Space of modular forms of level 48, weight 10, and trivial character

• Label: 48.10.a
• Level: $48$
• Weight: $10$
• Character orbit: 48.a
• Rep. character: $\chi_{48}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $8$
• Sturm bound: $80$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(80\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(48))\).

	Total	New	Old
Modular forms	78	9	69
Cusp forms	66	9	57
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
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(5\)

Trace form

\( 9 q + 81 q^{3} + 718 q^{5} - 3776 q^{7} + 59049 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(48))\) 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
48.10.a.a	$48$	$10$	48.a	1.a	$1$	$1$	$1$	$24.722$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-81\)	\(-1530\)	\(-9128\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}-1530q^{5}-9128q^{7}+3^{8}q^{9}+\cdots\)
48.10.a.b	$48$	$10$	48.a	1.a	$1$	$1$	$1$	$24.722$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-81\)	\(-794\)	\(5880\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}-794q^{5}+5880q^{7}+3^{8}q^{9}+\cdots\)
48.10.a.c	$48$	$10$	48.a	1.a	$1$	$1$	$1$	$24.722$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-81\)	\(614\)	\(-2184\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+614q^{5}-2184q^{7}+3^{8}q^{9}+\cdots\)
48.10.a.d	$48$	$10$	48.a	1.a	$1$	$1$	$1$	$24.722$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-81\)	\(2694\)	\(3544\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+2694q^{5}+3544q^{7}+3^{8}q^{9}+\cdots\)
48.10.a.e	$48$	$10$	48.a	1.a	$1$	$1$	$1$	$24.722$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(81\)	\(-1314\)	\(4480\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}-1314q^{5}+4480q^{7}+3^{8}q^{9}+\cdots\)
48.10.a.f	$48$	$10$	48.a	1.a	$1$	$1$	$1$	$24.722$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(81\)	\(830\)	\(-672\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+830q^{5}-672q^{7}+3^{8}q^{9}+\cdots\)
48.10.a.g	$48$	$10$	48.a	1.a	$1$	$1$	$1$	$24.722$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(81\)	\(990\)	\(-8576\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+990q^{5}-8576q^{7}+3^{8}q^{9}+\cdots\)
48.10.a.h	$48$	$10$	48.a	1.a	$1$	$2$	$2$	$24.722$	\(\Q(\sqrt{109}) \)	None	✓	$1$	$0$	\(0\)	\(162\)	\(-772\)	\(2880\)		$+$	$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+(-386-\beta )q^{5}+(1440-5\beta )q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(48)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)