Space of modular forms of level 48, weight 12, and trivial character

• Label: 48.12.a
• Level: $48$
• Weight: $12$
• Character orbit: 48.a
• Rep. character: $\chi_{48}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $10$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(48))\).

	Total	New	Old
Modular forms	94	11	83
Cusp forms	82	11	71
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
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(5\)

Trace form

\( 11 q - 243 q^{3} + 2642 q^{5} + 84960 q^{7} + 649539 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(48))\) 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
48.12.a.a	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-243\)	\(-11730\)	\(50008\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}-11730q^{5}+50008q^{7}+\cdots\)
48.12.a.b	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-243\)	\(1870\)	\(72312\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+1870q^{5}+72312q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.c	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-243\)	\(2862\)	\(-9128\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+2862q^{5}-9128q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.d	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-243\)	\(3630\)	\(-32936\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+3630q^{5}-32936q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.e	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(243\)	\(-7130\)	\(19536\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-7130q^{5}+19536q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.f	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(243\)	\(-5370\)	\(27760\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}-5370q^{5}+27760q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.g	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(243\)	\(1190\)	\(-18480\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+1190q^{5}-18480q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.h	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(243\)	\(5766\)	\(-72464\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+5766q^{5}-72464q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.i	$48$	$12$	48.a	1.a	$1$	$1$	$1$	$36.880$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(243\)	\(9990\)	\(86128\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+9990q^{5}+86128q^{7}+3^{10}q^{9}+\cdots\)
48.12.a.j	$48$	$12$	48.a	1.a	$1$	$2$	$2$	$36.880$	\(\Q(\sqrt{3061}) \)	None	✓	$1$	$0$	\(0\)	\(-486\)	\(1564\)	\(-37776\)		$+$	$+$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+(782-\beta )q^{5}+(-18888+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(48)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)