↑

Properties

Label 405.3.h.k
Level $405$
Weight $3$
Character orbit 405.h
Analytic conductor $11.035$
Analytic rank $0$
Dimension $48$
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] = [405,3,Mod(134,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.134");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.h (of order \(6\), degree \(2\), 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{4} + 24 q^{10} - 96 q^{16} - 48 q^{25} - 144 q^{34} - 72 q^{40} - 336 q^{46} + 288 q^{49} - 264 q^{55} + 360 q^{61} - 144 q^{64} + 156 q^{70} - 48 q^{76} + 480 q^{79} + 456 q^{85} - 96 q^{91} - 384 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1 −1.78706 − 3.09527i 0 −4.38714 + 7.59875i −1.88580 − 4.63074i 0 5.21507 − 3.01092i 17.0638 0 −10.9634 + 14.1125i
134.2 −1.78706 − 3.09527i 0 −4.38714 + 7.59875i 4.95324 − 0.682218i 0 −5.21507 + 3.01092i 17.0638 0 −10.9634 − 14.1125i
134.3 −1.74200 − 3.01723i 0 −4.06913 + 7.04793i −3.55914 + 3.51177i 0 −11.0647 + 6.38818i 14.4177 0 16.7958 + 4.62123i
134.4 −1.74200 − 3.01723i 0 −4.06913 + 7.04793i −1.26172 + 4.83819i 0 11.0647 − 6.38818i 14.4177 0 16.7958 − 4.62123i
134.5 −1.14196 − 1.97794i 0 −0.608153 + 1.05335i −4.77039 − 1.49780i 0 2.98475 − 1.72325i −6.35774 0 2.48505 + 11.1459i
134.6 −1.14196 − 1.97794i 0 −0.608153 + 1.05335i 3.68233 + 3.38238i 0 −2.98475 + 1.72325i −6.35774 0 2.48505 − 11.1459i
134.7 −1.06375 − 1.84247i 0 −0.263128 + 0.455750i −0.323905 − 4.98950i 0 1.01789 − 0.587680i −7.39039 0 −8.84844 + 5.90436i
134.8 −1.06375 − 1.84247i 0 −0.263128 + 0.455750i 4.48298 − 2.21424i 0 −1.01789 + 0.587680i −7.39039 0 −8.84844 − 5.90436i
134.9 −0.577818 − 1.00081i 0 1.33225 − 2.30753i −4.87073 + 1.12959i 0 −6.09671 + 3.51994i −7.70174 0 3.94490 + 4.22198i
134.10 −0.577818 − 1.00081i 0 1.33225 − 2.30753i 1.45712 + 4.78297i 0 6.09671 − 3.51994i −7.70174 0 3.94490 − 4.22198i
134.11 −0.0484988 − 0.0840023i 0 1.99530 − 3.45595i −0.121738 − 4.99852i 0 −8.81825 + 5.09122i −0.775068 0 −0.413983 + 0.252648i
134.12 −0.0484988 − 0.0840023i 0 1.99530 − 3.45595i 4.38971 − 2.39383i 0 8.81825 − 5.09122i −0.775068 0 −0.413983 − 0.252648i
134.13 0.0484988 + 0.0840023i 0 1.99530 − 3.45595i −4.38971 + 2.39383i 0 8.81825 − 5.09122i 0.775068 0 −0.413983 − 0.252648i
134.14 0.0484988 + 0.0840023i 0 1.99530 − 3.45595i 0.121738 + 4.99852i 0 −8.81825 + 5.09122i 0.775068 0 −0.413983 + 0.252648i
134.15 0.577818 + 1.00081i 0 1.33225 − 2.30753i −1.45712 − 4.78297i 0 6.09671 − 3.51994i 7.70174 0 3.94490 − 4.22198i
134.16 0.577818 + 1.00081i 0 1.33225 − 2.30753i 4.87073 − 1.12959i 0 −6.09671 + 3.51994i 7.70174 0 3.94490 + 4.22198i
134.17 1.06375 + 1.84247i 0 −0.263128 + 0.455750i −4.48298 + 2.21424i 0 −1.01789 + 0.587680i 7.39039 0 −8.84844 − 5.90436i
134.18 1.06375 + 1.84247i 0 −0.263128 + 0.455750i 0.323905 + 4.98950i 0 1.01789 − 0.587680i 7.39039 0 −8.84844 + 5.90436i
134.19 1.14196 + 1.97794i 0 −0.608153 + 1.05335i −3.68233 − 3.38238i 0 −2.98475 + 1.72325i 6.35774 0 2.48505 − 11.1459i
134.20 1.14196 + 1.97794i 0 −0.608153 + 1.05335i 4.77039 + 1.49780i 0 2.98475 − 1.72325i 6.35774 0 2.48505 + 11.1459i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 134.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.h.k 48
3.b odd 2 1 inner 405.3.h.k 48
5.b even 2 1 inner 405.3.h.k 48
9.c even 3 1 405.3.d.b ✓ 24
9.c even 3 1 inner 405.3.h.k 48
9.d odd 6 1 405.3.d.b ✓ 24
9.d odd 6 1 inner 405.3.h.k 48
15.d odd 2 1 inner 405.3.h.k 48
45.h odd 6 1 405.3.d.b ✓ 24
45.h odd 6 1 inner 405.3.h.k 48
45.j even 6 1 405.3.d.b ✓ 24
45.j even 6 1 inner 405.3.h.k 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.3.d.b ✓ 24 9.c even 3 1
405.3.d.b ✓ 24 9.d odd 6 1
405.3.d.b ✓ 24 45.h odd 6 1
405.3.d.b ✓ 24 45.j even 6 1
405.3.h.k 48 1.a even 1 1 trivial
405.3.h.k 48 3.b odd 2 1 inner
405.3.h.k 48 5.b even 2 1 inner
405.3.h.k 48 9.c even 3 1 inner
405.3.h.k 48 9.d odd 6 1 inner
405.3.h.k 48 15.d odd 2 1 inner
405.3.h.k 48 45.h odd 6 1 inner
405.3.h.k 48 45.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{24} + 36 T_{2}^{22} + 828 T_{2}^{20} + 11516 T_{2}^{18} + 116559 T_{2}^{16} + 785430 T_{2}^{14} + \cdots + 2116 \) Copy content Toggle raw display
\( T_{7}^{24} - 366 T_{7}^{22} + 87633 T_{7}^{20} - 11974230 T_{7}^{18} + 1177762275 T_{7}^{16} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display