Space of modular forms of level 576, weight 2, and trivial character

• Label: 576.2.a
• Level: $576$
• Weight: $2$
• Character orbit: 576.a
• Rep. character: $\chi_{576}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $9$
• Sturm bound: $192$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(192\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(576))\).

	Total	New	Old
Modular forms	120	11	109
Cusp forms	73	9	64
Eisenstein series	47	2	45

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
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(6\)

Trace form

\( 9 q - 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(576))\) 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
576.2.a.a	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				288.2.a.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}+6q^{13}+8q^{17}+11q^{25}+\cdots\)
576.2.a.b	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	None	✓				24.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{11}+2q^{13}-2q^{17}-4q^{19}+\cdots\)
576.2.a.c	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.2.a.a	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{5}-6q^{13}-2q^{17}-q^{25}-10q^{29}+\cdots\)
576.2.a.d	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	None	✓				24.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{11}+2q^{13}-2q^{17}+4q^{19}+\cdots\)
576.2.a.e	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓		✓	✓	36.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{7}-2q^{13}-8q^{19}-5q^{25}-4q^{31}+\cdots\)
576.2.a.f	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				36.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{7}-2q^{13}+8q^{19}-5q^{25}+4q^{31}+\cdots\)
576.2.a.g	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	None	✓				96.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
576.2.a.h	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	None	✓				96.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
576.2.a.i	$576$	$2$	576.a	1.a	$1$	$1$	$1$	$4.599$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				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 \(S_{2}^{\mathrm{old}}(\Gamma_0(576))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(576)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)