Space of modular forms of level 138, weight 6, and trivial character

• Label: 138.6.a
• Level: $138$
• Weight: $6$
• Character orbit: 138.a
• Rep. character: $\chi_{138}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $9$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(138))\).

	Total	New	Old
Modular forms	124	18	106
Cusp forms	116	18	98
Eisenstein series	8	0	8

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
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(7\)
Minus space			\(-\)	\(11\)

Trace form

\( 18 q - 8 q^{2} + 18 q^{3} + 288 q^{4} + 44 q^{5} + 72 q^{6} + 120 q^{7} - 128 q^{8} + 1458 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\) 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
138.6.a.a	$138$	$6$	138.a	1.a	$1$	$1$	$1$	$22.133$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(-9\)	\(-44\)	\(-70\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}-44q^{5}+6^{2}q^{6}+\cdots\)
138.6.a.b	$138$	$6$	138.a	1.a	$1$	$1$	$1$	$22.133$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(-9\)	\(76\)	\(210\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+76q^{5}+6^{2}q^{6}+\cdots\)
138.6.a.c	$138$	$6$	138.a	1.a	$1$	$1$	$1$	$22.133$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(-9\)	\(-10\)	\(-8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}-10q^{5}-6^{2}q^{6}+\cdots\)
138.6.a.d	$138$	$6$	138.a	1.a	$1$	$1$	$1$	$22.133$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(9\)	\(-46\)	\(-136\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}-46q^{5}+6^{2}q^{6}+\cdots\)
138.6.a.e	$138$	$6$	138.a	1.a	$1$	$2$	$2$	$22.133$	\(\Q(\sqrt{154}) \)	None	✓	$1$	$1$	\(-8\)	\(18\)	\(-4\)	\(-40\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-2+3\beta )q^{5}+\cdots\)
138.6.a.f	$138$	$6$	138.a	1.a	$1$	$2$	$2$	$22.133$	\(\Q(\sqrt{514}) \)	None	✓	$1$	$0$	\(8\)	\(-18\)	\(40\)	\(-100\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+(20+3\beta )q^{5}+\cdots\)
138.6.a.g	$138$	$6$	138.a	1.a	$1$	$3$	$3$	$22.133$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-12\)	\(-27\)	\(-18\)	\(-50\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-6-\beta _{2})q^{5}+\cdots\)
138.6.a.h	$138$	$6$	138.a	1.a	$1$	$3$	$3$	$22.133$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-12\)	\(27\)	\(-4\)	\(-34\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
138.6.a.i	$138$	$6$	138.a	1.a	$1$	$4$	$4$	$22.133$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(16\)	\(36\)	\(54\)	\(348\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+(13-\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(138))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(138)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)