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Properties

Label 297.2.k.b
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
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{4} + 32 q^{16} - 30 q^{22} + 26 q^{25} - 10 q^{28} - 16 q^{31} + 44 q^{34} + 18 q^{37} + 30 q^{40} + 40 q^{46} - 24 q^{49} - 80 q^{52} - 16 q^{55} - 48 q^{58} - 60 q^{61} - 96 q^{67} - 82 q^{70} - 70 q^{73} - 120 q^{79} - 66 q^{82} + 160 q^{85} + 48 q^{88} - 60 q^{91} + 200 q^{94} + 48 q^{97}+O(q^{100}) \) 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 −1.97425 + 1.43438i 0 1.22220 − 3.76155i 0.123910 − 0.170547i 0 −0.542094 − 0.176137i 1.47436 + 4.53760i 0 0.514437i
107.2 −1.46478 + 1.06422i 0 0.394966 − 1.21558i −2.52447 + 3.47463i 0 −2.74051 − 0.890446i −0.403879 − 1.24301i 0 − 7.77615i
107.3 −1.17155 + 0.851178i 0 0.0299823 − 0.0922759i 1.29134 − 1.77737i 0 3.47214 + 1.12817i −0.851564 − 2.62084i 0 3.18143i
107.4 −0.662067 + 0.481020i 0 −0.411081 + 1.26518i 0.919440 − 1.26550i 0 0.928501 + 0.301688i −0.842187 − 2.59198i 0 1.28012i
107.5 0.662067 − 0.481020i 0 −0.411081 + 1.26518i −0.919440 + 1.26550i 0 0.928501 + 0.301688i 0.842187 + 2.59198i 0 1.28012i
107.6 1.17155 − 0.851178i 0 0.0299823 − 0.0922759i −1.29134 + 1.77737i 0 3.47214 + 1.12817i 0.851564 + 2.62084i 0 3.18143i
107.7 1.46478 − 1.06422i 0 0.394966 − 1.21558i 2.52447 − 3.47463i 0 −2.74051 − 0.890446i 0.403879 + 1.24301i 0 − 7.77615i
107.8 1.97425 − 1.43438i 0 1.22220 − 3.76155i −0.123910 + 0.170547i 0 −0.542094 − 0.176137i −1.47436 − 4.53760i 0 0.514437i
134.1 −0.824960 + 2.53896i 0 −4.14775 − 3.01352i 3.38136 − 1.09867i 0 −0.102445 + 0.141003i 6.75339 − 4.90663i 0 9.49150i
134.2 −0.634332 + 1.95227i 0 −1.79096 − 1.30121i −1.69624 + 0.551140i 0 −0.180329 + 0.248202i 0.354978 − 0.257906i 0 − 3.66112i
134.3 −0.244968 + 0.753934i 0 1.10963 + 0.806191i 3.15331 − 1.02457i 0 −2.80363 + 3.85887i −2.16231 + 1.57101i 0 2.62838i
134.4 −0.0543407 + 0.167243i 0 1.59302 + 1.15739i 1.75915 − 0.571582i 0 1.96837 − 2.70923i −0.564664 + 0.410252i 0 0.325266i
134.5 0.0543407 − 0.167243i 0 1.59302 + 1.15739i −1.75915 + 0.571582i 0 1.96837 − 2.70923i 0.564664 − 0.410252i 0 0.325266i
134.6 0.244968 − 0.753934i 0 1.10963 + 0.806191i −3.15331 + 1.02457i 0 −2.80363 + 3.85887i 2.16231 − 1.57101i 0 2.62838i
134.7 0.634332 − 1.95227i 0 −1.79096 − 1.30121i 1.69624 − 0.551140i 0 −0.180329 + 0.248202i −0.354978 + 0.257906i 0 − 3.66112i
134.8 0.824960 − 2.53896i 0 −4.14775 − 3.01352i −3.38136 + 1.09867i 0 −0.102445 + 0.141003i −6.75339 + 4.90663i 0 9.49150i
161.1 −1.97425 − 1.43438i 0 1.22220 + 3.76155i 0.123910 + 0.170547i 0 −0.542094 + 0.176137i 1.47436 − 4.53760i 0 − 0.514437i
161.2 −1.46478 − 1.06422i 0 0.394966 + 1.21558i −2.52447 − 3.47463i 0 −2.74051 + 0.890446i −0.403879 + 1.24301i 0 7.77615i
161.3 −1.17155 − 0.851178i 0 0.0299823 + 0.0922759i 1.29134 + 1.77737i 0 3.47214 − 1.12817i −0.851564 + 2.62084i 0 − 3.18143i
161.4 −0.662067 − 0.481020i 0 −0.411081 − 1.26518i 0.919440 + 1.26550i 0 0.928501 − 0.301688i −0.842187 + 2.59198i 0 − 1.28012i
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.b ✓ 32
3.b odd 2 1 inner 297.2.k.b ✓ 32
9.c even 3 1 891.2.u.b 32
9.c even 3 1 891.2.u.d 32
9.d odd 6 1 891.2.u.b 32
9.d odd 6 1 891.2.u.d 32
11.d odd 10 1 inner 297.2.k.b ✓ 32
33.f even 10 1 inner 297.2.k.b ✓ 32
99.o odd 30 1 891.2.u.b 32
99.o odd 30 1 891.2.u.d 32
99.p even 30 1 891.2.u.b 32
99.p even 30 1 891.2.u.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.k.b ✓ 32 1.a even 1 1 trivial
297.2.k.b ✓ 32 3.b odd 2 1 inner
297.2.k.b ✓ 32 11.d odd 10 1 inner
297.2.k.b ✓ 32 33.f even 10 1 inner
891.2.u.b 32 9.c even 3 1
891.2.u.b 32 9.d odd 6 1
891.2.u.b 32 99.o odd 30 1
891.2.u.b 32 99.p even 30 1
891.2.u.d 32 9.c even 3 1
891.2.u.d 32 9.d odd 6 1
891.2.u.d 32 99.o odd 30 1
891.2.u.d 32 99.p even 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 12 T_{2}^{30} + 92 T_{2}^{28} + 576 T_{2}^{26} + 4036 T_{2}^{24} + 15072 T_{2}^{22} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display