Space of modular forms of level 322, weight 2, and trivial character

• Label: 322.2.a
• Level: $322$
• Weight: $2$
• Character orbit: 322.a
• Rep. character: $\chi_{322}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $7$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(322))\).

	Total	New	Old
Modular forms	52	11	41
Cusp forms	45	11	34
Eisenstein series	7	0	7

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

\(2\)	\(7\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(7\)

Trace form

\( 11 q + 3 q^{2} + 11 q^{4} - 2 q^{5} - q^{7} + 3 q^{8} + 15 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7	23
322.2.a.a	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
322.2.a.b	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-2q^{6}+q^{7}-q^{8}+\cdots\)
322.2.a.c	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-2\)	\(-2\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{5}-2q^{6}-q^{7}+\cdots\)
322.2.a.d	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(2\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+q^{7}+\cdots\)
322.2.a.e	$322$	$2$	322.a	1.a	$1$	$2$	$2$	$2.571$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
322.2.a.f	$322$	$2$	322.a	1.a	$1$	$2$	$2$	$2.571$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
322.2.a.g	$322$	$2$	322.a	1.a	$1$	$3$	$3$	$2.571$	3.3.316.1	None	✓	$1$	$0$	\(3\)	\(2\)	\(4\)	\(-3\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(322))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(322)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)