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Properties

Label 540.2.k.f
Level $540$
Weight $2$
Character orbit 540.k
Analytic conductor $4.312$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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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: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{10} + 8 q^{16} - 8 q^{22} - 16 q^{28} + 16 q^{37} - 48 q^{40} + 8 q^{46} - 32 q^{52} - 64 q^{58} + 64 q^{61} - 88 q^{70} + 16 q^{73} - 48 q^{76} - 16 q^{85} - 80 q^{88} + 80 q^{97}+O(q^{100}) \) 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 −1.40361 − 0.172822i 0 1.94027 + 0.485151i 0.916879 + 2.03944i 0 −3.47124 − 3.47124i −2.63954 − 1.01629i 0 −0.934483 − 3.02105i
163.2 −1.37934 + 0.312138i 0 1.80514 − 0.861087i −1.57572 − 1.58654i 0 −0.806142 − 0.806142i −2.22112 + 1.75118i 0 2.66866 + 1.69654i
163.3 −1.37191 − 0.343303i 0 1.76429 + 0.941963i 0.116853 − 2.23301i 0 1.06377 + 1.06377i −2.09707 − 1.89798i 0 −0.926912 + 3.02338i
163.4 −1.23686 + 0.685686i 0 1.05967 − 1.69620i −1.35730 + 1.77700i 0 2.92892 + 2.92892i −0.147607 + 2.82457i 0 0.460337 − 3.12859i
163.5 −1.20980 + 0.732381i 0 0.927237 − 1.77207i 2.23408 + 0.0942881i 0 1.25928 + 1.25928i 0.176057 + 2.82294i 0 −2.77185 + 1.52213i
163.6 −1.14403 − 0.831375i 0 0.617631 + 1.90224i 2.19759 − 0.413020i 0 0.0683103 + 0.0683103i 0.874887 − 2.68972i 0 −2.85750 − 1.35451i
163.7 −0.831375 − 1.14403i 0 −0.617631 + 1.90224i −2.19759 + 0.413020i 0 −0.0683103 − 0.0683103i 2.68972 − 0.874887i 0 2.29953 + 2.17075i
163.8 −0.732381 + 1.20980i 0 −0.927237 − 1.77207i 2.23408 + 0.0942881i 0 −1.25928 − 1.25928i 2.82294 + 0.176057i 0 −1.75027 + 2.63374i
163.9 −0.685686 + 1.23686i 0 −1.05967 − 1.69620i −1.35730 + 1.77700i 0 −2.92892 − 2.92892i 2.82457 − 0.147607i 0 −1.26723 − 2.89726i
163.10 −0.343303 − 1.37191i 0 −1.76429 + 0.941963i −0.116853 + 2.23301i 0 −1.06377 − 1.06377i 1.89798 + 2.09707i 0 3.10361 − 0.606288i
163.11 −0.312138 + 1.37934i 0 −1.80514 − 0.861087i −1.57572 − 1.58654i 0 0.806142 + 0.806142i 1.75118 − 2.22112i 0 2.68022 − 1.67822i
163.12 −0.172822 − 1.40361i 0 −1.94027 + 0.485151i −0.916879 − 2.03944i 0 3.47124 + 3.47124i 1.01629 + 2.63954i 0 −2.70414 + 1.63940i
163.13 0.172822 + 1.40361i 0 −1.94027 + 0.485151i 0.916879 + 2.03944i 0 3.47124 + 3.47124i −1.01629 − 2.63954i 0 −2.70414 + 1.63940i
163.14 0.312138 − 1.37934i 0 −1.80514 − 0.861087i 1.57572 + 1.58654i 0 0.806142 + 0.806142i −1.75118 + 2.22112i 0 2.68022 − 1.67822i
163.15 0.343303 + 1.37191i 0 −1.76429 + 0.941963i 0.116853 − 2.23301i 0 −1.06377 − 1.06377i −1.89798 − 2.09707i 0 3.10361 − 0.606288i
163.16 0.685686 − 1.23686i 0 −1.05967 − 1.69620i 1.35730 − 1.77700i 0 −2.92892 − 2.92892i −2.82457 + 0.147607i 0 −1.26723 − 2.89726i
163.17 0.732381 − 1.20980i 0 −0.927237 − 1.77207i −2.23408 − 0.0942881i 0 −1.25928 − 1.25928i −2.82294 − 0.176057i 0 −1.75027 + 2.63374i
163.18 0.831375 + 1.14403i 0 −0.617631 + 1.90224i 2.19759 − 0.413020i 0 −0.0683103 − 0.0683103i −2.68972 + 0.874887i 0 2.29953 + 2.17075i
163.19 1.14403 + 0.831375i 0 0.617631 + 1.90224i −2.19759 + 0.413020i 0 0.0683103 + 0.0683103i −0.874887 + 2.68972i 0 −2.85750 − 1.35451i
163.20 1.20980 − 0.732381i 0 0.927237 − 1.77207i −2.23408 − 0.0942881i 0 1.25928 + 1.25928i −0.176057 − 2.82294i 0 −2.77185 + 1.52213i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.24
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
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.f ✓ 48
3.b odd 2 1 inner 540.2.k.f ✓ 48
4.b odd 2 1 inner 540.2.k.f ✓ 48
5.c odd 4 1 inner 540.2.k.f ✓ 48
12.b even 2 1 inner 540.2.k.f ✓ 48
15.e even 4 1 inner 540.2.k.f ✓ 48
20.e even 4 1 inner 540.2.k.f ✓ 48
60.l odd 4 1 inner 540.2.k.f ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.k.f ✓ 48 1.a even 1 1 trivial
540.2.k.f ✓ 48 3.b odd 2 1 inner
540.2.k.f ✓ 48 4.b odd 2 1 inner
540.2.k.f ✓ 48 5.c odd 4 1 inner
540.2.k.f ✓ 48 12.b even 2 1 inner
540.2.k.f ✓ 48 15.e even 4 1 inner
540.2.k.f ✓ 48 20.e even 4 1 inner
540.2.k.f ✓ 48 60.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 892T_{7}^{20} + 185798T_{7}^{16} + 2951748T_{7}^{12} + 13269041T_{7}^{8} + 14881096T_{7}^{4} + 1296 \) acting on \(S_{2}^{\mathrm{new}}(540, [\chi])\). Copy content Toggle raw display