Space of modular forms of level 360, weight 2, and trivial character

• Label: 360.2.a
• Level: $360$
• Weight: $2$
• Character orbit: 360.a
• Rep. character: $\chi_{360}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $144$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(144\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(360))\).

	Total	New	Old
Modular forms	88	5	83
Cusp forms	57	5	52
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.

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(10\)	\(0\)	\(10\)		\(7\)	\(0\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(12\)	\(1\)	\(11\)		\(8\)	\(1\)	\(7\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(12\)	\(1\)	\(11\)		\(8\)	\(1\)	\(7\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(10\)	\(0\)	\(10\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(12\)	\(1\)	\(11\)		\(8\)	\(1\)	\(7\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(10\)	\(0\)	\(10\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(6\)	\(1\)	\(5\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(12\)	\(1\)	\(11\)		\(8\)	\(1\)	\(7\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(40\)	\(1\)	\(39\)		\(25\)	\(1\)	\(24\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(48\)	\(4\)	\(44\)		\(32\)	\(4\)	\(28\)		\(16\)	\(0\)	\(16\)

Trace form

5 * q - q^5 + 4 * q^7 + 6 * q^13 + 6 * q^17 + 12 * q^19 + 4 * q^23 + 5 * q^25 + 10 * q^29 - 8 * q^31 + 8 * q^35 - 2 * q^37 + 2 * q^41 - 16 * q^43 - 20 * q^47 + 5 * q^49 - 22 * q^53 - 4 * q^55 - 8 * q^59 - 10 * q^61 - 10 * q^65 - 8 * q^71 - 14 * q^73 + 16 * q^77 - 16 * q^79 + 16 * q^83 + 2 * q^85 - 6 * q^89 + 4 * q^95 - 38 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(360))\) 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
360.2.a.a	$360$	$2$	360.a	1.a	$1$	$1$	$1$	$2.875$	\(\Q\)	None	✓		✓	✓	40.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{7}-4q^{11}-2q^{13}-2q^{17}+\cdots\)
360.2.a.b	$360$	$2$	360.a	1.a	$1$	$1$	$1$	$2.875$	\(\Q\)	None	✓		✓	✓	120.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{11}+6q^{13}+6q^{17}-4q^{19}+\cdots\)
360.2.a.c	$360$	$2$	360.a	1.a	$1$	$1$	$1$	$2.875$	\(\Q\)	None	✓	✓	✓	✓	360.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}+2q^{11}+4q^{13}-2q^{17}+\cdots\)
360.2.a.d	$360$	$2$	360.a	1.a	$1$	$1$	$1$	$2.875$	\(\Q\)	None	✓	✓	✓	✓	360.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}-2q^{11}+4q^{13}+2q^{17}+\cdots\)
360.2.a.e	$360$	$2$	360.a	1.a	$1$	$1$	$1$	$2.875$	\(\Q\)	None	✓		✓	✓	120.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}-6q^{13}+2q^{17}+4q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(360))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(360)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)