Properties

Label 1323.2.h.h
Level $1323$
Weight $2$
Character orbit 1323.h
Analytic conductor $10.564$
Analytic rank $0$
Dimension $24$
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] = [1323,2,Mod(226,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.h (of order \(3\), 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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 441)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{2} + 24 q^{4} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{2} + 24 q^{4} + 24 q^{8} - 20 q^{11} + 24 q^{16} - 32 q^{23} - 12 q^{25} - 16 q^{29} + 96 q^{32} - 12 q^{37} - 56 q^{44} + 24 q^{46} + 4 q^{50} - 32 q^{53} + 96 q^{64} + 120 q^{65} + 24 q^{67} + 112 q^{71} - 68 q^{74} - 24 q^{79} + 12 q^{85} - 76 q^{86} - 16 q^{92} + 128 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1 −2.17631 0 2.73633 −0.634145 1.09837i 0 0 −1.60248 0 1.38010 + 2.39040i
226.2 −2.17631 0 2.73633 0.634145 + 1.09837i 0 0 −1.60248 0 −1.38010 2.39040i
226.3 −1.29987 0 −0.310333 1.76292 + 3.05347i 0 0 3.00314 0 −2.29157 3.96912i
226.4 −1.29987 0 −0.310333 −1.76292 3.05347i 0 0 3.00314 0 2.29157 + 3.96912i
226.5 −0.0683740 0 −1.99532 1.33190 + 2.30691i 0 0 0.273176 0 −0.0910670 0.157733i
226.6 −0.0683740 0 −1.99532 −1.33190 2.30691i 0 0 0.273176 0 0.0910670 + 0.157733i
226.7 1.10281 0 −0.783802 −0.0527330 0.0913363i 0 0 −3.07001 0 −0.0581547 0.100727i
226.8 1.10281 0 −0.783802 0.0527330 + 0.0913363i 0 0 −3.07001 0 0.0581547 + 0.100727i
226.9 1.72661 0 0.981184 1.75616 + 3.04175i 0 0 −1.75910 0 3.03220 + 5.25192i
226.10 1.72661 0 0.981184 −1.75616 3.04175i 0 0 −1.75910 0 −3.03220 5.25192i
226.11 2.71513 0 5.37195 0.793197 + 1.37386i 0 0 9.15528 0 2.15363 + 3.73020i
226.12 2.71513 0 5.37195 −0.793197 1.37386i 0 0 9.15528 0 −2.15363 3.73020i
802.1 −2.17631 0 2.73633 −0.634145 + 1.09837i 0 0 −1.60248 0 1.38010 2.39040i
802.2 −2.17631 0 2.73633 0.634145 1.09837i 0 0 −1.60248 0 −1.38010 + 2.39040i
802.3 −1.29987 0 −0.310333 1.76292 3.05347i 0 0 3.00314 0 −2.29157 + 3.96912i
802.4 −1.29987 0 −0.310333 −1.76292 + 3.05347i 0 0 3.00314 0 2.29157 3.96912i
802.5 −0.0683740 0 −1.99532 1.33190 2.30691i 0 0 0.273176 0 −0.0910670 + 0.157733i
802.6 −0.0683740 0 −1.99532 −1.33190 + 2.30691i 0 0 0.273176 0 0.0910670 0.157733i
802.7 1.10281 0 −0.783802 −0.0527330 + 0.0913363i 0 0 −3.07001 0 −0.0581547 + 0.100727i
802.8 1.10281 0 −0.783802 0.0527330 0.0913363i 0 0 −3.07001 0 0.0581547 0.100727i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.h even 3 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.h 24
3.b odd 2 1 441.2.h.h 24
7.b odd 2 1 inner 1323.2.h.h 24
7.c even 3 1 1323.2.f.h 24
7.c even 3 1 1323.2.g.h 24
7.d odd 6 1 1323.2.f.h 24
7.d odd 6 1 1323.2.g.h 24
9.c even 3 1 1323.2.g.h 24
9.d odd 6 1 441.2.g.h 24
21.c even 2 1 441.2.h.h 24
21.g even 6 1 441.2.f.h 24
21.g even 6 1 441.2.g.h 24
21.h odd 6 1 441.2.f.h 24
21.h odd 6 1 441.2.g.h 24
63.g even 3 1 1323.2.f.h 24
63.h even 3 1 inner 1323.2.h.h 24
63.h even 3 1 3969.2.a.bi 12
63.i even 6 1 441.2.h.h 24
63.i even 6 1 3969.2.a.bh 12
63.j odd 6 1 441.2.h.h 24
63.j odd 6 1 3969.2.a.bh 12
63.k odd 6 1 1323.2.f.h 24
63.l odd 6 1 1323.2.g.h 24
63.n odd 6 1 441.2.f.h 24
63.o even 6 1 441.2.g.h 24
63.s even 6 1 441.2.f.h 24
63.t odd 6 1 inner 1323.2.h.h 24
63.t odd 6 1 3969.2.a.bi 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.h 24 21.g even 6 1
441.2.f.h 24 21.h odd 6 1
441.2.f.h 24 63.n odd 6 1
441.2.f.h 24 63.s even 6 1
441.2.g.h 24 9.d odd 6 1
441.2.g.h 24 21.g even 6 1
441.2.g.h 24 21.h odd 6 1
441.2.g.h 24 63.o even 6 1
441.2.h.h 24 3.b odd 2 1
441.2.h.h 24 21.c even 2 1
441.2.h.h 24 63.i even 6 1
441.2.h.h 24 63.j odd 6 1
1323.2.f.h 24 7.c even 3 1
1323.2.f.h 24 7.d odd 6 1
1323.2.f.h 24 63.g even 3 1
1323.2.f.h 24 63.k odd 6 1
1323.2.g.h 24 7.c even 3 1
1323.2.g.h 24 7.d odd 6 1
1323.2.g.h 24 9.c even 3 1
1323.2.g.h 24 63.l odd 6 1
1323.2.h.h 24 1.a even 1 1 trivial
1323.2.h.h 24 7.b odd 2 1 inner
1323.2.h.h 24 63.h even 3 1 inner
1323.2.h.h 24 63.t odd 6 1 inner
3969.2.a.bh 12 63.i even 6 1
3969.2.a.bh 12 63.j odd 6 1
3969.2.a.bi 12 63.h even 3 1
3969.2.a.bi 12 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\( T_{2}^{6} - 2T_{2}^{5} - 7T_{2}^{4} + 12T_{2}^{3} + 10T_{2}^{2} - 14T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{24} + 36 T_{5}^{22} + 831 T_{5}^{20} + 11580 T_{5}^{18} + 117495 T_{5}^{16} + 782970 T_{5}^{14} + 3775328 T_{5}^{12} + 10937664 T_{5}^{10} + 22667115 T_{5}^{8} + 25896660 T_{5}^{6} + 19694250 T_{5}^{4} + \cdots + 2401 \) Copy content Toggle raw display