Space of modular forms of level 288, weight 2, and trivial character

• Label: 288.2.a
• Level: $288$
• Weight: $2$
• Character orbit: 288.a
• Rep. character: $\chi_{288}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $96$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(96\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(288))\).

	Total	New	Old
Modular forms	64	5	59
Cusp forms	33	5	28
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.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(14\)	\(1\)	\(13\)		\(7\)	\(1\)	\(6\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)		\(16\)	\(2\)	\(14\)		\(8\)	\(2\)	\(6\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(18\)	\(1\)	\(17\)		\(10\)	\(1\)	\(9\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(16\)	\(1\)	\(15\)		\(8\)	\(1\)	\(7\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(30\)	\(2\)	\(28\)		\(15\)	\(2\)	\(13\)		\(15\)	\(0\)	\(15\)
Minus space		\(-\)		\(34\)	\(3\)	\(31\)		\(18\)	\(3\)	\(15\)		\(16\)	\(0\)	\(16\)

Trace form

\( 5 q - 2 q^{5} - 10 q^{13} + 10 q^{17} + 19 q^{25} + 6 q^{29} - 10 q^{37} - 14 q^{41} - 3 q^{49} - 34 q^{53} - 18 q^{61} + 20 q^{65} - 6 q^{73} + 32 q^{77} + 36 q^{85} - 30 q^{89} - 46 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(288))\) 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
288.2.a.a	$288$	$2$	288.a	1.a	$1$	$1$	$1$	$2.300$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓	✓	✓	288.2.a.a	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}-6q^{13}-8q^{17}+11q^{25}+\cdots\)
288.2.a.b	$288$	$2$	288.a	1.a	$1$	$1$	$1$	$2.300$	\(\Q\)	None	✓		✓	✓	96.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
288.2.a.c	$288$	$2$	288.a	1.a	$1$	$1$	$1$	$2.300$	\(\Q\)	None	✓				96.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{7}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
288.2.a.d	$288$	$2$	288.a	1.a	$1$	$1$	$1$	$2.300$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.2.a.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{5}+6q^{13}-2q^{17}-q^{25}+10q^{29}+\cdots\)
288.2.a.e	$288$	$2$	288.a	1.a	$1$	$1$	$1$	$2.300$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓	✓	✓	288.2.a.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}-6q^{13}+8q^{17}+11q^{25}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(288))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(288)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)