Space of modular forms of level 328, weight 2, and trivial character

• Label: 328.2.a
• Level: $328$
• Weight: $2$
• Character orbit: 328.a
• Rep. character: $\chi_{328}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $5$
• Sturm bound: $84$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 328 = 2^{3} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	328.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(84\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(328))\).

	Total	New	Old
Modular forms	46	10	36
Cusp forms	39	10	29
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.

\(2\)	\(41\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(6\)

Trace form

\( 10 q - 2 q^{3} + 10 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(328))\) 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	41
328.2.a.a	$328$	$2$	328.a	1.a	$1$	$1$	$1$	$2.619$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}-3q^{9}-4q^{13}-2q^{17}+\cdots\)
328.2.a.b	$328$	$2$	328.a	1.a	$1$	$1$	$1$	$2.619$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}-2q^{7}+q^{9}+2q^{11}+\cdots\)
328.2.a.c	$328$	$2$	328.a	1.a	$1$	$2$	$2$	$2.619$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
328.2.a.d	$328$	$2$	328.a	1.a	$1$	$3$	$3$	$2.619$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
328.2.a.e	$328$	$2$	328.a	1.a	$1$	$3$	$3$	$2.619$	3.3.788.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{2})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(328))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(328)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(164))\)\(^{\oplus 2}\)