Space of modular forms of level 864, weight 4, and trivial character

• Label: 864.4.a
• Level: $864$
• Weight: $4$
• Character orbit: 864.a
• Rep. character: $\chi_{864}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $20$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(576\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(864))\).

	Total	New	Old
Modular forms	456	48	408
Cusp forms	408	48	360
Eisenstein series	48	0	48

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
\(+\)	\(+\)	$+$	\(13\)
\(+\)	\(-\)	$-$	\(11\)
\(-\)	\(+\)	$-$	\(11\)
\(-\)	\(-\)	$+$	\(13\)
Plus space		\(+\)	\(26\)
Minus space		\(-\)	\(22\)

Trace form

\( 48 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(864))\) 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
864.4.a.a	$864$	$4$	864.a	1.a	$1$	$1$	$1$	$50.978$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(0\)	\(-19\)	\(-13\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-19q^{5}-13q^{7}-65q^{11}-56q^{13}+\cdots\)
864.4.a.b	$864$	$4$	864.a	1.a	$1$	$1$	$1$	$50.978$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-19\)	\(13\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-19q^{5}+13q^{7}+65q^{11}-56q^{13}+\cdots\)
864.4.a.c	$864$	$4$	864.a	1.a	$1$	$1$	$1$	$50.978$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(19\)	\(-13\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+19q^{5}-13q^{7}+65q^{11}-56q^{13}+\cdots\)
864.4.a.d	$864$	$4$	864.a	1.a	$1$	$1$	$1$	$50.978$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(19\)	\(13\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+19q^{5}+13q^{7}-65q^{11}-56q^{13}+\cdots\)
864.4.a.e	$864$	$4$	864.a	1.a	$1$	$2$	$2$	$50.978$	\(\Q(\sqrt{37}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-5\beta q^{7}-43q^{11}+52q^{13}+\cdots\)
864.4.a.f	$864$	$4$	864.a	1.a	$1$	$2$	$2$	$50.978$	\(\Q(\sqrt{73}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{7}-q^{11}-8q^{13}+12\beta q^{17}+\cdots\)
864.4.a.g	$864$	$4$	864.a	1.a	$1$	$2$	$2$	$50.978$	\(\Q(\sqrt{73}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-\beta q^{7}+q^{11}-8q^{13}+12\beta q^{17}+\cdots\)
864.4.a.h	$864$	$4$	864.a	1.a	$1$	$2$	$2$	$50.978$	\(\Q(\sqrt{37}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+5\beta q^{7}+43q^{11}+52q^{13}+\cdots\)
864.4.a.i	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.2708.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-19\)		$-$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(-6-\beta _{2})q^{7}+(13+\cdots)q^{11}+\cdots\)
864.4.a.j	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.2708.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(19\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(-13+\cdots)q^{11}+\cdots\)
864.4.a.k	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(-9\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{5}+(-3-\beta _{1}+\beta _{2})q^{7}+\cdots\)
864.4.a.l	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(9\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{5}+(3+\beta _{1}-\beta _{2})q^{7}+\cdots\)
864.4.a.m	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-3\)		$+$	$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(-7+\cdots)q^{11}+\cdots\)
864.4.a.n	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(3\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(1-\beta _{2})q^{7}+(7-3\beta _{1}+\cdots)q^{11}+\cdots\)
864.4.a.o	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(7-3\beta _{1}+\cdots)q^{11}+\cdots\)
864.4.a.p	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(3\)	\(3\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+(-7+3\beta _{1}+\cdots)q^{11}+\cdots\)
864.4.a.q	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-9\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{2})q^{5}+(-3-\beta _{1}+\beta _{2})q^{7}+\cdots\)
864.4.a.r	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(9\)		$+$	$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{2})q^{5}+(3+\beta _{1}-\beta _{2})q^{7}+(6+\cdots)q^{11}+\cdots\)
864.4.a.s	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.2708.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-19\)		$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(-6-\beta _{2})q^{7}+(-13+\cdots)q^{11}+\cdots\)
864.4.a.t	$864$	$4$	864.a	1.a	$1$	$3$	$3$	$50.978$	3.3.2708.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(19\)		$-$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(13+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(864))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(864)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)