Space of modular forms of level 240, weight 4, and trivial character

• Label: 240.4.a
• Level: $240$
• Weight: $4$
• Character orbit: 240.a
• Rep. character: $\chi_{240}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $12$
• Sturm bound: $192$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(192\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(240))\).

	Total	New	Old
Modular forms	156	12	144
Cusp forms	132	12	120
Eisenstein series	24	0	24

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(22\)	\(1\)	\(21\)		\(19\)	\(1\)	\(18\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(17\)	\(1\)	\(16\)		\(14\)	\(1\)	\(13\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(19\)	\(2\)	\(17\)		\(16\)	\(2\)	\(14\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(20\)	\(2\)	\(18\)		\(17\)	\(2\)	\(15\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(19\)	\(1\)	\(18\)		\(16\)	\(1\)	\(15\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(20\)	\(2\)	\(18\)		\(17\)	\(2\)	\(15\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(18\)	\(2\)	\(16\)		\(15\)	\(2\)	\(13\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(21\)	\(1\)	\(20\)		\(18\)	\(1\)	\(17\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(80\)	\(7\)	\(73\)		\(68\)	\(7\)	\(61\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(76\)	\(5\)	\(71\)		\(64\)	\(5\)	\(59\)		\(12\)	\(0\)	\(12\)

Trace form

12 * q + 6 * q^3 - 64 * q^7 + 108 * q^9 + 40 * q^11 - 30 * q^15 - 192 * q^19 + 328 * q^23 + 300 * q^25 + 54 * q^27 + 400 * q^29 + 192 * q^31 + 16 * q^37 - 420 * q^39 - 296 * q^41 + 1240 * q^43 + 744 * q^47 + 844 * q^49 + 372 * q^51 - 1536 * q^53 - 440 * q^55 + 168 * q^57 - 808 * q^59 + 56 * q^61 - 576 * q^63 + 280 * q^65 - 360 * q^67 + 264 * q^69 + 2688 * q^71 - 216 * q^73 + 150 * q^75 - 1536 * q^77 - 720 * q^79 + 972 * q^81 + 456 * q^83 - 120 * q^85 - 1044 * q^87 - 504 * q^89 - 1504 * q^91 - 1768 * q^97 + 360 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(240))\) 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
240.4.a.a	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓		✓	✓	120.4.a.e	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(-20\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-20q^{7}+9q^{9}-2^{4}q^{11}+\cdots\)
240.4.a.b	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓		✓	✓	30.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+4q^{7}+9q^{9}+48q^{11}+\cdots\)
240.4.a.c	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				30.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(-32\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}+60q^{11}+\cdots\)
240.4.a.d	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓		✓	✓	120.4.a.f	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(-8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-8q^{7}+9q^{9}-20q^{11}+\cdots\)
240.4.a.e	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				15.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(24\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+24q^{7}+9q^{9}-52q^{11}+\cdots\)
240.4.a.f	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				15.4.a.b	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(-20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-20q^{7}+9q^{9}+24q^{11}+\cdots\)
240.4.a.g	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				120.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(-20\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-20q^{7}+9q^{9}+56q^{11}+\cdots\)
240.4.a.h	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				120.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-4q^{7}+9q^{9}-72q^{11}+\cdots\)
240.4.a.i	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				60.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+28q^{7}+9q^{9}+24q^{11}+\cdots\)
240.4.a.j	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓		✓	✓	60.4.a.b	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-32\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
240.4.a.k	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				120.4.a.d	$1$	$0$	\(0\)	\(3\)	\(5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+9q^{9}-4q^{11}+54q^{13}+\cdots\)
240.4.a.l	$240$	$4$	240.a	1.a	$1$	$1$	$1$	$14.160$	\(\Q\)	None	✓				120.4.a.c	$1$	$0$	\(0\)	\(3\)	\(5\)	\(16\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+2^{4}q^{7}+9q^{9}+28q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(240)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)