Space of modular forms of level 324, weight 5, and Character 324.d

• Label: 324.5.d
• Level: $324$
• Weight: $5$
• Character orbit: 324.d
• Rep. character: $\chi_{324}(163,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $7$
• Sturm bound: $270$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(270\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(324, [\chi])\).

	Total	New	Old
Modular forms	228	100	128
Cusp forms	204	92	112
Eisenstein series	24	8	16

Trace form

\( 92 q + 2 q^{4} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.5.d.a	$324$	$5$	324.d	4.b	$2$	$2$	$2$	$33.492$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(-8\)	\(0\)	\(14\)	\(0\)	$2^{3}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q-4q^{2}+2^{4}q^{4}+(7+\beta )q^{5}-2^{6}q^{8}+\cdots\)
324.5.d.b	$324$	$5$	324.d	4.b	$2$	$2$	$2$	$33.492$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(8\)	\(0\)	\(-14\)	\(0\)	$2^{3}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+4q^{2}+2^{4}q^{4}+(-7+\beta )q^{5}+2^{6}q^{8}+\cdots\)
324.5.d.c	$324$	$5$	324.d	4.b	$2$	$10$	$10$	$33.492$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(-11\)	\(0\)	\(2\)	\(0\)	$2^{16}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{2}+(-4+\beta _{1}+\beta _{6})q^{4}+\cdots\)
324.5.d.d	$324$	$5$	324.d	4.b	$2$	$10$	$10$	$33.492$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(11\)	\(0\)	\(-2\)	\(0\)	$2^{16}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{2}+(-4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
324.5.d.e	$324$	$5$	324.d	4.b	$2$	$22$	$22$	$33.492$		None		$2$	$0$	\(-1\)	\(0\)	\(-2\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
324.5.d.f	$324$	$5$	324.d	4.b	$2$	$22$	$22$	$33.492$		None		$2$	$0$	\(1\)	\(0\)	\(2\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
324.5.d.g	$324$	$5$	324.d	4.b	$2$	$24$	$24$	$33.492$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{5}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)