# 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 $4$

# Related objects

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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{2} + 24 q^{4} + 24 q^{8}+O(q^{10})$$ 24 * q + 8 * q^2 + 24 * q^4 + 24 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{2} + 24 q^{4} + 24 q^{8} - 20 q^{11} + 24 q^{16} - 32 q^{23} - 12 q^{25} - 16 q^{29} + 96 q^{32} - 12 q^{37} - 56 q^{44} + 24 q^{46} + 4 q^{50} - 32 q^{53} + 96 q^{64} + 120 q^{65} + 24 q^{67} + 112 q^{71} - 68 q^{74} - 24 q^{79} + 12 q^{85} - 76 q^{86} - 16 q^{92} + 128 q^{95}+O(q^{100})$$ 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

 $$T_{2}^{6} - 2T_{2}^{5} - 7T_{2}^{4} + 12T_{2}^{3} + 10T_{2}^{2} - 14T_{2} - 1$$ T2^6 - 2*T2^5 - 7*T2^4 + 12*T2^3 + 10*T2^2 - 14*T2 - 1 $$T_{5}^{24} + 36 T_{5}^{22} + 831 T_{5}^{20} + 11580 T_{5}^{18} + 117495 T_{5}^{16} + 782970 T_{5}^{14} + 3775328 T_{5}^{12} + 10937664 T_{5}^{10} + 22667115 T_{5}^{8} + 25896660 T_{5}^{6} + 19694250 T_{5}^{4} + \cdots + 2401$$ T5^24 + 36*T5^22 + 831*T5^20 + 11580*T5^18 + 117495*T5^16 + 782970*T5^14 + 3775328*T5^12 + 10937664*T5^10 + 22667115*T5^8 + 25896660*T5^6 + 19694250*T5^4 + 219030*T5^2 + 2401