Properties

Label 507.4.b.k.337.15
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.15
Root \(4.83218i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.k.337.4

$q$-expansion

\(f(q)\) \(=\) \(q+4.03025i q^{2} +3.00000 q^{3} -8.24289 q^{4} +8.08864i q^{5} +12.0907i q^{6} -5.95078i q^{7} -0.978887i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.03025i q^{2} +3.00000 q^{3} -8.24289 q^{4} +8.08864i q^{5} +12.0907i q^{6} -5.95078i q^{7} -0.978887i q^{8} +9.00000 q^{9} -32.5992 q^{10} -17.2359i q^{11} -24.7287 q^{12} +23.9831 q^{14} +24.2659i q^{15} -61.9979 q^{16} -92.9299 q^{17} +36.2722i q^{18} +13.3832i q^{19} -66.6738i q^{20} -17.8523i q^{21} +69.4648 q^{22} -219.710 q^{23} -2.93666i q^{24} +59.5738 q^{25} +27.0000 q^{27} +49.0516i q^{28} -199.485 q^{29} -97.7977 q^{30} +307.777i q^{31} -257.698i q^{32} -51.7076i q^{33} -374.531i q^{34} +48.1337 q^{35} -74.1860 q^{36} +333.777i q^{37} -53.9376 q^{38} +7.91787 q^{40} -200.689i q^{41} +71.9493 q^{42} -116.806 q^{43} +142.073i q^{44} +72.7978i q^{45} -885.487i q^{46} -338.610i q^{47} -185.994 q^{48} +307.588 q^{49} +240.097i q^{50} -278.790 q^{51} -26.6215 q^{53} +108.817i q^{54} +139.415 q^{55} -5.82514 q^{56} +40.1496i q^{57} -803.976i q^{58} +280.058i q^{59} -200.021i q^{60} -207.084 q^{61} -1240.42 q^{62} -53.5570i q^{63} +542.603 q^{64} +208.394 q^{66} -285.981i q^{67} +766.011 q^{68} -659.131 q^{69} +193.991i q^{70} -317.673i q^{71} -8.80999i q^{72} +63.0668i q^{73} -1345.20 q^{74} +178.721 q^{75} -110.316i q^{76} -102.567 q^{77} -623.835 q^{79} -501.479i q^{80} +81.0000 q^{81} +808.824 q^{82} +659.874i q^{83} +147.155i q^{84} -751.677i q^{85} -470.755i q^{86} -598.456 q^{87} -16.8720 q^{88} -1273.19i q^{89} -293.393 q^{90} +1811.05 q^{92} +923.331i q^{93} +1364.68 q^{94} -108.252 q^{95} -773.094i q^{96} +603.746i q^{97} +1239.66i q^{98} -155.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.03025i 1.42491i 0.701719 + 0.712454i \(0.252416\pi\)
−0.701719 + 0.712454i \(0.747584\pi\)
\(3\) 3.00000 0.577350
\(4\) −8.24289 −1.03036
\(5\) 8.08864i 0.723470i 0.932281 + 0.361735i \(0.117816\pi\)
−0.932281 + 0.361735i \(0.882184\pi\)
\(6\) 12.0907i 0.822671i
\(7\) − 5.95078i − 0.321312i −0.987010 0.160656i \(-0.948639\pi\)
0.987010 0.160656i \(-0.0513610\pi\)
\(8\) − 0.978887i − 0.0432611i
\(9\) 9.00000 0.333333
\(10\) −32.5992 −1.03088
\(11\) − 17.2359i − 0.472437i −0.971700 0.236219i \(-0.924092\pi\)
0.971700 0.236219i \(-0.0759082\pi\)
\(12\) −24.7287 −0.594879
\(13\) 0 0
\(14\) 23.9831 0.457840
\(15\) 24.2659i 0.417696i
\(16\) −61.9979 −0.968718
\(17\) −92.9299 −1.32581 −0.662907 0.748702i \(-0.730677\pi\)
−0.662907 + 0.748702i \(0.730677\pi\)
\(18\) 36.2722i 0.474969i
\(19\) 13.3832i 0.161596i 0.996731 + 0.0807979i \(0.0257468\pi\)
−0.996731 + 0.0807979i \(0.974253\pi\)
\(20\) − 66.6738i − 0.745435i
\(21\) − 17.8523i − 0.185510i
\(22\) 69.4648 0.673179
\(23\) −219.710 −1.99186 −0.995930 0.0901293i \(-0.971272\pi\)
−0.995930 + 0.0901293i \(0.971272\pi\)
\(24\) − 2.93666i − 0.0249768i
\(25\) 59.5738 0.476591
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 49.0516i 0.331067i
\(29\) −199.485 −1.27736 −0.638681 0.769471i \(-0.720520\pi\)
−0.638681 + 0.769471i \(0.720520\pi\)
\(30\) −97.7977 −0.595178
\(31\) 307.777i 1.78317i 0.452851 + 0.891586i \(0.350407\pi\)
−0.452851 + 0.891586i \(0.649593\pi\)
\(32\) − 257.698i − 1.42359i
\(33\) − 51.7076i − 0.272762i
\(34\) − 374.531i − 1.88916i
\(35\) 48.1337 0.232460
\(36\) −74.1860 −0.343454
\(37\) 333.777i 1.48304i 0.670930 + 0.741521i \(0.265895\pi\)
−0.670930 + 0.741521i \(0.734105\pi\)
\(38\) −53.9376 −0.230259
\(39\) 0 0
\(40\) 7.91787 0.0312981
\(41\) − 200.689i − 0.764446i −0.924070 0.382223i \(-0.875159\pi\)
0.924070 0.382223i \(-0.124841\pi\)
\(42\) 71.9493 0.264334
\(43\) −116.806 −0.414248 −0.207124 0.978315i \(-0.566410\pi\)
−0.207124 + 0.978315i \(0.566410\pi\)
\(44\) 142.073i 0.486781i
\(45\) 72.7978i 0.241157i
\(46\) − 885.487i − 2.83822i
\(47\) − 338.610i − 1.05088i −0.850831 0.525440i \(-0.823901\pi\)
0.850831 0.525440i \(-0.176099\pi\)
\(48\) −185.994 −0.559289
\(49\) 307.588 0.896759
\(50\) 240.097i 0.679097i
\(51\) −278.790 −0.765459
\(52\) 0 0
\(53\) −26.6215 −0.0689951 −0.0344975 0.999405i \(-0.510983\pi\)
−0.0344975 + 0.999405i \(0.510983\pi\)
\(54\) 108.817i 0.274224i
\(55\) 139.415 0.341794
\(56\) −5.82514 −0.0139003
\(57\) 40.1496i 0.0932973i
\(58\) − 803.976i − 1.82012i
\(59\) 280.058i 0.617973i 0.951066 + 0.308987i \(0.0999899\pi\)
−0.951066 + 0.308987i \(0.900010\pi\)
\(60\) − 200.021i − 0.430377i
\(61\) −207.084 −0.434663 −0.217332 0.976098i \(-0.569735\pi\)
−0.217332 + 0.976098i \(0.569735\pi\)
\(62\) −1240.42 −2.54086
\(63\) − 53.5570i − 0.107104i
\(64\) 542.603 1.05977
\(65\) 0 0
\(66\) 208.394 0.388660
\(67\) − 285.981i − 0.521465i −0.965411 0.260732i \(-0.916036\pi\)
0.965411 0.260732i \(-0.0839640\pi\)
\(68\) 766.011 1.36607
\(69\) −659.131 −1.15000
\(70\) 193.991i 0.331233i
\(71\) − 317.673i − 0.530998i −0.964111 0.265499i \(-0.914463\pi\)
0.964111 0.265499i \(-0.0855368\pi\)
\(72\) − 8.80999i − 0.0144204i
\(73\) 63.0668i 0.101115i 0.998721 + 0.0505576i \(0.0160999\pi\)
−0.998721 + 0.0505576i \(0.983900\pi\)
\(74\) −1345.20 −2.11320
\(75\) 178.721 0.275160
\(76\) − 110.316i − 0.166502i
\(77\) −102.567 −0.151800
\(78\) 0 0
\(79\) −623.835 −0.888443 −0.444221 0.895917i \(-0.646520\pi\)
−0.444221 + 0.895917i \(0.646520\pi\)
\(80\) − 501.479i − 0.700838i
\(81\) 81.0000 0.111111
\(82\) 808.824 1.08926
\(83\) 659.874i 0.872658i 0.899787 + 0.436329i \(0.143722\pi\)
−0.899787 + 0.436329i \(0.856278\pi\)
\(84\) 147.155i 0.191142i
\(85\) − 751.677i − 0.959186i
\(86\) − 470.755i − 0.590265i
\(87\) −598.456 −0.737486
\(88\) −16.8720 −0.0204382
\(89\) − 1273.19i − 1.51638i −0.652032 0.758191i \(-0.726083\pi\)
0.652032 0.758191i \(-0.273917\pi\)
\(90\) −293.393 −0.343626
\(91\) 0 0
\(92\) 1811.05 2.05233
\(93\) 923.331i 1.02952i
\(94\) 1364.68 1.49741
\(95\) −108.252 −0.116910
\(96\) − 773.094i − 0.821912i
\(97\) 603.746i 0.631970i 0.948764 + 0.315985i \(0.102335\pi\)
−0.948764 + 0.315985i \(0.897665\pi\)
\(98\) 1239.66i 1.27780i
\(99\) − 155.123i − 0.157479i
\(100\) −491.060 −0.491060
\(101\) 740.588 0.729616 0.364808 0.931083i \(-0.381135\pi\)
0.364808 + 0.931083i \(0.381135\pi\)
\(102\) − 1123.59i − 1.09071i
\(103\) 1888.57 1.80666 0.903332 0.428942i \(-0.141113\pi\)
0.903332 + 0.428942i \(0.141113\pi\)
\(104\) 0 0
\(105\) 144.401 0.134211
\(106\) − 107.291i − 0.0983116i
\(107\) −919.463 −0.830728 −0.415364 0.909655i \(-0.636346\pi\)
−0.415364 + 0.909655i \(0.636346\pi\)
\(108\) −222.558 −0.198293
\(109\) 1570.40i 1.37998i 0.723821 + 0.689988i \(0.242384\pi\)
−0.723821 + 0.689988i \(0.757616\pi\)
\(110\) 561.876i 0.487025i
\(111\) 1001.33i 0.856234i
\(112\) 368.936i 0.311260i
\(113\) −1324.66 −1.10277 −0.551386 0.834250i \(-0.685901\pi\)
−0.551386 + 0.834250i \(0.685901\pi\)
\(114\) −161.813 −0.132940
\(115\) − 1777.16i − 1.44105i
\(116\) 1644.34 1.31614
\(117\) 0 0
\(118\) −1128.70 −0.880555
\(119\) 553.006i 0.426000i
\(120\) 23.7536 0.0180700
\(121\) 1033.92 0.776803
\(122\) − 834.601i − 0.619354i
\(123\) − 602.066i − 0.441353i
\(124\) − 2536.97i − 1.83731i
\(125\) 1492.95i 1.06827i
\(126\) 215.848 0.152613
\(127\) 2350.39 1.64223 0.821115 0.570763i \(-0.193352\pi\)
0.821115 + 0.570763i \(0.193352\pi\)
\(128\) 125.240i 0.0864824i
\(129\) −350.417 −0.239166
\(130\) 0 0
\(131\) 2308.69 1.53978 0.769890 0.638177i \(-0.220311\pi\)
0.769890 + 0.638177i \(0.220311\pi\)
\(132\) 426.220i 0.281043i
\(133\) 79.6405 0.0519226
\(134\) 1152.57 0.743039
\(135\) 218.393i 0.139232i
\(136\) 90.9680i 0.0573562i
\(137\) 374.912i 0.233802i 0.993144 + 0.116901i \(0.0372961\pi\)
−0.993144 + 0.116901i \(0.962704\pi\)
\(138\) − 2656.46i − 1.63864i
\(139\) 487.711 0.297605 0.148802 0.988867i \(-0.452458\pi\)
0.148802 + 0.988867i \(0.452458\pi\)
\(140\) −396.761 −0.239517
\(141\) − 1015.83i − 0.606726i
\(142\) 1280.30 0.756623
\(143\) 0 0
\(144\) −557.981 −0.322906
\(145\) − 1613.57i − 0.924134i
\(146\) −254.175 −0.144080
\(147\) 922.765 0.517744
\(148\) − 2751.28i − 1.52807i
\(149\) 1055.91i 0.580558i 0.956942 + 0.290279i \(0.0937481\pi\)
−0.956942 + 0.290279i \(0.906252\pi\)
\(150\) 720.292i 0.392077i
\(151\) 888.560i 0.478874i 0.970912 + 0.239437i \(0.0769629\pi\)
−0.970912 + 0.239437i \(0.923037\pi\)
\(152\) 13.1007 0.00699081
\(153\) −836.369 −0.441938
\(154\) − 413.370i − 0.216301i
\(155\) −2489.50 −1.29007
\(156\) 0 0
\(157\) −3648.56 −1.85469 −0.927346 0.374206i \(-0.877915\pi\)
−0.927346 + 0.374206i \(0.877915\pi\)
\(158\) − 2514.21i − 1.26595i
\(159\) −79.8644 −0.0398343
\(160\) 2084.43 1.02993
\(161\) 1307.45i 0.640008i
\(162\) 326.450i 0.158323i
\(163\) − 1386.49i − 0.666249i −0.942883 0.333125i \(-0.891897\pi\)
0.942883 0.333125i \(-0.108103\pi\)
\(164\) 1654.25i 0.787655i
\(165\) 418.244 0.197335
\(166\) −2659.46 −1.24346
\(167\) 3376.19i 1.56442i 0.623017 + 0.782209i \(0.285907\pi\)
−0.623017 + 0.782209i \(0.714093\pi\)
\(168\) −17.4754 −0.00802535
\(169\) 0 0
\(170\) 3029.44 1.36675
\(171\) 120.449i 0.0538652i
\(172\) 962.815 0.426825
\(173\) −341.817 −0.150219 −0.0751095 0.997175i \(-0.523931\pi\)
−0.0751095 + 0.997175i \(0.523931\pi\)
\(174\) − 2411.93i − 1.05085i
\(175\) − 354.511i − 0.153134i
\(176\) 1068.59i 0.457658i
\(177\) 840.174i 0.356787i
\(178\) 5131.28 2.16070
\(179\) 2885.55 1.20489 0.602446 0.798159i \(-0.294193\pi\)
0.602446 + 0.798159i \(0.294193\pi\)
\(180\) − 600.064i − 0.248478i
\(181\) −3795.62 −1.55871 −0.779354 0.626584i \(-0.784453\pi\)
−0.779354 + 0.626584i \(0.784453\pi\)
\(182\) 0 0
\(183\) −621.253 −0.250953
\(184\) 215.072i 0.0861701i
\(185\) −2699.80 −1.07294
\(186\) −3721.25 −1.46696
\(187\) 1601.73i 0.626363i
\(188\) 2791.12i 1.08278i
\(189\) − 160.671i − 0.0618365i
\(190\) − 436.282i − 0.166586i
\(191\) −2805.90 −1.06297 −0.531486 0.847067i \(-0.678366\pi\)
−0.531486 + 0.847067i \(0.678366\pi\)
\(192\) 1627.81 0.611859
\(193\) 2485.64i 0.927048i 0.886084 + 0.463524i \(0.153415\pi\)
−0.886084 + 0.463524i \(0.846585\pi\)
\(194\) −2433.24 −0.900499
\(195\) 0 0
\(196\) −2535.41 −0.923985
\(197\) 2750.70i 0.994818i 0.867516 + 0.497409i \(0.165715\pi\)
−0.867516 + 0.497409i \(0.834285\pi\)
\(198\) 625.183 0.224393
\(199\) −3998.53 −1.42436 −0.712182 0.701994i \(-0.752293\pi\)
−0.712182 + 0.701994i \(0.752293\pi\)
\(200\) − 58.3161i − 0.0206178i
\(201\) − 857.943i − 0.301068i
\(202\) 2984.75i 1.03964i
\(203\) 1187.09i 0.410432i
\(204\) 2298.03 0.788698
\(205\) 1623.30 0.553054
\(206\) 7611.41i 2.57433i
\(207\) −1977.39 −0.663953
\(208\) 0 0
\(209\) 230.671 0.0763438
\(210\) 581.973i 0.191238i
\(211\) −1375.80 −0.448882 −0.224441 0.974488i \(-0.572056\pi\)
−0.224441 + 0.974488i \(0.572056\pi\)
\(212\) 219.438 0.0710898
\(213\) − 953.020i − 0.306572i
\(214\) − 3705.66i − 1.18371i
\(215\) − 944.799i − 0.299696i
\(216\) − 26.4300i − 0.00832561i
\(217\) 1831.51 0.572955
\(218\) −6329.12 −1.96634
\(219\) 189.200i 0.0583789i
\(220\) −1149.18 −0.352171
\(221\) 0 0
\(222\) −4035.61 −1.22005
\(223\) 1694.37i 0.508805i 0.967098 + 0.254403i \(0.0818788\pi\)
−0.967098 + 0.254403i \(0.918121\pi\)
\(224\) −1533.50 −0.457418
\(225\) 536.164 0.158864
\(226\) − 5338.70i − 1.57135i
\(227\) − 405.400i − 0.118534i −0.998242 0.0592672i \(-0.981124\pi\)
0.998242 0.0592672i \(-0.0188764\pi\)
\(228\) − 330.949i − 0.0961299i
\(229\) − 2359.54i − 0.680884i −0.940265 0.340442i \(-0.889423\pi\)
0.940265 0.340442i \(-0.110577\pi\)
\(230\) 7162.39 2.05337
\(231\) −307.701 −0.0876416
\(232\) 195.274i 0.0552601i
\(233\) −938.507 −0.263878 −0.131939 0.991258i \(-0.542120\pi\)
−0.131939 + 0.991258i \(0.542120\pi\)
\(234\) 0 0
\(235\) 2738.90 0.760280
\(236\) − 2308.49i − 0.636736i
\(237\) −1871.51 −0.512943
\(238\) −2228.75 −0.607010
\(239\) 4664.25i 1.26237i 0.775634 + 0.631183i \(0.217430\pi\)
−0.775634 + 0.631183i \(0.782570\pi\)
\(240\) − 1504.44i − 0.404629i
\(241\) 3774.94i 1.00898i 0.863416 + 0.504492i \(0.168320\pi\)
−0.863416 + 0.504492i \(0.831680\pi\)
\(242\) 4166.97i 1.10687i
\(243\) 243.000 0.0641500
\(244\) 1706.97 0.447860
\(245\) 2487.97i 0.648778i
\(246\) 2426.47 0.628887
\(247\) 0 0
\(248\) 301.279 0.0771421
\(249\) 1979.62i 0.503829i
\(250\) −6016.96 −1.52219
\(251\) 960.695 0.241588 0.120794 0.992678i \(-0.461456\pi\)
0.120794 + 0.992678i \(0.461456\pi\)
\(252\) 441.464i 0.110356i
\(253\) 3786.90i 0.941029i
\(254\) 9472.64i 2.34003i
\(255\) − 2255.03i − 0.553787i
\(256\) 3836.08 0.936542
\(257\) 16.7302 0.00406070 0.00203035 0.999998i \(-0.499354\pi\)
0.00203035 + 0.999998i \(0.499354\pi\)
\(258\) − 1412.27i − 0.340790i
\(259\) 1986.23 0.476519
\(260\) 0 0
\(261\) −1795.37 −0.425788
\(262\) 9304.58i 2.19404i
\(263\) −400.736 −0.0939560 −0.0469780 0.998896i \(-0.514959\pi\)
−0.0469780 + 0.998896i \(0.514959\pi\)
\(264\) −50.6159 −0.0118000
\(265\) − 215.332i − 0.0499159i
\(266\) 320.971i 0.0739849i
\(267\) − 3819.57i − 0.875484i
\(268\) 2357.31i 0.537297i
\(269\) 2236.97 0.507027 0.253514 0.967332i \(-0.418414\pi\)
0.253514 + 0.967332i \(0.418414\pi\)
\(270\) −880.179 −0.198393
\(271\) 4018.89i 0.900850i 0.892814 + 0.450425i \(0.148727\pi\)
−0.892814 + 0.450425i \(0.851273\pi\)
\(272\) 5761.46 1.28434
\(273\) 0 0
\(274\) −1510.99 −0.333147
\(275\) − 1026.81i − 0.225159i
\(276\) 5433.14 1.18492
\(277\) −6792.95 −1.47346 −0.736731 0.676186i \(-0.763632\pi\)
−0.736731 + 0.676186i \(0.763632\pi\)
\(278\) 1965.59i 0.424059i
\(279\) 2769.99i 0.594391i
\(280\) − 47.1175i − 0.0100565i
\(281\) 7286.80i 1.54695i 0.633824 + 0.773477i \(0.281484\pi\)
−0.633824 + 0.773477i \(0.718516\pi\)
\(282\) 4094.04 0.864528
\(283\) 2429.77 0.510369 0.255185 0.966892i \(-0.417864\pi\)
0.255185 + 0.966892i \(0.417864\pi\)
\(284\) 2618.54i 0.547120i
\(285\) −324.756 −0.0674979
\(286\) 0 0
\(287\) −1194.25 −0.245626
\(288\) − 2319.28i − 0.474531i
\(289\) 3722.97 0.757780
\(290\) 6503.07 1.31681
\(291\) 1811.24i 0.364868i
\(292\) − 519.853i − 0.104185i
\(293\) − 4211.25i − 0.839672i −0.907600 0.419836i \(-0.862088\pi\)
0.907600 0.419836i \(-0.137912\pi\)
\(294\) 3718.97i 0.737737i
\(295\) −2265.29 −0.447085
\(296\) 326.730 0.0641580
\(297\) − 465.368i − 0.0909206i
\(298\) −4255.56 −0.827242
\(299\) 0 0
\(300\) −1473.18 −0.283514
\(301\) 695.084i 0.133103i
\(302\) −3581.12 −0.682351
\(303\) 2221.76 0.421244
\(304\) − 829.731i − 0.156541i
\(305\) − 1675.03i − 0.314466i
\(306\) − 3370.78i − 0.629720i
\(307\) − 8212.27i − 1.52671i −0.645982 0.763353i \(-0.723552\pi\)
0.645982 0.763353i \(-0.276448\pi\)
\(308\) 845.447 0.156408
\(309\) 5665.71 1.04308
\(310\) − 10033.3i − 1.83823i
\(311\) 4238.55 0.772818 0.386409 0.922328i \(-0.373715\pi\)
0.386409 + 0.922328i \(0.373715\pi\)
\(312\) 0 0
\(313\) 3807.82 0.687638 0.343819 0.939036i \(-0.388279\pi\)
0.343819 + 0.939036i \(0.388279\pi\)
\(314\) − 14704.6i − 2.64276i
\(315\) 433.204 0.0774865
\(316\) 5142.20 0.915416
\(317\) − 10497.7i − 1.85996i −0.367608 0.929981i \(-0.619823\pi\)
0.367608 0.929981i \(-0.380177\pi\)
\(318\) − 321.873i − 0.0567602i
\(319\) 3438.31i 0.603474i
\(320\) 4388.92i 0.766713i
\(321\) −2758.39 −0.479621
\(322\) −5269.34 −0.911953
\(323\) − 1243.70i − 0.214246i
\(324\) −667.674 −0.114485
\(325\) 0 0
\(326\) 5587.91 0.949343
\(327\) 4711.21i 0.796730i
\(328\) −196.452 −0.0330708
\(329\) −2014.99 −0.337660
\(330\) 1685.63i 0.281184i
\(331\) 11905.2i 1.97695i 0.151389 + 0.988474i \(0.451625\pi\)
−0.151389 + 0.988474i \(0.548375\pi\)
\(332\) − 5439.27i − 0.899152i
\(333\) 3003.99i 0.494347i
\(334\) −13606.9 −2.22915
\(335\) 2313.20 0.377264
\(336\) 1106.81i 0.179706i
\(337\) −8981.18 −1.45174 −0.725869 0.687833i \(-0.758562\pi\)
−0.725869 + 0.687833i \(0.758562\pi\)
\(338\) 0 0
\(339\) −3973.97 −0.636686
\(340\) 6195.99i 0.988308i
\(341\) 5304.80 0.842437
\(342\) −485.439 −0.0767530
\(343\) − 3871.51i − 0.609451i
\(344\) 114.339i 0.0179208i
\(345\) − 5331.48i − 0.831992i
\(346\) − 1377.61i − 0.214048i
\(347\) −5712.87 −0.883813 −0.441906 0.897061i \(-0.645698\pi\)
−0.441906 + 0.897061i \(0.645698\pi\)
\(348\) 4933.01 0.759876
\(349\) − 9507.28i − 1.45820i −0.684406 0.729102i \(-0.739938\pi\)
0.684406 0.729102i \(-0.260062\pi\)
\(350\) 1428.77 0.218202
\(351\) 0 0
\(352\) −4441.65 −0.672559
\(353\) − 3972.96i − 0.599035i −0.954091 0.299518i \(-0.903174\pi\)
0.954091 0.299518i \(-0.0968257\pi\)
\(354\) −3386.11 −0.508389
\(355\) 2569.55 0.384162
\(356\) 10494.8i 1.56242i
\(357\) 1659.02i 0.245951i
\(358\) 11629.5i 1.71686i
\(359\) 109.866i 0.0161519i 0.999967 + 0.00807594i \(0.00257068\pi\)
−0.999967 + 0.00807594i \(0.997429\pi\)
\(360\) 71.2609 0.0104327
\(361\) 6679.89 0.973887
\(362\) − 15297.3i − 2.22101i
\(363\) 3101.77 0.448487
\(364\) 0 0
\(365\) −510.125 −0.0731539
\(366\) − 2503.80i − 0.357584i
\(367\) 7020.02 0.998480 0.499240 0.866464i \(-0.333613\pi\)
0.499240 + 0.866464i \(0.333613\pi\)
\(368\) 13621.6 1.92955
\(369\) − 1806.20i − 0.254815i
\(370\) − 10880.9i − 1.52884i
\(371\) 158.418i 0.0221689i
\(372\) − 7610.91i − 1.06077i
\(373\) 2227.18 0.309166 0.154583 0.987980i \(-0.450597\pi\)
0.154583 + 0.987980i \(0.450597\pi\)
\(374\) −6455.36 −0.892510
\(375\) 4478.86i 0.616766i
\(376\) −331.461 −0.0454622
\(377\) 0 0
\(378\) 647.544 0.0881113
\(379\) − 11004.6i − 1.49147i −0.666244 0.745734i \(-0.732099\pi\)
0.666244 0.745734i \(-0.267901\pi\)
\(380\) 892.309 0.120459
\(381\) 7051.16 0.948142
\(382\) − 11308.5i − 1.51464i
\(383\) − 4436.85i − 0.591938i −0.955198 0.295969i \(-0.904357\pi\)
0.955198 0.295969i \(-0.0956426\pi\)
\(384\) 375.720i 0.0499306i
\(385\) − 829.627i − 0.109823i
\(386\) −10017.7 −1.32096
\(387\) −1051.25 −0.138083
\(388\) − 4976.61i − 0.651157i
\(389\) 2561.58 0.333874 0.166937 0.985968i \(-0.446612\pi\)
0.166937 + 0.985968i \(0.446612\pi\)
\(390\) 0 0
\(391\) 20417.7 2.64083
\(392\) − 301.094i − 0.0387948i
\(393\) 6926.07 0.888992
\(394\) −11086.0 −1.41752
\(395\) − 5045.98i − 0.642762i
\(396\) 1278.66i 0.162260i
\(397\) − 2206.90i − 0.278996i −0.990222 0.139498i \(-0.955451\pi\)
0.990222 0.139498i \(-0.0445488\pi\)
\(398\) − 16115.1i − 2.02959i
\(399\) 238.922 0.0299775
\(400\) −3693.45 −0.461682
\(401\) 8071.02i 1.00511i 0.864546 + 0.502553i \(0.167606\pi\)
−0.864546 + 0.502553i \(0.832394\pi\)
\(402\) 3457.72 0.428994
\(403\) 0 0
\(404\) −6104.58 −0.751768
\(405\) 655.180i 0.0803856i
\(406\) −4784.28 −0.584827
\(407\) 5752.93 0.700644
\(408\) 272.904i 0.0331146i
\(409\) 6298.36i 0.761452i 0.924688 + 0.380726i \(0.124326\pi\)
−0.924688 + 0.380726i \(0.875674\pi\)
\(410\) 6542.29i 0.788051i
\(411\) 1124.74i 0.134986i
\(412\) −15567.3 −1.86152
\(413\) 1666.56 0.198562
\(414\) − 7969.38i − 0.946072i
\(415\) −5337.49 −0.631342
\(416\) 0 0
\(417\) 1463.13 0.171822
\(418\) 929.662i 0.108783i
\(419\) −1432.01 −0.166965 −0.0834824 0.996509i \(-0.526604\pi\)
−0.0834824 + 0.996509i \(0.526604\pi\)
\(420\) −1190.28 −0.138285
\(421\) 9641.43i 1.11614i 0.829794 + 0.558069i \(0.188458\pi\)
−0.829794 + 0.558069i \(0.811542\pi\)
\(422\) − 5544.82i − 0.639615i
\(423\) − 3047.49i − 0.350293i
\(424\) 26.0594i 0.00298480i
\(425\) −5536.19 −0.631870
\(426\) 3840.90 0.436837
\(427\) 1232.31i 0.139662i
\(428\) 7579.03 0.855949
\(429\) 0 0
\(430\) 3807.77 0.427040
\(431\) 3370.61i 0.376698i 0.982102 + 0.188349i \(0.0603136\pi\)
−0.982102 + 0.188349i \(0.939686\pi\)
\(432\) −1673.94 −0.186430
\(433\) −6248.91 −0.693541 −0.346770 0.937950i \(-0.612722\pi\)
−0.346770 + 0.937950i \(0.612722\pi\)
\(434\) 7381.45i 0.816407i
\(435\) − 4840.70i − 0.533549i
\(436\) − 12944.7i − 1.42187i
\(437\) − 2940.43i − 0.321876i
\(438\) −762.524 −0.0831845
\(439\) −2866.16 −0.311604 −0.155802 0.987788i \(-0.549796\pi\)
−0.155802 + 0.987788i \(0.549796\pi\)
\(440\) − 136.471i − 0.0147864i
\(441\) 2768.29 0.298920
\(442\) 0 0
\(443\) 10776.3 1.15575 0.577877 0.816124i \(-0.303881\pi\)
0.577877 + 0.816124i \(0.303881\pi\)
\(444\) − 8253.85i − 0.882230i
\(445\) 10298.4 1.09706
\(446\) −6828.74 −0.725000
\(447\) 3167.72i 0.335185i
\(448\) − 3228.91i − 0.340517i
\(449\) − 13834.9i − 1.45414i −0.686564 0.727069i \(-0.740882\pi\)
0.686564 0.727069i \(-0.259118\pi\)
\(450\) 2160.87i 0.226366i
\(451\) −3459.04 −0.361153
\(452\) 10919.0 1.13625
\(453\) 2665.68i 0.276478i
\(454\) 1633.86 0.168901
\(455\) 0 0
\(456\) 39.3020 0.00403615
\(457\) 17233.0i 1.76395i 0.471295 + 0.881976i \(0.343787\pi\)
−0.471295 + 0.881976i \(0.656213\pi\)
\(458\) 9509.51 0.970197
\(459\) −2509.11 −0.255153
\(460\) 14648.9i 1.48480i
\(461\) 11485.0i 1.16033i 0.814500 + 0.580164i \(0.197012\pi\)
−0.814500 + 0.580164i \(0.802988\pi\)
\(462\) − 1240.11i − 0.124881i
\(463\) − 1613.61i − 0.161967i −0.996715 0.0809837i \(-0.974194\pi\)
0.996715 0.0809837i \(-0.0258062\pi\)
\(464\) 12367.7 1.23740
\(465\) −7468.49 −0.744824
\(466\) − 3782.41i − 0.376002i
\(467\) 8149.20 0.807495 0.403747 0.914871i \(-0.367707\pi\)
0.403747 + 0.914871i \(0.367707\pi\)
\(468\) 0 0
\(469\) −1701.81 −0.167553
\(470\) 11038.4i 1.08333i
\(471\) −10945.7 −1.07081
\(472\) 274.145 0.0267342
\(473\) 2013.24i 0.195706i
\(474\) − 7542.63i − 0.730896i
\(475\) 797.289i 0.0770150i
\(476\) − 4558.36i − 0.438933i
\(477\) −239.593 −0.0229984
\(478\) −18798.1 −1.79875
\(479\) − 1827.62i − 0.174334i −0.996194 0.0871670i \(-0.972219\pi\)
0.996194 0.0871670i \(-0.0277814\pi\)
\(480\) 6253.28 0.594629
\(481\) 0 0
\(482\) −15213.9 −1.43771
\(483\) 3922.34i 0.369509i
\(484\) −8522.52 −0.800387
\(485\) −4883.49 −0.457212
\(486\) 979.350i 0.0914078i
\(487\) − 7838.95i − 0.729398i −0.931125 0.364699i \(-0.881172\pi\)
0.931125 0.364699i \(-0.118828\pi\)
\(488\) 202.712i 0.0188040i
\(489\) − 4159.48i − 0.384659i
\(490\) −10027.1 −0.924449
\(491\) −17196.1 −1.58055 −0.790273 0.612755i \(-0.790061\pi\)
−0.790273 + 0.612755i \(0.790061\pi\)
\(492\) 4962.76i 0.454753i
\(493\) 18538.2 1.69354
\(494\) 0 0
\(495\) 1254.73 0.113931
\(496\) − 19081.5i − 1.72739i
\(497\) −1890.40 −0.170616
\(498\) −7978.37 −0.717910
\(499\) 5355.28i 0.480431i 0.970720 + 0.240216i \(0.0772182\pi\)
−0.970720 + 0.240216i \(0.922782\pi\)
\(500\) − 12306.2i − 1.10070i
\(501\) 10128.6i 0.903217i
\(502\) 3871.84i 0.344240i
\(503\) 2979.80 0.264141 0.132071 0.991240i \(-0.457837\pi\)
0.132071 + 0.991240i \(0.457837\pi\)
\(504\) −52.4263 −0.00463344
\(505\) 5990.35i 0.527856i
\(506\) −15262.1 −1.34088
\(507\) 0 0
\(508\) −19374.0 −1.69209
\(509\) 15353.7i 1.33701i 0.743706 + 0.668507i \(0.233066\pi\)
−0.743706 + 0.668507i \(0.766934\pi\)
\(510\) 9088.33 0.789094
\(511\) 375.297 0.0324895
\(512\) 16462.3i 1.42097i
\(513\) 361.347i 0.0310991i
\(514\) 67.4267i 0.00578611i
\(515\) 15276.0i 1.30707i
\(516\) 2888.44 0.246428
\(517\) −5836.24 −0.496475
\(518\) 8005.00i 0.678995i
\(519\) −1025.45 −0.0867289
\(520\) 0 0
\(521\) 433.801 0.0364783 0.0182391 0.999834i \(-0.494194\pi\)
0.0182391 + 0.999834i \(0.494194\pi\)
\(522\) − 7235.78i − 0.606708i
\(523\) 18900.3 1.58021 0.790106 0.612970i \(-0.210026\pi\)
0.790106 + 0.612970i \(0.210026\pi\)
\(524\) −19030.3 −1.58653
\(525\) − 1063.53i − 0.0884121i
\(526\) − 1615.06i − 0.133879i
\(527\) − 28601.7i − 2.36415i
\(528\) 3205.76i 0.264229i
\(529\) 36105.7 2.96751
\(530\) 867.839 0.0711255
\(531\) 2520.52i 0.205991i
\(532\) −656.468 −0.0534990
\(533\) 0 0
\(534\) 15393.8 1.24748
\(535\) − 7437.21i − 0.601007i
\(536\) −279.943 −0.0225592
\(537\) 8656.64 0.695645
\(538\) 9015.53i 0.722467i
\(539\) − 5301.55i − 0.423662i
\(540\) − 1800.19i − 0.143459i
\(541\) − 13247.5i − 1.05278i −0.850243 0.526391i \(-0.823545\pi\)
0.850243 0.526391i \(-0.176455\pi\)
\(542\) −16197.1 −1.28363
\(543\) −11386.9 −0.899920
\(544\) 23947.9i 1.88742i
\(545\) −12702.4 −0.998372
\(546\) 0 0
\(547\) −5543.52 −0.433316 −0.216658 0.976248i \(-0.569516\pi\)
−0.216658 + 0.976248i \(0.569516\pi\)
\(548\) − 3090.36i − 0.240901i
\(549\) −1863.76 −0.144888
\(550\) 4138.28 0.320831
\(551\) − 2669.76i − 0.206416i
\(552\) 645.215i 0.0497503i
\(553\) 3712.31i 0.285467i
\(554\) − 27377.3i − 2.09955i
\(555\) −8099.40 −0.619460
\(556\) −4020.14 −0.306640
\(557\) − 6733.19i − 0.512198i −0.966651 0.256099i \(-0.917563\pi\)
0.966651 0.256099i \(-0.0824374\pi\)
\(558\) −11163.7 −0.846952
\(559\) 0 0
\(560\) −2984.19 −0.225188
\(561\) 4805.18i 0.361631i
\(562\) −29367.6 −2.20427
\(563\) −24282.4 −1.81773 −0.908863 0.417096i \(-0.863048\pi\)
−0.908863 + 0.417096i \(0.863048\pi\)
\(564\) 8373.37i 0.625146i
\(565\) − 10714.7i − 0.797823i
\(566\) 9792.55i 0.727229i
\(567\) − 482.013i − 0.0357013i
\(568\) −310.966 −0.0229716
\(569\) 19876.2 1.46441 0.732207 0.681082i \(-0.238490\pi\)
0.732207 + 0.681082i \(0.238490\pi\)
\(570\) − 1308.85i − 0.0961782i
\(571\) −225.014 −0.0164913 −0.00824567 0.999966i \(-0.502625\pi\)
−0.00824567 + 0.999966i \(0.502625\pi\)
\(572\) 0 0
\(573\) −8417.70 −0.613707
\(574\) − 4813.14i − 0.349994i
\(575\) −13089.0 −0.949302
\(576\) 4883.43 0.353257
\(577\) − 12258.1i − 0.884424i −0.896910 0.442212i \(-0.854194\pi\)
0.896910 0.442212i \(-0.145806\pi\)
\(578\) 15004.5i 1.07977i
\(579\) 7456.92i 0.535231i
\(580\) 13300.4i 0.952191i
\(581\) 3926.77 0.280395
\(582\) −7299.73 −0.519903
\(583\) 458.844i 0.0325958i
\(584\) 61.7353 0.00437436
\(585\) 0 0
\(586\) 16972.4 1.19645
\(587\) 10282.3i 0.722992i 0.932374 + 0.361496i \(0.117734\pi\)
−0.932374 + 0.361496i \(0.882266\pi\)
\(588\) −7606.24 −0.533463
\(589\) −4119.04 −0.288153
\(590\) − 9129.67i − 0.637055i
\(591\) 8252.10i 0.574359i
\(592\) − 20693.5i − 1.43665i
\(593\) − 12355.1i − 0.855586i −0.903877 0.427793i \(-0.859291\pi\)
0.903877 0.427793i \(-0.140709\pi\)
\(594\) 1875.55 0.129553
\(595\) −4473.07 −0.308198
\(596\) − 8703.71i − 0.598184i
\(597\) −11995.6 −0.822357
\(598\) 0 0
\(599\) 13035.2 0.889157 0.444578 0.895740i \(-0.353354\pi\)
0.444578 + 0.895740i \(0.353354\pi\)
\(600\) − 174.948i − 0.0119037i
\(601\) −5153.36 −0.349767 −0.174884 0.984589i \(-0.555955\pi\)
−0.174884 + 0.984589i \(0.555955\pi\)
\(602\) −2801.36 −0.189659
\(603\) − 2573.83i − 0.173822i
\(604\) − 7324.30i − 0.493413i
\(605\) 8363.05i 0.561994i
\(606\) 8954.25i 0.600234i
\(607\) −1406.55 −0.0940530 −0.0470265 0.998894i \(-0.514975\pi\)
−0.0470265 + 0.998894i \(0.514975\pi\)
\(608\) 3448.83 0.230047
\(609\) 3561.28i 0.236963i
\(610\) 6750.79 0.448085
\(611\) 0 0
\(612\) 6894.10 0.455355
\(613\) − 15984.3i − 1.05318i −0.850120 0.526590i \(-0.823470\pi\)
0.850120 0.526590i \(-0.176530\pi\)
\(614\) 33097.5 2.17541
\(615\) 4869.90 0.319306
\(616\) 100.401i 0.00656703i
\(617\) − 10324.4i − 0.673652i −0.941567 0.336826i \(-0.890647\pi\)
0.941567 0.336826i \(-0.109353\pi\)
\(618\) 22834.2i 1.48629i
\(619\) 7423.00i 0.481996i 0.970526 + 0.240998i \(0.0774748\pi\)
−0.970526 + 0.240998i \(0.922525\pi\)
\(620\) 20520.6 1.32924
\(621\) −5932.18 −0.383334
\(622\) 17082.4i 1.10119i
\(623\) −7576.48 −0.487232
\(624\) 0 0
\(625\) −4629.23 −0.296271
\(626\) 15346.4i 0.979820i
\(627\) 692.014 0.0440771
\(628\) 30074.6 1.91100
\(629\) − 31017.8i − 1.96624i
\(630\) 1745.92i 0.110411i
\(631\) − 1190.30i − 0.0750954i −0.999295 0.0375477i \(-0.988045\pi\)
0.999295 0.0375477i \(-0.0119546\pi\)
\(632\) 610.665i 0.0384350i
\(633\) −4127.41 −0.259162
\(634\) 42308.2 2.65027
\(635\) 19011.5i 1.18810i
\(636\) 658.313 0.0410437
\(637\) 0 0
\(638\) −13857.2 −0.859894
\(639\) − 2859.06i − 0.176999i
\(640\) −1013.02 −0.0625674
\(641\) −705.064 −0.0434451 −0.0217226 0.999764i \(-0.506915\pi\)
−0.0217226 + 0.999764i \(0.506915\pi\)
\(642\) − 11117.0i − 0.683415i
\(643\) 14641.4i 0.897976i 0.893538 + 0.448988i \(0.148215\pi\)
−0.893538 + 0.448988i \(0.851785\pi\)
\(644\) − 10777.1i − 0.659439i
\(645\) − 2834.40i − 0.173030i
\(646\) 5012.42 0.305280
\(647\) −1520.83 −0.0924111 −0.0462056 0.998932i \(-0.514713\pi\)
−0.0462056 + 0.998932i \(0.514713\pi\)
\(648\) − 79.2899i − 0.00480679i
\(649\) 4827.04 0.291954
\(650\) 0 0
\(651\) 5494.54 0.330796
\(652\) 11428.7i 0.686477i
\(653\) −22054.4 −1.32168 −0.660838 0.750529i \(-0.729799\pi\)
−0.660838 + 0.750529i \(0.729799\pi\)
\(654\) −18987.3 −1.13527
\(655\) 18674.2i 1.11398i
\(656\) 12442.3i 0.740532i
\(657\) 567.601i 0.0337051i
\(658\) − 8120.92i − 0.481134i
\(659\) −12652.3 −0.747895 −0.373947 0.927450i \(-0.621996\pi\)
−0.373947 + 0.927450i \(0.621996\pi\)
\(660\) −3447.54 −0.203326
\(661\) 10893.2i 0.640994i 0.947250 + 0.320497i \(0.103850\pi\)
−0.947250 + 0.320497i \(0.896150\pi\)
\(662\) −47981.0 −2.81697
\(663\) 0 0
\(664\) 645.942 0.0377522
\(665\) 644.184i 0.0375645i
\(666\) −12106.8 −0.704399
\(667\) 43829.0 2.54433
\(668\) − 27829.6i − 1.61191i
\(669\) 5083.12i 0.293759i
\(670\) 9322.76i 0.537567i
\(671\) 3569.28i 0.205351i
\(672\) −4600.51 −0.264090
\(673\) 9019.75 0.516621 0.258310 0.966062i \(-0.416834\pi\)
0.258310 + 0.966062i \(0.416834\pi\)
\(674\) − 36196.4i − 2.06859i
\(675\) 1608.49 0.0917199
\(676\) 0 0
\(677\) −24923.6 −1.41491 −0.707454 0.706760i \(-0.750156\pi\)
−0.707454 + 0.706760i \(0.750156\pi\)
\(678\) − 16016.1i − 0.907218i
\(679\) 3592.76 0.203060
\(680\) −735.807 −0.0414955
\(681\) − 1216.20i − 0.0684359i
\(682\) 21379.7i 1.20039i
\(683\) 24634.2i 1.38009i 0.723767 + 0.690044i \(0.242409\pi\)
−0.723767 + 0.690044i \(0.757591\pi\)
\(684\) − 992.846i − 0.0555006i
\(685\) −3032.53 −0.169149
\(686\) 15603.1 0.868411
\(687\) − 7078.61i − 0.393109i
\(688\) 7241.70 0.401290
\(689\) 0 0
\(690\) 21487.2 1.18551
\(691\) − 10340.2i − 0.569264i −0.958637 0.284632i \(-0.908129\pi\)
0.958637 0.284632i \(-0.0918714\pi\)
\(692\) 2817.56 0.154780
\(693\) −923.102 −0.0505999
\(694\) − 23024.3i − 1.25935i
\(695\) 3944.92i 0.215308i
\(696\) 585.821i 0.0319045i
\(697\) 18650.0i 1.01351i
\(698\) 38316.7 2.07780
\(699\) −2815.52 −0.152350
\(700\) 2922.19i 0.157784i
\(701\) 20833.9 1.12252 0.561258 0.827641i \(-0.310317\pi\)
0.561258 + 0.827641i \(0.310317\pi\)
\(702\) 0 0
\(703\) −4467.00 −0.239653
\(704\) − 9352.23i − 0.500676i
\(705\) 8216.69 0.438948
\(706\) 16012.0 0.853569
\(707\) − 4407.07i − 0.234434i
\(708\) − 6925.46i − 0.367619i
\(709\) − 9189.56i − 0.486772i −0.969930 0.243386i \(-0.921742\pi\)
0.969930 0.243386i \(-0.0782581\pi\)
\(710\) 10355.9i 0.547395i
\(711\) −5614.52 −0.296148
\(712\) −1246.31 −0.0656004
\(713\) − 67621.8i − 3.55183i
\(714\) −6686.25 −0.350457
\(715\) 0 0
\(716\) −23785.2 −1.24147
\(717\) 13992.8i 0.728827i
\(718\) −442.788 −0.0230149
\(719\) −7503.22 −0.389183 −0.194592 0.980884i \(-0.562338\pi\)
−0.194592 + 0.980884i \(0.562338\pi\)
\(720\) − 4513.31i − 0.233613i
\(721\) − 11238.5i − 0.580503i
\(722\) 26921.6i 1.38770i
\(723\) 11324.8i 0.582537i
\(724\) 31286.8 1.60603
\(725\) −11884.1 −0.608779
\(726\) 12500.9i 0.639053i
\(727\) 20727.2 1.05740 0.528700 0.848809i \(-0.322680\pi\)
0.528700 + 0.848809i \(0.322680\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 2055.93i − 0.104237i
\(731\) 10854.7 0.549216
\(732\) 5120.92 0.258572
\(733\) 17361.3i 0.874836i 0.899258 + 0.437418i \(0.144107\pi\)
−0.899258 + 0.437418i \(0.855893\pi\)
\(734\) 28292.4i 1.42274i
\(735\) 7463.92i 0.374572i
\(736\) 56618.9i 2.83560i
\(737\) −4929.13 −0.246359
\(738\) 7279.42 0.363088
\(739\) − 18093.8i − 0.900663i −0.892861 0.450331i \(-0.851306\pi\)
0.892861 0.450331i \(-0.148694\pi\)
\(740\) 22254.1 1.10551
\(741\) 0 0
\(742\) −638.466 −0.0315887
\(743\) − 14875.3i − 0.734482i −0.930126 0.367241i \(-0.880302\pi\)
0.930126 0.367241i \(-0.119698\pi\)
\(744\) 903.837 0.0445380
\(745\) −8540.85 −0.420017
\(746\) 8976.07i 0.440533i
\(747\) 5938.87i 0.290886i
\(748\) − 13202.9i − 0.645380i
\(749\) 5471.52i 0.266923i
\(750\) −18050.9 −0.878834
\(751\) 17750.9 0.862502 0.431251 0.902232i \(-0.358072\pi\)
0.431251 + 0.902232i \(0.358072\pi\)
\(752\) 20993.1i 1.01801i
\(753\) 2882.09 0.139481
\(754\) 0 0
\(755\) −7187.25 −0.346451
\(756\) 1324.39i 0.0637139i
\(757\) −10220.3 −0.490702 −0.245351 0.969434i \(-0.578903\pi\)
−0.245351 + 0.969434i \(0.578903\pi\)
\(758\) 44351.1 2.12520
\(759\) 11360.7i 0.543303i
\(760\) 105.967i 0.00505765i
\(761\) 14112.0i 0.672221i 0.941823 + 0.336110i \(0.109112\pi\)
−0.941823 + 0.336110i \(0.890888\pi\)
\(762\) 28417.9i 1.35101i
\(763\) 9345.13 0.443403
\(764\) 23128.7 1.09525
\(765\) − 6765.10i − 0.319729i
\(766\) 17881.6 0.843457
\(767\) 0 0
\(768\) 11508.2 0.540713
\(769\) − 22736.7i − 1.06620i −0.846052 0.533100i \(-0.821027\pi\)
0.846052 0.533100i \(-0.178973\pi\)
\(770\) 3343.60 0.156487
\(771\) 50.1905 0.00234444
\(772\) − 20488.8i − 0.955194i
\(773\) − 343.173i − 0.0159678i −0.999968 0.00798388i \(-0.997459\pi\)
0.999968 0.00798388i \(-0.00254137\pi\)
\(774\) − 4236.80i − 0.196755i
\(775\) 18335.4i 0.849844i
\(776\) 590.999 0.0273397
\(777\) 5958.69 0.275118
\(778\) 10323.8i 0.475740i
\(779\) 2685.86 0.123531
\(780\) 0 0
\(781\) −5475.37 −0.250863
\(782\) 82288.3i 3.76294i
\(783\) −5386.11 −0.245829
\(784\) −19069.8 −0.868706
\(785\) − 29511.9i − 1.34181i
\(786\) 27913.8i 1.26673i
\(787\) 19086.4i 0.864496i 0.901755 + 0.432248i \(0.142279\pi\)
−0.901755 + 0.432248i \(0.857721\pi\)
\(788\) − 22673.7i − 1.02502i
\(789\) −1202.21 −0.0542455
\(790\) 20336.6 0.915876
\(791\) 7882.74i 0.354334i
\(792\) −151.848 −0.00681272
\(793\) 0 0
\(794\) 8894.36 0.397543
\(795\) − 645.995i − 0.0288190i
\(796\) 32959.5 1.46761
\(797\) −12031.1 −0.534709 −0.267354 0.963598i \(-0.586149\pi\)
−0.267354 + 0.963598i \(0.586149\pi\)
\(798\) 962.913i 0.0427152i
\(799\) 31467.0i 1.39327i
\(800\) − 15352.1i − 0.678471i
\(801\) − 11458.7i − 0.505461i
\(802\) −32528.2 −1.43218
\(803\) 1087.01 0.0477706
\(804\) 7071.93i 0.310208i
\(805\) −10575.5 −0.463027
\(806\) 0 0
\(807\) 6710.91 0.292732
\(808\) − 724.952i − 0.0315640i
\(809\) 10175.4 0.442210 0.221105 0.975250i \(-0.429034\pi\)
0.221105 + 0.975250i \(0.429034\pi\)
\(810\) −2640.54 −0.114542
\(811\) 26754.1i 1.15840i 0.815185 + 0.579201i \(0.196635\pi\)
−0.815185 + 0.579201i \(0.803365\pi\)
\(812\) − 9785.08i − 0.422893i
\(813\) 12056.7i 0.520106i
\(814\) 23185.7i 0.998353i
\(815\) 11214.9 0.482011
\(816\) 17284.4 0.741513
\(817\) − 1563.23i − 0.0669408i
\(818\) −25383.9 −1.08500
\(819\) 0 0
\(820\) −13380.7 −0.569845
\(821\) − 15777.5i − 0.670693i −0.942095 0.335347i \(-0.891147\pi\)
0.942095 0.335347i \(-0.108853\pi\)
\(822\) −4532.97 −0.192342
\(823\) 13863.9 0.587197 0.293599 0.955929i \(-0.405147\pi\)
0.293599 + 0.955929i \(0.405147\pi\)
\(824\) − 1848.70i − 0.0781583i
\(825\) − 3080.42i − 0.129996i
\(826\) 6716.66i 0.282933i
\(827\) 26835.0i 1.12835i 0.825655 + 0.564175i \(0.190806\pi\)
−0.825655 + 0.564175i \(0.809194\pi\)
\(828\) 16299.4 0.684111
\(829\) −625.251 −0.0261953 −0.0130976 0.999914i \(-0.504169\pi\)
−0.0130976 + 0.999914i \(0.504169\pi\)
\(830\) − 21511.4i − 0.899604i
\(831\) −20378.9 −0.850704
\(832\) 0 0
\(833\) −28584.2 −1.18893
\(834\) 5896.78i 0.244831i
\(835\) −27308.8 −1.13181
\(836\) −1901.40 −0.0786617
\(837\) 8309.98i 0.343172i
\(838\) − 5771.35i − 0.237909i
\(839\) 27307.4i 1.12367i 0.827251 + 0.561833i \(0.189904\pi\)
−0.827251 + 0.561833i \(0.810096\pi\)
\(840\) − 141.353i − 0.00580610i
\(841\) 15405.4 0.631656
\(842\) −38857.3 −1.59039
\(843\) 21860.4i 0.893135i
\(844\) 11340.6 0.462510
\(845\) 0 0
\(846\) 12282.1 0.499135
\(847\) − 6152.66i − 0.249596i
\(848\) 1650.48 0.0668368
\(849\) 7289.30 0.294662
\(850\) − 22312.2i − 0.900356i
\(851\) − 73334.2i − 2.95401i
\(852\) 7855.63i 0.315880i
\(853\) − 44801.9i − 1.79834i −0.437595 0.899172i \(-0.644170\pi\)
0.437595 0.899172i \(-0.355830\pi\)
\(854\) −4966.53 −0.199006
\(855\) −974.268 −0.0389699
\(856\) 900.051i 0.0359382i
\(857\) 25167.1 1.00314 0.501571 0.865116i \(-0.332756\pi\)
0.501571 + 0.865116i \(0.332756\pi\)
\(858\) 0 0
\(859\) 4059.10 0.161228 0.0806138 0.996745i \(-0.474312\pi\)
0.0806138 + 0.996745i \(0.474312\pi\)
\(860\) 7787.87i 0.308795i
\(861\) −3582.76 −0.141812
\(862\) −13584.4 −0.536759
\(863\) 818.924i 0.0323019i 0.999870 + 0.0161509i \(0.00514122\pi\)
−0.999870 + 0.0161509i \(0.994859\pi\)
\(864\) − 6957.85i − 0.273971i
\(865\) − 2764.84i − 0.108679i
\(866\) − 25184.6i − 0.988231i
\(867\) 11168.9 0.437505
\(868\) −15096.9 −0.590350
\(869\) 10752.3i 0.419733i
\(870\) 19509.2 0.760258
\(871\) 0 0
\(872\) 1537.25 0.0596993
\(873\) 5433.71i 0.210657i
\(874\) 11850.7 0.458644
\(875\) 8884.23 0.343248
\(876\) − 1559.56i − 0.0601513i
\(877\) 190.251i 0.00732533i 0.999993 + 0.00366267i \(0.00116587\pi\)
−0.999993 + 0.00366267i \(0.998834\pi\)
\(878\) − 11551.3i − 0.444007i
\(879\) − 12633.8i − 0.484785i
\(880\) −8643.43 −0.331102
\(881\) 19803.4 0.757315 0.378657 0.925537i \(-0.376386\pi\)
0.378657 + 0.925537i \(0.376386\pi\)
\(882\) 11156.9i 0.425933i
\(883\) −19652.1 −0.748974 −0.374487 0.927232i \(-0.622181\pi\)
−0.374487 + 0.927232i \(0.622181\pi\)
\(884\) 0 0
\(885\) −6795.87 −0.258125
\(886\) 43431.2i 1.64684i
\(887\) −26295.0 −0.995379 −0.497689 0.867355i \(-0.665818\pi\)
−0.497689 + 0.867355i \(0.665818\pi\)
\(888\) 980.189 0.0370417
\(889\) − 13986.6i − 0.527668i
\(890\) 41505.1i 1.56321i
\(891\) − 1396.11i − 0.0524930i
\(892\) − 13966.5i − 0.524253i
\(893\) 4531.69 0.169818
\(894\) −12766.7 −0.477608
\(895\) 23340.2i 0.871704i
\(896\) 745.275 0.0277878
\(897\) 0 0
\(898\) 55758.0 2.07201
\(899\) − 61397.0i − 2.27776i
\(900\) −4419.54 −0.163687
\(901\) 2473.93 0.0914746
\(902\) − 13940.8i − 0.514609i
\(903\) 2085.25i 0.0768470i
\(904\) 1296.69i 0.0477072i
\(905\) − 30701.4i − 1.12768i
\(906\) −10743.3 −0.393956
\(907\) −42417.3 −1.55286 −0.776429 0.630204i \(-0.782971\pi\)
−0.776429 + 0.630204i \(0.782971\pi\)
\(908\) 3341.66i 0.122133i
\(909\) 6665.29 0.243205
\(910\) 0 0
\(911\) −872.245 −0.0317220 −0.0158610 0.999874i \(-0.505049\pi\)
−0.0158610 + 0.999874i \(0.505049\pi\)
\(912\) − 2489.19i − 0.0903788i
\(913\) 11373.5 0.412276
\(914\) −69453.2 −2.51347
\(915\) − 5025.10i − 0.181557i
\(916\) 19449.4i 0.701557i
\(917\) − 13738.5i − 0.494749i
\(918\) − 10112.3i − 0.363569i
\(919\) −17181.4 −0.616716 −0.308358 0.951270i \(-0.599779\pi\)
−0.308358 + 0.951270i \(0.599779\pi\)
\(920\) −1739.64 −0.0623415
\(921\) − 24636.8i − 0.881444i
\(922\) −46287.5 −1.65336
\(923\) 0 0
\(924\) 2536.34 0.0903025
\(925\) 19884.4i 0.706804i
\(926\) 6503.25 0.230788
\(927\) 16997.1 0.602221
\(928\) 51407.0i 1.81845i
\(929\) − 56042.9i − 1.97923i −0.143728 0.989617i \(-0.545909\pi\)
0.143728 0.989617i \(-0.454091\pi\)
\(930\) − 30099.9i − 1.06130i
\(931\) 4116.52i 0.144912i
\(932\) 7736.00 0.271890
\(933\) 12715.7 0.446187
\(934\) 32843.3i 1.15060i
\(935\) −12955.8 −0.453155
\(936\) 0 0
\(937\) −36672.5 −1.27859 −0.639295 0.768961i \(-0.720774\pi\)
−0.639295 + 0.768961i \(0.720774\pi\)
\(938\) − 6858.71i − 0.238747i
\(939\) 11423.5 0.397008
\(940\) −22576.4 −0.783363
\(941\) 21069.0i 0.729895i 0.931028 + 0.364947i \(0.118913\pi\)
−0.931028 + 0.364947i \(0.881087\pi\)
\(942\) − 44113.7i − 1.52580i
\(943\) 44093.4i 1.52267i
\(944\) − 17363.0i − 0.598642i
\(945\) 1299.61 0.0447369
\(946\) −8113.87 −0.278863
\(947\) − 1838.70i − 0.0630938i −0.999502 0.0315469i \(-0.989957\pi\)
0.999502 0.0315469i \(-0.0100434\pi\)
\(948\) 15426.6 0.528516
\(949\) 0 0
\(950\) −3213.27 −0.109739
\(951\) − 31493.0i − 1.07385i
\(952\) 541.330 0.0184292
\(953\) −13599.8 −0.462266 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(954\) − 965.620i − 0.0327705i
\(955\) − 22695.9i − 0.769029i
\(956\) − 38446.9i − 1.30069i
\(957\) 10314.9i 0.348416i
\(958\) 7365.75 0.248410
\(959\) 2231.02 0.0751235
\(960\) 13166.8i 0.442662i
\(961\) −64935.6 −2.17971
\(962\) 0 0
\(963\) −8275.17 −0.276909
\(964\) − 31116.4i − 1.03962i
\(965\) −20105.5 −0.670692
\(966\) −15808.0 −0.526516
\(967\) − 2081.30i − 0.0692141i −0.999401 0.0346070i \(-0.988982\pi\)
0.999401 0.0346070i \(-0.0110180\pi\)
\(968\) − 1012.10i − 0.0336054i
\(969\) − 3731.10i − 0.123695i
\(970\) − 19681.6i − 0.651484i
\(971\) 1636.62 0.0540904 0.0270452 0.999634i \(-0.491390\pi\)
0.0270452 + 0.999634i \(0.491390\pi\)
\(972\) −2003.02 −0.0660977
\(973\) − 2902.26i − 0.0956240i
\(974\) 31592.9 1.03932
\(975\) 0 0
\(976\) 12838.8 0.421066
\(977\) 29387.1i 0.962311i 0.876635 + 0.481156i \(0.159783\pi\)
−0.876635 + 0.481156i \(0.840217\pi\)
\(978\) 16763.7 0.548104
\(979\) −21944.6 −0.716396
\(980\) − 20508.1i − 0.668476i
\(981\) 14133.6i 0.459992i
\(982\) − 69304.4i − 2.25213i
\(983\) 24084.4i 0.781457i 0.920506 + 0.390728i \(0.127777\pi\)
−0.920506 + 0.390728i \(0.872223\pi\)
\(984\) −589.355 −0.0190934
\(985\) −22249.4 −0.719722
\(986\) 74713.4i 2.41314i
\(987\) −6044.98 −0.194948
\(988\) 0 0
\(989\) 25663.4 0.825125
\(990\) 5056.88i 0.162342i
\(991\) 1413.43 0.0453068 0.0226534 0.999743i \(-0.492789\pi\)
0.0226534 + 0.999743i \(0.492789\pi\)
\(992\) 79313.5 2.53851
\(993\) 35715.7i 1.14139i
\(994\) − 7618.79i − 0.243112i
\(995\) − 32342.7i − 1.03049i
\(996\) − 16317.8i − 0.519126i
\(997\) −33357.4 −1.05962 −0.529809 0.848117i \(-0.677737\pi\)
−0.529809 + 0.848117i \(0.677737\pi\)
\(998\) −21583.1 −0.684570
\(999\) 9011.97i 0.285411i
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.k.337.15 18
13.5 odd 4 507.4.a.p.1.7 yes 9
13.8 odd 4 507.4.a.o.1.3 9
13.12 even 2 inner 507.4.b.k.337.4 18
39.5 even 4 1521.4.a.bf.1.3 9
39.8 even 4 1521.4.a.bi.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.3 9 13.8 odd 4
507.4.a.p.1.7 yes 9 13.5 odd 4
507.4.b.k.337.4 18 13.12 even 2 inner
507.4.b.k.337.15 18 1.1 even 1 trivial
1521.4.a.bf.1.3 9 39.5 even 4
1521.4.a.bi.1.7 9 39.8 even 4