Space of modular forms of level 48, weight 4, and trivial character

• Label: 48.4.a
• Level: $48$
• Weight: $4$
• Character orbit: 48.a
• Rep. character: $\chi_{48}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $32$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(32\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(48))\).

	Total	New	Old
Modular forms	30	3	27
Cusp forms	18	3	15
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
\(+\)	\(+\)	$+$	\(1\)
\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(1\)

Trace form

\( 3 q - 3 q^{3} + 2 q^{5} + 32 q^{7} + 27 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(48))\) 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
48.4.a.a	$48$	$4$	48.a	1.a	$1$	$1$	$1$	$2.832$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-18\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-18q^{5}-8q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
48.4.a.b	$48$	$4$	48.a	1.a	$1$	$1$	$1$	$2.832$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(14\)	\(24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+14q^{5}+24q^{7}+9q^{9}+28q^{11}+\cdots\)
48.4.a.c	$48$	$4$	48.a	1.a	$1$	$1$	$1$	$2.832$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(6\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{5}+2^{4}q^{7}+9q^{9}-12q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

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