# Properties

 Label 585.2.bs.c Level $585$ Weight $2$ Character orbit 585.bs Analytic conductor $4.671$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.bs (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 20 q^{4}+O(q^{10})$$ 32 * q + 20 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 20 q^{4} - 6 q^{10} - 28 q^{16} - 8 q^{19} + 28 q^{25} + 8 q^{31} - 8 q^{34} - 20 q^{40} - 8 q^{46} + 44 q^{49} + 20 q^{55} - 56 q^{61} - 136 q^{64} - 80 q^{70} + 88 q^{76} - 72 q^{79} - 50 q^{85} - 28 q^{91} + 48 q^{94}+O(q^{100})$$ 32 * q + 20 * q^4 - 6 * q^10 - 28 * q^16 - 8 * q^19 + 28 * q^25 + 8 * q^31 - 8 * q^34 - 20 * q^40 - 8 * q^46 + 44 * q^49 + 20 * q^55 - 56 * q^61 - 136 * q^64 - 80 * q^70 + 88 * q^76 - 72 * q^79 - 50 * q^85 - 28 * q^91 + 48 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1 −2.32723 + 1.34363i 0 2.61067 4.52182i −2.23433 + 0.0881535i 0 −3.14272 1.81445i 8.65659i 0 5.08136 3.20726i
289.2 −2.32723 + 1.34363i 0 2.61067 4.52182i 2.23433 + 0.0881535i 0 3.14272 + 1.81445i 8.65659i 0 −5.31825 + 2.79696i
289.3 −1.57126 + 0.907167i 0 0.645904 1.11874i −1.19685 + 1.88880i 0 4.06197 + 2.34518i 1.28490i 0 0.167109 4.05353i
289.4 −1.57126 + 0.907167i 0 0.645904 1.11874i 1.19685 + 1.88880i 0 −4.06197 2.34518i 1.28490i 0 −3.59402 1.88204i
289.5 −1.35625 + 0.783029i 0 0.226269 0.391909i −1.90777 1.16637i 0 0.331025 + 0.191118i 2.42342i 0 3.50070 + 0.0880529i
289.6 −1.35625 + 0.783029i 0 0.226269 0.391909i 1.90777 1.16637i 0 −0.331025 0.191118i 2.42342i 0 −1.67410 + 3.07573i
289.7 −0.160403 + 0.0926085i 0 −0.982847 + 1.70234i −1.29836 1.82051i 0 1.66258 + 0.959889i 0.734514i 0 0.376855 + 0.171775i
289.8 −0.160403 + 0.0926085i 0 −0.982847 + 1.70234i 1.29836 1.82051i 0 −1.66258 0.959889i 0.734514i 0 −0.0396663 + 0.412254i
289.9 0.160403 0.0926085i 0 −0.982847 + 1.70234i −1.29836 + 1.82051i 0 −1.66258 0.959889i 0.734514i 0 −0.0396663 + 0.412254i
289.10 0.160403 0.0926085i 0 −0.982847 + 1.70234i 1.29836 + 1.82051i 0 1.66258 + 0.959889i 0.734514i 0 0.376855 + 0.171775i
289.11 1.35625 0.783029i 0 0.226269 0.391909i −1.90777 + 1.16637i 0 −0.331025 0.191118i 2.42342i 0 −1.67410 + 3.07573i
289.12 1.35625 0.783029i 0 0.226269 0.391909i 1.90777 + 1.16637i 0 0.331025 + 0.191118i 2.42342i 0 3.50070 + 0.0880529i
289.13 1.57126 0.907167i 0 0.645904 1.11874i −1.19685 1.88880i 0 −4.06197 2.34518i 1.28490i 0 −3.59402 1.88204i
289.14 1.57126 0.907167i 0 0.645904 1.11874i 1.19685 1.88880i 0 4.06197 + 2.34518i 1.28490i 0 0.167109 4.05353i
289.15 2.32723 1.34363i 0 2.61067 4.52182i −2.23433 0.0881535i 0 3.14272 + 1.81445i 8.65659i 0 −5.31825 + 2.79696i
289.16 2.32723 1.34363i 0 2.61067 4.52182i 2.23433 0.0881535i 0 −3.14272 1.81445i 8.65659i 0 5.08136 3.20726i
334.1 −2.32723 1.34363i 0 2.61067 + 4.52182i −2.23433 0.0881535i 0 −3.14272 + 1.81445i 8.65659i 0 5.08136 + 3.20726i
334.2 −2.32723 1.34363i 0 2.61067 + 4.52182i 2.23433 0.0881535i 0 3.14272 1.81445i 8.65659i 0 −5.31825 2.79696i
334.3 −1.57126 0.907167i 0 0.645904 + 1.11874i −1.19685 1.88880i 0 4.06197 2.34518i 1.28490i 0 0.167109 + 4.05353i
334.4 −1.57126 0.907167i 0 0.645904 + 1.11874i 1.19685 1.88880i 0 −4.06197 + 2.34518i 1.28490i 0 −3.59402 + 1.88204i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 334.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
13.c even 3 1 inner
15.d odd 2 1 inner
39.i odd 6 1 inner
65.n even 6 1 inner
195.x odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bs.c 32
3.b odd 2 1 inner 585.2.bs.c 32
5.b even 2 1 inner 585.2.bs.c 32
13.c even 3 1 inner 585.2.bs.c 32
15.d odd 2 1 inner 585.2.bs.c 32
39.i odd 6 1 inner 585.2.bs.c 32
65.n even 6 1 inner 585.2.bs.c 32
195.x odd 6 1 inner 585.2.bs.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.bs.c 32 1.a even 1 1 trivial
585.2.bs.c 32 3.b odd 2 1 inner
585.2.bs.c 32 5.b even 2 1 inner
585.2.bs.c 32 13.c even 3 1 inner
585.2.bs.c 32 15.d odd 2 1 inner
585.2.bs.c 32 39.i odd 6 1 inner
585.2.bs.c 32 65.n even 6 1 inner
585.2.bs.c 32 195.x odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 13T_{2}^{14} + 119T_{2}^{12} - 530T_{2}^{10} + 1718T_{2}^{8} - 2948T_{2}^{6} + 3500T_{2}^{4} - 120T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.