Space of modular forms of level 323, weight 2, and trivial character

• Label: 323.2.a
• Level: $323$
• Weight: $2$
• Character orbit: 323.a
• Rep. character: $\chi_{323}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $6$
• Sturm bound: $60$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 323 = 17 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	323.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(60\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(323))\).

	Total	New	Old
Modular forms	32	25	7
Cusp forms	29	25	4
Eisenstein series	3	0	3

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

\(17\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(16\)

Trace form

\( 25 q - q^{2} + 27 q^{4} - 2 q^{5} - 12 q^{6} - 4 q^{7} - 9 q^{8} + 25 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(323))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		17	19
323.2.a.a	$323$	$2$	323.a	1.a	$1$	$1$	$1$	$2.579$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{4}-2q^{5}+4q^{7}+6q^{9}+\cdots\)
323.2.a.b	$323$	$2$	323.a	1.a	$1$	$2$	$2$	$2.579$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1-\beta )q^{3}+(2+\beta )q^{4}+2q^{5}+\cdots\)
323.2.a.c	$323$	$2$	323.a	1.a	$1$	$4$	$4$	$2.579$	4.4.1957.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-7\)	\(-11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2}+\beta _{3})q^{2}+(-\beta _{1}-\beta _{3})q^{3}+\cdots\)
323.2.a.d	$323$	$2$	323.a	1.a	$1$	$5$	$5$	$2.579$	5.5.106069.1	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(-3\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{3})q^{3}+(1+\cdots)q^{4}+\cdots\)
323.2.a.e	$323$	$2$	323.a	1.a	$1$	$6$	$6$	$2.579$	6.6.28145473.1	None	✓	$1$	$0$	\(2\)	\(-3\)	\(-1\)	\(5\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}-\beta _{1}q^{3}+(1+\beta _{4}-\beta _{5})q^{4}+\cdots\)
323.2.a.f	$323$	$2$	323.a	1.a	$1$	$7$	$7$	$2.579$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(3\)	\(7\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{6})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(323))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(323)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)