# Properties

 Label 288.2.s.a Level $288$ Weight $2$ Character orbit 288.s Analytic conductor $2.300$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{9} + 8q^{21} + 12q^{25} + 24q^{29} - 20q^{33} - 36q^{41} - 8q^{45} + 12q^{49} - 36q^{57} - 48q^{65} - 16q^{69} + 24q^{73} - 48q^{77} - 20q^{81} - 64q^{93} + 12q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1 0 −1.71910 0.211392i 0 −0.135038 0.0779642i 0 0.349281 0.201658i 0 2.91063 + 0.726809i 0
95.2 0 −1.55381 + 0.765299i 0 1.81740 + 1.04928i 0 −0.143714 + 0.0829731i 0 1.82863 2.37826i 0
95.3 0 −1.20567 1.24353i 0 −0.398132 0.229862i 0 −4.28309 + 2.47284i 0 −0.0927324 + 2.99857i 0
95.4 0 −1.05094 + 1.37678i 0 −3.01113 1.73848i 0 3.12309 1.80312i 0 −0.791040 2.89383i 0
95.5 0 −0.683478 1.59150i 0 3.40926 + 1.96834i 0 0.961325 0.555021i 0 −2.06572 + 2.17550i 0
95.6 0 −0.324214 1.70144i 0 −1.68236 0.971313i 0 2.61432 1.50938i 0 −2.78977 + 1.10326i 0
95.7 0 0.324214 + 1.70144i 0 −1.68236 0.971313i 0 −2.61432 + 1.50938i 0 −2.78977 + 1.10326i 0
95.8 0 0.683478 + 1.59150i 0 3.40926 + 1.96834i 0 −0.961325 + 0.555021i 0 −2.06572 + 2.17550i 0
95.9 0 1.05094 1.37678i 0 −3.01113 1.73848i 0 −3.12309 + 1.80312i 0 −0.791040 2.89383i 0
95.10 0 1.20567 + 1.24353i 0 −0.398132 0.229862i 0 4.28309 2.47284i 0 −0.0927324 + 2.99857i 0
95.11 0 1.55381 0.765299i 0 1.81740 + 1.04928i 0 0.143714 0.0829731i 0 1.82863 2.37826i 0
95.12 0 1.71910 + 0.211392i 0 −0.135038 0.0779642i 0 −0.349281 + 0.201658i 0 2.91063 + 0.726809i 0
191.1 0 −1.71910 + 0.211392i 0 −0.135038 + 0.0779642i 0 0.349281 + 0.201658i 0 2.91063 0.726809i 0
191.2 0 −1.55381 0.765299i 0 1.81740 1.04928i 0 −0.143714 0.0829731i 0 1.82863 + 2.37826i 0
191.3 0 −1.20567 + 1.24353i 0 −0.398132 + 0.229862i 0 −4.28309 2.47284i 0 −0.0927324 2.99857i 0
191.4 0 −1.05094 1.37678i 0 −3.01113 + 1.73848i 0 3.12309 + 1.80312i 0 −0.791040 + 2.89383i 0
191.5 0 −0.683478 + 1.59150i 0 3.40926 1.96834i 0 0.961325 + 0.555021i 0 −2.06572 2.17550i 0
191.6 0 −0.324214 + 1.70144i 0 −1.68236 + 0.971313i 0 2.61432 + 1.50938i 0 −2.78977 1.10326i 0
191.7 0 0.324214 1.70144i 0 −1.68236 + 0.971313i 0 −2.61432 1.50938i 0 −2.78977 1.10326i 0
191.8 0 0.683478 1.59150i 0 3.40926 1.96834i 0 −0.961325 0.555021i 0 −2.06572 2.17550i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.12 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 288.2.s.a 24
3.b odd 2 1 864.2.s.a 24
4.b odd 2 1 inner 288.2.s.a 24
8.b even 2 1 576.2.s.g 24
8.d odd 2 1 576.2.s.g 24
9.c even 3 1 864.2.s.a 24
9.c even 3 1 2592.2.c.c 24
9.d odd 6 1 inner 288.2.s.a 24
9.d odd 6 1 2592.2.c.c 24
12.b even 2 1 864.2.s.a 24
24.f even 2 1 1728.2.s.g 24
24.h odd 2 1 1728.2.s.g 24
36.f odd 6 1 864.2.s.a 24
36.f odd 6 1 2592.2.c.c 24
36.h even 6 1 inner 288.2.s.a 24
36.h even 6 1 2592.2.c.c 24
72.j odd 6 1 576.2.s.g 24
72.j odd 6 1 5184.2.c.m 24
72.l even 6 1 576.2.s.g 24
72.l even 6 1 5184.2.c.m 24
72.n even 6 1 1728.2.s.g 24
72.n even 6 1 5184.2.c.m 24
72.p odd 6 1 1728.2.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 1.a even 1 1 trivial
288.2.s.a 24 4.b odd 2 1 inner
288.2.s.a 24 9.d odd 6 1 inner
288.2.s.a 24 36.h even 6 1 inner
576.2.s.g 24 8.b even 2 1
576.2.s.g 24 8.d odd 2 1
576.2.s.g 24 72.j odd 6 1
576.2.s.g 24 72.l even 6 1
864.2.s.a 24 3.b odd 2 1
864.2.s.a 24 9.c even 3 1
864.2.s.a 24 12.b even 2 1
864.2.s.a 24 36.f odd 6 1
1728.2.s.g 24 24.f even 2 1
1728.2.s.g 24 24.h odd 2 1
1728.2.s.g 24 72.n even 6 1
1728.2.s.g 24 72.p odd 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 $$S_{2}^{\mathrm{new}}(288, [\chi])$$.