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Properties

Label 237.2.b.a
Level $237$
Weight $2$
Character orbit 237.b
Analytic conductor $1.892$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [237,2,Mod(236,237)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(237, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("237.236");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 237 = 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 237.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: \(1.89245452790\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} + 2 q^{9} + 4 q^{10} - 16 q^{13} + 12 q^{16} - 22 q^{18} - 8 q^{19} + 22 q^{21} - 8 q^{22} - 4 q^{25} + 16 q^{31} - 22 q^{36} + 32 q^{40} - 26 q^{42} + 10 q^{45} + 44 q^{46} - 64 q^{49} - 40 q^{51} + 16 q^{52} + 40 q^{55} + 28 q^{64} - 32 q^{67} + 58 q^{72} - 52 q^{73} + 4 q^{76} + 16 q^{79} + 2 q^{81} - 6 q^{84} + 16 q^{87} - 4 q^{88} + 98 q^{90} + 56 q^{97} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
236.1 − 2.50217i −1.64937 + 0.528753i −4.26087 1.01684i 1.32303 + 4.12701i 4.05862i 5.65710i 2.44084 − 1.74422i 2.54430
236.2 − 2.50217i 1.64937 − 0.528753i −4.26087 1.01684i −1.32303 − 4.12701i − 4.05862i 5.65710i 2.44084 − 1.74422i 2.54430
236.3 − 2.36268i −0.275859 + 1.70994i −3.58225 − 1.77810i 4.04004 + 0.651767i − 2.68530i 3.73835i −2.84780 − 0.943406i −4.20109
236.4 − 2.36268i 0.275859 − 1.70994i −3.58225 − 1.77810i −4.04004 − 0.651767i 2.68530i 3.73835i −2.84780 − 0.943406i −4.20109
236.5 − 1.52453i −1.67608 + 0.436754i −0.324186 2.96900i 0.665844 + 2.55523i − 3.48169i − 2.55483i 2.61849 − 1.46407i 4.52632
236.6 − 1.52453i 1.67608 − 0.436754i −0.324186 2.96900i −0.665844 − 2.55523i 3.48169i − 2.55483i 2.61849 − 1.46407i 4.52632
236.7 − 1.48890i −1.42217 − 0.988655i −0.216833 − 2.25576i −1.47201 + 2.11747i − 0.748343i − 2.65496i 1.04512 + 2.81207i −3.35860
236.8 − 1.48890i 1.42217 + 0.988655i −0.216833 − 2.25576i 1.47201 − 2.11747i 0.748343i − 2.65496i 1.04512 + 2.81207i −3.35860
236.9 − 0.752910i −0.622466 + 1.61633i 1.43313 0.960379i 1.21695 + 0.468661i 2.03283i − 2.58484i −2.22507 − 2.01223i 0.723079
236.10 − 0.752910i 0.622466 − 1.61633i 1.43313 0.960379i −1.21695 − 0.468661i − 2.03283i − 2.58484i −2.22507 − 2.01223i 0.723079
236.11 − 0.221318i −1.11095 − 1.32883i 1.95102 3.46102i −0.294095 + 0.245874i 4.18356i − 0.874433i −0.531581 + 2.95253i 0.765988
236.12 − 0.221318i 1.11095 + 1.32883i 1.95102 3.46102i 0.294095 − 0.245874i − 4.18356i − 0.874433i −0.531581 + 2.95253i 0.765988
236.13 0.221318i −1.11095 + 1.32883i 1.95102 − 3.46102i −0.294095 − 0.245874i − 4.18356i 0.874433i −0.531581 − 2.95253i 0.765988
236.14 0.221318i 1.11095 − 1.32883i 1.95102 − 3.46102i 0.294095 + 0.245874i 4.18356i 0.874433i −0.531581 − 2.95253i 0.765988
236.15 0.752910i −0.622466 − 1.61633i 1.43313 − 0.960379i 1.21695 − 0.468661i − 2.03283i 2.58484i −2.22507 + 2.01223i 0.723079
236.16 0.752910i 0.622466 + 1.61633i 1.43313 − 0.960379i −1.21695 + 0.468661i 2.03283i 2.58484i −2.22507 + 2.01223i 0.723079
236.17 1.48890i −1.42217 + 0.988655i −0.216833 2.25576i −1.47201 − 2.11747i 0.748343i 2.65496i 1.04512 − 2.81207i −3.35860
236.18 1.48890i 1.42217 − 0.988655i −0.216833 2.25576i 1.47201 + 2.11747i − 0.748343i 2.65496i 1.04512 − 2.81207i −3.35860
236.19 1.52453i −1.67608 − 0.436754i −0.324186 − 2.96900i 0.665844 − 2.55523i 3.48169i 2.55483i 2.61849 + 1.46407i 4.52632
236.20 1.52453i 1.67608 + 0.436754i −0.324186 − 2.96900i −0.665844 + 2.55523i − 3.48169i 2.55483i 2.61849 + 1.46407i 4.52632
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 236.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 237.2.b.a ✓ 24
3.b odd 2 1 inner 237.2.b.a ✓ 24
79.b odd 2 1 inner 237.2.b.a ✓ 24
237.b even 2 1 inner 237.2.b.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.2.b.a ✓ 24 1.a even 1 1 trivial
237.2.b.a ✓ 24 3.b odd 2 1 inner
237.2.b.a ✓ 24 79.b odd 2 1 inner
237.2.b.a ✓ 24 237.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(237, [\chi])\).