# Properties

 Label 864.2.s.a Level 864 Weight 2 Character orbit 864.s Analytic conductor 6.899 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 288) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{25} - 24q^{29} + 36q^{41} + 12q^{49} + 48q^{65} + 24q^{73} + 48q^{77} + 12q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
287.1 0 0 0 −3.40926 1.96834i 0 −0.961325 + 0.555021i 0 0 0
287.2 0 0 0 −3.40926 1.96834i 0 0.961325 0.555021i 0 0 0
287.3 0 0 0 −1.81740 1.04928i 0 −0.143714 + 0.0829731i 0 0 0
287.4 0 0 0 −1.81740 1.04928i 0 0.143714 0.0829731i 0 0 0
287.5 0 0 0 0.135038 + 0.0779642i 0 −0.349281 + 0.201658i 0 0 0
287.6 0 0 0 0.135038 + 0.0779642i 0 0.349281 0.201658i 0 0 0
287.7 0 0 0 0.398132 + 0.229862i 0 −4.28309 + 2.47284i 0 0 0
287.8 0 0 0 0.398132 + 0.229862i 0 4.28309 2.47284i 0 0 0
287.9 0 0 0 1.68236 + 0.971313i 0 −2.61432 + 1.50938i 0 0 0
287.10 0 0 0 1.68236 + 0.971313i 0 2.61432 1.50938i 0 0 0
287.11 0 0 0 3.01113 + 1.73848i 0 −3.12309 + 1.80312i 0 0 0
287.12 0 0 0 3.01113 + 1.73848i 0 3.12309 1.80312i 0 0 0
575.1 0 0 0 −3.40926 + 1.96834i 0 −0.961325 0.555021i 0 0 0
575.2 0 0 0 −3.40926 + 1.96834i 0 0.961325 + 0.555021i 0 0 0
575.3 0 0 0 −1.81740 + 1.04928i 0 −0.143714 0.0829731i 0 0 0
575.4 0 0 0 −1.81740 + 1.04928i 0 0.143714 + 0.0829731i 0 0 0
575.5 0 0 0 0.135038 0.0779642i 0 −0.349281 0.201658i 0 0 0
575.6 0 0 0 0.135038 0.0779642i 0 0.349281 + 0.201658i 0 0 0
575.7 0 0 0 0.398132 0.229862i 0 −4.28309 2.47284i 0 0 0
575.8 0 0 0 0.398132 0.229862i 0 4.28309 + 2.47284i 0 0 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 575.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.s.a 24
3.b odd 2 1 288.2.s.a 24
4.b odd 2 1 inner 864.2.s.a 24
8.b even 2 1 1728.2.s.g 24
8.d odd 2 1 1728.2.s.g 24
9.c even 3 1 288.2.s.a 24
9.c even 3 1 2592.2.c.c 24
9.d odd 6 1 inner 864.2.s.a 24
9.d odd 6 1 2592.2.c.c 24
12.b even 2 1 288.2.s.a 24
24.f even 2 1 576.2.s.g 24
24.h odd 2 1 576.2.s.g 24
36.f odd 6 1 288.2.s.a 24
36.f odd 6 1 2592.2.c.c 24
36.h even 6 1 inner 864.2.s.a 24
36.h even 6 1 2592.2.c.c 24
72.j odd 6 1 1728.2.s.g 24
72.j odd 6 1 5184.2.c.m 24
72.l even 6 1 1728.2.s.g 24
72.l even 6 1 5184.2.c.m 24
72.n even 6 1 576.2.s.g 24
72.n even 6 1 5184.2.c.m 24
72.p odd 6 1 576.2.s.g 24
72.p odd 6 1 5184.2.c.m 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.s.a 24 3.b odd 2 1
288.2.s.a 24 9.c even 3 1
288.2.s.a 24 12.b even 2 1
288.2.s.a 24 36.f odd 6 1
576.2.s.g 24 24.f even 2 1
576.2.s.g 24 24.h odd 2 1
576.2.s.g 24 72.n even 6 1
576.2.s.g 24 72.p odd 6 1
864.2.s.a 24 1.a even 1 1 trivial
864.2.s.a 24 4.b odd 2 1 inner
864.2.s.a 24 9.d odd 6 1 inner
864.2.s.a 24 36.h even 6 1 inner
1728.2.s.g 24 8.b even 2 1
1728.2.s.g 24 8.d odd 2 1
1728.2.s.g 24 72.j odd 6 1
1728.2.s.g 24 72.l even 6 1
2592.2.c.c 24 9.c even 3 1
2592.2.c.c 24 9.d odd 6 1
2592.2.c.c 24 36.f odd 6 1
2592.2.c.c 24 36.h even 6 1
5184.2.c.m 24 72.j odd 6 1
5184.2.c.m 24 72.l even 6 1
5184.2.c.m 24 72.n even 6 1
5184.2.c.m 24 72.p odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database