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Properties

Label 693.2.be.a
Level $693$
Weight $2$
Character orbit 693.be
Analytic conductor $5.534$
Analytic rank $0$
Dimension $56$
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] = [693,2,Mod(89,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.be (of order \(6\), degree \(2\), 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 32 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 32 q^{4} - 4 q^{7} - 40 q^{16} - 12 q^{19} - 36 q^{25} + 48 q^{28} + 12 q^{31} - 12 q^{37} - 120 q^{40} - 8 q^{43} - 56 q^{46} + 20 q^{49} + 72 q^{52} + 32 q^{58} + 24 q^{61} - 64 q^{64} + 44 q^{67} + 40 q^{70} + 12 q^{73} - 20 q^{79} + 168 q^{82} - 16 q^{85} + 24 q^{88} - 92 q^{91} - 96 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 −2.42408 − 1.39954i 0 2.91745 + 5.05318i 1.56843 − 2.71660i 0 1.69282 − 2.03331i − 10.7342i 0 −7.60401 + 4.39018i
89.2 −2.21566 − 1.27921i 0 2.27276 + 3.93654i −0.535435 + 0.927400i 0 1.41126 + 2.23793i − 6.51252i 0 2.37268 − 1.36987i
89.3 −2.15043 − 1.24155i 0 2.08289 + 3.60768i 1.05405 − 1.82567i 0 −2.60541 − 0.460280i − 5.37787i 0 −4.53331 + 2.61731i
89.4 −1.93141 − 1.11510i 0 1.48689 + 2.57537i −1.51558 + 2.62506i 0 −0.977022 − 2.45875i − 2.17172i 0 5.85440 − 3.38004i
89.5 −1.86110 − 1.07451i 0 1.30912 + 2.26747i −1.08975 + 1.88750i 0 2.13285 + 1.56556i − 1.32862i 0 4.05626 − 2.34188i
89.6 −1.76440 − 1.01868i 0 1.07541 + 1.86266i 1.30421 − 2.25895i 0 −1.13713 + 2.38892i − 0.307271i 0 −4.60229 + 2.65713i
89.7 −1.58572 − 0.915517i 0 0.676343 + 1.17146i −0.217660 + 0.376998i 0 2.02228 − 1.70598i 1.18525i 0 0.690296 − 0.398542i
89.8 −1.44598 − 0.834837i 0 0.393904 + 0.682263i −1.74573 + 3.02369i 0 −2.48702 − 0.902629i 2.02396i 0 5.04857 − 2.91480i
89.9 −0.983916 − 0.568064i 0 −0.354607 − 0.614197i −0.228920 + 0.396501i 0 −2.37684 + 1.16217i 3.07801i 0 0.450476 − 0.260082i
89.10 −0.907190 − 0.523766i 0 −0.451338 − 0.781740i 1.88313 − 3.26167i 0 −0.995906 − 2.45116i 3.04065i 0 −3.41671 + 1.97264i
89.11 −0.700268 − 0.404300i 0 −0.673083 − 1.16581i 0.647797 − 1.12202i 0 0.829946 + 2.51221i 2.70571i 0 −0.907263 + 0.523809i
89.12 −0.512545 − 0.295918i 0 −0.824865 − 1.42871i −1.09505 + 1.89668i 0 2.51613 − 0.817980i 2.16004i 0 1.12253 − 0.648091i
89.13 −0.341422 − 0.197120i 0 −0.922288 − 1.59745i −2.15508 + 3.73271i 0 1.56016 + 2.13680i 1.51568i 0 1.47158 − 0.849620i
89.14 −0.130783 − 0.0755078i 0 −0.988597 − 1.71230i 0.0122719 − 0.0212556i 0 −2.58612 + 0.558556i 0.600618i 0 −0.00320993 + 0.00185325i
89.15 0.130783 + 0.0755078i 0 −0.988597 − 1.71230i −0.0122719 + 0.0212556i 0 −2.58612 + 0.558556i − 0.600618i 0 −0.00320993 + 0.00185325i
89.16 0.341422 + 0.197120i 0 −0.922288 − 1.59745i 2.15508 − 3.73271i 0 1.56016 + 2.13680i − 1.51568i 0 1.47158 − 0.849620i
89.17 0.512545 + 0.295918i 0 −0.824865 − 1.42871i 1.09505 − 1.89668i 0 2.51613 − 0.817980i − 2.16004i 0 1.12253 − 0.648091i
89.18 0.700268 + 0.404300i 0 −0.673083 − 1.16581i −0.647797 + 1.12202i 0 0.829946 + 2.51221i − 2.70571i 0 −0.907263 + 0.523809i
89.19 0.907190 + 0.523766i 0 −0.451338 − 0.781740i −1.88313 + 3.26167i 0 −0.995906 − 2.45116i − 3.04065i 0 −3.41671 + 1.97264i
89.20 0.983916 + 0.568064i 0 −0.354607 − 0.614197i 0.228920 − 0.396501i 0 −2.37684 + 1.16217i − 3.07801i 0 0.450476 − 0.260082i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.be.a ✓ 56
3.b odd 2 1 inner 693.2.be.a ✓ 56
7.d odd 6 1 inner 693.2.be.a ✓ 56
21.g even 6 1 inner 693.2.be.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.be.a ✓ 56 1.a even 1 1 trivial
693.2.be.a ✓ 56 3.b odd 2 1 inner
693.2.be.a ✓ 56 7.d odd 6 1 inner
693.2.be.a ✓ 56 21.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).