Space of modular forms of level 324, weight 4, and trivial character

• Label: 324.4.a
• Level: $324$
• Weight: $4$
• Character orbit: 324.a
• Rep. character: $\chi_{324}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $5$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(216\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(324))\).

	Total	New	Old
Modular forms	180	12	168
Cusp forms	144	12	132
Eisenstein series	36	0	36

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	Fricke	Dim
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(5\)

Trace form

\( 12 q - 12 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(324))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
324.4.a.a	$324$	$4$	324.a	1.a	$1$	$1$	$1$	$19.117$	\(\Q\)	None	✓	✓			324.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-4q^{7}+24q^{11}-5^{2}q^{13}+\cdots\)
324.4.a.b	$324$	$4$	324.a	1.a	$1$	$1$	$1$	$19.117$	\(\Q\)	None	✓	✓			324.4.a.a	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-4q^{7}-24q^{11}-5^{2}q^{13}+\cdots\)
324.4.a.c	$324$	$4$	324.a	1.a	$1$	$3$	$3$	$19.117$	3.3.1509.1	None	✓		✓		36.4.e.a	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(6\)		$-$	$+$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
324.4.a.d	$324$	$4$	324.a	1.a	$1$	$3$	$3$	$19.117$	3.3.1509.1	None	✓				36.4.e.a	$1$	$0$	\(0\)	\(0\)	\(6\)	\(6\)		$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+(17+\cdots)q^{11}+\cdots\)
324.4.a.e	$324$	$4$	324.a	1.a	$1$	$4$	$4$	$19.117$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	✓	✓		324.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)		$-$	$-$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+(-4+\beta _{3})q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(324)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)