Properties

Label 507.4.a.g.1.2
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -8.00000 q^{4} +5.19615 q^{5} +10.3923 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -8.00000 q^{4} +5.19615 q^{5} +10.3923 q^{7} +9.00000 q^{9} -51.9615 q^{11} +24.0000 q^{12} -15.5885 q^{15} +64.0000 q^{16} +117.000 q^{17} +24.2487 q^{19} -41.5692 q^{20} -31.1769 q^{21} -18.0000 q^{23} -98.0000 q^{25} -27.0000 q^{27} -83.1384 q^{28} -99.0000 q^{29} +193.990 q^{31} +155.885 q^{33} +54.0000 q^{35} -72.0000 q^{36} -112.583 q^{37} -36.3731 q^{41} +82.0000 q^{43} +415.692 q^{44} +46.7654 q^{45} +72.7461 q^{47} -192.000 q^{48} -235.000 q^{49} -351.000 q^{51} -261.000 q^{53} -270.000 q^{55} -72.7461 q^{57} -789.815 q^{59} +124.708 q^{60} -719.000 q^{61} +93.5307 q^{63} -512.000 q^{64} -703.213 q^{67} -936.000 q^{68} +54.0000 q^{69} +467.654 q^{71} -684.160 q^{73} +294.000 q^{75} -193.990 q^{76} -540.000 q^{77} -440.000 q^{79} +332.554 q^{80} +81.0000 q^{81} +1195.12 q^{83} +249.415 q^{84} +607.950 q^{85} +297.000 q^{87} -1517.28 q^{89} +144.000 q^{92} -581.969 q^{93} +126.000 q^{95} +1157.01 q^{97} -467.654 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 16 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 16 q^{4} + 18 q^{9} + 48 q^{12} + 128 q^{16} + 234 q^{17} - 36 q^{23} - 196 q^{25} - 54 q^{27} - 198 q^{29} + 108 q^{35} - 144 q^{36} + 164 q^{43} - 384 q^{48} - 470 q^{49} - 702 q^{51} - 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} - 1872 q^{68} + 108 q^{69} + 588 q^{75} - 1080 q^{77} - 880 q^{79} + 162 q^{81} + 594 q^{87} + 288 q^{92} + 252 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −3.00000 −0.577350
\(4\) −8.00000 −1.00000
\(5\) 5.19615 0.464758 0.232379 0.972625i \(-0.425349\pi\)
0.232379 + 0.972625i \(0.425349\pi\)
\(6\) 0 0
\(7\) 10.3923 0.561132 0.280566 0.959835i \(-0.409478\pi\)
0.280566 + 0.959835i \(0.409478\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −51.9615 −1.42427 −0.712136 0.702042i \(-0.752272\pi\)
−0.712136 + 0.702042i \(0.752272\pi\)
\(12\) 24.0000 0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) −15.5885 −0.268328
\(16\) 64.0000 1.00000
\(17\) 117.000 1.66922 0.834608 0.550845i \(-0.185694\pi\)
0.834608 + 0.550845i \(0.185694\pi\)
\(18\) 0 0
\(19\) 24.2487 0.292791 0.146396 0.989226i \(-0.453233\pi\)
0.146396 + 0.989226i \(0.453233\pi\)
\(20\) −41.5692 −0.464758
\(21\) −31.1769 −0.323970
\(22\) 0 0
\(23\) −18.0000 −0.163185 −0.0815926 0.996666i \(-0.526001\pi\)
−0.0815926 + 0.996666i \(0.526001\pi\)
\(24\) 0 0
\(25\) −98.0000 −0.784000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) −83.1384 −0.561132
\(29\) −99.0000 −0.633925 −0.316963 0.948438i \(-0.602663\pi\)
−0.316963 + 0.948438i \(0.602663\pi\)
\(30\) 0 0
\(31\) 193.990 1.12392 0.561961 0.827164i \(-0.310047\pi\)
0.561961 + 0.827164i \(0.310047\pi\)
\(32\) 0 0
\(33\) 155.885 0.822304
\(34\) 0 0
\(35\) 54.0000 0.260790
\(36\) −72.0000 −0.333333
\(37\) −112.583 −0.500232 −0.250116 0.968216i \(-0.580469\pi\)
−0.250116 + 0.968216i \(0.580469\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −36.3731 −0.138549 −0.0692746 0.997598i \(-0.522068\pi\)
−0.0692746 + 0.997598i \(0.522068\pi\)
\(42\) 0 0
\(43\) 82.0000 0.290811 0.145406 0.989372i \(-0.453551\pi\)
0.145406 + 0.989372i \(0.453551\pi\)
\(44\) 415.692 1.42427
\(45\) 46.7654 0.154919
\(46\) 0 0
\(47\) 72.7461 0.225768 0.112884 0.993608i \(-0.463991\pi\)
0.112884 + 0.993608i \(0.463991\pi\)
\(48\) −192.000 −0.577350
\(49\) −235.000 −0.685131
\(50\) 0 0
\(51\) −351.000 −0.963722
\(52\) 0 0
\(53\) −261.000 −0.676436 −0.338218 0.941068i \(-0.609824\pi\)
−0.338218 + 0.941068i \(0.609824\pi\)
\(54\) 0 0
\(55\) −270.000 −0.661942
\(56\) 0 0
\(57\) −72.7461 −0.169043
\(58\) 0 0
\(59\) −789.815 −1.74280 −0.871400 0.490574i \(-0.836787\pi\)
−0.871400 + 0.490574i \(0.836787\pi\)
\(60\) 124.708 0.268328
\(61\) −719.000 −1.50916 −0.754578 0.656210i \(-0.772158\pi\)
−0.754578 + 0.656210i \(0.772158\pi\)
\(62\) 0 0
\(63\) 93.5307 0.187044
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −703.213 −1.28226 −0.641128 0.767434i \(-0.721533\pi\)
−0.641128 + 0.767434i \(0.721533\pi\)
\(68\) −936.000 −1.66922
\(69\) 54.0000 0.0942150
\(70\) 0 0
\(71\) 467.654 0.781694 0.390847 0.920456i \(-0.372182\pi\)
0.390847 + 0.920456i \(0.372182\pi\)
\(72\) 0 0
\(73\) −684.160 −1.09692 −0.548458 0.836178i \(-0.684785\pi\)
−0.548458 + 0.836178i \(0.684785\pi\)
\(74\) 0 0
\(75\) 294.000 0.452643
\(76\) −193.990 −0.292791
\(77\) −540.000 −0.799204
\(78\) 0 0
\(79\) −440.000 −0.626631 −0.313316 0.949649i \(-0.601440\pi\)
−0.313316 + 0.949649i \(0.601440\pi\)
\(80\) 332.554 0.464758
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1195.12 1.58049 0.790247 0.612789i \(-0.209952\pi\)
0.790247 + 0.612789i \(0.209952\pi\)
\(84\) 249.415 0.323970
\(85\) 607.950 0.775781
\(86\) 0 0
\(87\) 297.000 0.365997
\(88\) 0 0
\(89\) −1517.28 −1.80709 −0.903545 0.428493i \(-0.859045\pi\)
−0.903545 + 0.428493i \(0.859045\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 144.000 0.163185
\(93\) −581.969 −0.648897
\(94\) 0 0
\(95\) 126.000 0.136077
\(96\) 0 0
\(97\) 1157.01 1.21110 0.605549 0.795808i \(-0.292953\pi\)
0.605549 + 0.795808i \(0.292953\pi\)
\(98\) 0 0
\(99\) −467.654 −0.474757
\(100\) 784.000 0.784000
\(101\) −1575.00 −1.55167 −0.775833 0.630938i \(-0.782670\pi\)
−0.775833 + 0.630938i \(0.782670\pi\)
\(102\) 0 0
\(103\) −794.000 −0.759565 −0.379782 0.925076i \(-0.624001\pi\)
−0.379782 + 0.925076i \(0.624001\pi\)
\(104\) 0 0
\(105\) −162.000 −0.150567
\(106\) 0 0
\(107\) 450.000 0.406571 0.203286 0.979119i \(-0.434838\pi\)
0.203286 + 0.979119i \(0.434838\pi\)
\(108\) 216.000 0.192450
\(109\) 595.825 0.523576 0.261788 0.965125i \(-0.415688\pi\)
0.261788 + 0.965125i \(0.415688\pi\)
\(110\) 0 0
\(111\) 337.750 0.288809
\(112\) 665.108 0.561132
\(113\) −1701.00 −1.41608 −0.708038 0.706174i \(-0.750420\pi\)
−0.708038 + 0.706174i \(0.750420\pi\)
\(114\) 0 0
\(115\) −93.5307 −0.0758416
\(116\) 792.000 0.633925
\(117\) 0 0
\(118\) 0 0
\(119\) 1215.90 0.936650
\(120\) 0 0
\(121\) 1369.00 1.02855
\(122\) 0 0
\(123\) 109.119 0.0799914
\(124\) −1551.92 −1.12392
\(125\) −1158.74 −0.829128
\(126\) 0 0
\(127\) 1664.00 1.16265 0.581323 0.813673i \(-0.302535\pi\)
0.581323 + 0.813673i \(0.302535\pi\)
\(128\) 0 0
\(129\) −246.000 −0.167900
\(130\) 0 0
\(131\) −1476.00 −0.984418 −0.492209 0.870477i \(-0.663810\pi\)
−0.492209 + 0.870477i \(0.663810\pi\)
\(132\) −1247.08 −0.822304
\(133\) 252.000 0.164295
\(134\) 0 0
\(135\) −140.296 −0.0894427
\(136\) 0 0
\(137\) 1013.25 0.631882 0.315941 0.948779i \(-0.397680\pi\)
0.315941 + 0.948779i \(0.397680\pi\)
\(138\) 0 0
\(139\) 1124.00 0.685874 0.342937 0.939358i \(-0.388578\pi\)
0.342937 + 0.939358i \(0.388578\pi\)
\(140\) −432.000 −0.260790
\(141\) −218.238 −0.130347
\(142\) 0 0
\(143\) 0 0
\(144\) 576.000 0.333333
\(145\) −514.419 −0.294622
\(146\) 0 0
\(147\) 705.000 0.395561
\(148\) 900.666 0.500232
\(149\) −3268.38 −1.79702 −0.898510 0.438952i \(-0.855350\pi\)
−0.898510 + 0.438952i \(0.855350\pi\)
\(150\) 0 0
\(151\) 1638.52 0.883052 0.441526 0.897248i \(-0.354437\pi\)
0.441526 + 0.897248i \(0.354437\pi\)
\(152\) 0 0
\(153\) 1053.00 0.556405
\(154\) 0 0
\(155\) 1008.00 0.522352
\(156\) 0 0
\(157\) 1259.00 0.639995 0.319997 0.947418i \(-0.396318\pi\)
0.319997 + 0.947418i \(0.396318\pi\)
\(158\) 0 0
\(159\) 783.000 0.390540
\(160\) 0 0
\(161\) −187.061 −0.0915684
\(162\) 0 0
\(163\) 2951.41 1.41824 0.709118 0.705089i \(-0.249093\pi\)
0.709118 + 0.705089i \(0.249093\pi\)
\(164\) 290.985 0.138549
\(165\) 810.000 0.382172
\(166\) 0 0
\(167\) 3138.48 1.45427 0.727133 0.686496i \(-0.240852\pi\)
0.727133 + 0.686496i \(0.240852\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 218.238 0.0975971
\(172\) −656.000 −0.290811
\(173\) −4266.00 −1.87479 −0.937393 0.348273i \(-0.886768\pi\)
−0.937393 + 0.348273i \(0.886768\pi\)
\(174\) 0 0
\(175\) −1018.45 −0.439927
\(176\) −3325.54 −1.42427
\(177\) 2369.45 1.00621
\(178\) 0 0
\(179\) −3006.00 −1.25519 −0.627595 0.778540i \(-0.715961\pi\)
−0.627595 + 0.778540i \(0.715961\pi\)
\(180\) −374.123 −0.154919
\(181\) 1873.00 0.769166 0.384583 0.923090i \(-0.374345\pi\)
0.384583 + 0.923090i \(0.374345\pi\)
\(182\) 0 0
\(183\) 2157.00 0.871312
\(184\) 0 0
\(185\) −585.000 −0.232487
\(186\) 0 0
\(187\) −6079.50 −2.37742
\(188\) −581.969 −0.225768
\(189\) −280.592 −0.107990
\(190\) 0 0
\(191\) −2736.00 −1.03649 −0.518246 0.855232i \(-0.673415\pi\)
−0.518246 + 0.855232i \(0.673415\pi\)
\(192\) 1536.00 0.577350
\(193\) 2603.27 0.970920 0.485460 0.874259i \(-0.338652\pi\)
0.485460 + 0.874259i \(0.338652\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1880.00 0.685131
\(197\) 3720.45 1.34554 0.672768 0.739853i \(-0.265105\pi\)
0.672768 + 0.739853i \(0.265105\pi\)
\(198\) 0 0
\(199\) −1198.00 −0.426754 −0.213377 0.976970i \(-0.568446\pi\)
−0.213377 + 0.976970i \(0.568446\pi\)
\(200\) 0 0
\(201\) 2109.64 0.740310
\(202\) 0 0
\(203\) −1028.84 −0.355716
\(204\) 2808.00 0.963722
\(205\) −189.000 −0.0643919
\(206\) 0 0
\(207\) −162.000 −0.0543951
\(208\) 0 0
\(209\) −1260.00 −0.417014
\(210\) 0 0
\(211\) 2392.00 0.780436 0.390218 0.920722i \(-0.372400\pi\)
0.390218 + 0.920722i \(0.372400\pi\)
\(212\) 2088.00 0.676436
\(213\) −1402.96 −0.451311
\(214\) 0 0
\(215\) 426.084 0.135157
\(216\) 0 0
\(217\) 2016.00 0.630668
\(218\) 0 0
\(219\) 2052.48 0.633305
\(220\) 2160.00 0.661942
\(221\) 0 0
\(222\) 0 0
\(223\) −2036.89 −0.611661 −0.305830 0.952086i \(-0.598934\pi\)
−0.305830 + 0.952086i \(0.598934\pi\)
\(224\) 0 0
\(225\) −882.000 −0.261333
\(226\) 0 0
\(227\) −2151.21 −0.628990 −0.314495 0.949259i \(-0.601835\pi\)
−0.314495 + 0.949259i \(0.601835\pi\)
\(228\) 581.969 0.169043
\(229\) −3471.03 −1.00162 −0.500812 0.865556i \(-0.666965\pi\)
−0.500812 + 0.865556i \(0.666965\pi\)
\(230\) 0 0
\(231\) 1620.00 0.461421
\(232\) 0 0
\(233\) 1854.00 0.521286 0.260643 0.965435i \(-0.416065\pi\)
0.260643 + 0.965435i \(0.416065\pi\)
\(234\) 0 0
\(235\) 378.000 0.104928
\(236\) 6318.52 1.74280
\(237\) 1320.00 0.361786
\(238\) 0 0
\(239\) −4458.30 −1.20662 −0.603312 0.797505i \(-0.706153\pi\)
−0.603312 + 0.797505i \(0.706153\pi\)
\(240\) −997.661 −0.268328
\(241\) −417.424 −0.111571 −0.0557856 0.998443i \(-0.517766\pi\)
−0.0557856 + 0.998443i \(0.517766\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 5752.00 1.50916
\(245\) −1221.10 −0.318420
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −3585.35 −0.912498
\(250\) 0 0
\(251\) −4104.00 −1.03204 −0.516020 0.856576i \(-0.672587\pi\)
−0.516020 + 0.856576i \(0.672587\pi\)
\(252\) −748.246 −0.187044
\(253\) 935.307 0.232420
\(254\) 0 0
\(255\) −1823.85 −0.447898
\(256\) 4096.00 1.00000
\(257\) −1989.00 −0.482764 −0.241382 0.970430i \(-0.577601\pi\)
−0.241382 + 0.970430i \(0.577601\pi\)
\(258\) 0 0
\(259\) −1170.00 −0.280696
\(260\) 0 0
\(261\) −891.000 −0.211308
\(262\) 0 0
\(263\) 738.000 0.173031 0.0865153 0.996251i \(-0.472427\pi\)
0.0865153 + 0.996251i \(0.472427\pi\)
\(264\) 0 0
\(265\) −1356.20 −0.314379
\(266\) 0 0
\(267\) 4551.83 1.04332
\(268\) 5625.70 1.28226
\(269\) −2106.00 −0.477342 −0.238671 0.971100i \(-0.576712\pi\)
−0.238671 + 0.971100i \(0.576712\pi\)
\(270\) 0 0
\(271\) −685.892 −0.153745 −0.0768727 0.997041i \(-0.524493\pi\)
−0.0768727 + 0.997041i \(0.524493\pi\)
\(272\) 7488.00 1.66922
\(273\) 0 0
\(274\) 0 0
\(275\) 5092.23 1.11663
\(276\) −432.000 −0.0942150
\(277\) −3665.00 −0.794977 −0.397488 0.917607i \(-0.630118\pi\)
−0.397488 + 0.917607i \(0.630118\pi\)
\(278\) 0 0
\(279\) 1745.91 0.374641
\(280\) 0 0
\(281\) 1719.93 0.365132 0.182566 0.983194i \(-0.441560\pi\)
0.182566 + 0.983194i \(0.441560\pi\)
\(282\) 0 0
\(283\) 1826.00 0.383549 0.191775 0.981439i \(-0.438576\pi\)
0.191775 + 0.981439i \(0.438576\pi\)
\(284\) −3741.23 −0.781694
\(285\) −378.000 −0.0785642
\(286\) 0 0
\(287\) −378.000 −0.0777444
\(288\) 0 0
\(289\) 8776.00 1.78628
\(290\) 0 0
\(291\) −3471.03 −0.699228
\(292\) 5473.28 1.09692
\(293\) −504.027 −0.100497 −0.0502484 0.998737i \(-0.516001\pi\)
−0.0502484 + 0.998737i \(0.516001\pi\)
\(294\) 0 0
\(295\) −4104.00 −0.809980
\(296\) 0 0
\(297\) 1402.96 0.274101
\(298\) 0 0
\(299\) 0 0
\(300\) −2352.00 −0.452643
\(301\) 852.169 0.163183
\(302\) 0 0
\(303\) 4725.00 0.895855
\(304\) 1551.92 0.292791
\(305\) −3736.03 −0.701392
\(306\) 0 0
\(307\) −1950.29 −0.362570 −0.181285 0.983431i \(-0.558026\pi\)
−0.181285 + 0.983431i \(0.558026\pi\)
\(308\) 4320.00 0.799204
\(309\) 2382.00 0.438535
\(310\) 0 0
\(311\) 3798.00 0.692491 0.346246 0.938144i \(-0.387456\pi\)
0.346246 + 0.938144i \(0.387456\pi\)
\(312\) 0 0
\(313\) 1378.00 0.248847 0.124424 0.992229i \(-0.460292\pi\)
0.124424 + 0.992229i \(0.460292\pi\)
\(314\) 0 0
\(315\) 486.000 0.0869302
\(316\) 3520.00 0.626631
\(317\) 7103.14 1.25852 0.629262 0.777193i \(-0.283357\pi\)
0.629262 + 0.777193i \(0.283357\pi\)
\(318\) 0 0
\(319\) 5144.19 0.902882
\(320\) −2660.43 −0.464758
\(321\) −1350.00 −0.234734
\(322\) 0 0
\(323\) 2837.10 0.488732
\(324\) −648.000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) −1787.48 −0.302286
\(328\) 0 0
\(329\) 756.000 0.126686
\(330\) 0 0
\(331\) −10073.6 −1.67280 −0.836398 0.548122i \(-0.815343\pi\)
−0.836398 + 0.548122i \(0.815343\pi\)
\(332\) −9560.92 −1.58049
\(333\) −1013.25 −0.166744
\(334\) 0 0
\(335\) −3654.00 −0.595938
\(336\) −1995.32 −0.323970
\(337\) −9001.00 −1.45494 −0.727471 0.686138i \(-0.759305\pi\)
−0.727471 + 0.686138i \(0.759305\pi\)
\(338\) 0 0
\(339\) 5103.00 0.817572
\(340\) −4863.60 −0.775781
\(341\) −10080.0 −1.60077
\(342\) 0 0
\(343\) −6006.75 −0.945581
\(344\) 0 0
\(345\) 280.592 0.0437872
\(346\) 0 0
\(347\) 3294.00 0.509600 0.254800 0.966994i \(-0.417990\pi\)
0.254800 + 0.966994i \(0.417990\pi\)
\(348\) −2376.00 −0.365997
\(349\) 10544.7 1.61732 0.808662 0.588273i \(-0.200192\pi\)
0.808662 + 0.588273i \(0.200192\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2478.56 −0.373713 −0.186856 0.982387i \(-0.559830\pi\)
−0.186856 + 0.982387i \(0.559830\pi\)
\(354\) 0 0
\(355\) 2430.00 0.363299
\(356\) 12138.2 1.80709
\(357\) −3647.70 −0.540775
\(358\) 0 0
\(359\) 5414.39 0.795991 0.397995 0.917387i \(-0.369706\pi\)
0.397995 + 0.917387i \(0.369706\pi\)
\(360\) 0 0
\(361\) −6271.00 −0.914273
\(362\) 0 0
\(363\) −4107.00 −0.593834
\(364\) 0 0
\(365\) −3555.00 −0.509801
\(366\) 0 0
\(367\) −9946.00 −1.41465 −0.707326 0.706888i \(-0.750099\pi\)
−0.707326 + 0.706888i \(0.750099\pi\)
\(368\) −1152.00 −0.163185
\(369\) −327.358 −0.0461831
\(370\) 0 0
\(371\) −2712.39 −0.379570
\(372\) 4655.75 0.648897
\(373\) −7301.00 −1.01349 −0.506745 0.862096i \(-0.669151\pi\)
−0.506745 + 0.862096i \(0.669151\pi\)
\(374\) 0 0
\(375\) 3476.23 0.478697
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3422.53 −0.463862 −0.231931 0.972732i \(-0.574504\pi\)
−0.231931 + 0.972732i \(0.574504\pi\)
\(380\) −1008.00 −0.136077
\(381\) −4992.00 −0.671254
\(382\) 0 0
\(383\) 5778.12 0.770883 0.385442 0.922732i \(-0.374049\pi\)
0.385442 + 0.922732i \(0.374049\pi\)
\(384\) 0 0
\(385\) −2805.92 −0.371436
\(386\) 0 0
\(387\) 738.000 0.0969371
\(388\) −9256.08 −1.21110
\(389\) −9153.00 −1.19300 −0.596498 0.802614i \(-0.703442\pi\)
−0.596498 + 0.802614i \(0.703442\pi\)
\(390\) 0 0
\(391\) −2106.00 −0.272391
\(392\) 0 0
\(393\) 4428.00 0.568354
\(394\) 0 0
\(395\) −2286.31 −0.291232
\(396\) 3741.23 0.474757
\(397\) 2023.04 0.255751 0.127876 0.991790i \(-0.459184\pi\)
0.127876 + 0.991790i \(0.459184\pi\)
\(398\) 0 0
\(399\) −756.000 −0.0948555
\(400\) −6272.00 −0.784000
\(401\) −8308.65 −1.03470 −0.517349 0.855774i \(-0.673081\pi\)
−0.517349 + 0.855774i \(0.673081\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 12600.0 1.55167
\(405\) 420.888 0.0516398
\(406\) 0 0
\(407\) 5850.00 0.712466
\(408\) 0 0
\(409\) 10418.3 1.25954 0.629769 0.776782i \(-0.283150\pi\)
0.629769 + 0.776782i \(0.283150\pi\)
\(410\) 0 0
\(411\) −3039.75 −0.364817
\(412\) 6352.00 0.759565
\(413\) −8208.00 −0.977940
\(414\) 0 0
\(415\) 6210.00 0.734547
\(416\) 0 0
\(417\) −3372.00 −0.395989
\(418\) 0 0
\(419\) 4176.00 0.486900 0.243450 0.969913i \(-0.421721\pi\)
0.243450 + 0.969913i \(0.421721\pi\)
\(420\) 1296.00 0.150567
\(421\) 14471.3 1.67527 0.837633 0.546233i \(-0.183939\pi\)
0.837633 + 0.546233i \(0.183939\pi\)
\(422\) 0 0
\(423\) 654.715 0.0752561
\(424\) 0 0
\(425\) −11466.0 −1.30867
\(426\) 0 0
\(427\) −7472.07 −0.846835
\(428\) −3600.00 −0.406571
\(429\) 0 0
\(430\) 0 0
\(431\) −6578.33 −0.735190 −0.367595 0.929986i \(-0.619819\pi\)
−0.367595 + 0.929986i \(0.619819\pi\)
\(432\) −1728.00 −0.192450
\(433\) 6605.00 0.733062 0.366531 0.930406i \(-0.380545\pi\)
0.366531 + 0.930406i \(0.380545\pi\)
\(434\) 0 0
\(435\) 1543.26 0.170100
\(436\) −4766.60 −0.523576
\(437\) −436.477 −0.0477792
\(438\) 0 0
\(439\) 8542.00 0.928673 0.464336 0.885659i \(-0.346293\pi\)
0.464336 + 0.885659i \(0.346293\pi\)
\(440\) 0 0
\(441\) −2115.00 −0.228377
\(442\) 0 0
\(443\) −14328.0 −1.53667 −0.768334 0.640049i \(-0.778914\pi\)
−0.768334 + 0.640049i \(0.778914\pi\)
\(444\) −2702.00 −0.288809
\(445\) −7884.00 −0.839859
\(446\) 0 0
\(447\) 9805.14 1.03751
\(448\) −5320.86 −0.561132
\(449\) 3013.77 0.316767 0.158384 0.987378i \(-0.449372\pi\)
0.158384 + 0.987378i \(0.449372\pi\)
\(450\) 0 0
\(451\) 1890.00 0.197332
\(452\) 13608.0 1.41608
\(453\) −4915.56 −0.509830
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2887.33 −0.295544 −0.147772 0.989021i \(-0.547210\pi\)
−0.147772 + 0.989021i \(0.547210\pi\)
\(458\) 0 0
\(459\) −3159.00 −0.321241
\(460\) 748.246 0.0758416
\(461\) −3600.93 −0.363801 −0.181900 0.983317i \(-0.558225\pi\)
−0.181900 + 0.983317i \(0.558225\pi\)
\(462\) 0 0
\(463\) −2677.75 −0.268781 −0.134391 0.990928i \(-0.542908\pi\)
−0.134391 + 0.990928i \(0.542908\pi\)
\(464\) −6336.00 −0.633925
\(465\) −3024.00 −0.301580
\(466\) 0 0
\(467\) −13878.0 −1.37515 −0.687577 0.726111i \(-0.741326\pi\)
−0.687577 + 0.726111i \(0.741326\pi\)
\(468\) 0 0
\(469\) −7308.00 −0.719514
\(470\) 0 0
\(471\) −3777.00 −0.369501
\(472\) 0 0
\(473\) −4260.84 −0.414194
\(474\) 0 0
\(475\) −2376.37 −0.229548
\(476\) −9727.20 −0.936650
\(477\) −2349.00 −0.225479
\(478\) 0 0
\(479\) −1101.58 −0.105079 −0.0525393 0.998619i \(-0.516731\pi\)
−0.0525393 + 0.998619i \(0.516731\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 561.184 0.0528670
\(484\) −10952.0 −1.02855
\(485\) 6012.00 0.562868
\(486\) 0 0
\(487\) 17123.1 1.59326 0.796632 0.604464i \(-0.206613\pi\)
0.796632 + 0.604464i \(0.206613\pi\)
\(488\) 0 0
\(489\) −8854.24 −0.818820
\(490\) 0 0
\(491\) −450.000 −0.0413609 −0.0206805 0.999786i \(-0.506583\pi\)
−0.0206805 + 0.999786i \(0.506583\pi\)
\(492\) −872.954 −0.0799914
\(493\) −11583.0 −1.05816
\(494\) 0 0
\(495\) −2430.00 −0.220647
\(496\) 12415.3 1.12392
\(497\) 4860.00 0.438633
\(498\) 0 0
\(499\) 13219.0 1.18590 0.592950 0.805239i \(-0.297963\pi\)
0.592950 + 0.805239i \(0.297963\pi\)
\(500\) 9269.94 0.829128
\(501\) −9415.43 −0.839621
\(502\) 0 0
\(503\) 5346.00 0.473889 0.236945 0.971523i \(-0.423854\pi\)
0.236945 + 0.971523i \(0.423854\pi\)
\(504\) 0 0
\(505\) −8183.94 −0.721150
\(506\) 0 0
\(507\) 0 0
\(508\) −13312.0 −1.16265
\(509\) −5866.46 −0.510857 −0.255428 0.966828i \(-0.582217\pi\)
−0.255428 + 0.966828i \(0.582217\pi\)
\(510\) 0 0
\(511\) −7110.00 −0.615514
\(512\) 0 0
\(513\) −654.715 −0.0563477
\(514\) 0 0
\(515\) −4125.75 −0.353014
\(516\) 1968.00 0.167900
\(517\) −3780.00 −0.321556
\(518\) 0 0
\(519\) 12798.0 1.08241
\(520\) 0 0
\(521\) −9657.00 −0.812055 −0.406028 0.913861i \(-0.633086\pi\)
−0.406028 + 0.913861i \(0.633086\pi\)
\(522\) 0 0
\(523\) 21626.0 1.80811 0.904053 0.427421i \(-0.140578\pi\)
0.904053 + 0.427421i \(0.140578\pi\)
\(524\) 11808.0 0.984418
\(525\) 3055.34 0.253992
\(526\) 0 0
\(527\) 22696.8 1.87607
\(528\) 9976.61 0.822304
\(529\) −11843.0 −0.973371
\(530\) 0 0
\(531\) −7108.34 −0.580933
\(532\) −2016.00 −0.164295
\(533\) 0 0
\(534\) 0 0
\(535\) 2338.27 0.188957
\(536\) 0 0
\(537\) 9018.00 0.724684
\(538\) 0 0
\(539\) 12211.0 0.975813
\(540\) 1122.37 0.0894427
\(541\) 5371.09 0.426841 0.213421 0.976960i \(-0.431540\pi\)
0.213421 + 0.976960i \(0.431540\pi\)
\(542\) 0 0
\(543\) −5619.00 −0.444078
\(544\) 0 0
\(545\) 3096.00 0.243336
\(546\) 0 0
\(547\) 16946.0 1.32460 0.662302 0.749237i \(-0.269579\pi\)
0.662302 + 0.749237i \(0.269579\pi\)
\(548\) −8106.00 −0.631882
\(549\) −6471.00 −0.503052
\(550\) 0 0
\(551\) −2400.62 −0.185608
\(552\) 0 0
\(553\) −4572.61 −0.351623
\(554\) 0 0
\(555\) 1755.00 0.134226
\(556\) −8992.00 −0.685874
\(557\) −3860.74 −0.293689 −0.146845 0.989160i \(-0.546912\pi\)
−0.146845 + 0.989160i \(0.546912\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3456.00 0.260790
\(561\) 18238.5 1.37260
\(562\) 0 0
\(563\) −21672.0 −1.62232 −0.811160 0.584825i \(-0.801163\pi\)
−0.811160 + 0.584825i \(0.801163\pi\)
\(564\) 1745.91 0.130347
\(565\) −8838.66 −0.658133
\(566\) 0 0
\(567\) 841.777 0.0623480
\(568\) 0 0
\(569\) 1386.00 0.102116 0.0510581 0.998696i \(-0.483741\pi\)
0.0510581 + 0.998696i \(0.483741\pi\)
\(570\) 0 0
\(571\) −1162.00 −0.0851632 −0.0425816 0.999093i \(-0.513558\pi\)
−0.0425816 + 0.999093i \(0.513558\pi\)
\(572\) 0 0
\(573\) 8208.00 0.598419
\(574\) 0 0
\(575\) 1764.00 0.127937
\(576\) −4608.00 −0.333333
\(577\) −8045.38 −0.580474 −0.290237 0.956955i \(-0.593734\pi\)
−0.290237 + 0.956955i \(0.593734\pi\)
\(578\) 0 0
\(579\) −7809.82 −0.560561
\(580\) 4115.35 0.294622
\(581\) 12420.0 0.886865
\(582\) 0 0
\(583\) 13562.0 0.963429
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27622.7 −1.94227 −0.971135 0.238530i \(-0.923335\pi\)
−0.971135 + 0.238530i \(0.923335\pi\)
\(588\) −5640.00 −0.395561
\(589\) 4704.00 0.329075
\(590\) 0 0
\(591\) −11161.3 −0.776846
\(592\) −7205.33 −0.500232
\(593\) 275.396 0.0190711 0.00953555 0.999955i \(-0.496965\pi\)
0.00953555 + 0.999955i \(0.496965\pi\)
\(594\) 0 0
\(595\) 6318.00 0.435316
\(596\) 26147.0 1.79702
\(597\) 3594.00 0.246386
\(598\) 0 0
\(599\) 22356.0 1.52494 0.762472 0.647021i \(-0.223986\pi\)
0.762472 + 0.647021i \(0.223986\pi\)
\(600\) 0 0
\(601\) −18083.0 −1.22732 −0.613661 0.789569i \(-0.710304\pi\)
−0.613661 + 0.789569i \(0.710304\pi\)
\(602\) 0 0
\(603\) −6328.91 −0.427418
\(604\) −13108.2 −0.883052
\(605\) 7113.53 0.478027
\(606\) 0 0
\(607\) 5480.00 0.366435 0.183218 0.983072i \(-0.441349\pi\)
0.183218 + 0.983072i \(0.441349\pi\)
\(608\) 0 0
\(609\) 3086.51 0.205373
\(610\) 0 0
\(611\) 0 0
\(612\) −8424.00 −0.556405
\(613\) 17737.9 1.16872 0.584362 0.811493i \(-0.301345\pi\)
0.584362 + 0.811493i \(0.301345\pi\)
\(614\) 0 0
\(615\) 567.000 0.0371767
\(616\) 0 0
\(617\) −9867.49 −0.643842 −0.321921 0.946767i \(-0.604328\pi\)
−0.321921 + 0.946767i \(0.604328\pi\)
\(618\) 0 0
\(619\) −4115.35 −0.267221 −0.133611 0.991034i \(-0.542657\pi\)
−0.133611 + 0.991034i \(0.542657\pi\)
\(620\) −8064.00 −0.522352
\(621\) 486.000 0.0314050
\(622\) 0 0
\(623\) −15768.0 −1.01402
\(624\) 0 0
\(625\) 6229.00 0.398656
\(626\) 0 0
\(627\) 3780.00 0.240763
\(628\) −10072.0 −0.639995
\(629\) −13172.2 −0.834995
\(630\) 0 0
\(631\) −12664.8 −0.799011 −0.399506 0.916731i \(-0.630818\pi\)
−0.399506 + 0.916731i \(0.630818\pi\)
\(632\) 0 0
\(633\) −7176.00 −0.450585
\(634\) 0 0
\(635\) 8646.40 0.540349
\(636\) −6264.00 −0.390540
\(637\) 0 0
\(638\) 0 0
\(639\) 4208.88 0.260565
\(640\) 0 0
\(641\) 3789.00 0.233473 0.116737 0.993163i \(-0.462757\pi\)
0.116737 + 0.993163i \(0.462757\pi\)
\(642\) 0 0
\(643\) −16911.7 −1.03722 −0.518611 0.855010i \(-0.673551\pi\)
−0.518611 + 0.855010i \(0.673551\pi\)
\(644\) 1496.49 0.0915684
\(645\) −1278.25 −0.0780328
\(646\) 0 0
\(647\) 27792.0 1.68874 0.844371 0.535759i \(-0.179974\pi\)
0.844371 + 0.535759i \(0.179974\pi\)
\(648\) 0 0
\(649\) 41040.0 2.48222
\(650\) 0 0
\(651\) −6048.00 −0.364116
\(652\) −23611.3 −1.41824
\(653\) −594.000 −0.0355973 −0.0177986 0.999842i \(-0.505666\pi\)
−0.0177986 + 0.999842i \(0.505666\pi\)
\(654\) 0 0
\(655\) −7669.52 −0.457516
\(656\) −2327.88 −0.138549
\(657\) −6157.44 −0.365639
\(658\) 0 0
\(659\) 17748.0 1.04911 0.524555 0.851376i \(-0.324232\pi\)
0.524555 + 0.851376i \(0.324232\pi\)
\(660\) −6480.00 −0.382172
\(661\) −15791.1 −0.929203 −0.464601 0.885520i \(-0.653802\pi\)
−0.464601 + 0.885520i \(0.653802\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1309.43 0.0763572
\(666\) 0 0
\(667\) 1782.00 0.103447
\(668\) −25107.8 −1.45427
\(669\) 6110.68 0.353143
\(670\) 0 0
\(671\) 37360.3 2.14945
\(672\) 0 0
\(673\) −20933.0 −1.19897 −0.599486 0.800385i \(-0.704628\pi\)
−0.599486 + 0.800385i \(0.704628\pi\)
\(674\) 0 0
\(675\) 2646.00 0.150881
\(676\) 0 0
\(677\) 3402.00 0.193131 0.0965653 0.995327i \(-0.469214\pi\)
0.0965653 + 0.995327i \(0.469214\pi\)
\(678\) 0 0
\(679\) 12024.0 0.679586
\(680\) 0 0
\(681\) 6453.62 0.363147
\(682\) 0 0
\(683\) 24983.1 1.39964 0.699818 0.714321i \(-0.253264\pi\)
0.699818 + 0.714321i \(0.253264\pi\)
\(684\) −1745.91 −0.0975971
\(685\) 5265.00 0.293672
\(686\) 0 0
\(687\) 10413.1 0.578288
\(688\) 5248.00 0.290811
\(689\) 0 0
\(690\) 0 0
\(691\) 13866.8 0.763412 0.381706 0.924284i \(-0.375337\pi\)
0.381706 + 0.924284i \(0.375337\pi\)
\(692\) 34128.0 1.87479
\(693\) −4860.00 −0.266401
\(694\) 0 0
\(695\) 5840.48 0.318765
\(696\) 0 0
\(697\) −4255.65 −0.231269
\(698\) 0 0
\(699\) −5562.00 −0.300964
\(700\) 8147.57 0.439927
\(701\) 21906.0 1.18028 0.590141 0.807300i \(-0.299072\pi\)
0.590141 + 0.807300i \(0.299072\pi\)
\(702\) 0 0
\(703\) −2730.00 −0.146464
\(704\) 26604.3 1.42427
\(705\) −1134.00 −0.0605800
\(706\) 0 0
\(707\) −16367.9 −0.870690
\(708\) −18955.6 −1.00621
\(709\) −13057.9 −0.691680 −0.345840 0.938294i \(-0.612406\pi\)
−0.345840 + 0.938294i \(0.612406\pi\)
\(710\) 0 0
\(711\) −3960.00 −0.208877
\(712\) 0 0
\(713\) −3491.81 −0.183407
\(714\) 0 0
\(715\) 0 0
\(716\) 24048.0 1.25519
\(717\) 13374.9 0.696645
\(718\) 0 0
\(719\) 14220.0 0.737575 0.368788 0.929514i \(-0.379773\pi\)
0.368788 + 0.929514i \(0.379773\pi\)
\(720\) 2992.98 0.154919
\(721\) −8251.49 −0.426216
\(722\) 0 0
\(723\) 1252.27 0.0644157
\(724\) −14984.0 −0.769166
\(725\) 9702.00 0.496998
\(726\) 0 0
\(727\) 5282.00 0.269462 0.134731 0.990882i \(-0.456983\pi\)
0.134731 + 0.990882i \(0.456983\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 9594.00 0.485427
\(732\) −17256.0 −0.871312
\(733\) 11419.4 0.575424 0.287712 0.957717i \(-0.407105\pi\)
0.287712 + 0.957717i \(0.407105\pi\)
\(734\) 0 0
\(735\) 3663.29 0.183840
\(736\) 0 0
\(737\) 36540.0 1.82628
\(738\) 0 0
\(739\) 20535.2 1.02219 0.511096 0.859524i \(-0.329240\pi\)
0.511096 + 0.859524i \(0.329240\pi\)
\(740\) 4680.00 0.232487
\(741\) 0 0
\(742\) 0 0
\(743\) −20826.2 −1.02832 −0.514158 0.857696i \(-0.671895\pi\)
−0.514158 + 0.857696i \(0.671895\pi\)
\(744\) 0 0
\(745\) −16983.0 −0.835180
\(746\) 0 0
\(747\) 10756.0 0.526831
\(748\) 48636.0 2.37742
\(749\) 4676.54 0.228140
\(750\) 0 0
\(751\) 4834.00 0.234880 0.117440 0.993080i \(-0.462531\pi\)
0.117440 + 0.993080i \(0.462531\pi\)
\(752\) 4655.75 0.225768
\(753\) 12312.0 0.595849
\(754\) 0 0
\(755\) 8514.00 0.410406
\(756\) 2244.74 0.107990
\(757\) 9046.00 0.434323 0.217161 0.976136i \(-0.430320\pi\)
0.217161 + 0.976136i \(0.430320\pi\)
\(758\) 0 0
\(759\) −2805.92 −0.134188
\(760\) 0 0
\(761\) −12034.3 −0.573249 −0.286625 0.958043i \(-0.592533\pi\)
−0.286625 + 0.958043i \(0.592533\pi\)
\(762\) 0 0
\(763\) 6192.00 0.293795
\(764\) 21888.0 1.03649
\(765\) 5471.55 0.258594
\(766\) 0 0
\(767\) 0 0
\(768\) −12288.0 −0.577350
\(769\) 37543.9 1.76056 0.880279 0.474457i \(-0.157355\pi\)
0.880279 + 0.474457i \(0.157355\pi\)
\(770\) 0 0
\(771\) 5967.00 0.278724
\(772\) −20826.2 −0.970920
\(773\) 15713.2 0.731130 0.365565 0.930786i \(-0.380876\pi\)
0.365565 + 0.930786i \(0.380876\pi\)
\(774\) 0 0
\(775\) −19011.0 −0.881155
\(776\) 0 0
\(777\) 3510.00 0.162060
\(778\) 0 0
\(779\) −882.000 −0.0405660
\(780\) 0 0
\(781\) −24300.0 −1.11334
\(782\) 0 0
\(783\) 2673.00 0.121999
\(784\) −15040.0 −0.685131
\(785\) 6541.96 0.297443
\(786\) 0 0
\(787\) −3755.09 −0.170082 −0.0850409 0.996377i \(-0.527102\pi\)
−0.0850409 + 0.996377i \(0.527102\pi\)
\(788\) −29763.6 −1.34554
\(789\) −2214.00 −0.0998992
\(790\) 0 0
\(791\) −17677.3 −0.794605
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 4068.59 0.181507
\(796\) 9584.00 0.426754
\(797\) 7830.00 0.347996 0.173998 0.984746i \(-0.444331\pi\)
0.173998 + 0.984746i \(0.444331\pi\)
\(798\) 0 0
\(799\) 8511.30 0.376856
\(800\) 0 0
\(801\) −13655.5 −0.602363
\(802\) 0 0
\(803\) 35550.0 1.56231
\(804\) −16877.1 −0.740310
\(805\) −972.000 −0.0425571
\(806\) 0 0
\(807\) 6318.00 0.275594
\(808\) 0 0
\(809\) −6165.00 −0.267923 −0.133962 0.990987i \(-0.542770\pi\)
−0.133962 + 0.990987i \(0.542770\pi\)
\(810\) 0 0
\(811\) −29839.8 −1.29201 −0.646003 0.763335i \(-0.723560\pi\)
−0.646003 + 0.763335i \(0.723560\pi\)
\(812\) 8230.71 0.355716
\(813\) 2057.68 0.0887649
\(814\) 0 0
\(815\) 15336.0 0.659137
\(816\) −22464.0 −0.963722
\(817\) 1988.39 0.0851470
\(818\) 0 0
\(819\) 0 0
\(820\) 1512.00 0.0643919
\(821\) −29763.6 −1.26523 −0.632616 0.774466i \(-0.718019\pi\)
−0.632616 + 0.774466i \(0.718019\pi\)
\(822\) 0 0
\(823\) 8920.00 0.377803 0.188901 0.981996i \(-0.439507\pi\)
0.188901 + 0.981996i \(0.439507\pi\)
\(824\) 0 0
\(825\) −15276.7 −0.644686
\(826\) 0 0
\(827\) 18041.0 0.758583 0.379292 0.925277i \(-0.376168\pi\)
0.379292 + 0.925277i \(0.376168\pi\)
\(828\) 1296.00 0.0543951
\(829\) 21023.0 0.880771 0.440385 0.897809i \(-0.354842\pi\)
0.440385 + 0.897809i \(0.354842\pi\)
\(830\) 0 0
\(831\) 10995.0 0.458980
\(832\) 0 0
\(833\) −27495.0 −1.14363
\(834\) 0 0
\(835\) 16308.0 0.675882
\(836\) 10080.0 0.417014
\(837\) −5237.72 −0.216299
\(838\) 0 0
\(839\) −22073.3 −0.908288 −0.454144 0.890928i \(-0.650055\pi\)
−0.454144 + 0.890928i \(0.650055\pi\)
\(840\) 0 0
\(841\) −14588.0 −0.598139
\(842\) 0 0
\(843\) −5159.78 −0.210809
\(844\) −19136.0 −0.780436
\(845\) 0 0
\(846\) 0 0
\(847\) 14227.1 0.577152
\(848\) −16704.0 −0.676436
\(849\) −5478.00 −0.221442
\(850\) 0 0
\(851\) 2026.50 0.0816304
\(852\) 11223.7 0.451311
\(853\) 26609.5 1.06810 0.534051 0.845452i \(-0.320669\pi\)
0.534051 + 0.845452i \(0.320669\pi\)
\(854\) 0 0
\(855\) 1134.00 0.0453590
\(856\) 0 0
\(857\) 12771.0 0.509042 0.254521 0.967067i \(-0.418082\pi\)
0.254521 + 0.967067i \(0.418082\pi\)
\(858\) 0 0
\(859\) 17134.0 0.680564 0.340282 0.940323i \(-0.389477\pi\)
0.340282 + 0.940323i \(0.389477\pi\)
\(860\) −3408.68 −0.135157
\(861\) 1134.00 0.0448857
\(862\) 0 0
\(863\) −7929.33 −0.312766 −0.156383 0.987696i \(-0.549983\pi\)
−0.156383 + 0.987696i \(0.549983\pi\)
\(864\) 0 0
\(865\) −22166.8 −0.871322
\(866\) 0 0
\(867\) −26328.0 −1.03131
\(868\) −16128.0 −0.630668
\(869\) 22863.1 0.892493
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10413.1 0.403700
\(874\) 0 0
\(875\) −12042.0 −0.465250
\(876\) −16419.8 −0.633305
\(877\) −9864.03 −0.379800 −0.189900 0.981803i \(-0.560816\pi\)
−0.189900 + 0.981803i \(0.560816\pi\)
\(878\) 0 0
\(879\) 1512.08 0.0580218
\(880\) −17280.0 −0.661942
\(881\) −29169.0 −1.11547 −0.557735 0.830019i \(-0.688329\pi\)
−0.557735 + 0.830019i \(0.688329\pi\)
\(882\) 0 0
\(883\) −928.000 −0.0353677 −0.0176839 0.999844i \(-0.505629\pi\)
−0.0176839 + 0.999844i \(0.505629\pi\)
\(884\) 0 0
\(885\) 12312.0 0.467642
\(886\) 0 0
\(887\) −14400.0 −0.545101 −0.272551 0.962141i \(-0.587867\pi\)
−0.272551 + 0.962141i \(0.587867\pi\)
\(888\) 0 0
\(889\) 17292.8 0.652398
\(890\) 0 0
\(891\) −4208.88 −0.158252
\(892\) 16295.1 0.611661
\(893\) 1764.00 0.0661030
\(894\) 0 0
\(895\) −15619.6 −0.583360
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −19205.0 −0.712483
\(900\) 7056.00 0.261333
\(901\) −30537.0 −1.12912
\(902\) 0 0
\(903\) −2556.51 −0.0942140
\(904\) 0 0
\(905\) 9732.39 0.357476
\(906\) 0 0
\(907\) −19684.0 −0.720614 −0.360307 0.932834i \(-0.617328\pi\)
−0.360307 + 0.932834i \(0.617328\pi\)
\(908\) 17209.7 0.628990
\(909\) −14175.0 −0.517222
\(910\) 0 0
\(911\) 24480.0 0.890295 0.445147 0.895457i \(-0.353151\pi\)
0.445147 + 0.895457i \(0.353151\pi\)
\(912\) −4655.75 −0.169043
\(913\) −62100.0 −2.25105
\(914\) 0 0
\(915\) 11208.1 0.404949
\(916\) 27768.2 1.00162
\(917\) −15339.0 −0.552388
\(918\) 0 0
\(919\) 38608.0 1.38581 0.692906 0.721028i \(-0.256330\pi\)
0.692906 + 0.721028i \(0.256330\pi\)
\(920\) 0 0
\(921\) 5850.87 0.209330
\(922\) 0 0
\(923\) 0 0
\(924\) −12960.0 −0.461421
\(925\) 11033.2 0.392182
\(926\) 0 0
\(927\) −7146.00 −0.253188
\(928\) 0 0
\(929\) −10210.4 −0.360596 −0.180298 0.983612i \(-0.557706\pi\)
−0.180298 + 0.983612i \(0.557706\pi\)
\(930\) 0 0
\(931\) −5698.45 −0.200600
\(932\) −14832.0 −0.521286
\(933\) −11394.0 −0.399810
\(934\) 0 0
\(935\) −31590.0 −1.10492
\(936\) 0 0
\(937\) 28495.0 0.993480 0.496740 0.867899i \(-0.334530\pi\)
0.496740 + 0.867899i \(0.334530\pi\)
\(938\) 0 0
\(939\) −4134.00 −0.143672
\(940\) −3024.00 −0.104928
\(941\) 10724.9 0.371541 0.185771 0.982593i \(-0.440522\pi\)
0.185771 + 0.982593i \(0.440522\pi\)
\(942\) 0 0
\(943\) 654.715 0.0226092
\(944\) −50548.2 −1.74280
\(945\) −1458.00 −0.0501891
\(946\) 0 0
\(947\) −33962.1 −1.16538 −0.582692 0.812693i \(-0.698001\pi\)
−0.582692 + 0.812693i \(0.698001\pi\)
\(948\) −10560.0 −0.361786
\(949\) 0 0
\(950\) 0 0
\(951\) −21309.4 −0.726609
\(952\) 0 0
\(953\) −23814.0 −0.809456 −0.404728 0.914437i \(-0.632634\pi\)
−0.404728 + 0.914437i \(0.632634\pi\)
\(954\) 0 0
\(955\) −14216.7 −0.481718
\(956\) 35666.4 1.20662
\(957\) −15432.6 −0.521279
\(958\) 0 0
\(959\) 10530.0 0.354569
\(960\) 7981.29 0.268328
\(961\) 7841.00 0.263200
\(962\) 0 0
\(963\) 4050.00 0.135524
\(964\) 3339.39 0.111571
\(965\) 13527.0 0.451243
\(966\) 0 0
\(967\) 51549.3 1.71429 0.857143 0.515079i \(-0.172238\pi\)
0.857143 + 0.515079i \(0.172238\pi\)
\(968\) 0 0
\(969\) −8511.30 −0.282170
\(970\) 0 0
\(971\) −12312.0 −0.406911 −0.203456 0.979084i \(-0.565217\pi\)
−0.203456 + 0.979084i \(0.565217\pi\)
\(972\) 1944.00 0.0641500
\(973\) 11681.0 0.384865
\(974\) 0 0
\(975\) 0 0
\(976\) −46016.0 −1.50916
\(977\) 21538.1 0.705285 0.352642 0.935758i \(-0.385283\pi\)
0.352642 + 0.935758i \(0.385283\pi\)
\(978\) 0 0
\(979\) 78840.0 2.57379
\(980\) 9768.77 0.318420
\(981\) 5362.43 0.174525
\(982\) 0 0
\(983\) 32611.1 1.05812 0.529060 0.848585i \(-0.322545\pi\)
0.529060 + 0.848585i \(0.322545\pi\)
\(984\) 0 0
\(985\) 19332.0 0.625349
\(986\) 0 0
\(987\) −2268.00 −0.0731421
\(988\) 0 0
\(989\) −1476.00 −0.0474561
\(990\) 0 0
\(991\) −22330.0 −0.715778 −0.357889 0.933764i \(-0.616503\pi\)
−0.357889 + 0.933764i \(0.616503\pi\)
\(992\) 0 0
\(993\) 30220.8 0.965789
\(994\) 0 0
\(995\) −6224.99 −0.198337
\(996\) 28682.8 0.912498
\(997\) −24931.0 −0.791949 −0.395974 0.918262i \(-0.629593\pi\)
−0.395974 + 0.918262i \(0.629593\pi\)
\(998\) 0 0
\(999\) 3039.75 0.0962697
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 507.4.a.g.1.2 2
3.2 odd 2 1521.4.a.m.1.1 2
13.2 odd 12 39.4.j.a.4.1 2
13.5 odd 4 507.4.b.a.337.1 2
13.7 odd 12 39.4.j.a.10.1 yes 2
13.8 odd 4 507.4.b.a.337.2 2
13.12 even 2 inner 507.4.a.g.1.1 2
39.2 even 12 117.4.q.b.82.1 2
39.20 even 12 117.4.q.b.10.1 2
39.38 odd 2 1521.4.a.m.1.2 2
52.7 even 12 624.4.bv.a.49.1 2
52.15 even 12 624.4.bv.a.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.a.4.1 2 13.2 odd 12
39.4.j.a.10.1 yes 2 13.7 odd 12
117.4.q.b.10.1 2 39.20 even 12
117.4.q.b.82.1 2 39.2 even 12
507.4.a.g.1.1 2 13.12 even 2 inner
507.4.a.g.1.2 2 1.1 even 1 trivial
507.4.b.a.337.1 2 13.5 odd 4
507.4.b.a.337.2 2 13.8 odd 4
624.4.bv.a.49.1 2 52.7 even 12
624.4.bv.a.433.1 2 52.15 even 12
1521.4.a.m.1.1 2 3.2 odd 2
1521.4.a.m.1.2 2 39.38 odd 2