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 + 8 * q^11 + 2 * q^15 + 8 * q^23 + 4 * q^25 - 2 * q^27 - 16 * q^29 - 16 * q^37 - 4 * q^39 + 8 * q^41 - 8 * q^43 - 24 * q^47 + 4 * q^49 + 4 * q^51 + 8 * q^55 - 8 * q^57 - 8 * q^59 + 8 * q^61 + 8 * q^65 - 8 * q^67 - 8 * q^69 - 8 * q^73 - 2 * q^75 - 16 * q^79 + 4 * q^81 - 24 * q^83 - 8 * q^85 + 12 * q^87 + 24 * q^89 + 32 * q^91 + 8 * q^97 + 8 * q^99

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	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	5
240.2.a.a	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓		✓	✓	120.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}-6q^{13}+q^{15}+\cdots\)
240.2.a.b	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓		✓	✓	30.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+4q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
240.2.a.c	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓		✓	✓	120.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
240.2.a.d	$240$	$2$	240.a	1.a	$1$	$1$	$1$	$1.916$	\(\Q\)	None	✓		✓	✓	15.2.a.a	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)

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

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