Properties

Label 289.4.b.f.288.9
Level $289$
Weight $4$
Character 289.288
Analytic conductor $17.052$
Analytic rank $0$
Dimension $24$
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: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 288.9
Character \(\chi\) \(=\) 289.288
Dual form 289.4.b.f.288.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.26162 q^{2} +6.09538i q^{3} -6.40830 q^{4} -13.2627i q^{5} -7.69007i q^{6} -27.2593i q^{7} +18.1779 q^{8} -10.1536 q^{9} +O(q^{10})\) \(q-1.26162 q^{2} +6.09538i q^{3} -6.40830 q^{4} -13.2627i q^{5} -7.69007i q^{6} -27.2593i q^{7} +18.1779 q^{8} -10.1536 q^{9} +16.7325i q^{10} +45.5149i q^{11} -39.0610i q^{12} -52.5379 q^{13} +34.3910i q^{14} +80.8409 q^{15} +28.3328 q^{16} +12.8100 q^{18} -3.08418 q^{19} +84.9911i q^{20} +166.156 q^{21} -57.4227i q^{22} +112.369i q^{23} +110.801i q^{24} -50.8981 q^{25} +66.2831 q^{26} +102.685i q^{27} +174.686i q^{28} +18.6294i q^{29} -101.991 q^{30} -238.763i q^{31} -181.168 q^{32} -277.430 q^{33} -361.531 q^{35} +65.0674 q^{36} +162.717i q^{37} +3.89108 q^{38} -320.238i q^{39} -241.087i q^{40} +383.816i q^{41} -209.626 q^{42} -468.761 q^{43} -291.673i q^{44} +134.664i q^{45} -141.767i q^{46} -199.727 q^{47} +172.699i q^{48} -400.071 q^{49} +64.2142 q^{50} +336.679 q^{52} +105.679 q^{53} -129.550i q^{54} +603.649 q^{55} -495.516i q^{56} -18.7993i q^{57} -23.5033i q^{58} -207.142 q^{59} -518.053 q^{60} +586.101i q^{61} +301.230i q^{62} +276.780i q^{63} +1.90375 q^{64} +696.792i q^{65} +350.013 q^{66} +401.953 q^{67} -684.930 q^{69} +456.116 q^{70} +481.559i q^{71} -184.571 q^{72} +725.281i q^{73} -205.287i q^{74} -310.243i q^{75} +19.7644 q^{76} +1240.71 q^{77} +404.020i q^{78} -382.621i q^{79} -375.768i q^{80} -900.052 q^{81} -484.231i q^{82} +182.391 q^{83} -1064.78 q^{84} +591.401 q^{86} -113.553 q^{87} +827.364i q^{88} -623.009 q^{89} -169.895i q^{90} +1432.15i q^{91} -720.094i q^{92} +1455.35 q^{93} +251.981 q^{94} +40.9045i q^{95} -1104.29i q^{96} +369.544i q^{97} +504.739 q^{98} -462.140i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26162 −0.446051 −0.223026 0.974813i \(-0.571593\pi\)
−0.223026 + 0.974813i \(0.571593\pi\)
\(3\) 6.09538i 1.17306i 0.809929 + 0.586528i \(0.199506\pi\)
−0.809929 + 0.586528i \(0.800494\pi\)
\(4\) −6.40830 −0.801038
\(5\) − 13.2627i − 1.18625i −0.805111 0.593124i \(-0.797894\pi\)
0.805111 0.593124i \(-0.202106\pi\)
\(6\) − 7.69007i − 0.523243i
\(7\) − 27.2593i − 1.47186i −0.677055 0.735932i \(-0.736744\pi\)
0.677055 0.735932i \(-0.263256\pi\)
\(8\) 18.1779 0.803356
\(9\) −10.1536 −0.376059
\(10\) 16.7325i 0.529128i
\(11\) 45.5149i 1.24757i 0.781596 + 0.623785i \(0.214406\pi\)
−0.781596 + 0.623785i \(0.785594\pi\)
\(12\) − 39.0610i − 0.939662i
\(13\) −52.5379 −1.12088 −0.560438 0.828196i \(-0.689367\pi\)
−0.560438 + 0.828196i \(0.689367\pi\)
\(14\) 34.3910i 0.656527i
\(15\) 80.8409 1.39153
\(16\) 28.3328 0.442700
\(17\) 0 0
\(18\) 12.8100 0.167742
\(19\) −3.08418 −0.0372400 −0.0186200 0.999827i \(-0.505927\pi\)
−0.0186200 + 0.999827i \(0.505927\pi\)
\(20\) 84.9911i 0.950230i
\(21\) 166.156 1.72658
\(22\) − 57.4227i − 0.556480i
\(23\) 112.369i 1.01872i 0.860554 + 0.509359i \(0.170118\pi\)
−0.860554 + 0.509359i \(0.829882\pi\)
\(24\) 110.801i 0.942381i
\(25\) −50.8981 −0.407185
\(26\) 66.2831 0.499968
\(27\) 102.685i 0.731917i
\(28\) 174.686i 1.17902i
\(29\) 18.6294i 0.119290i 0.998220 + 0.0596448i \(0.0189968\pi\)
−0.998220 + 0.0596448i \(0.981003\pi\)
\(30\) −101.991 −0.620696
\(31\) − 238.763i − 1.38333i −0.722219 0.691664i \(-0.756878\pi\)
0.722219 0.691664i \(-0.243122\pi\)
\(32\) −181.168 −1.00082
\(33\) −277.430 −1.46347
\(34\) 0 0
\(35\) −361.531 −1.74600
\(36\) 65.0674 0.301238
\(37\) 162.717i 0.722986i 0.932375 + 0.361493i \(0.117733\pi\)
−0.932375 + 0.361493i \(0.882267\pi\)
\(38\) 3.89108 0.0166110
\(39\) − 320.238i − 1.31485i
\(40\) − 241.087i − 0.952979i
\(41\) 383.816i 1.46200i 0.682378 + 0.730999i \(0.260946\pi\)
−0.682378 + 0.730999i \(0.739054\pi\)
\(42\) −209.626 −0.770143
\(43\) −468.761 −1.66245 −0.831226 0.555935i \(-0.812360\pi\)
−0.831226 + 0.555935i \(0.812360\pi\)
\(44\) − 291.673i − 0.999350i
\(45\) 134.664i 0.446100i
\(46\) − 141.767i − 0.454401i
\(47\) −199.727 −0.619856 −0.309928 0.950760i \(-0.600305\pi\)
−0.309928 + 0.950760i \(0.600305\pi\)
\(48\) 172.699i 0.519312i
\(49\) −400.071 −1.16639
\(50\) 64.2142 0.181625
\(51\) 0 0
\(52\) 336.679 0.897864
\(53\) 105.679 0.273889 0.136944 0.990579i \(-0.456272\pi\)
0.136944 + 0.990579i \(0.456272\pi\)
\(54\) − 129.550i − 0.326473i
\(55\) 603.649 1.47993
\(56\) − 495.516i − 1.18243i
\(57\) − 18.7993i − 0.0436846i
\(58\) − 23.5033i − 0.0532093i
\(59\) −207.142 −0.457078 −0.228539 0.973535i \(-0.573395\pi\)
−0.228539 + 0.973535i \(0.573395\pi\)
\(60\) −518.053 −1.11467
\(61\) 586.101i 1.23021i 0.788447 + 0.615103i \(0.210886\pi\)
−0.788447 + 0.615103i \(0.789114\pi\)
\(62\) 301.230i 0.617036i
\(63\) 276.780i 0.553508i
\(64\) 1.90375 0.00371826
\(65\) 696.792i 1.32964i
\(66\) 350.013 0.652782
\(67\) 401.953 0.732931 0.366466 0.930432i \(-0.380568\pi\)
0.366466 + 0.930432i \(0.380568\pi\)
\(68\) 0 0
\(69\) −684.930 −1.19501
\(70\) 456.116 0.778805
\(71\) 481.559i 0.804937i 0.915434 + 0.402468i \(0.131848\pi\)
−0.915434 + 0.402468i \(0.868152\pi\)
\(72\) −184.571 −0.302109
\(73\) 725.281i 1.16285i 0.813602 + 0.581423i \(0.197504\pi\)
−0.813602 + 0.581423i \(0.802496\pi\)
\(74\) − 205.287i − 0.322489i
\(75\) − 310.243i − 0.477650i
\(76\) 19.7644 0.0298307
\(77\) 1240.71 1.83625
\(78\) 404.020i 0.586491i
\(79\) − 382.621i − 0.544914i −0.962168 0.272457i \(-0.912164\pi\)
0.962168 0.272457i \(-0.0878363\pi\)
\(80\) − 375.768i − 0.525152i
\(81\) −900.052 −1.23464
\(82\) − 484.231i − 0.652127i
\(83\) 182.391 0.241204 0.120602 0.992701i \(-0.461517\pi\)
0.120602 + 0.992701i \(0.461517\pi\)
\(84\) −1064.78 −1.38306
\(85\) 0 0
\(86\) 591.401 0.741539
\(87\) −113.553 −0.139933
\(88\) 827.364i 1.00224i
\(89\) −623.009 −0.742010 −0.371005 0.928631i \(-0.620987\pi\)
−0.371005 + 0.928631i \(0.620987\pi\)
\(90\) − 169.895i − 0.198983i
\(91\) 1432.15i 1.64978i
\(92\) − 720.094i − 0.816032i
\(93\) 1455.35 1.62272
\(94\) 251.981 0.276488
\(95\) 40.9045i 0.0441759i
\(96\) − 1104.29i − 1.17402i
\(97\) 369.544i 0.386820i 0.981118 + 0.193410i \(0.0619547\pi\)
−0.981118 + 0.193410i \(0.938045\pi\)
\(98\) 504.739 0.520268
\(99\) − 462.140i − 0.469160i
\(100\) 326.170 0.326170
\(101\) 810.381 0.798375 0.399188 0.916869i \(-0.369292\pi\)
0.399188 + 0.916869i \(0.369292\pi\)
\(102\) 0 0
\(103\) 772.683 0.739172 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(104\) −955.026 −0.900462
\(105\) − 2203.67i − 2.04815i
\(106\) −133.327 −0.122169
\(107\) 996.966i 0.900751i 0.892839 + 0.450375i \(0.148710\pi\)
−0.892839 + 0.450375i \(0.851290\pi\)
\(108\) − 658.038i − 0.586293i
\(109\) 700.996i 0.615993i 0.951388 + 0.307996i \(0.0996585\pi\)
−0.951388 + 0.307996i \(0.900342\pi\)
\(110\) −761.578 −0.660123
\(111\) −991.820 −0.848102
\(112\) − 772.333i − 0.651595i
\(113\) 1459.26i 1.21483i 0.794384 + 0.607415i \(0.207794\pi\)
−0.794384 + 0.607415i \(0.792206\pi\)
\(114\) 23.7176i 0.0194856i
\(115\) 1490.31 1.20845
\(116\) − 119.383i − 0.0955554i
\(117\) 533.449 0.421516
\(118\) 261.335 0.203880
\(119\) 0 0
\(120\) 1469.51 1.11790
\(121\) −740.607 −0.556429
\(122\) − 739.439i − 0.548735i
\(123\) −2339.50 −1.71501
\(124\) 1530.07i 1.10810i
\(125\) − 982.788i − 0.703226i
\(126\) − 349.193i − 0.246893i
\(127\) −426.418 −0.297941 −0.148970 0.988842i \(-0.547596\pi\)
−0.148970 + 0.988842i \(0.547596\pi\)
\(128\) 1446.94 0.999164
\(129\) − 2857.28i − 1.95015i
\(130\) − 879.089i − 0.593086i
\(131\) 19.9037i 0.0132747i 0.999978 + 0.00663737i \(0.00211276\pi\)
−0.999978 + 0.00663737i \(0.997887\pi\)
\(132\) 1777.86 1.17229
\(133\) 84.0728i 0.0548123i
\(134\) −507.114 −0.326925
\(135\) 1361.88 0.868235
\(136\) 0 0
\(137\) −1824.89 −1.13804 −0.569019 0.822324i \(-0.692677\pi\)
−0.569019 + 0.822324i \(0.692677\pi\)
\(138\) 864.125 0.533037
\(139\) − 1803.87i − 1.10074i −0.834922 0.550368i \(-0.814488\pi\)
0.834922 0.550368i \(-0.185512\pi\)
\(140\) 2316.80 1.39861
\(141\) − 1217.41i − 0.727126i
\(142\) − 607.546i − 0.359043i
\(143\) − 2391.26i − 1.39837i
\(144\) −287.680 −0.166482
\(145\) 247.076 0.141507
\(146\) − 915.032i − 0.518689i
\(147\) − 2438.58i − 1.36824i
\(148\) − 1042.74i − 0.579139i
\(149\) 2742.46 1.50786 0.753929 0.656955i \(-0.228156\pi\)
0.753929 + 0.656955i \(0.228156\pi\)
\(150\) 391.410i 0.213057i
\(151\) −2085.22 −1.12379 −0.561896 0.827208i \(-0.689928\pi\)
−0.561896 + 0.827208i \(0.689928\pi\)
\(152\) −56.0639 −0.0299170
\(153\) 0 0
\(154\) −1565.30 −0.819063
\(155\) −3166.64 −1.64097
\(156\) 2052.18i 1.05324i
\(157\) −2703.53 −1.37430 −0.687149 0.726516i \(-0.741138\pi\)
−0.687149 + 0.726516i \(0.741138\pi\)
\(158\) 482.724i 0.243060i
\(159\) 644.152i 0.321287i
\(160\) 2402.77i 1.18722i
\(161\) 3063.10 1.49942
\(162\) 1135.53 0.550712
\(163\) 2923.57i 1.40486i 0.711755 + 0.702428i \(0.247901\pi\)
−0.711755 + 0.702428i \(0.752099\pi\)
\(164\) − 2459.61i − 1.17112i
\(165\) 3679.46i 1.73604i
\(166\) −230.108 −0.107590
\(167\) − 2419.31i − 1.12103i −0.828145 0.560514i \(-0.810603\pi\)
0.828145 0.560514i \(-0.189397\pi\)
\(168\) 3020.36 1.38706
\(169\) 563.229 0.256363
\(170\) 0 0
\(171\) 31.3156 0.0140045
\(172\) 3003.97 1.33169
\(173\) − 393.655i − 0.173000i −0.996252 0.0865000i \(-0.972432\pi\)
0.996252 0.0865000i \(-0.0275683\pi\)
\(174\) 143.262 0.0624174
\(175\) 1387.45i 0.599321i
\(176\) 1289.57i 0.552299i
\(177\) − 1262.61i − 0.536178i
\(178\) 786.004 0.330975
\(179\) −1918.60 −0.801133 −0.400567 0.916268i \(-0.631187\pi\)
−0.400567 + 0.916268i \(0.631187\pi\)
\(180\) − 862.966i − 0.357343i
\(181\) 1677.03i 0.688688i 0.938844 + 0.344344i \(0.111899\pi\)
−0.938844 + 0.344344i \(0.888101\pi\)
\(182\) − 1806.83i − 0.735886i
\(183\) −3572.51 −1.44310
\(184\) 2042.63i 0.818393i
\(185\) 2158.06 0.857640
\(186\) −1836.11 −0.723817
\(187\) 0 0
\(188\) 1279.91 0.496529
\(189\) 2799.13 1.07728
\(190\) − 51.6061i − 0.0197047i
\(191\) 2347.19 0.889196 0.444598 0.895730i \(-0.353347\pi\)
0.444598 + 0.895730i \(0.353347\pi\)
\(192\) 11.6041i 0.00436173i
\(193\) − 650.982i − 0.242791i −0.992604 0.121396i \(-0.961263\pi\)
0.992604 0.121396i \(-0.0387369\pi\)
\(194\) − 466.226i − 0.172542i
\(195\) −4247.21 −1.55974
\(196\) 2563.77 0.934320
\(197\) 2681.01i 0.969614i 0.874621 + 0.484807i \(0.161110\pi\)
−0.874621 + 0.484807i \(0.838890\pi\)
\(198\) 583.047i 0.209269i
\(199\) − 1784.74i − 0.635762i −0.948131 0.317881i \(-0.897029\pi\)
0.948131 0.317881i \(-0.102971\pi\)
\(200\) −925.218 −0.327114
\(201\) 2450.06i 0.859769i
\(202\) −1022.40 −0.356116
\(203\) 507.825 0.175578
\(204\) 0 0
\(205\) 5090.42 1.73429
\(206\) −974.835 −0.329709
\(207\) − 1140.95i − 0.383099i
\(208\) −1488.55 −0.496212
\(209\) − 140.376i − 0.0464595i
\(210\) 2780.20i 0.913581i
\(211\) − 68.5651i − 0.0223707i −0.999937 0.0111853i \(-0.996440\pi\)
0.999937 0.0111853i \(-0.00356048\pi\)
\(212\) −677.222 −0.219395
\(213\) −2935.28 −0.944235
\(214\) − 1257.80i − 0.401781i
\(215\) 6217.02i 1.97208i
\(216\) 1866.60i 0.587990i
\(217\) −6508.53 −2.03607
\(218\) − 884.393i − 0.274765i
\(219\) −4420.86 −1.36408
\(220\) −3868.36 −1.18548
\(221\) 0 0
\(222\) 1251.30 0.378297
\(223\) −4734.22 −1.42164 −0.710822 0.703372i \(-0.751677\pi\)
−0.710822 + 0.703372i \(0.751677\pi\)
\(224\) 4938.52i 1.47308i
\(225\) 516.799 0.153126
\(226\) − 1841.04i − 0.541877i
\(227\) − 1722.86i − 0.503745i −0.967760 0.251873i \(-0.918954\pi\)
0.967760 0.251873i \(-0.0810464\pi\)
\(228\) 120.471i 0.0349930i
\(229\) 313.375 0.0904297 0.0452149 0.998977i \(-0.485603\pi\)
0.0452149 + 0.998977i \(0.485603\pi\)
\(230\) −1880.21 −0.539032
\(231\) 7562.57i 2.15403i
\(232\) 338.643i 0.0958319i
\(233\) − 2101.61i − 0.590905i −0.955357 0.295453i \(-0.904530\pi\)
0.955357 0.295453i \(-0.0954705\pi\)
\(234\) −673.012 −0.188018
\(235\) 2648.92i 0.735303i
\(236\) 1327.43 0.366137
\(237\) 2332.22 0.639215
\(238\) 0 0
\(239\) −5525.44 −1.49544 −0.747721 0.664013i \(-0.768852\pi\)
−0.747721 + 0.664013i \(0.768852\pi\)
\(240\) 2290.45 0.616033
\(241\) 2005.92i 0.536153i 0.963398 + 0.268076i \(0.0863880\pi\)
−0.963398 + 0.268076i \(0.913612\pi\)
\(242\) 934.367 0.248196
\(243\) − 2713.65i − 0.716383i
\(244\) − 3755.91i − 0.985442i
\(245\) 5306.00i 1.38362i
\(246\) 2951.57 0.764981
\(247\) 162.037 0.0417414
\(248\) − 4340.21i − 1.11130i
\(249\) 1111.74i 0.282946i
\(250\) 1239.91i 0.313675i
\(251\) 4427.81 1.11347 0.556735 0.830690i \(-0.312054\pi\)
0.556735 + 0.830690i \(0.312054\pi\)
\(252\) − 1773.69i − 0.443381i
\(253\) −5114.46 −1.27092
\(254\) 537.979 0.132897
\(255\) 0 0
\(256\) −1840.73 −0.449397
\(257\) 6198.01 1.50436 0.752182 0.658956i \(-0.229002\pi\)
0.752182 + 0.658956i \(0.229002\pi\)
\(258\) 3604.81i 0.869867i
\(259\) 4435.55 1.06414
\(260\) − 4465.26i − 1.06509i
\(261\) − 189.156i − 0.0448599i
\(262\) − 25.1109i − 0.00592122i
\(263\) −477.072 −0.111854 −0.0559269 0.998435i \(-0.517811\pi\)
−0.0559269 + 0.998435i \(0.517811\pi\)
\(264\) −5043.09 −1.17569
\(265\) − 1401.58i − 0.324900i
\(266\) − 106.068i − 0.0244491i
\(267\) − 3797.48i − 0.870419i
\(268\) −2575.84 −0.587106
\(269\) − 7583.93i − 1.71896i −0.511168 0.859481i \(-0.670787\pi\)
0.511168 0.859481i \(-0.329213\pi\)
\(270\) −1718.18 −0.387278
\(271\) −5431.92 −1.21759 −0.608793 0.793329i \(-0.708346\pi\)
−0.608793 + 0.793329i \(0.708346\pi\)
\(272\) 0 0
\(273\) −8729.47 −1.93528
\(274\) 2302.33 0.507624
\(275\) − 2316.62i − 0.507991i
\(276\) 4389.24 0.957251
\(277\) − 4091.98i − 0.887594i −0.896127 0.443797i \(-0.853631\pi\)
0.896127 0.443797i \(-0.146369\pi\)
\(278\) 2275.81i 0.490985i
\(279\) 2424.31i 0.520213i
\(280\) −6571.86 −1.40266
\(281\) 3410.26 0.723983 0.361991 0.932181i \(-0.382097\pi\)
0.361991 + 0.932181i \(0.382097\pi\)
\(282\) 1535.92i 0.324336i
\(283\) − 4322.86i − 0.908011i −0.890999 0.454006i \(-0.849995\pi\)
0.890999 0.454006i \(-0.150005\pi\)
\(284\) − 3085.97i − 0.644785i
\(285\) −249.328 −0.0518208
\(286\) 3016.87i 0.623745i
\(287\) 10462.6 2.15186
\(288\) 1839.51 0.376369
\(289\) 0 0
\(290\) −311.716 −0.0631194
\(291\) −2252.51 −0.453761
\(292\) − 4647.82i − 0.931484i
\(293\) 3073.56 0.612831 0.306416 0.951898i \(-0.400870\pi\)
0.306416 + 0.951898i \(0.400870\pi\)
\(294\) 3076.57i 0.610304i
\(295\) 2747.26i 0.542208i
\(296\) 2957.84i 0.580815i
\(297\) −4673.70 −0.913117
\(298\) −3459.95 −0.672583
\(299\) − 5903.62i − 1.14186i
\(300\) 1988.13i 0.382616i
\(301\) 12778.1i 2.44690i
\(302\) 2630.76 0.501269
\(303\) 4939.57i 0.936538i
\(304\) −87.3836 −0.0164862
\(305\) 7773.26 1.45933
\(306\) 0 0
\(307\) −3954.45 −0.735154 −0.367577 0.929993i \(-0.619813\pi\)
−0.367577 + 0.929993i \(0.619813\pi\)
\(308\) −7950.82 −1.47091
\(309\) 4709.79i 0.867089i
\(310\) 3995.11 0.731957
\(311\) − 8984.40i − 1.63813i −0.573701 0.819065i \(-0.694493\pi\)
0.573701 0.819065i \(-0.305507\pi\)
\(312\) − 5821.24i − 1.05629i
\(313\) 1737.01i 0.313680i 0.987624 + 0.156840i \(0.0501306\pi\)
−0.987624 + 0.156840i \(0.949869\pi\)
\(314\) 3410.83 0.613008
\(315\) 3670.84 0.656598
\(316\) 2451.95i 0.436497i
\(317\) 8232.55i 1.45863i 0.684177 + 0.729316i \(0.260162\pi\)
−0.684177 + 0.729316i \(0.739838\pi\)
\(318\) − 812.678i − 0.143310i
\(319\) −847.916 −0.148822
\(320\) − 25.2488i − 0.00441078i
\(321\) −6076.88 −1.05663
\(322\) −3864.48 −0.668817
\(323\) 0 0
\(324\) 5767.81 0.988993
\(325\) 2674.08 0.456403
\(326\) − 3688.44i − 0.626638i
\(327\) −4272.83 −0.722594
\(328\) 6976.95i 1.17451i
\(329\) 5444.44i 0.912345i
\(330\) − 4642.10i − 0.774361i
\(331\) 4153.12 0.689655 0.344828 0.938666i \(-0.387937\pi\)
0.344828 + 0.938666i \(0.387937\pi\)
\(332\) −1168.81 −0.193214
\(333\) − 1652.16i − 0.271885i
\(334\) 3052.26i 0.500037i
\(335\) − 5330.97i − 0.869439i
\(336\) 4707.66 0.764357
\(337\) − 6569.49i − 1.06191i −0.847401 0.530954i \(-0.821834\pi\)
0.847401 0.530954i \(-0.178166\pi\)
\(338\) −710.584 −0.114351
\(339\) −8894.75 −1.42506
\(340\) 0 0
\(341\) 10867.3 1.72580
\(342\) −39.5085 −0.00624671
\(343\) 1555.70i 0.244898i
\(344\) −8521.08 −1.33554
\(345\) 9084.00i 1.41758i
\(346\) 496.644i 0.0771669i
\(347\) − 2038.58i − 0.315380i −0.987489 0.157690i \(-0.949595\pi\)
0.987489 0.157690i \(-0.0504047\pi\)
\(348\) 727.684 0.112092
\(349\) 1791.21 0.274731 0.137365 0.990520i \(-0.456137\pi\)
0.137365 + 0.990520i \(0.456137\pi\)
\(350\) − 1750.44i − 0.267328i
\(351\) − 5394.86i − 0.820388i
\(352\) − 8245.86i − 1.24860i
\(353\) −5928.71 −0.893919 −0.446959 0.894554i \(-0.647493\pi\)
−0.446959 + 0.894554i \(0.647493\pi\)
\(354\) 1592.94i 0.239163i
\(355\) 6386.75 0.954854
\(356\) 3992.43 0.594378
\(357\) 0 0
\(358\) 2420.55 0.357347
\(359\) −413.998 −0.0608634 −0.0304317 0.999537i \(-0.509688\pi\)
−0.0304317 + 0.999537i \(0.509688\pi\)
\(360\) 2447.90i 0.358377i
\(361\) −6849.49 −0.998613
\(362\) − 2115.78i − 0.307190i
\(363\) − 4514.28i − 0.652722i
\(364\) − 9177.63i − 1.32154i
\(365\) 9619.15 1.37942
\(366\) 4507.16 0.643697
\(367\) 6979.15i 0.992667i 0.868132 + 0.496334i \(0.165321\pi\)
−0.868132 + 0.496334i \(0.834679\pi\)
\(368\) 3183.73i 0.450987i
\(369\) − 3897.11i − 0.549798i
\(370\) −2722.66 −0.382552
\(371\) − 2880.73i − 0.403127i
\(372\) −9326.34 −1.29986
\(373\) 6855.64 0.951667 0.475833 0.879535i \(-0.342147\pi\)
0.475833 + 0.879535i \(0.342147\pi\)
\(374\) 0 0
\(375\) 5990.46 0.824923
\(376\) −3630.62 −0.497965
\(377\) − 978.750i − 0.133709i
\(378\) −3531.45 −0.480524
\(379\) 8206.78i 1.11228i 0.831089 + 0.556140i \(0.187718\pi\)
−0.831089 + 0.556140i \(0.812282\pi\)
\(380\) − 262.128i − 0.0353866i
\(381\) − 2599.18i − 0.349501i
\(382\) −2961.27 −0.396627
\(383\) 889.401 0.118659 0.0593293 0.998238i \(-0.481104\pi\)
0.0593293 + 0.998238i \(0.481104\pi\)
\(384\) 8819.67i 1.17208i
\(385\) − 16455.1i − 2.17825i
\(386\) 821.294i 0.108297i
\(387\) 4759.62 0.625180
\(388\) − 2368.15i − 0.309857i
\(389\) 12992.8 1.69347 0.846736 0.532013i \(-0.178564\pi\)
0.846736 + 0.532013i \(0.178564\pi\)
\(390\) 5358.38 0.695723
\(391\) 0 0
\(392\) −7272.43 −0.937023
\(393\) −121.320 −0.0155720
\(394\) − 3382.42i − 0.432498i
\(395\) −5074.57 −0.646403
\(396\) 2961.54i 0.375815i
\(397\) 4325.29i 0.546801i 0.961900 + 0.273400i \(0.0881484\pi\)
−0.961900 + 0.273400i \(0.911852\pi\)
\(398\) 2251.67i 0.283583i
\(399\) −512.455 −0.0642979
\(400\) −1442.09 −0.180261
\(401\) − 11849.1i − 1.47560i −0.675022 0.737798i \(-0.735866\pi\)
0.675022 0.737798i \(-0.264134\pi\)
\(402\) − 3091.05i − 0.383501i
\(403\) 12544.1i 1.55054i
\(404\) −5193.17 −0.639529
\(405\) 11937.1i 1.46459i
\(406\) −640.685 −0.0783168
\(407\) −7406.04 −0.901975
\(408\) 0 0
\(409\) 7677.24 0.928154 0.464077 0.885795i \(-0.346386\pi\)
0.464077 + 0.885795i \(0.346386\pi\)
\(410\) −6422.19 −0.773584
\(411\) − 11123.4i − 1.33498i
\(412\) −4951.59 −0.592105
\(413\) 5646.55i 0.672757i
\(414\) 1439.45i 0.170882i
\(415\) − 2418.98i − 0.286128i
\(416\) 9518.20 1.12180
\(417\) 10995.3 1.29122
\(418\) 177.102i 0.0207233i
\(419\) − 3235.83i − 0.377280i −0.982046 0.188640i \(-0.939592\pi\)
0.982046 0.188640i \(-0.0604080\pi\)
\(420\) 14121.8i 1.64065i
\(421\) 14496.6 1.67820 0.839100 0.543977i \(-0.183082\pi\)
0.839100 + 0.543977i \(0.183082\pi\)
\(422\) 86.5033i 0.00997848i
\(423\) 2027.95 0.233103
\(424\) 1921.02 0.220030
\(425\) 0 0
\(426\) 3703.22 0.421178
\(427\) 15976.7 1.81070
\(428\) − 6388.86i − 0.721536i
\(429\) 14575.6 1.64037
\(430\) − 7843.54i − 0.879649i
\(431\) 7136.25i 0.797543i 0.917050 + 0.398771i \(0.130563\pi\)
−0.917050 + 0.398771i \(0.869437\pi\)
\(432\) 2909.36i 0.324020i
\(433\) −11434.3 −1.26904 −0.634521 0.772905i \(-0.718803\pi\)
−0.634521 + 0.772905i \(0.718803\pi\)
\(434\) 8211.32 0.908193
\(435\) 1506.02i 0.165996i
\(436\) − 4492.20i − 0.493434i
\(437\) − 346.566i − 0.0379371i
\(438\) 5577.46 0.608451
\(439\) 13308.0i 1.44682i 0.690417 + 0.723412i \(0.257427\pi\)
−0.690417 + 0.723412i \(0.742573\pi\)
\(440\) 10973.0 1.18891
\(441\) 4062.16 0.438630
\(442\) 0 0
\(443\) −10897.9 −1.16880 −0.584398 0.811467i \(-0.698669\pi\)
−0.584398 + 0.811467i \(0.698669\pi\)
\(444\) 6355.88 0.679362
\(445\) 8262.76i 0.880208i
\(446\) 5972.80 0.634126
\(447\) 16716.3i 1.76880i
\(448\) − 51.8949i − 0.00547278i
\(449\) − 16756.4i − 1.76121i −0.473855 0.880603i \(-0.657138\pi\)
0.473855 0.880603i \(-0.342862\pi\)
\(450\) −652.006 −0.0683019
\(451\) −17469.3 −1.82394
\(452\) − 9351.40i − 0.973126i
\(453\) − 12710.2i − 1.31827i
\(454\) 2173.60i 0.224696i
\(455\) 18994.1 1.95705
\(456\) − 341.730i − 0.0350943i
\(457\) −13561.8 −1.38817 −0.694087 0.719891i \(-0.744192\pi\)
−0.694087 + 0.719891i \(0.744192\pi\)
\(458\) −395.361 −0.0403363
\(459\) 0 0
\(460\) −9550.36 −0.968017
\(461\) −15516.4 −1.56762 −0.783810 0.621000i \(-0.786727\pi\)
−0.783810 + 0.621000i \(0.786727\pi\)
\(462\) − 9541.11i − 0.960807i
\(463\) −15186.4 −1.52435 −0.762174 0.647372i \(-0.775868\pi\)
−0.762174 + 0.647372i \(0.775868\pi\)
\(464\) 527.824i 0.0528095i
\(465\) − 19301.8i − 1.92495i
\(466\) 2651.44i 0.263574i
\(467\) −10544.3 −1.04482 −0.522412 0.852693i \(-0.674968\pi\)
−0.522412 + 0.852693i \(0.674968\pi\)
\(468\) −3418.50 −0.337650
\(469\) − 10957.0i − 1.07878i
\(470\) − 3341.94i − 0.327983i
\(471\) − 16479.0i − 1.61213i
\(472\) −3765.40 −0.367196
\(473\) − 21335.6i − 2.07402i
\(474\) −2942.38 −0.285123
\(475\) 156.979 0.0151636
\(476\) 0 0
\(477\) −1073.02 −0.102998
\(478\) 6971.02 0.667044
\(479\) 5762.22i 0.549650i 0.961494 + 0.274825i \(0.0886200\pi\)
−0.961494 + 0.274825i \(0.911380\pi\)
\(480\) −14645.8 −1.39268
\(481\) − 8548.79i − 0.810377i
\(482\) − 2530.72i − 0.239152i
\(483\) 18670.7i 1.75890i
\(484\) 4746.03 0.445721
\(485\) 4901.14 0.458864
\(486\) 3423.61i 0.319544i
\(487\) 5369.36i 0.499608i 0.968296 + 0.249804i \(0.0803661\pi\)
−0.968296 + 0.249804i \(0.919634\pi\)
\(488\) 10654.1i 0.988293i
\(489\) −17820.2 −1.64797
\(490\) − 6694.18i − 0.617167i
\(491\) −17878.0 −1.64322 −0.821611 0.570049i \(-0.806924\pi\)
−0.821611 + 0.570049i \(0.806924\pi\)
\(492\) 14992.2 1.37378
\(493\) 0 0
\(494\) −204.429 −0.0186188
\(495\) −6129.21 −0.556540
\(496\) − 6764.84i − 0.612400i
\(497\) 13127.0 1.18476
\(498\) − 1402.60i − 0.126209i
\(499\) − 3507.03i − 0.314622i −0.987549 0.157311i \(-0.949718\pi\)
0.987549 0.157311i \(-0.0502825\pi\)
\(500\) 6298.01i 0.563311i
\(501\) 14746.6 1.31503
\(502\) −5586.23 −0.496665
\(503\) 1207.56i 0.107043i 0.998567 + 0.0535214i \(0.0170445\pi\)
−0.998567 + 0.0535214i \(0.982955\pi\)
\(504\) 5031.27i 0.444664i
\(505\) − 10747.8i − 0.947071i
\(506\) 6452.52 0.566897
\(507\) 3433.09i 0.300728i
\(508\) 2732.62 0.238662
\(509\) −14092.1 −1.22716 −0.613578 0.789634i \(-0.710270\pi\)
−0.613578 + 0.789634i \(0.710270\pi\)
\(510\) 0 0
\(511\) 19770.7 1.71155
\(512\) −9253.25 −0.798710
\(513\) − 316.700i − 0.0272566i
\(514\) −7819.56 −0.671023
\(515\) − 10247.8i − 0.876841i
\(516\) 18310.3i 1.56214i
\(517\) − 9090.58i − 0.773314i
\(518\) −5595.99 −0.474660
\(519\) 2399.47 0.202939
\(520\) 12666.2i 1.06817i
\(521\) 10080.1i 0.847635i 0.905748 + 0.423818i \(0.139310\pi\)
−0.905748 + 0.423818i \(0.860690\pi\)
\(522\) 238.643i 0.0200098i
\(523\) 12330.2 1.03090 0.515452 0.856919i \(-0.327624\pi\)
0.515452 + 0.856919i \(0.327624\pi\)
\(524\) − 127.549i − 0.0106336i
\(525\) −8457.01 −0.703036
\(526\) 601.886 0.0498925
\(527\) 0 0
\(528\) −7860.38 −0.647877
\(529\) −459.760 −0.0377875
\(530\) 1768.27i 0.144922i
\(531\) 2103.24 0.171888
\(532\) − 538.764i − 0.0439067i
\(533\) − 20164.9i − 1.63872i
\(534\) 4790.99i 0.388252i
\(535\) 13222.4 1.06851
\(536\) 7306.65 0.588805
\(537\) − 11694.6i − 0.939774i
\(538\) 9568.07i 0.766745i
\(539\) − 18209.2i − 1.45515i
\(540\) −8727.33 −0.695490
\(541\) 7122.72i 0.566044i 0.959113 + 0.283022i \(0.0913369\pi\)
−0.959113 + 0.283022i \(0.908663\pi\)
\(542\) 6853.04 0.543106
\(543\) −10222.1 −0.807869
\(544\) 0 0
\(545\) 9297.07 0.730720
\(546\) 11013.3 0.863235
\(547\) 14919.7i 1.16622i 0.812394 + 0.583109i \(0.198164\pi\)
−0.812394 + 0.583109i \(0.801836\pi\)
\(548\) 11694.5 0.911612
\(549\) − 5951.04i − 0.462630i
\(550\) 2922.70i 0.226590i
\(551\) − 57.4566i − 0.00444234i
\(552\) −12450.6 −0.960021
\(553\) −10430.0 −0.802040
\(554\) 5162.55i 0.395913i
\(555\) 13154.2i 1.00606i
\(556\) 11559.7i 0.881731i
\(557\) −18935.0 −1.44039 −0.720197 0.693769i \(-0.755949\pi\)
−0.720197 + 0.693769i \(0.755949\pi\)
\(558\) − 3058.57i − 0.232042i
\(559\) 24627.7 1.86340
\(560\) −10243.2 −0.772953
\(561\) 0 0
\(562\) −4302.47 −0.322934
\(563\) −8150.42 −0.610123 −0.305062 0.952333i \(-0.598677\pi\)
−0.305062 + 0.952333i \(0.598677\pi\)
\(564\) 7801.56i 0.582456i
\(565\) 19353.7 1.44109
\(566\) 5453.82i 0.405020i
\(567\) 24534.8i 1.81722i
\(568\) 8753.71i 0.646650i
\(569\) −18218.2 −1.34226 −0.671131 0.741338i \(-0.734191\pi\)
−0.671131 + 0.741338i \(0.734191\pi\)
\(570\) 314.558 0.0231147
\(571\) − 2443.35i − 0.179073i −0.995984 0.0895367i \(-0.971461\pi\)
0.995984 0.0895367i \(-0.0285386\pi\)
\(572\) 15323.9i 1.12015i
\(573\) 14307.0i 1.04308i
\(574\) −13199.8 −0.959842
\(575\) − 5719.36i − 0.414806i
\(576\) −19.3299 −0.00139829
\(577\) −852.752 −0.0615261 −0.0307630 0.999527i \(-0.509794\pi\)
−0.0307630 + 0.999527i \(0.509794\pi\)
\(578\) 0 0
\(579\) 3967.98 0.284807
\(580\) −1583.34 −0.113352
\(581\) − 4971.84i − 0.355020i
\(582\) 2841.82 0.202401
\(583\) 4809.96i 0.341695i
\(584\) 13184.1i 0.934179i
\(585\) − 7074.95i − 0.500022i
\(586\) −3877.68 −0.273354
\(587\) −900.272 −0.0633019 −0.0316509 0.999499i \(-0.510076\pi\)
−0.0316509 + 0.999499i \(0.510076\pi\)
\(588\) 15627.2i 1.09601i
\(589\) 736.390i 0.0515152i
\(590\) − 3466.00i − 0.241853i
\(591\) −16341.7 −1.13741
\(592\) 4610.22i 0.320066i
\(593\) 25007.2 1.73174 0.865870 0.500268i \(-0.166765\pi\)
0.865870 + 0.500268i \(0.166765\pi\)
\(594\) 5896.46 0.407297
\(595\) 0 0
\(596\) −17574.5 −1.20785
\(597\) 10878.6 0.745784
\(598\) 7448.15i 0.509327i
\(599\) −3448.82 −0.235251 −0.117625 0.993058i \(-0.537528\pi\)
−0.117625 + 0.993058i \(0.537528\pi\)
\(600\) − 5639.55i − 0.383723i
\(601\) − 942.873i − 0.0639943i −0.999488 0.0319972i \(-0.989813\pi\)
0.999488 0.0319972i \(-0.0101868\pi\)
\(602\) − 16121.2i − 1.09145i
\(603\) −4081.27 −0.275626
\(604\) 13362.7 0.900201
\(605\) 9822.41i 0.660063i
\(606\) − 6231.89i − 0.417744i
\(607\) − 10173.9i − 0.680306i −0.940370 0.340153i \(-0.889521\pi\)
0.940370 0.340153i \(-0.110479\pi\)
\(608\) 558.756 0.0372707
\(609\) 3095.39i 0.205963i
\(610\) −9806.93 −0.650936
\(611\) 10493.3 0.694782
\(612\) 0 0
\(613\) −17327.0 −1.14165 −0.570824 0.821073i \(-0.693376\pi\)
−0.570824 + 0.821073i \(0.693376\pi\)
\(614\) 4989.03 0.327916
\(615\) 31028.0i 2.03442i
\(616\) 22553.4 1.47516
\(617\) − 19782.8i − 1.29080i −0.763844 0.645401i \(-0.776690\pi\)
0.763844 0.645401i \(-0.223310\pi\)
\(618\) − 5941.98i − 0.386766i
\(619\) 15033.2i 0.976147i 0.872802 + 0.488074i \(0.162300\pi\)
−0.872802 + 0.488074i \(0.837700\pi\)
\(620\) 20292.8 1.31448
\(621\) −11538.6 −0.745618
\(622\) 11334.9i 0.730690i
\(623\) 16982.8i 1.09214i
\(624\) − 9073.25i − 0.582084i
\(625\) −19396.6 −1.24139
\(626\) − 2191.46i − 0.139917i
\(627\) 855.647 0.0544996
\(628\) 17325.0 1.10087
\(629\) 0 0
\(630\) −4631.22 −0.292877
\(631\) 24060.5 1.51796 0.758979 0.651115i \(-0.225698\pi\)
0.758979 + 0.651115i \(0.225698\pi\)
\(632\) − 6955.23i − 0.437760i
\(633\) 417.930 0.0262421
\(634\) − 10386.4i − 0.650625i
\(635\) 5655.44i 0.353432i
\(636\) − 4127.92i − 0.257363i
\(637\) 21018.9 1.30737
\(638\) 1069.75 0.0663822
\(639\) − 4889.55i − 0.302704i
\(640\) − 19190.3i − 1.18526i
\(641\) 4886.07i 0.301074i 0.988604 + 0.150537i \(0.0481002\pi\)
−0.988604 + 0.150537i \(0.951900\pi\)
\(642\) 7666.74 0.471312
\(643\) 24508.4i 1.50314i 0.659655 + 0.751568i \(0.270702\pi\)
−0.659655 + 0.751568i \(0.729298\pi\)
\(644\) −19629.3 −1.20109
\(645\) −37895.1 −2.31336
\(646\) 0 0
\(647\) 2932.50 0.178189 0.0890947 0.996023i \(-0.471603\pi\)
0.0890947 + 0.996023i \(0.471603\pi\)
\(648\) −16361.0 −0.991854
\(649\) − 9428.06i − 0.570237i
\(650\) −3373.68 −0.203579
\(651\) − 39671.9i − 2.38843i
\(652\) − 18735.1i − 1.12534i
\(653\) 19977.8i 1.19723i 0.801035 + 0.598617i \(0.204283\pi\)
−0.801035 + 0.598617i \(0.795717\pi\)
\(654\) 5390.71 0.322314
\(655\) 263.975 0.0157471
\(656\) 10874.6i 0.647227i
\(657\) − 7364.21i − 0.437299i
\(658\) − 6868.83i − 0.406953i
\(659\) 28222.7 1.66829 0.834143 0.551548i \(-0.185963\pi\)
0.834143 + 0.551548i \(0.185963\pi\)
\(660\) − 23579.1i − 1.39063i
\(661\) −20712.8 −1.21881 −0.609405 0.792859i \(-0.708592\pi\)
−0.609405 + 0.792859i \(0.708592\pi\)
\(662\) −5239.67 −0.307622
\(663\) 0 0
\(664\) 3315.47 0.193773
\(665\) 1115.03 0.0650210
\(666\) 2084.41i 0.121275i
\(667\) −2093.37 −0.121522
\(668\) 15503.7i 0.897987i
\(669\) − 28856.8i − 1.66767i
\(670\) 6725.68i 0.387814i
\(671\) −26676.3 −1.53477
\(672\) −30102.2 −1.72800
\(673\) 395.468i 0.0226511i 0.999936 + 0.0113255i \(0.00360511\pi\)
−0.999936 + 0.0113255i \(0.996395\pi\)
\(674\) 8288.23i 0.473666i
\(675\) − 5226.48i − 0.298025i
\(676\) −3609.34 −0.205356
\(677\) 2297.47i 0.130427i 0.997871 + 0.0652135i \(0.0207729\pi\)
−0.997871 + 0.0652135i \(0.979227\pi\)
\(678\) 11221.8 0.635652
\(679\) 10073.5 0.569347
\(680\) 0 0
\(681\) 10501.5 0.590921
\(682\) −13710.4 −0.769794
\(683\) 28611.7i 1.60292i 0.598048 + 0.801461i \(0.295943\pi\)
−0.598048 + 0.801461i \(0.704057\pi\)
\(684\) −200.680 −0.0112181
\(685\) 24202.9i 1.35000i
\(686\) − 1962.71i − 0.109237i
\(687\) 1910.14i 0.106079i
\(688\) −13281.3 −0.735968
\(689\) −5552.14 −0.306995
\(690\) − 11460.6i − 0.632315i
\(691\) 4902.87i 0.269919i 0.990851 + 0.134959i \(0.0430904\pi\)
−0.990851 + 0.134959i \(0.956910\pi\)
\(692\) 2522.66i 0.138580i
\(693\) −12597.6 −0.690540
\(694\) 2571.93i 0.140676i
\(695\) −23924.1 −1.30575
\(696\) −2064.16 −0.112416
\(697\) 0 0
\(698\) −2259.83 −0.122544
\(699\) 12810.1 0.693165
\(700\) − 8891.18i − 0.480079i
\(701\) 1714.99 0.0924027 0.0462014 0.998932i \(-0.485288\pi\)
0.0462014 + 0.998932i \(0.485288\pi\)
\(702\) 6806.28i 0.365935i
\(703\) − 501.849i − 0.0269240i
\(704\) 86.6490i 0.00463879i
\(705\) −16146.1 −0.862552
\(706\) 7479.80 0.398734
\(707\) − 22090.4i − 1.17510i
\(708\) 8091.18i 0.429499i
\(709\) 13939.8i 0.738393i 0.929351 + 0.369196i \(0.120367\pi\)
−0.929351 + 0.369196i \(0.879633\pi\)
\(710\) −8057.67 −0.425914
\(711\) 3884.98i 0.204920i
\(712\) −11325.0 −0.596098
\(713\) 26829.6 1.40922
\(714\) 0 0
\(715\) −31714.4 −1.65881
\(716\) 12295.0 0.641738
\(717\) − 33679.6i − 1.75424i
\(718\) 522.310 0.0271482
\(719\) 1779.18i 0.0922839i 0.998935 + 0.0461419i \(0.0146926\pi\)
−0.998935 + 0.0461419i \(0.985307\pi\)
\(720\) 3815.40i 0.197488i
\(721\) − 21062.8i − 1.08796i
\(722\) 8641.48 0.445433
\(723\) −12226.9 −0.628937
\(724\) − 10746.9i − 0.551665i
\(725\) − 948.201i − 0.0485729i
\(726\) 5695.32i 0.291148i
\(727\) 9141.02 0.466330 0.233165 0.972437i \(-0.425092\pi\)
0.233165 + 0.972437i \(0.425092\pi\)
\(728\) 26033.4i 1.32536i
\(729\) −7760.65 −0.394282
\(730\) −12135.8 −0.615294
\(731\) 0 0
\(732\) 22893.7 1.15598
\(733\) −12176.0 −0.613547 −0.306773 0.951783i \(-0.599249\pi\)
−0.306773 + 0.951783i \(0.599249\pi\)
\(734\) − 8805.07i − 0.442781i
\(735\) −32342.1 −1.62307
\(736\) − 20357.7i − 1.01956i
\(737\) 18294.9i 0.914383i
\(738\) 4916.69i 0.245238i
\(739\) −25025.4 −1.24570 −0.622852 0.782339i \(-0.714026\pi\)
−0.622852 + 0.782339i \(0.714026\pi\)
\(740\) −13829.5 −0.687003
\(741\) 987.673i 0.0489650i
\(742\) 3634.40i 0.179816i
\(743\) 10781.3i 0.532339i 0.963926 + 0.266170i \(0.0857581\pi\)
−0.963926 + 0.266170i \(0.914242\pi\)
\(744\) 26455.2 1.30362
\(745\) − 36372.3i − 1.78869i
\(746\) −8649.24 −0.424492
\(747\) −1851.92 −0.0907072
\(748\) 0 0
\(749\) 27176.6 1.32578
\(750\) −7557.71 −0.367958
\(751\) − 33910.7i − 1.64769i −0.566812 0.823847i \(-0.691823\pi\)
0.566812 0.823847i \(-0.308177\pi\)
\(752\) −5658.84 −0.274410
\(753\) 26989.2i 1.30616i
\(754\) 1234.81i 0.0596410i
\(755\) 27655.5i 1.33310i
\(756\) −17937.7 −0.862945
\(757\) 31545.0 1.51456 0.757281 0.653090i \(-0.226528\pi\)
0.757281 + 0.653090i \(0.226528\pi\)
\(758\) − 10353.9i − 0.496134i
\(759\) − 31174.5i − 1.49086i
\(760\) 743.556i 0.0354890i
\(761\) 9150.51 0.435881 0.217941 0.975962i \(-0.430066\pi\)
0.217941 + 0.975962i \(0.430066\pi\)
\(762\) 3279.19i 0.155895i
\(763\) 19108.7 0.906658
\(764\) −15041.5 −0.712280
\(765\) 0 0
\(766\) −1122.09 −0.0529279
\(767\) 10882.8 0.512328
\(768\) − 11219.9i − 0.527168i
\(769\) 33797.5 1.58488 0.792438 0.609952i \(-0.208811\pi\)
0.792438 + 0.609952i \(0.208811\pi\)
\(770\) 20760.1i 0.971612i
\(771\) 37779.2i 1.76470i
\(772\) 4171.69i 0.194485i
\(773\) 33969.4 1.58059 0.790295 0.612727i \(-0.209927\pi\)
0.790295 + 0.612727i \(0.209927\pi\)
\(774\) −6004.85 −0.278863
\(775\) 12152.6i 0.563270i
\(776\) 6717.52i 0.310754i
\(777\) 27036.3i 1.24829i
\(778\) −16392.0 −0.755376
\(779\) − 1183.76i − 0.0544449i
\(780\) 27217.4 1.24941
\(781\) −21918.1 −1.00421
\(782\) 0 0
\(783\) −1912.96 −0.0873100
\(784\) −11335.1 −0.516359
\(785\) 35856.0i 1.63026i
\(786\) 153.061 0.00694592
\(787\) 10962.7i 0.496543i 0.968691 + 0.248271i \(0.0798624\pi\)
−0.968691 + 0.248271i \(0.920138\pi\)
\(788\) − 17180.7i − 0.776698i
\(789\) − 2907.93i − 0.131211i
\(790\) 6402.20 0.288329
\(791\) 39778.5 1.78807
\(792\) − 8400.72i − 0.376902i
\(793\) − 30792.5i − 1.37891i
\(794\) − 5456.89i − 0.243901i
\(795\) 8543.17 0.381126
\(796\) 11437.1i 0.509270i
\(797\) 8041.29 0.357386 0.178693 0.983905i \(-0.442813\pi\)
0.178693 + 0.983905i \(0.442813\pi\)
\(798\) 646.526 0.0286802
\(799\) 0 0
\(800\) 9221.12 0.407520
\(801\) 6325.79 0.279040
\(802\) 14949.1i 0.658192i
\(803\) −33011.1 −1.45073
\(804\) − 15700.7i − 0.688708i
\(805\) − 40624.8i − 1.77868i
\(806\) − 15826.0i − 0.691620i
\(807\) 46226.9 2.01644
\(808\) 14731.0 0.641379
\(809\) 36568.0i 1.58920i 0.607134 + 0.794599i \(0.292319\pi\)
−0.607134 + 0.794599i \(0.707681\pi\)
\(810\) − 15060.1i − 0.653282i
\(811\) − 37315.7i − 1.61570i −0.589390 0.807849i \(-0.700632\pi\)
0.589390 0.807849i \(-0.299368\pi\)
\(812\) −3254.30 −0.140645
\(813\) − 33109.6i − 1.42830i
\(814\) 9343.64 0.402327
\(815\) 38774.3 1.66651
\(816\) 0 0
\(817\) 1445.75 0.0619097
\(818\) −9685.79 −0.414005
\(819\) − 14541.4i − 0.620414i
\(820\) −32620.9 −1.38924
\(821\) 27365.1i 1.16328i 0.813448 + 0.581638i \(0.197588\pi\)
−0.813448 + 0.581638i \(0.802412\pi\)
\(822\) 14033.6i 0.595471i
\(823\) − 21138.4i − 0.895308i −0.894207 0.447654i \(-0.852260\pi\)
0.894207 0.447654i \(-0.147740\pi\)
\(824\) 14045.7 0.593818
\(825\) 14120.7 0.595902
\(826\) − 7123.83i − 0.300084i
\(827\) − 43453.3i − 1.82711i −0.406717 0.913554i \(-0.633327\pi\)
0.406717 0.913554i \(-0.366673\pi\)
\(828\) 7311.55i 0.306877i
\(829\) 36460.2 1.52752 0.763760 0.645501i \(-0.223351\pi\)
0.763760 + 0.645501i \(0.223351\pi\)
\(830\) 3051.85i 0.127628i
\(831\) 24942.2 1.04120
\(832\) −100.019 −0.00416771
\(833\) 0 0
\(834\) −13871.9 −0.575952
\(835\) −32086.5 −1.32982
\(836\) 899.575i 0.0372158i
\(837\) 24517.4 1.01248
\(838\) 4082.39i 0.168286i
\(839\) 20236.0i 0.832687i 0.909207 + 0.416344i \(0.136689\pi\)
−0.909207 + 0.416344i \(0.863311\pi\)
\(840\) − 40058.0i − 1.64539i
\(841\) 24041.9 0.985770
\(842\) −18289.3 −0.748564
\(843\) 20786.8i 0.849272i
\(844\) 439.386i 0.0179198i
\(845\) − 7469.92i − 0.304110i
\(846\) −2558.51 −0.103976
\(847\) 20188.4i 0.818988i
\(848\) 2994.18 0.121251
\(849\) 26349.4 1.06515
\(850\) 0 0
\(851\) −18284.3 −0.736519
\(852\) 18810.2 0.756368
\(853\) 6934.19i 0.278338i 0.990269 + 0.139169i \(0.0444431\pi\)
−0.990269 + 0.139169i \(0.955557\pi\)
\(854\) −20156.6 −0.807664
\(855\) − 415.328i − 0.0166128i
\(856\) 18122.7i 0.723623i
\(857\) − 19207.8i − 0.765607i −0.923830 0.382803i \(-0.874959\pi\)
0.923830 0.382803i \(-0.125041\pi\)
\(858\) −18388.9 −0.731688
\(859\) 9954.55 0.395395 0.197698 0.980263i \(-0.436654\pi\)
0.197698 + 0.980263i \(0.436654\pi\)
\(860\) − 39840.6i − 1.57971i
\(861\) 63773.2i 2.52426i
\(862\) − 9003.26i − 0.355745i
\(863\) −33698.0 −1.32919 −0.664596 0.747203i \(-0.731396\pi\)
−0.664596 + 0.747203i \(0.731396\pi\)
\(864\) − 18603.3i − 0.732519i
\(865\) −5220.91 −0.205221
\(866\) 14425.7 0.566058
\(867\) 0 0
\(868\) 41708.6 1.63097
\(869\) 17415.0 0.679818
\(870\) − 1900.03i − 0.0740425i
\(871\) −21117.8 −0.821525
\(872\) 12742.6i 0.494861i
\(873\) − 3752.20i − 0.145467i
\(874\) 437.236i 0.0169219i
\(875\) −26790.1 −1.03505
\(876\) 28330.2 1.09268
\(877\) − 4469.65i − 0.172097i −0.996291 0.0860487i \(-0.972576\pi\)
0.996291 0.0860487i \(-0.0274241\pi\)
\(878\) − 16789.7i − 0.645358i
\(879\) 18734.5i 0.718885i
\(880\) 17103.1 0.655164
\(881\) 2081.32i 0.0795930i 0.999208 + 0.0397965i \(0.0126710\pi\)
−0.999208 + 0.0397965i \(0.987329\pi\)
\(882\) −5124.91 −0.195652
\(883\) 10124.4 0.385857 0.192929 0.981213i \(-0.438201\pi\)
0.192929 + 0.981213i \(0.438201\pi\)
\(884\) 0 0
\(885\) −16745.6 −0.636040
\(886\) 13749.1 0.521343
\(887\) − 42725.3i − 1.61733i −0.588268 0.808666i \(-0.700190\pi\)
0.588268 0.808666i \(-0.299810\pi\)
\(888\) −18029.2 −0.681328
\(889\) 11623.9i 0.438529i
\(890\) − 10424.5i − 0.392618i
\(891\) − 40965.8i − 1.54030i
\(892\) 30338.3 1.13879
\(893\) 615.996 0.0230835
\(894\) − 21089.7i − 0.788977i
\(895\) 25445.7i 0.950343i
\(896\) − 39442.7i − 1.47063i
\(897\) 35984.8 1.33946
\(898\) 21140.2i 0.785588i
\(899\) 4448.02 0.165017
\(900\) −3311.80 −0.122659
\(901\) 0 0
\(902\) 22039.7 0.813573
\(903\) −77887.4 −2.87036
\(904\) 26526.3i 0.975941i
\(905\) 22241.8 0.816954
\(906\) 16035.5i 0.588017i
\(907\) 21767.6i 0.796893i 0.917192 + 0.398446i \(0.130450\pi\)
−0.917192 + 0.398446i \(0.869550\pi\)
\(908\) 11040.6i 0.403519i
\(909\) −8228.28 −0.300236
\(910\) −23963.4 −0.872943
\(911\) 21654.2i 0.787524i 0.919212 + 0.393762i \(0.128827\pi\)
−0.919212 + 0.393762i \(0.871173\pi\)
\(912\) − 532.636i − 0.0193392i
\(913\) 8301.49i 0.300919i
\(914\) 17109.9 0.619197
\(915\) 47380.9i 1.71187i
\(916\) −2008.20 −0.0724376
\(917\) 542.560 0.0195386
\(918\) 0 0
\(919\) 21445.1 0.769760 0.384880 0.922967i \(-0.374243\pi\)
0.384880 + 0.922967i \(0.374243\pi\)
\(920\) 27090.6 0.970818
\(921\) − 24103.8i − 0.862376i
\(922\) 19575.9 0.699239
\(923\) − 25300.1i − 0.902234i
\(924\) − 48463.2i − 1.72546i
\(925\) − 8281.97i − 0.294389i
\(926\) 19159.6 0.679938
\(927\) −7845.51 −0.277972
\(928\) − 3375.06i − 0.119388i
\(929\) − 24271.4i − 0.857177i −0.903500 0.428589i \(-0.859011\pi\)
0.903500 0.428589i \(-0.140989\pi\)
\(930\) 24351.7i 0.858626i
\(931\) 1233.89 0.0434363
\(932\) 13467.7i 0.473338i
\(933\) 54763.3 1.92162
\(934\) 13303.0 0.466046
\(935\) 0 0
\(936\) 9696.96 0.338627
\(937\) −38491.5 −1.34201 −0.671005 0.741453i \(-0.734137\pi\)
−0.671005 + 0.741453i \(0.734137\pi\)
\(938\) 13823.6i 0.481190i
\(939\) −10587.7 −0.367964
\(940\) − 16975.1i − 0.589006i
\(941\) − 31323.4i − 1.08514i −0.840012 0.542568i \(-0.817452\pi\)
0.840012 0.542568i \(-0.182548\pi\)
\(942\) 20790.3i 0.719092i
\(943\) −43128.9 −1.48937
\(944\) −5868.92 −0.202349
\(945\) − 37123.9i − 1.27793i
\(946\) 26917.5i 0.925121i
\(947\) 32096.4i 1.10136i 0.834715 + 0.550682i \(0.185632\pi\)
−0.834715 + 0.550682i \(0.814368\pi\)
\(948\) −14945.6 −0.512035
\(949\) − 38104.7i − 1.30341i
\(950\) −198.049 −0.00676373
\(951\) −50180.5 −1.71106
\(952\) 0 0
\(953\) 10493.3 0.356675 0.178338 0.983969i \(-0.442928\pi\)
0.178338 + 0.983969i \(0.442928\pi\)
\(954\) 1353.75 0.0459426
\(955\) − 31129.9i − 1.05481i
\(956\) 35408.7 1.19791
\(957\) − 5168.37i − 0.174576i
\(958\) − 7269.75i − 0.245172i
\(959\) 49745.4i 1.67504i
\(960\) 153.901 0.00517409
\(961\) −27217.0 −0.913597
\(962\) 10785.4i 0.361470i
\(963\) − 10122.8i − 0.338736i
\(964\) − 12854.6i − 0.429479i
\(965\) −8633.74 −0.288010
\(966\) − 23555.4i − 0.784559i
\(967\) −31468.2 −1.04648 −0.523242 0.852184i \(-0.675278\pi\)
−0.523242 + 0.852184i \(0.675278\pi\)
\(968\) −13462.6 −0.447010
\(969\) 0 0
\(970\) −6183.39 −0.204677
\(971\) 46923.8 1.55083 0.775415 0.631452i \(-0.217541\pi\)
0.775415 + 0.631452i \(0.217541\pi\)
\(972\) 17389.9i 0.573850i
\(973\) −49172.3 −1.62013
\(974\) − 6774.11i − 0.222851i
\(975\) 16299.5i 0.535387i
\(976\) 16605.9i 0.544612i
\(977\) −8666.25 −0.283785 −0.141893 0.989882i \(-0.545319\pi\)
−0.141893 + 0.989882i \(0.545319\pi\)
\(978\) 22482.4 0.735081
\(979\) − 28356.2i − 0.925709i
\(980\) − 34002.5i − 1.10834i
\(981\) − 7117.63i − 0.231650i
\(982\) 22555.3 0.732961
\(983\) − 35155.3i − 1.14067i −0.821412 0.570336i \(-0.806813\pi\)
0.821412 0.570336i \(-0.193187\pi\)
\(984\) −42527.1 −1.37776
\(985\) 35557.3 1.15020
\(986\) 0 0
\(987\) −33185.9 −1.07023
\(988\) −1038.38 −0.0334365
\(989\) − 52674.2i − 1.69357i
\(990\) 7732.75 0.248246
\(991\) 11236.2i 0.360170i 0.983651 + 0.180085i \(0.0576374\pi\)
−0.983651 + 0.180085i \(0.942363\pi\)
\(992\) 43256.3i 1.38447i
\(993\) 25314.8i 0.809004i
\(994\) −16561.3 −0.528463
\(995\) −23670.4 −0.754172
\(996\) − 7124.36i − 0.226651i
\(997\) 18825.0i 0.597987i 0.954255 + 0.298994i \(0.0966509\pi\)
−0.954255 + 0.298994i \(0.903349\pi\)
\(998\) 4424.56i 0.140338i
\(999\) −16708.6 −0.529166
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.f.288.9 24
17.4 even 4 289.4.a.h.1.8 12
17.13 even 4 289.4.a.i.1.8 yes 12
17.16 even 2 inner 289.4.b.f.288.10 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
289.4.a.h.1.8 12 17.4 even 4
289.4.a.i.1.8 yes 12 17.13 even 4
289.4.b.f.288.9 24 1.1 even 1 trivial
289.4.b.f.288.10 24 17.16 even 2 inner