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Properties

Label 729.3.b.b
Level $729$
Weight $3$
Character orbit 729.b
Analytic conductor $19.864$
Analytic rank $0$
Dimension $36$
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] = [729,3,Mod(728,729)]
 
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([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.728");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 729.b (of order \(2\), degree \(1\), 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: \(19.8638112719\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 72 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} - 108 q^{82} + 216 q^{85} + 396 q^{91} - 432 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
728.1 − 3.85205i 0 −10.8383 − 2.06631i 0 10.6474 26.3416i 0 −7.95953
728.2 − 3.71133i 0 −9.77400 − 3.85345i 0 −8.70385 21.4292i 0 −14.3014
728.3 − 3.69293i 0 −9.63771 8.75713i 0 −3.86733 20.8197i 0 32.3395
728.4 − 3.32908i 0 −7.08275 − 7.70869i 0 −2.62390 10.2627i 0 −25.6628
728.5 − 3.11732i 0 −5.71768 3.82628i 0 10.3874 5.35455i 0 11.9277
728.6 − 2.94436i 0 −4.66923 − 1.37953i 0 −10.3075 1.97046i 0 −4.06183
728.7 − 2.69780i 0 −3.27814 7.63359i 0 −1.11740 − 1.94743i 0 20.5939
728.8 − 2.61300i 0 −2.82775 − 0.428239i 0 1.95676 − 3.06309i 0 −1.11899
728.9 − 2.55819i 0 −2.54436 2.21256i 0 −3.59671 − 3.72381i 0 5.66015
728.10 − 2.12871i 0 −0.531419 − 9.61201i 0 7.75139 − 7.38361i 0 −20.4612
728.11 − 2.12535i 0 −0.517131 − 2.26378i 0 5.68888 − 7.40233i 0 −4.81133
728.12 − 1.28710i 0 2.34336 2.41449i 0 −8.91843 − 8.16457i 0 3.10770
728.13 − 1.18486i 0 2.59612 7.10181i 0 13.5944 − 7.81545i 0 8.41463
728.14 − 1.17952i 0 2.60874 − 4.92036i 0 −0.141555 − 7.79513i 0 −5.80365
728.15 − 1.00766i 0 2.98463 7.84508i 0 −8.01140 − 7.03811i 0 7.90516
728.16 − 0.834667i 0 3.30333 − 6.88905i 0 −10.4274 − 6.09585i 0 −5.75006
728.17 − 0.645888i 0 3.58283 0.119331i 0 2.17315 − 4.89766i 0 0.0770747
728.18 − 0.0224626i 0 3.99950 − 4.22690i 0 5.51607 − 0.179690i 0 −0.0949474
728.19 0.0224626i 0 3.99950 4.22690i 0 5.51607 0.179690i 0 −0.0949474
728.20 0.645888i 0 3.58283 − 0.119331i 0 2.17315 4.89766i 0 0.0770747
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 728.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.3.b.b ✓ 36
3.b odd 2 1 inner 729.3.b.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.3.b.b ✓ 36 1.a even 1 1 trivial
729.3.b.b ✓ 36 3.b odd 2 1 inner