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Properties

Label 867.2.h.l
Level $867$
Weight $2$
Character orbit 867.h
Analytic conductor $6.923$
Analytic rank $0$
Dimension $24$
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] = [867,2,Mod(688,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.688");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.h (of order \(8\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{16} - 24 q^{18} - 24 q^{33} - 48 q^{35} - 48 q^{50} - 48 q^{52} + 144 q^{67} + 24 q^{69} + 168 q^{84} + 240 q^{86}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
688.1 −0.621819 + 0.621819i −0.382683 + 0.923880i 1.22668i 1.24474 + 0.515588i −0.336526 − 0.812446i −1.13331 + 0.469431i −2.00641 − 2.00641i −0.707107 − 0.707107i −1.09461 + 0.453400i
688.2 −0.621819 + 0.621819i 0.382683 − 0.923880i 1.22668i −1.24474 − 0.515588i 0.336526 + 0.812446i 1.13331 − 0.469431i −2.00641 − 2.00641i −0.707107 − 0.707107i 1.09461 − 0.453400i
688.3 0.952682 − 0.952682i −0.382683 + 0.923880i 0.184793i 2.33935 + 0.968988i 0.515588 + 1.24474i −0.170726 + 0.0707170i 2.08141 + 2.08141i −0.707107 − 0.707107i 3.15179 − 1.30551i
688.4 0.952682 − 0.952682i 0.382683 − 0.923880i 0.184793i −2.33935 − 0.968988i −0.515588 − 1.24474i 0.170726 − 0.0707170i 2.08141 + 2.08141i −0.707107 − 0.707107i −3.15179 + 1.30551i
688.5 1.79046 − 1.79046i −0.382683 + 0.923880i − 4.41147i −0.812446 − 0.336526i 0.968988 + 2.33935i 4.07567 − 1.68820i −4.31764 − 4.31764i −0.707107 − 0.707107i −2.05719 + 0.852114i
688.6 1.79046 − 1.79046i 0.382683 − 0.923880i − 4.41147i 0.812446 + 0.336526i −0.968988 − 2.33935i −4.07567 + 1.68820i −4.31764 − 4.31764i −0.707107 − 0.707107i 2.05719 − 0.852114i
712.1 −0.621819 − 0.621819i −0.382683 − 0.923880i − 1.22668i 1.24474 − 0.515588i −0.336526 + 0.812446i −1.13331 − 0.469431i −2.00641 + 2.00641i −0.707107 + 0.707107i −1.09461 − 0.453400i
712.2 −0.621819 − 0.621819i 0.382683 + 0.923880i − 1.22668i −1.24474 + 0.515588i 0.336526 − 0.812446i 1.13331 + 0.469431i −2.00641 + 2.00641i −0.707107 + 0.707107i 1.09461 + 0.453400i
712.3 0.952682 + 0.952682i −0.382683 − 0.923880i − 0.184793i 2.33935 − 0.968988i 0.515588 − 1.24474i −0.170726 − 0.0707170i 2.08141 − 2.08141i −0.707107 + 0.707107i 3.15179 + 1.30551i
712.4 0.952682 + 0.952682i 0.382683 + 0.923880i − 0.184793i −2.33935 + 0.968988i −0.515588 + 1.24474i 0.170726 + 0.0707170i 2.08141 − 2.08141i −0.707107 + 0.707107i −3.15179 − 1.30551i
712.5 1.79046 + 1.79046i −0.382683 − 0.923880i 4.41147i −0.812446 + 0.336526i 0.968988 − 2.33935i 4.07567 + 1.68820i −4.31764 + 4.31764i −0.707107 + 0.707107i −2.05719 − 0.852114i
712.6 1.79046 + 1.79046i 0.382683 + 0.923880i 4.41147i 0.812446 − 0.336526i −0.968988 + 2.33935i −4.07567 − 1.68820i −4.31764 + 4.31764i −0.707107 + 0.707107i 2.05719 + 0.852114i
733.1 −1.79046 − 1.79046i −0.923880 + 0.382683i 4.41147i 0.336526 + 0.812446i 2.33935 + 0.968988i −1.68820 + 4.07567i 4.31764 − 4.31764i 0.707107 − 0.707107i 0.852114 − 2.05719i
733.2 −1.79046 − 1.79046i 0.923880 − 0.382683i 4.41147i −0.336526 − 0.812446i −2.33935 − 0.968988i 1.68820 − 4.07567i 4.31764 − 4.31764i 0.707107 − 0.707107i −0.852114 + 2.05719i
733.3 −0.952682 − 0.952682i −0.923880 + 0.382683i − 0.184793i −0.968988 − 2.33935i 1.24474 + 0.515588i 0.0707170 − 0.170726i −2.08141 + 2.08141i 0.707107 − 0.707107i −1.30551 + 3.15179i
733.4 −0.952682 − 0.952682i 0.923880 − 0.382683i − 0.184793i 0.968988 + 2.33935i −1.24474 − 0.515588i −0.0707170 + 0.170726i −2.08141 + 2.08141i 0.707107 − 0.707107i 1.30551 − 3.15179i
733.5 0.621819 + 0.621819i −0.923880 + 0.382683i − 1.22668i −0.515588 − 1.24474i −0.812446 − 0.336526i 0.469431 − 1.13331i 2.00641 − 2.00641i 0.707107 − 0.707107i 0.453400 − 1.09461i
733.6 0.621819 + 0.621819i 0.923880 − 0.382683i − 1.22668i 0.515588 + 1.24474i 0.812446 + 0.336526i −0.469431 + 1.13331i 2.00641 − 2.00641i 0.707107 − 0.707107i −0.453400 + 1.09461i
757.1 −1.79046 + 1.79046i −0.923880 − 0.382683i − 4.41147i 0.336526 − 0.812446i 2.33935 − 0.968988i −1.68820 − 4.07567i 4.31764 + 4.31764i 0.707107 + 0.707107i 0.852114 + 2.05719i
757.2 −1.79046 + 1.79046i 0.923880 + 0.382683i − 4.41147i −0.336526 + 0.812446i −2.33935 + 0.968988i 1.68820 + 4.07567i 4.31764 + 4.31764i 0.707107 + 0.707107i −0.852114 − 2.05719i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 688.6
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
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.h.l 24
17.b even 2 1 inner 867.2.h.l 24
17.c even 4 2 inner 867.2.h.l 24
17.d even 8 4 inner 867.2.h.l 24
17.e odd 16 1 867.2.a.i ✓ 3
17.e odd 16 1 867.2.a.j yes 3
17.e odd 16 2 867.2.d.d 6
17.e odd 16 4 867.2.e.j 12
51.i even 16 1 2601.2.a.y 3
51.i even 16 1 2601.2.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.i ✓ 3 17.e odd 16 1
867.2.a.j yes 3 17.e odd 16 1
867.2.d.d 6 17.e odd 16 2
867.2.e.j 12 17.e odd 16 4
867.2.h.l 24 1.a even 1 1 trivial
867.2.h.l 24 17.b even 2 1 inner
867.2.h.l 24 17.c even 4 2 inner
867.2.h.l 24 17.d even 8 4 inner
2601.2.a.y 3 51.i even 16 1
2601.2.a.z 3 51.i even 16 1

Hecke kernels

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

\( T_{2}^{12} + 45T_{2}^{8} + 162T_{2}^{4} + 81 \) Copy content Toggle raw display
\( T_{5}^{24} + 1701T_{5}^{16} + 18954T_{5}^{8} + 6561 \) Copy content Toggle raw display