Space of modular forms of level 338, weight 2, and Character 338.i

• Label: 338.2.i
• Level: $338$
• Weight: $2$
• Character orbit: 338.i
• Rep. character: $\chi_{338}(3,\cdot)$
• Character field: $\Q(\zeta_{39})$
• Dimension: $360$
• Newform subspaces: $2$
• Sturm bound: $91$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.i (of order \(39\) and degree \(24\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 169 \)
Character field:			\(\Q(\zeta_{39})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(91\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(338, [\chi])\).

	Total	New	Old
Modular forms	1128	360	768
Cusp forms	1032	360	672
Eisenstein series	96	0	96

Trace form

\( 360 q + q^{2} + 15 q^{4} + 2 q^{5} + 4 q^{7} - 2 q^{8} + 11 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(338, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
338.2.i.a	$338$	$2$	338.i	169.i	$39$	$168$	$7$	$2.699$		None		✓			338.2.i.a	$2$	$0$	\(-7\)	\(0\)	\(0\)	\(14\)		$\mathrm{SU}(2)[C_{39}]$
338.2.i.b	$338$	$2$	338.i	169.i	$39$	$192$	$8$	$2.699$		None		✓	✓		338.2.i.b	$2$	$0$	\(8\)	\(0\)	\(2\)	\(-10\)		$\mathrm{SU}(2)[C_{39}]$

Decomposition of \(S_{2}^{\mathrm{old}}(338, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(338, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)