 0 0
\(625\) 168468. 0.431278
\(626\) 55285.4 0.141079
\(627\) 0 0
\(628\) 420308.i 1.06573i
\(629\) 282828.i 0.714860i
\(630\) 0 0
\(631\) 58189.1 0.146145 0.0730723 0.997327i \(-0.476720\pi\)
0.0730723 + 0.997327i \(0.476720\pi\)
\(632\) 63084.5i 0.157939i
\(633\) 0 0
\(634\) 5177.60i 0.0128810i
\(635\) −120373. −0.298525
\(636\) 0 0
\(637\) 323625.i 0.797561i
\(638\) 8725.60 0.0214365
\(639\) 0 0
\(640\) 139783.i 0.341268i
\(641\) −774979. −1.88614 −0.943070 0.332595i \(-0.892076\pi\)
−0.943070 + 0.332595i \(0.892076\pi\)
\(642\) 0 0
\(643\) −69007.6 −0.166907 −0.0834536 0.996512i \(-0.526595\pi\)
−0.0834536 + 0.996512i \(0.526595\pi\)
\(644\) 66794.1i 0.161052i
\(645\) 0 0
\(646\) 39861.9i 0.0955198i
\(647\) −278274. −0.664760 −0.332380 0.943146i \(-0.607852\pi\)
−0.332380 + 0.943146i \(0.607852\pi\)
\(648\) 0 0
\(649\) −65311.9 24067.3i −0.155061 0.0571397i
\(650\) 80519.5 0.190579
\(651\) 0 0
\(652\) 442178. 1.04017
\(653\) 158117. 0.370811 0.185406 0.982662i \(-0.440640\pi\)
0.185406 + 0.982662i \(0.440640\pi\)
\(654\) 0 0
\(655\) 155324.i 0.362040i
\(656\) 457046. 1.06207
\(657\) 0 0
\(658\) −95653.1 −0.220926
\(659\) 619035.i 1.42543i −0.701456 0.712713i \(-0.747466\pi\)
0.701456 0.712713i \(-0.252534\pi\)
\(660\) 0 0
\(661\) −109703. −0.251081 −0.125541 0.992088i \(-0.540067\pi\)
−0.125541 + 0.992088i \(0.540067\pi\)
\(662\) 135978.i 0.310280i
\(663\) 0 0
\(664\) −122419. −0.277660
\(665\) 87677.2 0.198264
\(666\) 0 0
\(667\) 84841.6i 0.190703i
\(668\) −243011. −0.544594
\(669\) 0 0
\(670\) 37205.9 0.0828824
\(671\) 29884.4 0.0663743
\(672\) 0 0
\(673\) 496628.i 1.09648i 0.836321 + 0.548240i \(0.184702\pi\)
−0.836321 + 0.548240i \(0.815298\pi\)
\(674\) 102055. 0.224654
\(675\) 0 0
\(676\) −116382. −0.254678
\(677\) 273451. 0.596626 0.298313 0.954468i \(-0.403576\pi\)
0.298313 + 0.954468i \(0.403576\pi\)
\(678\) 0 0
\(679\) 36355.2i 0.0788545i
\(680\) 47803.2i 0.103381i
\(681\) 0 0
\(682\) 27675.4i 0.0595010i
\(683\) 80479.6i 0.172522i 0.996273 + 0.0862609i \(0.0274919\pi\)
−0.996273 + 0.0862609i \(0.972508\pi\)
\(684\) 0 0
\(685\) −197501. −0.420910
\(686\) 92254.2i 0.196037i
\(687\) 0 0
\(688\) 483766.i 1.02202i
\(689\) 46962.5i 0.0989266i
\(690\) 0 0
\(691\) 700168.i 1.46638i −0.680025 0.733189i \(-0.738031\pi\)
0.680025 0.733189i \(-0.261969\pi\)
\(692\) 482642.i 1.00789i
\(693\) 0 0
\(694\) 156487. 0.324907
\(695\) −384867. −0.796785
\(696\) 0 0
\(697\) 328533. 0.676260
\(698\) 152375. 0.312753
\(699\) 0 0
\(700\) 201187. 0.410586
\(701\) 441649.i 0.898754i 0.893342 + 0.449377i \(0.148354\pi\)
−0.893342 + 0.449377i \(0.851646\pi\)
\(702\) 0 0
\(703\) 521546.i 1.05531i
\(704\) 60529.1i 0.122129i
\(705\) 0 0
\(706\) 106054. 0.212773
\(707\) 87746.5i 0.175546i
\(708\) 0 0
\(709\) 110109. 0.219043 0.109522 0.993984i \(-0.465068\pi\)
0.109522 + 0.993984i \(0.465068\pi\)
\(710\) 17857.0i 0.0354234i
\(711\) 0 0
\(712\) −237069. −0.467644
\(713\) 269096. 0.529331
\(714\) 0 0
\(715\) −42878.3 −0.0838736
\(716\) 945816.i 1.84493i
\(717\) 0 0
\(718\) 65848.6i 0.127731i
\(719\) 66331.4i 0.128310i 0.997940 + 0.0641551i \(0.0204353\pi\)
−0.997940 + 0.0641551i \(0.979565\pi\)
\(720\) 0 0
\(721\) 345651.i 0.664916i
\(722\) 37294.0i 0.0715427i
\(723\) 0 0
\(724\) 71242.0 0.135912
\(725\) 255547. 0.486178
\(726\) 0 0
\(727\) −199842. −0.378110 −0.189055 0.981967i \(-0.560542\pi\)
−0.189055 + 0.981967i \(0.560542\pi\)
\(728\) 133783. 0.252428
\(729\) 0 0
\(730\) −5935.46 −0.0111380
\(731\) 347740.i 0.650758i
\(732\) 0 0
\(733\) −582907. −1.08490 −0.542452 0.840087i \(-0.682504\pi\)
−0.542452 + 0.840087i \(0.682504\pi\)
\(734\) −36181.7 −0.0671578
\(735\) 0 0
\(736\) 101509. 0.187390
\(737\) 77615.5 0.142894
\(738\) 0 0
\(739\) 45515.8i 0.0833438i −0.999131 0.0416719i \(-0.986732\pi\)
0.999131 0.0416719i \(-0.0132684\pi\)
\(740\) 305496.i 0.557882i
\(741\) 0 0
\(742\) 5552.24i 0.0100846i
\(743\) 156217. 0.282977 0.141488 0.989940i \(-0.454811\pi\)
0.141488 + 0.989940i \(0.454811\pi\)
\(744\) 0 0
\(745\) 244795.i 0.441053i
\(746\) 73814.2i 0.132636i
\(747\) 0 0
\(748\) 48708.9i 0.0870574i
\(749\) −323262. −0.576223
\(750\) 0 0
\(751\) 335449.i 0.594767i −0.954758 0.297383i \(-0.903886\pi\)
0.954758 0.297383i \(-0.0961140\pi\)
\(752\) 943548.i 1.66851i
\(753\) 0 0
\(754\) 83001.5 0.145997
\(755\) 260602.i 0.457177i
\(756\) 0 0
\(757\) 797908. 1.39239 0.696196 0.717852i \(-0.254875\pi\)
0.696196 + 0.717852i \(0.254875\pi\)
\(758\) 38325.4i 0.0667035i
\(759\) 0 0
\(760\) 88151.1i 0.152616i
\(761\) 23661.3 0.0408572 0.0204286 0.999791i \(-0.493497\pi\)
0.0204286 + 0.999791i \(0.493497\pi\)
\(762\) 0 0
\(763\) 150334.i 0.258231i
\(764\) 431924.i 0.739981i
\(765\) 0 0
\(766\) 234239.i 0.399209i
\(767\) −621274. 228938.i −1.05607 0.389159i
\(768\) 0 0
\(769\) 698020.i 1.18036i 0.807271 + 0.590181i \(0.200944\pi\)
−0.807271 + 0.590181i \(0.799056\pi\)
\(770\) 5069.37 0.00855013
\(771\) 0 0
\(772\) −318310. −0.534091
\(773\) 531513.i 0.889519i 0.895650 + 0.444759i \(0.146711\pi\)
−0.895650 + 0.444759i \(0.853289\pi\)
\(774\) 0 0
\(775\) 810530.i 1.34948i
\(776\) 36551.6 0.0606993
\(777\) 0 0
\(778\) 88558.0i 0.146308i
\(779\) −605829. −0.998332
\(780\) 0 0
\(781\) 37251.5i 0.0610720i
\(782\) 22409.9 0.0366460
\(783\) 0 0
\(784\) −377419. −0.614034
\(785\) 310168.i 0.503336i
\(786\) 0 0
\(787\) −127904. −0.206506 −0.103253 0.994655i \(-0.532925\pi\)
−0.103253 + 0.994655i \(0.532925\pi\)
\(788\) 787246. 1.26782
\(789\) 0 0
\(790\) 22738.8i 0.0364345i
\(791\) 438507.i 0.700847i
\(792\) 0 0
\(793\) 284273. 0.452053
\(794\) 25599.2 0.0406055
\(795\) 0 0
\(796\) −498133. −0.786175
\(797\)