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Properties

Label 945.2.cs.a
Level $945$
Weight $2$
Character orbit 945.cs
Analytic conductor $7.546$
Analytic rank $0$
Dimension $24$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(104,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([11, 9, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.104");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cs (of order \(18\), degree \(6\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{9} - 18 q^{11} - 24 q^{39} + 96 q^{64} - 180 q^{65} - 12 q^{79} + 102 q^{81} + 84 q^{84} + 60 q^{85} + 228 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
104.1 0 −1.62968 − 0.586627i −1.53209 − 1.28558i −2.20210 − 0.388289i 0 2.02676 − 1.70066i 0 2.31174 + 1.91203i 0
104.2 0 −0.306808 + 1.70466i −1.53209 − 1.28558i −2.20210 − 0.388289i 0 −2.02676 + 1.70066i 0 −2.81174 − 1.04601i 0
104.3 0 0.306808 − 1.70466i −1.53209 − 1.28558i 2.20210 + 0.388289i 0 2.02676 − 1.70066i 0 −2.81174 − 1.04601i 0
104.4 0 1.62968 + 0.586627i −1.53209 − 1.28558i 2.20210 + 0.388289i 0 −2.02676 + 1.70066i 0 2.31174 + 1.91203i 0
209.1 0 −1.62968 + 0.586627i −1.53209 + 1.28558i −2.20210 + 0.388289i 0 2.02676 + 1.70066i 0 2.31174 − 1.91203i 0
209.2 0 −0.306808 − 1.70466i −1.53209 + 1.28558i −2.20210 + 0.388289i 0 −2.02676 − 1.70066i 0 −2.81174 + 1.04601i 0
209.3 0 0.306808 + 1.70466i −1.53209 + 1.28558i 2.20210 − 0.388289i 0 2.02676 + 1.70066i 0 −2.81174 + 1.04601i 0
209.4 0 1.62968 − 0.586627i −1.53209 + 1.28558i 2.20210 − 0.388289i 0 −2.02676 − 1.70066i 0 2.31174 − 1.91203i 0
419.1 0 −1.62968 − 0.586627i 1.87939 − 0.684040i 1.43732 − 1.71293i 0 −2.48619 − 0.904900i 0 2.31174 + 1.91203i 0
419.2 0 −0.306808 + 1.70466i 1.87939 − 0.684040i 1.43732 − 1.71293i 0 2.48619 + 0.904900i 0 −2.81174 − 1.04601i 0
419.3 0 0.306808 − 1.70466i 1.87939 − 0.684040i −1.43732 + 1.71293i 0 −2.48619 − 0.904900i 0 −2.81174 − 1.04601i 0
419.4 0 1.62968 + 0.586627i 1.87939 − 0.684040i −1.43732 + 1.71293i 0 2.48619 + 0.904900i 0 2.31174 + 1.91203i 0
524.1 0 −1.62968 + 0.586627i −0.347296 − 1.96962i 0.764780 − 2.10122i 0 0.459430 − 2.60556i 0 2.31174 − 1.91203i 0
524.2 0 −0.306808 − 1.70466i −0.347296 − 1.96962i 0.764780 − 2.10122i 0 −0.459430 + 2.60556i 0 −2.81174 + 1.04601i 0
524.3 0 0.306808 + 1.70466i −0.347296 − 1.96962i −0.764780 + 2.10122i 0 0.459430 − 2.60556i 0 −2.81174 + 1.04601i 0
524.4 0 1.62968 − 0.586627i −0.347296 − 1.96962i −0.764780 + 2.10122i 0 −0.459430 + 2.60556i 0 2.31174 − 1.91203i 0
734.1 0 −1.62968 − 0.586627i −0.347296 + 1.96962i 0.764780 + 2.10122i 0 0.459430 + 2.60556i 0 2.31174 + 1.91203i 0
734.2 0 −0.306808 + 1.70466i −0.347296 + 1.96962i 0.764780 + 2.10122i 0 −0.459430 − 2.60556i 0 −2.81174 − 1.04601i 0
734.3 0 0.306808 − 1.70466i −0.347296 + 1.96962i −0.764780 − 2.10122i 0 0.459430 + 2.60556i 0 −2.81174 − 1.04601i 0
734.4 0 1.62968 + 0.586627i −0.347296 + 1.96962i −0.764780 − 2.10122i 0 −0.459430 − 2.60556i 0 2.31174 + 1.91203i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 104.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
27.f odd 18 1 inner
135.n odd 18 1 inner
189.be even 18 1 inner
945.cs even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.cs.a ✓ 24
5.b even 2 1 inner 945.2.cs.a ✓ 24
7.b odd 2 1 inner 945.2.cs.a ✓ 24
27.f odd 18 1 inner 945.2.cs.a ✓ 24
35.c odd 2 1 CM 945.2.cs.a ✓ 24
135.n odd 18 1 inner 945.2.cs.a ✓ 24
189.be even 18 1 inner 945.2.cs.a ✓ 24
945.cs even 18 1 inner 945.2.cs.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.cs.a ✓ 24 1.a even 1 1 trivial
945.2.cs.a ✓ 24 5.b even 2 1 inner
945.2.cs.a ✓ 24 7.b odd 2 1 inner
945.2.cs.a ✓ 24 27.f odd 18 1 inner
945.2.cs.a ✓ 24 35.c odd 2 1 CM
945.2.cs.a ✓ 24 135.n odd 18 1 inner
945.2.cs.a ✓ 24 189.be even 18 1 inner
945.2.cs.a ✓ 24 945.cs even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display