# Properties

 Label 1323.2.f.h Level $1323$ Weight $2$ Character orbit 1323.f Analytic conductor $10.564$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{2} - 12q^{4} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{2} - 12q^{4} + 24q^{8} - 20q^{11} - 12q^{16} - 32q^{23} - 12q^{25} - 16q^{29} - 48q^{32} + 24q^{37} + 112q^{44} - 48q^{46} + 4q^{50} + 64q^{53} + 96q^{64} - 60q^{65} - 12q^{67} + 112q^{71} - 68q^{74} + 12q^{79} + 12q^{85} - 76q^{86} - 16q^{92} - 64q^{95} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
442.1 −1.35757 2.35137i 0 −2.68597 + 4.65224i 0.793197 1.37386i 0 0 9.15528 0 −4.30727
442.2 −1.35757 2.35137i 0 −2.68597 + 4.65224i −0.793197 + 1.37386i 0 0 9.15528 0 4.30727
442.3 −0.863305 1.49529i 0 −0.490592 + 0.849731i −1.75616 + 3.04175i 0 0 −1.75910 0 6.06439
442.4 −0.863305 1.49529i 0 −0.490592 + 0.849731i 1.75616 3.04175i 0 0 −1.75910 0 −6.06439
442.5 −0.551407 0.955065i 0 0.391901 0.678793i 0.0527330 0.0913363i 0 0 −3.07001 0 −0.116309
442.6 −0.551407 0.955065i 0 0.391901 0.678793i −0.0527330 + 0.0913363i 0 0 −3.07001 0 0.116309
442.7 0.0341870 + 0.0592136i 0 0.997662 1.72800i −1.33190 + 2.30691i 0 0 0.273176 0 −0.182134
442.8 0.0341870 + 0.0592136i 0 0.997662 1.72800i 1.33190 2.30691i 0 0 0.273176 0 0.182134
442.9 0.649936 + 1.12572i 0 0.155166 0.268756i −1.76292 + 3.05347i 0 0 3.00314 0 −4.58314
442.10 0.649936 + 1.12572i 0 0.155166 0.268756i 1.76292 3.05347i 0 0 3.00314 0 4.58314
442.11 1.08816 + 1.88474i 0 −1.36816 + 2.36973i −0.634145 + 1.09837i 0 0 −1.60248 0 −2.76019
442.12 1.08816 + 1.88474i 0 −1.36816 + 2.36973i 0.634145 1.09837i 0 0 −1.60248 0 2.76019
883.1 −1.35757 + 2.35137i 0 −2.68597 4.65224i 0.793197 + 1.37386i 0 0 9.15528 0 −4.30727
883.2 −1.35757 + 2.35137i 0 −2.68597 4.65224i −0.793197 1.37386i 0 0 9.15528 0 4.30727
883.3 −0.863305 + 1.49529i 0 −0.490592 0.849731i −1.75616 3.04175i 0 0 −1.75910 0 6.06439
883.4 −0.863305 + 1.49529i 0 −0.490592 0.849731i 1.75616 + 3.04175i 0 0 −1.75910 0 −6.06439
883.5 −0.551407 + 0.955065i 0 0.391901 + 0.678793i 0.0527330 + 0.0913363i 0 0 −3.07001 0 −0.116309
883.6 −0.551407 + 0.955065i 0 0.391901 + 0.678793i −0.0527330 0.0913363i 0 0 −3.07001 0 0.116309
883.7 0.0341870 0.0592136i 0 0.997662 + 1.72800i −1.33190 2.30691i 0 0 0.273176 0 −0.182134
883.8 0.0341870 0.0592136i 0 0.997662 + 1.72800i 1.33190 + 2.30691i 0 0 0.273176 0 0.182134
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.12 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
9.c even 3 1 inner
63.l 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.f.h 24
3.b odd 2 1 441.2.f.h 24
7.b odd 2 1 inner 1323.2.f.h 24
7.c even 3 1 1323.2.g.h 24
7.c even 3 1 1323.2.h.h 24
7.d odd 6 1 1323.2.g.h 24
7.d odd 6 1 1323.2.h.h 24
9.c even 3 1 inner 1323.2.f.h 24
9.c even 3 1 3969.2.a.bi 12
9.d odd 6 1 441.2.f.h 24
9.d odd 6 1 3969.2.a.bh 12
21.c even 2 1 441.2.f.h 24
21.g even 6 1 441.2.g.h 24
21.g even 6 1 441.2.h.h 24
21.h odd 6 1 441.2.g.h 24
21.h odd 6 1 441.2.h.h 24
63.g even 3 1 1323.2.h.h 24
63.h even 3 1 1323.2.g.h 24
63.i even 6 1 441.2.g.h 24
63.j odd 6 1 441.2.g.h 24
63.k odd 6 1 1323.2.h.h 24
63.l odd 6 1 inner 1323.2.f.h 24
63.l odd 6 1 3969.2.a.bi 12
63.n odd 6 1 441.2.h.h 24
63.o even 6 1 441.2.f.h 24
63.o even 6 1 3969.2.a.bh 12
63.s even 6 1 441.2.h.h 24
63.t odd 6 1 1323.2.g.h 24

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

## Hecke kernels

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

 $$T_{2}^{12} + \cdots$$ $$T_{5}^{24} + \cdots$$