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Properties

Label 207.2.k.a
Level $207$
Weight $2$
Character orbit 207.k
Analytic conductor $1.653$
Analytic rank $0$
Dimension $80$
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] = [207,2,Mod(17,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.k (of order \(22\), degree \(10\), 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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{13} + 8 q^{16} - 32 q^{25} + 32 q^{31} - 44 q^{34} - 88 q^{37} - 44 q^{40} - 88 q^{43} - 144 q^{46} + 16 q^{49} - 244 q^{52} - 44 q^{55} + 36 q^{58} + 88 q^{61} + 144 q^{64} + 44 q^{67} + 104 q^{70} + 28 q^{73} + 176 q^{76} + 88 q^{79} + 176 q^{82} + 116 q^{85} - 104 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −2.21384 − 1.01103i 0 2.56919 + 2.96501i 0.384200 − 0.246910i 0 −0.813086 − 0.116904i −1.31873 − 4.49119i 0 −1.10019 + 0.158183i
17.2 −1.13300 − 0.517426i 0 −0.293751 − 0.339007i −2.14139 + 1.37619i 0 0.879451 + 0.126446i 0.859242 + 2.92631i 0 3.13828 − 0.451217i
17.3 −0.925562 − 0.422690i 0 −0.631724 − 0.729048i −0.252085 + 0.162005i 0 4.39696 + 0.632187i 0.849871 + 2.89439i 0 0.301798 − 0.0433920i
17.4 −0.321553 − 0.146848i 0 −1.22789 − 1.41706i 2.94387 − 1.89191i 0 −4.46333 − 0.641729i 0.385922 + 1.31433i 0 −1.22443 + 0.176047i
17.5 0.321553 + 0.146848i 0 −1.22789 − 1.41706i −2.94387 + 1.89191i 0 −4.46333 − 0.641729i −0.385922 − 1.31433i 0 −1.22443 + 0.176047i
17.6 0.925562 + 0.422690i 0 −0.631724 − 0.729048i 0.252085 − 0.162005i 0 4.39696 + 0.632187i −0.849871 − 2.89439i 0 0.301798 − 0.0433920i
17.7 1.13300 + 0.517426i 0 −0.293751 − 0.339007i 2.14139 − 1.37619i 0 0.879451 + 0.126446i −0.859242 − 2.92631i 0 3.13828 − 0.451217i
17.8 2.21384 + 1.01103i 0 2.56919 + 2.96501i −0.384200 + 0.246910i 0 −0.813086 − 0.116904i 1.31873 + 4.49119i 0 −1.10019 + 0.158183i
44.1 −1.48398 + 2.30911i 0 −2.29899 − 5.03408i 2.07941 + 0.610569i 0 2.24532 + 1.94558i 9.60207 + 1.38057i 0 −4.49567 + 3.89552i
44.2 −1.29685 + 2.01794i 0 −1.55942 − 3.41466i −2.95246 − 0.866919i 0 −0.0494561 − 0.0428540i 4.16429 + 0.598735i 0 5.57828 − 4.83361i
44.3 −0.791907 + 1.23223i 0 −0.0604480 − 0.132363i −1.86419 − 0.547375i 0 −3.48039 − 3.01578i −2.68872 − 0.386580i 0 2.15076 − 1.86364i
44.4 −0.343797 + 0.534958i 0 0.662846 + 1.45143i 0.481444 + 0.141365i 0 1.28453 + 1.11305i −2.26320 − 0.325400i 0 −0.241143 + 0.208951i
44.5 0.343797 − 0.534958i 0 0.662846 + 1.45143i −0.481444 − 0.141365i 0 1.28453 + 1.11305i 2.26320 + 0.325400i 0 −0.241143 + 0.208951i
44.6 0.791907 − 1.23223i 0 −0.0604480 − 0.132363i 1.86419 + 0.547375i 0 −3.48039 − 3.01578i 2.68872 + 0.386580i 0 2.15076 − 1.86364i
44.7 1.29685 − 2.01794i 0 −1.55942 − 3.41466i 2.95246 + 0.866919i 0 −0.0494561 − 0.0428540i −4.16429 − 0.598735i 0 5.57828 − 4.83361i
44.8 1.48398 − 2.30911i 0 −2.29899 − 5.03408i −2.07941 − 0.610569i 0 2.24532 + 1.94558i −9.60207 − 1.38057i 0 −4.49567 + 3.89552i
53.1 −1.67820 − 1.45417i 0 0.417123 + 2.90116i −0.566584 − 1.24065i 0 −0.443004 − 1.50873i 1.11769 − 1.73916i 0 −0.853270 + 2.90597i
53.2 −1.49323 − 1.29389i 0 0.270950 + 1.88450i 0.126149 + 0.276229i 0 1.18970 + 4.05175i −0.102678 + 0.159769i 0 0.169040 − 0.575696i
53.3 −0.712154 − 0.617085i 0 −0.158260 − 1.10072i 1.70779 + 3.73954i 0 0.437410 + 1.48968i −1.58544 + 2.46700i 0 1.09140 − 3.71697i
53.4 −0.592921 − 0.513769i 0 −0.197033 − 1.37039i 0.379818 + 0.831685i 0 −1.18411 − 4.03270i −1.43556 + 2.23377i 0 0.202092 − 0.688263i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.2.k.a ✓ 80
3.b odd 2 1 inner 207.2.k.a ✓ 80
23.d odd 22 1 inner 207.2.k.a ✓ 80
69.g even 22 1 inner 207.2.k.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.k.a ✓ 80 1.a even 1 1 trivial
207.2.k.a ✓ 80 3.b odd 2 1 inner
207.2.k.a ✓ 80 23.d odd 22 1 inner
207.2.k.a ✓ 80 69.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(207, [\chi])\).