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Properties

Label 81.2.g.a.13.1
Level $81$
Weight $2$
Character 81.13
Analytic conductor $0.647$
Analytic rank $0$
Dimension $144$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,2,Mod(4,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 81.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(8\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 81.13
Dual form 81.2.g.a.25.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.38576 - 1.19817i) q^{2} +(1.71560 - 0.238151i) q^{3} +(3.06192 + 4.11287i) q^{4} +(0.727126 + 2.42877i) q^{5} +(-4.37836 - 1.48742i) q^{6} +(0.202369 + 0.469143i) q^{7} +(-1.44988 - 8.22270i) q^{8} +(2.88657 - 0.817145i) q^{9} +(1.17534 - 6.66569i) q^{10} +(1.49541 - 0.354419i) q^{11} +(6.23252 + 6.32684i) q^{12} +(-4.71325 - 3.09996i) q^{13} +(0.0793122 - 1.36174i) q^{14} +(1.82587 + 3.99363i) q^{15} +(-3.45199 + 11.5305i) q^{16} +(-1.33086 + 0.484393i) q^{17} +(-7.86575 - 1.50910i) q^{18} +(0.986977 + 0.359230i) q^{19} +(-7.76282 + 10.4273i) q^{20} +(0.458911 + 0.756668i) q^{21} +(-3.99235 - 0.946204i) q^{22} +(0.103125 - 0.239071i) q^{23} +(-4.44567 - 13.7616i) q^{24} +(-1.19278 + 0.784502i) q^{25} +(7.53041 + 13.0431i) q^{26} +(4.75759 - 2.08933i) q^{27} +(-1.30989 + 2.26880i) q^{28} +(-0.103412 - 1.77552i) q^{29} +(0.428972 - 11.7156i) q^{30} +(-5.13181 + 0.599822i) q^{31} +(10.5915 - 11.2264i) q^{32} +(2.48112 - 0.964174i) q^{33} +(3.75550 + 0.438955i) q^{34} +(-0.992294 + 0.832634i) q^{35} +(12.1993 + 9.37005i) q^{36} +(-5.04999 - 4.23745i) q^{37} +(-1.92427 - 2.03961i) q^{38} +(-8.82432 - 4.19582i) q^{39} +(18.9168 - 9.50037i) q^{40} +(-6.03866 + 3.03273i) q^{41} +(-0.188232 - 2.35509i) q^{42} +(4.19282 + 4.44412i) q^{43} +(6.03650 + 5.06523i) q^{44} +(4.08356 + 6.41665i) q^{45} +(-0.532481 + 0.446804i) q^{46} +(-7.26206 - 0.848813i) q^{47} +(-3.17624 + 20.6038i) q^{48} +(4.62455 - 4.90174i) q^{49} +(3.78565 - 0.442479i) q^{50} +(-2.16786 + 1.14797i) q^{51} +(-1.68188 - 28.8768i) q^{52} +(4.74440 - 8.21755i) q^{53} +(-13.8539 - 0.715772i) q^{54} +(1.94815 + 3.37430i) q^{55} +(3.56421 - 2.34422i) q^{56} +(1.77881 + 0.381246i) q^{57} +(-1.88066 + 4.35987i) q^{58} +(-4.82144 - 1.14270i) q^{59} +(-10.8346 + 19.7378i) q^{60} +(-7.70381 + 10.3480i) q^{61} +(12.9620 + 4.71777i) q^{62} +(0.967509 + 1.18885i) q^{63} +(-16.0995 + 5.85975i) q^{64} +(4.10195 - 13.7015i) q^{65} +(-7.07461 - 0.672524i) q^{66} +(-0.373231 + 6.40814i) q^{67} +(-6.06723 - 3.99048i) q^{68} +(0.119986 - 0.434709i) q^{69} +(3.36502 - 0.797524i) q^{70} +(-0.896716 + 5.08553i) q^{71} +(-10.9043 - 22.5506i) q^{72} +(-1.03528 - 5.87137i) q^{73} +(6.97088 + 16.1603i) q^{74} +(-1.85950 + 1.62995i) q^{75} +(1.54458 + 5.15924i) q^{76} +(0.468897 + 0.629838i) q^{77} +(16.0254 + 20.5833i) q^{78} +(8.88455 + 4.46199i) q^{79} -30.5149 q^{80} +(7.66455 - 4.71749i) q^{81} +18.0405 q^{82} +(11.5744 + 5.81289i) q^{83} +(-1.70693 + 4.20430i) q^{84} +(-2.14418 - 2.88014i) q^{85} +(-4.67822 - 15.6263i) q^{86} +(-0.600256 - 3.02145i) q^{87} +(-5.08245 - 11.7824i) q^{88} +(-1.71260 - 9.71264i) q^{89} +(-2.05414 - 20.2014i) q^{90} +(0.500509 - 2.83853i) q^{91} +(1.29903 - 0.307875i) q^{92} +(-8.66128 + 2.25120i) q^{93} +(16.3085 + 10.7263i) q^{94} +(-0.154831 + 2.65835i) q^{95} +(15.4972 - 21.7823i) q^{96} +(1.03163 - 3.44589i) q^{97} +(-16.9062 + 6.15336i) q^{98} +(4.02699 - 2.24502i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9} - 18 q^{10} - 18 q^{11} - 18 q^{12} - 18 q^{13} - 18 q^{14} - 18 q^{15} - 18 q^{16} - 18 q^{17} - 9 q^{18} - 18 q^{19}+ \cdots + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{27}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38576 − 1.19817i −1.68699 − 0.847237i −0.991710 − 0.128497i \(-0.958985\pi\)
−0.695278 − 0.718741i \(-0.744719\pi\)
\(3\) 1.71560 − 0.238151i 0.990502 − 0.137497i
\(4\) 3.06192 + 4.11287i 1.53096 + 2.05644i
\(5\) 0.727126 + 2.42877i 0.325181 + 1.08618i 0.951766 + 0.306826i \(0.0992670\pi\)
−0.626585 + 0.779353i \(0.715548\pi\)
\(6\) −4.37836 − 1.48742i −1.78746 − 0.607235i
\(7\) 0.202369 + 0.469143i 0.0764882 + 0.177320i 0.952170 − 0.305569i \(-0.0988468\pi\)
−0.875682 + 0.482889i \(0.839588\pi\)
\(8\) −1.44988 − 8.22270i −0.512611 − 2.90716i
\(9\) 2.88657 − 0.817145i 0.962189 − 0.272382i
\(10\) 1.17534 − 6.66569i 0.371676 − 2.10788i
\(11\) 1.49541 − 0.354419i 0.450883 − 0.106861i 0.00110057 − 0.999999i \(-0.499650\pi\)
0.449782 + 0.893138i \(0.351502\pi\)
\(12\) 6.23252 + 6.32684i 1.79917 + 1.82640i
\(13\) −4.71325 − 3.09996i −1.30722 − 0.859773i −0.311327 − 0.950303i \(-0.600774\pi\)
−0.995894 + 0.0905298i \(0.971144\pi\)
\(14\) 0.0793122 − 1.36174i 0.0211971 − 0.363940i
\(15\) 1.82587 + 3.99363i 0.471438 + 1.03115i
\(16\) −3.45199 + 11.5305i −0.862998 + 2.88261i
\(17\) −1.33086 + 0.484393i −0.322781 + 0.117483i −0.498328 − 0.866988i \(-0.666053\pi\)
0.175548 + 0.984471i \(0.443830\pi\)
\(18\) −7.86575 − 1.50910i −1.85397 − 0.355698i
\(19\) 0.986977 + 0.359230i 0.226428 + 0.0824131i 0.452743 − 0.891641i \(-0.350446\pi\)
−0.226315 + 0.974054i \(0.572668\pi\)
\(20\) −7.76282 + 10.4273i −1.73582 + 2.33161i
\(21\) 0.458911 + 0.756668i 0.100143 + 0.165119i
\(22\) −3.99235 − 0.946204i −0.851171 − 0.201731i
\(23\) 0.103125 − 0.239071i 0.0215031 − 0.0498497i −0.907126 − 0.420859i \(-0.861729\pi\)
0.928629 + 0.371009i \(0.120988\pi\)
\(24\) −4.44567 − 13.7616i −0.907468 − 2.80907i
\(25\) −1.19278 + 0.784502i −0.238555 + 0.156900i
\(26\) 7.53041 + 13.0431i 1.47684 + 2.55795i
\(27\) 4.75759 − 2.08933i 0.915599 − 0.402093i
\(28\) −1.30989 + 2.26880i −0.247546 + 0.428762i
\(29\) −0.103412 − 1.77552i −0.0192031 − 0.329705i −0.994204 − 0.107506i \(-0.965713\pi\)
0.975001 − 0.222199i \(-0.0713235\pi\)
\(30\) 0.428972 − 11.7156i 0.0783193 − 2.13896i
\(31\) −5.13181 + 0.599822i −0.921700 + 0.107731i −0.563690 − 0.825986i \(-0.690619\pi\)
−0.358010 + 0.933718i \(0.616545\pi\)
\(32\) 10.5915 − 11.2264i 1.87233 − 1.98456i
\(33\) 2.48112 − 0.964174i 0.431908 − 0.167841i
\(34\) 3.75550 + 0.438955i 0.644063 + 0.0752802i
\(35\) −0.992294 + 0.832634i −0.167728 + 0.140741i
\(36\) 12.1993 + 9.37005i 2.03321 + 1.56168i
\(37\) −5.04999 − 4.23745i −0.830214 − 0.696632i 0.125126 − 0.992141i \(-0.460066\pi\)
−0.955340 + 0.295509i \(0.904511\pi\)
\(38\) −1.92427 − 2.03961i −0.312158 − 0.330868i
\(39\) −8.82432 − 4.19582i −1.41302 − 0.671868i
\(40\) 18.9168 − 9.50037i 2.99101 − 1.50214i
\(41\) −6.03866 + 3.03273i −0.943080 + 0.473633i −0.852773 − 0.522281i \(-0.825081\pi\)
−0.0903071 + 0.995914i \(0.528785\pi\)
\(42\) −0.188232 − 2.35509i −0.0290448 − 0.363398i
\(43\) 4.19282 + 4.44412i 0.639398 + 0.677723i 0.963318 − 0.268362i \(-0.0864823\pi\)
−0.323920 + 0.946085i \(0.605001\pi\)
\(44\) 6.03650 + 5.06523i 0.910037 + 0.763612i
\(45\) 4.08356 + 6.41665i 0.608741 + 0.956537i
\(46\) −0.532481 + 0.446804i −0.0785100 + 0.0658777i
\(47\) −7.26206 − 0.848813i −1.05928 − 0.123812i −0.431430 − 0.902146i \(-0.641991\pi\)
−0.627850 + 0.778334i \(0.716065\pi\)
\(48\) −3.17624 + 20.6038i −0.458451 + 2.97390i
\(49\) 4.62455 − 4.90174i 0.660650 − 0.700248i
\(50\) 3.78565 − 0.442479i 0.535372 − 0.0625760i
\(51\) −2.16786 + 1.14797i −0.303562 + 0.160748i
\(52\) −1.68188 − 28.8768i −0.233235 − 4.00449i
\(53\) 4.74440 − 8.21755i 0.651694 − 1.12877i −0.331018 − 0.943624i \(-0.607392\pi\)
0.982712 − 0.185142i \(-0.0592746\pi\)
\(54\) −13.8539 − 0.715772i −1.88527 − 0.0974043i
\(55\) 1.94815 + 3.37430i 0.262689 + 0.454991i
\(56\) 3.56421 − 2.34422i 0.476288 − 0.313259i
\(57\) 1.77881 + 0.381246i 0.235609 + 0.0504972i
\(58\) −1.88066 + 4.35987i −0.246943 + 0.572479i
\(59\) −4.82144 − 1.14270i −0.627698 − 0.148767i −0.0955586 − 0.995424i \(-0.530464\pi\)
−0.532140 + 0.846657i \(0.678612\pi\)
\(60\) −10.8346 + 19.7378i −1.39874 + 2.54813i
\(61\) −7.70381 + 10.3480i −0.986372 + 1.32493i −0.0409219 + 0.999162i \(0.513029\pi\)
−0.945451 + 0.325766i \(0.894378\pi\)
\(62\) 12.9620 + 4.71777i 1.64617 + 0.599157i
\(63\) 0.967509 + 1.18885i 0.121895 + 0.149781i
\(64\) −16.0995 + 5.85975i −2.01244 + 0.732468i
\(65\) 4.10195 − 13.7015i 0.508784 − 1.69946i
\(66\) −7.07461 − 0.672524i −0.870824 − 0.0827820i
\(67\) −0.373231 + 6.40814i −0.0455975 + 0.782878i 0.895566 + 0.444928i \(0.146771\pi\)
−0.941164 + 0.337951i \(0.890266\pi\)
\(68\) −6.06723 − 3.99048i −0.735760 − 0.483917i
\(69\) 0.119986 − 0.434709i 0.0144447 − 0.0523329i
\(70\) 3.36502 − 0.797524i 0.402197 − 0.0953223i
\(71\) −0.896716 + 5.08553i −0.106421 + 0.603541i 0.884223 + 0.467065i \(0.154689\pi\)
−0.990643 + 0.136476i \(0.956422\pi\)
\(72\) −10.9043 − 22.5506i −1.28509 − 2.65761i
\(73\) −1.03528 − 5.87137i −0.121171 − 0.687192i −0.983509 − 0.180860i \(-0.942112\pi\)
0.862338 − 0.506332i \(-0.168999\pi\)
\(74\) 6.97088 + 16.1603i 0.810348 + 1.87860i
\(75\) −1.85950 + 1.62995i −0.214716 + 0.188211i
\(76\) 1.54458 + 5.15924i 0.177175 + 0.591806i
\(77\) 0.468897 + 0.629838i 0.0534358 + 0.0717767i
\(78\) 16.0254 + 20.5833i 1.81452 + 2.33060i
\(79\) 8.88455 + 4.46199i 0.999590 + 0.502013i 0.871838 − 0.489793i \(-0.162928\pi\)
0.127751 + 0.991806i \(0.459224\pi\)
\(80\) −30.5149 −3.41167
\(81\) 7.66455 − 4.71749i 0.851616 − 0.524165i
\(82\) 18.0405 1.99224
\(83\) 11.5744 + 5.81289i 1.27046 + 0.638048i 0.950894 − 0.309517i \(-0.100167\pi\)
0.319564 + 0.947565i \(0.396464\pi\)
\(84\) −1.70693 + 4.20430i −0.186241 + 0.458727i
\(85\) −2.14418 − 2.88014i −0.232569 − 0.312395i
\(86\) −4.67822 − 15.6263i −0.504466 − 1.68503i
\(87\) −0.600256 − 3.02145i −0.0643542 − 0.323933i
\(88\) −5.08245 − 11.7824i −0.541790 − 1.25601i
\(89\) −1.71260 − 9.71264i −0.181535 − 1.02954i −0.930327 − 0.366732i \(-0.880477\pi\)
0.748791 − 0.662806i \(-0.230634\pi\)
\(90\) −2.05414 − 20.2014i −0.216525 − 2.12941i
\(91\) 0.500509 − 2.83853i 0.0524676 − 0.297558i
\(92\) 1.29903 − 0.307875i 0.135433 − 0.0320982i
\(93\) −8.66128 + 2.25120i −0.898133 + 0.233439i
\(94\) 16.3085 + 10.7263i 1.68209 + 1.10633i
\(95\) −0.154831 + 2.65835i −0.0158853 + 0.272741i
\(96\) 15.4972 − 21.7823i 1.58168 − 2.22315i
\(97\) 1.03163 − 3.44589i 0.104746 − 0.349877i −0.889508 − 0.456919i \(-0.848953\pi\)
0.994254 + 0.107042i \(0.0341380\pi\)
\(98\) −16.9062 + 6.15336i −1.70778 + 0.621583i
\(99\) 4.02699 − 2.24502i 0.404728 − 0.225633i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 81.2.g.a.13.1 ✓ 144
3.2 odd 2 243.2.g.a.10.8 144
9.2 odd 6 729.2.g.a.514.1 144
9.4 even 3 729.2.g.c.28.8 144
9.5 odd 6 729.2.g.b.28.1 144
9.7 even 3 729.2.g.d.514.8 144
81.2 odd 54 729.2.g.b.703.1 144
81.5 odd 54 6561.2.a.d.1.71 72
81.25 even 27 inner 81.2.g.a.25.1 yes 144
81.29 odd 54 729.2.g.a.217.1 144
81.52 even 27 729.2.g.d.217.8 144
81.56 odd 54 243.2.g.a.73.8 144
81.76 even 27 6561.2.a.c.1.2 72
81.79 even 27 729.2.g.c.703.8 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.g.a.13.1 ✓ 144 1.1 even 1 trivial
81.2.g.a.25.1 yes 144 81.25 even 27 inner
243.2.g.a.10.8 144 3.2 odd 2
243.2.g.a.73.8 144 81.56 odd 54
729.2.g.a.217.1 144 81.29 odd 54
729.2.g.a.514.1 144 9.2 odd 6
729.2.g.b.28.1 144 9.5 odd 6
729.2.g.b.703.1 144 81.2 odd 54
729.2.g.c.28.8 144 9.4 even 3
729.2.g.c.703.8 144 81.79 even 27
729.2.g.d.217.8 144 81.52 even 27
729.2.g.d.514.8 144 9.7 even 3
6561.2.a.c.1.2 72 81.76 even 27
6561.2.a.d.1.71 72 81.5 odd 54