# 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 $4$

# Related objects

## Newspace parameters

 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

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{2} - 12q^{4} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{2} - 12q^{4} + 24q^{8} + 40q^{11} - 12q^{16} + 64q^{23} + 24q^{25} - 16q^{29} - 48q^{32} - 12q^{37} - 56q^{44} + 24q^{46} + 4q^{50} - 32q^{53} + 96q^{64} - 60q^{65} - 12q^{67} + 112q^{71} + 136q^{74} + 12q^{79} + 12q^{85} + 152q^{86} - 16q^{92} - 64q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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 667.12 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.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 $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

 $$T_{2}^{12} + \cdots$$ $$T_{5}^{12} - 36 T_{5}^{10} + 465 T_{5}^{8} - 2580 T_{5}^{6} + 5850 T_{5}^{4} - 4470 T_{5}^{2} + 49$$