Space of modular forms of level 288, weight 4, and trivial character

• Label: 288.4.a
• Level: $288$
• Weight: $4$
• Character orbit: 288.a
• Rep. character: $\chi_{288}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $13$
• Sturm bound: $192$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	288.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(192\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(288))\).

	Total	New	Old
Modular forms	160	15	145
Cusp forms	128	15	113
Eisenstein series	32	0	32

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	Dim
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	$+$	\(5\)
Plus space		\(+\)	\(8\)
Minus space		\(-\)	\(7\)

Trace form

\( 15 q + 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(288))\) 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
288.4.a.a	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-22\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-22q^{5}-18q^{13}+94q^{17}+359q^{25}+\cdots\)
288.4.a.b	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{5}-4q^{7}-20q^{11}+70q^{13}+\cdots\)
288.4.a.c	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{5}+4q^{7}+20q^{11}+70q^{13}+\cdots\)
288.4.a.d	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}+18q^{13}-104q^{17}-109q^{25}+\cdots\)
288.4.a.e	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-12q^{7}+60q^{11}-42q^{13}+\cdots\)
288.4.a.f	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+12q^{7}-60q^{11}-42q^{13}+\cdots\)
288.4.a.g	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}+18q^{13}+104q^{17}-109q^{25}+\cdots\)
288.4.a.h	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{5}-2^{4}q^{7}-40q^{11}-50q^{13}+\cdots\)
288.4.a.i	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{5}+2^{4}q^{7}+40q^{11}-50q^{13}+\cdots\)
288.4.a.j	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(14\)	\(-36\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}-6^{2}q^{7}+6^{2}q^{11}+54q^{13}+\cdots\)
288.4.a.k	$288$	$4$	288.a	1.a	$1$	$1$	$1$	$16.993$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(14\)	\(36\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}+6^{2}q^{7}-6^{2}q^{11}+54q^{13}+\cdots\)
288.4.a.l	$288$	$4$	288.a	1.a	$1$	$2$	$2$	$16.993$	\(\Q(\sqrt{13}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+2\beta q^{7}-2^{5}q^{11}-14q^{13}+\cdots\)
288.4.a.m	$288$	$4$	288.a	1.a	$1$	$2$	$2$	$16.993$	\(\Q(\sqrt{13}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-2\beta q^{7}+2^{5}q^{11}-14q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(288))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(288)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)