Space of modular forms of level 338, weight 2, and trivial character

• Label: 338.2.a
• Level: $338$
• Weight: $2$
• Character orbit: 338.a
• Rep. character: $\chi_{338}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $8$
• Sturm bound: $91$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(91\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(338))\).

	Total	New	Old
Modular forms	59	12	47
Cusp forms	32	12	20
Eisenstein series	27	0	27

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

\(2\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(5\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(9\)

Trace form

\( 12 q + 2 q^{3} + 12 q^{4} + 4 q^{5} + 4 q^{6} + 10 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(338))\) 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	13
338.2.a.a	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
338.2.a.b	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(3\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
338.2.a.c	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots\)
338.2.a.d	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
338.2.a.e	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots\)
338.2.a.f	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
338.2.a.g	$338$	$2$	338.a	1.a	$1$	$3$	$3$	$2.699$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(-3\)	\(3\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots\)
338.2.a.h	$338$	$2$	338.a	1.a	$1$	$3$	$3$	$2.699$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(3\)	\(3\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(338)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)