Properties

Label 624.4.c.c.337.1
Level $624$
Weight $4$
Character 624.337
Analytic conductor $36.817$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1362828.1
Defining polynomial: \( x^{4} + 23x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(4.54739i\) of defining polynomial
Character \(\chi\) \(=\) 624.337
Dual form 624.4.c.c.337.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -12.9118i q^{5} -16.7289i q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -12.9118i q^{5} -16.7289i q^{7} +9.00000 q^{9} -24.9280i q^{11} +(33.7151 - 32.5621i) q^{13} +38.7355i q^{15} +134.145 q^{17} -14.9376i q^{19} +50.1866i q^{21} +72.0000 q^{23} -41.7151 q^{25} -27.0000 q^{27} -206.145 q^{29} -249.142i q^{31} +74.7841i q^{33} -216.000 q^{35} +293.955i q^{37} +(-101.145 + 97.6863i) q^{39} -250.506i q^{41} +432.145 q^{43} -116.206i q^{45} +159.889i q^{47} +63.1454 q^{49} -402.436 q^{51} -194.581 q^{53} -321.866 q^{55} +44.8129i q^{57} -232.647i q^{59} -185.006 q^{61} -150.560i q^{63} +(-420.436 - 435.324i) q^{65} -39.4393i q^{67} -216.000 q^{69} +920.460i q^{71} -549.078i q^{73} +125.145 q^{75} -417.018 q^{77} -933.140 q^{79} +81.0000 q^{81} -1095.38i q^{83} -1732.06i q^{85} +618.436 q^{87} -532.114i q^{89} +(-544.727 - 564.015i) q^{91} +747.425i q^{93} -192.872 q^{95} +362.661i q^{97} -224.352i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 12.9118i 1.15487i −0.816437 0.577434i \(-0.804054\pi\)
0.816437 0.577434i \(-0.195946\pi\)
\(6\) 0 0
\(7\) 16.7289i 0.903273i −0.892202 0.451637i \(-0.850840\pi\)
0.892202 0.451637i \(-0.149160\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 24.9280i 0.683280i −0.939831 0.341640i \(-0.889018\pi\)
0.939831 0.341640i \(-0.110982\pi\)
\(12\) 0 0
\(13\) 33.7151 32.5621i 0.719299 0.694700i
\(14\) 0 0
\(15\) 38.7355i 0.666764i
\(16\) 0 0
\(17\) 134.145 1.91383 0.956913 0.290376i \(-0.0937804\pi\)
0.956913 + 0.290376i \(0.0937804\pi\)
\(18\) 0 0
\(19\) 14.9376i 0.180365i −0.995925 0.0901824i \(-0.971255\pi\)
0.995925 0.0901824i \(-0.0287450\pi\)
\(20\) 0 0
\(21\) 50.1866i 0.521505i
\(22\) 0 0
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) −41.7151 −0.333721
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −206.145 −1.32001 −0.660004 0.751262i \(-0.729445\pi\)
−0.660004 + 0.751262i \(0.729445\pi\)
\(30\) 0 0
\(31\) 249.142i 1.44346i −0.692176 0.721728i \(-0.743348\pi\)
0.692176 0.721728i \(-0.256652\pi\)
\(32\) 0 0
\(33\) 74.7841i 0.394492i
\(34\) 0 0
\(35\) −216.000 −1.04316
\(36\) 0 0
\(37\) 293.955i 1.30610i 0.757313 + 0.653052i \(0.226512\pi\)
−0.757313 + 0.653052i \(0.773488\pi\)
\(38\) 0 0
\(39\) −101.145 + 97.6863i −0.415288 + 0.401085i
\(40\) 0 0
\(41\) 250.506i 0.954208i −0.878847 0.477104i \(-0.841686\pi\)
0.878847 0.477104i \(-0.158314\pi\)
\(42\) 0 0
\(43\) 432.145 1.53259 0.766297 0.642486i \(-0.222097\pi\)
0.766297 + 0.642486i \(0.222097\pi\)
\(44\) 0 0
\(45\) 116.206i 0.384956i
\(46\) 0 0
\(47\) 159.889i 0.496217i 0.968732 + 0.248109i \(0.0798090\pi\)
−0.968732 + 0.248109i \(0.920191\pi\)
\(48\) 0 0
\(49\) 63.1454 0.184097
\(50\) 0 0
\(51\) −402.436 −1.10495
\(52\) 0 0
\(53\) −194.581 −0.504298 −0.252149 0.967688i \(-0.581137\pi\)
−0.252149 + 0.967688i \(0.581137\pi\)
\(54\) 0 0
\(55\) −321.866 −0.789099
\(56\) 0 0
\(57\) 44.8129i 0.104134i
\(58\) 0 0
\(59\) 232.647i 0.513358i −0.966497 0.256679i \(-0.917372\pi\)
0.966497 0.256679i \(-0.0826283\pi\)
\(60\) 0 0
\(61\) −185.006 −0.388321 −0.194160 0.980970i \(-0.562198\pi\)
−0.194160 + 0.980970i \(0.562198\pi\)
\(62\) 0 0
\(63\) 150.560i 0.301091i
\(64\) 0 0
\(65\) −420.436 435.324i −0.802287 0.830696i
\(66\) 0 0
\(67\) 39.4393i 0.0719145i −0.999353 0.0359573i \(-0.988552\pi\)
0.999353 0.0359573i \(-0.0114480\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) 920.460i 1.53857i 0.638905 + 0.769286i \(0.279388\pi\)
−0.638905 + 0.769286i \(0.720612\pi\)
\(72\) 0 0
\(73\) 549.078i 0.880338i −0.897915 0.440169i \(-0.854918\pi\)
0.897915 0.440169i \(-0.145082\pi\)
\(74\) 0 0
\(75\) 125.145 0.192674
\(76\) 0 0
\(77\) −417.018 −0.617189
\(78\) 0 0
\(79\) −933.140 −1.32894 −0.664471 0.747314i \(-0.731343\pi\)
−0.664471 + 0.747314i \(0.731343\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1095.38i 1.44860i −0.689484 0.724301i \(-0.742163\pi\)
0.689484 0.724301i \(-0.257837\pi\)
\(84\) 0 0
\(85\) 1732.06i 2.21022i
\(86\) 0 0
\(87\) 618.436 0.762107
\(88\) 0 0
\(89\) 532.114i 0.633753i −0.948467 0.316876i \(-0.897366\pi\)
0.948467 0.316876i \(-0.102634\pi\)
\(90\) 0 0
\(91\) −544.727 564.015i −0.627504 0.649724i
\(92\) 0 0
\(93\) 747.425i 0.833380i
\(94\) 0 0
\(95\) −192.872 −0.208298
\(96\) 0 0
\(97\) 362.661i 0.379615i 0.981821 + 0.189808i \(0.0607864\pi\)
−0.981821 + 0.189808i \(0.939214\pi\)
\(98\) 0 0
\(99\) 224.352i 0.227760i
\(100\) 0 0
\(101\) −1490.58 −1.46850 −0.734249 0.678880i \(-0.762466\pi\)
−0.734249 + 0.678880i \(0.762466\pi\)
\(102\) 0 0
\(103\) 628.436 0.601181 0.300591 0.953753i \(-0.402816\pi\)
0.300591 + 0.953753i \(0.402816\pi\)
\(104\) 0 0
\(105\) 648.000 0.602270
\(106\) 0 0
\(107\) −477.454 −0.431376 −0.215688 0.976462i \(-0.569199\pi\)
−0.215688 + 0.976462i \(0.569199\pi\)
\(108\) 0 0
\(109\) 378.207i 0.332345i 0.986097 + 0.166173i \(0.0531409\pi\)
−0.986097 + 0.166173i \(0.946859\pi\)
\(110\) 0 0
\(111\) 881.864i 0.754079i
\(112\) 0 0
\(113\) 13.2732 0.0110499 0.00552495 0.999985i \(-0.498241\pi\)
0.00552495 + 0.999985i \(0.498241\pi\)
\(114\) 0 0
\(115\) 929.651i 0.753830i
\(116\) 0 0
\(117\) 303.436 293.059i 0.239766 0.231567i
\(118\) 0 0
\(119\) 2244.10i 1.72871i
\(120\) 0 0
\(121\) 709.593 0.533128
\(122\) 0 0
\(123\) 751.519i 0.550912i
\(124\) 0 0
\(125\) 1075.36i 0.769465i
\(126\) 0 0
\(127\) 145.988 0.102003 0.0510015 0.998699i \(-0.483759\pi\)
0.0510015 + 0.998699i \(0.483759\pi\)
\(128\) 0 0
\(129\) −1296.44 −0.884844
\(130\) 0 0
\(131\) −317.163 −0.211532 −0.105766 0.994391i \(-0.533729\pi\)
−0.105766 + 0.994391i \(0.533729\pi\)
\(132\) 0 0
\(133\) −249.890 −0.162919
\(134\) 0 0
\(135\) 348.619i 0.222255i
\(136\) 0 0
\(137\) 443.149i 0.276356i 0.990407 + 0.138178i \(0.0441247\pi\)
−0.990407 + 0.138178i \(0.955875\pi\)
\(138\) 0 0
\(139\) −785.018 −0.479024 −0.239512 0.970893i \(-0.576987\pi\)
−0.239512 + 0.970893i \(0.576987\pi\)
\(140\) 0 0
\(141\) 479.667i 0.286491i
\(142\) 0 0
\(143\) −811.709 840.452i −0.474675 0.491483i
\(144\) 0 0
\(145\) 2661.71i 1.52444i
\(146\) 0 0
\(147\) −189.436 −0.106289
\(148\) 0 0
\(149\) 135.420i 0.0744566i −0.999307 0.0372283i \(-0.988147\pi\)
0.999307 0.0372283i \(-0.0118529\pi\)
\(150\) 0 0
\(151\) 2373.74i 1.27929i 0.768672 + 0.639643i \(0.220918\pi\)
−0.768672 + 0.639643i \(0.779082\pi\)
\(152\) 0 0
\(153\) 1207.31 0.637942
\(154\) 0 0
\(155\) −3216.87 −1.66700
\(156\) 0 0
\(157\) −1166.73 −0.593089 −0.296544 0.955019i \(-0.595834\pi\)
−0.296544 + 0.955019i \(0.595834\pi\)
\(158\) 0 0
\(159\) 583.744 0.291157
\(160\) 0 0
\(161\) 1204.48i 0.589603i
\(162\) 0 0
\(163\) 2309.19i 1.10963i 0.831974 + 0.554815i \(0.187211\pi\)
−0.831974 + 0.554815i \(0.812789\pi\)
\(164\) 0 0
\(165\) 965.599 0.455587
\(166\) 0 0
\(167\) 600.788i 0.278386i 0.990265 + 0.139193i \(0.0444508\pi\)
−0.990265 + 0.139193i \(0.955549\pi\)
\(168\) 0 0
\(169\) 76.4186 2195.67i 0.0347831 0.999395i
\(170\) 0 0
\(171\) 134.439i 0.0601216i
\(172\) 0 0
\(173\) −3430.36 −1.50755 −0.753773 0.657135i \(-0.771768\pi\)
−0.753773 + 0.657135i \(0.771768\pi\)
\(174\) 0 0
\(175\) 697.846i 0.301441i
\(176\) 0 0
\(177\) 697.942i 0.296387i
\(178\) 0 0
\(179\) 978.837 0.408725 0.204362 0.978895i \(-0.434488\pi\)
0.204362 + 0.978895i \(0.434488\pi\)
\(180\) 0 0
\(181\) 3839.09 1.57656 0.788279 0.615318i \(-0.210972\pi\)
0.788279 + 0.615318i \(0.210972\pi\)
\(182\) 0 0
\(183\) 555.018 0.224197
\(184\) 0 0
\(185\) 3795.49 1.50838
\(186\) 0 0
\(187\) 3343.98i 1.30768i
\(188\) 0 0
\(189\) 451.679i 0.173835i
\(190\) 0 0
\(191\) 487.709 0.184761 0.0923806 0.995724i \(-0.470552\pi\)
0.0923806 + 0.995724i \(0.470552\pi\)
\(192\) 0 0
\(193\) 4245.61i 1.58345i 0.610878 + 0.791725i \(0.290817\pi\)
−0.610878 + 0.791725i \(0.709183\pi\)
\(194\) 0 0
\(195\) 1261.31 + 1305.97i 0.463201 + 0.479603i
\(196\) 0 0
\(197\) 2712.71i 0.981079i 0.871419 + 0.490539i \(0.163200\pi\)
−0.871419 + 0.490539i \(0.836800\pi\)
\(198\) 0 0
\(199\) 3116.90 1.11031 0.555153 0.831748i \(-0.312660\pi\)
0.555153 + 0.831748i \(0.312660\pi\)
\(200\) 0 0
\(201\) 118.318i 0.0415199i
\(202\) 0 0
\(203\) 3448.58i 1.19233i
\(204\) 0 0
\(205\) −3234.49 −1.10198
\(206\) 0 0
\(207\) 648.000 0.217580
\(208\) 0 0
\(209\) −372.366 −0.123240
\(210\) 0 0
\(211\) 1051.22 0.342981 0.171491 0.985186i \(-0.445142\pi\)
0.171491 + 0.985186i \(0.445142\pi\)
\(212\) 0 0
\(213\) 2761.38i 0.888294i
\(214\) 0 0
\(215\) 5579.78i 1.76994i
\(216\) 0 0
\(217\) −4167.85 −1.30384
\(218\) 0 0
\(219\) 1647.23i 0.508264i
\(220\) 0 0
\(221\) 4522.73 4368.06i 1.37661 1.32953i
\(222\) 0 0
\(223\) 5496.12i 1.65044i 0.564814 + 0.825218i \(0.308948\pi\)
−0.564814 + 0.825218i \(0.691052\pi\)
\(224\) 0 0
\(225\) −375.436 −0.111240
\(226\) 0 0
\(227\) 921.570i 0.269457i −0.990883 0.134729i \(-0.956984\pi\)
0.990883 0.134729i \(-0.0430162\pi\)
\(228\) 0 0
\(229\) 192.941i 0.0556764i 0.999612 + 0.0278382i \(0.00886232\pi\)
−0.999612 + 0.0278382i \(0.991138\pi\)
\(230\) 0 0
\(231\) 1251.05 0.356334
\(232\) 0 0
\(233\) −913.779 −0.256926 −0.128463 0.991714i \(-0.541004\pi\)
−0.128463 + 0.991714i \(0.541004\pi\)
\(234\) 0 0
\(235\) 2064.46 0.573066
\(236\) 0 0
\(237\) 2799.42 0.767265
\(238\) 0 0
\(239\) 1976.86i 0.535032i −0.963553 0.267516i \(-0.913797\pi\)
0.963553 0.267516i \(-0.0862028\pi\)
\(240\) 0 0
\(241\) 3904.45i 1.04360i 0.853068 + 0.521800i \(0.174739\pi\)
−0.853068 + 0.521800i \(0.825261\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 815.322i 0.212608i
\(246\) 0 0
\(247\) −486.401 503.624i −0.125299 0.129736i
\(248\) 0 0
\(249\) 3286.15i 0.836350i
\(250\) 0 0
\(251\) −942.035 −0.236895 −0.118448 0.992960i \(-0.537792\pi\)
−0.118448 + 0.992960i \(0.537792\pi\)
\(252\) 0 0
\(253\) 1794.82i 0.446005i
\(254\) 0 0
\(255\) 5196.18i 1.27607i
\(256\) 0 0
\(257\) 812.616 0.197236 0.0986179 0.995125i \(-0.468558\pi\)
0.0986179 + 0.995125i \(0.468558\pi\)
\(258\) 0 0
\(259\) 4917.52 1.17977
\(260\) 0 0
\(261\) −1855.31 −0.440003
\(262\) 0 0
\(263\) 2608.29 0.611536 0.305768 0.952106i \(-0.401087\pi\)
0.305768 + 0.952106i \(0.401087\pi\)
\(264\) 0 0
\(265\) 2512.40i 0.582398i
\(266\) 0 0
\(267\) 1596.34i 0.365897i
\(268\) 0 0
\(269\) −4791.02 −1.08592 −0.542962 0.839757i \(-0.682697\pi\)
−0.542962 + 0.839757i \(0.682697\pi\)
\(270\) 0 0
\(271\) 3663.62i 0.821214i 0.911812 + 0.410607i \(0.134683\pi\)
−0.911812 + 0.410607i \(0.865317\pi\)
\(272\) 0 0
\(273\) 1634.18 + 1692.05i 0.362290 + 0.375118i
\(274\) 0 0
\(275\) 1039.88i 0.228025i
\(276\) 0 0
\(277\) 624.326 0.135423 0.0677114 0.997705i \(-0.478430\pi\)
0.0677114 + 0.997705i \(0.478430\pi\)
\(278\) 0 0
\(279\) 2242.27i 0.481152i
\(280\) 0 0
\(281\) 5535.12i 1.17508i 0.809195 + 0.587540i \(0.199904\pi\)
−0.809195 + 0.587540i \(0.800096\pi\)
\(282\) 0 0
\(283\) 175.151 0.0367903 0.0183952 0.999831i \(-0.494144\pi\)
0.0183952 + 0.999831i \(0.494144\pi\)
\(284\) 0 0
\(285\) 578.616 0.120261
\(286\) 0 0
\(287\) −4190.69 −0.861911
\(288\) 0 0
\(289\) 13082.0 2.66273
\(290\) 0 0
\(291\) 1087.98i 0.219171i
\(292\) 0 0
\(293\) 7774.33i 1.55011i −0.631895 0.775054i \(-0.717723\pi\)
0.631895 0.775054i \(-0.282277\pi\)
\(294\) 0 0
\(295\) −3003.90 −0.592861
\(296\) 0 0
\(297\) 673.057i 0.131497i
\(298\) 0 0
\(299\) 2427.49 2344.47i 0.469516 0.453459i
\(300\) 0 0
\(301\) 7229.30i 1.38435i
\(302\) 0 0
\(303\) 4471.74 0.847838
\(304\) 0 0
\(305\) 2388.76i 0.448459i
\(306\) 0 0
\(307\) 8022.85i 1.49149i −0.666230 0.745746i \(-0.732093\pi\)
0.666230 0.745746i \(-0.267907\pi\)
\(308\) 0 0
\(309\) −1885.31 −0.347092
\(310\) 0 0
\(311\) −9264.87 −1.68927 −0.844635 0.535343i \(-0.820182\pi\)
−0.844635 + 0.535343i \(0.820182\pi\)
\(312\) 0 0
\(313\) 7423.57 1.34059 0.670296 0.742094i \(-0.266167\pi\)
0.670296 + 0.742094i \(0.266167\pi\)
\(314\) 0 0
\(315\) −1944.00 −0.347721
\(316\) 0 0
\(317\) 2641.04i 0.467935i −0.972244 0.233968i \(-0.924829\pi\)
0.972244 0.233968i \(-0.0751710\pi\)
\(318\) 0 0
\(319\) 5138.80i 0.901936i
\(320\) 0 0
\(321\) 1432.36 0.249055
\(322\) 0 0
\(323\) 2003.82i 0.345187i
\(324\) 0 0
\(325\) −1406.43 + 1358.33i −0.240045 + 0.231836i
\(326\) 0 0
\(327\) 1134.62i 0.191880i
\(328\) 0 0
\(329\) 2674.76 0.448220
\(330\) 0 0
\(331\) 10779.5i 1.79001i 0.446052 + 0.895007i \(0.352830\pi\)
−0.446052 + 0.895007i \(0.647170\pi\)
\(332\) 0 0
\(333\) 2645.59i 0.435368i
\(334\) 0 0
\(335\) −509.233 −0.0830518
\(336\) 0 0
\(337\) 313.465 0.0506693 0.0253346 0.999679i \(-0.491935\pi\)
0.0253346 + 0.999679i \(0.491935\pi\)
\(338\) 0 0
\(339\) −39.8196 −0.00637966
\(340\) 0 0
\(341\) −6210.61 −0.986286
\(342\) 0 0
\(343\) 6794.35i 1.06956i
\(344\) 0 0
\(345\) 2788.95i 0.435224i
\(346\) 0 0
\(347\) 2849.23 0.440792 0.220396 0.975410i \(-0.429265\pi\)
0.220396 + 0.975410i \(0.429265\pi\)
\(348\) 0 0
\(349\) 6466.94i 0.991883i 0.868356 + 0.495941i \(0.165177\pi\)
−0.868356 + 0.495941i \(0.834823\pi\)
\(350\) 0 0
\(351\) −910.308 + 879.177i −0.138429 + 0.133695i
\(352\) 0 0
\(353\) 2773.10i 0.418122i 0.977903 + 0.209061i \(0.0670408\pi\)
−0.977903 + 0.209061i \(0.932959\pi\)
\(354\) 0 0
\(355\) 11884.8 1.77685
\(356\) 0 0
\(357\) 6732.30i 0.998070i
\(358\) 0 0
\(359\) 1467.11i 0.215685i 0.994168 + 0.107843i \(0.0343942\pi\)
−0.994168 + 0.107843i \(0.965606\pi\)
\(360\) 0 0
\(361\) 6635.87 0.967469
\(362\) 0 0
\(363\) −2128.78 −0.307801
\(364\) 0 0
\(365\) −7089.59 −1.01667
\(366\) 0 0
\(367\) −4648.22 −0.661130 −0.330565 0.943783i \(-0.607239\pi\)
−0.330565 + 0.943783i \(0.607239\pi\)
\(368\) 0 0
\(369\) 2254.56i 0.318069i
\(370\) 0 0
\(371\) 3255.12i 0.455519i
\(372\) 0 0
\(373\) 1763.72 0.244831 0.122416 0.992479i \(-0.460936\pi\)
0.122416 + 0.992479i \(0.460936\pi\)
\(374\) 0 0
\(375\) 3226.08i 0.444251i
\(376\) 0 0
\(377\) −6950.22 + 6712.53i −0.949481 + 0.917010i
\(378\) 0 0
\(379\) 1930.47i 0.261640i 0.991406 + 0.130820i \(0.0417611\pi\)
−0.991406 + 0.130820i \(0.958239\pi\)
\(380\) 0 0
\(381\) −437.965 −0.0588914
\(382\) 0 0
\(383\) 8845.93i 1.18017i −0.807340 0.590086i \(-0.799094\pi\)
0.807340 0.590086i \(-0.200906\pi\)
\(384\) 0 0
\(385\) 5384.46i 0.712772i
\(386\) 0 0
\(387\) 3889.31 0.510865
\(388\) 0 0
\(389\) 1598.08 0.208292 0.104146 0.994562i \(-0.466789\pi\)
0.104146 + 0.994562i \(0.466789\pi\)
\(390\) 0 0
\(391\) 9658.47 1.24923
\(392\) 0 0
\(393\) 951.489 0.122128
\(394\) 0 0
\(395\) 12048.5i 1.53475i
\(396\) 0 0
\(397\) 3578.82i 0.452433i 0.974077 + 0.226217i \(0.0726357\pi\)
−0.974077 + 0.226217i \(0.927364\pi\)
\(398\) 0 0
\(399\) 749.669 0.0940611
\(400\) 0 0
\(401\) 3485.99i 0.434120i −0.976158 0.217060i \(-0.930353\pi\)
0.976158 0.217060i \(-0.0696467\pi\)
\(402\) 0 0
\(403\) −8112.58 8399.84i −1.00277 1.03828i
\(404\) 0 0
\(405\) 1045.86i 0.128319i
\(406\) 0 0
\(407\) 7327.71 0.892435
\(408\) 0 0
\(409\) 14709.1i 1.77828i −0.457637 0.889139i \(-0.651304\pi\)
0.457637 0.889139i \(-0.348696\pi\)
\(410\) 0 0
\(411\) 1329.45i 0.159554i
\(412\) 0 0
\(413\) −3891.92 −0.463702
\(414\) 0 0
\(415\) −14143.4 −1.67294
\(416\) 0 0
\(417\) 2355.05 0.276565
\(418\) 0 0
\(419\) −3709.01 −0.432451 −0.216226 0.976343i \(-0.569375\pi\)
−0.216226 + 0.976343i \(0.569375\pi\)
\(420\) 0 0
\(421\) 794.029i 0.0919207i −0.998943 0.0459603i \(-0.985365\pi\)
0.998943 0.0459603i \(-0.0146348\pi\)
\(422\) 0 0
\(423\) 1439.00i 0.165406i
\(424\) 0 0
\(425\) −5595.89 −0.638684
\(426\) 0 0
\(427\) 3094.94i 0.350760i
\(428\) 0 0
\(429\) 2435.13 + 2521.35i 0.274054 + 0.283758i
\(430\) 0 0
\(431\) 2891.52i 0.323155i 0.986860 + 0.161577i \(0.0516582\pi\)
−0.986860 + 0.161577i \(0.948342\pi\)
\(432\) 0 0
\(433\) −5560.94 −0.617186 −0.308593 0.951194i \(-0.599858\pi\)
−0.308593 + 0.951194i \(0.599858\pi\)
\(434\) 0 0
\(435\) 7985.14i 0.880133i
\(436\) 0 0
\(437\) 1075.51i 0.117731i
\(438\) 0 0
\(439\) 15127.2 1.64460 0.822302 0.569051i \(-0.192689\pi\)
0.822302 + 0.569051i \(0.192689\pi\)
\(440\) 0 0
\(441\) 568.308 0.0613658
\(442\) 0 0
\(443\) −2357.89 −0.252883 −0.126441 0.991974i \(-0.540356\pi\)
−0.126441 + 0.991974i \(0.540356\pi\)
\(444\) 0 0
\(445\) −6870.56 −0.731901
\(446\) 0 0
\(447\) 406.260i 0.0429876i
\(448\) 0 0
\(449\) 7165.06i 0.753096i −0.926397 0.376548i \(-0.877111\pi\)
0.926397 0.376548i \(-0.122889\pi\)
\(450\) 0 0
\(451\) −6244.63 −0.651992
\(452\) 0 0
\(453\) 7121.22i 0.738596i
\(454\) 0 0
\(455\) −7282.47 + 7033.41i −0.750346 + 0.724685i
\(456\) 0 0
\(457\) 8020.96i 0.821017i −0.911857 0.410508i \(-0.865351\pi\)
0.911857 0.410508i \(-0.134649\pi\)
\(458\) 0 0
\(459\) −3621.92 −0.368316
\(460\) 0 0
\(461\) 4146.59i 0.418928i 0.977816 + 0.209464i \(0.0671720\pi\)
−0.977816 + 0.209464i \(0.932828\pi\)
\(462\) 0 0
\(463\) 7118.21i 0.714495i −0.934010 0.357248i \(-0.883715\pi\)
0.934010 0.357248i \(-0.116285\pi\)
\(464\) 0 0
\(465\) 9650.62 0.962444
\(466\) 0 0
\(467\) −2128.22 −0.210883 −0.105441 0.994426i \(-0.533626\pi\)
−0.105441 + 0.994426i \(0.533626\pi\)
\(468\) 0 0
\(469\) −659.774 −0.0649585
\(470\) 0 0
\(471\) 3500.18 0.342420
\(472\) 0 0
\(473\) 10772.5i 1.04719i
\(474\) 0 0
\(475\) 623.125i 0.0601915i
\(476\) 0 0
\(477\) −1751.23 −0.168099
\(478\) 0 0
\(479\) 3715.30i 0.354397i 0.984175 + 0.177199i \(0.0567035\pi\)
−0.984175 + 0.177199i \(0.943296\pi\)
\(480\) 0 0
\(481\) 9571.78 + 9910.71i 0.907350 + 0.939479i
\(482\) 0 0
\(483\) 3613.43i 0.340408i
\(484\) 0 0
\(485\) 4682.62 0.438405
\(486\) 0 0
\(487\) 8139.28i 0.757343i −0.925531 0.378671i \(-0.876381\pi\)
0.925531 0.378671i \(-0.123619\pi\)
\(488\) 0 0
\(489\) 6927.57i 0.640645i
\(490\) 0 0
\(491\) 18081.7 1.66194 0.830972 0.556315i \(-0.187785\pi\)
0.830972 + 0.556315i \(0.187785\pi\)
\(492\) 0 0
\(493\) −27653.4 −2.52626
\(494\) 0 0
\(495\) −2896.80 −0.263033
\(496\) 0 0
\(497\) 15398.3 1.38975
\(498\) 0 0
\(499\) 11031.5i 0.989659i 0.868990 + 0.494829i \(0.164769\pi\)
−0.868990 + 0.494829i \(0.835231\pi\)
\(500\) 0 0
\(501\) 1802.37i 0.160726i
\(502\) 0 0
\(503\) −8016.14 −0.710581 −0.355290 0.934756i \(-0.615618\pi\)
−0.355290 + 0.934756i \(0.615618\pi\)
\(504\) 0 0
\(505\) 19246.1i 1.69592i
\(506\) 0 0
\(507\) −229.256 + 6587.01i −0.0200821 + 0.577001i
\(508\) 0 0
\(509\) 20173.9i 1.75676i −0.477959 0.878382i \(-0.658623\pi\)
0.477959 0.878382i \(-0.341377\pi\)
\(510\) 0 0
\(511\) −9185.44 −0.795186
\(512\) 0 0
\(513\) 403.316i 0.0347112i
\(514\) 0 0
\(515\) 8114.25i 0.694285i
\(516\) 0 0
\(517\) 3985.72 0.339056
\(518\) 0 0
\(519\) 10291.1 0.870382
\(520\) 0 0
\(521\) 9746.95 0.819619 0.409810 0.912171i \(-0.365595\pi\)
0.409810 + 0.912171i \(0.365595\pi\)
\(522\) 0 0
\(523\) 18929.3 1.58264 0.791320 0.611402i \(-0.209394\pi\)
0.791320 + 0.611402i \(0.209394\pi\)
\(524\) 0 0
\(525\) 2093.54i 0.174037i
\(526\) 0 0
\(527\) 33421.2i 2.76252i
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 2093.83i 0.171119i
\(532\) 0 0
\(533\) −8157.02 8445.85i −0.662889 0.686361i
\(534\) 0 0
\(535\) 6164.80i 0.498182i
\(536\) 0 0
\(537\) −2936.51 −0.235977
\(538\) 0 0
\(539\) 1574.09i 0.125790i
\(540\) 0 0
\(541\) 11366.8i 0.903321i −0.892190 0.451661i \(-0.850832\pi\)
0.892190 0.451661i \(-0.149168\pi\)
\(542\) 0 0
\(543\) −11517.3 −0.910227
\(544\) 0 0
\(545\) 4883.34 0.383815
\(546\) 0 0
\(547\) −17495.4 −1.36755 −0.683775 0.729693i \(-0.739663\pi\)
−0.683775 + 0.729693i \(0.739663\pi\)
\(548\) 0 0
\(549\) −1665.05 −0.129440
\(550\) 0 0
\(551\) 3079.33i 0.238083i
\(552\) 0 0
\(553\) 15610.4i 1.20040i
\(554\) 0 0
\(555\) −11386.5 −0.870862
\(556\) 0 0
\(557\) 11873.1i 0.903192i −0.892223 0.451596i \(-0.850855\pi\)
0.892223 0.451596i \(-0.149145\pi\)
\(558\) 0 0
\(559\) 14569.8 14071.6i 1.10239 1.06469i
\(560\) 0 0
\(561\) 10031.9i 0.754989i
\(562\) 0 0
\(563\) −2829.31 −0.211796 −0.105898 0.994377i \(-0.533772\pi\)
−0.105898 + 0.994377i \(0.533772\pi\)
\(564\) 0 0
\(565\) 171.381i 0.0127612i
\(566\) 0 0
\(567\) 1355.04i 0.100364i
\(568\) 0 0
\(569\) −16136.8 −1.18891 −0.594453 0.804130i \(-0.702632\pi\)
−0.594453 + 0.804130i \(0.702632\pi\)
\(570\) 0 0
\(571\) −17840.5 −1.30754 −0.653769 0.756695i \(-0.726813\pi\)
−0.653769 + 0.756695i \(0.726813\pi\)
\(572\) 0 0
\(573\) −1463.13 −0.106672
\(574\) 0 0
\(575\) −3003.49 −0.217833
\(576\) 0 0
\(577\) 8516.17i 0.614442i −0.951638 0.307221i \(-0.900601\pi\)
0.951638 0.307221i \(-0.0993990\pi\)
\(578\) 0 0
\(579\) 12736.8i 0.914205i
\(580\) 0 0
\(581\) −18324.5 −1.30848
\(582\) 0 0
\(583\) 4850.53i 0.344577i
\(584\) 0 0
\(585\) −3783.92 3917.91i −0.267429 0.276899i
\(586\) 0 0
\(587\) 20688.3i 1.45468i −0.686277 0.727340i \(-0.740756\pi\)
0.686277 0.727340i \(-0.259244\pi\)
\(588\) 0 0
\(589\) −3721.59 −0.260349
\(590\) 0 0
\(591\) 8138.13i 0.566426i
\(592\) 0 0
\(593\) 11435.9i 0.791933i 0.918265 + 0.395966i \(0.129590\pi\)
−0.918265 + 0.395966i \(0.870410\pi\)
\(594\) 0 0
\(595\) −28975.4 −1.99643
\(596\) 0 0
\(597\) −9350.69 −0.641035
\(598\) 0 0
\(599\) −1260.80 −0.0860016 −0.0430008 0.999075i \(-0.513692\pi\)
−0.0430008 + 0.999075i \(0.513692\pi\)
\(600\) 0 0
\(601\) 6261.10 0.424951 0.212476 0.977166i \(-0.431847\pi\)
0.212476 + 0.977166i \(0.431847\pi\)
\(602\) 0 0
\(603\) 354.954i 0.0239715i
\(604\) 0 0
\(605\) 9162.14i 0.615692i
\(606\) 0 0
\(607\) −3230.33 −0.216005 −0.108003 0.994151i \(-0.534445\pi\)
−0.108003 + 0.994151i \(0.534445\pi\)
\(608\) 0 0
\(609\) 10345.7i 0.688391i
\(610\) 0 0
\(611\) 5206.33 + 5390.68i 0.344722 + 0.356929i
\(612\) 0 0
\(613\) 14868.5i 0.979660i −0.871818 0.489830i \(-0.837059\pi\)
0.871818 0.489830i \(-0.162941\pi\)
\(614\) 0 0
\(615\) 9703.48 0.636231
\(616\) 0 0
\(617\) 19952.8i 1.30190i 0.759121 + 0.650949i \(0.225629\pi\)
−0.759121 + 0.650949i \(0.774371\pi\)
\(618\) 0 0
\(619\) 8316.48i 0.540012i −0.962859 0.270006i \(-0.912974\pi\)
0.962859 0.270006i \(-0.0870257\pi\)
\(620\) 0 0
\(621\) −1944.00 −0.125620
\(622\) 0 0
\(623\) −8901.66 −0.572452
\(624\) 0 0
\(625\) −19099.2 −1.22235
\(626\) 0 0
\(627\) 1117.10 0.0711525
\(628\) 0 0
\(629\) 39432.6i 2.49965i
\(630\) 0 0
\(631\) 12605.9i 0.795299i 0.917537 + 0.397649i \(0.130174\pi\)
−0.917537 + 0.397649i \(0.869826\pi\)
\(632\) 0 0
\(633\) −3153.66 −0.198020
\(634\) 0 0
\(635\) 1884.98i 0.117800i
\(636\) 0 0
\(637\) 2128.95 2056.15i 0.132421 0.127892i
\(638\) 0 0
\(639\) 8284.14i 0.512857i
\(640\) 0 0
\(641\) 9224.04 0.568374 0.284187 0.958769i \(-0.408276\pi\)
0.284187 + 0.958769i \(0.408276\pi\)
\(642\) 0 0
\(643\) 4439.16i 0.272260i 0.990691 + 0.136130i \(0.0434665\pi\)
−0.990691 + 0.136130i \(0.956533\pi\)
\(644\) 0 0
\(645\) 16739.4i 1.02188i
\(646\) 0 0
\(647\) −9601.15 −0.583401 −0.291700 0.956510i \(-0.594221\pi\)
−0.291700 + 0.956510i \(0.594221\pi\)
\(648\) 0 0
\(649\) −5799.44 −0.350767
\(650\) 0 0
\(651\) 12503.6 0.752770
\(652\) 0 0
\(653\) 27112.8 1.62482 0.812410 0.583087i \(-0.198155\pi\)
0.812410 + 0.583087i \(0.198155\pi\)
\(654\) 0 0
\(655\) 4095.15i 0.244291i
\(656\) 0 0
\(657\) 4941.70i 0.293446i
\(658\) 0 0
\(659\) −5587.26 −0.330271 −0.165136 0.986271i \(-0.552806\pi\)
−0.165136 + 0.986271i \(0.552806\pi\)
\(660\) 0 0
\(661\) 3060.13i 0.180069i −0.995939 0.0900343i \(-0.971302\pi\)
0.995939 0.0900343i \(-0.0286977\pi\)
\(662\) 0 0
\(663\) −13568.2 + 13104.2i −0.794788 + 0.767607i
\(664\) 0 0
\(665\) 3226.53i 0.188150i
\(666\) 0 0
\(667\) −14842.5 −0.861623
\(668\) 0 0
\(669\) 16488.4i 0.952880i
\(670\) 0 0
\(671\) 4611.83i 0.265332i
\(672\) 0 0
\(673\) −4121.55 −0.236069 −0.118034 0.993010i \(-0.537659\pi\)
−0.118034 + 0.993010i \(0.537659\pi\)
\(674\) 0 0
\(675\) 1126.31 0.0642246
\(676\) 0 0
\(677\) 22889.5 1.29943 0.649715 0.760178i \(-0.274888\pi\)
0.649715 + 0.760178i \(0.274888\pi\)
\(678\) 0 0
\(679\) 6066.91 0.342896
\(680\) 0 0
\(681\) 2764.71i 0.155571i
\(682\) 0 0
\(683\) 19297.9i 1.08113i 0.841301 + 0.540566i \(0.181790\pi\)
−0.841301 + 0.540566i \(0.818210\pi\)
\(684\) 0 0
\(685\) 5721.87 0.319155
\(686\) 0 0
\(687\) 578.823i 0.0321448i
\(688\) 0 0
\(689\) −6560.34 + 6335.98i −0.362742 + 0.350336i
\(690\) 0 0
\(691\) 30317.8i 1.66910i −0.550935 0.834548i \(-0.685729\pi\)
0.550935 0.834548i \(-0.314271\pi\)
\(692\) 0 0
\(693\) −3753.16 −0.205730
\(694\) 0 0
\(695\) 10136.0i 0.553210i
\(696\) 0 0
\(697\) 33604.3i 1.82619i
\(698\) 0 0
\(699\) 2741.34 0.148336
\(700\) 0 0
\(701\) −9606.16 −0.517574 −0.258787 0.965934i \(-0.583323\pi\)
−0.258787 + 0.965934i \(0.583323\pi\)
\(702\) 0 0
\(703\) 4390.99 0.235575
\(704\) 0 0
\(705\) −6193.38 −0.330860
\(706\) 0 0
\(707\) 24935.7i 1.32646i
\(708\) 0 0
\(709\) 23398.8i 1.23944i −0.784825 0.619718i \(-0.787247\pi\)
0.784825 0.619718i \(-0.212753\pi\)
\(710\) 0 0
\(711\) −8398.26 −0.442981
\(712\) 0 0
\(713\) 17938.2i 0.942203i
\(714\) 0 0
\(715\) −10851.8 + 10480.6i −0.567598 + 0.548187i
\(716\) 0 0
\(717\) 5930.59i 0.308901i
\(718\) 0 0
\(719\) −23588.7 −1.22352 −0.611758 0.791045i \(-0.709537\pi\)
−0.611758 + 0.791045i \(0.709537\pi\)
\(720\) 0 0
\(721\) 10513.0i 0.543031i
\(722\) 0 0
\(723\) 11713.3i 0.602522i
\(724\) 0 0
\(725\) 8599.38 0.440514
\(726\) 0 0
\(727\) 15733.5 0.802644 0.401322 0.915937i \(-0.368551\pi\)
0.401322 + 0.915937i \(0.368551\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 57970.3 2.93312
\(732\) 0 0
\(733\) 17297.1i 0.871598i 0.900044 + 0.435799i \(0.143534\pi\)
−0.900044 + 0.435799i \(0.856466\pi\)
\(734\) 0 0
\(735\) 2445.96i 0.122749i
\(736\) 0 0
\(737\) −983.144 −0.0491378
\(738\) 0 0
\(739\) 38749.2i 1.92884i −0.264377 0.964419i \(-0.585166\pi\)
0.264377 0.964419i \(-0.414834\pi\)
\(740\) 0 0
\(741\) 1459.20 + 1510.87i 0.0723417 + 0.0749033i
\(742\) 0 0
\(743\) 3139.76i 0.155029i −0.996991 0.0775144i \(-0.975302\pi\)
0.996991 0.0775144i \(-0.0246984\pi\)
\(744\) 0 0
\(745\) −1748.52 −0.0859876
\(746\) 0 0
\(747\) 9858.45i 0.482867i
\(748\) 0 0
\(749\) 7987.25i 0.389650i
\(750\) 0 0
\(751\) 40628.6 1.97411 0.987055 0.160380i \(-0.0512721\pi\)
0.987055 + 0.160380i \(0.0512721\pi\)
\(752\) 0 0
\(753\) 2826.11 0.136772
\(754\) 0 0
\(755\) 30649.3 1.47741
\(756\) 0 0
\(757\) 24004.9 1.15254 0.576271 0.817259i \(-0.304507\pi\)
0.576271 + 0.817259i \(0.304507\pi\)
\(758\) 0 0
\(759\) 5384.46i 0.257501i
\(760\) 0 0
\(761\) 29540.1i 1.40713i 0.710630 + 0.703566i \(0.248410\pi\)
−0.710630 + 0.703566i \(0.751590\pi\)
\(762\) 0 0
\(763\) 6326.97 0.300199
\(764\) 0 0
\(765\) 15588.5i 0.736739i
\(766\) 0 0
\(767\) −7575.49 7843.73i −0.356630 0.369258i
\(768\) 0 0
\(769\) 7585.63i 0.355715i −0.984056 0.177857i \(-0.943083\pi\)
0.984056 0.177857i \(-0.0569166\pi\)
\(770\) 0 0
\(771\) −2437.85 −0.113874
\(772\) 0 0
\(773\) 3284.29i 0.152817i −0.997077 0.0764086i \(-0.975655\pi\)
0.997077 0.0764086i \(-0.0243453\pi\)
\(774\) 0 0
\(775\) 10393.0i 0.481712i
\(776\) 0 0
\(777\) −14752.6 −0.681140
\(778\) 0 0
\(779\) −3741.98 −0.172106
\(780\) 0 0
\(781\) 22945.3 1.05128
\(782\) 0 0
\(783\) 5565.92 0.254036
\(784\) 0 0
\(785\) 15064.6i 0.684939i
\(786\) 0 0
\(787\) 41624.1i 1.88531i 0.333772 + 0.942654i \(0.391679\pi\)
−0.333772 + 0.942654i \(0.608321\pi\)
\(788\) 0 0
\(789\) −7824.87 −0.353071
\(790\) 0 0
\(791\) 222.046i 0.00998108i
\(792\) 0 0
\(793\) −6237.49 + 6024.18i −0.279319 + 0.269767i
\(794\) 0 0
\(795\) 7537.20i 0.336248i
\(796\) 0 0
\(797\) 30333.3 1.34813 0.674066 0.738671i \(-0.264546\pi\)
0.674066 + 0.738671i \(0.264546\pi\)