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Properties

Label 322.3.b.a
Level $322$
Weight $3$
Character orbit 322.b
Analytic conductor $8.774$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,3,Mod(139,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.139");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 322.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: \(8.77386451240\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 12 q^{7} - 128 q^{9} + 24 q^{11} - 8 q^{14} + 16 q^{15} + 128 q^{16} + 84 q^{21} + 16 q^{22} - 224 q^{25} - 24 q^{28} + 96 q^{30} + 32 q^{35} - 256 q^{36} - 128 q^{39} - 80 q^{42} + 144 q^{43}+ \cdots - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 −1.41421 − 5.78135i 2.00000 − 5.81041i 8.17607i 2.60526 + 6.49712i −2.82843 −24.4240 8.21716i
139.2 −1.41421 − 5.04789i 2.00000 0.579996i 7.13880i 5.70567 − 4.05528i −2.82843 −16.4812 − 0.820238i
139.3 −1.41421 − 3.82674i 2.00000 7.51239i 5.41183i −4.12978 + 5.65198i −2.82843 −5.64396 − 10.6241i
139.4 −1.41421 − 3.46826i 2.00000 2.85039i 4.90486i −6.91362 + 1.09631i −2.82843 −3.02883 − 4.03106i
139.5 −1.41421 − 3.38816i 2.00000 − 5.67755i 4.79158i −6.01256 − 3.58456i −2.82843 −2.47960 8.02927i
139.6 −1.41421 − 2.45755i 2.00000 1.94523i 3.47550i 6.91966 + 1.05748i −2.82843 2.96047 − 2.75098i
139.7 −1.41421 − 0.808444i 2.00000 − 8.92170i 1.14331i 0.0318752 + 6.99993i −2.82843 8.34642 12.6172i
139.8 −1.41421 − 0.499241i 2.00000 1.45903i 0.706034i 0.207708 − 6.99692i −2.82843 8.75076 − 2.06338i
139.9 −1.41421 0.499241i 2.00000 − 1.45903i − 0.706034i 0.207708 + 6.99692i −2.82843 8.75076 2.06338i
139.10 −1.41421 0.808444i 2.00000 8.92170i − 1.14331i 0.0318752 − 6.99993i −2.82843 8.34642 − 12.6172i
139.11 −1.41421 2.45755i 2.00000 − 1.94523i − 3.47550i 6.91966 − 1.05748i −2.82843 2.96047 2.75098i
139.12 −1.41421 3.38816i 2.00000 5.67755i − 4.79158i −6.01256 + 3.58456i −2.82843 −2.47960 − 8.02927i
139.13 −1.41421 3.46826i 2.00000 − 2.85039i − 4.90486i −6.91362 − 1.09631i −2.82843 −3.02883 4.03106i
139.14 −1.41421 3.82674i 2.00000 − 7.51239i − 5.41183i −4.12978 − 5.65198i −2.82843 −5.64396 10.6241i
139.15 −1.41421 5.04789i 2.00000 − 0.579996i − 7.13880i 5.70567 + 4.05528i −2.82843 −16.4812 0.820238i
139.16 −1.41421 5.78135i 2.00000 5.81041i − 8.17607i 2.60526 − 6.49712i −2.82843 −24.4240 − 8.21716i
139.17 1.41421 − 5.72743i 2.00000 1.63680i − 8.09981i −6.95198 + 0.818518i 2.82843 −23.8034 2.31478i
139.18 1.41421 − 4.95679i 2.00000 9.92413i − 7.00996i 6.78730 + 1.71249i 2.82843 −15.5698 14.0348i
139.19 1.41421 − 4.52542i 2.00000 − 5.90217i − 6.39991i 4.60838 − 5.26905i 2.82843 −11.4794 − 8.34694i
139.20 1.41421 − 3.31400i 2.00000 − 9.29762i − 4.68671i −4.96652 + 4.93292i 2.82843 −1.98262 − 13.1488i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.3.b.a ✓ 32
7.b odd 2 1 inner 322.3.b.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.3.b.a ✓ 32 1.a even 1 1 trivial
322.3.b.a ✓ 32 7.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).