↑

Properties

Label 684.2.bs.a
Level $684$
Weight $2$
Character orbit 684.bs
Analytic conductor $5.462$
Analytic rank $0$
Dimension $696$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,2,Mod(23,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 15, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.bs (of order \(18\), degree \(6\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(696\)
Relative dimension: \(116\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 9 q^{2} - 3 q^{4} - 18 q^{5} - 6 q^{6} - 18 q^{9} - 12 q^{10} - 3 q^{12} - 6 q^{13} - 9 q^{14} - 3 q^{16} - 12 q^{18} - 18 q^{20} + 24 q^{21} + 9 q^{22} + 45 q^{24} - 6 q^{25} + 72 q^{26} - 18 q^{29}+ \cdots + 45 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.41202 + 0.0786649i −1.38928 + 1.03436i 1.98762 − 0.222153i −4.15853 + 0.733261i 1.88033 − 1.56983i − 3.64991i −2.78910 + 0.470042i 0.860198 − 2.87403i 5.81426 − 1.36251i
23.2 −1.40911 + 0.120010i −0.775266 + 1.54886i 1.97120 − 0.338215i 3.87264 − 0.682851i 0.906558 − 2.27556i 0.103868i −2.73705 + 0.713147i −1.79793 − 2.40155i −5.37504 + 1.42697i
23.3 −1.40868 − 0.125026i −0.226717 + 1.71715i 1.96874 + 0.352243i 1.04087 − 0.183533i 0.534060 − 2.39056i 1.52016i −2.72927 − 0.742339i −2.89720 − 0.778615i −1.48919 + 0.128403i
23.4 −1.40831 + 0.129087i −0.356216 − 1.69503i 1.96667 − 0.363589i −1.53473 + 0.270614i 0.720469 + 2.34114i − 0.923751i −2.72275 + 0.765918i −2.74622 + 1.20759i 2.12644 − 0.579223i
23.5 −1.40681 + 0.144565i 1.40038 + 1.01929i 1.95820 − 0.406750i −1.83392 + 0.323370i −2.11741 − 1.23149i − 0.495207i −2.69601 + 0.855305i 0.922103 + 2.85477i 2.53322 − 0.720039i
23.6 −1.40179 − 0.187006i 1.60726 − 0.645527i 1.93006 + 0.524287i −0.300003 + 0.0528987i −2.37377 + 0.604329i − 4.64420i −2.60750 − 1.09588i 2.16659 − 2.07506i 0.430435 − 0.0180508i
23.7 −1.39710 − 0.219372i 1.71213 − 0.261969i 1.90375 + 0.612968i 1.82315 − 0.321470i −2.44947 − 0.00959719i 3.37875i −2.52525 − 1.27401i 2.86274 − 0.897047i −2.61763 + 0.0491764i
23.8 −1.37799 − 0.318047i −1.72498 − 0.156369i 1.79769 + 0.876530i 0.975388 − 0.171987i 2.32726 + 0.764099i − 0.0214153i −2.19842 − 1.77960i 2.95110 + 0.539468i −1.39877 − 0.0732235i
23.9 −1.37141 + 0.345309i −1.40038 − 1.01929i 1.76152 − 0.947120i −1.83392 + 0.323370i 2.27246 + 0.914298i 0.495207i −2.08872 + 1.90716i 0.922103 + 2.85477i 2.40339 − 1.07674i
23.10 −1.36862 − 0.356187i −1.19183 − 1.25680i 1.74626 + 0.974972i 2.67723 − 0.472068i 1.18351 + 2.14460i − 4.72128i −2.04270 − 1.95637i −0.159081 + 2.99578i −3.83226 − 0.307512i
23.11 −1.36753 + 0.360368i 0.356216 + 1.69503i 1.74027 − 0.985628i −1.53473 + 0.270614i −1.09797 − 2.18963i 0.923751i −2.02468 + 1.97501i −2.74622 + 1.20759i 2.00127 − 0.923141i
23.12 −1.36518 + 0.369172i 0.775266 − 1.54886i 1.72742 − 1.00797i 3.87264 − 0.682851i −0.486581 + 2.40067i − 0.103868i −1.98613 + 2.01378i −1.79793 − 2.40155i −5.03475 + 2.36188i
23.13 −1.35377 + 0.409020i 1.38928 − 1.03436i 1.66541 − 1.10744i −4.15853 + 0.733261i −1.45770 + 1.96853i 3.64991i −1.80162 + 2.18041i 0.860198 − 2.87403i 5.32979 − 2.69359i
23.14 −1.35302 − 0.411502i 0.219865 − 1.71804i 1.66133 + 1.11354i −0.859989 + 0.151639i −1.00446 + 2.23407i 3.05338i −1.78959 − 2.19029i −2.90332 − 0.755474i 1.22598 + 0.148716i
23.15 −1.28409 − 0.592547i −0.707659 + 1.58089i 1.29778 + 1.52177i −3.14727 + 0.554948i 1.84545 − 1.61069i 5.13171i −0.764742 − 2.72308i −1.99844 − 2.23747i 4.37021 + 1.15230i
23.16 −1.28096 + 0.599282i 0.226717 − 1.71715i 1.28172 − 1.53531i 1.04087 − 0.183533i 0.738640 + 2.33547i − 1.52016i −0.721752 + 2.73479i −2.89720 − 0.778615i −1.22332 + 0.858872i
23.17 −1.25666 − 0.648697i −1.62202 − 0.607495i 1.15839 + 1.63038i −2.64599 + 0.466559i 1.64425 + 1.81561i 1.11605i −0.398073 − 2.80027i 2.26190 + 1.97074i 3.62776 + 1.13014i
23.18 −1.25330 + 0.655170i −1.60726 + 0.645527i 1.14150 − 1.64224i −0.300003 + 0.0528987i 1.59145 − 1.86207i 4.64420i −0.354695 + 2.80610i 2.16659 − 2.07506i 0.341336 − 0.262851i
23.19 −1.24014 − 0.679748i 1.64569 + 0.540095i 1.07589 + 1.68596i −3.25234 + 0.573476i −1.67376 − 1.78845i − 0.838984i −0.188220 − 2.82216i 2.41659 + 1.77766i 4.42317 + 1.49958i
23.20 −1.23781 + 0.683977i −1.71213 + 0.261969i 1.06435 − 1.69327i 1.82315 − 0.321470i 1.94011 − 1.49532i − 3.37875i −0.159307 + 2.82394i 2.86274 − 0.897047i −2.03683 + 1.64491i
See next 80 embeddings (of 696 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
171.bf odd 18 1 inner
684.bs even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.bs.a ✓ 696
4.b odd 2 1 inner 684.2.bs.a ✓ 696
9.d odd 6 1 684.2.ch.a yes 696
19.e even 9 1 684.2.ch.a yes 696
36.h even 6 1 684.2.ch.a yes 696
76.l odd 18 1 684.2.ch.a yes 696
171.bf odd 18 1 inner 684.2.bs.a ✓ 696
684.bs even 18 1 inner 684.2.bs.a ✓ 696
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.bs.a ✓ 696 1.a even 1 1 trivial
684.2.bs.a ✓ 696 4.b odd 2 1 inner
684.2.bs.a ✓ 696 171.bf odd 18 1 inner
684.2.bs.a ✓ 696 684.bs even 18 1 inner
684.2.ch.a yes 696 9.d odd 6 1
684.2.ch.a yes 696 19.e even 9 1
684.2.ch.a yes 696 36.h even 6 1
684.2.ch.a yes 696 76.l odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).