Space of modular forms of level 324, weight 9, and Character 324.g

• Label: 324.9.g
• Level: $324$
• Weight: $9$
• Character orbit: 324.g
• Rep. character: $\chi_{324}(53,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $64$
• Newform subspaces: $8$
• Sturm bound: $486$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(486\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(324, [\chi])\).

	Total	New	Old
Modular forms	900	64	836
Cusp forms	828	64	764
Eisenstein series	72	0	72

Trace form

64 * q - 4615 * q^7 - 8425 * q^13 + 108626 * q^19 + 2838394 * q^25 + 970790 * q^31 + 1702370 * q^37 + 2714960 * q^43 - 32815383 * q^49 + 74574828 * q^55 - 21835225 * q^61 + 46321427 * q^67 - 30700306 * q^73 - 187831237 * q^79 - 22955364 * q^85 - 304930790 * q^91 + 117284045 * q^97

Decomposition of \(S_{9}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.9.g.a	$324$	$9$	324.g	9.d	$6$	$2$	$1$	$131.991$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					12.9.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4034\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-4034+4034\zeta_{6})q^{7}+35806\zeta_{6}q^{13}+\cdots\)
324.9.g.b	$324$	$9$	324.g	9.d	$6$	$2$	$1$	$131.991$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					108.9.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4273\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(4273-4273\zeta_{6})q^{7}+20641\zeta_{6}q^{13}+\cdots\)
324.9.g.c	$324$	$9$	324.g	9.d	$6$	$4$	$2$	$131.991$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					108.9.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-5278\)	$2^{10}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{5}+(-2639-2639\beta _{1})q^{7}+\cdots\)
324.9.g.d	$324$	$9$	324.g	9.d	$6$	$4$	$2$	$131.991$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					36.9.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-616\)	$3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}-308\beta _{2}q^{7}+(-244\beta _{1}+244\beta _{3})q^{11}+\cdots\)
324.9.g.e	$324$	$9$	324.g	9.d	$6$	$4$	$2$	$131.991$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None					108.9.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{2}\cdot 3^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{5}+(7+7\beta _{1})q^{7}+(19\beta _{2}-19\beta _{3})q^{11}+\cdots\)
324.9.g.f	$324$	$9$	324.g	9.d	$6$	$4$	$2$	$131.991$	\(\Q(\sqrt{-3}, \sqrt{-110})\)	None					12.9.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(6188\)	$2^{4}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+3094\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)
324.9.g.g	$324$	$9$	324.g	9.d	$6$	$12$	$6$	$131.991$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					108.9.c.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1470\)	$2^{18}\cdot 3^{48}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{5}+(-245+245\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
324.9.g.h	$324$	$9$	324.g	9.d	$6$	$32$	$16$	$131.991$		None		✓	✓		324.9.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-3692\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{9}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

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