Space of modular forms of level 240, weight 2, and trivial character

• Label: 240.2.a
• Level: $240$
• Weight: $2$
• Character orbit: 240.a
• Rep. character: $\chi_{240}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $96$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(96\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(240))\).

	Total	New	Old
Modular forms	60	4	56
Cusp forms	37	4	33
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(1\)
Plus space			\(+\)	\(1\)
Minus space			\(-\)	\(3\)

Trace form

\( 4 q - 2 q^{3} + 4 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(240))\) 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	5
240.2.a.a	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}-6q^{13}+q^{15}+\cdots\)
240.2.a.b	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+4q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
240.2.a.c	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
240.2.a.d	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(240)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)