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Properties

Label 867.2.e.k
Level $867$
Weight $2$
Character orbit 867.e
Analytic conductor $6.923$
Analytic rank $0$
Dimension $24$
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] = [867,2,Mod(616,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.616");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.e (of order \(4\), degree \(2\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 36 q^{4} - 36 q^{13} + 60 q^{16} + 12 q^{18} - 12 q^{21} - 48 q^{30} - 36 q^{33} + 24 q^{38} - 96 q^{47} + 48 q^{50} - 72 q^{52} + 96 q^{55} - 96 q^{64} - 24 q^{67} - 36 q^{69} - 48 q^{72} - 24 q^{81} - 60 q^{84} + 24 q^{86} + 228 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
616.1 − 2.09548i −0.707107 + 0.707107i −2.39104 −1.10178 + 1.10178i 1.48173 + 1.48173i 2.92418 + 2.92418i 0.819422i − 1.00000i 2.30876 + 2.30876i
616.2 − 2.09548i 0.707107 − 0.707107i −2.39104 1.10178 − 1.10178i −1.48173 − 1.48173i −2.92418 − 2.92418i 0.819422i − 1.00000i −2.30876 − 2.30876i
616.3 − 1.43915i −0.707107 + 0.707107i −0.0711653 −1.63621 + 1.63621i 1.01764 + 1.01764i −3.14078 − 3.14078i − 2.77589i − 1.00000i 2.35475 + 2.35475i
616.4 − 1.43915i 0.707107 − 0.707107i −0.0711653 1.63621 − 1.63621i −1.01764 − 1.01764i 3.14078 + 3.14078i − 2.77589i − 1.00000i −2.35475 − 2.35475i
616.5 0.435433i −0.707107 + 0.707107i 1.81040 −2.98454 + 2.98454i −0.307898 − 0.307898i 2.05755 + 2.05755i 1.65917i − 1.00000i −1.29957 − 1.29957i
616.6 0.435433i 0.707107 − 0.707107i 1.81040 2.98454 − 2.98454i 0.307898 + 0.307898i −2.05755 − 2.05755i 1.65917i − 1.00000i 1.29957 + 1.29957i
616.7 0.907065i −0.707107 + 0.707107i 1.17723 2.25802 − 2.25802i −0.641392 − 0.641392i 2.51896 + 2.51896i 2.88196i − 1.00000i 2.04818 + 2.04818i
616.8 0.907065i 0.707107 − 0.707107i 1.17723 −2.25802 + 2.25802i 0.641392 + 0.641392i −2.51896 − 2.51896i 2.88196i − 1.00000i −2.04818 − 2.04818i
616.9 2.44395i −0.707107 + 0.707107i −3.97290 2.03186 − 2.03186i −1.72814 − 1.72814i −1.10487 − 1.10487i − 4.82168i − 1.00000i 4.96577 + 4.96577i
616.10 2.44395i 0.707107 − 0.707107i −3.97290 −2.03186 + 2.03186i 1.72814 + 1.72814i 1.10487 + 1.10487i − 4.82168i − 1.00000i −4.96577 − 4.96577i
616.11 2.74819i −0.707107 + 0.707107i −5.55252 −0.688676 + 0.688676i −1.94326 − 1.94326i −1.13372 − 1.13372i − 9.76298i − 1.00000i −1.89261 − 1.89261i
616.12 2.74819i 0.707107 − 0.707107i −5.55252 0.688676 − 0.688676i 1.94326 + 1.94326i 1.13372 + 1.13372i − 9.76298i − 1.00000i 1.89261 + 1.89261i
829.1 − 2.74819i −0.707107 − 0.707107i −5.55252 −0.688676 − 0.688676i −1.94326 + 1.94326i −1.13372 + 1.13372i 9.76298i 1.00000i −1.89261 + 1.89261i
829.2 − 2.74819i 0.707107 + 0.707107i −5.55252 0.688676 + 0.688676i 1.94326 − 1.94326i 1.13372 − 1.13372i 9.76298i 1.00000i 1.89261 − 1.89261i
829.3 − 2.44395i −0.707107 − 0.707107i −3.97290 2.03186 + 2.03186i −1.72814 + 1.72814i −1.10487 + 1.10487i 4.82168i 1.00000i 4.96577 − 4.96577i
829.4 − 2.44395i 0.707107 + 0.707107i −3.97290 −2.03186 − 2.03186i 1.72814 − 1.72814i 1.10487 − 1.10487i 4.82168i 1.00000i −4.96577 + 4.96577i
829.5 − 0.907065i −0.707107 − 0.707107i 1.17723 2.25802 + 2.25802i −0.641392 + 0.641392i 2.51896 − 2.51896i − 2.88196i 1.00000i 2.04818 − 2.04818i
829.6 − 0.907065i 0.707107 + 0.707107i 1.17723 −2.25802 − 2.25802i 0.641392 − 0.641392i −2.51896 + 2.51896i − 2.88196i 1.00000i −2.04818 + 2.04818i
829.7 − 0.435433i −0.707107 − 0.707107i 1.81040 −2.98454 − 2.98454i −0.307898 + 0.307898i 2.05755 − 2.05755i − 1.65917i 1.00000i −1.29957 + 1.29957i
829.8 − 0.435433i 0.707107 + 0.707107i 1.81040 2.98454 + 2.98454i 0.307898 − 0.307898i −2.05755 + 2.05755i − 1.65917i 1.00000i 1.29957 − 1.29957i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 616.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.e.k 24
17.b even 2 1 inner 867.2.e.k 24
17.c even 4 2 inner 867.2.e.k 24
17.d even 8 1 867.2.a.o ✓ 6
17.d even 8 1 867.2.a.p yes 6
17.d even 8 2 867.2.d.g 12
17.e odd 16 8 867.2.h.m 48
51.g odd 8 1 2601.2.a.bh 6
51.g odd 8 1 2601.2.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.o ✓ 6 17.d even 8 1
867.2.a.p yes 6 17.d even 8 1
867.2.d.g 12 17.d even 8 2
867.2.e.k 24 1.a even 1 1 trivial
867.2.e.k 24 17.b even 2 1 inner
867.2.e.k 24 17.c even 4 2 inner
867.2.h.m 48 17.e odd 16 8
2601.2.a.bh 6 51.g odd 8 1
2601.2.a.bi 6 51.g odd 8 1

Hecke kernels

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

\( T_{2}^{12} + 21T_{2}^{10} + 162T_{2}^{8} + 561T_{2}^{6} + 852T_{2}^{4} + 480T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{5}^{24} + 525T_{5}^{20} + 79290T_{5}^{16} + 4537217T_{5}^{12} + 92217504T_{5}^{8} + 459576576T_{5}^{4} + 342102016 \) Copy content Toggle raw display