↑

Properties

Label 540.2.k.e
Level $540$
Weight $2$
Character orbit 540.k
Analytic conductor $4.312$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [540,2,Mod(163,540)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(540, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("540.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 540.k (of order \(4\), 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: \(4.31192170915\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{10} - 40 q^{13} - 68 q^{16} - 24 q^{22} - 64 q^{25} + 16 q^{28} + 52 q^{40} + 4 q^{46} + 4 q^{52} + 16 q^{58} - 16 q^{61} - 20 q^{70} + 48 q^{73} + 100 q^{76} - 12 q^{82} + 88 q^{85} + 72 q^{88} + 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −1.33414 + 0.469122i 0 1.55985 − 1.25175i −1.24162 + 1.85967i 0 −0.794282 − 0.794282i −1.49383 + 2.40176i 0 0.784077 − 3.06353i
163.2 −1.22954 − 0.698742i 0 1.02352 + 1.71826i 0.252746 + 2.22174i 0 1.10809 + 1.10809i −0.0578372 − 2.82784i 0 1.24166 − 2.90831i
163.3 −1.10629 + 0.880980i 0 0.447748 − 1.94924i 0.190805 − 2.22791i 0 3.08026 + 3.08026i 1.22190 + 2.55087i 0 1.75166 + 2.63281i
163.4 −1.06355 − 0.932123i 0 0.262294 + 1.98273i −2.08761 − 0.801188i 0 2.76645 + 2.76645i 1.56918 − 2.35323i 0 1.47348 + 2.79801i
163.5 −0.932123 − 1.06355i 0 −0.262294 + 1.98273i 2.08761 + 0.801188i 0 −2.76645 − 2.76645i 2.35323 − 1.56918i 0 −1.09380 − 2.96709i
163.6 −0.880980 + 1.10629i 0 −0.447748 − 1.94924i 0.190805 − 2.22791i 0 −3.08026 − 3.08026i 2.55087 + 1.22190i 0 2.29662 + 2.17383i
163.7 −0.698742 − 1.22954i 0 −1.02352 + 1.71826i −0.252746 − 2.22174i 0 −1.10809 − 1.10809i 2.82784 + 0.0578372i 0 −2.55510 + 1.86318i
163.8 −0.469122 + 1.33414i 0 −1.55985 − 1.25175i −1.24162 + 1.85967i 0 0.794282 + 0.794282i 2.40176 − 1.49383i 0 −1.89859 − 2.52890i
163.9 0.469122 − 1.33414i 0 −1.55985 − 1.25175i 1.24162 − 1.85967i 0 0.794282 + 0.794282i −2.40176 + 1.49383i 0 −1.89859 − 2.52890i
163.10 0.698742 + 1.22954i 0 −1.02352 + 1.71826i 0.252746 + 2.22174i 0 −1.10809 − 1.10809i −2.82784 − 0.0578372i 0 −2.55510 + 1.86318i
163.11 0.880980 − 1.10629i 0 −0.447748 − 1.94924i −0.190805 + 2.22791i 0 −3.08026 − 3.08026i −2.55087 − 1.22190i 0 2.29662 + 2.17383i
163.12 0.932123 + 1.06355i 0 −0.262294 + 1.98273i −2.08761 − 0.801188i 0 −2.76645 − 2.76645i −2.35323 + 1.56918i 0 −1.09380 − 2.96709i
163.13 1.06355 + 0.932123i 0 0.262294 + 1.98273i 2.08761 + 0.801188i 0 2.76645 + 2.76645i −1.56918 + 2.35323i 0 1.47348 + 2.79801i
163.14 1.10629 − 0.880980i 0 0.447748 − 1.94924i −0.190805 + 2.22791i 0 3.08026 + 3.08026i −1.22190 − 2.55087i 0 1.75166 + 2.63281i
163.15 1.22954 + 0.698742i 0 1.02352 + 1.71826i −0.252746 − 2.22174i 0 1.10809 + 1.10809i 0.0578372 + 2.82784i 0 1.24166 − 2.90831i
163.16 1.33414 − 0.469122i 0 1.55985 − 1.25175i 1.24162 − 1.85967i 0 −0.794282 − 0.794282i 1.49383 − 2.40176i 0 0.784077 − 3.06353i
487.1 −1.33414 − 0.469122i 0 1.55985 + 1.25175i −1.24162 − 1.85967i 0 −0.794282 + 0.794282i −1.49383 − 2.40176i 0 0.784077 + 3.06353i
487.2 −1.22954 + 0.698742i 0 1.02352 − 1.71826i 0.252746 − 2.22174i 0 1.10809 − 1.10809i −0.0578372 + 2.82784i 0 1.24166 + 2.90831i
487.3 −1.10629 − 0.880980i 0 0.447748 + 1.94924i 0.190805 + 2.22791i 0 3.08026 − 3.08026i 1.22190 − 2.55087i 0 1.75166 − 2.63281i
487.4 −1.06355 + 0.932123i 0 0.262294 − 1.98273i −2.08761 + 0.801188i 0 2.76645 − 2.76645i 1.56918 + 2.35323i 0 1.47348 − 2.79801i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.16
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
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.2.k.e ✓ 32
3.b odd 2 1 inner 540.2.k.e ✓ 32
4.b odd 2 1 inner 540.2.k.e ✓ 32
5.c odd 4 1 inner 540.2.k.e ✓ 32
12.b even 2 1 inner 540.2.k.e ✓ 32
15.e even 4 1 inner 540.2.k.e ✓ 32
20.e even 4 1 inner 540.2.k.e ✓ 32
60.l odd 4 1 inner 540.2.k.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.k.e ✓ 32 1.a even 1 1 trivial
540.2.k.e ✓ 32 3.b odd 2 1 inner
540.2.k.e ✓ 32 4.b odd 2 1 inner
540.2.k.e ✓ 32 5.c odd 4 1 inner
540.2.k.e ✓ 32 12.b even 2 1 inner
540.2.k.e ✓ 32 15.e even 4 1 inner
540.2.k.e ✓ 32 20.e even 4 1 inner
540.2.k.e ✓ 32 60.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 602T_{7}^{12} + 88905T_{7}^{8} + 648796T_{7}^{4} + 810000 \) acting on \(S_{2}^{\mathrm{new}}(540, [\chi])\). Copy content Toggle raw display