Space of modular forms of level 324, weight 3, and Character 324.d

• Label: 324.3.d
• Level: $324$
• Weight: $3$
• Character orbit: 324.d
• Rep. character: $\chi_{324}(163,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $9$
• Sturm bound: $162$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(162\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(324, [\chi])\).

	Total	New	Old
Modular forms	120	52	68
Cusp forms	96	44	52
Eisenstein series	24	8	16

Trace form

\( 44 q + 2 q^{4} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(324, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.3.d.a	$324$	$3$	324.d	4.b	$2$	$2$	$2$	$8.828$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(-4\)	\(0\)	\(-8\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}+(-4+\beta )q^{5}-8q^{8}+\cdots\)
324.3.d.b	$324$	$3$	324.d	4.b	$2$	$2$	$2$	$8.828$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}+4q^{5}+\cdots\)
324.3.d.c	$324$	$3$	324.d	4.b	$2$	$2$	$2$	$8.828$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(2\)	\(0\)	\(-8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}-4q^{5}+\cdots\)
324.3.d.d	$324$	$3$	324.d	4.b	$2$	$2$	$2$	$8.828$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(4\)	\(0\)	\(8\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}+(4+\beta )q^{5}+8q^{8}+(8+\cdots)q^{10}+\cdots\)
324.3.d.e	$324$	$3$	324.d	4.b	$2$	$6$	$6$	$8.828$	6.0.3636603.2	None		$2$	$0$	\(-1\)	\(0\)	\(-2\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-\beta _{1}q^{4}+(-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
324.3.d.f	$324$	$3$	324.d	4.b	$2$	$6$	$6$	$8.828$	6.0.3636603.2	None		$2$	$0$	\(1\)	\(0\)	\(2\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-\beta _{2}q^{4}+(-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
324.3.d.g	$324$	$3$	324.d	4.b	$2$	$8$	$8$	$8.828$	8.0.\(\cdots\).1	None		$2$	$0$	\(-3\)	\(0\)	\(6\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{5}-\beta _{6})q^{4}+\cdots\)
324.3.d.h	$324$	$3$	324.d	4.b	$2$	$8$	$8$	$8.828$	8.0.\(\cdots\).4	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(-1-\beta _{6})q^{4}+(-2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
324.3.d.i	$324$	$3$	324.d	4.b	$2$	$8$	$8$	$8.828$	8.0.\(\cdots\).1	None		$2$	$0$	\(3\)	\(0\)	\(-6\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{4}+\beta _{6})q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

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