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Properties

Label 1323.2.g.h
Level $1323$
Weight $2$
Character orbit 1323.g
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(361,1323)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.g (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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 12 q^{4} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 12 q^{4} + 24 q^{8} + 40 q^{11} - 12 q^{16} + 64 q^{23} + 24 q^{25} - 16 q^{29} - 48 q^{32} - 12 q^{37} - 56 q^{44} + 24 q^{46} + 4 q^{50} - 32 q^{53} + 96 q^{64} - 60 q^{65} - 12 q^{67} + 112 q^{71} + 136 q^{74} + 12 q^{79} + 12 q^{85} + 152 q^{86} - 16 q^{92} - 64 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1 −1.35757 + 2.35137i 0 −2.68597 − 4.65224i 1.58639 0 0 9.15528 0 −2.15363 + 3.73020i
361.2 −1.35757 + 2.35137i 0 −2.68597 − 4.65224i −1.58639 0 0 9.15528 0 2.15363 − 3.73020i
361.3 −0.863305 + 1.49529i 0 −0.490592 − 0.849731i −3.51231 0 0 −1.75910 0 3.03220 − 5.25192i
361.4 −0.863305 + 1.49529i 0 −0.490592 − 0.849731i 3.51231 0 0 −1.75910 0 −3.03220 + 5.25192i
361.5 −0.551407 + 0.955065i 0 0.391901 + 0.678793i −0.105466 0 0 −3.07001 0 0.0581547 − 0.100727i
361.6 −0.551407 + 0.955065i 0 0.391901 + 0.678793i 0.105466 0 0 −3.07001 0 −0.0581547 + 0.100727i
361.7 0.0341870 − 0.0592136i 0 0.997662 + 1.72800i −2.66379 0 0 0.273176 0 −0.0910670 + 0.157733i
361.8 0.0341870 − 0.0592136i 0 0.997662 + 1.72800i 2.66379 0 0 0.273176 0 0.0910670 − 0.157733i
361.9 0.649936 − 1.12572i 0 0.155166 + 0.268756i 3.52584 0 0 3.00314 0 2.29157 − 3.96912i
361.10 0.649936 − 1.12572i 0 0.155166 + 0.268756i −3.52584 0 0 3.00314 0 −2.29157 + 3.96912i
361.11 1.08816 − 1.88474i 0 −1.36816 − 2.36973i −1.26829 0 0 −1.60248 0 −1.38010 + 2.39040i
361.12 1.08816 − 1.88474i 0 −1.36816 − 2.36973i 1.26829 0 0 −1.60248 0 1.38010 − 2.39040i
667.1 −1.35757 − 2.35137i 0 −2.68597 + 4.65224i 1.58639 0 0 9.15528 0 −2.15363 − 3.73020i
667.2 −1.35757 − 2.35137i 0 −2.68597 + 4.65224i −1.58639 0 0 9.15528 0 2.15363 + 3.73020i
667.3 −0.863305 − 1.49529i 0 −0.490592 + 0.849731i −3.51231 0 0 −1.75910 0 3.03220 + 5.25192i
667.4 −0.863305 − 1.49529i 0 −0.490592 + 0.849731i 3.51231 0 0 −1.75910 0 −3.03220 − 5.25192i
667.5 −0.551407 − 0.955065i 0 0.391901 − 0.678793i −0.105466 0 0 −3.07001 0 0.0581547 + 0.100727i
667.6 −0.551407 − 0.955065i 0 0.391901 − 0.678793i 0.105466 0 0 −3.07001 0 −0.0581547 − 0.100727i
667.7 0.0341870 + 0.0592136i 0 0.997662 − 1.72800i −2.66379 0 0 0.273176 0 −0.0910670 − 0.157733i
667.8 0.0341870 + 0.0592136i 0 0.997662 − 1.72800i 2.66379 0 0 0.273176 0 0.0910670 + 0.157733i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{12} + 2 T_{2}^{11} + 11 T_{2}^{10} + 10 T_{2}^{9} + 63 T_{2}^{8} + 58 T_{2}^{7} + 184 T_{2}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{12} - 36T_{5}^{10} + 465T_{5}^{8} - 2580T_{5}^{6} + 5850T_{5}^{4} - 4470T_{5}^{2} + 49 \) Copy content Toggle raw display