# Properties

 Label 1323.2.o.e Level $1323$ Weight $2$ Character orbit 1323.o 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.o (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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 24q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 24q^{4} + 24q^{11} - 24q^{16} + 48q^{23} - 24q^{25} - 120q^{32} - 48q^{50} - 48q^{64} - 120q^{65} + 168q^{74} - 24q^{79} - 24q^{85} - 24q^{86} + 144q^{92} - 96q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1 −2.23278 1.28910i 0 2.32354 + 4.02449i 1.16595 + 2.01948i 0 0 6.82470i 0 6.01207i
440.2 −2.23278 1.28910i 0 2.32354 + 4.02449i −1.16595 2.01948i 0 0 6.82470i 0 6.01207i
440.3 −1.80506 1.04215i 0 1.17216 + 2.03024i −1.65233 2.86191i 0 0 0.717672i 0 6.88790i
440.4 −1.80506 1.04215i 0 1.17216 + 2.03024i 1.65233 + 2.86191i 0 0 0.717672i 0 6.88790i
440.5 −1.61855 0.934468i 0 0.746462 + 1.29291i −1.25287 2.17003i 0 0 0.947692i 0 4.68306i
440.6 −1.61855 0.934468i 0 0.746462 + 1.29291i 1.25287 + 2.17003i 0 0 0.947692i 0 4.68306i
440.7 −1.58658 0.916012i 0 0.678156 + 1.17460i 0.322784 + 0.559079i 0 0 1.17925i 0 1.18270i
440.8 −1.58658 0.916012i 0 0.678156 + 1.17460i −0.322784 0.559079i 0 0 1.17925i 0 1.18270i
440.9 −0.575298 0.332148i 0 −0.779355 1.34988i −0.0141520 0.0245119i 0 0 2.36404i 0 0.0188022i
440.10 −0.575298 0.332148i 0 −0.779355 1.34988i 0.0141520 + 0.0245119i 0 0 2.36404i 0 0.0188022i
440.11 −0.105953 0.0611722i 0 −0.992516 1.71909i −0.264715 0.458500i 0 0 0.487547i 0 0.0647728i
440.12 −0.105953 0.0611722i 0 −0.992516 1.71909i 0.264715 + 0.458500i 0 0 0.487547i 0 0.0647728i
440.13 0.367369 + 0.212101i 0 −0.910027 1.57621i 1.80381 + 3.12430i 0 0 1.62047i 0 1.53036i
440.14 0.367369 + 0.212101i 0 −0.910027 1.57621i −1.80381 3.12430i 0 0 1.62047i 0 1.53036i
440.15 0.850109 + 0.490811i 0 −0.518210 0.897565i 0.940599 + 1.62916i 0 0 2.98061i 0 1.84662i
440.16 0.850109 + 0.490811i 0 −0.518210 0.897565i −0.940599 1.62916i 0 0 2.98061i 0 1.84662i
440.17 1.02035 + 0.589100i 0 −0.305921 0.529871i −2.16601 3.75164i 0 0 3.07728i 0 5.10399i
440.18 1.02035 + 0.589100i 0 −0.305921 0.529871i 2.16601 + 3.75164i 0 0 3.07728i 0 5.10399i
440.19 1.28562 + 0.742253i 0 0.101880 + 0.176462i −0.154215 0.267109i 0 0 2.66653i 0 0.457868i
440.20 1.28562 + 0.742253i 0 0.101880 + 0.176462i 0.154215 + 0.267109i 0 0 2.66653i 0 0.457868i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.24 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
9.d odd 6 1 inner
63.o 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.o.e 48
3.b odd 2 1 441.2.o.e 48
7.b odd 2 1 inner 1323.2.o.e 48
7.c even 3 1 1323.2.i.d 48
7.c even 3 1 1323.2.s.d 48
7.d odd 6 1 1323.2.i.d 48
7.d odd 6 1 1323.2.s.d 48
9.c even 3 1 441.2.o.e 48
9.d odd 6 1 inner 1323.2.o.e 48
21.c even 2 1 441.2.o.e 48
21.g even 6 1 441.2.i.d 48
21.g even 6 1 441.2.s.d 48
21.h odd 6 1 441.2.i.d 48
21.h odd 6 1 441.2.s.d 48
63.g even 3 1 441.2.i.d 48
63.h even 3 1 441.2.s.d 48
63.i even 6 1 1323.2.s.d 48
63.j odd 6 1 1323.2.s.d 48
63.k odd 6 1 441.2.i.d 48
63.l odd 6 1 441.2.o.e 48
63.n odd 6 1 1323.2.i.d 48
63.o even 6 1 inner 1323.2.o.e 48
63.s even 6 1 1323.2.i.d 48
63.t odd 6 1 441.2.s.d 48

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

## Hecke kernels

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

 $$T_{2}^{24} - \cdots$$ $$11\!\cdots\!90$$$$T_{5}^{28} +$$$$49\!\cdots\!44$$$$T_{5}^{26} +$$$$16\!\cdots\!48$$$$T_{5}^{24} +$$$$42\!\cdots\!76$$$$T_{5}^{22} +$$$$79\!\cdots\!44$$$$T_{5}^{20} +$$$$10\!\cdots\!12$$$$T_{5}^{18} +$$$$95\!\cdots\!91$$$$T_{5}^{16} +$$$$52\!\cdots\!64$$$$T_{5}^{14} +$$$$20\!\cdots\!80$$$$T_{5}^{12} +$$$$50\!\cdots\!24$$$$T_{5}^{10} + 859511233110 T_{5}^{8} + 68412222392 T_{5}^{6} + 3795958686 T_{5}^{4} + 3040548 T_{5}^{2} + 2401$$">$$T_{5}^{48} + \cdots$$