Properties

Label 289.4.b.d.288.8
Level $289$
Weight $4$
Character 289.288
Analytic conductor $17.052$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(288,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.288");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 43x^{6} + 505x^{4} + 1528x^{2} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 288.8
Root \(3.68488i\) of defining polynomial
Character \(\chi\) \(=\) 289.288
Dual form 289.4.b.d.288.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.68488 q^{2} +9.05894i q^{3} +5.57832 q^{4} -7.08909i q^{5} +33.3811i q^{6} +28.1854i q^{7} -8.92359 q^{8} -55.0643 q^{9} +O(q^{10})\) \(q+3.68488 q^{2} +9.05894i q^{3} +5.57832 q^{4} -7.08909i q^{5} +33.3811i q^{6} +28.1854i q^{7} -8.92359 q^{8} -55.0643 q^{9} -26.1224i q^{10} -15.3047i q^{11} +50.5336i q^{12} +2.51532 q^{13} +103.860i q^{14} +64.2196 q^{15} -77.5089 q^{16} -202.905 q^{18} -14.3453 q^{19} -39.5452i q^{20} -255.330 q^{21} -56.3958i q^{22} +180.191i q^{23} -80.8383i q^{24} +74.7448 q^{25} +9.26864 q^{26} -254.233i q^{27} +157.227i q^{28} -41.2425i q^{29} +236.641 q^{30} -155.242i q^{31} -214.222 q^{32} +138.644 q^{33} +199.809 q^{35} -307.167 q^{36} +225.676i q^{37} -52.8606 q^{38} +22.7861i q^{39} +63.2602i q^{40} +234.306i q^{41} -940.860 q^{42} +321.875 q^{43} -85.3743i q^{44} +390.356i q^{45} +663.983i q^{46} -326.183 q^{47} -702.148i q^{48} -451.418 q^{49} +275.425 q^{50} +14.0313 q^{52} +57.1579 q^{53} -936.818i q^{54} -108.496 q^{55} -251.515i q^{56} -129.953i q^{57} -151.974i q^{58} +241.623 q^{59} +358.238 q^{60} +460.925i q^{61} -572.046i q^{62} -1552.01i q^{63} -169.311 q^{64} -17.8313i q^{65} +510.886 q^{66} +392.696 q^{67} -1632.34 q^{69} +736.272 q^{70} -615.558i q^{71} +491.372 q^{72} +697.452i q^{73} +831.587i q^{74} +677.109i q^{75} -80.0226 q^{76} +431.369 q^{77} +83.9641i q^{78} +991.317i q^{79} +549.468i q^{80} +816.345 q^{81} +863.387i q^{82} +98.9057 q^{83} -1424.31 q^{84} +1186.07 q^{86} +373.613 q^{87} +136.573i q^{88} +698.470 q^{89} +1438.41i q^{90} +70.8954i q^{91} +1005.16i q^{92} +1406.32 q^{93} -1201.94 q^{94} +101.695i q^{95} -1940.62i q^{96} +1428.66i q^{97} -1663.42 q^{98} +842.741i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35} - 282 q^{36} - 1048 q^{38} - 2910 q^{42} + 228 q^{43} + 20 q^{47} - 2012 q^{49} + 1610 q^{50} - 3074 q^{52} - 100 q^{53} - 2632 q^{55} - 1992 q^{59} + 434 q^{60} - 300 q^{64} + 2180 q^{66} - 1736 q^{67} - 2256 q^{69} - 2104 q^{70} - 78 q^{72} + 1746 q^{76} + 1788 q^{77} + 2160 q^{81} - 1700 q^{83} + 886 q^{84} + 4822 q^{86} + 768 q^{87} + 1568 q^{89} + 3100 q^{93} - 2238 q^{94} - 3754 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.68488 1.30280 0.651400 0.758734i \(-0.274182\pi\)
0.651400 + 0.758734i \(0.274182\pi\)
\(3\) 9.05894i 1.74339i 0.490046 + 0.871697i \(0.336980\pi\)
−0.490046 + 0.871697i \(0.663020\pi\)
\(4\) 5.57832 0.697290
\(5\) − 7.08909i − 0.634067i −0.948414 0.317034i \(-0.897313\pi\)
0.948414 0.317034i \(-0.102687\pi\)
\(6\) 33.3811i 2.27129i
\(7\) 28.1854i 1.52187i 0.648828 + 0.760935i \(0.275259\pi\)
−0.648828 + 0.760935i \(0.724741\pi\)
\(8\) −8.92359 −0.394371
\(9\) −55.0643 −2.03942
\(10\) − 26.1224i − 0.826064i
\(11\) − 15.3047i − 0.419503i −0.977755 0.209751i \(-0.932735\pi\)
0.977755 0.209751i \(-0.0672655\pi\)
\(12\) 50.5336i 1.21565i
\(13\) 2.51532 0.0536634 0.0268317 0.999640i \(-0.491458\pi\)
0.0268317 + 0.999640i \(0.491458\pi\)
\(14\) 103.860i 1.98269i
\(15\) 64.2196 1.10543
\(16\) −77.5089 −1.21108
\(17\) 0 0
\(18\) −202.905 −2.65696
\(19\) −14.3453 −0.173212 −0.0866062 0.996243i \(-0.527602\pi\)
−0.0866062 + 0.996243i \(0.527602\pi\)
\(20\) − 39.5452i − 0.442129i
\(21\) −255.330 −2.65322
\(22\) − 56.3958i − 0.546529i
\(23\) 180.191i 1.63359i 0.576931 + 0.816793i \(0.304250\pi\)
−0.576931 + 0.816793i \(0.695750\pi\)
\(24\) − 80.8383i − 0.687544i
\(25\) 74.7448 0.597958
\(26\) 9.26864 0.0699127
\(27\) − 254.233i − 1.81212i
\(28\) 157.227i 1.06118i
\(29\) − 41.2425i − 0.264088i −0.991244 0.132044i \(-0.957846\pi\)
0.991244 0.132044i \(-0.0421540\pi\)
\(30\) 236.641 1.44015
\(31\) − 155.242i − 0.899426i −0.893173 0.449713i \(-0.851526\pi\)
0.893173 0.449713i \(-0.148474\pi\)
\(32\) −214.222 −1.18342
\(33\) 138.644 0.731358
\(34\) 0 0
\(35\) 199.809 0.964968
\(36\) −307.167 −1.42207
\(37\) 225.676i 1.00273i 0.865237 + 0.501363i \(0.167168\pi\)
−0.865237 + 0.501363i \(0.832832\pi\)
\(38\) −52.8606 −0.225661
\(39\) 22.7861i 0.0935564i
\(40\) 63.2602i 0.250058i
\(41\) 234.306i 0.892497i 0.894909 + 0.446249i \(0.147240\pi\)
−0.894909 + 0.446249i \(0.852760\pi\)
\(42\) −940.860 −3.45661
\(43\) 321.875 1.14152 0.570761 0.821117i \(-0.306648\pi\)
0.570761 + 0.821117i \(0.306648\pi\)
\(44\) − 85.3743i − 0.292515i
\(45\) 390.356i 1.29313i
\(46\) 663.983i 2.12824i
\(47\) −326.183 −1.01231 −0.506156 0.862442i \(-0.668934\pi\)
−0.506156 + 0.862442i \(0.668934\pi\)
\(48\) − 702.148i − 2.11138i
\(49\) −451.418 −1.31609
\(50\) 275.425 0.779021
\(51\) 0 0
\(52\) 14.0313 0.0374189
\(53\) 57.1579 0.148137 0.0740683 0.997253i \(-0.476402\pi\)
0.0740683 + 0.997253i \(0.476402\pi\)
\(54\) − 936.818i − 2.36083i
\(55\) −108.496 −0.265993
\(56\) − 251.515i − 0.600181i
\(57\) − 129.953i − 0.301977i
\(58\) − 151.974i − 0.344054i
\(59\) 241.623 0.533164 0.266582 0.963812i \(-0.414106\pi\)
0.266582 + 0.963812i \(0.414106\pi\)
\(60\) 358.238 0.770805
\(61\) 460.925i 0.967465i 0.875216 + 0.483732i \(0.160719\pi\)
−0.875216 + 0.483732i \(0.839281\pi\)
\(62\) − 572.046i − 1.17177i
\(63\) − 1552.01i − 3.10373i
\(64\) −169.311 −0.330685
\(65\) − 17.8313i − 0.0340262i
\(66\) 510.886 0.952814
\(67\) 392.696 0.716052 0.358026 0.933712i \(-0.383450\pi\)
0.358026 + 0.933712i \(0.383450\pi\)
\(68\) 0 0
\(69\) −1632.34 −2.84798
\(70\) 736.272 1.25716
\(71\) − 615.558i − 1.02892i −0.857515 0.514460i \(-0.827993\pi\)
0.857515 0.514460i \(-0.172007\pi\)
\(72\) 491.372 0.804288
\(73\) 697.452i 1.11823i 0.829091 + 0.559114i \(0.188858\pi\)
−0.829091 + 0.559114i \(0.811142\pi\)
\(74\) 831.587i 1.30635i
\(75\) 677.109i 1.04248i
\(76\) −80.0226 −0.120779
\(77\) 431.369 0.638429
\(78\) 83.9641i 0.121885i
\(79\) 991.317i 1.41180i 0.708313 + 0.705898i \(0.249456\pi\)
−0.708313 + 0.705898i \(0.750544\pi\)
\(80\) 549.468i 0.767904i
\(81\) 816.345 1.11981
\(82\) 863.387i 1.16275i
\(83\) 98.9057 0.130799 0.0653995 0.997859i \(-0.479168\pi\)
0.0653995 + 0.997859i \(0.479168\pi\)
\(84\) −1424.31 −1.85006
\(85\) 0 0
\(86\) 1186.07 1.48717
\(87\) 373.613 0.460409
\(88\) 136.573i 0.165440i
\(89\) 698.470 0.831884 0.415942 0.909391i \(-0.363452\pi\)
0.415942 + 0.909391i \(0.363452\pi\)
\(90\) 1438.41i 1.68469i
\(91\) 70.8954i 0.0816687i
\(92\) 1005.16i 1.13908i
\(93\) 1406.32 1.56805
\(94\) −1201.94 −1.31884
\(95\) 101.695i 0.109828i
\(96\) − 1940.62i − 2.06317i
\(97\) 1428.66i 1.49545i 0.664009 + 0.747725i \(0.268854\pi\)
−0.664009 + 0.747725i \(0.731146\pi\)
\(98\) −1663.42 −1.71460
\(99\) 842.741i 0.855542i
\(100\) 416.950 0.416950
\(101\) 408.671 0.402617 0.201308 0.979528i \(-0.435481\pi\)
0.201308 + 0.979528i \(0.435481\pi\)
\(102\) 0 0
\(103\) 1649.55 1.57801 0.789007 0.614384i \(-0.210596\pi\)
0.789007 + 0.614384i \(0.210596\pi\)
\(104\) −22.4457 −0.0211633
\(105\) 1810.06i 1.68232i
\(106\) 210.620 0.192993
\(107\) − 2055.01i − 1.85668i −0.371731 0.928340i \(-0.621236\pi\)
0.371731 0.928340i \(-0.378764\pi\)
\(108\) − 1418.19i − 1.26357i
\(109\) − 445.064i − 0.391095i −0.980694 0.195548i \(-0.937352\pi\)
0.980694 0.195548i \(-0.0626485\pi\)
\(110\) −399.795 −0.346536
\(111\) −2044.38 −1.74814
\(112\) − 2184.62i − 1.84310i
\(113\) 1226.02i 1.02066i 0.859979 + 0.510330i \(0.170477\pi\)
−0.859979 + 0.510330i \(0.829523\pi\)
\(114\) − 478.861i − 0.393416i
\(115\) 1277.39 1.03580
\(116\) − 230.064i − 0.184146i
\(117\) −138.504 −0.109442
\(118\) 890.352 0.694607
\(119\) 0 0
\(120\) −573.070 −0.435949
\(121\) 1096.77 0.824017
\(122\) 1698.45i 1.26041i
\(123\) −2122.56 −1.55597
\(124\) − 865.987i − 0.627161i
\(125\) − 1416.01i − 1.01321i
\(126\) − 5718.97i − 4.04354i
\(127\) 591.124 0.413022 0.206511 0.978444i \(-0.433789\pi\)
0.206511 + 0.978444i \(0.433789\pi\)
\(128\) 1089.89 0.752604
\(129\) 2915.84i 1.99012i
\(130\) − 65.7063i − 0.0443294i
\(131\) − 2025.27i − 1.35075i −0.737474 0.675375i \(-0.763982\pi\)
0.737474 0.675375i \(-0.236018\pi\)
\(132\) 773.401 0.509969
\(133\) − 404.328i − 0.263607i
\(134\) 1447.04 0.932873
\(135\) −1802.28 −1.14901
\(136\) 0 0
\(137\) 623.919 0.389087 0.194544 0.980894i \(-0.437677\pi\)
0.194544 + 0.980894i \(0.437677\pi\)
\(138\) −6014.98 −3.71036
\(139\) 1033.14i 0.630429i 0.949020 + 0.315214i \(0.102076\pi\)
−0.949020 + 0.315214i \(0.897924\pi\)
\(140\) 1114.60 0.672863
\(141\) − 2954.87i − 1.76486i
\(142\) − 2268.25i − 1.34048i
\(143\) − 38.4961i − 0.0225119i
\(144\) 4267.98 2.46989
\(145\) −292.372 −0.167449
\(146\) 2570.02i 1.45683i
\(147\) − 4089.37i − 2.29446i
\(148\) 1258.89i 0.699190i
\(149\) −284.476 −0.156411 −0.0782054 0.996937i \(-0.524919\pi\)
−0.0782054 + 0.996937i \(0.524919\pi\)
\(150\) 2495.06i 1.35814i
\(151\) −3153.25 −1.69939 −0.849696 0.527273i \(-0.823214\pi\)
−0.849696 + 0.527273i \(0.823214\pi\)
\(152\) 128.012 0.0683099
\(153\) 0 0
\(154\) 1589.54 0.831745
\(155\) −1100.52 −0.570297
\(156\) 127.108i 0.0652359i
\(157\) −482.335 −0.245188 −0.122594 0.992457i \(-0.539121\pi\)
−0.122594 + 0.992457i \(0.539121\pi\)
\(158\) 3652.88i 1.83929i
\(159\) 517.790i 0.258261i
\(160\) 1518.64i 0.750369i
\(161\) −5078.77 −2.48611
\(162\) 3008.13 1.45890
\(163\) − 3717.39i − 1.78631i −0.449749 0.893155i \(-0.648487\pi\)
0.449749 0.893155i \(-0.351513\pi\)
\(164\) 1307.03i 0.622329i
\(165\) − 982.860i − 0.463731i
\(166\) 364.455 0.170405
\(167\) − 2128.93i − 0.986475i −0.869895 0.493237i \(-0.835813\pi\)
0.869895 0.493237i \(-0.164187\pi\)
\(168\) 2278.46 1.04635
\(169\) −2190.67 −0.997120
\(170\) 0 0
\(171\) 789.914 0.353253
\(172\) 1795.52 0.795971
\(173\) 1197.86i 0.526426i 0.964738 + 0.263213i \(0.0847822\pi\)
−0.964738 + 0.263213i \(0.915218\pi\)
\(174\) 1376.72 0.599821
\(175\) 2106.71i 0.910015i
\(176\) 1186.25i 0.508050i
\(177\) 2188.85i 0.929515i
\(178\) 2573.78 1.08378
\(179\) −4095.63 −1.71018 −0.855089 0.518482i \(-0.826497\pi\)
−0.855089 + 0.518482i \(0.826497\pi\)
\(180\) 2177.53i 0.901687i
\(181\) 1297.08i 0.532660i 0.963882 + 0.266330i \(0.0858110\pi\)
−0.963882 + 0.266330i \(0.914189\pi\)
\(182\) 261.241i 0.106398i
\(183\) −4175.49 −1.68667
\(184\) − 1607.95i − 0.644239i
\(185\) 1599.83 0.635795
\(186\) 5182.13 2.04286
\(187\) 0 0
\(188\) −1819.55 −0.705876
\(189\) 7165.67 2.75781
\(190\) 374.734i 0.143084i
\(191\) 4381.72 1.65995 0.829974 0.557803i \(-0.188355\pi\)
0.829974 + 0.557803i \(0.188355\pi\)
\(192\) − 1533.77i − 0.576514i
\(193\) − 1022.20i − 0.381240i −0.981664 0.190620i \(-0.938950\pi\)
0.981664 0.190620i \(-0.0610498\pi\)
\(194\) 5264.44i 1.94827i
\(195\) 161.533 0.0593211
\(196\) −2518.15 −0.917695
\(197\) 2202.50i 0.796555i 0.917265 + 0.398278i \(0.130392\pi\)
−0.917265 + 0.398278i \(0.869608\pi\)
\(198\) 3105.40i 1.11460i
\(199\) − 76.0165i − 0.0270787i −0.999908 0.0135394i \(-0.995690\pi\)
0.999908 0.0135394i \(-0.00430985\pi\)
\(200\) −666.992 −0.235817
\(201\) 3557.41i 1.24836i
\(202\) 1505.90 0.524529
\(203\) 1162.44 0.401907
\(204\) 0 0
\(205\) 1661.01 0.565904
\(206\) 6078.40 2.05584
\(207\) − 9922.12i − 3.33157i
\(208\) −194.960 −0.0649905
\(209\) 219.550i 0.0726631i
\(210\) 6669.84i 2.19173i
\(211\) − 1972.63i − 0.643609i −0.946806 0.321804i \(-0.895711\pi\)
0.946806 0.321804i \(-0.104289\pi\)
\(212\) 318.845 0.103294
\(213\) 5576.30 1.79381
\(214\) − 7572.44i − 2.41889i
\(215\) − 2281.80i − 0.723801i
\(216\) 2268.67i 0.714647i
\(217\) 4375.55 1.36881
\(218\) − 1640.01i − 0.509520i
\(219\) −6318.17 −1.94951
\(220\) −605.226 −0.185474
\(221\) 0 0
\(222\) −7533.29 −2.27748
\(223\) −1177.47 −0.353584 −0.176792 0.984248i \(-0.556572\pi\)
−0.176792 + 0.984248i \(0.556572\pi\)
\(224\) − 6037.94i − 1.80101i
\(225\) −4115.77 −1.21949
\(226\) 4517.75i 1.32972i
\(227\) − 2845.44i − 0.831976i −0.909370 0.415988i \(-0.863436\pi\)
0.909370 0.415988i \(-0.136564\pi\)
\(228\) − 724.920i − 0.210566i
\(229\) −3605.93 −1.04055 −0.520277 0.853998i \(-0.674171\pi\)
−0.520277 + 0.853998i \(0.674171\pi\)
\(230\) 4707.03 1.34945
\(231\) 3907.74i 1.11303i
\(232\) 368.032i 0.104149i
\(233\) − 835.204i − 0.234833i −0.993083 0.117416i \(-0.962539\pi\)
0.993083 0.117416i \(-0.0374612\pi\)
\(234\) −510.372 −0.142581
\(235\) 2312.34i 0.641875i
\(236\) 1347.85 0.371770
\(237\) −8980.28 −2.46132
\(238\) 0 0
\(239\) −4057.52 −1.09816 −0.549078 0.835771i \(-0.685021\pi\)
−0.549078 + 0.835771i \(0.685021\pi\)
\(240\) −4977.59 −1.33876
\(241\) − 2130.64i − 0.569487i −0.958604 0.284743i \(-0.908092\pi\)
0.958604 0.284743i \(-0.0919084\pi\)
\(242\) 4041.45 1.07353
\(243\) 530.923i 0.140159i
\(244\) 2571.19i 0.674604i
\(245\) 3200.14i 0.834489i
\(246\) −7821.37 −2.02712
\(247\) −36.0830 −0.00929516
\(248\) 1385.31i 0.354708i
\(249\) 895.981i 0.228034i
\(250\) − 5217.82i − 1.32002i
\(251\) 2164.17 0.544227 0.272113 0.962265i \(-0.412277\pi\)
0.272113 + 0.962265i \(0.412277\pi\)
\(252\) − 8657.62i − 2.16420i
\(253\) 2757.77 0.685294
\(254\) 2178.22 0.538085
\(255\) 0 0
\(256\) 5370.59 1.31118
\(257\) −6991.31 −1.69691 −0.848456 0.529267i \(-0.822467\pi\)
−0.848456 + 0.529267i \(0.822467\pi\)
\(258\) 10744.5i 2.59273i
\(259\) −6360.76 −1.52602
\(260\) − 99.4688i − 0.0237261i
\(261\) 2270.99i 0.538586i
\(262\) − 7462.85i − 1.75976i
\(263\) 2985.57 0.699993 0.349996 0.936751i \(-0.386183\pi\)
0.349996 + 0.936751i \(0.386183\pi\)
\(264\) −1237.20 −0.288426
\(265\) − 405.198i − 0.0939287i
\(266\) − 1489.90i − 0.343427i
\(267\) 6327.40i 1.45030i
\(268\) 2190.58 0.499296
\(269\) − 4063.25i − 0.920969i −0.887668 0.460484i \(-0.847676\pi\)
0.887668 0.460484i \(-0.152324\pi\)
\(270\) −6641.19 −1.49693
\(271\) 6295.85 1.41124 0.705619 0.708591i \(-0.250669\pi\)
0.705619 + 0.708591i \(0.250669\pi\)
\(272\) 0 0
\(273\) −642.237 −0.142381
\(274\) 2299.06 0.506903
\(275\) − 1143.94i − 0.250845i
\(276\) −9105.72 −1.98587
\(277\) − 4710.13i − 1.02168i −0.859677 0.510838i \(-0.829335\pi\)
0.859677 0.510838i \(-0.170665\pi\)
\(278\) 3806.99i 0.821323i
\(279\) 8548.28i 1.83431i
\(280\) −1783.01 −0.380555
\(281\) 2458.26 0.521878 0.260939 0.965355i \(-0.415968\pi\)
0.260939 + 0.965355i \(0.415968\pi\)
\(282\) − 10888.3i − 2.29926i
\(283\) 9204.70i 1.93344i 0.255843 + 0.966718i \(0.417647\pi\)
−0.255843 + 0.966718i \(0.582353\pi\)
\(284\) − 3433.78i − 0.717455i
\(285\) −921.249 −0.191474
\(286\) − 141.854i − 0.0293286i
\(287\) −6604.00 −1.35826
\(288\) 11796.0 2.41349
\(289\) 0 0
\(290\) −1077.35 −0.218153
\(291\) −12942.2 −2.60716
\(292\) 3890.61i 0.779729i
\(293\) 423.010 0.0843431 0.0421716 0.999110i \(-0.486572\pi\)
0.0421716 + 0.999110i \(0.486572\pi\)
\(294\) − 15068.8i − 2.98922i
\(295\) − 1712.89i − 0.338062i
\(296\) − 2013.84i − 0.395446i
\(297\) −3890.95 −0.760189
\(298\) −1048.26 −0.203772
\(299\) 453.239i 0.0876638i
\(300\) 3777.13i 0.726909i
\(301\) 9072.17i 1.73725i
\(302\) −11619.4 −2.21397
\(303\) 3702.12i 0.701919i
\(304\) 1111.89 0.209773
\(305\) 3267.54 0.613438
\(306\) 0 0
\(307\) 632.590 0.117602 0.0588010 0.998270i \(-0.481272\pi\)
0.0588010 + 0.998270i \(0.481272\pi\)
\(308\) 2406.31 0.445170
\(309\) 14943.2i 2.75110i
\(310\) −4055.29 −0.742983
\(311\) 5293.07i 0.965089i 0.875872 + 0.482544i \(0.160287\pi\)
−0.875872 + 0.482544i \(0.839713\pi\)
\(312\) − 203.334i − 0.0368959i
\(313\) − 3817.93i − 0.689464i −0.938701 0.344732i \(-0.887970\pi\)
0.938701 0.344732i \(-0.112030\pi\)
\(314\) −1777.35 −0.319431
\(315\) −11002.4 −1.96798
\(316\) 5529.88i 0.984431i
\(317\) − 1389.54i − 0.246197i −0.992394 0.123098i \(-0.960717\pi\)
0.992394 0.123098i \(-0.0392830\pi\)
\(318\) 1907.99i 0.336462i
\(319\) −631.203 −0.110786
\(320\) 1200.26i 0.209677i
\(321\) 18616.2 3.23692
\(322\) −18714.6 −3.23890
\(323\) 0 0
\(324\) 4553.83 0.780836
\(325\) 188.007 0.0320885
\(326\) − 13698.1i − 2.32721i
\(327\) 4031.81 0.681833
\(328\) − 2090.85i − 0.351975i
\(329\) − 9193.61i − 1.54061i
\(330\) − 3621.72i − 0.604149i
\(331\) 5736.36 0.952565 0.476282 0.879292i \(-0.341984\pi\)
0.476282 + 0.879292i \(0.341984\pi\)
\(332\) 551.728 0.0912048
\(333\) − 12426.7i − 2.04498i
\(334\) − 7844.83i − 1.28518i
\(335\) − 2783.86i − 0.454025i
\(336\) 19790.3 3.21325
\(337\) 3997.46i 0.646159i 0.946372 + 0.323080i \(0.104718\pi\)
−0.946372 + 0.323080i \(0.895282\pi\)
\(338\) −8072.36 −1.29905
\(339\) −11106.5 −1.77941
\(340\) 0 0
\(341\) −2375.92 −0.377312
\(342\) 2910.74 0.460218
\(343\) − 3055.81i − 0.481045i
\(344\) −2872.28 −0.450183
\(345\) 11571.8i 1.80581i
\(346\) 4413.97i 0.685828i
\(347\) − 2485.37i − 0.384500i −0.981346 0.192250i \(-0.938422\pi\)
0.981346 0.192250i \(-0.0615785\pi\)
\(348\) 2084.14 0.321038
\(349\) 9518.10 1.45986 0.729932 0.683520i \(-0.239552\pi\)
0.729932 + 0.683520i \(0.239552\pi\)
\(350\) 7762.98i 1.18557i
\(351\) − 639.478i − 0.0972444i
\(352\) 3278.60i 0.496448i
\(353\) 174.509 0.0263121 0.0131561 0.999913i \(-0.495812\pi\)
0.0131561 + 0.999913i \(0.495812\pi\)
\(354\) 8065.65i 1.21097i
\(355\) −4363.74 −0.652404
\(356\) 3896.29 0.580065
\(357\) 0 0
\(358\) −15091.9 −2.22802
\(359\) 8112.87 1.19270 0.596352 0.802723i \(-0.296616\pi\)
0.596352 + 0.802723i \(0.296616\pi\)
\(360\) − 3483.38i − 0.509973i
\(361\) −6653.21 −0.969997
\(362\) 4779.59i 0.693949i
\(363\) 9935.55i 1.43659i
\(364\) 395.477i 0.0569468i
\(365\) 4944.30 0.709031
\(366\) −15386.2 −2.19740
\(367\) 41.1424i 0.00585181i 0.999996 + 0.00292591i \(0.000931346\pi\)
−0.999996 + 0.00292591i \(0.999069\pi\)
\(368\) − 13966.4i − 1.97840i
\(369\) − 12901.9i − 1.82018i
\(370\) 5895.19 0.828315
\(371\) 1611.02i 0.225445i
\(372\) 7844.93 1.09339
\(373\) 1275.50 0.177059 0.0885296 0.996074i \(-0.471783\pi\)
0.0885296 + 0.996074i \(0.471783\pi\)
\(374\) 0 0
\(375\) 12827.5 1.76643
\(376\) 2910.73 0.399227
\(377\) − 103.738i − 0.0141718i
\(378\) 26404.6 3.59288
\(379\) − 4101.12i − 0.555832i −0.960605 0.277916i \(-0.910356\pi\)
0.960605 0.277916i \(-0.0896438\pi\)
\(380\) 567.287i 0.0765822i
\(381\) 5354.95i 0.720059i
\(382\) 16146.1 2.16258
\(383\) 10390.1 1.38619 0.693094 0.720848i \(-0.256247\pi\)
0.693094 + 0.720848i \(0.256247\pi\)
\(384\) 9873.22i 1.31209i
\(385\) − 3058.01i − 0.404807i
\(386\) − 3766.67i − 0.496679i
\(387\) −17723.8 −2.32804
\(388\) 7969.53i 1.04276i
\(389\) −6213.37 −0.809846 −0.404923 0.914351i \(-0.632702\pi\)
−0.404923 + 0.914351i \(0.632702\pi\)
\(390\) 595.229 0.0772835
\(391\) 0 0
\(392\) 4028.27 0.519027
\(393\) 18346.8 2.35489
\(394\) 8115.93i 1.03775i
\(395\) 7027.54 0.895174
\(396\) 4701.08i 0.596561i
\(397\) 171.318i 0.0216580i 0.999941 + 0.0108290i \(0.00344704\pi\)
−0.999941 + 0.0108290i \(0.996553\pi\)
\(398\) − 280.111i − 0.0352782i
\(399\) 3662.78 0.459570
\(400\) −5793.39 −0.724174
\(401\) 9328.60i 1.16172i 0.814005 + 0.580858i \(0.197283\pi\)
−0.814005 + 0.580858i \(0.802717\pi\)
\(402\) 13108.6i 1.62636i
\(403\) − 390.482i − 0.0482663i
\(404\) 2279.70 0.280741
\(405\) − 5787.14i − 0.710038i
\(406\) 4283.44 0.523605
\(407\) 3453.89 0.420646
\(408\) 0 0
\(409\) −6370.74 −0.770202 −0.385101 0.922874i \(-0.625833\pi\)
−0.385101 + 0.922874i \(0.625833\pi\)
\(410\) 6120.63 0.737260
\(411\) 5652.04i 0.678332i
\(412\) 9201.74 1.10033
\(413\) 6810.26i 0.811406i
\(414\) − 36561.8i − 4.34037i
\(415\) − 701.152i − 0.0829354i
\(416\) −538.837 −0.0635064
\(417\) −9359.13 −1.09909
\(418\) 809.014i 0.0946655i
\(419\) 8067.75i 0.940657i 0.882491 + 0.470329i \(0.155865\pi\)
−0.882491 + 0.470329i \(0.844135\pi\)
\(420\) 10097.1i 1.17306i
\(421\) −6254.81 −0.724088 −0.362044 0.932161i \(-0.617921\pi\)
−0.362044 + 0.932161i \(0.617921\pi\)
\(422\) − 7268.90i − 0.838494i
\(423\) 17961.1 2.06453
\(424\) −510.054 −0.0584208
\(425\) 0 0
\(426\) 20548.0 2.33698
\(427\) −12991.4 −1.47236
\(428\) − 11463.5i − 1.29464i
\(429\) 348.734 0.0392472
\(430\) − 8408.14i − 0.942969i
\(431\) 5910.99i 0.660609i 0.943874 + 0.330305i \(0.107151\pi\)
−0.943874 + 0.330305i \(0.892849\pi\)
\(432\) 19705.3i 2.19461i
\(433\) −11833.5 −1.31335 −0.656677 0.754172i \(-0.728038\pi\)
−0.656677 + 0.754172i \(0.728038\pi\)
\(434\) 16123.4 1.78329
\(435\) − 2648.58i − 0.291930i
\(436\) − 2482.71i − 0.272707i
\(437\) − 2584.90i − 0.282957i
\(438\) −23281.7 −2.53982
\(439\) − 7520.94i − 0.817665i −0.912609 0.408833i \(-0.865936\pi\)
0.912609 0.408833i \(-0.134064\pi\)
\(440\) 968.176 0.104900
\(441\) 24857.0 2.68406
\(442\) 0 0
\(443\) 8478.34 0.909296 0.454648 0.890671i \(-0.349765\pi\)
0.454648 + 0.890671i \(0.349765\pi\)
\(444\) −11404.2 −1.21896
\(445\) − 4951.52i − 0.527471i
\(446\) −4338.83 −0.460649
\(447\) − 2577.05i − 0.272685i
\(448\) − 4772.09i − 0.503259i
\(449\) − 1078.67i − 0.113375i −0.998392 0.0566877i \(-0.981946\pi\)
0.998392 0.0566877i \(-0.0180539\pi\)
\(450\) −15166.1 −1.58875
\(451\) 3585.97 0.374405
\(452\) 6839.15i 0.711696i
\(453\) − 28565.1i − 2.96271i
\(454\) − 10485.1i − 1.08390i
\(455\) 502.584 0.0517835
\(456\) 1159.65i 0.119091i
\(457\) −5664.74 −0.579837 −0.289918 0.957051i \(-0.593628\pi\)
−0.289918 + 0.957051i \(0.593628\pi\)
\(458\) −13287.4 −1.35563
\(459\) 0 0
\(460\) 7125.70 0.722256
\(461\) −710.717 −0.0718034 −0.0359017 0.999355i \(-0.511430\pi\)
−0.0359017 + 0.999355i \(0.511430\pi\)
\(462\) 14399.5i 1.45006i
\(463\) 8328.23 0.835952 0.417976 0.908458i \(-0.362740\pi\)
0.417976 + 0.908458i \(0.362740\pi\)
\(464\) 3196.66i 0.319831i
\(465\) − 9969.56i − 0.994252i
\(466\) − 3077.62i − 0.305940i
\(467\) 14132.8 1.40040 0.700201 0.713946i \(-0.253094\pi\)
0.700201 + 0.713946i \(0.253094\pi\)
\(468\) −772.622 −0.0763130
\(469\) 11068.3i 1.08974i
\(470\) 8520.69i 0.836235i
\(471\) − 4369.44i − 0.427459i
\(472\) −2156.15 −0.210264
\(473\) − 4926.18i − 0.478871i
\(474\) −33091.2 −3.20660
\(475\) −1072.24 −0.103574
\(476\) 0 0
\(477\) −3147.36 −0.302113
\(478\) −14951.5 −1.43068
\(479\) − 6522.89i − 0.622210i −0.950376 0.311105i \(-0.899301\pi\)
0.950376 0.311105i \(-0.100699\pi\)
\(480\) −13757.3 −1.30819
\(481\) 567.646i 0.0538096i
\(482\) − 7851.13i − 0.741928i
\(483\) − 46008.3i − 4.33426i
\(484\) 6118.12 0.574579
\(485\) 10127.9 0.948216
\(486\) 1956.39i 0.182600i
\(487\) − 2178.49i − 0.202704i −0.994851 0.101352i \(-0.967683\pi\)
0.994851 0.101352i \(-0.0323168\pi\)
\(488\) − 4113.11i − 0.381540i
\(489\) 33675.6 3.11424
\(490\) 11792.1i 1.08717i
\(491\) 12618.8 1.15983 0.579915 0.814677i \(-0.303086\pi\)
0.579915 + 0.814677i \(0.303086\pi\)
\(492\) −11840.3 −1.08496
\(493\) 0 0
\(494\) −132.961 −0.0121097
\(495\) 5974.27 0.542472
\(496\) 12032.6i 1.08927i
\(497\) 17349.8 1.56588
\(498\) 3301.58i 0.297083i
\(499\) 13762.8i 1.23469i 0.786694 + 0.617343i \(0.211791\pi\)
−0.786694 + 0.617343i \(0.788209\pi\)
\(500\) − 7898.95i − 0.706504i
\(501\) 19285.8 1.71981
\(502\) 7974.69 0.709019
\(503\) 491.671i 0.0435836i 0.999763 + 0.0217918i \(0.00693709\pi\)
−0.999763 + 0.0217918i \(0.993063\pi\)
\(504\) 13849.5i 1.22402i
\(505\) − 2897.11i − 0.255286i
\(506\) 10162.0 0.892802
\(507\) − 19845.2i − 1.73837i
\(508\) 3297.48 0.287996
\(509\) 576.514 0.0502034 0.0251017 0.999685i \(-0.492009\pi\)
0.0251017 + 0.999685i \(0.492009\pi\)
\(510\) 0 0
\(511\) −19658.0 −1.70180
\(512\) 11070.9 0.955600
\(513\) 3647.05i 0.313881i
\(514\) −25762.1 −2.21074
\(515\) − 11693.8i − 1.00057i
\(516\) 16265.5i 1.38769i
\(517\) 4992.12i 0.424668i
\(518\) −23438.6 −1.98810
\(519\) −10851.4 −0.917768
\(520\) 159.120i 0.0134189i
\(521\) 6102.95i 0.513196i 0.966518 + 0.256598i \(0.0826016\pi\)
−0.966518 + 0.256598i \(0.917398\pi\)
\(522\) 8368.33i 0.701670i
\(523\) 13593.3 1.13651 0.568255 0.822853i \(-0.307619\pi\)
0.568255 + 0.822853i \(0.307619\pi\)
\(524\) − 11297.6i − 0.941864i
\(525\) −19084.6 −1.58651
\(526\) 11001.5 0.911951
\(527\) 0 0
\(528\) −10746.1 −0.885731
\(529\) −20301.9 −1.66860
\(530\) − 1493.10i − 0.122370i
\(531\) −13304.8 −1.08735
\(532\) − 2255.47i − 0.183810i
\(533\) 589.354i 0.0478944i
\(534\) 23315.7i 1.88945i
\(535\) −14568.1 −1.17726
\(536\) −3504.26 −0.282390
\(537\) − 37102.1i − 2.98151i
\(538\) − 14972.6i − 1.19984i
\(539\) 6908.80i 0.552103i
\(540\) −10053.7 −0.801190
\(541\) 20360.3i 1.61803i 0.587786 + 0.809017i \(0.300000\pi\)
−0.587786 + 0.809017i \(0.700000\pi\)
\(542\) 23199.4 1.83856
\(543\) −11750.2 −0.928635
\(544\) 0 0
\(545\) −3155.10 −0.247981
\(546\) −2366.56 −0.185494
\(547\) 253.860i 0.0198433i 0.999951 + 0.00992164i \(0.00315821\pi\)
−0.999951 + 0.00992164i \(0.996842\pi\)
\(548\) 3480.42 0.271307
\(549\) − 25380.5i − 1.97307i
\(550\) − 4215.29i − 0.326801i
\(551\) 591.636i 0.0457433i
\(552\) 14566.4 1.12316
\(553\) −27940.7 −2.14857
\(554\) − 17356.2i − 1.33104i
\(555\) 14492.8i 1.10844i
\(556\) 5763.17i 0.439592i
\(557\) 1519.94 0.115623 0.0578116 0.998328i \(-0.481588\pi\)
0.0578116 + 0.998328i \(0.481588\pi\)
\(558\) 31499.4i 2.38974i
\(559\) 809.617 0.0612579
\(560\) −15487.0 −1.16865
\(561\) 0 0
\(562\) 9058.40 0.679903
\(563\) 25112.9 1.87990 0.939948 0.341317i \(-0.110873\pi\)
0.939948 + 0.341317i \(0.110873\pi\)
\(564\) − 16483.2i − 1.23062i
\(565\) 8691.39 0.647168
\(566\) 33918.2i 2.51888i
\(567\) 23009.0i 1.70421i
\(568\) 5492.99i 0.405776i
\(569\) −3649.16 −0.268859 −0.134430 0.990923i \(-0.542920\pi\)
−0.134430 + 0.990923i \(0.542920\pi\)
\(570\) −3394.69 −0.249452
\(571\) 4111.91i 0.301362i 0.988582 + 0.150681i \(0.0481467\pi\)
−0.988582 + 0.150681i \(0.951853\pi\)
\(572\) − 214.744i − 0.0156974i
\(573\) 39693.7i 2.89394i
\(574\) −24334.9 −1.76955
\(575\) 13468.4i 0.976817i
\(576\) 9322.98 0.674405
\(577\) 25307.1 1.82591 0.912954 0.408062i \(-0.133795\pi\)
0.912954 + 0.408062i \(0.133795\pi\)
\(578\) 0 0
\(579\) 9260.01 0.664651
\(580\) −1630.94 −0.116761
\(581\) 2787.70i 0.199059i
\(582\) −47690.3 −3.39661
\(583\) − 874.783i − 0.0621438i
\(584\) − 6223.78i − 0.440996i
\(585\) 981.870i 0.0693938i
\(586\) 1558.74 0.109882
\(587\) −6734.14 −0.473506 −0.236753 0.971570i \(-0.576083\pi\)
−0.236753 + 0.971570i \(0.576083\pi\)
\(588\) − 22811.8i − 1.59990i
\(589\) 2226.99i 0.155792i
\(590\) − 6311.79i − 0.440427i
\(591\) −19952.3 −1.38871
\(592\) − 17491.9i − 1.21438i
\(593\) −885.596 −0.0613273 −0.0306636 0.999530i \(-0.509762\pi\)
−0.0306636 + 0.999530i \(0.509762\pi\)
\(594\) −14337.7 −0.990374
\(595\) 0 0
\(596\) −1586.90 −0.109064
\(597\) 688.629 0.0472089
\(598\) 1670.13i 0.114208i
\(599\) 7411.88 0.505578 0.252789 0.967521i \(-0.418652\pi\)
0.252789 + 0.967521i \(0.418652\pi\)
\(600\) − 6042.24i − 0.411123i
\(601\) 19232.0i 1.30531i 0.757657 + 0.652653i \(0.226344\pi\)
−0.757657 + 0.652653i \(0.773656\pi\)
\(602\) 33429.8i 2.26329i
\(603\) −21623.6 −1.46033
\(604\) −17589.9 −1.18497
\(605\) − 7775.08i − 0.522483i
\(606\) 13641.9i 0.914461i
\(607\) 3485.07i 0.233039i 0.993188 + 0.116519i \(0.0371737\pi\)
−0.993188 + 0.116519i \(0.962826\pi\)
\(608\) 3073.08 0.204983
\(609\) 10530.5i 0.700682i
\(610\) 12040.5 0.799188
\(611\) −820.455 −0.0543241
\(612\) 0 0
\(613\) −27347.9 −1.80191 −0.900954 0.433915i \(-0.857132\pi\)
−0.900954 + 0.433915i \(0.857132\pi\)
\(614\) 2331.02 0.153212
\(615\) 15047.0i 0.986592i
\(616\) −3849.36 −0.251778
\(617\) − 12625.9i − 0.823822i −0.911224 0.411911i \(-0.864861\pi\)
0.911224 0.411911i \(-0.135139\pi\)
\(618\) 55063.9i 3.58413i
\(619\) 16469.8i 1.06943i 0.845033 + 0.534714i \(0.179581\pi\)
−0.845033 + 0.534714i \(0.820419\pi\)
\(620\) −6139.06 −0.397662
\(621\) 45810.6 2.96025
\(622\) 19504.3i 1.25732i
\(623\) 19686.7i 1.26602i
\(624\) − 1766.13i − 0.113304i
\(625\) −695.113 −0.0444872
\(626\) − 14068.6i − 0.898235i
\(627\) −1988.89 −0.126680
\(628\) −2690.62 −0.170967
\(629\) 0 0
\(630\) −40542.3 −2.56388
\(631\) −6691.13 −0.422139 −0.211069 0.977471i \(-0.567695\pi\)
−0.211069 + 0.977471i \(0.567695\pi\)
\(632\) − 8846.11i − 0.556771i
\(633\) 17869.9 1.12206
\(634\) − 5120.28i − 0.320745i
\(635\) − 4190.53i − 0.261884i
\(636\) 2888.40i 0.180082i
\(637\) −1135.46 −0.0706258
\(638\) −2325.91 −0.144332
\(639\) 33895.3i 2.09840i
\(640\) − 7726.31i − 0.477202i
\(641\) − 14335.5i − 0.883335i −0.897179 0.441668i \(-0.854387\pi\)
0.897179 0.441668i \(-0.145613\pi\)
\(642\) 68598.3 4.21707
\(643\) − 30421.9i − 1.86582i −0.360109 0.932910i \(-0.617260\pi\)
0.360109 0.932910i \(-0.382740\pi\)
\(644\) −28331.0 −1.73354
\(645\) 20670.7 1.26187
\(646\) 0 0
\(647\) −9221.00 −0.560301 −0.280151 0.959956i \(-0.590384\pi\)
−0.280151 + 0.959956i \(0.590384\pi\)
\(648\) −7284.73 −0.441622
\(649\) − 3697.96i − 0.223664i
\(650\) 692.783 0.0418049
\(651\) 39637.8i 2.38637i
\(652\) − 20736.8i − 1.24558i
\(653\) − 24958.5i − 1.49572i −0.663859 0.747858i \(-0.731082\pi\)
0.663859 0.747858i \(-0.268918\pi\)
\(654\) 14856.7 0.888293
\(655\) −14357.3 −0.856467
\(656\) − 18160.8i − 1.08088i
\(657\) − 38404.7i − 2.28054i
\(658\) − 33877.3i − 2.00711i
\(659\) −3367.32 −0.199047 −0.0995236 0.995035i \(-0.531732\pi\)
−0.0995236 + 0.995035i \(0.531732\pi\)
\(660\) − 5482.71i − 0.323355i
\(661\) 6389.67 0.375990 0.187995 0.982170i \(-0.439801\pi\)
0.187995 + 0.982170i \(0.439801\pi\)
\(662\) 21137.8 1.24100
\(663\) 0 0
\(664\) −882.595 −0.0515833
\(665\) −2866.32 −0.167144
\(666\) − 45790.8i − 2.66420i
\(667\) 7431.55 0.431410
\(668\) − 11875.8i − 0.687859i
\(669\) − 10666.6i − 0.616436i
\(670\) − 10258.2i − 0.591504i
\(671\) 7054.30 0.405854
\(672\) 54697.3 3.13987
\(673\) − 27441.4i − 1.57175i −0.618383 0.785877i \(-0.712212\pi\)
0.618383 0.785877i \(-0.287788\pi\)
\(674\) 14730.2i 0.841817i
\(675\) − 19002.6i − 1.08357i
\(676\) −12220.3 −0.695282
\(677\) 16931.5i 0.961194i 0.876941 + 0.480597i \(0.159580\pi\)
−0.876941 + 0.480597i \(0.840420\pi\)
\(678\) −40926.0 −2.31822
\(679\) −40267.4 −2.27588
\(680\) 0 0
\(681\) 25776.7 1.45046
\(682\) −8754.98 −0.491562
\(683\) − 69.2829i − 0.00388146i −0.999998 0.00194073i \(-0.999382\pi\)
0.999998 0.00194073i \(-0.000617754\pi\)
\(684\) 4406.39 0.246320
\(685\) − 4423.01i − 0.246708i
\(686\) − 11260.3i − 0.626705i
\(687\) − 32665.9i − 1.81409i
\(688\) −24948.1 −1.38247
\(689\) 143.770 0.00794952
\(690\) 42640.7i 2.35262i
\(691\) − 1509.96i − 0.0831282i −0.999136 0.0415641i \(-0.986766\pi\)
0.999136 0.0415641i \(-0.0132341\pi\)
\(692\) 6682.05i 0.367072i
\(693\) −23753.0 −1.30202
\(694\) − 9158.27i − 0.500927i
\(695\) 7324.01 0.399734
\(696\) −3333.98 −0.181572
\(697\) 0 0
\(698\) 35073.0 1.90191
\(699\) 7566.06 0.409406
\(700\) 11751.9i 0.634544i
\(701\) −16452.5 −0.886449 −0.443224 0.896411i \(-0.646165\pi\)
−0.443224 + 0.896411i \(0.646165\pi\)
\(702\) − 2356.40i − 0.126690i
\(703\) − 3237.38i − 0.173684i
\(704\) 2591.24i 0.138723i
\(705\) −20947.4 −1.11904
\(706\) 643.045 0.0342795
\(707\) 11518.6i 0.612730i
\(708\) 12210.1i 0.648141i
\(709\) 4602.45i 0.243792i 0.992543 + 0.121896i \(0.0388975\pi\)
−0.992543 + 0.121896i \(0.961103\pi\)
\(710\) −16079.9 −0.849953
\(711\) − 54586.2i − 2.87925i
\(712\) −6232.87 −0.328071
\(713\) 27973.2 1.46929
\(714\) 0 0
\(715\) −272.903 −0.0142741
\(716\) −22846.7 −1.19249
\(717\) − 36756.8i − 1.91452i
\(718\) 29894.9 1.55386
\(719\) 23515.7i 1.21973i 0.792505 + 0.609865i \(0.208776\pi\)
−0.792505 + 0.609865i \(0.791224\pi\)
\(720\) − 30256.1i − 1.56608i
\(721\) 46493.4i 2.40153i
\(722\) −24516.3 −1.26371
\(723\) 19301.3 0.992839
\(724\) 7235.54i 0.371418i
\(725\) − 3082.66i − 0.157914i
\(726\) 36611.3i 1.87159i
\(727\) −8389.04 −0.427968 −0.213984 0.976837i \(-0.568644\pi\)
−0.213984 + 0.976837i \(0.568644\pi\)
\(728\) − 632.641i − 0.0322078i
\(729\) 17231.7 0.875462
\(730\) 18219.1 0.923727
\(731\) 0 0
\(732\) −23292.2 −1.17610
\(733\) 7677.37 0.386863 0.193431 0.981114i \(-0.438038\pi\)
0.193431 + 0.981114i \(0.438038\pi\)
\(734\) 151.605i 0.00762374i
\(735\) −28989.9 −1.45484
\(736\) − 38601.0i − 1.93322i
\(737\) − 6010.08i − 0.300386i
\(738\) − 47541.9i − 2.37133i
\(739\) −26617.3 −1.32494 −0.662472 0.749087i \(-0.730493\pi\)
−0.662472 + 0.749087i \(0.730493\pi\)
\(740\) 8924.39 0.443334
\(741\) − 326.873i − 0.0162051i
\(742\) 5936.41i 0.293710i
\(743\) 20951.6i 1.03451i 0.855832 + 0.517253i \(0.173046\pi\)
−0.855832 + 0.517253i \(0.826954\pi\)
\(744\) −12549.5 −0.618395
\(745\) 2016.68i 0.0991750i
\(746\) 4700.07 0.230673
\(747\) −5446.18 −0.266754
\(748\) 0 0
\(749\) 57921.2 2.82563
\(750\) 47267.9 2.30131
\(751\) 8252.35i 0.400975i 0.979696 + 0.200488i \(0.0642527\pi\)
−0.979696 + 0.200488i \(0.935747\pi\)
\(752\) 25282.1 1.22599
\(753\) 19605.0i 0.948801i
\(754\) − 382.262i − 0.0184631i
\(755\) 22353.7i 1.07753i
\(756\) 39972.4 1.92299
\(757\) 21203.1 1.01802 0.509010 0.860760i \(-0.330012\pi\)
0.509010 + 0.860760i \(0.330012\pi\)
\(758\) − 15112.1i − 0.724139i
\(759\) 24982.4i 1.19474i
\(760\) − 907.485i − 0.0433131i
\(761\) 29378.2 1.39942 0.699711 0.714426i \(-0.253312\pi\)
0.699711 + 0.714426i \(0.253312\pi\)
\(762\) 19732.3i 0.938094i
\(763\) 12544.3 0.595196
\(764\) 24442.6 1.15746
\(765\) 0 0
\(766\) 38286.3 1.80593
\(767\) 607.760 0.0286114
\(768\) 48651.8i 2.28590i
\(769\) −16559.6 −0.776532 −0.388266 0.921547i \(-0.626926\pi\)
−0.388266 + 0.921547i \(0.626926\pi\)
\(770\) − 11268.4i − 0.527383i
\(771\) − 63333.9i − 2.95838i
\(772\) − 5702.13i − 0.265835i
\(773\) 39537.3 1.83966 0.919830 0.392317i \(-0.128327\pi\)
0.919830 + 0.392317i \(0.128327\pi\)
\(774\) −65310.1 −3.03297
\(775\) − 11603.5i − 0.537820i
\(776\) − 12748.8i − 0.589762i
\(777\) − 57621.7i − 2.66045i
\(778\) −22895.5 −1.05507
\(779\) − 3361.18i − 0.154592i
\(780\) 901.082 0.0413640
\(781\) −9420.91 −0.431634
\(782\) 0 0
\(783\) −10485.2 −0.478558
\(784\) 34988.9 1.59388
\(785\) 3419.32i 0.155466i
\(786\) 67605.5 3.06795
\(787\) 17801.2i 0.806280i 0.915138 + 0.403140i \(0.132081\pi\)
−0.915138 + 0.403140i \(0.867919\pi\)
\(788\) 12286.2i 0.555430i
\(789\) 27046.1i 1.22036i
\(790\) 25895.6 1.16623
\(791\) −34556.0 −1.55331
\(792\) − 7520.28i − 0.337401i
\(793\) 1159.37i 0.0519175i
\(794\) 631.287i 0.0282161i
\(795\) 3670.66 0.163755
\(796\) − 424.044i − 0.0188817i
\(797\) −34006.4 −1.51138 −0.755688 0.654932i \(-0.772697\pi\)
−0.755688 + 0.654932i \(0.772697\pi\)
\(798\) 13496.9 0.598728
\(799\) 0 0
\(800\) −16012.0 −0.707636
\(801\) −38460.8 −1.69656
\(802\) 34374.7i 1.51348i
\(803\) 10674.3 0.469099
\(804\) 19844.4i 0.870469i
\(805\) 36003.8i 1.57636i
\(806\) − 1438.88i − 0.0628813i
\(807\) 36808.7 1.60561
\(808\) −3646.81 −0.158780
\(809\) 7943.98i 0.345235i 0.984989 + 0.172618i \(0.0552225\pi\)
−0.984989 + 0.172618i \(0.944777\pi\)
\(810\) − 21324.9i − 0.925038i
\(811\) 29182.7i 1.26355i 0.775150 + 0.631777i \(0.217674\pi\)
−0.775150 + 0.631777i \(0.782326\pi\)
\(812\) 6484.45 0.280246
\(813\) 57033.7i 2.46034i
\(814\) 12727.2 0.548018
\(815\) −26352.9 −1.13264
\(816\) 0 0
\(817\) −4617.38 −0.197726
\(818\) −23475.4 −1.00342
\(819\) − 3903.81i − 0.166557i
\(820\) 9265.66 0.394599
\(821\) − 29589.4i − 1.25783i −0.777475 0.628913i \(-0.783500\pi\)
0.777475 0.628913i \(-0.216500\pi\)
\(822\) 20827.1i 0.883732i
\(823\) − 11121.5i − 0.471046i −0.971869 0.235523i \(-0.924320\pi\)
0.971869 0.235523i \(-0.0756803\pi\)
\(824\) −14720.0 −0.622323
\(825\) 10362.9 0.437322
\(826\) 25095.0i 1.05710i
\(827\) − 12401.7i − 0.521461i −0.965412 0.260730i \(-0.916037\pi\)
0.965412 0.260730i \(-0.0839634\pi\)
\(828\) − 55348.7i − 2.32307i
\(829\) −6224.06 −0.260761 −0.130380 0.991464i \(-0.541620\pi\)
−0.130380 + 0.991464i \(0.541620\pi\)
\(830\) − 2583.66i − 0.108048i
\(831\) 42668.8 1.78118
\(832\) −425.870 −0.0177457
\(833\) 0 0
\(834\) −34487.3 −1.43189
\(835\) −15092.1 −0.625491
\(836\) 1224.72i 0.0506672i
\(837\) −39467.6 −1.62987
\(838\) 29728.7i 1.22549i
\(839\) 36811.6i 1.51475i 0.652978 + 0.757377i \(0.273519\pi\)
−0.652978 + 0.757377i \(0.726481\pi\)
\(840\) − 16152.2i − 0.663458i
\(841\) 22688.1 0.930258
\(842\) −23048.2 −0.943342
\(843\) 22269.3i 0.909839i
\(844\) − 11004.0i − 0.448782i
\(845\) 15529.9i 0.632241i
\(846\) 66184.3 2.68967
\(847\) 30912.8i 1.25405i
\(848\) −4430.25 −0.179405
\(849\) −83384.8 −3.37074
\(850\) 0 0
\(851\) −40664.8 −1.63804
\(852\) 31106.4 1.25081
\(853\) − 3319.10i − 0.133228i −0.997779 0.0666141i \(-0.978780\pi\)
0.997779 0.0666141i \(-0.0212196\pi\)
\(854\) −47871.6 −1.91819
\(855\) − 5599.77i − 0.223986i
\(856\) 18338.0i 0.732221i
\(857\) − 8084.66i − 0.322248i −0.986934 0.161124i \(-0.948488\pi\)
0.986934 0.161124i \(-0.0515119\pi\)
\(858\) 1285.04 0.0511313
\(859\) 23619.8 0.938181 0.469091 0.883150i \(-0.344582\pi\)
0.469091 + 0.883150i \(0.344582\pi\)
\(860\) − 12728.6i − 0.504699i
\(861\) − 59825.3i − 2.36799i
\(862\) 21781.3i 0.860642i
\(863\) −34938.3 −1.37812 −0.689058 0.724706i \(-0.741975\pi\)
−0.689058 + 0.724706i \(0.741975\pi\)
\(864\) 54462.4i 2.14450i
\(865\) 8491.75 0.333790
\(866\) −43605.0 −1.71104
\(867\) 0 0
\(868\) 24408.2 0.954457
\(869\) 15171.8 0.592252
\(870\) − 9759.69i − 0.380327i
\(871\) 987.756 0.0384258
\(872\) 3971.57i 0.154237i
\(873\) − 78668.3i − 3.04985i
\(874\) − 9525.02i − 0.368637i
\(875\) 39910.8 1.54198
\(876\) −35244.8 −1.35937
\(877\) 20700.0i 0.797022i 0.917164 + 0.398511i \(0.130473\pi\)
−0.917164 + 0.398511i \(0.869527\pi\)
\(878\) − 27713.8i − 1.06525i
\(879\) 3832.02i 0.147043i
\(880\) 8409.42 0.322138
\(881\) 5183.11i 0.198211i 0.995077 + 0.0991053i \(0.0315981\pi\)
−0.995077 + 0.0991053i \(0.968402\pi\)
\(882\) 91595.2 3.49679
\(883\) 7357.34 0.280401 0.140201 0.990123i \(-0.455225\pi\)
0.140201 + 0.990123i \(0.455225\pi\)
\(884\) 0 0
\(885\) 15517.0 0.589375
\(886\) 31241.6 1.18463
\(887\) 35702.2i 1.35148i 0.737140 + 0.675740i \(0.236176\pi\)
−0.737140 + 0.675740i \(0.763824\pi\)
\(888\) 18243.2 0.689417
\(889\) 16661.1i 0.628565i
\(890\) − 18245.7i − 0.687189i
\(891\) − 12493.9i − 0.469765i
\(892\) −6568.30 −0.246550
\(893\) 4679.19 0.175345
\(894\) − 9496.12i − 0.355255i
\(895\) 29034.3i 1.08437i
\(896\) 30718.9i 1.14537i
\(897\) −4105.86 −0.152832
\(898\) − 3974.77i − 0.147706i
\(899\) −6402.56 −0.237528
\(900\) −22959.1 −0.850337
\(901\) 0 0
\(902\) 13213.9 0.487775
\(903\) −82184.2 −3.02870
\(904\) − 10940.5i − 0.402519i
\(905\) 9195.14 0.337742
\(906\) − 105259.i − 3.85982i
\(907\) − 5660.56i − 0.207228i −0.994618 0.103614i \(-0.966959\pi\)
0.994618 0.103614i \(-0.0330407\pi\)
\(908\) − 15872.8i − 0.580128i
\(909\) −22503.2 −0.821105
\(910\) 1851.96 0.0674636
\(911\) − 11754.9i − 0.427506i −0.976888 0.213753i \(-0.931431\pi\)
0.976888 0.213753i \(-0.0685687\pi\)
\(912\) 10072.5i 0.365718i
\(913\) − 1513.72i − 0.0548705i
\(914\) −20873.9 −0.755412
\(915\) 29600.4i 1.06946i
\(916\) −20115.0 −0.725567
\(917\) 57083.0 2.05567
\(918\) 0 0
\(919\) 4513.88 0.162023 0.0810116 0.996713i \(-0.474185\pi\)
0.0810116 + 0.996713i \(0.474185\pi\)
\(920\) −11398.9 −0.408491
\(921\) 5730.59i 0.205027i
\(922\) −2618.90 −0.0935455
\(923\) − 1548.32i − 0.0552153i
\(924\) 21798.6i 0.776106i
\(925\) 16868.1i 0.599588i
\(926\) 30688.5 1.08908
\(927\) −90831.6 −3.21823
\(928\) 8835.06i 0.312527i
\(929\) − 15124.4i − 0.534138i −0.963677 0.267069i \(-0.913945\pi\)
0.963677 0.267069i \(-0.0860552\pi\)
\(930\) − 36736.6i − 1.29531i
\(931\) 6475.72 0.227963
\(932\) − 4659.03i − 0.163746i
\(933\) −47949.6 −1.68253
\(934\) 52077.6 1.82444
\(935\) 0 0
\(936\) 1235.96 0.0431608
\(937\) 19918.8 0.694470 0.347235 0.937778i \(-0.387121\pi\)
0.347235 + 0.937778i \(0.387121\pi\)
\(938\) 40785.4i 1.41971i
\(939\) 34586.4 1.20201
\(940\) 12899.0i 0.447573i
\(941\) 2072.11i 0.0717843i 0.999356 + 0.0358921i \(0.0114273\pi\)
−0.999356 + 0.0358921i \(0.988573\pi\)
\(942\) − 16100.9i − 0.556894i
\(943\) −42219.8 −1.45797
\(944\) −18728.0 −0.645703
\(945\) − 50798.1i − 1.74864i
\(946\) − 18152.4i − 0.623874i
\(947\) 5522.83i 0.189512i 0.995501 + 0.0947560i \(0.0302071\pi\)
−0.995501 + 0.0947560i \(0.969793\pi\)
\(948\) −50094.9 −1.71625
\(949\) 1754.31i 0.0600079i
\(950\) −3951.06 −0.134936
\(951\) 12587.8 0.429217
\(952\) 0 0
\(953\) −39192.3 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(954\) −11597.6 −0.393593
\(955\) − 31062.4i − 1.05252i
\(956\) −22634.1 −0.765733
\(957\) − 5718.03i − 0.193143i
\(958\) − 24036.1i − 0.810616i
\(959\) 17585.4i 0.592140i
\(960\) −10873.1 −0.365549
\(961\) 5691.04 0.191032
\(962\) 2091.71i 0.0701033i
\(963\) 113158.i 3.78655i
\(964\) − 11885.4i − 0.397097i
\(965\) −7246.44 −0.241732
\(966\) − 169535.i − 5.64668i
\(967\) −53414.4 −1.77631 −0.888155 0.459543i \(-0.848013\pi\)
−0.888155 + 0.459543i \(0.848013\pi\)
\(968\) −9787.11 −0.324968
\(969\) 0 0
\(970\) 37320.1 1.23534
\(971\) 39644.8 1.31026 0.655130 0.755516i \(-0.272614\pi\)
0.655130 + 0.755516i \(0.272614\pi\)
\(972\) 2961.66i 0.0977317i
\(973\) −29119.4 −0.959431
\(974\) − 8027.47i − 0.264083i
\(975\) 1703.14i 0.0559428i
\(976\) − 35725.8i − 1.17167i
\(977\) −40232.8 −1.31746 −0.658732 0.752378i \(-0.728907\pi\)
−0.658732 + 0.752378i \(0.728907\pi\)
\(978\) 124090. 4.05723
\(979\) − 10689.9i − 0.348978i
\(980\) 17851.4i 0.581880i
\(981\) 24507.2i 0.797608i
\(982\) 46498.6 1.51103
\(983\) 40449.2i 1.31244i 0.754569 + 0.656220i \(0.227846\pi\)
−0.754569 + 0.656220i \(0.772154\pi\)
\(984\) 18940.9 0.613631
\(985\) 15613.7 0.505070
\(986\) 0 0
\(987\) 83284.3 2.68589
\(988\) −201.282 −0.00648142
\(989\) 57999.0i 1.86477i
\(990\) 22014.5 0.706733
\(991\) − 41635.0i − 1.33459i −0.744793 0.667296i \(-0.767452\pi\)
0.744793 0.667296i \(-0.232548\pi\)
\(992\) 33256.2i 1.06440i
\(993\) 51965.4i 1.66070i
\(994\) 63931.7 2.04003
\(995\) −538.888 −0.0171697
\(996\) 4998.07i 0.159006i
\(997\) 33986.6i 1.07960i 0.841792 + 0.539802i \(0.181501\pi\)
−0.841792 + 0.539802i \(0.818499\pi\)
\(998\) 50714.3i 1.60855i
\(999\) 57374.2 1.81706
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 289.4.b.d.288.8 8
17.4 even 4 289.4.a.d.1.1 yes 4
17.13 even 4 289.4.a.c.1.1 4
17.16 even 2 inner 289.4.b.d.288.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
289.4.a.c.1.1 4 17.13 even 4
289.4.a.d.1.1 yes 4 17.4 even 4
289.4.b.d.288.7 8 17.16 even 2 inner
289.4.b.d.288.8 8 1.1 even 1 trivial