Properties

Label 507.4.b.a.337.1
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.a.337.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +8.00000 q^{4} -5.19615i q^{5} +10.3923i q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +8.00000 q^{4} -5.19615i q^{5} +10.3923i q^{7} +9.00000 q^{9} -51.9615i q^{11} -24.0000 q^{12} +15.5885i q^{15} +64.0000 q^{16} -117.000 q^{17} -24.2487i q^{19} -41.5692i q^{20} -31.1769i q^{21} +18.0000 q^{23} +98.0000 q^{25} -27.0000 q^{27} +83.1384i q^{28} -99.0000 q^{29} -193.990i q^{31} +155.885i q^{33} +54.0000 q^{35} +72.0000 q^{36} -112.583i q^{37} +36.3731i q^{41} -82.0000 q^{43} -415.692i q^{44} -46.7654i q^{45} +72.7461i q^{47} -192.000 q^{48} +235.000 q^{49} +351.000 q^{51} -261.000 q^{53} -270.000 q^{55} +72.7461i q^{57} -789.815i q^{59} +124.708i q^{60} -719.000 q^{61} +93.5307i q^{63} +512.000 q^{64} +703.213i q^{67} -936.000 q^{68} -54.0000 q^{69} -467.654i q^{71} -684.160i q^{73} -294.000 q^{75} -193.990i q^{76} +540.000 q^{77} -440.000 q^{79} -332.554i q^{80} +81.0000 q^{81} -1195.12i q^{83} -249.415i q^{84} +607.950i q^{85} +297.000 q^{87} -1517.28i q^{89} +144.000 q^{92} +581.969i q^{93} -126.000 q^{95} -1157.01i q^{97} -467.654i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9}+O(q^{10}) \) 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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −3.00000 −0.577350
\(4\) 8.00000 1.00000
\(5\) − 5.19615i − 0.464758i −0.972625 0.232379i \(-0.925349\pi\)
0.972625 0.232379i \(-0.0746510\pi\)
\(6\) 0 0
\(7\) 10.3923i 0.561132i 0.959835 + 0.280566i \(0.0905221\pi\)
−0.959835 + 0.280566i \(0.909478\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) − 51.9615i − 1.42427i −0.702042 0.712136i \(-0.747728\pi\)
0.702042 0.712136i \(-0.252272\pi\)
\(12\) −24.0000 −0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) 15.5885i 0.268328i
\(16\) 64.0000 1.00000
\(17\) −117.000 −1.66922 −0.834608 0.550845i \(-0.814306\pi\)
−0.834608 + 0.550845i \(0.814306\pi\)
\(18\) 0 0
\(19\) − 24.2487i − 0.292791i −0.989226 0.146396i \(-0.953233\pi\)
0.989226 0.146396i \(-0.0467673\pi\)
\(20\) − 41.5692i − 0.464758i
\(21\) − 31.1769i − 0.323970i
\(22\) 0 0
\(23\) 18.0000 0.163185 0.0815926 0.996666i \(-0.473999\pi\)
0.0815926 + 0.996666i \(0.473999\pi\)
\(24\) 0 0
\(25\) 98.0000 0.784000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 83.1384i 0.561132i
\(29\) −99.0000 −0.633925 −0.316963 0.948438i \(-0.602663\pi\)
−0.316963 + 0.948438i \(0.602663\pi\)
\(30\) 0 0
\(31\) − 193.990i − 1.12392i −0.827164 0.561961i \(-0.810047\pi\)
0.827164 0.561961i \(-0.189953\pi\)
\(32\) 0 0
\(33\) 155.885i 0.822304i
\(34\) 0 0
\(35\) 54.0000 0.260790
\(36\) 72.0000 0.333333
\(37\) − 112.583i − 0.500232i −0.968216 0.250116i \(-0.919531\pi\)
0.968216 0.250116i \(-0.0804687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 36.3731i 0.138549i 0.997598 + 0.0692746i \(0.0220685\pi\)
−0.997598 + 0.0692746i \(0.977932\pi\)
\(42\) 0 0
\(43\) −82.0000 −0.290811 −0.145406 0.989372i \(-0.546449\pi\)
−0.145406 + 0.989372i \(0.546449\pi\)
\(44\) − 415.692i − 1.42427i
\(45\) − 46.7654i − 0.154919i
\(46\) 0 0
\(47\) 72.7461i 0.225768i 0.993608 + 0.112884i \(0.0360089\pi\)
−0.993608 + 0.112884i \(0.963991\pi\)
\(48\) −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
\(57\) 72.7461i 0.169043i
\(58\) 0 0
\(59\) − 789.815i − 1.74280i −0.490574 0.871400i \(-0.663213\pi\)
0.490574 0.871400i \(-0.336787\pi\)
\(60\) 124.708i 0.268328i
\(61\) −719.000 −1.50916 −0.754578 0.656210i \(-0.772158\pi\)
−0.754578 + 0.656210i \(0.772158\pi\)
\(62\) 0 0
\(63\) 93.5307i 0.187044i
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 703.213i 1.28226i 0.767434 + 0.641128i \(0.221533\pi\)
−0.767434 + 0.641128i \(0.778467\pi\)
\(68\) −936.000 −1.66922
\(69\) −54.0000 −0.0942150
\(70\) 0 0
\(71\) − 467.654i − 0.781694i −0.920456 0.390847i \(-0.872182\pi\)
0.920456 0.390847i \(-0.127818\pi\)
\(72\) 0 0
\(73\) − 684.160i − 1.09692i −0.836178 0.548458i \(-0.815215\pi\)
0.836178 0.548458i \(-0.184785\pi\)
\(74\) 0 0
\(75\) −294.000 −0.452643
\(76\) − 193.990i − 0.292791i
\(77\) 540.000 0.799204
\(78\) 0 0
\(79\) −440.000 −0.626631 −0.313316 0.949649i \(-0.601440\pi\)
−0.313316 + 0.949649i \(0.601440\pi\)
\(80\) − 332.554i − 0.464758i
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 1195.12i − 1.58049i −0.612789 0.790247i \(-0.709952\pi\)
0.612789 0.790247i \(-0.290048\pi\)
\(84\) − 249.415i − 0.323970i
\(85\) 607.950i 0.775781i
\(86\) 0 0
\(87\) 297.000 0.365997
\(88\) 0 0
\(89\) − 1517.28i − 1.80709i −0.428493 0.903545i \(-0.640955\pi\)
0.428493 0.903545i \(-0.359045\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 144.000 0.163185
\(93\) 581.969i 0.648897i
\(94\) 0 0
\(95\) −126.000 −0.136077
\(96\) 0 0
\(97\) − 1157.01i − 1.21110i −0.795808 0.605549i \(-0.792953\pi\)
0.795808 0.605549i \(-0.207047\pi\)
\(98\) 0 0
\(99\) − 467.654i − 0.474757i
\(100\) 784.000 0.784000
\(101\) 1575.00 1.55167 0.775833 0.630938i \(-0.217330\pi\)
0.775833 + 0.630938i \(0.217330\pi\)
\(102\) 0 0
\(103\) 794.000 0.759565 0.379782 0.925076i \(-0.375999\pi\)
0.379782 + 0.925076i \(0.375999\pi\)
\(104\) 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
\(109\) − 595.825i − 0.523576i −0.965125 0.261788i \(-0.915688\pi\)
0.965125 0.261788i \(-0.0843120\pi\)
\(110\) 0 0
\(111\) 337.750i 0.288809i
\(112\) 665.108i 0.561132i
\(113\) −1701.00 −1.41608 −0.708038 0.706174i \(-0.750420\pi\)
−0.708038 + 0.706174i \(0.750420\pi\)
\(114\) 0 0
\(115\) − 93.5307i − 0.0758416i
\(116\) −792.000 −0.633925
\(117\) 0 0
\(118\) 0 0
\(119\) − 1215.90i − 0.936650i
\(120\) 0 0
\(121\) −1369.00 −1.02855
\(122\) 0 0
\(123\) − 109.119i − 0.0799914i
\(124\) − 1551.92i − 1.12392i
\(125\) − 1158.74i − 0.829128i
\(126\) 0 0
\(127\) −1664.00 −1.16265 −0.581323 0.813673i \(-0.697465\pi\)
−0.581323 + 0.813673i \(0.697465\pi\)
\(128\) 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 \(0.663810\pi\)
\(132\) 1247.08i 0.822304i
\(133\) 252.000 0.164295
\(134\) 0 0
\(135\) 140.296i 0.0894427i
\(136\) 0 0
\(137\) 1013.25i 0.631882i 0.948779 + 0.315941i \(0.102320\pi\)
−0.948779 + 0.315941i \(0.897680\pi\)
\(138\) 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
\(141\) − 218.238i − 0.130347i
\(142\) 0 0
\(143\) 0 0
\(144\) 576.000 0.333333
\(145\) 514.419i 0.294622i
\(146\) 0 0
\(147\) −705.000 −0.395561
\(148\) − 900.666i − 0.500232i
\(149\) 3268.38i 1.79702i 0.438952 + 0.898510i \(0.355350\pi\)
−0.438952 + 0.898510i \(0.644650\pi\)
\(150\) 0 0
\(151\) 1638.52i 0.883052i 0.897248 + 0.441526i \(0.145563\pi\)
−0.897248 + 0.441526i \(0.854437\pi\)
\(152\) 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
\(161\) 187.061i 0.0915684i
\(162\) 0 0
\(163\) 2951.41i 1.41824i 0.705089 + 0.709118i \(0.250907\pi\)
−0.705089 + 0.709118i \(0.749093\pi\)
\(164\) 290.985i 0.138549i
\(165\) 810.000 0.382172
\(166\) 0 0
\(167\) 3138.48i 1.45427i 0.686496 + 0.727133i \(0.259148\pi\)
−0.686496 + 0.727133i \(0.740852\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 218.238i − 0.0975971i
\(172\) −656.000 −0.290811
\(173\) 4266.00 1.87479 0.937393 0.348273i \(-0.113232\pi\)
0.937393 + 0.348273i \(0.113232\pi\)
\(174\) 0 0
\(175\) 1018.45i 0.439927i
\(176\) − 3325.54i − 1.42427i
\(177\) 2369.45i 1.00621i
\(178\) 0 0
\(179\) 3006.00 1.25519 0.627595 0.778540i \(-0.284039\pi\)
0.627595 + 0.778540i \(0.284039\pi\)
\(180\) − 374.123i − 0.154919i
\(181\) −1873.00 −0.769166 −0.384583 0.923090i \(-0.625655\pi\)
−0.384583 + 0.923090i \(0.625655\pi\)
\(182\) 0 0
\(183\) 2157.00 0.871312
\(184\) 0 0
\(185\) −585.000 −0.232487
\(186\) 0 0
\(187\) 6079.50i 2.37742i
\(188\) 581.969i 0.225768i
\(189\) − 280.592i − 0.107990i
\(190\) 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
\(193\) 2603.27i 0.970920i 0.874259 + 0.485460i \(0.161348\pi\)
−0.874259 + 0.485460i \(0.838652\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1880.00 0.685131
\(197\) − 3720.45i − 1.34554i −0.739853 0.672768i \(-0.765105\pi\)
0.739853 0.672768i \(-0.234895\pi\)
\(198\) 0 0
\(199\) 1198.00 0.426754 0.213377 0.976970i \(-0.431554\pi\)
0.213377 + 0.976970i \(0.431554\pi\)
\(200\) 0 0
\(201\) − 2109.64i − 0.740310i
\(202\) 0 0
\(203\) − 1028.84i − 0.355716i
\(204\) 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
\(213\) 1402.96i 0.451311i
\(214\) 0 0
\(215\) 426.084i 0.135157i
\(216\) 0 0
\(217\) 2016.00 0.630668
\(218\) 0 0
\(219\) 2052.48i 0.633305i
\(220\) −2160.00 −0.661942
\(221\) 0 0
\(222\) 0 0
\(223\) 2036.89i 0.611661i 0.952086 + 0.305830i \(0.0989341\pi\)
−0.952086 + 0.305830i \(0.901066\pi\)
\(224\) 0 0
\(225\) 882.000 0.261333
\(226\) 0 0
\(227\) 2151.21i 0.628990i 0.949259 + 0.314495i \(0.101835\pi\)
−0.949259 + 0.314495i \(0.898165\pi\)
\(228\) 581.969i 0.169043i
\(229\) − 3471.03i − 1.00162i −0.865556 0.500812i \(-0.833035\pi\)
0.865556 0.500812i \(-0.166965\pi\)
\(230\) 0 0
\(231\) −1620.00 −0.461421
\(232\) 0 0
\(233\) −1854.00 −0.521286 −0.260643 0.965435i \(-0.583935\pi\)
−0.260643 + 0.965435i \(0.583935\pi\)
\(234\) 0 0
\(235\) 378.000 0.104928
\(236\) − 6318.52i − 1.74280i
\(237\) 1320.00 0.361786
\(238\) 0 0
\(239\) 4458.30i 1.20662i 0.797505 + 0.603312i \(0.206153\pi\)
−0.797505 + 0.603312i \(0.793847\pi\)
\(240\) 997.661i 0.268328i
\(241\) − 417.424i − 0.111571i −0.998443 0.0557856i \(-0.982234\pi\)
0.998443 0.0557856i \(-0.0177663\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) −5752.00 −1.50916
\(245\) − 1221.10i − 0.318420i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3585.35i 0.912498i
\(250\) 0 0
\(251\) 4104.00 1.03204 0.516020 0.856576i \(-0.327413\pi\)
0.516020 + 0.856576i \(0.327413\pi\)
\(252\) 748.246i 0.187044i
\(253\) − 935.307i − 0.232420i
\(254\) 0 0
\(255\) − 1823.85i − 0.447898i
\(256\) 4096.00 1.00000
\(257\) 1989.00 0.482764 0.241382 0.970430i \(-0.422399\pi\)
0.241382 + 0.970430i \(0.422399\pi\)
\(258\) 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
\(265\) 1356.20i 0.314379i
\(266\) 0 0
\(267\) 4551.83i 1.04332i
\(268\) 5625.70i 1.28226i
\(269\) −2106.00 −0.477342 −0.238671 0.971100i \(-0.576712\pi\)
−0.238671 + 0.971100i \(0.576712\pi\)
\(270\) 0 0
\(271\) − 685.892i − 0.153745i −0.997041 0.0768727i \(-0.975507\pi\)
0.997041 0.0768727i \(-0.0244935\pi\)
\(272\) −7488.00 −1.66922
\(273\) 0 0
\(274\) 0 0
\(275\) − 5092.23i − 1.11663i
\(276\) −432.000 −0.0942150
\(277\) 3665.00 0.794977 0.397488 0.917607i \(-0.369882\pi\)
0.397488 + 0.917607i \(0.369882\pi\)
\(278\) 0 0
\(279\) − 1745.91i − 0.374641i
\(280\) 0 0
\(281\) 1719.93i 0.365132i 0.983194 + 0.182566i \(0.0584404\pi\)
−0.983194 + 0.182566i \(0.941560\pi\)
\(282\) 0 0
\(283\) −1826.00 −0.383549 −0.191775 0.981439i \(-0.561424\pi\)
−0.191775 + 0.981439i \(0.561424\pi\)
\(284\) − 3741.23i − 0.781694i
\(285\) 378.000 0.0785642
\(286\) 0 0
\(287\) −378.000 −0.0777444
\(288\) 0 0
\(289\) 8776.00 1.78628
\(290\) 0 0
\(291\) 3471.03i 0.699228i
\(292\) − 5473.28i − 1.09692i
\(293\) − 504.027i − 0.100497i −0.998737 0.0502484i \(-0.983999\pi\)
0.998737 0.0502484i \(-0.0160013\pi\)
\(294\) 0 0
\(295\) −4104.00 −0.809980
\(296\) 0 0
\(297\) 1402.96i 0.274101i
\(298\) 0 0
\(299\) 0 0
\(300\) −2352.00 −0.452643
\(301\) − 852.169i − 0.163183i
\(302\) 0 0
\(303\) −4725.00 −0.895855
\(304\) − 1551.92i − 0.292791i
\(305\) 3736.03i 0.701392i
\(306\) 0 0
\(307\) − 1950.29i − 0.362570i −0.983431 0.181285i \(-0.941974\pi\)
0.983431 0.181285i \(-0.0580256\pi\)
\(308\) 4320.00 0.799204
\(309\) −2382.00 −0.438535
\(310\) 0 0
\(311\) −3798.00 −0.692491 −0.346246 0.938144i \(-0.612544\pi\)
−0.346246 + 0.938144i \(0.612544\pi\)
\(312\) 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
\(317\) − 7103.14i − 1.25852i −0.777193 0.629262i \(-0.783357\pi\)
0.777193 0.629262i \(-0.216643\pi\)
\(318\) 0 0
\(319\) 5144.19i 0.902882i
\(320\) − 2660.43i − 0.464758i
\(321\) −1350.00 −0.234734
\(322\) 0 0
\(323\) 2837.10i 0.488732i
\(324\) 648.000 0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) 1787.48i 0.302286i
\(328\) 0 0
\(329\) −756.000 −0.126686
\(330\) 0 0
\(331\) 10073.6i 1.67280i 0.548122 + 0.836398i \(0.315343\pi\)
−0.548122 + 0.836398i \(0.684657\pi\)
\(332\) − 9560.92i − 1.58049i
\(333\) − 1013.25i − 0.166744i
\(334\) 0 0
\(335\) 3654.00 0.595938
\(336\) − 1995.32i − 0.323970i
\(337\) 9001.00 1.45494 0.727471 0.686138i \(-0.240695\pi\)
0.727471 + 0.686138i \(0.240695\pi\)
\(338\) 0 0
\(339\) 5103.00 0.817572
\(340\) 4863.60i 0.775781i
\(341\) −10080.0 −1.60077
\(342\) 0 0
\(343\) 6006.75i 0.945581i
\(344\) 0 0
\(345\) 280.592i 0.0437872i
\(346\) 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
\(349\) 10544.7i 1.61732i 0.588273 + 0.808662i \(0.299808\pi\)
−0.588273 + 0.808662i \(0.700192\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2478.56i 0.373713i 0.982387 + 0.186856i \(0.0598299\pi\)
−0.982387 + 0.186856i \(0.940170\pi\)
\(354\) 0 0
\(355\) −2430.00 −0.363299
\(356\) − 12138.2i − 1.80709i
\(357\) 3647.70i 0.540775i
\(358\) 0 0
\(359\) 5414.39i 0.795991i 0.917387 + 0.397995i \(0.130294\pi\)
−0.917387 + 0.397995i \(0.869706\pi\)
\(360\) 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
\(369\) 327.358i 0.0461831i
\(370\) 0 0
\(371\) − 2712.39i − 0.379570i
\(372\) 4655.75i 0.648897i
\(373\) −7301.00 −1.01349 −0.506745 0.862096i \(-0.669151\pi\)
−0.506745 + 0.862096i \(0.669151\pi\)
\(374\) 0 0
\(375\) 3476.23i 0.478697i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3422.53i 0.463862i 0.972732 + 0.231931i \(0.0745044\pi\)
−0.972732 + 0.231931i \(0.925496\pi\)
\(380\) −1008.00 −0.136077
\(381\) 4992.00 0.671254
\(382\) 0 0
\(383\) − 5778.12i − 0.770883i −0.922732 0.385442i \(-0.874049\pi\)
0.922732 0.385442i \(-0.125951\pi\)
\(384\) 0 0
\(385\) − 2805.92i − 0.371436i
\(386\) 0 0
\(387\) −738.000 −0.0969371
\(388\) − 9256.08i − 1.21110i
\(389\) 9153.00 1.19300 0.596498 0.802614i \(-0.296558\pi\)
0.596498 + 0.802614i \(0.296558\pi\)
\(390\) 0 0
\(391\) −2106.00 −0.272391
\(392\) 0 0
\(393\) 4428.00 0.568354
\(394\) 0 0
\(395\) 2286.31i 0.291232i
\(396\) − 3741.23i − 0.474757i
\(397\) 2023.04i 0.255751i 0.991790 + 0.127876i \(0.0408158\pi\)
−0.991790 + 0.127876i \(0.959184\pi\)
\(398\) 0 0
\(399\) −756.000 −0.0948555
\(400\) 6272.00 0.784000
\(401\) − 8308.65i − 1.03470i −0.855774 0.517349i \(-0.826919\pi\)
0.855774 0.517349i \(-0.173081\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 12600.0 1.55167
\(405\) − 420.888i − 0.0516398i
\(406\) 0 0
\(407\) −5850.00 −0.712466
\(408\) 0 0
\(409\) − 10418.3i − 1.25954i −0.776782 0.629769i \(-0.783150\pi\)
0.776782 0.629769i \(-0.216850\pi\)
\(410\) 0 0
\(411\) − 3039.75i − 0.364817i
\(412\) 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
\(421\) − 14471.3i − 1.67527i −0.546233 0.837633i \(-0.683939\pi\)
0.546233 0.837633i \(-0.316061\pi\)
\(422\) 0 0
\(423\) 654.715i 0.0752561i
\(424\) 0 0
\(425\) −11466.0 −1.30867
\(426\) 0 0
\(427\) − 7472.07i − 0.846835i
\(428\) 3600.00 0.406571
\(429\) 0 0
\(430\) 0 0
\(431\) 6578.33i 0.735190i 0.929986 + 0.367595i \(0.119819\pi\)
−0.929986 + 0.367595i \(0.880181\pi\)
\(432\) −1728.00 −0.192450
\(433\) −6605.00 −0.733062 −0.366531 0.930406i \(-0.619455\pi\)
−0.366531 + 0.930406i \(0.619455\pi\)
\(434\) 0 0
\(435\) − 1543.26i − 0.170100i
\(436\) − 4766.60i − 0.523576i
\(437\) − 436.477i − 0.0477792i
\(438\) 0 0
\(439\) −8542.00 −0.928673 −0.464336 0.885659i \(-0.653707\pi\)
−0.464336 + 0.885659i \(0.653707\pi\)
\(440\) 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 \(0.778914\pi\)
\(444\) 2702.00i 0.288809i
\(445\) −7884.00 −0.839859
\(446\) 0 0
\(447\) − 9805.14i − 1.03751i
\(448\) 5320.86i 0.561132i
\(449\) 3013.77i 0.316767i 0.987378 + 0.158384i \(0.0506283\pi\)
−0.987378 + 0.158384i \(0.949372\pi\)
\(450\) 0 0
\(451\) 1890.00 0.197332
\(452\) −13608.0 −1.41608
\(453\) − 4915.56i − 0.509830i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2887.33i 0.295544i 0.989021 + 0.147772i \(0.0472102\pi\)
−0.989021 + 0.147772i \(0.952790\pi\)
\(458\) 0 0
\(459\) 3159.00 0.321241
\(460\) − 748.246i − 0.0758416i
\(461\) 3600.93i 0.363801i 0.983317 + 0.181900i \(0.0582249\pi\)
−0.983317 + 0.181900i \(0.941775\pi\)
\(462\) 0 0
\(463\) − 2677.75i − 0.268781i −0.990928 0.134391i \(-0.957092\pi\)
0.990928 0.134391i \(-0.0429077\pi\)
\(464\) −6336.00 −0.633925
\(465\) 3024.00 0.301580
\(466\) 0 0
\(467\) 13878.0 1.37515 0.687577 0.726111i \(-0.258674\pi\)
0.687577 + 0.726111i \(0.258674\pi\)
\(468\) 0 0
\(469\) −7308.00 −0.719514
\(470\) 0 0
\(471\) −3777.00 −0.369501
\(472\) 0 0
\(473\) 4260.84i 0.414194i
\(474\) 0 0
\(475\) − 2376.37i − 0.229548i
\(476\) − 9727.20i − 0.936650i
\(477\) −2349.00 −0.225479
\(478\) 0 0
\(479\) − 1101.58i − 0.105079i −0.998619 0.0525393i \(-0.983269\pi\)
0.998619 0.0525393i \(-0.0167315\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 561.184i − 0.0528670i
\(484\) −10952.0 −1.02855
\(485\) −6012.00 −0.562868
\(486\) 0 0
\(487\) − 17123.1i − 1.59326i −0.604464 0.796632i \(-0.706613\pi\)
0.604464 0.796632i \(-0.293387\pi\)
\(488\) 0 0
\(489\) − 8854.24i − 0.818820i
\(490\) 0 0
\(491\) 450.000 0.0413609 0.0206805 0.999786i \(-0.493417\pi\)
0.0206805 + 0.999786i \(0.493417\pi\)
\(492\) − 872.954i − 0.0799914i
\(493\) 11583.0 1.05816
\(494\) 0 0
\(495\) −2430.00 −0.220647
\(496\) − 12415.3i − 1.12392i
\(497\) 4860.00 0.438633
\(498\) 0 0
\(499\) − 13219.0i − 1.18590i −0.805239 0.592950i \(-0.797963\pi\)
0.805239 0.592950i \(-0.202037\pi\)
\(500\) − 9269.94i − 0.829128i
\(501\) − 9415.43i − 0.839621i
\(502\) 0 0
\(503\) 5346.00 0.473889 0.236945 0.971523i \(-0.423854\pi\)
0.236945 + 0.971523i \(0.423854\pi\)
\(504\) 0 0
\(505\) − 8183.94i − 0.721150i
\(506\) 0 0
\(507\) 0 0
\(508\) −13312.0 −1.16265
\(509\) 5866.46i 0.510857i 0.966828 + 0.255428i \(0.0822165\pi\)
−0.966828 + 0.255428i \(0.917783\pi\)
\(510\) 0 0
\(511\) 7110.00 0.615514
\(512\) 0 0
\(513\) 654.715i 0.0563477i
\(514\) 0 0
\(515\) − 4125.75i − 0.353014i
\(516\) 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
\(525\) − 3055.34i − 0.253992i
\(526\) 0 0
\(527\) 22696.8i 1.87607i
\(528\) 9976.61i 0.822304i
\(529\) −11843.0 −0.973371
\(530\) 0 0
\(531\) − 7108.34i − 0.580933i
\(532\) 2016.00 0.164295
\(533\) 0 0
\(534\) 0 0
\(535\) − 2338.27i − 0.188957i
\(536\) 0 0
\(537\) −9018.00 −0.724684
\(538\) 0 0
\(539\) − 12211.0i − 0.975813i
\(540\) 1122.37i 0.0894427i
\(541\) 5371.09i 0.426841i 0.976960 + 0.213421i \(0.0684605\pi\)
−0.976960 + 0.213421i \(0.931540\pi\)
\(542\) 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 \(0.269579\pi\)
\(548\) 8106.00i 0.631882i
\(549\) −6471.00 −0.503052
\(550\) 0 0
\(551\) 2400.62i 0.185608i
\(552\) 0 0
\(553\) − 4572.61i − 0.351623i
\(554\) 0 0
\(555\) 1755.00 0.134226
\(556\) 8992.00 0.685874
\(557\) − 3860.74i − 0.293689i −0.989160 0.146845i \(-0.953088\pi\)
0.989160 0.146845i \(-0.0469117\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3456.00 0.260790
\(561\) − 18238.5i − 1.37260i
\(562\) 0 0
\(563\) 21672.0 1.62232 0.811160 0.584825i \(-0.198837\pi\)
0.811160 + 0.584825i \(0.198837\pi\)
\(564\) − 1745.91i − 0.130347i
\(565\) 8838.66i 0.658133i
\(566\) 0 0
\(567\) 841.777i 0.0623480i
\(568\) 0 0
\(569\) −1386.00 −0.102116 −0.0510581 0.998696i \(-0.516259\pi\)
−0.0510581 + 0.998696i \(0.516259\pi\)
\(570\) 0 0
\(571\) 1162.00 0.0851632 0.0425816 0.999093i \(-0.486442\pi\)
0.0425816 + 0.999093i \(0.486442\pi\)
\(572\) 0 0
\(573\) 8208.00 0.598419
\(574\) 0 0
\(575\) 1764.00 0.127937
\(576\) 4608.00 0.333333
\(577\) 8045.38i 0.580474i 0.956955 + 0.290237i \(0.0937341\pi\)
−0.956955 + 0.290237i \(0.906266\pi\)
\(578\) 0 0
\(579\) − 7809.82i − 0.560561i
\(580\) 4115.35i 0.294622i
\(581\) 12420.0 0.886865
\(582\) 0 0
\(583\) 13562.0i 0.963429i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27622.7i 1.94227i 0.238530 + 0.971135i \(0.423335\pi\)
−0.238530 + 0.971135i \(0.576665\pi\)
\(588\) −5640.00 −0.395561
\(589\) −4704.00 −0.329075
\(590\) 0 0
\(591\) 11161.3i 0.776846i
\(592\) − 7205.33i − 0.500232i
\(593\) 275.396i 0.0190711i 0.999955 + 0.00953555i \(0.00303531\pi\)
−0.999955 + 0.00953555i \(0.996965\pi\)
\(594\) 0 0
\(595\) −6318.00 −0.435316
\(596\) 26147.0i 1.79702i
\(597\) −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
\(603\) 6328.91i 0.427418i
\(604\) 13108.2i 0.883052i
\(605\) 7113.53i 0.478027i
\(606\) 0 0
\(607\) 5480.00 0.366435 0.183218 0.983072i \(-0.441349\pi\)
0.183218 + 0.983072i \(0.441349\pi\)
\(608\) 0 0
\(609\) 3086.51i 0.205373i
\(610\) 0 0
\(611\) 0 0
\(612\) −8424.00 −0.556405
\(613\) − 17737.9i − 1.16872i −0.811493 0.584362i \(-0.801345\pi\)
0.811493 0.584362i \(-0.198655\pi\)
\(614\) 0 0
\(615\) −567.000 −0.0371767
\(616\) 0 0
\(617\) 9867.49i 0.643842i 0.946767 + 0.321921i \(0.104328\pi\)
−0.946767 + 0.321921i \(0.895672\pi\)
\(618\) 0 0
\(619\) − 4115.35i − 0.267221i −0.991034 0.133611i \(-0.957343\pi\)
0.991034 0.133611i \(-0.0426572\pi\)
\(620\) −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
\(629\) 13172.2i 0.834995i
\(630\) 0 0
\(631\) − 12664.8i − 0.799011i −0.916731 0.399506i \(-0.869182\pi\)
0.916731 0.399506i \(-0.130818\pi\)
\(632\) 0 0
\(633\) −7176.00 −0.450585
\(634\) 0 0
\(635\) 8646.40i 0.540349i
\(636\) 6264.00 0.390540
\(637\) 0 0
\(638\) 0 0
\(639\) − 4208.88i − 0.260565i
\(640\) 0 0
\(641\) −3789.00 −0.233473 −0.116737 0.993163i \(-0.537243\pi\)
−0.116737 + 0.993163i \(0.537243\pi\)
\(642\) 0 0
\(643\) 16911.7i 1.03722i 0.855010 + 0.518611i \(0.173551\pi\)
−0.855010 + 0.518611i \(0.826449\pi\)
\(644\) 1496.49i 0.0915684i
\(645\) − 1278.25i − 0.0780328i
\(646\) 0 0
\(647\) −27792.0 −1.68874 −0.844371 0.535759i \(-0.820026\pi\)
−0.844371 + 0.535759i \(0.820026\pi\)
\(648\) 0 0
\(649\) −41040.0 −2.48222
\(650\) 0 0
\(651\) −6048.00 −0.364116
\(652\) 23611.3i 1.41824i
\(653\) −594.000 −0.0355973 −0.0177986 0.999842i \(-0.505666\pi\)
−0.0177986 + 0.999842i \(0.505666\pi\)
\(654\) 0 0
\(655\) 7669.52i 0.457516i
\(656\) 2327.88i 0.138549i
\(657\) − 6157.44i − 0.365639i
\(658\) 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
\(661\) − 15791.1i − 0.929203i −0.885520 0.464601i \(-0.846198\pi\)
0.885520 0.464601i \(-0.153802\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1309.43i − 0.0763572i
\(666\) 0 0
\(667\) −1782.00 −0.103447
\(668\) 25107.8i 1.45427i
\(669\) − 6110.68i − 0.353143i
\(670\) 0 0
\(671\) 37360.3i 2.14945i
\(672\) 0 0
\(673\) 20933.0 1.19897 0.599486 0.800385i \(-0.295372\pi\)
0.599486 + 0.800385i \(0.295372\pi\)
\(674\) 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
\(681\) − 6453.62i − 0.363147i
\(682\) 0 0
\(683\) 24983.1i 1.39964i 0.714321 + 0.699818i \(0.246736\pi\)
−0.714321 + 0.699818i \(0.753264\pi\)
\(684\) − 1745.91i − 0.0975971i
\(685\) 5265.00 0.293672
\(686\) 0 0
\(687\) 10413.1i 0.578288i
\(688\) −5248.00 −0.290811
\(689\) 0 0
\(690\) 0 0
\(691\) − 13866.8i − 0.763412i −0.924284 0.381706i \(-0.875337\pi\)
0.924284 0.381706i \(-0.124663\pi\)
\(692\) 34128.0 1.87479
\(693\) 4860.00 0.266401
\(694\) 0 0
\(695\) − 5840.48i − 0.318765i
\(696\) 0 0
\(697\) − 4255.65i − 0.231269i
\(698\) 0 0
\(699\) 5562.00 0.300964
\(700\) 8147.57i 0.439927i
\(701\) −21906.0 −1.18028 −0.590141 0.807300i \(-0.700928\pi\)
−0.590141 + 0.807300i \(0.700928\pi\)
\(702\) 0 0
\(703\) −2730.00 −0.146464
\(704\) − 26604.3i − 1.42427i
\(705\) −1134.00 −0.0605800
\(706\) 0 0
\(707\) 16367.9i 0.870690i
\(708\) 18955.6i 1.00621i
\(709\) − 13057.9i − 0.691680i −0.938294 0.345840i \(-0.887594\pi\)
0.938294 0.345840i \(-0.112406\pi\)
\(710\) 0 0
\(711\) −3960.00 −0.208877
\(712\) 0 0
\(713\) − 3491.81i − 0.183407i
\(714\) 0 0
\(715\) 0 0
\(716\) 24048.0 1.25519
\(717\) − 13374.9i − 0.696645i
\(718\) 0 0
\(719\) −14220.0 −0.737575 −0.368788 0.929514i \(-0.620227\pi\)
−0.368788 + 0.929514i \(0.620227\pi\)
\(720\) − 2992.98i − 0.154919i
\(721\) 8251.49i 0.426216i
\(722\) 0 0
\(723\) 1252.27i 0.0644157i
\(724\) −14984.0 −0.769166
\(725\) −9702.00 −0.496998
\(726\) 0 0
\(727\) −5282.00 −0.269462 −0.134731 0.990882i \(-0.543017\pi\)
−0.134731 + 0.990882i \(0.543017\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 9594.00 0.485427
\(732\) 17256.0 0.871312
\(733\) − 11419.4i − 0.575424i −0.957717 0.287712i \(-0.907105\pi\)
0.957717 0.287712i \(-0.0928945\pi\)
\(734\) 0 0
\(735\) 3663.29i 0.183840i
\(736\) 0 0
\(737\) 36540.0 1.82628
\(738\) 0 0
\(739\) 20535.2i 1.02219i 0.859524 + 0.511096i \(0.170760\pi\)
−0.859524 + 0.511096i \(0.829240\pi\)
\(740\) −4680.00 −0.232487
\(741\) 0 0
\(742\) 0 0
\(743\) 20826.2i 1.02832i 0.857696 + 0.514158i \(0.171895\pi\)
−0.857696 + 0.514158i \(0.828105\pi\)
\(744\) 0 0
\(745\) 16983.0 0.835180
\(746\) 0 0
\(747\) − 10756.0i − 0.526831i
\(748\) 48636.0i 2.37742i
\(749\) 4676.54i 0.228140i
\(750\) 0 0
\(751\) −4834.00 −0.234880 −0.117440 0.993080i \(-0.537469\pi\)
−0.117440 + 0.993080i \(0.537469\pi\)
\(752\) 4655.75i 0.225768i
\(753\) −12312.0 −0.595849
\(754\) 0 0
\(755\) 8514.00 0.410406
\(756\) − 2244.74i − 0.107990i
\(757\) 9046.00 0.434323 0.217161 0.976136i \(-0.430320\pi\)
0.217161 + 0.976136i \(0.430320\pi\)
\(758\) 0 0
\(759\) 2805.92i 0.134188i
\(760\) 0 0
\(761\) − 12034.3i − 0.573249i −0.958043 0.286625i \(-0.907467\pi\)
0.958043 0.286625i \(-0.0925332\pi\)
\(762\) 0 0
\(763\) 6192.00 0.293795
\(764\) −21888.0 −1.03649
\(765\) 5471.55i 0.258594i
\(766\) 0 0
\(767\) 0 0
\(768\) −12288.0 −0.577350
\(769\) − 37543.9i − 1.76056i −0.474457 0.880279i \(-0.657355\pi\)
0.474457 0.880279i \(-0.342645\pi\)
\(770\) 0 0
\(771\) −5967.00 −0.278724
\(772\) 20826.2i 0.970920i
\(773\) − 15713.2i − 0.731130i −0.930786 0.365565i \(-0.880876\pi\)
0.930786 0.365565i \(-0.119124\pi\)
\(774\) 0 0
\(775\) − 19011.0i − 0.881155i
\(776\) 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
\(785\) − 6541.96i − 0.297443i
\(786\) 0 0
\(787\) − 3755.09i − 0.170082i −0.996377 0.0850409i \(-0.972898\pi\)
0.996377 0.0850409i \(-0.0271021\pi\)
\(788\) − 29763.6i − 1.34554i
\(789\) −2214.00 −0.0998992
\(790\) 0 0
\(791\) − 17677.3i − 0.794605i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 4068.59i − 0.181507i
\(796\) 9584.00 0.426754
\(797\) −7830.00 −0.347996 −0.173998 0.984746i \(-0.555669\pi\)
−0.173998 + 0.984746i \(0.555669\pi\)
\(798\) 0 0
\(799\) − 8511.30i − 0.376856i
\(800\) 0 0
\(801\) − 13655.5i − 0.602363i
\(802\) 0 0
\(803\) −35550.0 −1.56231
\(804\) − 16877.1i − 0.740310i
\(805\) 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
\(811\) 29839.8i 1.29201i 0.763335 + 0.646003i \(0.223560\pi\)
−0.763335 + 0.646003i \(0.776440\pi\)
\(812\) − 8230.71i − 0.355716i
\(813\) 2057.68i 0.0887649i
\(814\) 0 0
\(815\) 15336.0 0.659137
\(816\) 22464.0 0.963722
\(817\) 1988.39i 0.0851470i
\(818\) 0 0
\(819\) 0 0
\(820\) 1512.00 0.0643919
\(821\) 29763.6i 1.26523i 0.774466 + 0.632616i \(0.218019\pi\)
−0.774466 + 0.632616i \(0.781981\pi\)
\(822\) 0 0
\(823\) −8920.00 −0.377803 −0.188901 0.981996i \(-0.560493\pi\)
−0.188901 + 0.981996i \(0.560493\pi\)
\(824\) 0 0
\(825\) 15276.7i 0.644686i
\(826\) 0 0
\(827\) 18041.0i 0.758583i 0.925277 + 0.379292i \(0.123832\pi\)
−0.925277 + 0.379292i \(0.876168\pi\)
\(828\) 1296.00 0.0543951
\(829\) −21023.0 −0.880771 −0.440385 0.897809i \(-0.645158\pi\)
−0.440385 + 0.897809i \(0.645158\pi\)
\(830\) 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
\(837\) 5237.72i 0.216299i
\(838\) 0 0
\(839\) − 22073.3i − 0.908288i −0.890928 0.454144i \(-0.849945\pi\)
0.890928 0.454144i \(-0.150055\pi\)
\(840\) 0 0
\(841\) −14588.0 −0.598139
\(842\) 0 0
\(843\) − 5159.78i − 0.210809i
\(844\) 19136.0 0.780436
\(845\) 0 0
\(846\) 0 0
\(847\) − 14227.1i − 0.577152i
\(848\) −16704.0 −0.676436
\(849\) 5478.00 0.221442
\(850\) 0 0
\(851\) − 2026.50i − 0.0816304i
\(852\) 11223.7i 0.451311i
\(853\) 26609.5i 1.06810i 0.845452 + 0.534051i \(0.179331\pi\)
−0.845452 + 0.534051i \(0.820669\pi\)
\(854\) 0 0
\(855\) −1134.00 −0.0453590
\(856\) 0 0
\(857\) −12771.0 −0.509042 −0.254521 0.967067i \(-0.581918\pi\)
−0.254521 + 0.967067i \(0.581918\pi\)
\(858\) 0 0
\(859\) 17134.0 0.680564 0.340282 0.940323i \(-0.389477\pi\)
0.340282 + 0.940323i \(0.389477\pi\)
\(860\) 3408.68i 0.135157i
\(861\) 1134.00 0.0448857
\(862\) 0 0
\(863\) 7929.33i 0.312766i 0.987696 + 0.156383i \(0.0499835\pi\)
−0.987696 + 0.156383i \(0.950017\pi\)
\(864\) 0 0
\(865\) − 22166.8i − 0.871322i
\(866\) 0 0
\(867\) −26328.0 −1.03131
\(868\) 16128.0 0.630668
\(869\) 22863.1i 0.892493i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 10413.1i − 0.403700i
\(874\) 0 0
\(875\) 12042.0 0.465250
\(876\) 16419.8i 0.633305i
\(877\) 9864.03i 0.379800i 0.981803 + 0.189900i \(0.0608164\pi\)
−0.981803 + 0.189900i \(0.939184\pi\)
\(878\) 0 0
\(879\) 1512.08i 0.0580218i
\(880\) −17280.0 −0.661942
\(881\) 29169.0 1.11547 0.557735 0.830019i \(-0.311671\pi\)
0.557735 + 0.830019i \(0.311671\pi\)
\(882\) 0 0
\(883\) 928.000 0.0353677 0.0176839 0.999844i \(-0.494371\pi\)
0.0176839 + 0.999844i \(0.494371\pi\)
\(884\) 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
\(889\) − 17292.8i − 0.652398i
\(890\) 0 0
\(891\) − 4208.88i − 0.158252i
\(892\) 16295.1i 0.611661i
\(893\) 1764.00 0.0661030
\(894\) 0 0
\(895\) − 15619.6i − 0.583360i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 19205.0i 0.712483i
\(900\) 7056.00 0.261333
\(901\) 30537.0 1.12912
\(902\) 0 0
\(903\) 2556.51i 0.0942140i
\(904\) 0 0
\(905\) 9732.39i 0.357476i
\(906\) 0 0
\(907\) 19684.0 0.720614 0.360307 0.932834i \(-0.382672\pi\)
0.360307 + 0.932834i \(0.382672\pi\)
\(908\) 17209.7i 0.628990i
\(909\) 14175.0 0.517222
\(910\) 0 0
\(911\) 24480.0 0.890295 0.445147 0.895457i \(-0.353151\pi\)
0.445147 + 0.895457i \(0.353151\pi\)
\(912\) 4655.75i 0.169043i
\(913\) −62100.0 −2.25105
\(914\) 0 0
\(915\) − 11208.1i − 0.404949i
\(916\) − 27768.2i − 1.00162i
\(917\) − 15339.0i − 0.552388i
\(918\) 0 0
\(919\) 38608.0 1.38581 0.692906 0.721028i \(-0.256330\pi\)
0.692906 + 0.721028i \(0.256330\pi\)
\(920\) 0 0
\(921\) 5850.87i 0.209330i
\(922\) 0 0
\(923\) 0 0
\(924\) −12960.0 −0.461421
\(925\) − 11033.2i − 0.392182i
\(926\) 0 0
\(927\) 7146.00 0.253188
\(928\) 0 0
\(929\) 10210.4i 0.360596i 0.983612 + 0.180298i \(0.0577062\pi\)
−0.983612 + 0.180298i \(0.942294\pi\)
\(930\) 0 0
\(931\) − 5698.45i − 0.200600i
\(932\) −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
\(941\) − 10724.9i − 0.371541i −0.982593 0.185771i \(-0.940522\pi\)
0.982593 0.185771i \(-0.0594782\pi\)
\(942\) 0 0
\(943\) 654.715i 0.0226092i
\(944\) − 50548.2i − 1.74280i
\(945\) −1458.00 −0.0501891
\(946\) 0 0
\(947\) − 33962.1i − 1.16538i −0.812693 0.582692i \(-0.801999\pi\)
0.812693 0.582692i \(-0.198001\pi\)
\(948\) 10560.0 0.361786
\(949\) 0 0
\(950\) 0 0
\(951\) 21309.4i 0.726609i
\(952\) 0 0
\(953\) 23814.0 0.809456 0.404728 0.914437i \(-0.367366\pi\)
0.404728 + 0.914437i \(0.367366\pi\)
\(954\) 0 0
\(955\) 14216.7i 0.481718i
\(956\) 35666.4i 1.20662i
\(957\) − 15432.6i − 0.521279i
\(958\) 0 0
\(959\) −10530.0 −0.354569
\(960\) 7981.29i 0.268328i
\(961\) −7841.00 −0.263200
\(962\) 0 0
\(963\) 4050.00 0.135524
\(964\) − 3339.39i − 0.111571i
\(965\) 13527.0 0.451243
\(966\) 0 0
\(967\) − 51549.3i − 1.71429i −0.515079 0.857143i \(-0.672238\pi\)
0.515079 0.857143i \(-0.327762\pi\)
\(968\) 0 0
\(969\) − 8511.30i − 0.282170i
\(970\) 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
\(973\) 11681.0i 0.384865i
\(974\) 0 0
\(975\) 0 0
\(976\) −46016.0 −1.50916
\(977\) − 21538.1i − 0.705285i −0.935758 0.352642i \(-0.885283\pi\)
0.935758 0.352642i \(-0.114717\pi\)
\(978\) 0 0
\(979\) −78840.0 −2.57379
\(980\) − 9768.77i − 0.318420i
\(981\) − 5362.43i − 0.174525i
\(982\) 0 0
\(983\) 32611.1i 1.05812i 0.848585 + 0.529060i \(0.177455\pi\)
−0.848585 + 0.529060i \(0.822545\pi\)
\(984\) 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
\(993\) − 30220.8i − 0.965789i
\(994\) 0 0
\(995\) − 6224.99i − 0.198337i
\(996\) 28682.8i 0.912498i
\(997\) −24931.0 −0.791949 −0.395974 0.918262i \(-0.629593\pi\)
−0.395974 + 0.918262i \(0.629593\pi\)
\(998\) 0 0
\(999\) 3039.75i 0.0962697i
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.b.a.337.1 2
13.3 even 3 39.4.j.a.4.1 2
13.4 even 6 39.4.j.a.10.1 yes 2
13.5 odd 4 507.4.a.g.1.1 2
13.8 odd 4 507.4.a.g.1.2 2
13.12 even 2 inner 507.4.b.a.337.2 2
39.5 even 4 1521.4.a.m.1.2 2
39.8 even 4 1521.4.a.m.1.1 2
39.17 odd 6 117.4.q.b.10.1 2
39.29 odd 6 117.4.q.b.82.1 2
52.3 odd 6 624.4.bv.a.433.1 2
52.43 odd 6 624.4.bv.a.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.a.4.1 2 13.3 even 3
39.4.j.a.10.1 yes 2 13.4 even 6
117.4.q.b.10.1 2 39.17 odd 6
117.4.q.b.82.1 2 39.29 odd 6
507.4.a.g.1.1 2 13.5 odd 4
507.4.a.g.1.2 2 13.8 odd 4
507.4.b.a.337.1 2 1.1 even 1 trivial
507.4.b.a.337.2 2 13.12 even 2 inner
624.4.bv.a.49.1 2 52.43 odd 6
624.4.bv.a.433.1 2 52.3 odd 6
1521.4.a.m.1.1 2 39.8 even 4
1521.4.a.m.1.2 2 39.5 even 4