Properties

Label 867.2.i.e
Level $867$
Weight $2$
Character orbit 867.i
Analytic conductor $6.923$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $32$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [867,2,Mod(65,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.i (of order \(16\), degree \(8\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 160 q^{18} + 384 q^{52} - 160 q^{69}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 −0.855706 2.06586i −1.00198 + 1.41281i −2.12132 + 2.12132i 1.57139 + 2.35175i 3.77606 + 0.861009i −2.62934 1.75687i 2.06586 + 0.855706i −0.992053 2.83122i 3.51373 5.25868i
65.2 −0.855706 2.06586i 1.00198 1.41281i −2.12132 + 2.12132i −1.57139 2.35175i −3.77606 0.861009i 2.62934 + 1.75687i 2.06586 + 0.855706i −0.992053 2.83122i −3.51373 + 5.25868i
65.3 0.855706 + 2.06586i −0.385055 1.68871i −2.12132 + 2.12132i 1.57139 + 2.35175i 3.15913 2.24051i 2.62934 + 1.75687i −2.06586 0.855706i −2.70347 + 1.30049i −3.51373 + 5.25868i
65.4 0.855706 + 2.06586i 0.385055 + 1.68871i −2.12132 + 2.12132i −1.57139 2.35175i −3.15913 + 2.24051i −2.62934 1.75687i −2.06586 0.855706i −2.70347 + 1.30049i 3.51373 5.25868i
131.1 −2.06586 + 0.855706i −1.70752 + 0.290496i 2.12132 2.12132i 0.551799 2.77408i 3.27891 2.06126i 3.10152 0.616930i −0.855706 + 2.06586i 2.83122 0.992053i 1.23386 + 6.20303i
131.2 −2.06586 + 0.855706i 1.70752 0.290496i 2.12132 2.12132i −0.551799 + 2.77408i −3.27891 + 2.06126i −3.10152 + 0.616930i −0.855706 + 2.06586i 2.83122 0.992053i −1.23386 6.20303i
131.3 2.06586 0.855706i −0.921821 + 1.46637i 2.12132 2.12132i −0.551799 + 2.77408i −0.649569 + 3.81812i 3.10152 0.616930i 0.855706 2.06586i −1.30049 2.70347i 1.23386 + 6.20303i
131.4 2.06586 0.855706i 0.921821 1.46637i 2.12132 2.12132i 0.551799 2.77408i 0.649569 3.81812i −3.10152 + 0.616930i 0.855706 2.06586i −1.30049 2.70347i −1.23386 6.20303i
158.1 −2.06586 + 0.855706i −1.46637 0.921821i 2.12132 2.12132i −2.77408 0.551799i 3.81812 + 0.649569i −0.616930 3.10152i −0.855706 + 2.06586i 1.30049 + 2.70347i 6.20303 1.23386i
158.2 −2.06586 + 0.855706i 1.46637 + 0.921821i 2.12132 2.12132i 2.77408 + 0.551799i −3.81812 0.649569i 0.616930 + 3.10152i −0.855706 + 2.06586i 1.30049 + 2.70347i −6.20303 + 1.23386i
158.3 2.06586 0.855706i −0.290496 1.70752i 2.12132 2.12132i 2.77408 + 0.551799i −2.06126 3.27891i −0.616930 3.10152i 0.855706 2.06586i −2.83122 + 0.992053i 6.20303 1.23386i
158.4 2.06586 0.855706i 0.290496 + 1.70752i 2.12132 2.12132i −2.77408 0.551799i 2.06126 + 3.27891i 0.616930 + 3.10152i 0.855706 2.06586i −2.83122 + 0.992053i −6.20303 + 1.23386i
224.1 −0.855706 2.06586i −1.68871 + 0.385055i −2.12132 + 2.12132i 2.35175 1.57139i 2.24051 + 3.15913i −1.75687 + 2.62934i 2.06586 + 0.855706i 2.70347 1.30049i −5.25868 3.51373i
224.2 −0.855706 2.06586i 1.68871 0.385055i −2.12132 + 2.12132i −2.35175 + 1.57139i −2.24051 3.15913i 1.75687 2.62934i 2.06586 + 0.855706i 2.70347 1.30049i 5.25868 + 3.51373i
224.3 0.855706 + 2.06586i −1.41281 1.00198i −2.12132 + 2.12132i −2.35175 + 1.57139i 0.861009 3.77606i −1.75687 + 2.62934i −2.06586 0.855706i 0.992053 + 2.83122i −5.25868 3.51373i
224.4 0.855706 + 2.06586i 1.41281 + 1.00198i −2.12132 + 2.12132i 2.35175 1.57139i −0.861009 + 3.77606i 1.75687 2.62934i −2.06586 0.855706i 0.992053 + 2.83122i 5.25868 + 3.51373i
329.1 −0.855706 + 2.06586i −1.68871 0.385055i −2.12132 2.12132i 2.35175 + 1.57139i 2.24051 3.15913i −1.75687 2.62934i 2.06586 0.855706i 2.70347 + 1.30049i −5.25868 + 3.51373i
329.2 −0.855706 + 2.06586i 1.68871 + 0.385055i −2.12132 2.12132i −2.35175 1.57139i −2.24051 + 3.15913i 1.75687 + 2.62934i 2.06586 0.855706i 2.70347 + 1.30049i 5.25868 3.51373i
329.3 0.855706 2.06586i −1.41281 + 1.00198i −2.12132 2.12132i −2.35175 1.57139i 0.861009 + 3.77606i −1.75687 2.62934i −2.06586 + 0.855706i 0.992053 2.83122i −5.25868 + 3.51373i
329.4 0.855706 2.06586i 1.41281 1.00198i −2.12132 2.12132i 2.35175 + 1.57139i −0.861009 3.77606i 1.75687 + 2.62934i −2.06586 + 0.855706i 0.992053 2.83122i 5.25868 3.51373i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner
17.e odd 16 8 inner
51.c odd 2 1 inner
51.f odd 4 2 inner
51.g odd 8 4 inner
51.i even 16 8 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.i.e 32
3.b odd 2 1 inner 867.2.i.e 32
17.b even 2 1 inner 867.2.i.e 32
17.c even 4 2 inner 867.2.i.e 32
17.d even 8 4 inner 867.2.i.e 32
17.e odd 16 8 inner 867.2.i.e 32
51.c odd 2 1 inner 867.2.i.e 32
51.f odd 4 2 inner 867.2.i.e 32
51.g odd 8 4 inner 867.2.i.e 32
51.i even 16 8 inner 867.2.i.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.i.e 32 1.a even 1 1 trivial
867.2.i.e 32 3.b odd 2 1 inner
867.2.i.e 32 17.b even 2 1 inner
867.2.i.e 32 17.c even 4 2 inner
867.2.i.e 32 17.d even 8 4 inner
867.2.i.e 32 17.e odd 16 8 inner
867.2.i.e 32 51.c odd 2 1 inner
867.2.i.e 32 51.f odd 4 2 inner
867.2.i.e 32 51.g odd 8 4 inner
867.2.i.e 32 51.i even 16 8 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{8} + 625 \) Copy content Toggle raw display
\( T_{5}^{16} + 16777216 \) Copy content Toggle raw display
\( T_{7}^{16} + 100000000 \) Copy content Toggle raw display