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

• Label: 324.5.g
• Level: $324$
• Weight: $5$
• Character orbit: 324.g
• Rep. character: $\chi_{324}(53,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $32$
• Newform subspaces: $6$
• Sturm bound: $270$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(270\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	468	32	436
Cusp forms	396	32	364
Eisenstein series	72	0	72

Trace form

\( 32 q - 65 q^{7} + O(q^{10}) \)

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.g.a	$324$	$5$	324.g	9.d	$6$	$2$	$1$	$33.492$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-23\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-23+23\zeta_{6})q^{7}-191\zeta_{6}q^{13}+\cdots\)
324.5.g.b	$324$	$5$	324.g	9.d	$6$	$2$	$1$	$33.492$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(94\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(94-94\zeta_{6})q^{7}-146\zeta_{6}q^{13}-46q^{19}+\cdots\)
324.5.g.c	$324$	$5$	324.g	9.d	$6$	$4$	$2$	$33.492$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-136\)	$3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}-68\beta _{2}q^{7}+(-4\beta _{1}+4\beta _{3})q^{11}+\cdots\)
324.5.g.d	$324$	$5$	324.g	9.d	$6$	$4$	$2$	$33.492$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{5}-5\zeta_{12}^{2}q^{7}+(-13\zeta_{12}+\cdots)q^{11}+\cdots\)
324.5.g.e	$324$	$5$	324.g	9.d	$6$	$4$	$2$	$33.492$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(62\)	$2^{2}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{5}+(31+31\beta _{1})q^{7}+(-5\beta _{2}+\cdots)q^{11}+\cdots\)
324.5.g.f	$324$	$5$	324.g	9.d	$6$	$16$	$8$	$33.492$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-52\)	$2^{4}\cdot 3^{44}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{5}-\beta _{9})q^{5}+(-7-7\beta _{4}+\beta _{7})q^{7}+\cdots\)

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

\( S_{5}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)