Properties

Label 561.2.h.d
Level $561$
Weight $2$
Character orbit 561.h
Analytic conductor $4.480$
Analytic rank $0$
Dimension $56$
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] = [561,2,Mod(560,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.560");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.h (of order \(2\), degree \(1\), 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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 72 q^{4} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 72 q^{4} - 44 q^{9} + 36 q^{15} + 24 q^{16} + 16 q^{25} + 12 q^{33} - 32 q^{34} - 56 q^{36} - 8 q^{42} + 112 q^{49} - 24 q^{55} - 48 q^{60} - 40 q^{64} + 64 q^{67} + 44 q^{69} - 160 q^{70} - 44 q^{81} + 68 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
560.1 −2.62229 −0.492467 1.66056i 4.87641 −1.87989 1.29139 + 4.35448i −4.40416 −7.54277 −2.51495 + 1.63555i 4.92962
560.2 −2.62229 −0.492467 + 1.66056i 4.87641 −1.87989 1.29139 4.35448i −4.40416 −7.54277 −2.51495 1.63555i 4.92962
560.3 −2.62229 0.492467 1.66056i 4.87641 1.87989 −1.29139 + 4.35448i 4.40416 −7.54277 −2.51495 1.63555i −4.92962
560.4 −2.62229 0.492467 + 1.66056i 4.87641 1.87989 −1.29139 4.35448i 4.40416 −7.54277 −2.51495 + 1.63555i −4.92962
560.5 −2.41427 −1.39719 1.02364i 3.82869 1.10630 3.37320 + 2.47135i −0.177376 −4.41495 0.904305 + 2.86046i −2.67090
560.6 −2.41427 −1.39719 + 1.02364i 3.82869 1.10630 3.37320 2.47135i −0.177376 −4.41495 0.904305 2.86046i −2.67090
560.7 −2.41427 1.39719 1.02364i 3.82869 −1.10630 −3.37320 + 2.47135i 0.177376 −4.41495 0.904305 2.86046i 2.67090
560.8 −2.41427 1.39719 + 1.02364i 3.82869 −1.10630 −3.37320 2.47135i 0.177376 −4.41495 0.904305 + 2.86046i 2.67090
560.9 −1.96484 −0.118967 1.72796i 1.86059 3.79231 0.233751 + 3.39516i −2.62787 0.273922 −2.97169 + 0.411141i −7.45127
560.10 −1.96484 −0.118967 + 1.72796i 1.86059 3.79231 0.233751 3.39516i −2.62787 0.273922 −2.97169 0.411141i −7.45127
560.11 −1.96484 0.118967 1.72796i 1.86059 −3.79231 −0.233751 + 3.39516i 2.62787 0.273922 −2.97169 0.411141i 7.45127
560.12 −1.96484 0.118967 + 1.72796i 1.86059 −3.79231 −0.233751 3.39516i 2.62787 0.273922 −2.97169 + 0.411141i 7.45127
560.13 −1.75720 −1.65768 0.502106i 1.08774 −2.02373 2.91286 + 0.882298i −0.533738 1.60303 2.49578 + 1.66466i 3.55610
560.14 −1.75720 −1.65768 + 0.502106i 1.08774 −2.02373 2.91286 0.882298i −0.533738 1.60303 2.49578 1.66466i 3.55610
560.15 −1.75720 1.65768 0.502106i 1.08774 2.02373 −2.91286 + 0.882298i 0.533738 1.60303 2.49578 1.66466i −3.55610
560.16 −1.75720 1.65768 + 0.502106i 1.08774 2.02373 −2.91286 0.882298i 0.533738 1.60303 2.49578 + 1.66466i −3.55610
560.17 −1.63012 −1.13840 1.30539i 0.657285 1.34971 1.85573 + 2.12793i 4.67831 2.18878 −0.408070 + 2.97212i −2.20018
560.18 −1.63012 −1.13840 + 1.30539i 0.657285 1.34971 1.85573 2.12793i 4.67831 2.18878 −0.408070 2.97212i −2.20018
560.19 −1.63012 1.13840 1.30539i 0.657285 −1.34971 −1.85573 + 2.12793i −4.67831 2.18878 −0.408070 2.97212i 2.20018
560.20 −1.63012 1.13840 + 1.30539i 0.657285 −1.34971 −1.85573 2.12793i −4.67831 2.18878 −0.408070 + 2.97212i 2.20018
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 560.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
17.b even 2 1 inner
33.d even 2 1 inner
51.c odd 2 1 inner
187.b odd 2 1 inner
561.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.h.d 56
3.b odd 2 1 inner 561.2.h.d 56
11.b odd 2 1 inner 561.2.h.d 56
17.b even 2 1 inner 561.2.h.d 56
33.d even 2 1 inner 561.2.h.d 56
51.c odd 2 1 inner 561.2.h.d 56
187.b odd 2 1 inner 561.2.h.d 56
561.h even 2 1 inner 561.2.h.d 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.h.d 56 1.a even 1 1 trivial
561.2.h.d 56 3.b odd 2 1 inner
561.2.h.d 56 11.b odd 2 1 inner
561.2.h.d 56 17.b even 2 1 inner
561.2.h.d 56 33.d even 2 1 inner
561.2.h.d 56 51.c odd 2 1 inner
561.2.h.d 56 187.b odd 2 1 inner
561.2.h.d 56 561.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 23T_{2}^{12} + 208T_{2}^{10} - 938T_{2}^{8} + 2196T_{2}^{6} - 2480T_{2}^{4} + 1064T_{2}^{2} - 148 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display