↑

Properties

Label 847.2.l.k
Level $847$
Weight $2$
Character orbit 847.l
Analytic conductor $6.763$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(118,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, 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("847.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.l (of order \(10\), degree \(4\), not 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: \(6.76332905120\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{9} - 12 q^{14} + 8 q^{16} - 96 q^{23} + 48 q^{25} + 40 q^{37} - 32 q^{49} + 24 q^{53} + 48 q^{56} - 16 q^{58} - 32 q^{64} - 32 q^{67} + 44 q^{70} + 32 q^{71} - 72 q^{81} - 80 q^{86} - 44 q^{91} - 24 q^{92}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −1.13551 − 1.56290i 0 −0.535233 + 1.64728i −1.94946 + 2.68321i 0 −1.88951 + 1.85196i −0.492303 + 0.159959i −2.42705 + 1.76336i 6.40723
118.2 −1.13551 − 1.56290i 0 −0.535233 + 1.64728i 1.94946 − 2.68321i 0 −0.440093 − 2.60889i −0.492303 + 0.159959i −2.42705 + 1.76336i −6.40723
118.3 −0.304260 − 0.418778i 0 0.535233 − 1.64728i −1.94946 + 2.68321i 0 −0.440093 − 2.60889i −1.83730 + 0.596975i −2.42705 + 1.76336i 1.71681
118.4 −0.304260 − 0.418778i 0 0.535233 − 1.64728i 1.94946 − 2.68321i 0 −1.88951 + 1.85196i −1.83730 + 0.596975i −2.42705 + 1.76336i −1.71681
118.5 0.304260 + 0.418778i 0 0.535233 − 1.64728i −1.94946 + 2.68321i 0 0.440093 + 2.60889i 1.83730 − 0.596975i −2.42705 + 1.76336i −1.71681
118.6 0.304260 + 0.418778i 0 0.535233 − 1.64728i 1.94946 − 2.68321i 0 1.88951 − 1.85196i 1.83730 − 0.596975i −2.42705 + 1.76336i 1.71681
118.7 1.13551 + 1.56290i 0 −0.535233 + 1.64728i −1.94946 + 2.68321i 0 1.88951 − 1.85196i 0.492303 − 0.159959i −2.42705 + 1.76336i −6.40723
118.8 1.13551 + 1.56290i 0 −0.535233 + 1.64728i 1.94946 − 2.68321i 0 0.440093 + 2.60889i 0.492303 − 0.159959i −2.42705 + 1.76336i 6.40723
475.1 −1.83730 − 0.596975i 0 1.40126 + 1.01807i −3.15430 + 1.02489i 0 2.61720 + 0.387639i 0.304260 + 0.418778i 0.927051 − 2.85317i 6.40723
475.2 −1.83730 − 0.596975i 0 1.40126 + 1.01807i 3.15430 − 1.02489i 0 −1.17743 − 2.36932i 0.304260 + 0.418778i 0.927051 − 2.85317i −6.40723
475.3 −0.492303 − 0.159959i 0 −1.40126 − 1.01807i −3.15430 + 1.02489i 0 −1.17743 − 2.36932i 1.13551 + 1.56290i 0.927051 − 2.85317i 1.71681
475.4 −0.492303 − 0.159959i 0 −1.40126 − 1.01807i 3.15430 − 1.02489i 0 2.61720 + 0.387639i 1.13551 + 1.56290i 0.927051 − 2.85317i −1.71681
475.5 0.492303 + 0.159959i 0 −1.40126 − 1.01807i −3.15430 + 1.02489i 0 1.17743 + 2.36932i −1.13551 − 1.56290i 0.927051 − 2.85317i −1.71681
475.6 0.492303 + 0.159959i 0 −1.40126 − 1.01807i 3.15430 − 1.02489i 0 −2.61720 − 0.387639i −1.13551 − 1.56290i 0.927051 − 2.85317i 1.71681
475.7 1.83730 + 0.596975i 0 1.40126 + 1.01807i −3.15430 + 1.02489i 0 −2.61720 − 0.387639i −0.304260 − 0.418778i 0.927051 − 2.85317i −6.40723
475.8 1.83730 + 0.596975i 0 1.40126 + 1.01807i 3.15430 − 1.02489i 0 1.17743 + 2.36932i −0.304260 − 0.418778i 0.927051 − 2.85317i 6.40723
524.1 −1.13551 + 1.56290i 0 −0.535233 − 1.64728i −1.94946 − 2.68321i 0 −1.88951 − 1.85196i −0.492303 − 0.159959i −2.42705 − 1.76336i 6.40723
524.2 −1.13551 + 1.56290i 0 −0.535233 − 1.64728i 1.94946 + 2.68321i 0 −0.440093 + 2.60889i −0.492303 − 0.159959i −2.42705 − 1.76336i −6.40723
524.3 −0.304260 + 0.418778i 0 0.535233 + 1.64728i −1.94946 − 2.68321i 0 −0.440093 + 2.60889i −1.83730 − 0.596975i −2.42705 − 1.76336i 1.71681
524.4 −0.304260 + 0.418778i 0 0.535233 + 1.64728i 1.94946 + 2.68321i 0 −1.88951 − 1.85196i −1.83730 − 0.596975i −2.42705 − 1.76336i −1.71681
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
77.b even 2 1 inner
77.j odd 10 3 inner
77.l even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.l.k 32
7.b odd 2 1 inner 847.2.l.k 32
11.b odd 2 1 inner 847.2.l.k 32
11.c even 5 1 847.2.b.d ✓ 8
11.c even 5 3 inner 847.2.l.k 32
11.d odd 10 1 847.2.b.d ✓ 8
11.d odd 10 3 inner 847.2.l.k 32
77.b even 2 1 inner 847.2.l.k 32
77.j odd 10 1 847.2.b.d ✓ 8
77.j odd 10 3 inner 847.2.l.k 32
77.l even 10 1 847.2.b.d ✓ 8
77.l even 10 3 inner 847.2.l.k 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.b.d ✓ 8 11.c even 5 1
847.2.b.d ✓ 8 11.d odd 10 1
847.2.b.d ✓ 8 77.j odd 10 1
847.2.b.d ✓ 8 77.l even 10 1
847.2.l.k 32 1.a even 1 1 trivial
847.2.l.k 32 7.b odd 2 1 inner
847.2.l.k 32 11.b odd 2 1 inner
847.2.l.k 32 11.c even 5 3 inner
847.2.l.k 32 11.d odd 10 3 inner
847.2.l.k 32 77.b even 2 1 inner
847.2.l.k 32 77.j odd 10 3 inner
847.2.l.k 32 77.l even 10 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 4T_{2}^{14} + 15T_{2}^{12} - 56T_{2}^{10} + 209T_{2}^{8} - 56T_{2}^{6} + 15T_{2}^{4} - 4T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display