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Properties

Label 297.2.k.a
Level $297$
Weight $2$
Character orbit 297.k
Analytic conductor $2.372$
Analytic rank $0$
Dimension $32$
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] = [297,2,Mod(107,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.k (of order \(10\), degree \(4\), 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: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{4} - 4 q^{16} + 36 q^{22} + 8 q^{25} - 100 q^{28} + 8 q^{31} - 64 q^{34} - 12 q^{37} - 60 q^{40} - 20 q^{46} + 100 q^{52} + 8 q^{55} + 24 q^{58} + 60 q^{61} + 36 q^{64} - 24 q^{67} + 8 q^{70} + 80 q^{73} - 60 q^{79} + 72 q^{82} - 20 q^{85} - 24 q^{88} + 24 q^{91} + 20 q^{94} - 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −2.08816 + 1.51714i 0 1.44068 − 4.43395i −0.271884 + 0.374217i 0 −1.81371 − 0.589311i 2.12333 + 6.53495i 0 − 1.19391i
107.2 −1.59565 + 1.15931i 0 0.584068 − 1.79758i −0.552976 + 0.761106i 0 2.86353 + 0.930419i −0.0669928 − 0.206183i 0 − 1.85553i
107.3 −0.824821 + 0.599267i 0 −0.296826 + 0.913536i 1.82477 − 2.51158i 0 −3.98157 − 1.29369i −0.932731 − 2.87065i 0 3.16513i
107.4 −0.516971 + 0.375601i 0 −0.491852 + 1.51376i −1.44302 + 1.98614i 0 0.695678 + 0.226039i −0.709229 − 2.18278i 0 − 1.56878i
107.5 0.516971 − 0.375601i 0 −0.491852 + 1.51376i 1.44302 − 1.98614i 0 0.695678 + 0.226039i 0.709229 + 2.18278i 0 − 1.56878i
107.6 0.824821 − 0.599267i 0 −0.296826 + 0.913536i −1.82477 + 2.51158i 0 −3.98157 − 1.29369i 0.932731 + 2.87065i 0 3.16513i
107.7 1.59565 − 1.15931i 0 0.584068 − 1.79758i 0.552976 − 0.761106i 0 2.86353 + 0.930419i 0.0669928 + 0.206183i 0 − 1.85553i
107.8 2.08816 − 1.51714i 0 1.44068 − 4.43395i 0.271884 − 0.374217i 0 −1.81371 − 0.589311i −2.12333 − 6.53495i 0 − 1.19391i
134.1 −0.842742 + 2.59369i 0 −4.39899 − 3.19605i −2.11449 + 0.687040i 0 1.97014 − 2.71166i 7.58414 − 5.51020i 0 − 6.06334i
134.2 −0.518211 + 1.59489i 0 −0.657092 − 0.477406i 1.43727 − 0.466997i 0 2.42198 − 3.33356i −1.61147 + 1.17080i 0 2.53429i
134.3 −0.361043 + 1.11118i 0 0.513674 + 0.373206i −0.251450 + 0.0817011i 0 −0.852105 + 1.17282i −2.49060 + 1.80953i 0 − 0.308903i
134.4 −0.191808 + 0.590323i 0 1.30634 + 0.949113i −3.55080 + 1.15372i 0 −1.30394 + 1.79472i −1.81517 + 1.31880i 0 − 2.31741i
134.5 0.191808 − 0.590323i 0 1.30634 + 0.949113i 3.55080 − 1.15372i 0 −1.30394 + 1.79472i 1.81517 − 1.31880i 0 − 2.31741i
134.6 0.361043 − 1.11118i 0 0.513674 + 0.373206i 0.251450 − 0.0817011i 0 −0.852105 + 1.17282i 2.49060 − 1.80953i 0 − 0.308903i
134.7 0.518211 − 1.59489i 0 −0.657092 − 0.477406i −1.43727 + 0.466997i 0 2.42198 − 3.33356i 1.61147 − 1.17080i 0 2.53429i
134.8 0.842742 − 2.59369i 0 −4.39899 − 3.19605i 2.11449 − 0.687040i 0 1.97014 − 2.71166i −7.58414 + 5.51020i 0 − 6.06334i
161.1 −2.08816 − 1.51714i 0 1.44068 + 4.43395i −0.271884 − 0.374217i 0 −1.81371 + 0.589311i 2.12333 − 6.53495i 0 1.19391i
161.2 −1.59565 − 1.15931i 0 0.584068 + 1.79758i −0.552976 − 0.761106i 0 2.86353 − 0.930419i −0.0669928 + 0.206183i 0 1.85553i
161.3 −0.824821 − 0.599267i 0 −0.296826 − 0.913536i 1.82477 + 2.51158i 0 −3.98157 + 1.29369i −0.932731 + 2.87065i 0 − 3.16513i
161.4 −0.516971 − 0.375601i 0 −0.491852 − 1.51376i −1.44302 − 1.98614i 0 0.695678 − 0.226039i −0.709229 + 2.18278i 0 1.56878i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.k.a ✓ 32
3.b odd 2 1 inner 297.2.k.a ✓ 32
9.c even 3 2 891.2.u.e 64
9.d odd 6 2 891.2.u.e 64
11.d odd 10 1 inner 297.2.k.a ✓ 32
33.f even 10 1 inner 297.2.k.a ✓ 32
99.o odd 30 2 891.2.u.e 64
99.p even 30 2 891.2.u.e 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.k.a ✓ 32 1.a even 1 1 trivial
297.2.k.a ✓ 32 3.b odd 2 1 inner
297.2.k.a ✓ 32 11.d odd 10 1 inner
297.2.k.a ✓ 32 33.f even 10 1 inner
891.2.u.e 64 9.c even 3 2
891.2.u.e 64 9.d odd 6 2
891.2.u.e 64 99.o odd 30 2
891.2.u.e 64 99.p even 30 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 12 T_{2}^{30} + 101 T_{2}^{28} + 738 T_{2}^{26} + 5521 T_{2}^{24} + 22086 T_{2}^{22} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display