Space of modular forms of level 1248, weight 2, and trivial character

• Label: 1248.2.a
• Level: $1248$
• Weight: $2$
• Character orbit: 1248.a
• Rep. character: $\chi_{1248}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $16$
• Sturm bound: $448$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(448\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1248))\).

	Total	New	Old
Modular forms	240	24	216
Cusp forms	209	24	185
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
Plus space			\(+\)	\(10\)
Minus space			\(-\)	\(14\)

Trace form

\( 24 q + 24 q^{9} + 8 q^{25} - 16 q^{33} + 8 q^{49} - 16 q^{53} - 16 q^{61} - 16 q^{73} + 80 q^{77} + 24 q^{81} + 64 q^{85} - 32 q^{93} + 48 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1248))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	13
1248.2.a.a	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-2q^{7}+q^{9}+6q^{11}+\cdots\)
1248.2.a.b	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+4q^{11}+q^{13}+\cdots\)
1248.2.a.c	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+q^{13}+2q^{17}+\cdots\)
1248.2.a.d	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+4q^{11}+q^{13}+\cdots\)
1248.2.a.e	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-2q^{7}+q^{9}+2q^{11}+\cdots\)
1248.2.a.f	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-4q^{11}+q^{13}+\cdots\)
1248.2.a.g	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+2q^{7}+q^{9}-6q^{11}+\cdots\)
1248.2.a.h	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.d	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
1248.2.a.i	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}+q^{13}+2q^{17}+\cdots\)
1248.2.a.j	$1248$	$2$	1248.a	1.a	$1$	$1$	$1$	$9.965$	\(\Q\)	None	✓	✓			1248.2.a.e	$1$	$0$	\(0\)	\(1\)	\(2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
1248.2.a.k	$1248$	$2$	1248.a	1.a	$1$	$2$	$2$	$9.965$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		1248.2.a.k	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(6\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta )q^{5}+(3-\beta )q^{7}+q^{9}+\cdots\)
1248.2.a.l	$1248$	$2$	1248.a	1.a	$1$	$2$	$2$	$9.965$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		1248.2.a.l	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{5}+(-1-\beta )q^{7}+q^{9}+\cdots\)
1248.2.a.m	$1248$	$2$	1248.a	1.a	$1$	$2$	$2$	$9.965$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		1248.2.a.k	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-6\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{5}+(-3+\beta )q^{7}+\cdots\)
1248.2.a.n	$1248$	$2$	1248.a	1.a	$1$	$2$	$2$	$9.965$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		1248.2.a.l	$1$	$0$	\(0\)	\(2\)	\(2\)	\(2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+(1+\beta )q^{7}+q^{9}+\cdots\)
1248.2.a.o	$1248$	$2$	1248.a	1.a	$1$	$3$	$3$	$9.965$	3.3.148.1	None	✓	✓	✓		1248.2.a.o	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta _{1})q^{5}+\beta _{2}q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1248.2.a.p	$1248$	$2$	1248.a	1.a	$1$	$3$	$3$	$9.965$	3.3.148.1	None	✓	✓	✓		1248.2.a.o	$1$	$0$	\(0\)	\(3\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{1})q^{5}-\beta _{2}q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1248))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1248)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(624))\)\(^{\oplus 2}\)