Properties

Label 1152.2.i.h.769.2
Level $1152$
Weight $2$
Character 1152.769
Analytic conductor $9.199$
Analytic rank $0$
Dimension $10$
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] = [1152,2,Mod(385,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.385");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.i (of order \(3\), degree \(2\), 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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.8528759163648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 769.2
Root \(0.756905 - 1.55791i\) of defining polynomial
Character \(\chi\) \(=\) 1152.769
Dual form 1152.2.i.h.385.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.970741 + 1.43446i) q^{3} +(-1.07447 + 1.86104i) q^{5} +(-0.153174 - 0.265305i) q^{7} +(-1.11533 - 2.78497i) q^{9} +O(q^{10})\) \(q+(-0.970741 + 1.43446i) q^{3} +(-1.07447 + 1.86104i) q^{5} +(-0.153174 - 0.265305i) q^{7} +(-1.11533 - 2.78497i) q^{9} +(2.50736 + 4.34288i) q^{11} +(-0.470741 + 0.815346i) q^{13} +(-1.62655 - 3.34787i) q^{15} -4.70838 q^{17} +1.61796 q^{19} +(0.529259 + 0.0378211i) q^{21} +(-4.08184 + 7.06995i) q^{23} +(0.191022 + 0.330859i) q^{25} +(5.07761 + 1.10360i) q^{27} +(-2.39504 - 4.14834i) q^{29} +(1.29776 - 2.24778i) q^{31} +(-8.66367 - 0.619109i) q^{33} +0.658323 q^{35} -10.2093 q^{37} +(-0.712611 - 1.46675i) q^{39} +(3.86537 - 6.69502i) q^{41} +(0.138140 + 0.239265i) q^{43} +(6.38132 + 0.916704i) q^{45} +(-1.92007 - 3.32566i) q^{47} +(3.45308 - 5.98090i) q^{49} +(4.57062 - 6.75396i) q^{51} -2.23508 q^{53} -10.7764 q^{55} +(-1.57062 + 2.32089i) q^{57} +(-4.95830 + 8.58802i) q^{59} +(-5.36414 - 9.29097i) q^{61} +(-0.568026 + 0.722485i) q^{63} +(-1.01159 - 1.75213i) q^{65} +(2.02117 - 3.50078i) q^{67} +(-6.17912 - 12.7183i) q^{69} +3.59379 q^{71} -5.43811 q^{73} +(-0.660035 - 0.0471664i) q^{75} +(0.768124 - 1.33043i) q^{77} +(-8.30403 - 14.3830i) q^{79} +(-6.51210 + 6.21229i) q^{81} +(2.91867 + 5.05528i) q^{83} +(5.05902 - 8.76248i) q^{85} +(8.27557 + 0.591376i) q^{87} -1.94577 q^{89} +0.288420 q^{91} +(1.96456 + 4.04359i) q^{93} +(-1.73845 + 3.01108i) q^{95} +(7.07283 + 12.2505i) q^{97} +(9.29826 - 11.8267i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 4 q^{7} - q^{9} - q^{11} + 6 q^{13} - 12 q^{15} - 6 q^{17} + 18 q^{19} + 16 q^{21} - 4 q^{23} + q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} - 13 q^{33} - 24 q^{35} - 20 q^{37} + 18 q^{39} - 5 q^{41} - 13 q^{43} - 12 q^{45} + 6 q^{47} + 3 q^{49} + 3 q^{51} - 12 q^{55} + 27 q^{57} - 13 q^{59} + 10 q^{61} + 20 q^{63} - 17 q^{67} - 10 q^{69} - 8 q^{71} - 34 q^{73} - 29 q^{75} + 8 q^{77} + 6 q^{79} - q^{81} + 12 q^{83} + 18 q^{85} - 10 q^{87} + 44 q^{89} + 36 q^{91} + 26 q^{93} + 6 q^{95} + 27 q^{97} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.970741 + 1.43446i −0.560457 + 0.828183i
\(4\) 0 0
\(5\) −1.07447 + 1.86104i −0.480518 + 0.832282i −0.999750 0.0223513i \(-0.992885\pi\)
0.519232 + 0.854633i \(0.326218\pi\)
\(6\) 0 0
\(7\) −0.153174 0.265305i −0.0578942 0.100276i 0.835626 0.549299i \(-0.185105\pi\)
−0.893520 + 0.449024i \(0.851772\pi\)
\(8\) 0 0
\(9\) −1.11533 2.78497i −0.371775 0.928323i
\(10\) 0 0
\(11\) 2.50736 + 4.34288i 0.755999 + 1.30943i 0.944876 + 0.327428i \(0.106182\pi\)
−0.188878 + 0.982001i \(0.560485\pi\)
\(12\) 0 0
\(13\) −0.470741 + 0.815346i −0.130560 + 0.226136i −0.923893 0.382652i \(-0.875011\pi\)
0.793333 + 0.608788i \(0.208344\pi\)
\(14\) 0 0
\(15\) −1.62655 3.34787i −0.419972 0.864416i
\(16\) 0 0
\(17\) −4.70838 −1.14195 −0.570975 0.820967i \(-0.693435\pi\)
−0.570975 + 0.820967i \(0.693435\pi\)
\(18\) 0 0
\(19\) 1.61796 0.371185 0.185592 0.982627i \(-0.440580\pi\)
0.185592 + 0.982627i \(0.440580\pi\)
\(20\) 0 0
\(21\) 0.529259 + 0.0378211i 0.115494 + 0.00825323i
\(22\) 0 0
\(23\) −4.08184 + 7.06995i −0.851122 + 1.47419i 0.0290754 + 0.999577i \(0.490744\pi\)
−0.880197 + 0.474609i \(0.842590\pi\)
\(24\) 0 0
\(25\) 0.191022 + 0.330859i 0.0382043 + 0.0661718i
\(26\) 0 0
\(27\) 5.07761 + 1.10360i 0.977186 + 0.212387i
\(28\) 0 0
\(29\) −2.39504 4.14834i −0.444749 0.770327i 0.553286 0.832991i \(-0.313374\pi\)
−0.998035 + 0.0626641i \(0.980040\pi\)
\(30\) 0 0
\(31\) 1.29776 2.24778i 0.233084 0.403714i −0.725630 0.688085i \(-0.758452\pi\)
0.958714 + 0.284371i \(0.0917848\pi\)
\(32\) 0 0
\(33\) −8.66367 0.619109i −1.50815 0.107773i
\(34\) 0 0
\(35\) 0.658323 0.111277
\(36\) 0 0
\(37\) −10.2093 −1.67840 −0.839199 0.543824i \(-0.816976\pi\)
−0.839199 + 0.543824i \(0.816976\pi\)
\(38\) 0 0
\(39\) −0.712611 1.46675i −0.114109 0.234867i
\(40\) 0 0
\(41\) 3.86537 6.69502i 0.603669 1.04559i −0.388591 0.921410i \(-0.627038\pi\)
0.992260 0.124175i \(-0.0396285\pi\)
\(42\) 0 0
\(43\) 0.138140 + 0.239265i 0.0210661 + 0.0364876i 0.876366 0.481645i \(-0.159961\pi\)
−0.855300 + 0.518133i \(0.826627\pi\)
\(44\) 0 0
\(45\) 6.38132 + 0.916704i 0.951271 + 0.136654i
\(46\) 0 0
\(47\) −1.92007 3.32566i −0.280072 0.485098i 0.691331 0.722539i \(-0.257025\pi\)
−0.971402 + 0.237441i \(0.923692\pi\)
\(48\) 0 0
\(49\) 3.45308 5.98090i 0.493297 0.854415i
\(50\) 0 0
\(51\) 4.57062 6.75396i 0.640014 0.945744i
\(52\) 0 0
\(53\) −2.23508 −0.307012 −0.153506 0.988148i \(-0.549056\pi\)
−0.153506 + 0.988148i \(0.549056\pi\)
\(54\) 0 0
\(55\) −10.7764 −1.45308
\(56\) 0 0
\(57\) −1.57062 + 2.32089i −0.208033 + 0.307409i
\(58\) 0 0
\(59\) −4.95830 + 8.58802i −0.645515 + 1.11807i 0.338667 + 0.940906i \(0.390024\pi\)
−0.984182 + 0.177159i \(0.943309\pi\)
\(60\) 0 0
\(61\) −5.36414 9.29097i −0.686808 1.18959i −0.972865 0.231374i \(-0.925678\pi\)
0.286057 0.958213i \(-0.407655\pi\)
\(62\) 0 0
\(63\) −0.568026 + 0.722485i −0.0715646 + 0.0910245i
\(64\) 0 0
\(65\) −1.01159 1.75213i −0.125473 0.217325i
\(66\) 0 0
\(67\) 2.02117 3.50078i 0.246926 0.427688i −0.715746 0.698361i \(-0.753913\pi\)
0.962671 + 0.270673i \(0.0872463\pi\)
\(68\) 0 0
\(69\) −6.17912 12.7183i −0.743879 1.53110i
\(70\) 0 0
\(71\) 3.59379 0.426505 0.213252 0.976997i \(-0.431594\pi\)
0.213252 + 0.976997i \(0.431594\pi\)
\(72\) 0 0
\(73\) −5.43811 −0.636483 −0.318242 0.948010i \(-0.603092\pi\)
−0.318242 + 0.948010i \(0.603092\pi\)
\(74\) 0 0
\(75\) −0.660035 0.0471664i −0.0762143 0.00544630i
\(76\) 0 0
\(77\) 0.768124 1.33043i 0.0875359 0.151617i
\(78\) 0 0
\(79\) −8.30403 14.3830i −0.934276 1.61821i −0.775920 0.630831i \(-0.782714\pi\)
−0.158356 0.987382i \(-0.550619\pi\)
\(80\) 0 0
\(81\) −6.51210 + 6.21229i −0.723566 + 0.690255i
\(82\) 0 0
\(83\) 2.91867 + 5.05528i 0.320365 + 0.554889i 0.980563 0.196203i \(-0.0628610\pi\)
−0.660198 + 0.751092i \(0.729528\pi\)
\(84\) 0 0
\(85\) 5.05902 8.76248i 0.548728 0.950425i
\(86\) 0 0
\(87\) 8.27557 + 0.591376i 0.887235 + 0.0634021i
\(88\) 0 0
\(89\) −1.94577 −0.206251 −0.103125 0.994668i \(-0.532884\pi\)
−0.103125 + 0.994668i \(0.532884\pi\)
\(90\) 0 0
\(91\) 0.288420 0.0302347
\(92\) 0 0
\(93\) 1.96456 + 4.04359i 0.203715 + 0.419301i
\(94\) 0 0
\(95\) −1.73845 + 3.01108i −0.178361 + 0.308930i
\(96\) 0 0
\(97\) 7.07283 + 12.2505i 0.718137 + 1.24385i 0.961737 + 0.273974i \(0.0883383\pi\)
−0.243600 + 0.969876i \(0.578328\pi\)
\(98\) 0 0
\(99\) 9.29826 11.8267i 0.934510 1.18862i
\(100\) 0 0
\(101\) 9.41272 + 16.3033i 0.936601 + 1.62224i 0.771754 + 0.635922i \(0.219380\pi\)
0.164847 + 0.986319i \(0.447287\pi\)
\(102\) 0 0
\(103\) 2.95014 5.10979i 0.290686 0.503483i −0.683286 0.730151i \(-0.739450\pi\)
0.973972 + 0.226668i \(0.0727831\pi\)
\(104\) 0 0
\(105\) −0.639061 + 0.944335i −0.0623659 + 0.0921577i
\(106\) 0 0
\(107\) 3.86061 0.373219 0.186609 0.982434i \(-0.440250\pi\)
0.186609 + 0.982434i \(0.440250\pi\)
\(108\) 0 0
\(109\) −10.8821 −1.04232 −0.521159 0.853459i \(-0.674500\pi\)
−0.521159 + 0.853459i \(0.674500\pi\)
\(110\) 0 0
\(111\) 9.91058 14.6448i 0.940671 1.39002i
\(112\) 0 0
\(113\) 3.15157 5.45869i 0.296475 0.513510i −0.678852 0.734275i \(-0.737522\pi\)
0.975327 + 0.220765i \(0.0708555\pi\)
\(114\) 0 0
\(115\) −8.77163 15.1929i −0.817959 1.41675i
\(116\) 0 0
\(117\) 2.79574 + 0.401621i 0.258467 + 0.0371298i
\(118\) 0 0
\(119\) 0.721200 + 1.24915i 0.0661123 + 0.114510i
\(120\) 0 0
\(121\) −7.07375 + 12.2521i −0.643068 + 1.11383i
\(122\) 0 0
\(123\) 5.85143 + 12.0438i 0.527606 + 1.08595i
\(124\) 0 0
\(125\) −11.5657 −1.03447
\(126\) 0 0
\(127\) −11.7659 −1.04406 −0.522028 0.852928i \(-0.674825\pi\)
−0.522028 + 0.852928i \(0.674825\pi\)
\(128\) 0 0
\(129\) −0.477313 0.0341090i −0.0420251 0.00300313i
\(130\) 0 0
\(131\) 2.64077 4.57395i 0.230725 0.399628i −0.727297 0.686323i \(-0.759223\pi\)
0.958022 + 0.286696i \(0.0925568\pi\)
\(132\) 0 0
\(133\) −0.247828 0.429251i −0.0214894 0.0372208i
\(134\) 0 0
\(135\) −7.50958 + 8.26384i −0.646322 + 0.711238i
\(136\) 0 0
\(137\) −7.23452 12.5306i −0.618087 1.07056i −0.989834 0.142224i \(-0.954575\pi\)
0.371748 0.928334i \(-0.378759\pi\)
\(138\) 0 0
\(139\) −10.7880 + 18.6854i −0.915026 + 1.58487i −0.108164 + 0.994133i \(0.534497\pi\)
−0.806863 + 0.590739i \(0.798836\pi\)
\(140\) 0 0
\(141\) 6.63441 + 0.474097i 0.558718 + 0.0399262i
\(142\) 0 0
\(143\) −4.72127 −0.394813
\(144\) 0 0
\(145\) 10.2936 0.854839
\(146\) 0 0
\(147\) 5.22730 + 10.7592i 0.431140 + 0.887403i
\(148\) 0 0
\(149\) −7.80471 + 13.5181i −0.639386 + 1.10745i 0.346181 + 0.938168i \(0.387478\pi\)
−0.985568 + 0.169282i \(0.945855\pi\)
\(150\) 0 0
\(151\) 8.58275 + 14.8658i 0.698455 + 1.20976i 0.969002 + 0.247052i \(0.0794620\pi\)
−0.270547 + 0.962707i \(0.587205\pi\)
\(152\) 0 0
\(153\) 5.25138 + 13.1127i 0.424549 + 1.06010i
\(154\) 0 0
\(155\) 2.78881 + 4.83036i 0.224003 + 0.387984i
\(156\) 0 0
\(157\) −2.59257 + 4.49046i −0.206909 + 0.358378i −0.950739 0.309991i \(-0.899674\pi\)
0.743830 + 0.668369i \(0.233007\pi\)
\(158\) 0 0
\(159\) 2.16968 3.20612i 0.172067 0.254262i
\(160\) 0 0
\(161\) 2.50092 0.197100
\(162\) 0 0
\(163\) −17.8955 −1.40168 −0.700842 0.713317i \(-0.747192\pi\)
−0.700842 + 0.713317i \(0.747192\pi\)
\(164\) 0 0
\(165\) 10.4611 15.4582i 0.814392 1.20342i
\(166\) 0 0
\(167\) −5.48714 + 9.50401i −0.424608 + 0.735442i −0.996384 0.0849677i \(-0.972921\pi\)
0.571776 + 0.820410i \(0.306255\pi\)
\(168\) 0 0
\(169\) 6.05681 + 10.4907i 0.465908 + 0.806977i
\(170\) 0 0
\(171\) −1.80455 4.50596i −0.137997 0.344579i
\(172\) 0 0
\(173\) 8.63146 + 14.9501i 0.656238 + 1.13664i 0.981582 + 0.191042i \(0.0611865\pi\)
−0.325344 + 0.945596i \(0.605480\pi\)
\(174\) 0 0
\(175\) 0.0585190 0.101358i 0.00442362 0.00766193i
\(176\) 0 0
\(177\) −7.50592 15.4492i −0.564179 1.16123i
\(178\) 0 0
\(179\) −15.0571 −1.12542 −0.562709 0.826655i \(-0.690241\pi\)
−0.562709 + 0.826655i \(0.690241\pi\)
\(180\) 0 0
\(181\) 17.6813 1.31424 0.657120 0.753786i \(-0.271774\pi\)
0.657120 + 0.753786i \(0.271774\pi\)
\(182\) 0 0
\(183\) 18.5347 + 1.32449i 1.37012 + 0.0979095i
\(184\) 0 0
\(185\) 10.9696 18.9999i 0.806501 1.39690i
\(186\) 0 0
\(187\) −11.8056 20.4479i −0.863313 1.49530i
\(188\) 0 0
\(189\) −0.484966 1.51615i −0.0352761 0.110284i
\(190\) 0 0
\(191\) 10.3168 + 17.8693i 0.746501 + 1.29298i 0.949490 + 0.313796i \(0.101601\pi\)
−0.202990 + 0.979181i \(0.565066\pi\)
\(192\) 0 0
\(193\) 11.6134 20.1149i 0.835948 1.44791i −0.0573076 0.998357i \(-0.518252\pi\)
0.893256 0.449548i \(-0.148415\pi\)
\(194\) 0 0
\(195\) 3.49535 + 0.249779i 0.250307 + 0.0178871i
\(196\) 0 0
\(197\) −4.78497 −0.340915 −0.170458 0.985365i \(-0.554525\pi\)
−0.170458 + 0.985365i \(0.554525\pi\)
\(198\) 0 0
\(199\) −11.5938 −0.821862 −0.410931 0.911666i \(-0.634796\pi\)
−0.410931 + 0.911666i \(0.634796\pi\)
\(200\) 0 0
\(201\) 3.05967 + 6.29763i 0.215813 + 0.444200i
\(202\) 0 0
\(203\) −0.733715 + 1.27083i −0.0514967 + 0.0891950i
\(204\) 0 0
\(205\) 8.30646 + 14.3872i 0.580148 + 1.00485i
\(206\) 0 0
\(207\) 24.2422 + 3.48249i 1.68495 + 0.242050i
\(208\) 0 0
\(209\) 4.05681 + 7.02660i 0.280615 + 0.486040i
\(210\) 0 0
\(211\) 0.888671 1.53922i 0.0611786 0.105964i −0.833814 0.552046i \(-0.813847\pi\)
0.894993 + 0.446081i \(0.147181\pi\)
\(212\) 0 0
\(213\) −3.48864 + 5.15513i −0.239038 + 0.353224i
\(214\) 0 0
\(215\) −0.593709 −0.0404906
\(216\) 0 0
\(217\) −0.795130 −0.0539769
\(218\) 0 0
\(219\) 5.27900 7.80073i 0.356722 0.527125i
\(220\) 0 0
\(221\) 2.21643 3.83896i 0.149093 0.258237i
\(222\) 0 0
\(223\) 5.02422 + 8.70221i 0.336447 + 0.582743i 0.983762 0.179480i \(-0.0574415\pi\)
−0.647315 + 0.762223i \(0.724108\pi\)
\(224\) 0 0
\(225\) 0.708381 0.901005i 0.0472254 0.0600670i
\(226\) 0 0
\(227\) 5.27671 + 9.13953i 0.350228 + 0.606612i 0.986289 0.165026i \(-0.0527709\pi\)
−0.636061 + 0.771638i \(0.719438\pi\)
\(228\) 0 0
\(229\) 11.3955 19.7377i 0.753039 1.30430i −0.193304 0.981139i \(-0.561920\pi\)
0.946343 0.323163i \(-0.104746\pi\)
\(230\) 0 0
\(231\) 1.16279 + 2.39334i 0.0765062 + 0.157470i
\(232\) 0 0
\(233\) 5.97108 0.391179 0.195589 0.980686i \(-0.437338\pi\)
0.195589 + 0.980686i \(0.437338\pi\)
\(234\) 0 0
\(235\) 8.25226 0.538318
\(236\) 0 0
\(237\) 28.6928 + 2.05040i 1.86380 + 0.133188i
\(238\) 0 0
\(239\) 5.77549 10.0034i 0.373585 0.647069i −0.616529 0.787332i \(-0.711462\pi\)
0.990114 + 0.140264i \(0.0447950\pi\)
\(240\) 0 0
\(241\) 7.75827 + 13.4377i 0.499754 + 0.865600i 1.00000 0.000283894i \(-9.03662e-5\pi\)
−0.500246 + 0.865883i \(0.666757\pi\)
\(242\) 0 0
\(243\) −2.58970 15.3718i −0.166129 0.986104i
\(244\) 0 0
\(245\) 7.42046 + 12.8526i 0.474076 + 0.821124i
\(246\) 0 0
\(247\) −0.761638 + 1.31920i −0.0484619 + 0.0839384i
\(248\) 0 0
\(249\) −10.0848 0.720666i −0.639101 0.0456704i
\(250\) 0 0
\(251\) −14.5685 −0.919557 −0.459778 0.888034i \(-0.652071\pi\)
−0.459778 + 0.888034i \(0.652071\pi\)
\(252\) 0 0
\(253\) −40.9386 −2.57379
\(254\) 0 0
\(255\) 7.65839 + 15.7630i 0.479587 + 0.987120i
\(256\) 0 0
\(257\) −7.78071 + 13.4766i −0.485347 + 0.840646i −0.999858 0.0168376i \(-0.994640\pi\)
0.514511 + 0.857484i \(0.327974\pi\)
\(258\) 0 0
\(259\) 1.56380 + 2.70857i 0.0971695 + 0.168303i
\(260\) 0 0
\(261\) −8.88174 + 11.2969i −0.549766 + 0.699259i
\(262\) 0 0
\(263\) 11.2231 + 19.4389i 0.692044 + 1.19866i 0.971167 + 0.238400i \(0.0766229\pi\)
−0.279123 + 0.960255i \(0.590044\pi\)
\(264\) 0 0
\(265\) 2.40153 4.15957i 0.147525 0.255521i
\(266\) 0 0
\(267\) 1.88884 2.79112i 0.115595 0.170814i
\(268\) 0 0
\(269\) 26.0256 1.58681 0.793403 0.608696i \(-0.208307\pi\)
0.793403 + 0.608696i \(0.208307\pi\)
\(270\) 0 0
\(271\) 5.59761 0.340031 0.170015 0.985441i \(-0.445618\pi\)
0.170015 + 0.985441i \(0.445618\pi\)
\(272\) 0 0
\(273\) −0.279981 + 0.413726i −0.0169452 + 0.0250398i
\(274\) 0 0
\(275\) −0.957922 + 1.65917i −0.0577648 + 0.100052i
\(276\) 0 0
\(277\) 1.57957 + 2.73589i 0.0949069 + 0.164384i 0.909570 0.415551i \(-0.136411\pi\)
−0.814663 + 0.579935i \(0.803078\pi\)
\(278\) 0 0
\(279\) −7.70743 1.10721i −0.461432 0.0662867i
\(280\) 0 0
\(281\) −8.02031 13.8916i −0.478452 0.828703i 0.521243 0.853408i \(-0.325468\pi\)
−0.999695 + 0.0247057i \(0.992135\pi\)
\(282\) 0 0
\(283\) 1.87142 3.24140i 0.111245 0.192681i −0.805028 0.593237i \(-0.797850\pi\)
0.916272 + 0.400556i \(0.131183\pi\)
\(284\) 0 0
\(285\) −2.63168 5.41671i −0.155887 0.320858i
\(286\) 0 0
\(287\) −2.36829 −0.139796
\(288\) 0 0
\(289\) 5.16885 0.304050
\(290\) 0 0
\(291\) −24.4387 1.74640i −1.43262 0.102376i
\(292\) 0 0
\(293\) −5.49886 + 9.52431i −0.321247 + 0.556416i −0.980746 0.195290i \(-0.937435\pi\)
0.659499 + 0.751706i \(0.270769\pi\)
\(294\) 0 0
\(295\) −10.6551 18.4552i −0.620364 1.07450i
\(296\) 0 0
\(297\) 7.93862 + 24.8186i 0.460645 + 1.44012i
\(298\) 0 0
\(299\) −3.84297 6.65622i −0.222245 0.384939i
\(300\) 0 0
\(301\) 0.0423188 0.0732983i 0.00243921 0.00422484i
\(302\) 0 0
\(303\) −32.5237 2.32416i −1.86844 0.133519i
\(304\) 0 0
\(305\) 23.0545 1.32010
\(306\) 0 0
\(307\) 6.08416 0.347241 0.173621 0.984813i \(-0.444453\pi\)
0.173621 + 0.984813i \(0.444453\pi\)
\(308\) 0 0
\(309\) 4.46595 + 9.19213i 0.254059 + 0.522922i
\(310\) 0 0
\(311\) −3.75633 + 6.50616i −0.213002 + 0.368930i −0.952653 0.304061i \(-0.901657\pi\)
0.739651 + 0.672991i \(0.234991\pi\)
\(312\) 0 0
\(313\) 6.18076 + 10.7054i 0.349357 + 0.605105i 0.986135 0.165942i \(-0.0530666\pi\)
−0.636778 + 0.771047i \(0.719733\pi\)
\(314\) 0 0
\(315\) −0.734245 1.83341i −0.0413700 0.103301i
\(316\) 0 0
\(317\) −12.3204 21.3395i −0.691980 1.19854i −0.971188 0.238314i \(-0.923405\pi\)
0.279208 0.960231i \(-0.409928\pi\)
\(318\) 0 0
\(319\) 12.0105 20.8028i 0.672459 1.16473i
\(320\) 0 0
\(321\) −3.74765 + 5.53787i −0.209173 + 0.309094i
\(322\) 0 0
\(323\) −7.61796 −0.423874
\(324\) 0 0
\(325\) −0.359686 −0.0199518
\(326\) 0 0
\(327\) 10.5637 15.6099i 0.584175 0.863231i
\(328\) 0 0
\(329\) −0.588209 + 1.01881i −0.0324290 + 0.0561687i
\(330\) 0 0
\(331\) −11.0695 19.1730i −0.608436 1.05384i −0.991498 0.130119i \(-0.958464\pi\)
0.383063 0.923722i \(-0.374869\pi\)
\(332\) 0 0
\(333\) 11.3867 + 28.4326i 0.623987 + 1.55810i
\(334\) 0 0
\(335\) 4.34339 + 7.52297i 0.237305 + 0.411024i
\(336\) 0 0
\(337\) −5.63803 + 9.76536i −0.307123 + 0.531953i −0.977732 0.209858i \(-0.932700\pi\)
0.670609 + 0.741811i \(0.266033\pi\)
\(338\) 0 0
\(339\) 4.77088 + 9.81976i 0.259119 + 0.533336i
\(340\) 0 0
\(341\) 13.0158 0.704846
\(342\) 0 0
\(343\) −4.26011 −0.230024
\(344\) 0 0
\(345\) 30.3085 + 2.16586i 1.63176 + 0.116606i
\(346\) 0 0
\(347\) −11.3903 + 19.7286i −0.611465 + 1.05909i 0.379529 + 0.925180i \(0.376086\pi\)
−0.990994 + 0.133908i \(0.957247\pi\)
\(348\) 0 0
\(349\) −1.44215 2.49788i −0.0771966 0.133708i 0.824843 0.565362i \(-0.191264\pi\)
−0.902039 + 0.431654i \(0.857930\pi\)
\(350\) 0 0
\(351\) −3.29005 + 3.62050i −0.175610 + 0.193248i
\(352\) 0 0
\(353\) 9.41192 + 16.3019i 0.500946 + 0.867663i 0.999999 + 0.00109240i \(0.000347720\pi\)
−0.499054 + 0.866571i \(0.666319\pi\)
\(354\) 0 0
\(355\) −3.86143 + 6.68819i −0.204943 + 0.354972i
\(356\) 0 0
\(357\) −2.49196 0.178076i −0.131888 0.00942478i
\(358\) 0 0
\(359\) 26.6316 1.40556 0.702782 0.711406i \(-0.251941\pi\)
0.702782 + 0.711406i \(0.251941\pi\)
\(360\) 0 0
\(361\) −16.3822 −0.862222
\(362\) 0 0
\(363\) −10.7083 22.0406i −0.562040 1.15683i
\(364\) 0 0
\(365\) 5.84310 10.1205i 0.305842 0.529733i
\(366\) 0 0
\(367\) 12.6413 + 21.8953i 0.659869 + 1.14293i 0.980649 + 0.195773i \(0.0627216\pi\)
−0.320780 + 0.947154i \(0.603945\pi\)
\(368\) 0 0
\(369\) −22.9566 3.29781i −1.19507 0.171677i
\(370\) 0 0
\(371\) 0.342356 + 0.592977i 0.0177742 + 0.0307858i
\(372\) 0 0
\(373\) −0.427926 + 0.741189i −0.0221571 + 0.0383773i −0.876891 0.480689i \(-0.840387\pi\)
0.854734 + 0.519066i \(0.173720\pi\)
\(374\) 0 0
\(375\) 11.2273 16.5905i 0.579775 0.856729i
\(376\) 0 0
\(377\) 4.50978 0.232265
\(378\) 0 0
\(379\) 5.34571 0.274591 0.137295 0.990530i \(-0.456159\pi\)
0.137295 + 0.990530i \(0.456159\pi\)
\(380\) 0 0
\(381\) 11.4217 16.8777i 0.585149 0.864671i
\(382\) 0 0
\(383\) 0.132433 0.229381i 0.00676702 0.0117208i −0.862622 0.505849i \(-0.831179\pi\)
0.869389 + 0.494128i \(0.164513\pi\)
\(384\) 0 0
\(385\) 1.65066 + 2.85902i 0.0841252 + 0.145709i
\(386\) 0 0
\(387\) 0.512275 0.651574i 0.0260404 0.0331214i
\(388\) 0 0
\(389\) −10.9697 19.0001i −0.556187 0.963343i −0.997810 0.0661429i \(-0.978931\pi\)
0.441624 0.897200i \(-0.354403\pi\)
\(390\) 0 0
\(391\) 19.2188 33.2880i 0.971938 1.68345i
\(392\) 0 0
\(393\) 3.99762 + 8.22819i 0.201653 + 0.415057i
\(394\) 0 0
\(395\) 35.6898 1.79575
\(396\) 0 0
\(397\) 2.19238 0.110032 0.0550161 0.998485i \(-0.482479\pi\)
0.0550161 + 0.998485i \(0.482479\pi\)
\(398\) 0 0
\(399\) 0.856319 + 0.0611929i 0.0428696 + 0.00306347i
\(400\) 0 0
\(401\) −18.8864 + 32.7121i −0.943140 + 1.63357i −0.183707 + 0.982981i \(0.558810\pi\)
−0.759433 + 0.650585i \(0.774524\pi\)
\(402\) 0 0
\(403\) 1.22182 + 2.11625i 0.0608630 + 0.105418i
\(404\) 0 0
\(405\) −4.56426 18.7942i −0.226800 0.933891i
\(406\) 0 0
\(407\) −25.5984 44.3378i −1.26887 2.19774i
\(408\) 0 0
\(409\) 15.9676 27.6566i 0.789546 1.36753i −0.136700 0.990612i \(-0.543650\pi\)
0.926246 0.376920i \(-0.123017\pi\)
\(410\) 0 0
\(411\) 24.9974 + 1.78632i 1.23303 + 0.0881127i
\(412\) 0 0
\(413\) 3.03792 0.149486
\(414\) 0 0
\(415\) −12.5441 −0.615766
\(416\) 0 0
\(417\) −16.3310 33.6136i −0.799731 1.64606i
\(418\) 0 0
\(419\) 1.83505 3.17840i 0.0896480 0.155275i −0.817714 0.575624i \(-0.804759\pi\)
0.907362 + 0.420349i \(0.138092\pi\)
\(420\) 0 0
\(421\) 7.72300 + 13.3766i 0.376396 + 0.651937i 0.990535 0.137261i \(-0.0438299\pi\)
−0.614139 + 0.789198i \(0.710497\pi\)
\(422\) 0 0
\(423\) −7.12036 + 9.05654i −0.346204 + 0.440344i
\(424\) 0 0
\(425\) −0.899403 1.55781i −0.0436274 0.0755649i
\(426\) 0 0
\(427\) −1.64329 + 2.84626i −0.0795244 + 0.137740i
\(428\) 0 0
\(429\) 4.58313 6.77245i 0.221276 0.326977i
\(430\) 0 0
\(431\) −18.9913 −0.914779 −0.457390 0.889266i \(-0.651216\pi\)
−0.457390 + 0.889266i \(0.651216\pi\)
\(432\) 0 0
\(433\) 12.7931 0.614796 0.307398 0.951581i \(-0.400542\pi\)
0.307398 + 0.951581i \(0.400542\pi\)
\(434\) 0 0
\(435\) −9.99244 + 14.7658i −0.479101 + 0.707964i
\(436\) 0 0
\(437\) −6.60423 + 11.4389i −0.315923 + 0.547195i
\(438\) 0 0
\(439\) 14.5259 + 25.1595i 0.693281 + 1.20080i 0.970757 + 0.240065i \(0.0771689\pi\)
−0.277476 + 0.960733i \(0.589498\pi\)
\(440\) 0 0
\(441\) −20.5079 2.94605i −0.976568 0.140288i
\(442\) 0 0
\(443\) −18.4010 31.8714i −0.874256 1.51426i −0.857554 0.514395i \(-0.828017\pi\)
−0.0167020 0.999861i \(-0.505317\pi\)
\(444\) 0 0
\(445\) 2.09067 3.62115i 0.0991073 0.171659i
\(446\) 0 0
\(447\) −11.8148 24.3181i −0.558823 1.15021i
\(448\) 0 0
\(449\) −18.4952 −0.872842 −0.436421 0.899743i \(-0.643754\pi\)
−0.436421 + 0.899743i \(0.643754\pi\)
\(450\) 0 0
\(451\) 38.7675 1.82549
\(452\) 0 0
\(453\) −29.6559 2.11922i −1.39336 0.0995697i
\(454\) 0 0
\(455\) −0.309899 + 0.536761i −0.0145283 + 0.0251638i
\(456\) 0 0
\(457\) 9.79321 + 16.9623i 0.458107 + 0.793465i 0.998861 0.0477162i \(-0.0151943\pi\)
−0.540754 + 0.841181i \(0.681861\pi\)
\(458\) 0 0
\(459\) −23.9073 5.19615i −1.11590 0.242536i
\(460\) 0 0
\(461\) −6.17311 10.6921i −0.287510 0.497983i 0.685704 0.727880i \(-0.259494\pi\)
−0.973215 + 0.229897i \(0.926161\pi\)
\(462\) 0 0
\(463\) −18.6089 + 32.2316i −0.864830 + 1.49793i 0.00238525 + 0.999997i \(0.499241\pi\)
−0.867216 + 0.497933i \(0.834093\pi\)
\(464\) 0 0
\(465\) −9.63615 0.688603i −0.446866 0.0319332i
\(466\) 0 0
\(467\) 10.6412 0.492415 0.246208 0.969217i \(-0.420815\pi\)
0.246208 + 0.969217i \(0.420815\pi\)
\(468\) 0 0
\(469\) −1.23836 −0.0571823
\(470\) 0 0
\(471\) −3.92465 8.07799i −0.180838 0.372214i
\(472\) 0 0
\(473\) −0.692734 + 1.19985i −0.0318519 + 0.0551692i
\(474\) 0 0
\(475\) 0.309065 + 0.535316i 0.0141809 + 0.0245620i
\(476\) 0 0
\(477\) 2.49284 + 6.22463i 0.114139 + 0.285006i
\(478\) 0 0
\(479\) 6.22894 + 10.7888i 0.284608 + 0.492955i 0.972514 0.232845i \(-0.0748034\pi\)
−0.687906 + 0.725799i \(0.741470\pi\)
\(480\) 0 0
\(481\) 4.80593 8.32412i 0.219132 0.379547i
\(482\) 0 0
\(483\) −2.42774 + 3.58746i −0.110466 + 0.163235i
\(484\) 0 0
\(485\) −30.3982 −1.38031
\(486\) 0 0
\(487\) 12.5254 0.567580 0.283790 0.958886i \(-0.408408\pi\)
0.283790 + 0.958886i \(0.408408\pi\)
\(488\) 0 0
\(489\) 17.3719 25.6703i 0.785584 1.16085i
\(490\) 0 0
\(491\) −0.581151 + 1.00658i −0.0262270 + 0.0454264i −0.878841 0.477115i \(-0.841683\pi\)
0.852614 + 0.522541i \(0.175016\pi\)
\(492\) 0 0
\(493\) 11.2768 + 19.5320i 0.507881 + 0.879675i
\(494\) 0 0
\(495\) 12.0192 + 30.0118i 0.540221 + 1.34893i
\(496\) 0 0
\(497\) −0.550474 0.953449i −0.0246921 0.0427680i
\(498\) 0 0
\(499\) −12.8699 + 22.2912i −0.576134 + 0.997893i 0.419784 + 0.907624i \(0.362106\pi\)
−0.995917 + 0.0902688i \(0.971227\pi\)
\(500\) 0 0
\(501\) −8.30648 17.0970i −0.371106 0.763837i
\(502\) 0 0
\(503\) −24.1469 −1.07666 −0.538328 0.842736i \(-0.680944\pi\)
−0.538328 + 0.842736i \(0.680944\pi\)
\(504\) 0 0
\(505\) −40.4548 −1.80022
\(506\) 0 0
\(507\) −20.9280 1.49552i −0.929446 0.0664186i
\(508\) 0 0
\(509\) 4.55763 7.89404i 0.202013 0.349897i −0.747164 0.664640i \(-0.768585\pi\)
0.949177 + 0.314743i \(0.101918\pi\)
\(510\) 0 0
\(511\) 0.832976 + 1.44276i 0.0368487 + 0.0638238i
\(512\) 0 0
\(513\) 8.21535 + 1.78557i 0.362716 + 0.0788349i
\(514\) 0 0
\(515\) 6.33969 + 10.9807i 0.279360 + 0.483866i
\(516\) 0 0
\(517\) 9.62865 16.6773i 0.423467 0.733467i
\(518\) 0 0
\(519\) −29.8242 2.13125i −1.30914 0.0935514i
\(520\) 0 0
\(521\) −9.18121 −0.402236 −0.201118 0.979567i \(-0.564458\pi\)
−0.201118 + 0.979567i \(0.564458\pi\)
\(522\) 0 0
\(523\) 8.94824 0.391279 0.195640 0.980676i \(-0.437322\pi\)
0.195640 + 0.980676i \(0.437322\pi\)
\(524\) 0 0
\(525\) 0.0885866 + 0.182335i 0.00386623 + 0.00795775i
\(526\) 0 0
\(527\) −6.11034 + 10.5834i −0.266171 + 0.461021i
\(528\) 0 0
\(529\) −21.8228 37.7981i −0.948816 1.64340i
\(530\) 0 0
\(531\) 29.4475 + 4.23026i 1.27791 + 0.183578i
\(532\) 0 0
\(533\) 3.63917 + 6.30323i 0.157630 + 0.273023i
\(534\) 0 0
\(535\) −4.14811 + 7.18474i −0.179338 + 0.310623i
\(536\) 0 0
\(537\) 14.6165 21.5987i 0.630749 0.932053i
\(538\) 0 0
\(539\) 34.6325 1.49173
\(540\) 0 0
\(541\) −26.6203 −1.14450 −0.572249 0.820080i \(-0.693929\pi\)
−0.572249 + 0.820080i \(0.693929\pi\)
\(542\) 0 0
\(543\) −17.1640 + 25.3630i −0.736576 + 1.08843i
\(544\) 0 0
\(545\) 11.6925 20.2521i 0.500853 0.867503i
\(546\) 0 0
\(547\) −2.18028 3.77635i −0.0932219 0.161465i 0.815643 0.578555i \(-0.196383\pi\)
−0.908865 + 0.417090i \(0.863050\pi\)
\(548\) 0 0
\(549\) −19.8923 + 25.3014i −0.848982 + 1.07984i
\(550\) 0 0
\(551\) −3.87508 6.71183i −0.165084 0.285934i
\(552\) 0 0
\(553\) −2.54392 + 4.40619i −0.108178 + 0.187370i
\(554\) 0 0
\(555\) 16.6059 + 34.1794i 0.704881 + 1.45083i
\(556\) 0 0
\(557\) −9.33947 −0.395726 −0.197863 0.980230i \(-0.563400\pi\)
−0.197863 + 0.980230i \(0.563400\pi\)
\(558\) 0 0
\(559\) −0.260112 −0.0110016
\(560\) 0 0
\(561\) 40.7919 + 2.91500i 1.72223 + 0.123071i
\(562\) 0 0
\(563\) −0.603050 + 1.04451i −0.0254155 + 0.0440210i −0.878453 0.477828i \(-0.841424\pi\)
0.853038 + 0.521849i \(0.174758\pi\)
\(564\) 0 0
\(565\) 6.77255 + 11.7304i 0.284923 + 0.493502i
\(566\) 0 0
\(567\) 2.64563 + 0.776129i 0.111106 + 0.0325944i
\(568\) 0 0
\(569\) −3.83998 6.65104i −0.160980 0.278826i 0.774240 0.632892i \(-0.218132\pi\)
−0.935221 + 0.354066i \(0.884799\pi\)
\(570\) 0 0
\(571\) 4.44038 7.69097i 0.185824 0.321857i −0.758030 0.652220i \(-0.773838\pi\)
0.943854 + 0.330363i \(0.107171\pi\)
\(572\) 0 0
\(573\) −35.6477 2.54740i −1.48920 0.106419i
\(574\) 0 0
\(575\) −3.11888 −0.130066
\(576\) 0 0
\(577\) −33.1358 −1.37946 −0.689732 0.724065i \(-0.742271\pi\)
−0.689732 + 0.724065i \(0.742271\pi\)
\(578\) 0 0
\(579\) 17.5804 + 36.1852i 0.730617 + 1.50381i
\(580\) 0 0
\(581\) 0.894126 1.54867i 0.0370946 0.0642497i
\(582\) 0 0
\(583\) −5.60416 9.70669i −0.232101 0.402010i
\(584\) 0 0
\(585\) −3.75138 + 4.77146i −0.155100 + 0.197276i
\(586\) 0 0
\(587\) −18.0712 31.3002i −0.745877 1.29190i −0.949784 0.312906i \(-0.898698\pi\)
0.203908 0.978990i \(-0.434636\pi\)
\(588\) 0 0
\(589\) 2.09972 3.63682i 0.0865174 0.149852i
\(590\) 0 0
\(591\) 4.64496 6.86383i 0.191068 0.282340i
\(592\) 0 0
\(593\) −34.4076 −1.41295 −0.706476 0.707737i \(-0.749716\pi\)
−0.706476 + 0.707737i \(0.749716\pi\)
\(594\) 0 0
\(595\) −3.09964 −0.127073
\(596\) 0 0
\(597\) 11.2546 16.6308i 0.460619 0.680652i
\(598\) 0 0
\(599\) 21.2939 36.8822i 0.870047 1.50697i 0.00809947 0.999967i \(-0.497422\pi\)
0.861947 0.506998i \(-0.169245\pi\)
\(600\) 0 0
\(601\) −11.6910 20.2495i −0.476888 0.825994i 0.522762 0.852479i \(-0.324902\pi\)
−0.999649 + 0.0264854i \(0.991568\pi\)
\(602\) 0 0
\(603\) −12.0038 1.72440i −0.488833 0.0702230i
\(604\) 0 0
\(605\) −15.2011 26.3291i −0.618012 1.07043i
\(606\) 0 0
\(607\) 12.6852 21.9714i 0.514875 0.891790i −0.484976 0.874527i \(-0.661172\pi\)
0.999851 0.0172622i \(-0.00549499\pi\)
\(608\) 0 0
\(609\) −1.11071 2.28613i −0.0450081 0.0926387i
\(610\) 0 0
\(611\) 3.61543 0.146264
\(612\) 0 0
\(613\) 12.9459 0.522882 0.261441 0.965220i \(-0.415802\pi\)
0.261441 + 0.965220i \(0.415802\pi\)
\(614\) 0 0
\(615\) −28.7012 2.05100i −1.15735 0.0827043i
\(616\) 0 0
\(617\) 6.01514 10.4185i 0.242161 0.419434i −0.719169 0.694835i \(-0.755477\pi\)
0.961329 + 0.275401i \(0.0888106\pi\)
\(618\) 0 0
\(619\) 21.9475 + 38.0142i 0.882144 + 1.52792i 0.848952 + 0.528469i \(0.177234\pi\)
0.0331916 + 0.999449i \(0.489433\pi\)
\(620\) 0 0
\(621\) −28.5283 + 31.3937i −1.14480 + 1.25979i
\(622\) 0 0
\(623\) 0.298040 + 0.516221i 0.0119407 + 0.0206820i
\(624\) 0 0
\(625\) 11.4719 19.8699i 0.458877 0.794797i
\(626\) 0 0
\(627\) −14.0174 1.00169i −0.559803 0.0400037i
\(628\) 0 0
\(629\) 48.0693 1.91665
\(630\) 0 0
\(631\) −25.8646 −1.02965 −0.514827 0.857294i \(-0.672144\pi\)
−0.514827 + 0.857294i \(0.672144\pi\)
\(632\) 0 0
\(633\) 1.34528 + 2.76895i 0.0534700 + 0.110056i
\(634\) 0 0
\(635\) 12.6422 21.8968i 0.501688 0.868950i
\(636\) 0 0
\(637\) 3.25101 + 5.63091i 0.128810 + 0.223105i
\(638\) 0 0
\(639\) −4.00825 10.0086i −0.158564 0.395934i
\(640\) 0 0
\(641\) 3.81826 + 6.61342i 0.150812 + 0.261215i 0.931526 0.363674i \(-0.118478\pi\)
−0.780714 + 0.624888i \(0.785144\pi\)
\(642\) 0 0
\(643\) −21.8623 + 37.8667i −0.862166 + 1.49332i 0.00766794 + 0.999971i \(0.497559\pi\)
−0.869834 + 0.493345i \(0.835774\pi\)
\(644\) 0 0
\(645\) 0.576338 0.851650i 0.0226933 0.0335337i
\(646\) 0 0
\(647\) 24.1808 0.950647 0.475324 0.879811i \(-0.342331\pi\)
0.475324 + 0.879811i \(0.342331\pi\)
\(648\) 0 0
\(649\) −49.7290 −1.95203
\(650\) 0 0
\(651\) 0.771865 1.14058i 0.0302518 0.0447028i
\(652\) 0 0
\(653\) 3.95033 6.84218i 0.154589 0.267755i −0.778321 0.627867i \(-0.783928\pi\)
0.932909 + 0.360112i \(0.117261\pi\)
\(654\) 0 0
\(655\) 5.67487 + 9.82916i 0.221735 + 0.384057i
\(656\) 0 0
\(657\) 6.06527 + 15.1450i 0.236629 + 0.590862i
\(658\) 0 0
\(659\) 12.9895 + 22.4985i 0.506001 + 0.876419i 0.999976 + 0.00694272i \(0.00220995\pi\)
−0.493975 + 0.869476i \(0.664457\pi\)
\(660\) 0 0
\(661\) −0.254233 + 0.440344i −0.00988850 + 0.0171274i −0.870927 0.491412i \(-0.836481\pi\)
0.861039 + 0.508539i \(0.169814\pi\)
\(662\) 0 0
\(663\) 3.35525 + 6.90600i 0.130307 + 0.268207i
\(664\) 0 0
\(665\) 1.06514 0.0413043
\(666\) 0 0
\(667\) 39.1047 1.51414
\(668\) 0 0
\(669\) −17.3601 1.24056i −0.671182 0.0479629i
\(670\) 0 0
\(671\) 26.8997 46.5917i 1.03845 1.79865i
\(672\) 0 0
\(673\) 11.2425 + 19.4726i 0.433367 + 0.750613i 0.997161 0.0753024i \(-0.0239922\pi\)
−0.563794 + 0.825915i \(0.690659\pi\)
\(674\) 0 0
\(675\) 0.604797 + 1.89078i 0.0232787 + 0.0727763i
\(676\) 0 0
\(677\) −2.34992 4.07018i −0.0903147 0.156430i 0.817329 0.576171i \(-0.195454\pi\)
−0.907644 + 0.419742i \(0.862121\pi\)
\(678\) 0 0
\(679\) 2.16674 3.75291i 0.0831520 0.144023i
\(680\) 0 0
\(681\) −18.2326 1.30291i −0.698674 0.0499275i
\(682\) 0 0
\(683\) −15.4013 −0.589315 −0.294657 0.955603i \(-0.595206\pi\)
−0.294657 + 0.955603i \(0.595206\pi\)
\(684\) 0 0
\(685\) 31.0932 1.18801
\(686\) 0 0
\(687\) 17.2507 + 35.5066i 0.658155 + 1.35466i
\(688\) 0 0
\(689\) 1.05214 1.82237i 0.0400835 0.0694266i
\(690\) 0 0
\(691\) 13.9618 + 24.1825i 0.531131 + 0.919945i 0.999340 + 0.0363275i \(0.0115660\pi\)
−0.468209 + 0.883618i \(0.655101\pi\)
\(692\) 0 0
\(693\) −4.56191 0.655339i −0.173293 0.0248943i
\(694\) 0 0
\(695\) −23.1828 40.1538i −0.879374 1.52312i
\(696\) 0 0
\(697\) −18.1996 + 31.5227i −0.689360 + 1.19401i
\(698\) 0 0
\(699\) −5.79637 + 8.56525i −0.219239 + 0.323968i
\(700\) 0 0
\(701\) −7.33870 −0.277179 −0.138589 0.990350i \(-0.544257\pi\)
−0.138589 + 0.990350i \(0.544257\pi\)
\(702\) 0 0
\(703\) −16.5182 −0.622996
\(704\) 0 0
\(705\) −8.01080 + 11.8375i −0.301704 + 0.445826i
\(706\) 0 0
\(707\) 2.88356 4.99448i 0.108448 0.187837i
\(708\) 0 0
\(709\) −7.31665 12.6728i −0.274783 0.475938i 0.695298 0.718722i \(-0.255273\pi\)
−0.970080 + 0.242784i \(0.921939\pi\)
\(710\) 0 0
\(711\) −30.7945 + 39.1682i −1.15488 + 1.46892i
\(712\) 0 0
\(713\) 10.5945 + 18.3502i 0.396766 + 0.687219i
\(714\) 0 0
\(715\) 5.07287 8.78647i 0.189715 0.328595i
\(716\) 0 0
\(717\) 8.74299 + 17.9954i 0.326513 + 0.672051i
\(718\) 0 0
\(719\) −28.7125 −1.07080 −0.535398 0.844600i \(-0.679838\pi\)
−0.535398 + 0.844600i \(0.679838\pi\)
\(720\) 0 0
\(721\) −1.80754 −0.0673161
\(722\) 0 0
\(723\) −26.8071 1.91564i −0.996966 0.0712435i
\(724\) 0 0
\(725\) 0.915011 1.58484i 0.0339826 0.0588597i
\(726\) 0 0
\(727\) 12.8923 + 22.3301i 0.478149 + 0.828178i 0.999686 0.0250502i \(-0.00797457\pi\)
−0.521537 + 0.853229i \(0.674641\pi\)
\(728\) 0 0
\(729\) 24.5641 + 11.2073i 0.909783 + 0.415083i
\(730\) 0 0
\(731\) −0.650415 1.12655i −0.0240565 0.0416670i
\(732\) 0 0
\(733\) 5.34424 9.25649i 0.197394 0.341896i −0.750289 0.661110i \(-0.770086\pi\)
0.947683 + 0.319214i \(0.103419\pi\)
\(734\) 0 0
\(735\) −25.6399 1.83223i −0.945740 0.0675829i
\(736\) 0 0
\(737\) 20.2713 0.746702
\(738\) 0 0
\(739\) 19.5440 0.718937 0.359469 0.933157i \(-0.382958\pi\)
0.359469 + 0.933157i \(0.382958\pi\)
\(740\) 0 0
\(741\) −1.15297 2.37313i −0.0423556 0.0871792i
\(742\) 0 0
\(743\) 0.396827 0.687325i 0.0145582 0.0252155i −0.858655 0.512555i \(-0.828699\pi\)
0.873213 + 0.487339i \(0.162032\pi\)
\(744\) 0 0
\(745\) −16.7719 29.0497i −0.614474 1.06430i
\(746\) 0 0
\(747\) 10.8235 13.7667i 0.396012 0.503696i
\(748\) 0 0
\(749\) −0.591343 1.02424i −0.0216072 0.0374248i
\(750\) 0 0
\(751\) −2.55092 + 4.41832i −0.0930844 + 0.161227i −0.908808 0.417216i \(-0.863006\pi\)
0.815723 + 0.578443i \(0.196339\pi\)
\(752\) 0 0
\(753\) 14.1423 20.8979i 0.515372 0.761562i
\(754\) 0 0
\(755\) −36.8877 −1.34248
\(756\) 0 0
\(757\) −19.8422 −0.721177 −0.360589 0.932725i \(-0.617424\pi\)
−0.360589 + 0.932725i \(0.617424\pi\)
\(758\) 0 0
\(759\) 39.7408 58.7246i 1.44250 2.13157i
\(760\) 0 0
\(761\) −21.3960 + 37.0590i −0.775605 + 1.34339i 0.158849 + 0.987303i \(0.449222\pi\)
−0.934454 + 0.356084i \(0.884112\pi\)
\(762\) 0 0
\(763\) 1.66686 + 2.88708i 0.0603442 + 0.104519i
\(764\) 0 0
\(765\) −30.0457 4.31619i −1.08630 0.156052i
\(766\) 0 0
\(767\) −4.66814 8.08546i −0.168557 0.291949i
\(768\) 0 0
\(769\) 17.8574 30.9300i 0.643955 1.11536i −0.340587 0.940213i \(-0.610626\pi\)
0.984542 0.175150i \(-0.0560410\pi\)
\(770\) 0 0
\(771\) −11.7785 24.2433i −0.424193 0.873103i
\(772\) 0 0
\(773\) −17.1522 −0.616923 −0.308461 0.951237i \(-0.599814\pi\)
−0.308461 + 0.951237i \(0.599814\pi\)
\(774\) 0 0
\(775\) 0.991600 0.0356193
\(776\) 0 0
\(777\) −5.40337 0.386127i −0.193845 0.0138522i
\(778\) 0 0
\(779\) 6.25400 10.8322i 0.224073 0.388105i
\(780\) 0 0
\(781\) 9.01094 + 15.6074i 0.322437 + 0.558477i
\(782\) 0 0
\(783\) −7.58300 23.7068i −0.270994 0.847212i
\(784\) 0 0
\(785\) −5.57128 9.64974i −0.198848 0.344414i
\(786\) 0 0
\(787\) 10.5790 18.3233i 0.377100 0.653156i −0.613539 0.789664i \(-0.710255\pi\)
0.990639 + 0.136509i \(0.0435881\pi\)
\(788\) 0 0
\(789\) −38.7790 2.77116i −1.38057 0.0986559i
\(790\) 0 0
\(791\) −1.93095 −0.0686568
\(792\) 0 0
\(793\) 10.1005 0.358679
\(794\) 0 0
\(795\) 3.63546 + 7.48276i 0.128936 + 0.265386i
\(796\) 0 0
\(797\) −5.45601 + 9.45009i −0.193262 + 0.334739i −0.946329 0.323204i \(-0.895240\pi\)
0.753067 + 0.657943i \(0.228573\pi\)
\(798\) 0 0
\(799\) 9.04044 + 15.6585i 0.319828 + 0.553958i
\(800\) 0 0
\(801\) 2.17016 + 5.41890i 0.0766790 + 0.191467i
\(802\) 0 0
\(803\) −13.6353 23.6171i −0.481180 0.833429i
\(804\) 0 0
\(805\) −2.68717 + 4.65431i −0.0947102 + 0.164043i
\(806\) 0 0
\(807\) −25.2641 + 37.3325i −0.889337 + 1.31417i
\(808\) 0 0
\(809\) 14.6688 0.515729 0.257865 0.966181i \(-0.416981\pi\)
0.257865 + 0.966181i \(0.416981\pi\)
\(810\) 0 0
\(811\) −7.96618 −0.279731 −0.139865 0.990171i \(-0.544667\pi\)
−0.139865 + 0.990171i \(0.544667\pi\)
\(812\) 0 0
\(813\) −5.43383 + 8.02952i −0.190573 + 0.281608i
\(814\) 0 0
\(815\) 19.2282 33.3042i 0.673535 1.16660i
\(816\) 0 0
\(817\) 0.223504 + 0.387121i 0.00781943 + 0.0135436i
\(818\) 0 0
\(819\) −0.321682 0.803241i −0.0112405 0.0280675i
\(820\) 0 0
\(821\) −15.9260 27.5847i −0.555822 0.962712i −0.997839 0.0657057i \(-0.979070\pi\)
0.442017 0.897007i \(-0.354263\pi\)
\(822\) 0 0
\(823\) −5.06901 + 8.77978i −0.176694 + 0.306044i −0.940746 0.339111i \(-0.889874\pi\)
0.764052 + 0.645155i \(0.223207\pi\)
\(824\) 0 0
\(825\) −1.45011 2.98472i −0.0504864 0.103915i
\(826\) 0 0
\(827\) 16.2236 0.564148 0.282074 0.959393i \(-0.408978\pi\)
0.282074 + 0.959393i \(0.408978\pi\)
\(828\) 0 0
\(829\) 21.2902 0.739439 0.369720 0.929143i \(-0.379454\pi\)
0.369720 + 0.929143i \(0.379454\pi\)
\(830\) 0 0
\(831\) −5.45786 0.390020i −0.189331 0.0135297i
\(832\) 0 0
\(833\) −16.2584 + 28.1604i −0.563320 + 0.975699i
\(834\) 0 0
\(835\) −11.7916 20.4236i −0.408063 0.706787i
\(836\) 0 0
\(837\) 9.07015 9.98116i 0.313510 0.344999i
\(838\) 0 0
\(839\) −0.962971 1.66791i −0.0332454 0.0575828i 0.848924 0.528515i \(-0.177251\pi\)
−0.882169 + 0.470932i \(0.843918\pi\)
\(840\) 0 0
\(841\) 3.02752 5.24382i 0.104397 0.180821i
\(842\) 0 0
\(843\) 27.7125 + 1.98035i 0.954469 + 0.0682067i
\(844\) 0 0
\(845\) −26.0315 −0.895510
\(846\) 0 0
\(847\) 4.33405 0.148920
\(848\) 0 0
\(849\) 2.83298 + 5.83103i 0.0972275 + 0.200120i
\(850\) 0 0
\(851\) 41.6727 72.1792i 1.42852 2.47427i
\(852\) 0 0
\(853\) 19.7403 + 34.1912i 0.675895 + 1.17069i 0.976206 + 0.216844i \(0.0695763\pi\)
−0.300311 + 0.953841i \(0.597090\pi\)
\(854\) 0 0
\(855\) 10.3247 + 1.48319i 0.353097 + 0.0507240i
\(856\) 0 0
\(857\) 15.0539 + 26.0742i 0.514233 + 0.890677i 0.999864 + 0.0165134i \(0.00525662\pi\)
−0.485631 + 0.874164i \(0.661410\pi\)
\(858\) 0 0
\(859\) −20.2994 + 35.1597i −0.692608 + 1.19963i 0.278373 + 0.960473i \(0.410205\pi\)
−0.970980 + 0.239159i \(0.923128\pi\)
\(860\) 0 0
\(861\) 2.29900 3.39721i 0.0783496 0.115777i
\(862\) 0 0
\(863\) −1.46113 −0.0497374 −0.0248687 0.999691i \(-0.507917\pi\)
−0.0248687 + 0.999691i \(0.507917\pi\)
\(864\) 0 0
\(865\) −37.0970 −1.26134
\(866\) 0 0
\(867\) −5.01761 + 7.41449i −0.170407 + 0.251809i
\(868\) 0 0
\(869\) 41.6424 72.1268i 1.41262 2.44673i
\(870\) 0 0
\(871\) 1.90290 + 3.29591i 0.0644772 + 0.111678i
\(872\) 0 0
\(873\) 26.2288 33.3609i 0.887709 1.12910i
\(874\) 0 0
\(875\) 1.77156 + 3.06843i 0.0598897 + 0.103732i
\(876\) 0 0
\(877\) 5.07742 8.79435i 0.171452 0.296964i −0.767475 0.641078i \(-0.778487\pi\)
0.938928 + 0.344114i \(0.111821\pi\)
\(878\) 0 0
\(879\) −8.32423 17.1335i −0.280769 0.577899i
\(880\) 0 0
\(881\) 12.6952 0.427711 0.213855 0.976865i \(-0.431398\pi\)
0.213855 + 0.976865i \(0.431398\pi\)
\(882\) 0 0
\(883\) 35.3754 1.19048 0.595238 0.803549i \(-0.297058\pi\)
0.595238 + 0.803549i \(0.297058\pi\)
\(884\) 0 0
\(885\) 36.8165 + 2.63092i 1.23757 + 0.0884373i
\(886\) 0 0
\(887\) −27.7610 + 48.0835i −0.932124 + 1.61449i −0.152439 + 0.988313i \(0.548713\pi\)
−0.779685 + 0.626172i \(0.784621\pi\)
\(888\) 0 0
\(889\) 1.80223 + 3.12155i 0.0604448 + 0.104694i
\(890\) 0 0
\(891\) −43.3075 12.7048i −1.45085 0.425626i
\(892\) 0 0
\(893\) −3.10660 5.38078i −0.103958 0.180061i
\(894\) 0 0
\(895\) 16.1784 28.0218i 0.540784 0.936666i
\(896\) 0 0
\(897\) 13.2786 + 0.948892i 0.443359 + 0.0316826i
\(898\) 0 0
\(899\) −12.4328 −0.414656
\(900\) 0 0
\(901\) 10.5236 0.350592
\(902\) 0 0
\(903\) 0.0640626 + 0.131858i 0.00213187 + 0.00438796i
\(904\) 0 0
\(905\) −18.9981 + 32.9056i −0.631517 + 1.09382i
\(906\) 0 0
\(907\) −17.1123 29.6393i −0.568204 0.984158i −0.996744 0.0806350i \(-0.974305\pi\)
0.428540 0.903523i \(-0.359028\pi\)
\(908\) 0 0
\(909\) 34.9060 44.3976i 1.15776 1.47258i
\(910\) 0 0
\(911\) 8.27615 + 14.3347i 0.274201 + 0.474930i 0.969933 0.243371i \(-0.0782533\pi\)
−0.695732 + 0.718301i \(0.744920\pi\)
\(912\) 0 0
\(913\) −14.6363 + 25.3509i −0.484392 + 0.838991i
\(914\) 0 0
\(915\) −22.3799 + 33.0706i −0.739857 + 1.09328i
\(916\) 0 0
\(917\) −1.61799 −0.0534306
\(918\) 0 0
\(919\) 45.3120 1.49470 0.747352 0.664428i \(-0.231325\pi\)
0.747352 + 0.664428i \(0.231325\pi\)
\(920\) 0 0
\(921\) −5.90614 + 8.72745i −0.194614 + 0.287579i
\(922\) 0 0
\(923\) −1.69174 + 2.93019i −0.0556844 + 0.0964482i
\(924\) 0 0
\(925\) −1.95020 3.37784i −0.0641221 0.111063i
\(926\) 0 0
\(927\) −17.5210 2.51696i −0.575465 0.0826680i
\(928\) 0 0
\(929\) −10.0403 17.3903i −0.329412 0.570558i 0.652984 0.757372i \(-0.273517\pi\)
−0.982395 + 0.186814i \(0.940184\pi\)
\(930\) 0 0
\(931\) 5.58693 9.67684i 0.183104 0.317146i
\(932\) 0 0
\(933\) −5.68637 11.7041i −0.186163 0.383174i
\(934\) 0 0
\(935\) 50.7392 1.65935
\(936\) 0 0
\(937\) 57.3842 1.87466 0.937330 0.348443i \(-0.113289\pi\)
0.937330 + 0.348443i \(0.113289\pi\)
\(938\) 0 0
\(939\) −21.3563 1.52613i −0.696937 0.0498034i
\(940\) 0 0
\(941\) 16.7615 29.0318i 0.546409 0.946409i −0.452107 0.891963i \(-0.649328\pi\)
0.998517 0.0544452i \(-0.0173390\pi\)
\(942\) 0 0
\(943\) 31.5556 + 54.6559i 1.02759 + 1.77984i
\(944\) 0 0
\(945\) 3.34270 + 0.726523i 0.108738 + 0.0236338i
\(946\) 0 0
\(947\) −3.66775 6.35274i −0.119186 0.206436i 0.800259 0.599654i \(-0.204695\pi\)
−0.919445 + 0.393218i \(0.871362\pi\)
\(948\) 0 0
\(949\) 2.55994 4.43395i 0.0830992 0.143932i
\(950\) 0 0
\(951\) 42.5704 + 3.04210i 1.38044 + 0.0986467i
\(952\) 0 0
\(953\) −7.30761 −0.236717 −0.118358 0.992971i \(-0.537763\pi\)
−0.118358 + 0.992971i \(0.537763\pi\)
\(954\) 0 0
\(955\) −44.3406 −1.43483
\(956\) 0 0
\(957\) 18.1816 + 37.4226i 0.587728 + 1.20970i
\(958\) 0 0
\(959\) −2.21628 + 3.83870i −0.0715673 + 0.123958i
\(960\) 0 0
\(961\) 12.1316 + 21.0126i 0.391343 + 0.677827i
\(962\) 0 0
\(963\) −4.30583 10.7517i −0.138754 0.346468i
\(964\) 0 0
\(965\) 24.9565 + 43.2259i 0.803377 + 1.39149i
\(966\) 0 0
\(967\) 13.1258 22.7346i 0.422097 0.731094i −0.574047 0.818822i \(-0.694627\pi\)
0.996144 + 0.0877283i \(0.0279607\pi\)
\(968\) 0 0
\(969\) 7.39506 10.9276i 0.237564 0.351046i
\(970\) 0 0
\(971\) −15.9643 −0.512318 −0.256159 0.966635i \(-0.582457\pi\)
−0.256159 + 0.966635i \(0.582457\pi\)
\(972\) 0 0
\(973\) 6.60975 0.211899
\(974\) 0 0
\(975\) 0.349162 0.515954i 0.0111821 0.0165238i
\(976\) 0 0
\(977\) 3.96119 6.86099i 0.126730 0.219502i −0.795678 0.605720i \(-0.792885\pi\)
0.922408 + 0.386217i \(0.126219\pi\)
\(978\) 0 0
\(979\) −4.87875 8.45024i −0.155925 0.270071i
\(980\) 0 0
\(981\) 12.1371 + 30.3064i 0.387508 + 0.967608i
\(982\) 0 0
\(983\) −6.86126 11.8841i −0.218840 0.379042i 0.735613 0.677402i \(-0.236894\pi\)
−0.954454 + 0.298359i \(0.903561\pi\)
\(984\) 0 0
\(985\) 5.14131 8.90502i 0.163816 0.283737i
\(986\) 0 0
\(987\) −0.890437 1.83276i −0.0283429 0.0583374i
\(988\) 0 0
\(989\) −2.25546 −0.0717194
\(990\) 0 0
\(991\) 36.0366 1.14474 0.572370 0.819996i \(-0.306024\pi\)
0.572370 + 0.819996i \(0.306024\pi\)
\(992\) 0 0
\(993\) 38.2484 + 2.73324i 1.21378 + 0.0867369i
\(994\) 0 0
\(995\) 12.4572 21.5765i 0.394920 0.684021i
\(996\) 0 0
\(997\) 11.9518 + 20.7011i 0.378517 + 0.655610i 0.990847 0.134992i \(-0.0431009\pi\)
−0.612330 + 0.790602i \(0.709768\pi\)
\(998\) 0 0
\(999\) −51.8388 11.2669i −1.64011 0.356470i
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 1152.2.i.h.769.2 yes 10
3.2 odd 2 3456.2.i.h.2305.4 10
4.3 odd 2 1152.2.i.e.769.4 yes 10
8.3 odd 2 1152.2.i.g.769.2 yes 10
8.5 even 2 1152.2.i.f.769.4 yes 10
9.2 odd 6 3456.2.i.h.1153.4 10
9.7 even 3 inner 1152.2.i.h.385.2 yes 10
12.11 even 2 3456.2.i.e.2305.4 10
24.5 odd 2 3456.2.i.g.2305.2 10
24.11 even 2 3456.2.i.f.2305.2 10
36.7 odd 6 1152.2.i.e.385.4 10
36.11 even 6 3456.2.i.e.1153.4 10
72.11 even 6 3456.2.i.f.1153.2 10
72.29 odd 6 3456.2.i.g.1153.2 10
72.43 odd 6 1152.2.i.g.385.2 yes 10
72.61 even 6 1152.2.i.f.385.4 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.i.e.385.4 10 36.7 odd 6
1152.2.i.e.769.4 yes 10 4.3 odd 2
1152.2.i.f.385.4 yes 10 72.61 even 6
1152.2.i.f.769.4 yes 10 8.5 even 2
1152.2.i.g.385.2 yes 10 72.43 odd 6
1152.2.i.g.769.2 yes 10 8.3 odd 2
1152.2.i.h.385.2 yes 10 9.7 even 3 inner
1152.2.i.h.769.2 yes 10 1.1 even 1 trivial
3456.2.i.e.1153.4 10 36.11 even 6
3456.2.i.e.2305.4 10 12.11 even 2
3456.2.i.f.1153.2 10 72.11 even 6
3456.2.i.f.2305.2 10 24.11 even 2
3456.2.i.g.1153.2 10 72.29 odd 6
3456.2.i.g.2305.2 10 24.5 odd 2
3456.2.i.h.1153.4 10 9.2 odd 6
3456.2.i.h.2305.4 10 3.2 odd 2