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Properties

Label 336.4.h.b
Level $336$
Weight $4$
Character orbit 336.h
Analytic conductor $19.825$
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] = [336,4,Mod(239,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.239");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.h (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: \(19.8246417619\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −5.19367 − 0.160438i 0 7.59851i 0 7.00000i 0 26.9485 + 1.66652i 0
239.2 0 −5.19367 + 0.160438i 0 − 7.59851i 0 − 7.00000i 0 26.9485 − 1.66652i 0
239.3 0 −4.13038 − 3.15276i 0 1.18345i 0 − 7.00000i 0 7.12016 + 26.0443i 0
239.4 0 −4.13038 + 3.15276i 0 − 1.18345i 0 7.00000i 0 7.12016 − 26.0443i 0
239.5 0 −3.03558 − 4.21726i 0 − 3.22455i 0 7.00000i 0 −8.57055 + 25.6036i 0
239.6 0 −3.03558 + 4.21726i 0 3.22455i 0 − 7.00000i 0 −8.57055 − 25.6036i 0
239.7 0 −2.33643 − 4.64124i 0 − 12.0055i 0 7.00000i 0 −16.0822 + 21.6878i 0
239.8 0 −2.33643 + 4.64124i 0 12.0055i 0 − 7.00000i 0 −16.0822 − 21.6878i 0
239.9 0 −1.99829 − 4.79655i 0 16.7287i 0 − 7.00000i 0 −19.0137 + 19.1697i 0
239.10 0 −1.99829 + 4.79655i 0 − 16.7287i 0 7.00000i 0 −19.0137 − 19.1697i 0
239.11 0 −1.13969 − 5.06963i 0 − 14.6453i 0 − 7.00000i 0 −24.4022 + 11.5556i 0
239.12 0 −1.13969 + 5.06963i 0 14.6453i 0 7.00000i 0 −24.4022 − 11.5556i 0
239.13 0 1.13969 − 5.06963i 0 14.6453i 0 − 7.00000i 0 −24.4022 − 11.5556i 0
239.14 0 1.13969 + 5.06963i 0 − 14.6453i 0 7.00000i 0 −24.4022 + 11.5556i 0
239.15 0 1.99829 − 4.79655i 0 − 16.7287i 0 − 7.00000i 0 −19.0137 − 19.1697i 0
239.16 0 1.99829 + 4.79655i 0 16.7287i 0 7.00000i 0 −19.0137 + 19.1697i 0
239.17 0 2.33643 − 4.64124i 0 12.0055i 0 7.00000i 0 −16.0822 − 21.6878i 0
239.18 0 2.33643 + 4.64124i 0 − 12.0055i 0 − 7.00000i 0 −16.0822 + 21.6878i 0
239.19 0 3.03558 − 4.21726i 0 3.22455i 0 7.00000i 0 −8.57055 − 25.6036i 0
239.20 0 3.03558 + 4.21726i 0 − 3.22455i 0 − 7.00000i 0 −8.57055 + 25.6036i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.h.b ✓ 24
3.b odd 2 1 inner 336.4.h.b ✓ 24
4.b odd 2 1 inner 336.4.h.b ✓ 24
12.b even 2 1 inner 336.4.h.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.h.b ✓ 24 1.a even 1 1 trivial
336.4.h.b ✓ 24 3.b odd 2 1 inner
336.4.h.b ✓ 24 4.b odd 2 1 inner
336.4.h.b ✓ 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 708T_{5}^{10} + 176364T_{5}^{8} + 18224416T_{5}^{6} + 693441840T_{5}^{4} + 6129603648T_{5}^{2} + 7274042944 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display