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Properties

Label 289.2.d.f
Level $289$
Weight $2$
Character orbit 289.d
Analytic conductor $2.308$
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] = [289,2,Mod(110,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.110");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.d (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: \(2.30767661842\)
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{16} - 72 q^{18} + 24 q^{35} - 168 q^{50} - 24 q^{52} + 72 q^{67} + 168 q^{69} + 24 q^{84} + 48 q^{86}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1 −1.08335 + 1.08335i −0.515588 + 1.24474i − 0.347296i −3.26322 − 1.35167i −0.789927 − 1.90705i 0.320860 − 0.132905i −1.79046 − 1.79046i 0.837775 + 0.837775i 4.99955 − 2.07088i
110.2 −1.08335 + 1.08335i 0.515588 − 1.24474i − 0.347296i 3.26322 + 1.35167i 0.789927 + 1.90705i −0.320860 + 0.132905i −1.79046 − 1.79046i 0.837775 + 0.837775i −4.99955 + 2.07088i
110.3 −0.245576 + 0.245576i −0.336526 + 0.812446i 1.87939i 2.16862 + 0.898271i −0.116874 − 0.282160i 1.73633 − 0.719210i −0.952682 − 0.952682i 1.57450 + 1.57450i −0.753153 + 0.311966i
110.4 −0.245576 + 0.245576i 0.336526 − 0.812446i 1.87939i −2.16862 − 0.898271i 0.116874 + 0.282160i −1.73633 + 0.719210i −0.952682 − 0.952682i 1.57450 + 1.57450i 0.753153 − 0.311966i
110.5 1.32893 − 1.32893i −0.968988 + 2.33935i − 1.53209i −0.111434 − 0.0461573i 1.82110 + 4.39653i 1.41547 − 0.586305i 0.621819 + 0.621819i −2.41228 − 2.41228i −0.209426 + 0.0867473i
110.6 1.32893 − 1.32893i 0.968988 − 2.33935i − 1.53209i 0.111434 + 0.0461573i −1.82110 − 4.39653i −1.41547 + 0.586305i 0.621819 + 0.621819i −2.41228 − 2.41228i 0.209426 − 0.0867473i
134.1 −1.08335 − 1.08335i −0.515588 − 1.24474i 0.347296i −3.26322 + 1.35167i −0.789927 + 1.90705i 0.320860 + 0.132905i −1.79046 + 1.79046i 0.837775 − 0.837775i 4.99955 + 2.07088i
134.2 −1.08335 − 1.08335i 0.515588 + 1.24474i 0.347296i 3.26322 − 1.35167i 0.789927 − 1.90705i −0.320860 − 0.132905i −1.79046 + 1.79046i 0.837775 − 0.837775i −4.99955 − 2.07088i
134.3 −0.245576 − 0.245576i −0.336526 − 0.812446i − 1.87939i 2.16862 − 0.898271i −0.116874 + 0.282160i 1.73633 + 0.719210i −0.952682 + 0.952682i 1.57450 − 1.57450i −0.753153 − 0.311966i
134.4 −0.245576 − 0.245576i 0.336526 + 0.812446i − 1.87939i −2.16862 + 0.898271i 0.116874 − 0.282160i −1.73633 − 0.719210i −0.952682 + 0.952682i 1.57450 − 1.57450i 0.753153 + 0.311966i
134.5 1.32893 + 1.32893i −0.968988 − 2.33935i 1.53209i −0.111434 + 0.0461573i 1.82110 − 4.39653i 1.41547 + 0.586305i 0.621819 − 0.621819i −2.41228 + 2.41228i −0.209426 − 0.0867473i
134.6 1.32893 + 1.32893i 0.968988 + 2.33935i 1.53209i 0.111434 − 0.0461573i −1.82110 + 4.39653i −1.41547 − 0.586305i 0.621819 − 0.621819i −2.41228 + 2.41228i 0.209426 + 0.0867473i
155.1 −1.32893 − 1.32893i −2.33935 + 0.968988i 1.53209i 0.0461573 + 0.111434i 4.39653 + 1.82110i −0.586305 + 1.41547i −0.621819 + 0.621819i 2.41228 − 2.41228i 0.0867473 − 0.209426i
155.2 −1.32893 − 1.32893i 2.33935 − 0.968988i 1.53209i −0.0461573 − 0.111434i −4.39653 − 1.82110i 0.586305 − 1.41547i −0.621819 + 0.621819i 2.41228 − 2.41228i −0.0867473 + 0.209426i
155.3 0.245576 + 0.245576i −0.812446 + 0.336526i − 1.87939i −0.898271 − 2.16862i −0.282160 − 0.116874i −0.719210 + 1.73633i 0.952682 − 0.952682i −1.57450 + 1.57450i 0.311966 − 0.753153i
155.4 0.245576 + 0.245576i 0.812446 − 0.336526i − 1.87939i 0.898271 + 2.16862i 0.282160 + 0.116874i 0.719210 − 1.73633i 0.952682 − 0.952682i −1.57450 + 1.57450i −0.311966 + 0.753153i
155.5 1.08335 + 1.08335i −1.24474 + 0.515588i 0.347296i 1.35167 + 3.26322i −1.90705 − 0.789927i −0.132905 + 0.320860i 1.79046 − 1.79046i −0.837775 + 0.837775i −2.07088 + 4.99955i
155.6 1.08335 + 1.08335i 1.24474 − 0.515588i 0.347296i −1.35167 − 3.26322i 1.90705 + 0.789927i 0.132905 − 0.320860i 1.79046 − 1.79046i −0.837775 + 0.837775i 2.07088 − 4.99955i
179.1 −1.32893 + 1.32893i −2.33935 − 0.968988i − 1.53209i 0.0461573 − 0.111434i 4.39653 − 1.82110i −0.586305 − 1.41547i −0.621819 − 0.621819i 2.41228 + 2.41228i 0.0867473 + 0.209426i
179.2 −1.32893 + 1.32893i 2.33935 + 0.968988i − 1.53209i −0.0461573 + 0.111434i −4.39653 + 1.82110i 0.586305 + 1.41547i −0.621819 − 0.621819i 2.41228 + 2.41228i −0.0867473 − 0.209426i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 110.6
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 289.2.d.f 24
17.b even 2 1 inner 289.2.d.f 24
17.c even 4 2 inner 289.2.d.f 24
17.d even 8 4 inner 289.2.d.f 24
17.e odd 16 1 289.2.a.d ✓ 3
17.e odd 16 1 289.2.a.e yes 3
17.e odd 16 2 289.2.b.d 6
17.e odd 16 4 289.2.c.d 12
51.i even 16 1 2601.2.a.w 3
51.i even 16 1 2601.2.a.x 3
68.i even 16 1 4624.2.a.bd 3
68.i even 16 1 4624.2.a.bg 3
85.p odd 16 1 7225.2.a.s 3
85.p odd 16 1 7225.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.2.a.d ✓ 3 17.e odd 16 1
289.2.a.e yes 3 17.e odd 16 1
289.2.b.d 6 17.e odd 16 2
289.2.c.d 12 17.e odd 16 4
289.2.d.f 24 1.a even 1 1 trivial
289.2.d.f 24 17.b even 2 1 inner
289.2.d.f 24 17.c even 4 2 inner
289.2.d.f 24 17.d even 8 4 inner
2601.2.a.w 3 51.i even 16 1
2601.2.a.x 3 51.i even 16 1
4624.2.a.bd 3 68.i even 16 1
4624.2.a.bg 3 68.i even 16 1
7225.2.a.s 3 85.p odd 16 1
7225.2.a.t 3 85.p odd 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(289, [\chi])\):

\( T_{2}^{12} + 18T_{2}^{8} + 69T_{2}^{4} + 1 \) Copy content Toggle raw display
\( T_{3}^{24} + 1701T_{3}^{16} + 18954T_{3}^{8} + 6561 \) Copy content Toggle raw display