↑

Properties

Label 729.6.a.a
Level $729$
Weight $6$
Character orbit 729.a
Self dual yes
Analytic conductor $116.920$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [729,6,Mod(1,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

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: yes
Analytic conductor: \(116.919804644\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} + 432 q^{4} - 33 q^{5} + 294 q^{7} - 843 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} + 432 q^{4} - 33 q^{5} + 294 q^{7} - 843 q^{8} + 600 q^{10} - 30 q^{11} + 1014 q^{13} + 3720 q^{14} + 8448 q^{16} + 2709 q^{17} + 4332 q^{19} - 861 q^{20} - 2193 q^{22} - 4020 q^{23} + 15417 q^{25} - 17574 q^{26} + 4539 q^{28} + 15279 q^{29} + 17331 q^{31} - 43983 q^{32} + 20412 q^{34} - 2514 q^{35} + 11397 q^{37} - 78873 q^{38} + 52185 q^{40} + 31566 q^{41} + 34422 q^{43} + 63213 q^{44} + 62907 q^{46} - 507 q^{47} + 104106 q^{49} - 64929 q^{50} + 99237 q^{52} - 72297 q^{53} + 106611 q^{55} + 213900 q^{56} + 107733 q^{58} - 59226 q^{59} + 112452 q^{61} - 32100 q^{62} + 287115 q^{64} - 117795 q^{65} + 86055 q^{67} + 439146 q^{68} + 208623 q^{70} - 34992 q^{71} + 89580 q^{73} - 114360 q^{74} + 302031 q^{76} - 87639 q^{77} + 225600 q^{79} + 122331 q^{80} + 74526 q^{82} + 141234 q^{83} - 9900 q^{85} - 336804 q^{86} + 209895 q^{88} + 236637 q^{89} + 24354 q^{91} - 191454 q^{92} + 256080 q^{94} + 254139 q^{95} + 39363 q^{97} - 944667 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −11.2521 0 94.6093 8.52580 0 152.526 −704.485 0 −95.9329
1.2 −11.0172 0 89.3790 −58.3688 0 −214.792 −632.156 0 643.061
1.3 −10.0448 0 68.8976 83.2630 0 −225.519 −370.628 0 −836.358
1.4 −9.43780 0 57.0721 −109.612 0 94.5333 −236.626 0 1034.50
1.5 −7.76070 0 28.2284 33.6629 0 205.544 29.2700 0 −261.248
1.6 −6.92211 0 15.9156 42.9376 0 −174.162 111.338 0 −297.218
1.7 −6.30275 0 7.72471 40.2114 0 163.750 153.001 0 −253.443
1.8 −6.03324 0 4.39994 −75.3717 0 64.8318 166.518 0 454.735
1.9 −5.30646 0 −3.84147 47.6126 0 −184.697 190.191 0 −252.654
1.10 −3.74508 0 −17.9744 −104.284 0 30.6711 187.158 0 390.551
1.11 −2.60656 0 −25.2058 −22.7076 0 −124.035 149.110 0 59.1888
1.12 −0.908243 0 −31.1751 83.9961 0 204.577 57.3784 0 −76.2889
1.13 1.09514 0 −30.8007 −27.6575 0 −27.4634 −68.7757 0 −30.2890
1.14 1.69057 0 −29.1420 41.0011 0 164.613 −103.365 0 69.3154
1.15 2.05892 0 −27.7608 −22.2951 0 85.7048 −123.043 0 −45.9038
1.16 2.25556 0 −26.9125 51.7887 0 −26.8498 −132.881 0 116.812
1.17 4.86323 0 −8.34897 8.85823 0 −189.829 −196.226 0 43.0796
1.18 5.35036 0 −3.37370 −94.5899 0 9.88816 −189.262 0 −506.089
1.19 6.48467 0 10.0509 57.2659 0 −99.4053 −142.332 0 371.350
1.20 7.67489 0 26.9040 −88.7585 0 168.615 −39.1112 0 −681.212
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.6.a.a ✓ 24
3.b odd 2 1 729.6.a.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.6.a.a ✓ 24 1.a even 1 1 trivial
729.6.a.b yes 24 3.b odd 2 1