Properties

Label 864.2.y.d
Level $864$
Weight $2$
Character orbit 864.y
Analytic conductor $6.899$
Analytic rank $0$
Dimension $60$
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] = [864,2,Mod(97,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.y (of order \(9\), 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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{9} - 12 q^{17} + 24 q^{21} - 24 q^{25} + 6 q^{29} - 12 q^{33} - 30 q^{37} - 30 q^{41} - 90 q^{45} + 42 q^{49} - 36 q^{53} - 60 q^{57} + 48 q^{61} + 12 q^{65} + 78 q^{69} - 48 q^{73} - 12 q^{77} + 12 q^{81} - 102 q^{85} - 12 q^{89} - 36 q^{93} + 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −1.71652 + 0.231448i 0 −3.16823 1.15314i 0 0.327751 1.85877i 0 2.89286 0.794570i 0
97.2 0 −1.65691 + 0.504627i 0 1.76220 + 0.641388i 0 −0.498467 + 2.82694i 0 2.49070 1.67224i 0
97.3 0 −1.01321 1.40478i 0 −1.03120 0.375327i 0 −0.257635 + 1.46112i 0 −0.946819 + 2.84667i 0
97.4 0 −0.790581 1.54110i 0 3.49481 + 1.27201i 0 0.718530 4.07499i 0 −1.74996 + 2.43673i 0
97.5 0 −0.523054 + 1.65119i 0 −0.117882 0.0429055i 0 0.455772 2.58481i 0 −2.45283 1.72732i 0
97.6 0 0.523054 1.65119i 0 −0.117882 0.0429055i 0 −0.455772 + 2.58481i 0 −2.45283 1.72732i 0
97.7 0 0.790581 + 1.54110i 0 3.49481 + 1.27201i 0 −0.718530 + 4.07499i 0 −1.74996 + 2.43673i 0
97.8 0 1.01321 + 1.40478i 0 −1.03120 0.375327i 0 0.257635 1.46112i 0 −0.946819 + 2.84667i 0
97.9 0 1.65691 0.504627i 0 1.76220 + 0.641388i 0 0.498467 2.82694i 0 2.49070 1.67224i 0
97.10 0 1.71652 0.231448i 0 −3.16823 1.15314i 0 −0.327751 + 1.85877i 0 2.89286 0.794570i 0
193.1 0 −1.72168 0.189239i 0 0.679279 3.85238i 0 −2.91962 + 2.44985i 0 2.92838 + 0.651618i 0
193.2 0 −1.68152 + 0.415309i 0 −0.564773 + 3.20298i 0 −2.06064 + 1.72908i 0 2.65504 1.39670i 0
193.3 0 −1.28924 + 1.15666i 0 −0.110022 + 0.623966i 0 3.44796 2.89318i 0 0.324271 2.98242i 0
193.4 0 −0.998006 1.41562i 0 −0.476400 + 2.70180i 0 0.930891 0.781110i 0 −1.00797 + 2.82560i 0
193.5 0 −0.141380 + 1.72627i 0 0.298268 1.69156i 0 −0.0848018 + 0.0711572i 0 −2.96002 0.488119i 0
193.6 0 0.141380 1.72627i 0 0.298268 1.69156i 0 0.0848018 0.0711572i 0 −2.96002 0.488119i 0
193.7 0 0.998006 + 1.41562i 0 −0.476400 + 2.70180i 0 −0.930891 + 0.781110i 0 −1.00797 + 2.82560i 0
193.8 0 1.28924 1.15666i 0 −0.110022 + 0.623966i 0 −3.44796 + 2.89318i 0 0.324271 2.98242i 0
193.9 0 1.68152 0.415309i 0 −0.564773 + 3.20298i 0 2.06064 1.72908i 0 2.65504 1.39670i 0
193.10 0 1.72168 + 0.189239i 0 0.679279 3.85238i 0 2.91962 2.44985i 0 2.92838 + 0.651618i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.e even 9 1 inner
108.j odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.y.d 60
4.b odd 2 1 inner 864.2.y.d 60
27.e even 9 1 inner 864.2.y.d 60
108.j odd 18 1 inner 864.2.y.d 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.d 60 1.a even 1 1 trivial
864.2.y.d 60 4.b odd 2 1 inner
864.2.y.d 60 27.e even 9 1 inner
864.2.y.d 60 108.j odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\):

\( T_{5}^{30} + 6 T_{5}^{28} - 19 T_{5}^{27} + 63 T_{5}^{26} - 675 T_{5}^{25} + 3456 T_{5}^{24} + 270 T_{5}^{23} + 12006 T_{5}^{22} + 32910 T_{5}^{21} - 110277 T_{5}^{20} - 36891 T_{5}^{19} + 5454057 T_{5}^{18} - 5670288 T_{5}^{17} + \cdots + 341056 \) Copy content Toggle raw display
\( T_{7}^{60} - 21 T_{7}^{58} + 405 T_{7}^{56} + 4647 T_{7}^{54} - 122247 T_{7}^{52} + 3429648 T_{7}^{50} + 36003726 T_{7}^{48} - 665047908 T_{7}^{46} + 14409106767 T_{7}^{44} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display