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Properties

Label 729.3.b.a
Level $729$
Weight $3$
Character orbit 729.b
Analytic conductor $19.864$
Analytic rank $0$
Dimension $30$
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: \(30\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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 − 3.75900i 0 −10.1300 − 0.534675i 0 2.78504 23.0428i 0 −2.00984
728.2 − 3.51633i 0 −8.36459 4.92652i 0 3.39378 15.3473i 0 17.3233
728.3 − 3.39067i 0 −7.49663 3.49531i 0 −10.8678 11.8559i 0 11.8514
728.4 − 3.32411i 0 −7.04970 − 6.87815i 0 6.38137 10.1375i 0 −22.8637
728.5 − 2.62787i 0 −2.90572 − 6.25111i 0 −8.11762 − 2.87564i 0 −16.4271
728.6 − 2.61983i 0 −2.86351 − 0.807612i 0 5.91720 − 2.97742i 0 −2.11581
728.7 − 2.43656i 0 −1.93682 9.03835i 0 −1.73106 − 5.02707i 0 22.0225
728.8 − 2.35177i 0 −1.53083 − 5.78250i 0 −4.87067 − 5.80692i 0 −13.5991
728.9 − 1.94017i 0 0.235738 4.32253i 0 8.88939 − 8.21805i 0 8.38645
728.10 − 1.53349i 0 1.64842 − 0.0504138i 0 9.85801 − 8.66177i 0 −0.0773090
728.11 − 1.30309i 0 2.30195 0.437029i 0 −6.08419 − 8.21202i 0 0.569489
728.12 − 1.16096i 0 2.65217 5.38300i 0 −10.5344 − 7.72291i 0 6.24946
728.13 − 0.581939i 0 3.66135 − 7.66608i 0 1.33395 − 4.45844i 0 −4.46119
728.14 − 0.455989i 0 3.79207 − 2.45401i 0 −4.32645 − 3.55310i 0 −1.11900
728.15 − 0.117696i 0 3.98615 − 6.19791i 0 7.97344 − 0.939939i 0 −0.729470
728.16 0.117696i 0 3.98615 6.19791i 0 7.97344 0.939939i 0 −0.729470
728.17 0.455989i 0 3.79207 2.45401i 0 −4.32645 3.55310i 0 −1.11900
728.18 0.581939i 0 3.66135 7.66608i 0 1.33395 4.45844i 0 −4.46119
728.19 1.16096i 0 2.65217 − 5.38300i 0 −10.5344 7.72291i 0 6.24946
728.20 1.30309i 0 2.30195 − 0.437029i 0 −6.08419 8.21202i 0 0.569489
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 728.30
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.a 30
3.b odd 2 1 inner 729.3.b.a 30
27.e even 9 1 27.3.f.a ✓ 30
27.e even 9 1 81.3.f.a 30
27.e even 9 1 243.3.f.a 30
27.e even 9 1 243.3.f.b 30
27.e even 9 1 243.3.f.c 30
27.e even 9 1 243.3.f.d 30
27.f odd 18 1 27.3.f.a ✓ 30
27.f odd 18 1 81.3.f.a 30
27.f odd 18 1 243.3.f.a 30
27.f odd 18 1 243.3.f.b 30
27.f odd 18 1 243.3.f.c 30
27.f odd 18 1 243.3.f.d 30
108.j odd 18 1 432.3.bc.a 30
108.l even 18 1 432.3.bc.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.f.a ✓ 30 27.e even 9 1
27.3.f.a ✓ 30 27.f odd 18 1
81.3.f.a 30 27.e even 9 1
81.3.f.a 30 27.f odd 18 1
243.3.f.a 30 27.e even 9 1
243.3.f.a 30 27.f odd 18 1
243.3.f.b 30 27.e even 9 1
243.3.f.b 30 27.f odd 18 1
243.3.f.c 30 27.e even 9 1
243.3.f.c 30 27.f odd 18 1
243.3.f.d 30 27.e even 9 1
243.3.f.d 30 27.f odd 18 1
432.3.bc.a 30 108.j odd 18 1
432.3.bc.a 30 108.l even 18 1
729.3.b.a 30 1.a even 1 1 trivial
729.3.b.a 30 3.b odd 2 1 inner