↑

Properties

Label 1008.2.ch.b
Level $1008$
Weight $2$
Character orbit 1008.ch
Analytic conductor $8.049$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(239,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.ch (of order \(6\), degree \(2\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{5} - 16 q^{9} - 4 q^{21} + 24 q^{25} + 36 q^{29} - 8 q^{33} + 24 q^{37} - 36 q^{41} - 44 q^{45} + 12 q^{49} - 4 q^{57} - 36 q^{65} - 8 q^{69} + 24 q^{73} - 32 q^{81} - 12 q^{85} + 36 q^{93}+O(q^{100}) \) 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 −1.68144 − 0.415635i 0 −3.33673 − 1.92646i 0 0.866025 − 0.500000i 0 2.65450 + 1.39773i 0
239.2 0 −1.27370 + 1.17375i 0 −1.38379 − 0.798929i 0 −0.866025 + 0.500000i 0 0.244640 − 2.99001i 0
239.3 0 −1.22104 − 1.22844i 0 1.03368 + 0.596793i 0 −0.866025 + 0.500000i 0 −0.0181332 + 2.99995i 0
239.4 0 −0.677415 − 1.59409i 0 −1.05527 − 0.609261i 0 0.866025 − 0.500000i 0 −2.08222 + 2.15972i 0
239.5 0 −0.611713 + 1.62043i 0 −1.84481 − 1.06510i 0 0.866025 − 0.500000i 0 −2.25161 − 1.98248i 0
239.6 0 −0.475830 + 1.66541i 0 3.58692 + 2.07091i 0 −0.866025 + 0.500000i 0 −2.54717 − 1.58490i 0
239.7 0 0.475830 − 1.66541i 0 3.58692 + 2.07091i 0 0.866025 − 0.500000i 0 −2.54717 − 1.58490i 0
239.8 0 0.611713 − 1.62043i 0 −1.84481 − 1.06510i 0 −0.866025 + 0.500000i 0 −2.25161 − 1.98248i 0
239.9 0 0.677415 + 1.59409i 0 −1.05527 − 0.609261i 0 −0.866025 + 0.500000i 0 −2.08222 + 2.15972i 0
239.10 0 1.22104 + 1.22844i 0 1.03368 + 0.596793i 0 0.866025 − 0.500000i 0 −0.0181332 + 2.99995i 0
239.11 0 1.27370 − 1.17375i 0 −1.38379 − 0.798929i 0 0.866025 − 0.500000i 0 0.244640 − 2.99001i 0
239.12 0 1.68144 + 0.415635i 0 −3.33673 − 1.92646i 0 −0.866025 + 0.500000i 0 2.65450 + 1.39773i 0
911.1 0 −1.68144 + 0.415635i 0 −3.33673 + 1.92646i 0 0.866025 + 0.500000i 0 2.65450 − 1.39773i 0
911.2 0 −1.27370 − 1.17375i 0 −1.38379 + 0.798929i 0 −0.866025 − 0.500000i 0 0.244640 + 2.99001i 0
911.3 0 −1.22104 + 1.22844i 0 1.03368 − 0.596793i 0 −0.866025 − 0.500000i 0 −0.0181332 − 2.99995i 0
911.4 0 −0.677415 + 1.59409i 0 −1.05527 + 0.609261i 0 0.866025 + 0.500000i 0 −2.08222 − 2.15972i 0
911.5 0 −0.611713 − 1.62043i 0 −1.84481 + 1.06510i 0 0.866025 + 0.500000i 0 −2.25161 + 1.98248i 0
911.6 0 −0.475830 − 1.66541i 0 3.58692 − 2.07091i 0 −0.866025 − 0.500000i 0 −2.54717 + 1.58490i 0
911.7 0 0.475830 + 1.66541i 0 3.58692 − 2.07091i 0 0.866025 + 0.500000i 0 −2.54717 + 1.58490i 0
911.8 0 0.611713 + 1.62043i 0 −1.84481 + 1.06510i 0 −0.866025 − 0.500000i 0 −2.25161 + 1.98248i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.ch.b ✓ 24
3.b odd 2 1 3024.2.ch.c 24
4.b odd 2 1 inner 1008.2.ch.b ✓ 24
9.c even 3 1 3024.2.ch.c 24
9.d odd 6 1 inner 1008.2.ch.b ✓ 24
12.b even 2 1 3024.2.ch.c 24
36.f odd 6 1 3024.2.ch.c 24
36.h even 6 1 inner 1008.2.ch.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.ch.b ✓ 24 1.a even 1 1 trivial
1008.2.ch.b ✓ 24 4.b odd 2 1 inner
1008.2.ch.b ✓ 24 9.d odd 6 1 inner
1008.2.ch.b ✓ 24 36.h even 6 1 inner
3024.2.ch.c 24 3.b odd 2 1
3024.2.ch.c 24 9.c even 3 1
3024.2.ch.c 24 12.b even 2 1
3024.2.ch.c 24 36.f odd 6 1