Space of modular forms of level 48, weight 6, and trivial character

• Label: 48.6.a
• Level: $48$
• Weight: $6$
• Character orbit: 48.a
• Rep. character: $\chi_{48}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(48))\).

	Total	New	Old
Modular forms	46	5	41
Cusp forms	34	5	29
Eisenstein series	12	0	12

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

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

Trace form

\( 5 q + 9 q^{3} + 38 q^{5} - 160 q^{7} + 405 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(48))\) 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
48.6.a.a	$48$	$6$	48.a	1.a	$1$	$1$	$1$	$7.698$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-9\)	\(6\)	\(40\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+6q^{5}+40q^{7}+3^{4}q^{9}+564q^{11}+\cdots\)
48.6.a.b	$48$	$6$	48.a	1.a	$1$	$1$	$1$	$7.698$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(38\)	\(-120\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+38q^{5}-120q^{7}+3^{4}q^{9}+\cdots\)
48.6.a.c	$48$	$6$	48.a	1.a	$1$	$1$	$1$	$7.698$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(9\)	\(-66\)	\(-176\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-66q^{5}-176q^{7}+3^{4}q^{9}+\cdots\)
48.6.a.d	$48$	$6$	48.a	1.a	$1$	$1$	$1$	$7.698$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(-34\)	\(240\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-34q^{5}+240q^{7}+3^{4}q^{9}+\cdots\)
48.6.a.e	$48$	$6$	48.a	1.a	$1$	$1$	$1$	$7.698$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(94\)	\(-144\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+94q^{5}-12^{2}q^{7}+3^{4}q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(48)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)