Space of modular forms of level 864, weight 2, and trivial character

• Label: 864.2.a
• Level: $864$
• Weight: $2$
• Character orbit: 864.a
• Rep. character: $\chi_{864}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $14$
• Sturm bound: $288$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(288\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(864))\).

	Total	New	Old
Modular forms	168	16	152
Cusp forms	121	16	105
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)		\(38\)	\(3\)	\(35\)		\(27\)	\(3\)	\(24\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(44\)	\(5\)	\(39\)		\(32\)	\(5\)	\(27\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(46\)	\(5\)	\(41\)		\(34\)	\(5\)	\(29\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(40\)	\(3\)	\(37\)		\(28\)	\(3\)	\(25\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(78\)	\(6\)	\(72\)		\(55\)	\(6\)	\(49\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(90\)	\(10\)	\(80\)		\(66\)	\(10\)	\(56\)		\(24\)	\(0\)	\(24\)

Trace form

\( 16 q + 8 q^{13} + 8 q^{25} + 40 q^{37} + 16 q^{49} + 40 q^{61} + 16 q^{85} + 32 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(864))\) 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
864.2.a.a	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-3q^{7}+6q^{11}-3q^{13}+2q^{17}+\cdots\)
864.2.a.b	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}+2q^{11}+q^{13}-6q^{17}+\cdots\)
864.2.a.c	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-2q^{11}+q^{13}-6q^{17}+\cdots\)
864.2.a.d	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+3q^{7}-6q^{11}-3q^{13}+2q^{17}+\cdots\)
864.2.a.e	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots\)
864.2.a.f	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots\)
864.2.a.g	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.e	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots\)
864.2.a.h	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.e	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots\)
864.2.a.i	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{7}-6q^{11}-3q^{13}-2q^{17}+\cdots\)
864.2.a.j	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-2q^{11}+q^{13}+6q^{17}+\cdots\)
864.2.a.k	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}+2q^{11}+q^{13}+6q^{17}+\cdots\)
864.2.a.l	$864$	$2$	864.a	1.a	$1$	$1$	$1$	$6.899$	\(\Q\)	None	✓	✓			864.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+3q^{7}+6q^{11}-3q^{13}-2q^{17}+\cdots\)
864.2.a.m	$864$	$2$	864.a	1.a	$1$	$2$	$2$	$6.899$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓		864.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{7}-q^{11}+4q^{13}-2\beta q^{19}+\cdots\)
864.2.a.n	$864$	$2$	864.a	1.a	$1$	$2$	$2$	$6.899$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓		864.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-\beta q^{7}+q^{11}+4q^{13}+2\beta q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(864))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(864)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)