Space of modular forms of level 138, weight 2, and trivial character

• Label: 138.2.a
• Level: $138$
• Weight: $2$
• Character orbit: 138.a
• Rep. character: $\chi_{138}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(138))\).

	Total	New	Old
Modular forms	28	5	23
Cusp forms	21	5	16
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.

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

Trace form

\( 5 q + q^{2} + q^{3} + 5 q^{4} - 2 q^{5} + q^{6} + q^{8} + 5 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(138))\) 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	3	23
138.2.a.a	$138$	$2$	138.a	1.a	$1$	$1$	$1$	$1.102$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}-2q^{7}+\cdots\)
138.2.a.b	$138$	$2$	138.a	1.a	$1$	$1$	$1$	$1.102$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
138.2.a.c	$138$	$2$	138.a	1.a	$1$	$1$	$1$	$1.102$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{8}+\cdots\)
138.2.a.d	$138$	$2$	138.a	1.a	$1$	$2$	$2$	$1.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(-1+\beta )q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(138))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(138)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)