Properties

Label 320.3.r.a.111.2
Level $320$
Weight $3$
Character 320.111
Analytic conductor $8.719$
Analytic rank $0$
Dimension $32$
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] = [320,3,Mod(111,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.111");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.r (of order \(4\), degree \(2\), 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: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 111.2
Character \(\chi\) \(=\) 320.111
Dual form 320.3.r.a.271.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.58499 + 3.58499i) q^{3} +(-1.58114 + 1.58114i) q^{5} -10.0090 q^{7} -16.7043i q^{9} +O(q^{10})\) \(q+(-3.58499 + 3.58499i) q^{3} +(-1.58114 + 1.58114i) q^{5} -10.0090 q^{7} -16.7043i q^{9} +(0.304644 + 0.304644i) q^{11} +(6.87350 + 6.87350i) q^{13} -11.3367i q^{15} -25.8031 q^{17} +(10.2562 - 10.2562i) q^{19} +(35.8821 - 35.8821i) q^{21} +33.7343 q^{23} -5.00000i q^{25} +(27.6197 + 27.6197i) q^{27} +(23.1650 + 23.1650i) q^{29} -7.31924i q^{31} -2.18429 q^{33} +(15.8256 - 15.8256i) q^{35} +(6.85014 - 6.85014i) q^{37} -49.2828 q^{39} -63.7855i q^{41} +(-15.0257 - 15.0257i) q^{43} +(26.4118 + 26.4118i) q^{45} -41.8744i q^{47} +51.1799 q^{49} +(92.5039 - 92.5039i) q^{51} +(-36.5375 + 36.5375i) q^{53} -0.963370 q^{55} +73.5369i q^{57} +(-58.1638 - 58.1638i) q^{59} +(-26.2928 - 26.2928i) q^{61} +167.193i q^{63} -21.7359 q^{65} +(4.48091 - 4.48091i) q^{67} +(-120.937 + 120.937i) q^{69} -11.5673 q^{71} -101.723i q^{73} +(17.9249 + 17.9249i) q^{75} +(-3.04918 - 3.04918i) q^{77} +73.1288i q^{79} -47.6943 q^{81} +(-39.9292 + 39.9292i) q^{83} +(40.7983 - 40.7983i) q^{85} -166.093 q^{87} +37.2154i q^{89} +(-68.7968 - 68.7968i) q^{91} +(26.2394 + 26.2394i) q^{93} +32.4331i q^{95} -114.749 q^{97} +(5.08886 - 5.08886i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 32 q^{11} + 32 q^{19} + 128 q^{23} + 96 q^{27} + 32 q^{29} - 96 q^{37} - 384 q^{39} - 96 q^{43} + 224 q^{49} + 256 q^{51} - 160 q^{53} + 352 q^{59} - 32 q^{61} - 160 q^{67} + 96 q^{69} - 256 q^{71} + 224 q^{77} - 288 q^{81} + 480 q^{83} + 160 q^{85} + 384 q^{91} + 96 q^{93} - 608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −3.58499 + 3.58499i −1.19500 + 1.19500i −0.219350 + 0.975646i \(0.570394\pi\)
−0.975646 + 0.219350i \(0.929606\pi\)
\(4\) 0 0
\(5\) −1.58114 + 1.58114i −0.316228 + 0.316228i
\(6\) 0 0
\(7\) −10.0090 −1.42986 −0.714928 0.699198i \(-0.753540\pi\)
−0.714928 + 0.699198i \(0.753540\pi\)
\(8\) 0 0
\(9\) 16.7043i 1.85603i
\(10\) 0 0
\(11\) 0.304644 + 0.304644i 0.0276949 + 0.0276949i 0.720819 0.693124i \(-0.243766\pi\)
−0.693124 + 0.720819i \(0.743766\pi\)
\(12\) 0 0
\(13\) 6.87350 + 6.87350i 0.528731 + 0.528731i 0.920194 0.391463i \(-0.128031\pi\)
−0.391463 + 0.920194i \(0.628031\pi\)
\(14\) 0 0
\(15\) 11.3367i 0.755782i
\(16\) 0 0
\(17\) −25.8031 −1.51783 −0.758916 0.651189i \(-0.774270\pi\)
−0.758916 + 0.651189i \(0.774270\pi\)
\(18\) 0 0
\(19\) 10.2562 10.2562i 0.539802 0.539802i −0.383669 0.923471i \(-0.625340\pi\)
0.923471 + 0.383669i \(0.125340\pi\)
\(20\) 0 0
\(21\) 35.8821 35.8821i 1.70867 1.70867i
\(22\) 0 0
\(23\) 33.7343 1.46671 0.733355 0.679845i \(-0.237953\pi\)
0.733355 + 0.679845i \(0.237953\pi\)
\(24\) 0 0
\(25\) 5.00000i 0.200000i
\(26\) 0 0
\(27\) 27.6197 + 27.6197i 1.02295 + 1.02295i
\(28\) 0 0
\(29\) 23.1650 + 23.1650i 0.798794 + 0.798794i 0.982905 0.184111i \(-0.0589407\pi\)
−0.184111 + 0.982905i \(0.558941\pi\)
\(30\) 0 0
\(31\) 7.31924i 0.236104i −0.993007 0.118052i \(-0.962335\pi\)
0.993007 0.118052i \(-0.0376650\pi\)
\(32\) 0 0
\(33\) −2.18429 −0.0661907
\(34\) 0 0
\(35\) 15.8256 15.8256i 0.452160 0.452160i
\(36\) 0 0
\(37\) 6.85014 6.85014i 0.185139 0.185139i −0.608452 0.793591i \(-0.708209\pi\)
0.793591 + 0.608452i \(0.208209\pi\)
\(38\) 0 0
\(39\) −49.2828 −1.26366
\(40\) 0 0
\(41\) 63.7855i 1.55574i −0.628423 0.777872i \(-0.716299\pi\)
0.628423 0.777872i \(-0.283701\pi\)
\(42\) 0 0
\(43\) −15.0257 15.0257i −0.349434 0.349434i 0.510465 0.859899i \(-0.329473\pi\)
−0.859899 + 0.510465i \(0.829473\pi\)
\(44\) 0 0
\(45\) 26.4118 + 26.4118i 0.586928 + 0.586928i
\(46\) 0 0
\(47\) 41.8744i 0.890944i −0.895296 0.445472i \(-0.853036\pi\)
0.895296 0.445472i \(-0.146964\pi\)
\(48\) 0 0
\(49\) 51.1799 1.04449
\(50\) 0 0
\(51\) 92.5039 92.5039i 1.81380 1.81380i
\(52\) 0 0
\(53\) −36.5375 + 36.5375i −0.689388 + 0.689388i −0.962097 0.272709i \(-0.912080\pi\)
0.272709 + 0.962097i \(0.412080\pi\)
\(54\) 0 0
\(55\) −0.963370 −0.0175158
\(56\) 0 0
\(57\) 73.5369i 1.29012i
\(58\) 0 0
\(59\) −58.1638 58.1638i −0.985828 0.985828i 0.0140734 0.999901i \(-0.495520\pi\)
−0.999901 + 0.0140734i \(0.995520\pi\)
\(60\) 0 0
\(61\) −26.2928 26.2928i −0.431030 0.431030i 0.457949 0.888979i \(-0.348584\pi\)
−0.888979 + 0.457949i \(0.848584\pi\)
\(62\) 0 0
\(63\) 167.193i 2.65386i
\(64\) 0 0
\(65\) −21.7359 −0.334399
\(66\) 0 0
\(67\) 4.48091 4.48091i 0.0668793 0.0668793i −0.672876 0.739755i \(-0.734941\pi\)
0.739755 + 0.672876i \(0.234941\pi\)
\(68\) 0 0
\(69\) −120.937 + 120.937i −1.75271 + 1.75271i
\(70\) 0 0
\(71\) −11.5673 −0.162919 −0.0814595 0.996677i \(-0.525958\pi\)
−0.0814595 + 0.996677i \(0.525958\pi\)
\(72\) 0 0
\(73\) 101.723i 1.39346i −0.717332 0.696732i \(-0.754637\pi\)
0.717332 0.696732i \(-0.245363\pi\)
\(74\) 0 0
\(75\) 17.9249 + 17.9249i 0.238999 + 0.238999i
\(76\) 0 0
\(77\) −3.04918 3.04918i −0.0395998 0.0395998i
\(78\) 0 0
\(79\) 73.1288i 0.925681i 0.886442 + 0.462840i \(0.153170\pi\)
−0.886442 + 0.462840i \(0.846830\pi\)
\(80\) 0 0
\(81\) −47.6943 −0.588819
\(82\) 0 0
\(83\) −39.9292 + 39.9292i −0.481075 + 0.481075i −0.905475 0.424400i \(-0.860485\pi\)
0.424400 + 0.905475i \(0.360485\pi\)
\(84\) 0 0
\(85\) 40.7983 40.7983i 0.479980 0.479980i
\(86\) 0 0
\(87\) −166.093 −1.90911
\(88\) 0 0
\(89\) 37.2154i 0.418151i 0.977899 + 0.209076i \(0.0670455\pi\)
−0.977899 + 0.209076i \(0.932955\pi\)
\(90\) 0 0
\(91\) −68.7968 68.7968i −0.756008 0.756008i
\(92\) 0 0
\(93\) 26.2394 + 26.2394i 0.282144 + 0.282144i
\(94\) 0 0
\(95\) 32.4331i 0.341401i
\(96\) 0 0
\(97\) −114.749 −1.18298 −0.591490 0.806312i \(-0.701460\pi\)
−0.591490 + 0.806312i \(0.701460\pi\)
\(98\) 0 0
\(99\) 5.08886 5.08886i 0.0514026 0.0514026i
\(100\) 0 0
\(101\) 3.86189 3.86189i 0.0382366 0.0382366i −0.687730 0.725967i \(-0.741393\pi\)
0.725967 + 0.687730i \(0.241393\pi\)
\(102\) 0 0
\(103\) −82.3074 −0.799101 −0.399551 0.916711i \(-0.630834\pi\)
−0.399551 + 0.916711i \(0.630834\pi\)
\(104\) 0 0
\(105\) 113.469i 1.08066i
\(106\) 0 0
\(107\) 109.631 + 109.631i 1.02459 + 1.02459i 0.999690 + 0.0248967i \(0.00792568\pi\)
0.0248967 + 0.999690i \(0.492074\pi\)
\(108\) 0 0
\(109\) −2.99231 2.99231i −0.0274524 0.0274524i 0.693247 0.720700i \(-0.256179\pi\)
−0.720700 + 0.693247i \(0.756179\pi\)
\(110\) 0 0
\(111\) 49.1153i 0.442480i
\(112\) 0 0
\(113\) 73.3488 0.649105 0.324552 0.945868i \(-0.394786\pi\)
0.324552 + 0.945868i \(0.394786\pi\)
\(114\) 0 0
\(115\) −53.3387 + 53.3387i −0.463815 + 0.463815i
\(116\) 0 0
\(117\) 114.817 114.817i 0.981340 0.981340i
\(118\) 0 0
\(119\) 258.263 2.17028
\(120\) 0 0
\(121\) 120.814i 0.998466i
\(122\) 0 0
\(123\) 228.670 + 228.670i 1.85911 + 1.85911i
\(124\) 0 0
\(125\) 7.90569 + 7.90569i 0.0632456 + 0.0632456i
\(126\) 0 0
\(127\) 212.909i 1.67645i −0.545323 0.838226i \(-0.683593\pi\)
0.545323 0.838226i \(-0.316407\pi\)
\(128\) 0 0
\(129\) 107.734 0.835145
\(130\) 0 0
\(131\) −34.5741 + 34.5741i −0.263924 + 0.263924i −0.826646 0.562722i \(-0.809754\pi\)
0.562722 + 0.826646i \(0.309754\pi\)
\(132\) 0 0
\(133\) −102.655 + 102.655i −0.771838 + 0.771838i
\(134\) 0 0
\(135\) −87.3413 −0.646972
\(136\) 0 0
\(137\) 156.675i 1.14362i −0.820388 0.571808i \(-0.806242\pi\)
0.820388 0.571808i \(-0.193758\pi\)
\(138\) 0 0
\(139\) −150.961 150.961i −1.08605 1.08605i −0.995931 0.0901190i \(-0.971275\pi\)
−0.0901190 0.995931i \(-0.528725\pi\)
\(140\) 0 0
\(141\) 150.119 + 150.119i 1.06467 + 1.06467i
\(142\) 0 0
\(143\) 4.18794i 0.0292863i
\(144\) 0 0
\(145\) −73.2543 −0.505202
\(146\) 0 0
\(147\) −183.479 + 183.479i −1.24816 + 1.24816i
\(148\) 0 0
\(149\) 96.4520 96.4520i 0.647329 0.647329i −0.305018 0.952347i \(-0.598662\pi\)
0.952347 + 0.305018i \(0.0986624\pi\)
\(150\) 0 0
\(151\) −29.4114 −0.194777 −0.0973886 0.995246i \(-0.531049\pi\)
−0.0973886 + 0.995246i \(0.531049\pi\)
\(152\) 0 0
\(153\) 431.023i 2.81714i
\(154\) 0 0
\(155\) 11.5727 + 11.5727i 0.0746628 + 0.0746628i
\(156\) 0 0
\(157\) −117.658 117.658i −0.749415 0.749415i 0.224954 0.974369i \(-0.427777\pi\)
−0.974369 + 0.224954i \(0.927777\pi\)
\(158\) 0 0
\(159\) 261.973i 1.64763i
\(160\) 0 0
\(161\) −337.647 −2.09718
\(162\) 0 0
\(163\) 7.81267 7.81267i 0.0479305 0.0479305i −0.682735 0.730666i \(-0.739210\pi\)
0.730666 + 0.682735i \(0.239210\pi\)
\(164\) 0 0
\(165\) 3.45367 3.45367i 0.0209313 0.0209313i
\(166\) 0 0
\(167\) 255.372 1.52918 0.764588 0.644520i \(-0.222943\pi\)
0.764588 + 0.644520i \(0.222943\pi\)
\(168\) 0 0
\(169\) 74.5100i 0.440888i
\(170\) 0 0
\(171\) −171.323 171.323i −1.00189 1.00189i
\(172\) 0 0
\(173\) 27.0161 + 27.0161i 0.156162 + 0.156162i 0.780864 0.624701i \(-0.214779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(174\) 0 0
\(175\) 50.0449i 0.285971i
\(176\) 0 0
\(177\) 417.033 2.35612
\(178\) 0 0
\(179\) −50.8893 + 50.8893i −0.284298 + 0.284298i −0.834820 0.550523i \(-0.814428\pi\)
0.550523 + 0.834820i \(0.314428\pi\)
\(180\) 0 0
\(181\) −13.6488 + 13.6488i −0.0754079 + 0.0754079i −0.743805 0.668397i \(-0.766981\pi\)
0.668397 + 0.743805i \(0.266981\pi\)
\(182\) 0 0
\(183\) 188.519 1.03016
\(184\) 0 0
\(185\) 21.6620i 0.117092i
\(186\) 0 0
\(187\) −7.86078 7.86078i −0.0420362 0.0420362i
\(188\) 0 0
\(189\) −276.446 276.446i −1.46268 1.46268i
\(190\) 0 0
\(191\) 263.019i 1.37706i 0.725206 + 0.688532i \(0.241745\pi\)
−0.725206 + 0.688532i \(0.758255\pi\)
\(192\) 0 0
\(193\) −58.9456 −0.305418 −0.152709 0.988271i \(-0.548800\pi\)
−0.152709 + 0.988271i \(0.548800\pi\)
\(194\) 0 0
\(195\) 77.9230 77.9230i 0.399605 0.399605i
\(196\) 0 0
\(197\) 9.54461 9.54461i 0.0484498 0.0484498i −0.682467 0.730917i \(-0.739093\pi\)
0.730917 + 0.682467i \(0.239093\pi\)
\(198\) 0 0
\(199\) −103.151 −0.518348 −0.259174 0.965831i \(-0.583450\pi\)
−0.259174 + 0.965831i \(0.583450\pi\)
\(200\) 0 0
\(201\) 32.1280i 0.159841i
\(202\) 0 0
\(203\) −231.859 231.859i −1.14216 1.14216i
\(204\) 0 0
\(205\) 100.854 + 100.854i 0.491969 + 0.491969i
\(206\) 0 0
\(207\) 563.508i 2.72226i
\(208\) 0 0
\(209\) 6.24900 0.0298995
\(210\) 0 0
\(211\) −4.93870 + 4.93870i −0.0234062 + 0.0234062i −0.718713 0.695307i \(-0.755268\pi\)
0.695307 + 0.718713i \(0.255268\pi\)
\(212\) 0 0
\(213\) 41.4685 41.4685i 0.194688 0.194688i
\(214\) 0 0
\(215\) 47.5153 0.221002
\(216\) 0 0
\(217\) 73.2582i 0.337595i
\(218\) 0 0
\(219\) 364.675 + 364.675i 1.66518 + 1.66518i
\(220\) 0 0
\(221\) −177.358 177.358i −0.802524 0.802524i
\(222\) 0 0
\(223\) 177.151i 0.794401i −0.917732 0.397200i \(-0.869982\pi\)
0.917732 0.397200i \(-0.130018\pi\)
\(224\) 0 0
\(225\) −83.5214 −0.371206
\(226\) 0 0
\(227\) 239.268 239.268i 1.05404 1.05404i 0.0555894 0.998454i \(-0.482296\pi\)
0.998454 0.0555894i \(-0.0177038\pi\)
\(228\) 0 0
\(229\) −106.699 + 106.699i −0.465933 + 0.465933i −0.900594 0.434661i \(-0.856868\pi\)
0.434661 + 0.900594i \(0.356868\pi\)
\(230\) 0 0
\(231\) 21.8626 0.0946431
\(232\) 0 0
\(233\) 132.961i 0.570649i 0.958431 + 0.285324i \(0.0921013\pi\)
−0.958431 + 0.285324i \(0.907899\pi\)
\(234\) 0 0
\(235\) 66.2092 + 66.2092i 0.281741 + 0.281741i
\(236\) 0 0
\(237\) −262.166 262.166i −1.10618 1.10618i
\(238\) 0 0
\(239\) 89.6649i 0.375167i 0.982249 + 0.187584i \(0.0600655\pi\)
−0.982249 + 0.187584i \(0.939934\pi\)
\(240\) 0 0
\(241\) 259.019 1.07477 0.537383 0.843338i \(-0.319413\pi\)
0.537383 + 0.843338i \(0.319413\pi\)
\(242\) 0 0
\(243\) −77.5940 + 77.5940i −0.319317 + 0.319317i
\(244\) 0 0
\(245\) −80.9225 + 80.9225i −0.330296 + 0.330296i
\(246\) 0 0
\(247\) 140.992 0.570819
\(248\) 0 0
\(249\) 286.292i 1.14977i
\(250\) 0 0
\(251\) −85.2338 85.2338i −0.339577 0.339577i 0.516631 0.856208i \(-0.327186\pi\)
−0.856208 + 0.516631i \(0.827186\pi\)
\(252\) 0 0
\(253\) 10.2770 + 10.2770i 0.0406205 + 0.0406205i
\(254\) 0 0
\(255\) 292.523i 1.14715i
\(256\) 0 0
\(257\) 451.683 1.75752 0.878760 0.477264i \(-0.158372\pi\)
0.878760 + 0.477264i \(0.158372\pi\)
\(258\) 0 0
\(259\) −68.5629 + 68.5629i −0.264722 + 0.264722i
\(260\) 0 0
\(261\) 386.955 386.955i 1.48259 1.48259i
\(262\) 0 0
\(263\) 0.939393 0.00357184 0.00178592 0.999998i \(-0.499432\pi\)
0.00178592 + 0.999998i \(0.499432\pi\)
\(264\) 0 0
\(265\) 115.542i 0.436007i
\(266\) 0 0
\(267\) −133.417 133.417i −0.499689 0.499689i
\(268\) 0 0
\(269\) 141.098 + 141.098i 0.524527 + 0.524527i 0.918935 0.394408i \(-0.129050\pi\)
−0.394408 + 0.918935i \(0.629050\pi\)
\(270\) 0 0
\(271\) 237.677i 0.877037i −0.898722 0.438518i \(-0.855503\pi\)
0.898722 0.438518i \(-0.144497\pi\)
\(272\) 0 0
\(273\) 493.271 1.80685
\(274\) 0 0
\(275\) 1.52322 1.52322i 0.00553899 0.00553899i
\(276\) 0 0
\(277\) −301.681 + 301.681i −1.08910 + 1.08910i −0.0934800 + 0.995621i \(0.529799\pi\)
−0.995621 + 0.0934800i \(0.970201\pi\)
\(278\) 0 0
\(279\) −122.263 −0.438217
\(280\) 0 0
\(281\) 209.865i 0.746850i 0.927660 + 0.373425i \(0.121817\pi\)
−0.927660 + 0.373425i \(0.878183\pi\)
\(282\) 0 0
\(283\) −87.1403 87.1403i −0.307916 0.307916i 0.536184 0.844101i \(-0.319865\pi\)
−0.844101 + 0.536184i \(0.819865\pi\)
\(284\) 0 0
\(285\) −116.272 116.272i −0.407972 0.407972i
\(286\) 0 0
\(287\) 638.428i 2.22449i
\(288\) 0 0
\(289\) 376.801 1.30381
\(290\) 0 0
\(291\) 411.374 411.374i 1.41366 1.41366i
\(292\) 0 0
\(293\) 210.545 210.545i 0.718583 0.718583i −0.249732 0.968315i \(-0.580343\pi\)
0.968315 + 0.249732i \(0.0803426\pi\)
\(294\) 0 0
\(295\) 183.930 0.623492
\(296\) 0 0
\(297\) 16.8284i 0.0566612i
\(298\) 0 0
\(299\) 231.873 + 231.873i 0.775495 + 0.775495i
\(300\) 0 0
\(301\) 150.392 + 150.392i 0.499640 + 0.499640i
\(302\) 0 0
\(303\) 27.6897i 0.0913851i
\(304\) 0 0
\(305\) 83.1452 0.272607
\(306\) 0 0
\(307\) −345.339 + 345.339i −1.12488 + 1.12488i −0.133888 + 0.990996i \(0.542746\pi\)
−0.990996 + 0.133888i \(0.957254\pi\)
\(308\) 0 0
\(309\) 295.071 295.071i 0.954923 0.954923i
\(310\) 0 0
\(311\) 99.7096 0.320610 0.160305 0.987068i \(-0.448752\pi\)
0.160305 + 0.987068i \(0.448752\pi\)
\(312\) 0 0
\(313\) 311.334i 0.994678i −0.867556 0.497339i \(-0.834311\pi\)
0.867556 0.497339i \(-0.165689\pi\)
\(314\) 0 0
\(315\) −264.355 264.355i −0.839223 0.839223i
\(316\) 0 0
\(317\) 19.2419 + 19.2419i 0.0607000 + 0.0607000i 0.736805 0.676105i \(-0.236334\pi\)
−0.676105 + 0.736805i \(0.736334\pi\)
\(318\) 0 0
\(319\) 14.1142i 0.0442451i
\(320\) 0 0
\(321\) −786.050 −2.44875
\(322\) 0 0
\(323\) −264.643 + 264.643i −0.819328 + 0.819328i
\(324\) 0 0
\(325\) 34.3675 34.3675i 0.105746 0.105746i
\(326\) 0 0
\(327\) 21.4548 0.0656110
\(328\) 0 0
\(329\) 419.120i 1.27392i
\(330\) 0 0
\(331\) −194.214 194.214i −0.586750 0.586750i 0.350000 0.936750i \(-0.386182\pi\)
−0.936750 + 0.350000i \(0.886182\pi\)
\(332\) 0 0
\(333\) −114.427 114.427i −0.343623 0.343623i
\(334\) 0 0
\(335\) 14.1699i 0.0422982i
\(336\) 0 0
\(337\) −552.303 −1.63888 −0.819441 0.573164i \(-0.805716\pi\)
−0.819441 + 0.573164i \(0.805716\pi\)
\(338\) 0 0
\(339\) −262.955 + 262.955i −0.775677 + 0.775677i
\(340\) 0 0
\(341\) 2.22976 2.22976i 0.00653890 0.00653890i
\(342\) 0 0
\(343\) −21.8182 −0.0636099
\(344\) 0 0
\(345\) 382.437i 1.10851i
\(346\) 0 0
\(347\) −119.162 119.162i −0.343407 0.343407i 0.514239 0.857647i \(-0.328074\pi\)
−0.857647 + 0.514239i \(0.828074\pi\)
\(348\) 0 0
\(349\) −170.249 170.249i −0.487821 0.487821i 0.419797 0.907618i \(-0.362101\pi\)
−0.907618 + 0.419797i \(0.862101\pi\)
\(350\) 0 0
\(351\) 379.688i 1.08173i
\(352\) 0 0
\(353\) −551.816 −1.56322 −0.781609 0.623769i \(-0.785601\pi\)
−0.781609 + 0.623769i \(0.785601\pi\)
\(354\) 0 0
\(355\) 18.2894 18.2894i 0.0515195 0.0515195i
\(356\) 0 0
\(357\) −925.871 + 925.871i −2.59347 + 2.59347i
\(358\) 0 0
\(359\) −432.275 −1.20411 −0.602054 0.798456i \(-0.705651\pi\)
−0.602054 + 0.798456i \(0.705651\pi\)
\(360\) 0 0
\(361\) 150.619i 0.417228i
\(362\) 0 0
\(363\) 433.118 + 433.118i 1.19316 + 1.19316i
\(364\) 0 0
\(365\) 160.838 + 160.838i 0.440652 + 0.440652i
\(366\) 0 0
\(367\) 223.982i 0.610306i −0.952303 0.305153i \(-0.901292\pi\)
0.952303 0.305153i \(-0.0987076\pi\)
\(368\) 0 0
\(369\) −1065.49 −2.88751
\(370\) 0 0
\(371\) 365.704 365.704i 0.985725 0.985725i
\(372\) 0 0
\(373\) −348.716 + 348.716i −0.934897 + 0.934897i −0.998007 0.0631100i \(-0.979898\pi\)
0.0631100 + 0.998007i \(0.479898\pi\)
\(374\) 0 0
\(375\) −56.6836 −0.151156
\(376\) 0 0
\(377\) 318.450i 0.844694i
\(378\) 0 0
\(379\) 374.578 + 374.578i 0.988332 + 0.988332i 0.999933 0.0116005i \(-0.00369264\pi\)
−0.0116005 + 0.999933i \(0.503693\pi\)
\(380\) 0 0
\(381\) 763.278 + 763.278i 2.00335 + 2.00335i
\(382\) 0 0
\(383\) 431.347i 1.12623i −0.826378 0.563116i \(-0.809602\pi\)
0.826378 0.563116i \(-0.190398\pi\)
\(384\) 0 0
\(385\) 9.64236 0.0250451
\(386\) 0 0
\(387\) −250.993 + 250.993i −0.648560 + 0.648560i
\(388\) 0 0
\(389\) 55.8528 55.8528i 0.143580 0.143580i −0.631663 0.775243i \(-0.717627\pi\)
0.775243 + 0.631663i \(0.217627\pi\)
\(390\) 0 0
\(391\) −870.452 −2.22622
\(392\) 0 0
\(393\) 247.895i 0.630777i
\(394\) 0 0
\(395\) −115.627 115.627i −0.292726 0.292726i
\(396\) 0 0
\(397\) −104.083 104.083i −0.262173 0.262173i 0.563763 0.825936i \(-0.309353\pi\)
−0.825936 + 0.563763i \(0.809353\pi\)
\(398\) 0 0
\(399\) 736.030i 1.84469i
\(400\) 0 0
\(401\) 32.9517 0.0821738 0.0410869 0.999156i \(-0.486918\pi\)
0.0410869 + 0.999156i \(0.486918\pi\)
\(402\) 0 0
\(403\) 50.3088 50.3088i 0.124836 0.124836i
\(404\) 0 0
\(405\) 75.4114 75.4114i 0.186201 0.186201i
\(406\) 0 0
\(407\) 4.17371 0.0102548
\(408\) 0 0
\(409\) 559.822i 1.36876i 0.729126 + 0.684379i \(0.239927\pi\)
−0.729126 + 0.684379i \(0.760073\pi\)
\(410\) 0 0
\(411\) 561.679 + 561.679i 1.36662 + 1.36662i
\(412\) 0 0
\(413\) 582.161 + 582.161i 1.40959 + 1.40959i
\(414\) 0 0
\(415\) 126.267i 0.304259i
\(416\) 0 0
\(417\) 1082.39 2.59565
\(418\) 0 0
\(419\) −168.614 + 168.614i −0.402420 + 0.402420i −0.879085 0.476665i \(-0.841845\pi\)
0.476665 + 0.879085i \(0.341845\pi\)
\(420\) 0 0
\(421\) 409.607 409.607i 0.972937 0.972937i −0.0267062 0.999643i \(-0.508502\pi\)
0.999643 + 0.0267062i \(0.00850184\pi\)
\(422\) 0 0
\(423\) −699.481 −1.65362
\(424\) 0 0
\(425\) 129.016i 0.303566i
\(426\) 0 0
\(427\) 263.164 + 263.164i 0.616310 + 0.616310i
\(428\) 0 0
\(429\) −15.0137 15.0137i −0.0349970 0.0349970i
\(430\) 0 0
\(431\) 307.273i 0.712930i 0.934309 + 0.356465i \(0.116018\pi\)
−0.934309 + 0.356465i \(0.883982\pi\)
\(432\) 0 0
\(433\) −573.906 −1.32542 −0.662709 0.748877i \(-0.730593\pi\)
−0.662709 + 0.748877i \(0.730593\pi\)
\(434\) 0 0
\(435\) 262.616 262.616i 0.603714 0.603714i
\(436\) 0 0
\(437\) 345.987 345.987i 0.791733 0.791733i
\(438\) 0 0
\(439\) 491.449 1.11947 0.559737 0.828670i \(-0.310902\pi\)
0.559737 + 0.828670i \(0.310902\pi\)
\(440\) 0 0
\(441\) 854.922i 1.93860i
\(442\) 0 0
\(443\) −343.974 343.974i −0.776466 0.776466i 0.202762 0.979228i \(-0.435008\pi\)
−0.979228 + 0.202762i \(0.935008\pi\)
\(444\) 0 0
\(445\) −58.8428 58.8428i −0.132231 0.132231i
\(446\) 0 0
\(447\) 691.558i 1.54711i
\(448\) 0 0
\(449\) −153.260 −0.341337 −0.170668 0.985329i \(-0.554593\pi\)
−0.170668 + 0.985329i \(0.554593\pi\)
\(450\) 0 0
\(451\) 19.4319 19.4319i 0.0430862 0.0430862i
\(452\) 0 0
\(453\) 105.439 105.439i 0.232758 0.232758i
\(454\) 0 0
\(455\) 217.554 0.478142
\(456\) 0 0
\(457\) 708.281i 1.54985i −0.632054 0.774924i \(-0.717788\pi\)
0.632054 0.774924i \(-0.282212\pi\)
\(458\) 0 0
\(459\) −712.676 712.676i −1.55267 1.55267i
\(460\) 0 0
\(461\) 20.0249 + 20.0249i 0.0434379 + 0.0434379i 0.728492 0.685054i \(-0.240221\pi\)
−0.685054 + 0.728492i \(0.740221\pi\)
\(462\) 0 0
\(463\) 30.9338i 0.0668117i 0.999442 + 0.0334059i \(0.0106354\pi\)
−0.999442 + 0.0334059i \(0.989365\pi\)
\(464\) 0 0
\(465\) −82.9762 −0.178443
\(466\) 0 0
\(467\) −89.9400 + 89.9400i −0.192591 + 0.192591i −0.796815 0.604224i \(-0.793483\pi\)
0.604224 + 0.796815i \(0.293483\pi\)
\(468\) 0 0
\(469\) −44.8494 + 44.8494i −0.0956277 + 0.0956277i
\(470\) 0 0
\(471\) 843.606 1.79110
\(472\) 0 0
\(473\) 9.15497i 0.0193551i
\(474\) 0 0
\(475\) −51.2812 51.2812i −0.107960 0.107960i
\(476\) 0 0
\(477\) 610.333 + 610.333i 1.27952 + 1.27952i
\(478\) 0 0
\(479\) 424.069i 0.885322i 0.896689 + 0.442661i \(0.145965\pi\)
−0.896689 + 0.442661i \(0.854035\pi\)
\(480\) 0 0
\(481\) 94.1688 0.195777
\(482\) 0 0
\(483\) 1210.46 1210.46i 2.50613 2.50613i
\(484\) 0 0
\(485\) 181.434 181.434i 0.374091 0.374091i
\(486\) 0 0
\(487\) −509.544 −1.04629 −0.523146 0.852243i \(-0.675242\pi\)
−0.523146 + 0.852243i \(0.675242\pi\)
\(488\) 0 0
\(489\) 56.0166i 0.114553i
\(490\) 0 0
\(491\) −232.469 232.469i −0.473461 0.473461i 0.429572 0.903033i \(-0.358665\pi\)
−0.903033 + 0.429572i \(0.858665\pi\)
\(492\) 0 0
\(493\) −597.730 597.730i −1.21243 1.21243i
\(494\) 0 0
\(495\) 16.0924i 0.0325099i
\(496\) 0 0
\(497\) 115.777 0.232951
\(498\) 0 0
\(499\) 609.443 609.443i 1.22133 1.22133i 0.254170 0.967160i \(-0.418198\pi\)
0.967160 0.254170i \(-0.0818022\pi\)
\(500\) 0 0
\(501\) −915.506 + 915.506i −1.82736 + 1.82736i
\(502\) 0 0
\(503\) 83.3154 0.165637 0.0828185 0.996565i \(-0.473608\pi\)
0.0828185 + 0.996565i \(0.473608\pi\)
\(504\) 0 0
\(505\) 12.2124i 0.0241829i
\(506\) 0 0
\(507\) 267.118 + 267.118i 0.526859 + 0.526859i
\(508\) 0 0
\(509\) 534.566 + 534.566i 1.05023 + 1.05023i 0.998670 + 0.0515585i \(0.0164189\pi\)
0.0515585 + 0.998670i \(0.483581\pi\)
\(510\) 0 0
\(511\) 1018.14i 1.99245i
\(512\) 0 0
\(513\) 566.549 1.10438
\(514\) 0 0
\(515\) 130.139 130.139i 0.252698 0.252698i
\(516\) 0 0
\(517\) 12.7568 12.7568i 0.0246746 0.0246746i
\(518\) 0 0
\(519\) −193.705 −0.373227
\(520\) 0 0
\(521\) 513.800i 0.986181i 0.869978 + 0.493091i \(0.164133\pi\)
−0.869978 + 0.493091i \(0.835867\pi\)
\(522\) 0 0
\(523\) 146.299 + 146.299i 0.279731 + 0.279731i 0.833001 0.553271i \(-0.186621\pi\)
−0.553271 + 0.833001i \(0.686621\pi\)
\(524\) 0 0
\(525\) −179.411 179.411i −0.341734 0.341734i
\(526\) 0 0
\(527\) 188.859i 0.358367i
\(528\) 0 0
\(529\) 609.006 1.15124
\(530\) 0 0
\(531\) −971.585 + 971.585i −1.82973 + 1.82973i
\(532\) 0 0
\(533\) 438.429 438.429i 0.822569 0.822569i
\(534\) 0 0
\(535\) −346.683 −0.648006
\(536\) 0 0
\(537\) 364.875i 0.679469i
\(538\) 0 0
\(539\) 15.5917 + 15.5917i 0.0289270 + 0.0289270i
\(540\) 0 0
\(541\) −610.827 610.827i −1.12907 1.12907i −0.990328 0.138743i \(-0.955694\pi\)
−0.138743 0.990328i \(-0.544306\pi\)
\(542\) 0 0
\(543\) 97.8618i 0.180224i
\(544\) 0 0
\(545\) 9.46252 0.0173624
\(546\) 0 0
\(547\) 392.410 392.410i 0.717386 0.717386i −0.250683 0.968069i \(-0.580655\pi\)
0.968069 + 0.250683i \(0.0806552\pi\)
\(548\) 0 0
\(549\) −439.202 + 439.202i −0.800004 + 0.800004i
\(550\) 0 0
\(551\) 475.172 0.862381
\(552\) 0 0
\(553\) 731.945i 1.32359i
\(554\) 0 0
\(555\) −77.6581 77.6581i −0.139925 0.139925i
\(556\) 0 0
\(557\) −587.051 587.051i −1.05395 1.05395i −0.998459 0.0554921i \(-0.982327\pi\)
−0.0554921 0.998459i \(-0.517673\pi\)
\(558\) 0 0
\(559\) 206.558i 0.369513i
\(560\) 0 0
\(561\) 56.3616 0.100466
\(562\) 0 0
\(563\) −168.514 + 168.514i −0.299314 + 0.299314i −0.840745 0.541431i \(-0.817883\pi\)
0.541431 + 0.840745i \(0.317883\pi\)
\(564\) 0 0
\(565\) −115.975 + 115.975i −0.205265 + 0.205265i
\(566\) 0 0
\(567\) 477.372 0.841926
\(568\) 0 0
\(569\) 550.627i 0.967711i −0.875148 0.483855i \(-0.839236\pi\)
0.875148 0.483855i \(-0.160764\pi\)
\(570\) 0 0
\(571\) 163.518 + 163.518i 0.286371 + 0.286371i 0.835644 0.549272i \(-0.185095\pi\)
−0.549272 + 0.835644i \(0.685095\pi\)
\(572\) 0 0
\(573\) −942.921 942.921i −1.64559 1.64559i
\(574\) 0 0
\(575\) 168.672i 0.293342i
\(576\) 0 0
\(577\) 15.6965 0.0272037 0.0136018 0.999907i \(-0.495670\pi\)
0.0136018 + 0.999907i \(0.495670\pi\)
\(578\) 0 0
\(579\) 211.319 211.319i 0.364973 0.364973i
\(580\) 0 0
\(581\) 399.651 399.651i 0.687868 0.687868i
\(582\) 0 0
\(583\) −22.2619 −0.0381851
\(584\) 0 0
\(585\) 363.083i 0.620654i
\(586\) 0 0
\(587\) −715.538 715.538i −1.21897 1.21897i −0.967992 0.250983i \(-0.919246\pi\)
−0.250983 0.967992i \(-0.580754\pi\)
\(588\) 0 0
\(589\) −75.0678 75.0678i −0.127450 0.127450i
\(590\) 0 0
\(591\) 68.4346i 0.115795i
\(592\) 0 0
\(593\) −557.100 −0.939460 −0.469730 0.882810i \(-0.655649\pi\)
−0.469730 + 0.882810i \(0.655649\pi\)
\(594\) 0 0
\(595\) −408.350 + 408.350i −0.686303 + 0.686303i
\(596\) 0 0
\(597\) 369.796 369.796i 0.619424 0.619424i
\(598\) 0 0
\(599\) −524.845 −0.876202 −0.438101 0.898926i \(-0.644349\pi\)
−0.438101 + 0.898926i \(0.644349\pi\)
\(600\) 0 0
\(601\) 182.301i 0.303330i −0.988432 0.151665i \(-0.951537\pi\)
0.988432 0.151665i \(-0.0484635\pi\)
\(602\) 0 0
\(603\) −74.8504 74.8504i −0.124130 0.124130i
\(604\) 0 0
\(605\) 191.024 + 191.024i 0.315743 + 0.315743i
\(606\) 0 0
\(607\) 987.505i 1.62686i −0.581662 0.813431i \(-0.697597\pi\)
0.581662 0.813431i \(-0.302403\pi\)
\(608\) 0 0
\(609\) 1662.42 2.72975
\(610\) 0 0
\(611\) 287.823 287.823i 0.471069 0.471069i
\(612\) 0 0
\(613\) 768.517 768.517i 1.25370 1.25370i 0.299648 0.954050i \(-0.403131\pi\)
0.954050 0.299648i \(-0.0968692\pi\)
\(614\) 0 0
\(615\) −723.119 −1.17580
\(616\) 0 0
\(617\) 467.462i 0.757636i 0.925471 + 0.378818i \(0.123669\pi\)
−0.925471 + 0.378818i \(0.876331\pi\)
\(618\) 0 0
\(619\) 272.667 + 272.667i 0.440496 + 0.440496i 0.892179 0.451682i \(-0.149176\pi\)
−0.451682 + 0.892179i \(0.649176\pi\)
\(620\) 0 0
\(621\) 931.734 + 931.734i 1.50038 + 1.50038i
\(622\) 0 0
\(623\) 372.489i 0.597896i
\(624\) 0 0
\(625\) −25.0000 −0.0400000
\(626\) 0 0
\(627\) −22.4026 + 22.4026i −0.0357298 + 0.0357298i
\(628\) 0 0
\(629\) −176.755 + 176.755i −0.281009 + 0.281009i
\(630\) 0 0
\(631\) −1107.18 −1.75465 −0.877324 0.479898i \(-0.840674\pi\)
−0.877324 + 0.479898i \(0.840674\pi\)
\(632\) 0 0
\(633\) 35.4104i 0.0559405i
\(634\) 0 0
\(635\) 336.639 + 336.639i 0.530141 + 0.530141i
\(636\) 0 0
\(637\) 351.785 + 351.785i 0.552252 + 0.552252i
\(638\) 0 0
\(639\) 193.223i 0.302383i
\(640\) 0 0
\(641\) −645.436 −1.00692 −0.503460 0.864018i \(-0.667940\pi\)
−0.503460 + 0.864018i \(0.667940\pi\)
\(642\) 0 0
\(643\) −397.860 + 397.860i −0.618756 + 0.618756i −0.945212 0.326456i \(-0.894145\pi\)
0.326456 + 0.945212i \(0.394145\pi\)
\(644\) 0 0
\(645\) −170.342 + 170.342i −0.264096 + 0.264096i
\(646\) 0 0
\(647\) 386.850 0.597913 0.298956 0.954267i \(-0.403361\pi\)
0.298956 + 0.954267i \(0.403361\pi\)
\(648\) 0 0
\(649\) 35.4386i 0.0546049i
\(650\) 0 0
\(651\) −262.630 262.630i −0.403425 0.403425i
\(652\) 0 0
\(653\) −557.600 557.600i −0.853905 0.853905i 0.136707 0.990612i \(-0.456348\pi\)
−0.990612 + 0.136707i \(0.956348\pi\)
\(654\) 0 0
\(655\) 109.333i 0.166920i
\(656\) 0 0
\(657\) −1699.21 −2.58631
\(658\) 0 0
\(659\) −85.5304 + 85.5304i −0.129788 + 0.129788i −0.769017 0.639229i \(-0.779254\pi\)
0.639229 + 0.769017i \(0.279254\pi\)
\(660\) 0 0
\(661\) −677.148 + 677.148i −1.02443 + 1.02443i −0.0247351 + 0.999694i \(0.507874\pi\)
−0.999694 + 0.0247351i \(0.992126\pi\)
\(662\) 0 0
\(663\) 1271.65 1.91803
\(664\) 0 0
\(665\) 324.622i 0.488153i
\(666\) 0 0
\(667\) 781.457 + 781.457i 1.17160 + 1.17160i
\(668\) 0 0
\(669\) 635.086 + 635.086i 0.949306 + 0.949306i
\(670\) 0 0
\(671\) 16.0199i 0.0238747i
\(672\) 0 0
\(673\) 1077.92 1.60167 0.800836 0.598884i \(-0.204389\pi\)
0.800836 + 0.598884i \(0.204389\pi\)
\(674\) 0 0
\(675\) 138.099 138.099i 0.204591 0.204591i
\(676\) 0 0
\(677\) 926.183 926.183i 1.36807 1.36807i 0.504881 0.863189i \(-0.331536\pi\)
0.863189 0.504881i \(-0.168464\pi\)
\(678\) 0 0
\(679\) 1148.52 1.69149
\(680\) 0 0
\(681\) 1715.54i 2.51915i
\(682\) 0 0
\(683\) −548.522 548.522i −0.803107 0.803107i 0.180473 0.983580i \(-0.442237\pi\)
−0.983580 + 0.180473i \(0.942237\pi\)
\(684\) 0 0
\(685\) 247.725 + 247.725i 0.361643 + 0.361643i
\(686\) 0 0
\(687\) 765.027i 1.11358i
\(688\) 0 0
\(689\) −502.281 −0.729001
\(690\) 0 0
\(691\) 21.6062 21.6062i 0.0312680 0.0312680i −0.691300 0.722568i \(-0.742962\pi\)
0.722568 + 0.691300i \(0.242962\pi\)
\(692\) 0 0
\(693\) −50.9344 + 50.9344i −0.0734984 + 0.0734984i
\(694\) 0 0
\(695\) 477.380 0.686878
\(696\) 0 0
\(697\) 1645.86i 2.36136i
\(698\) 0 0
\(699\) −476.664 476.664i −0.681923 0.681923i
\(700\) 0 0
\(701\) −462.868 462.868i −0.660296 0.660296i 0.295153 0.955450i \(-0.404629\pi\)
−0.955450 + 0.295153i \(0.904629\pi\)
\(702\) 0 0
\(703\) 140.513i 0.199877i
\(704\) 0 0
\(705\) −474.718 −0.673359
\(706\) 0 0
\(707\) −38.6536 + 38.6536i −0.0546728 + 0.0546728i
\(708\) 0 0
\(709\) −91.9564 + 91.9564i −0.129699 + 0.129699i −0.768976 0.639277i \(-0.779234\pi\)
0.639277 + 0.768976i \(0.279234\pi\)
\(710\) 0 0
\(711\) 1221.56 1.71809
\(712\) 0 0
\(713\) 246.910i 0.346297i
\(714\) 0 0
\(715\) −6.62172 6.62172i −0.00926115 0.00926115i
\(716\) 0 0
\(717\) −321.448 321.448i −0.448323 0.448323i
\(718\) 0 0
\(719\) 463.016i 0.643972i −0.946744 0.321986i \(-0.895650\pi\)
0.946744 0.321986i \(-0.104350\pi\)
\(720\) 0 0
\(721\) 823.814 1.14260
\(722\) 0 0
\(723\) −928.579 + 928.579i −1.28434 + 1.28434i
\(724\) 0 0
\(725\) 115.825 115.825i 0.159759 0.159759i
\(726\) 0 0
\(727\) 233.171 0.320730 0.160365 0.987058i \(-0.448733\pi\)
0.160365 + 0.987058i \(0.448733\pi\)
\(728\) 0 0
\(729\) 985.596i 1.35198i
\(730\) 0 0
\(731\) 387.709 + 387.709i 0.530382 + 0.530382i
\(732\) 0 0
\(733\) 873.035 + 873.035i 1.19104 + 1.19104i 0.976775 + 0.214269i \(0.0687368\pi\)
0.214269 + 0.976775i \(0.431263\pi\)
\(734\) 0 0
\(735\) 580.212i 0.789404i
\(736\) 0 0
\(737\) 2.73017 0.00370443
\(738\) 0 0
\(739\) 127.699 127.699i 0.172799 0.172799i −0.615409 0.788208i \(-0.711009\pi\)
0.788208 + 0.615409i \(0.211009\pi\)
\(740\) 0 0
\(741\) −505.456 + 505.456i −0.682127 + 0.682127i
\(742\) 0 0
\(743\) −226.592 −0.304968 −0.152484 0.988306i \(-0.548727\pi\)
−0.152484 + 0.988306i \(0.548727\pi\)
\(744\) 0 0
\(745\) 305.008i 0.409407i
\(746\) 0 0
\(747\) 666.989 + 666.989i 0.892890 + 0.892890i
\(748\) 0 0
\(749\) −1097.29 1097.29i −1.46501 1.46501i
\(750\) 0 0
\(751\) 516.811i 0.688164i −0.938940 0.344082i \(-0.888190\pi\)
0.938940 0.344082i \(-0.111810\pi\)
\(752\) 0 0
\(753\) 611.124 0.811586
\(754\) 0 0
\(755\) 46.5034 46.5034i 0.0615940 0.0615940i
\(756\) 0 0
\(757\) 135.597 135.597i 0.179124 0.179124i −0.611850 0.790974i \(-0.709574\pi\)
0.790974 + 0.611850i \(0.209574\pi\)
\(758\) 0 0
\(759\) −73.6857 −0.0970826
\(760\) 0 0
\(761\) 268.435i 0.352739i 0.984324 + 0.176370i \(0.0564354\pi\)
−0.984324 + 0.176370i \(0.943565\pi\)
\(762\) 0 0
\(763\) 29.9500 + 29.9500i 0.0392530 + 0.0392530i
\(764\) 0 0
\(765\) −681.507 681.507i −0.890858 0.890858i
\(766\) 0 0
\(767\) 799.578i 1.04247i
\(768\) 0 0
\(769\) −826.424 −1.07467 −0.537337 0.843368i \(-0.680570\pi\)
−0.537337 + 0.843368i \(0.680570\pi\)
\(770\) 0 0
\(771\) −1619.28 + 1619.28i −2.10023 + 2.10023i
\(772\) 0 0
\(773\) −1042.45 + 1042.45i −1.34857 + 1.34857i −0.461359 + 0.887213i \(0.652638\pi\)
−0.887213 + 0.461359i \(0.847362\pi\)
\(774\) 0 0
\(775\) −36.5962 −0.0472209
\(776\) 0 0
\(777\) 491.595i 0.632683i
\(778\) 0 0
\(779\) −654.199 654.199i −0.839793 0.839793i
\(780\) 0 0
\(781\) −3.52390 3.52390i −0.00451203 0.00451203i
\(782\) 0 0
\(783\) 1279.62i 1.63426i
\(784\) 0 0
\(785\) 372.068 0.473972
\(786\) 0 0
\(787\) 924.222 924.222i 1.17436 1.17436i 0.193201 0.981159i \(-0.438113\pi\)
0.981159 0.193201i \(-0.0618870\pi\)
\(788\) 0 0
\(789\) −3.36771 + 3.36771i −0.00426833 + 0.00426833i
\(790\) 0 0
\(791\) −734.148 −0.928126
\(792\) 0 0
\(793\) 361.447i 0.455797i
\(794\) 0 0
\(795\) 414.216 + 414.216i 0.521027 + 0.521027i
\(796\) 0 0
\(797\) 761.055 + 761.055i 0.954899 + 0.954899i 0.999026 0.0441268i \(-0.0140505\pi\)
−0.0441268 + 0.999026i \(0.514051\pi\)
\(798\) 0 0
\(799\) 1080.49i 1.35230i
\(800\) 0 0
\(801\) 621.657 0.776101
\(802\) 0 0
\(803\) 30.9893 30.9893i 0.0385919 0.0385919i
\(804\) 0 0
\(805\) 533.866 533.866i 0.663188 0.663188i
\(806\) 0 0
\(807\) −1011.67 −1.25362
\(808\) 0 0
\(809\) 106.311i 0.131410i −0.997839 0.0657052i \(-0.979070\pi\)
0.997839 0.0657052i \(-0.0209297\pi\)
\(810\) 0 0
\(811\) 602.747 + 602.747i 0.743214 + 0.743214i 0.973195 0.229981i \(-0.0738664\pi\)
−0.229981 + 0.973195i \(0.573866\pi\)
\(812\) 0 0
\(813\) 852.069 + 852.069i 1.04806 + 1.04806i
\(814\) 0 0
\(815\) 24.7058i 0.0303139i
\(816\) 0 0
\(817\) −308.213 −0.377250
\(818\) 0 0
\(819\) −1149.20 + 1149.20i −1.40317 + 1.40317i
\(820\) 0 0
\(821\) 236.461 236.461i 0.288016 0.288016i −0.548280 0.836295i \(-0.684717\pi\)
0.836295 + 0.548280i \(0.184717\pi\)
\(822\) 0 0
\(823\) 412.888 0.501687 0.250843 0.968028i \(-0.419292\pi\)
0.250843 + 0.968028i \(0.419292\pi\)
\(824\) 0 0
\(825\) 10.9215i 0.0132381i
\(826\) 0 0
\(827\) 772.140 + 772.140i 0.933664 + 0.933664i 0.997933 0.0642682i \(-0.0204713\pi\)
−0.0642682 + 0.997933i \(0.520471\pi\)
\(828\) 0 0
\(829\) −163.360 163.360i −0.197056 0.197056i 0.601681 0.798737i \(-0.294498\pi\)
−0.798737 + 0.601681i \(0.794498\pi\)
\(830\) 0 0
\(831\) 2163.05i 2.60294i
\(832\) 0 0
\(833\) −1320.60 −1.58535
\(834\) 0 0
\(835\) −403.779 + 403.779i −0.483568 + 0.483568i
\(836\) 0 0
\(837\) 202.155 202.155i 0.241524 0.241524i
\(838\) 0 0
\(839\) −156.751 −0.186831 −0.0934154 0.995627i \(-0.529778\pi\)
−0.0934154 + 0.995627i \(0.529778\pi\)
\(840\) 0 0
\(841\) 232.237i 0.276144i
\(842\) 0 0
\(843\) −752.363 752.363i −0.892482 0.892482i
\(844\) 0 0
\(845\) 117.811 + 117.811i 0.139421 + 0.139421i
\(846\) 0 0
\(847\) 1209.23i 1.42766i
\(848\) 0 0
\(849\) 624.794 0.735917
\(850\) 0 0
\(851\) 231.085 231.085i 0.271545 0.271545i
\(852\) 0 0
\(853\) −933.500 + 933.500i −1.09437 + 1.09437i −0.0993169 + 0.995056i \(0.531666\pi\)
−0.995056 + 0.0993169i \(0.968334\pi\)
\(854\) 0 0
\(855\) 541.771 0.633650
\(856\) 0 0
\(857\) 988.694i 1.15367i 0.816861 + 0.576834i \(0.195712\pi\)
−0.816861 + 0.576834i \(0.804288\pi\)
\(858\) 0 0
\(859\) −868.729 868.729i −1.01133 1.01133i −0.999935 0.0113913i \(-0.996374\pi\)
−0.0113913 0.999935i \(-0.503626\pi\)
\(860\) 0 0
\(861\) −2288.76 2288.76i −2.65825 2.65825i
\(862\) 0 0
\(863\) 1065.18i 1.23428i 0.786854 + 0.617139i \(0.211709\pi\)
−0.786854 + 0.617139i \(0.788291\pi\)
\(864\) 0 0
\(865\) −85.4324 −0.0987658
\(866\) 0 0
\(867\) −1350.83 + 1350.83i −1.55805 + 1.55805i
\(868\) 0 0
\(869\) −22.2783 + 22.2783i −0.0256367 + 0.0256367i
\(870\) 0 0
\(871\) 61.5991 0.0707222
\(872\) 0 0
\(873\) 1916.80i 2.19565i
\(874\) 0 0
\(875\) −79.1280 79.1280i −0.0904320 0.0904320i
\(876\) 0 0
\(877\) −107.637 107.637i −0.122733 0.122733i 0.643072 0.765805i \(-0.277659\pi\)
−0.765805 + 0.643072i \(0.777659\pi\)
\(878\) 0 0
\(879\) 1509.60i 1.71741i
\(880\) 0 0
\(881\) 1289.21 1.46334 0.731672 0.681657i \(-0.238740\pi\)
0.731672 + 0.681657i \(0.238740\pi\)
\(882\) 0 0
\(883\) 89.1493 89.1493i 0.100962 0.100962i −0.654822 0.755783i \(-0.727256\pi\)
0.755783 + 0.654822i \(0.227256\pi\)
\(884\) 0 0
\(885\) −659.387 + 659.387i −0.745071 + 0.745071i
\(886\) 0 0
\(887\) −1235.62 −1.39303 −0.696516 0.717541i \(-0.745268\pi\)
−0.696516 + 0.717541i \(0.745268\pi\)
\(888\) 0 0
\(889\) 2131.01i 2.39708i
\(890\) 0 0
\(891\) −14.5298 14.5298i −0.0163073 0.0163073i
\(892\) 0 0
\(893\) −429.473 429.473i −0.480933 0.480933i
\(894\) 0 0
\(895\) 160.926i 0.179806i
\(896\) 0 0
\(897\) −1662.52 −1.85343
\(898\) 0 0
\(899\) 169.550 169.550i 0.188599 0.188599i
\(900\) 0 0
\(901\) 942.783 942.783i 1.04637 1.04637i
\(902\) 0 0
\(903\) −1078.30 −1.19414
\(904\) 0 0
\(905\) 43.1614i 0.0476921i
\(906\) 0 0
\(907\) −1170.07 1170.07i −1.29004 1.29004i −0.934757 0.355287i \(-0.884383\pi\)
−0.355287 0.934757i \(-0.615617\pi\)
\(908\) 0 0
\(909\) −64.5101 64.5101i −0.0709682 0.0709682i
\(910\) 0 0
\(911\) 174.325i 0.191356i −0.995412 0.0956781i \(-0.969498\pi\)
0.995412 0.0956781i \(-0.0305019\pi\)
\(912\) 0 0
\(913\) −24.3284 −0.0266467
\(914\) 0 0
\(915\) −298.074 + 298.074i −0.325764 + 0.325764i
\(916\) 0 0
\(917\) 346.052 346.052i 0.377374 0.377374i
\(918\) 0 0
\(919\) 1108.52 1.20622 0.603110 0.797658i \(-0.293928\pi\)
0.603110 + 0.797658i \(0.293928\pi\)
\(920\) 0 0
\(921\) 2476.08i 2.68846i
\(922\) 0 0
\(923\) −79.5075 79.5075i −0.0861403 0.0861403i
\(924\) 0 0
\(925\) −34.2507 34.2507i −0.0370278 0.0370278i
\(926\) 0 0
\(927\) 1374.89i 1.48316i
\(928\) 0 0
\(929\) −410.945 −0.442352 −0.221176 0.975234i \(-0.570990\pi\)
−0.221176 + 0.975234i \(0.570990\pi\)
\(930\) 0 0
\(931\) 524.912 524.912i 0.563816 0.563816i
\(932\) 0 0
\(933\) −357.458 + 357.458i −0.383127 + 0.383127i
\(934\) 0 0
\(935\) 24.8580 0.0265860
\(936\) 0 0
\(937\) 181.492i 0.193694i −0.995299 0.0968472i \(-0.969124\pi\)
0.995299 0.0968472i \(-0.0308758\pi\)
\(938\) 0 0
\(939\) 1116.13 + 1116.13i 1.18864 + 1.18864i
\(940\) 0 0
\(941\) −363.737 363.737i −0.386543 0.386543i 0.486910 0.873452i \(-0.338124\pi\)
−0.873452 + 0.486910i \(0.838124\pi\)
\(942\) 0 0
\(943\) 2151.76i 2.28183i
\(944\) 0 0
\(945\) 874.198 0.925077
\(946\) 0 0
\(947\) 616.971 616.971i 0.651500 0.651500i −0.301854 0.953354i \(-0.597605\pi\)
0.953354 + 0.301854i \(0.0976055\pi\)
\(948\) 0 0
\(949\) 699.192 699.192i 0.736767 0.736767i
\(950\) 0 0
\(951\) −137.964 −0.145072
\(952\) 0 0
\(953\) 602.419i 0.632129i −0.948738 0.316064i \(-0.897638\pi\)
0.948738 0.316064i \(-0.102362\pi\)
\(954\) 0 0
\(955\) −415.870 415.870i −0.435466 0.435466i
\(956\) 0 0
\(957\) −50.5992 50.5992i −0.0528727 0.0528727i
\(958\) 0 0
\(959\) 1568.16i 1.63521i
\(960\) 0 0
\(961\) 907.429 0.944255
\(962\) 0 0
\(963\) 1831.30 1831.30i 1.90166 1.90166i
\(964\) 0 0
\(965\) 93.2012 93.2012i 0.0965815 0.0965815i
\(966\) 0 0
\(967\) −1277.51 −1.32111 −0.660553 0.750779i \(-0.729678\pi\)
−0.660553 + 0.750779i \(0.729678\pi\)
\(968\) 0 0
\(969\) 1897.48i 1.95819i
\(970\) 0 0
\(971\) −122.966 122.966i −0.126639 0.126639i 0.640947 0.767585i \(-0.278542\pi\)
−0.767585 + 0.640947i \(0.778542\pi\)
\(972\) 0 0
\(973\) 1510.97 + 1510.97i 1.55289 + 1.55289i
\(974\) 0 0
\(975\) 246.414i 0.252732i
\(976\) 0 0
\(977\) 809.129 0.828177 0.414088 0.910237i \(-0.364100\pi\)
0.414088 + 0.910237i \(0.364100\pi\)
\(978\) 0 0
\(979\) −11.3375 + 11.3375i −0.0115807 + 0.0115807i
\(980\) 0 0
\(981\) −49.9844 + 49.9844i −0.0509525 + 0.0509525i
\(982\) 0 0
\(983\) 986.542 1.00360 0.501802 0.864983i \(-0.332671\pi\)
0.501802 + 0.864983i \(0.332671\pi\)
\(984\) 0 0
\(985\) 30.1827i 0.0306423i
\(986\) 0 0
\(987\) −1502.54 1502.54i −1.52233 1.52233i
\(988\) 0 0
\(989\) −506.881 506.881i −0.512519 0.512519i
\(990\) 0 0
\(991\) 674.433i 0.680558i 0.940325 + 0.340279i \(0.110521\pi\)
−0.940325 + 0.340279i \(0.889479\pi\)
\(992\) 0 0
\(993\) 1392.51 1.40233
\(994\) 0 0
\(995\) 163.097 163.097i 0.163916 0.163916i
\(996\) 0 0
\(997\) −1315.09 + 1315.09i −1.31905 + 1.31905i −0.404519 + 0.914530i \(0.632561\pi\)
−0.914530 + 0.404519i \(0.867439\pi\)
\(998\) 0 0
\(999\) 378.398 0.378777
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 320.3.r.a.111.2 32
4.3 odd 2 80.3.r.a.51.7 yes 32
8.3 odd 2 640.3.r.a.351.2 32
8.5 even 2 640.3.r.b.351.15 32
16.3 odd 4 640.3.r.b.31.15 32
16.5 even 4 80.3.r.a.11.7 32
16.11 odd 4 inner 320.3.r.a.271.2 32
16.13 even 4 640.3.r.a.31.2 32
20.3 even 4 400.3.k.h.99.16 32
20.7 even 4 400.3.k.g.99.1 32
20.19 odd 2 400.3.r.f.51.10 32
80.37 odd 4 400.3.k.h.299.16 32
80.53 odd 4 400.3.k.g.299.1 32
80.69 even 4 400.3.r.f.251.10 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.3.r.a.11.7 32 16.5 even 4
80.3.r.a.51.7 yes 32 4.3 odd 2
320.3.r.a.111.2 32 1.1 even 1 trivial
320.3.r.a.271.2 32 16.11 odd 4 inner
400.3.k.g.99.1 32 20.7 even 4
400.3.k.g.299.1 32 80.53 odd 4
400.3.k.h.99.16 32 20.3 even 4
400.3.k.h.299.16 32 80.37 odd 4
400.3.r.f.51.10 32 20.19 odd 2
400.3.r.f.251.10 32 80.69 even 4
640.3.r.a.31.2 32 16.13 even 4
640.3.r.a.351.2 32 8.3 odd 2
640.3.r.b.31.15 32 16.3 odd 4
640.3.r.b.351.15 32 8.5 even 2