# Properties

 Label 1323.2.s.d Level $1323$ Weight $2$ Character orbit 1323.s Analytic conductor $10.564$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 441) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 24q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 24q^{4} - 24q^{16} + 48q^{25} + 120q^{32} - 96q^{44} - 48q^{50} - 48q^{53} - 48q^{64} + 120q^{65} - 24q^{79} - 24q^{85} + 144q^{92} + 96q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
656.1 −2.34591 + 1.35441i 0 2.66888 4.62263i 1.20293 0 0 9.04141i 0 −2.82197 + 1.62926i
656.2 −2.34591 + 1.35441i 0 2.66888 4.62263i −1.20293 0 0 9.04141i 0 2.82197 1.62926i
656.3 −2.05485 + 1.18637i 0 1.81495 3.14358i −3.43548 0 0 3.86732i 0 7.05942 4.07576i
656.4 −2.05485 + 1.18637i 0 1.81495 3.14358i 3.43548 0 0 3.86732i 0 −7.05942 + 4.07576i
656.5 −1.28562 + 0.742253i 0 0.101880 0.176462i 0.308431 0 0 2.66653i 0 −0.396525 + 0.228934i
656.6 −1.28562 + 0.742253i 0 0.101880 0.176462i −0.308431 0 0 2.66653i 0 0.396525 0.228934i
656.7 −1.02035 + 0.589100i 0 −0.305921 + 0.529871i −4.33202 0 0 3.07728i 0 4.42019 2.55200i
656.8 −1.02035 + 0.589100i 0 −0.305921 + 0.529871i 4.33202 0 0 3.07728i 0 −4.42019 + 2.55200i
656.9 −0.850109 + 0.490811i 0 −0.518210 + 0.897565i −1.88120 0 0 2.98061i 0 1.59922 0.923312i
656.10 −0.850109 + 0.490811i 0 −0.518210 + 0.897565i 1.88120 0 0 2.98061i 0 −1.59922 + 0.923312i
656.11 −0.367369 + 0.212101i 0 −0.910027 + 1.57621i 3.60763 0 0 1.62047i 0 −1.32533 + 0.765180i
656.12 −0.367369 + 0.212101i 0 −0.910027 + 1.57621i −3.60763 0 0 1.62047i 0 1.32533 0.765180i
656.13 0.105953 0.0611722i 0 −0.992516 + 1.71909i 0.529430 0 0 0.487547i 0 0.0560949 0.0323864i
656.14 0.105953 0.0611722i 0 −0.992516 + 1.71909i −0.529430 0 0 0.487547i 0 −0.0560949 + 0.0323864i
656.15 0.575298 0.332148i 0 −0.779355 + 1.34988i 0.0283039 0 0 2.36404i 0 0.0162832 0.00940110i
656.16 0.575298 0.332148i 0 −0.779355 + 1.34988i −0.0283039 0 0 2.36404i 0 −0.0162832 + 0.00940110i
656.17 1.58658 0.916012i 0 0.678156 1.17460i −0.645568 0 0 1.17925i 0 −1.02425 + 0.591348i
656.18 1.58658 0.916012i 0 0.678156 1.17460i 0.645568 0 0 1.17925i 0 1.02425 0.591348i
656.19 1.61855 0.934468i 0 0.746462 1.29291i −2.50573 0 0 0.947692i 0 −4.05565 + 2.34153i
656.20 1.61855 0.934468i 0 0.746462 1.29291i 2.50573 0 0 0.947692i 0 4.05565 2.34153i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 962.24 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.n odd 6 1 inner
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.s.d 48
3.b odd 2 1 441.2.s.d 48
7.b odd 2 1 inner 1323.2.s.d 48
7.c even 3 1 1323.2.i.d 48
7.c even 3 1 1323.2.o.e 48
7.d odd 6 1 1323.2.i.d 48
7.d odd 6 1 1323.2.o.e 48
9.c even 3 1 441.2.i.d 48
9.d odd 6 1 1323.2.i.d 48
21.c even 2 1 441.2.s.d 48
21.g even 6 1 441.2.i.d 48
21.g even 6 1 441.2.o.e 48
21.h odd 6 1 441.2.i.d 48
21.h odd 6 1 441.2.o.e 48
63.g even 3 1 441.2.s.d 48
63.h even 3 1 441.2.o.e 48
63.i even 6 1 1323.2.o.e 48
63.j odd 6 1 1323.2.o.e 48
63.k odd 6 1 441.2.s.d 48
63.l odd 6 1 441.2.i.d 48
63.n odd 6 1 inner 1323.2.s.d 48
63.o even 6 1 1323.2.i.d 48
63.s even 6 1 inner 1323.2.s.d 48
63.t odd 6 1 441.2.o.e 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.i.d 48 9.c even 3 1
441.2.i.d 48 21.g even 6 1
441.2.i.d 48 21.h odd 6 1
441.2.i.d 48 63.l odd 6 1
441.2.o.e 48 21.g even 6 1
441.2.o.e 48 21.h odd 6 1
441.2.o.e 48 63.h even 3 1
441.2.o.e 48 63.t odd 6 1
441.2.s.d 48 3.b odd 2 1
441.2.s.d 48 21.c even 2 1
441.2.s.d 48 63.g even 3 1
441.2.s.d 48 63.k odd 6 1
1323.2.i.d 48 7.c even 3 1
1323.2.i.d 48 7.d odd 6 1
1323.2.i.d 48 9.d odd 6 1
1323.2.i.d 48 63.o even 6 1
1323.2.o.e 48 7.c even 3 1
1323.2.o.e 48 7.d odd 6 1
1323.2.o.e 48 63.i even 6 1
1323.2.o.e 48 63.j odd 6 1
1323.2.s.d 48 1.a even 1 1 trivial
1323.2.s.d 48 7.b odd 2 1 inner
1323.2.s.d 48 63.n odd 6 1 inner
1323.2.s.d 48 63.s even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$.