↑

Properties

Label 165.3.l.g
Level $165$
Weight $3$
Character orbit 165.l
Analytic conductor $4.496$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,3,Mod(32,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.32");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 165.l (of order \(4\), 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: \(4.49592436194\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 16 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 16 q^{3} + 64 q^{12} - 136 q^{15} - 384 q^{16} - 124 q^{22} - 136 q^{25} + 80 q^{27} + 240 q^{31} + 36 q^{33} + 408 q^{36} - 472 q^{37} + 416 q^{42} + 56 q^{45} + 232 q^{48} + 68 q^{55} + 512 q^{58} + 248 q^{60} - 48 q^{66} + 944 q^{67} + 80 q^{70} + 616 q^{75} + 712 q^{78} - 1384 q^{81} - 488 q^{82} - 1276 q^{88} - 464 q^{91} - 224 q^{93} - 464 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −2.61936 − 2.61936i 0.734402 − 2.90872i 9.72210i 4.91675 − 0.908599i −9.54265 + 5.69532i 3.51454 − 3.51454i 14.9882 − 14.9882i −7.92131 − 4.27234i −15.2587 − 10.4988i
32.2 −2.61936 − 2.61936i 2.90872 − 0.734402i 9.72210i −4.91675 + 0.908599i −9.54265 − 5.69532i −3.51454 + 3.51454i 14.9882 − 14.9882i 7.92131 − 4.27234i 15.2587 + 10.4988i
32.3 −2.55873 − 2.55873i −2.39127 + 1.81158i 9.09417i −0.525685 − 4.97229i 10.7539 + 1.48327i 7.57206 − 7.57206i 13.0346 − 13.0346i 2.43635 − 8.66396i −11.3776 + 14.0678i
32.4 −2.55873 − 2.55873i −1.81158 + 2.39127i 9.09417i 0.525685 + 4.97229i 10.7539 − 1.48327i −7.57206 + 7.57206i 13.0346 − 13.0346i −2.43635 − 8.66396i 11.3776 − 14.0678i
32.5 −2.36241 − 2.36241i −2.29372 − 1.93360i 7.16199i −2.88421 + 4.08428i 0.850752 + 9.98670i 3.02593 − 3.02593i 7.46993 − 7.46993i 1.52235 + 8.87031i 16.4624 − 2.83507i
32.6 −2.36241 − 2.36241i 1.93360 + 2.29372i 7.16199i 2.88421 − 4.08428i 0.850752 − 9.98670i −3.02593 + 3.02593i 7.46993 − 7.46993i −1.52235 + 8.87031i −16.4624 + 2.83507i
32.7 −1.95755 − 1.95755i −2.61385 − 1.47235i 3.66400i 4.91172 − 0.935442i 2.23454 + 7.99893i −4.77945 + 4.77945i −0.657740 + 0.657740i 4.66439 + 7.69698i −11.4461 − 7.78375i
32.8 −1.95755 − 1.95755i 1.47235 + 2.61385i 3.66400i −4.91172 + 0.935442i 2.23454 − 7.99893i 4.77945 − 4.77945i −0.657740 + 0.657740i −4.66439 + 7.69698i 11.4461 + 7.78375i
32.9 −1.81129 − 1.81129i 1.00333 − 2.82725i 2.56152i −2.46356 − 4.35096i −6.93828 + 3.30364i −3.88148 + 3.88148i −2.60550 + 2.60550i −6.98666 − 5.67332i −3.41864 + 12.3431i
32.10 −1.81129 − 1.81129i 2.82725 − 1.00333i 2.56152i 2.46356 + 4.35096i −6.93828 − 3.30364i 3.88148 − 3.88148i −2.60550 + 2.60550i 6.98666 − 5.67332i 3.41864 − 12.3431i
32.11 −0.958750 − 0.958750i −2.16488 + 2.07685i − 2.16160i 4.27644 − 2.59076i 4.06675 + 0.0843958i −3.72033 + 3.72033i −5.90743 + 5.90743i 0.373386 − 8.99225i −6.58393 − 1.61614i
32.12 −0.958750 − 0.958750i −2.07685 + 2.16488i − 2.16160i −4.27644 + 2.59076i 4.06675 − 0.0843958i 3.72033 − 3.72033i −5.90743 + 5.90743i −0.373386 − 8.99225i 6.58393 + 1.61614i
32.13 −0.869697 − 0.869697i −1.51144 − 2.59144i − 2.48725i 2.25028 − 4.46500i −0.939267 + 3.56826i 8.66909 − 8.66909i −5.64195 + 5.64195i −4.43108 + 7.83361i −5.84026 + 1.92613i
32.14 −0.869697 − 0.869697i 2.59144 + 1.51144i − 2.48725i −2.25028 + 4.46500i −0.939267 − 3.56826i −8.66909 + 8.66909i −5.64195 + 5.64195i 4.43108 + 7.83361i 5.84026 − 1.92613i
32.15 −0.471737 − 0.471737i −2.97714 − 0.369645i − 3.55493i 2.27701 + 4.45143i 1.23005 + 1.57880i 6.94641 − 6.94641i −3.56394 + 3.56394i 8.72672 + 2.20097i 1.02575 − 3.17405i
32.16 −0.471737 − 0.471737i 0.369645 + 2.97714i − 3.55493i −2.27701 − 4.45143i 1.23005 − 1.57880i −6.94641 + 6.94641i −3.56394 + 3.56394i −8.72672 + 2.20097i −1.02575 + 3.17405i
32.17 0.471737 + 0.471737i −2.97714 − 0.369645i − 3.55493i 2.27701 + 4.45143i −1.23005 − 1.57880i −6.94641 + 6.94641i 3.56394 − 3.56394i 8.72672 + 2.20097i −1.02575 + 3.17405i
32.18 0.471737 + 0.471737i 0.369645 + 2.97714i − 3.55493i −2.27701 − 4.45143i −1.23005 + 1.57880i 6.94641 − 6.94641i 3.56394 − 3.56394i −8.72672 + 2.20097i 1.02575 − 3.17405i
32.19 0.869697 + 0.869697i −1.51144 − 2.59144i − 2.48725i 2.25028 − 4.46500i 0.939267 − 3.56826i −8.66909 + 8.66909i 5.64195 − 5.64195i −4.43108 + 7.83361i 5.84026 − 1.92613i
32.20 0.869697 + 0.869697i 2.59144 + 1.51144i − 2.48725i −2.25028 + 4.46500i 0.939267 + 3.56826i 8.66909 − 8.66909i 5.64195 − 5.64195i 4.43108 + 7.83361i −5.84026 + 1.92613i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
11.b odd 2 1 inner
15.e even 4 1 inner
33.d even 2 1 inner
55.e even 4 1 inner
165.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.l.g ✓ 64
3.b odd 2 1 inner 165.3.l.g ✓ 64
5.c odd 4 1 inner 165.3.l.g ✓ 64
11.b odd 2 1 inner 165.3.l.g ✓ 64
15.e even 4 1 inner 165.3.l.g ✓ 64
33.d even 2 1 inner 165.3.l.g ✓ 64
55.e even 4 1 inner 165.3.l.g ✓ 64
165.l odd 4 1 inner 165.3.l.g ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.l.g ✓ 64 1.a even 1 1 trivial
165.3.l.g ✓ 64 3.b odd 2 1 inner
165.3.l.g ✓ 64 5.c odd 4 1 inner
165.3.l.g ✓ 64 11.b odd 2 1 inner
165.3.l.g ✓ 64 15.e even 4 1 inner
165.3.l.g ✓ 64 33.d even 2 1 inner
165.3.l.g ✓ 64 55.e even 4 1 inner
165.3.l.g ✓ 64 165.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(165, [\chi])\):

\( T_{2}^{32} + 592 T_{2}^{28} + 132384 T_{2}^{24} + 13857436 T_{2}^{20} + 682392470 T_{2}^{16} + \cdots + 15583777225 \) Copy content Toggle raw display
\( T_{23}^{32} + 2073092 T_{23}^{28} + 1244881707380 T_{23}^{24} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display