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Properties

Label 729.2.c.a.487.4
Level $729$
Weight $2$
Character 729.487
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
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] = [729,2,Mod(244,729)] 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("729.244"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-3,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 487.4
Root \(1.37340i\) of defining polynomial
Character \(\chi\) \(=\) 729.487
Dual form 729.2.c.a.244.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0864880 - 0.149802i) q^{2} +(0.985040 + 1.70614i) q^{4} +(-1.86828 - 3.23596i) q^{5} +(-1.51575 + 2.62535i) q^{7} +0.686728 q^{8} -0.646335 q^{10} +(-1.24585 + 2.15787i) q^{11} +(0.382569 + 0.662630i) q^{13} +(0.262188 + 0.454123i) q^{14} +(-1.91069 + 3.30940i) q^{16} -4.62278 q^{17} -0.611844 q^{19} +(3.68066 - 6.37509i) q^{20} +(0.215502 + 0.373260i) q^{22} +(3.26219 + 5.65028i) q^{23} +(-4.48095 + 7.76123i) q^{25} +0.132351 q^{26} -5.97229 q^{28} +(-3.27545 + 5.67324i) q^{29} +(-3.27521 - 5.67283i) q^{31} +(1.01723 + 1.76190i) q^{32} +(-0.399815 + 0.692500i) q^{34} +11.3274 q^{35} +4.95969 q^{37} +(-0.0529171 + 0.0916552i) q^{38} +(-1.28300 - 2.22222i) q^{40} +(2.63012 + 4.55550i) q^{41} +(-2.78529 + 4.82426i) q^{43} -4.90884 q^{44} +1.12856 q^{46} +(0.553808 - 0.959223i) q^{47} +(-1.09499 - 1.89658i) q^{49} +(0.775096 + 1.34251i) q^{50} +(-0.753692 + 1.30543i) q^{52} -8.84310 q^{53} +9.31038 q^{55} +(-1.04091 + 1.80290i) q^{56} +(0.566573 + 0.981334i) q^{58} +(-5.92588 - 10.2639i) q^{59} +(-4.09350 + 7.09015i) q^{61} -1.13307 q^{62} -7.29083 q^{64} +(1.42949 - 2.47596i) q^{65} +(0.606169 + 1.04992i) q^{67} +(-4.55362 - 7.88711i) q^{68} +(0.979682 - 1.69686i) q^{70} +4.91946 q^{71} +4.29945 q^{73} +(0.428953 - 0.742969i) q^{74} +(-0.602691 - 1.04389i) q^{76} +(-3.77679 - 6.54158i) q^{77} +(5.89730 - 10.2144i) q^{79} +14.2788 q^{80} +0.909895 q^{82} +(4.50804 - 7.80815i) q^{83} +(8.63666 + 14.9591i) q^{85} +(0.481788 + 0.834481i) q^{86} +(-0.855559 + 1.48187i) q^{88} -7.53885 q^{89} -2.31952 q^{91} +(-6.42677 + 11.1315i) q^{92} +(-0.0957954 - 0.165923i) q^{94} +(1.14310 + 1.97990i) q^{95} +(0.474177 - 0.821299i) q^{97} -0.378814 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 9 q^{4} + 3 q^{5} - 6 q^{7} + 12 q^{8} + 12 q^{10} + 6 q^{11} - 6 q^{13} - 24 q^{14} - 15 q^{16} - 18 q^{17} + 24 q^{19} + 21 q^{20} - 3 q^{22} + 12 q^{23} - 9 q^{25} + 48 q^{26} + 6 q^{28}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0864880 − 0.149802i 0.0611562 − 0.105926i −0.833826 − 0.552027i \(-0.813855\pi\)
0.894982 + 0.446101i \(0.147188\pi\)
\(3\) 0 0
\(4\) 0.985040 + 1.70614i 0.492520 + 0.853069i
\(5\) −1.86828 − 3.23596i −0.835521 − 1.44716i −0.893606 − 0.448853i \(-0.851833\pi\)
0.0580849 − 0.998312i \(-0.481501\pi\)
\(6\) 0 0
\(7\) −1.51575 + 2.62535i −0.572899 + 0.992291i 0.423367 + 0.905958i \(0.360848\pi\)
−0.996266 + 0.0863324i \(0.972485\pi\)
\(8\) 0.686728 0.242795
\(9\) 0 0
\(10\) −0.646335 −0.204389
\(11\) −1.24585 + 2.15787i −0.375637 + 0.650623i −0.990422 − 0.138072i \(-0.955909\pi\)
0.614785 + 0.788695i \(0.289243\pi\)
\(12\) 0 0
\(13\) 0.382569 + 0.662630i 0.106106 + 0.183780i 0.914189 − 0.405287i \(-0.132828\pi\)
−0.808084 + 0.589068i \(0.799495\pi\)
\(14\) 0.262188 + 0.454123i 0.0700727 + 0.121369i
\(15\) 0 0
\(16\) −1.91069 + 3.30940i −0.477671 + 0.827351i
\(17\) −4.62278 −1.12119 −0.560595 − 0.828090i \(-0.689427\pi\)
−0.560595 + 0.828090i \(0.689427\pi\)
\(18\) 0 0
\(19\) −0.611844 −0.140367 −0.0701833 − 0.997534i \(-0.522358\pi\)
−0.0701833 + 0.997534i \(0.522358\pi\)
\(20\) 3.68066 − 6.37509i 0.823021 − 1.42551i
\(21\) 0 0
\(22\) 0.215502 + 0.373260i 0.0459451 + 0.0795793i
\(23\) 3.26219 + 5.65028i 0.680213 + 1.17816i 0.974916 + 0.222575i \(0.0714462\pi\)
−0.294702 + 0.955589i \(0.595221\pi\)
\(24\) 0 0
\(25\) −4.48095 + 7.76123i −0.896190 + 1.55225i
\(26\) 0.132351 0.0259561
\(27\) 0 0
\(28\) −5.97229 −1.12866
\(29\) −3.27545 + 5.67324i −0.608235 + 1.05349i 0.383296 + 0.923626i \(0.374789\pi\)
−0.991531 + 0.129869i \(0.958544\pi\)
\(30\) 0 0
\(31\) −3.27521 − 5.67283i −0.588246 − 1.01887i −0.994462 − 0.105094i \(-0.966486\pi\)
0.406217 − 0.913777i \(-0.366848\pi\)
\(32\) 1.01723 + 1.76190i 0.179823 + 0.311462i
\(33\) 0 0
\(34\) −0.399815 + 0.692500i −0.0685677 + 0.118763i
\(35\) 11.3274 1.91468
\(36\) 0 0
\(37\) 4.95969 0.815368 0.407684 − 0.913123i \(-0.366337\pi\)
0.407684 + 0.913123i \(0.366337\pi\)
\(38\) −0.0529171 + 0.0916552i −0.00858429 + 0.0148684i
\(39\) 0 0
\(40\) −1.28300 − 2.22222i −0.202860 − 0.351364i
\(41\) 2.63012 + 4.55550i 0.410756 + 0.711450i 0.994973 − 0.100148i \(-0.0319316\pi\)
−0.584217 + 0.811598i \(0.698598\pi\)
\(42\) 0 0
\(43\) −2.78529 + 4.82426i −0.424752 + 0.735692i −0.996397 − 0.0848086i \(-0.972972\pi\)
0.571645 + 0.820501i \(0.306305\pi\)
\(44\) −4.90884 −0.740035
\(45\) 0 0
\(46\) 1.12856 0.166397
\(47\) 0.553808 − 0.959223i 0.0807812 − 0.139917i −0.822805 − 0.568324i \(-0.807592\pi\)
0.903586 + 0.428407i \(0.140925\pi\)
\(48\) 0 0
\(49\) −1.09499 − 1.89658i −0.156427 − 0.270940i
\(50\) 0.775096 + 1.34251i 0.109615 + 0.189859i
\(51\) 0 0
\(52\) −0.753692 + 1.30543i −0.104518 + 0.181031i
\(53\) −8.84310 −1.21469 −0.607346 − 0.794437i \(-0.707766\pi\)
−0.607346 + 0.794437i \(0.707766\pi\)
\(54\) 0 0
\(55\) 9.31038 1.25541
\(56\) −1.04091 + 1.80290i −0.139097 + 0.240923i
\(57\) 0 0
\(58\) 0.566573 + 0.981334i 0.0743947 + 0.128855i
\(59\) −5.92588 − 10.2639i −0.771484 − 1.33625i −0.936750 − 0.350000i \(-0.886182\pi\)
0.165266 − 0.986249i \(-0.447152\pi\)
\(60\) 0 0
\(61\) −4.09350 + 7.09015i −0.524119 + 0.907801i 0.475486 + 0.879723i \(0.342272\pi\)
−0.999606 + 0.0280781i \(0.991061\pi\)
\(62\) −1.13307 −0.143900
\(63\) 0 0
\(64\) −7.29083 −0.911354
\(65\) 1.42949 − 2.47596i 0.177307 − 0.307105i
\(66\) 0 0
\(67\) 0.606169 + 1.04992i 0.0740553 + 0.128268i 0.900675 − 0.434493i \(-0.143073\pi\)
−0.826620 + 0.562761i \(0.809739\pi\)
\(68\) −4.55362 − 7.88711i −0.552208 − 0.956452i
\(69\) 0 0
\(70\) 0.979682 − 1.69686i 0.117094 − 0.202813i
\(71\) 4.91946 0.583833 0.291916 − 0.956444i \(-0.405707\pi\)
0.291916 + 0.956444i \(0.405707\pi\)
\(72\) 0 0
\(73\) 4.29945 0.503213 0.251606 − 0.967830i \(-0.419041\pi\)
0.251606 + 0.967830i \(0.419041\pi\)
\(74\) 0.428953 − 0.742969i 0.0498648 − 0.0863684i
\(75\) 0 0
\(76\) −0.602691 − 1.04389i −0.0691333 − 0.119742i
\(77\) −3.77679 − 6.54158i −0.430405 − 0.745483i
\(78\) 0 0
\(79\) 5.89730 − 10.2144i 0.663498 − 1.14921i −0.316192 − 0.948695i \(-0.602404\pi\)
0.979690 − 0.200518i \(-0.0642624\pi\)
\(80\) 14.2788 1.59642
\(81\) 0 0
\(82\) 0.909895 0.100481
\(83\) 4.50804 − 7.80815i 0.494821 − 0.857056i −0.505161 − 0.863025i \(-0.668567\pi\)
0.999982 + 0.00596962i \(0.00190020\pi\)
\(84\) 0 0
\(85\) 8.63666 + 14.9591i 0.936777 + 1.62255i
\(86\) 0.481788 + 0.834481i 0.0519525 + 0.0899844i
\(87\) 0 0
\(88\) −0.855559 + 1.48187i −0.0912029 + 0.157968i
\(89\) −7.53885 −0.799117 −0.399558 − 0.916708i \(-0.630837\pi\)
−0.399558 + 0.916708i \(0.630837\pi\)
\(90\) 0 0
\(91\) −2.31952 −0.243151
\(92\) −6.42677 + 11.1315i −0.670037 + 1.16054i
\(93\) 0 0
\(94\) −0.0957954 − 0.165923i −0.00988055 − 0.0171136i
\(95\) 1.14310 + 1.97990i 0.117279 + 0.203134i
\(96\) 0 0
\(97\) 0.474177 − 0.821299i 0.0481454 − 0.0833903i −0.840948 − 0.541115i \(-0.818002\pi\)
0.889094 + 0.457725i \(0.151336\pi\)
\(98\) −0.378814 −0.0382659
\(99\) 0 0
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 729.2.c.a.487.4 12
3.2 odd 2 729.2.c.d.487.3 12
9.2 odd 6 729.2.a.b.1.4 ✓ 6
9.4 even 3 inner 729.2.c.a.244.4 12
9.5 odd 6 729.2.c.d.244.3 12
9.7 even 3 729.2.a.e.1.3 yes 6
27.2 odd 18 729.2.e.j.325.2 12
27.4 even 9 729.2.e.k.163.2 12
27.5 odd 18 729.2.e.j.406.2 12
27.7 even 9 729.2.e.l.82.1 12
27.11 odd 18 729.2.e.t.568.1 12
27.13 even 9 729.2.e.l.649.1 12
27.14 odd 18 729.2.e.s.649.2 12
27.16 even 9 729.2.e.k.568.2 12
27.20 odd 18 729.2.e.s.82.2 12
27.22 even 9 729.2.e.u.406.1 12
27.23 odd 18 729.2.e.t.163.1 12
27.25 even 9 729.2.e.u.325.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.4 ✓ 6 9.2 odd 6
729.2.a.e.1.3 yes 6 9.7 even 3
729.2.c.a.244.4 12 9.4 even 3 inner
729.2.c.a.487.4 12 1.1 even 1 trivial
729.2.c.d.244.3 12 9.5 odd 6
729.2.c.d.487.3 12 3.2 odd 2
729.2.e.j.325.2 12 27.2 odd 18
729.2.e.j.406.2 12 27.5 odd 18
729.2.e.k.163.2 12 27.4 even 9
729.2.e.k.568.2 12 27.16 even 9
729.2.e.l.82.1 12 27.7 even 9
729.2.e.l.649.1 12 27.13 even 9
729.2.e.s.82.2 12 27.20 odd 18
729.2.e.s.649.2 12 27.14 odd 18
729.2.e.t.163.1 12 27.23 odd 18
729.2.e.t.568.1 12 27.11 odd 18
729.2.e.u.325.1 12 27.25 even 9
729.2.e.u.406.1 12 27.22 even 9