Space of modular forms of level 324, weight 2, and trivial character

• Label: 324.2.a
• Level: $324$
• Weight: $2$
• Character orbit: 324.a
• Rep. character: $\chi_{324}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $108$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(108\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(324))\).

	Total	New	Old
Modular forms	72	4	68
Cusp forms	37	4	33
Eisenstein series	35	0	35

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(18\)	\(0\)	\(18\)		\(7\)	\(0\)	\(7\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(21\)	\(0\)	\(21\)		\(9\)	\(0\)	\(9\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(18\)	\(3\)	\(15\)		\(12\)	\(3\)	\(9\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(15\)	\(1\)	\(14\)		\(9\)	\(1\)	\(8\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(33\)	\(1\)	\(32\)		\(16\)	\(1\)	\(15\)		\(17\)	\(0\)	\(17\)
Minus space		\(-\)		\(39\)	\(3\)	\(36\)		\(21\)	\(3\)	\(18\)		\(18\)	\(0\)	\(18\)

Trace form

\( 4 q + 2 q^{7} + 8 q^{13} - 4 q^{19} + 16 q^{25} + 2 q^{31} + 14 q^{37} - 22 q^{43} - 18 q^{49} - 18 q^{55} - 16 q^{61} - 10 q^{67} - 22 q^{73} + 2 q^{79} + 18 q^{85} + 22 q^{91} + 2 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(324))\) 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
324.2.a.a	$324$	$2$	324.a	1.a	$1$	$1$	$1$	$2.587$	\(\Q\)	None	✓		✓	✓	36.2.e.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-q^{7}-3q^{11}-q^{13}-6q^{17}+\cdots\)
324.2.a.b	$324$	$2$	324.a	1.a	$1$	$1$	$1$	$2.587$	\(\Q\)	None	✓	✓			324.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+2q^{7}+6q^{11}+5q^{13}+3q^{17}+\cdots\)
324.2.a.c	$324$	$2$	324.a	1.a	$1$	$1$	$1$	$2.587$	\(\Q\)	None	✓				36.2.e.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-q^{7}+3q^{11}-q^{13}+6q^{17}+\cdots\)
324.2.a.d	$324$	$2$	324.a	1.a	$1$	$1$	$1$	$2.587$	\(\Q\)	None	✓	✓			324.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+2q^{7}-6q^{11}+5q^{13}-3q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces

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