↑

Properties

Label 366.3.s.a
Level $366$
Weight $3$
Character orbit 366.s
Analytic conductor $9.973$
Analytic rank $0$
Dimension $80$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [366,3,Mod(37,366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(366, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("366.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 366 = 2 \cdot 3 \cdot 61 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 366.s (of order \(20\), degree \(8\), 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: \(9.97277767559\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 20 q^{2} - 40 q^{5} + 20 q^{7} + 40 q^{8} + 60 q^{9} + 60 q^{10} + 20 q^{11} - 32 q^{13} + 80 q^{16} + 80 q^{17} + 60 q^{18} - 100 q^{19} - 40 q^{20} + 12 q^{21} - 60 q^{22} + 56 q^{23} + 144 q^{25}+ \cdots - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −0.642040 + 1.26007i −1.64728 + 0.535233i −1.17557 − 1.61803i −4.68375 − 6.44663i 0.383185 − 2.41933i −3.06260 + 1.56047i 2.79360 − 0.442463i 2.42705 − 1.76336i 11.1304 − 1.76288i
37.2 −0.642040 + 1.26007i −1.64728 + 0.535233i −1.17557 − 1.61803i −3.06365 − 4.21676i 0.383185 − 2.41933i −6.63267 + 3.37952i 2.79360 − 0.442463i 2.42705 − 1.76336i 7.28041 − 1.15310i
37.3 −0.642040 + 1.26007i −1.64728 + 0.535233i −1.17557 − 1.61803i −0.695904 − 0.957829i 0.383185 − 2.41933i 8.24931 − 4.20323i 2.79360 − 0.442463i 2.42705 − 1.76336i 1.65373 − 0.261926i
37.4 −0.642040 + 1.26007i −1.64728 + 0.535233i −1.17557 − 1.61803i 2.30152 + 3.16777i 0.383185 − 2.41933i −1.21093 + 0.616999i 2.79360 − 0.442463i 2.42705 − 1.76336i −5.46930 + 0.866251i
37.5 −0.642040 + 1.26007i −1.64728 + 0.535233i −1.17557 − 1.61803i 3.44549 + 4.74231i 0.383185 − 2.41933i 3.66563 − 1.86773i 2.79360 − 0.442463i 2.42705 − 1.76336i −8.18781 + 1.29682i
37.6 −0.642040 + 1.26007i 1.64728 − 0.535233i −1.17557 − 1.61803i −5.41919 − 7.45887i −0.383185 + 2.41933i 0.532814 − 0.271482i 2.79360 − 0.442463i 2.42705 − 1.76336i 12.8781 − 2.03968i
37.7 −0.642040 + 1.26007i 1.64728 − 0.535233i −1.17557 − 1.61803i −1.24209 − 1.70959i −0.383185 + 2.41933i 5.13070 − 2.61422i 2.79360 − 0.442463i 2.42705 − 1.76336i 2.95169 − 0.467501i
37.8 −0.642040 + 1.26007i 1.64728 − 0.535233i −1.17557 − 1.61803i −1.15144 − 1.58482i −0.383185 + 2.41933i −6.58739 + 3.35644i 2.79360 − 0.442463i 2.42705 − 1.76336i 2.73627 − 0.433382i
37.9 −0.642040 + 1.26007i 1.64728 − 0.535233i −1.17557 − 1.61803i 2.16962 + 2.98622i −0.383185 + 2.41933i −9.86354 + 5.02573i 2.79360 − 0.442463i 2.42705 − 1.76336i −5.15583 + 0.816604i
37.10 −0.642040 + 1.26007i 1.64728 − 0.535233i −1.17557 − 1.61803i 2.94681 + 4.05594i −0.383185 + 2.41933i 6.29591 − 3.20793i 2.79360 − 0.442463i 2.42705 − 1.76336i −7.00276 + 1.10913i
85.1 1.26007 + 0.642040i −1.64728 + 0.535233i 1.17557 + 1.61803i −5.20482 − 7.16382i −2.41933 − 0.383185i 2.86610 + 5.62504i 0.442463 + 2.79360i 2.42705 − 1.76336i −1.95900 − 12.3686i
85.2 1.26007 + 0.642040i −1.64728 + 0.535233i 1.17557 + 1.61803i −2.70670 − 3.72546i −2.41933 − 0.383185i 0.0226989 + 0.0445491i 0.442463 + 2.79360i 2.42705 − 1.76336i −1.01875 − 6.43216i
85.3 1.26007 + 0.642040i −1.64728 + 0.535233i 1.17557 + 1.61803i 0.381580 + 0.525200i −2.41933 − 0.383185i −4.85898 − 9.53628i 0.442463 + 2.79360i 2.42705 − 1.76336i 0.143620 + 0.906779i
85.4 1.26007 + 0.642040i −1.64728 + 0.535233i 1.17557 + 1.61803i 3.57646 + 4.92258i −2.41933 − 0.383185i −2.25210 − 4.41999i 0.442463 + 2.79360i 2.42705 − 1.76336i 1.34612 + 8.49904i
85.5 1.26007 + 0.642040i −1.64728 + 0.535233i 1.17557 + 1.61803i 3.88584 + 5.34840i −2.41933 − 0.383185i 4.39558 + 8.62680i 0.442463 + 2.79360i 2.42705 − 1.76336i 1.46256 + 9.23423i
85.6 1.26007 + 0.642040i 1.64728 − 0.535233i 1.17557 + 1.61803i −3.85777 − 5.30976i 2.41933 + 0.383185i −3.06055 − 6.00667i 0.442463 + 2.79360i 2.42705 − 1.76336i −1.45199 − 9.16753i
85.7 1.26007 + 0.642040i 1.64728 − 0.535233i 1.17557 + 1.61803i −3.00896 − 4.14148i 2.41933 + 0.383185i 4.80197 + 9.42440i 0.442463 + 2.79360i 2.42705 − 1.76336i −1.13252 − 7.15044i
85.8 1.26007 + 0.642040i 1.64728 − 0.535233i 1.17557 + 1.61803i 0.415468 + 0.571843i 2.41933 + 0.383185i −3.82718 − 7.51126i 0.442463 + 2.79360i 2.42705 − 1.76336i 0.156375 + 0.987311i
85.9 1.26007 + 0.642040i 1.64728 − 0.535233i 1.17557 + 1.61803i 2.50035 + 3.44144i 2.41933 + 0.383185i 1.92026 + 3.76873i 0.442463 + 2.79360i 2.42705 − 1.76336i 0.941088 + 5.94179i
85.10 1.26007 + 0.642040i 1.64728 − 0.535233i 1.17557 + 1.61803i 3.88327 + 5.34486i 2.41933 + 0.383185i 1.76675 + 3.46745i 0.442463 + 2.79360i 2.42705 − 1.76336i 1.46159 + 9.22813i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.j odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 366.3.s.a ✓ 80
61.j odd 20 1 inner 366.3.s.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
366.3.s.a ✓ 80 1.a even 1 1 trivial
366.3.s.a ✓ 80 61.j odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{80} + 40 T_{5}^{79} + 478 T_{5}^{78} - 2560 T_{5}^{77} - 107113 T_{5}^{76} + \cdots + 55\!\cdots\!16 \) acting on \(S_{3}^{\mathrm{new}}(366, [\chi])\). Copy content Toggle raw display