Space of modular forms of level 216, weight 2, and trivial character

• Label: 216.2.a
• Level: $216$
• Weight: $2$
• Character orbit: 216.a
• Rep. character: $\chi_{216}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $72$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(72\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(216))\).

	Total	New	Old
Modular forms	48	4	44
Cusp forms	25	4	21
Eisenstein series	23	0	23

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
\(+\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(5\)	\(1\)	\(4\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(13\)	\(1\)	\(12\)		\(7\)	\(1\)	\(6\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(14\)	\(2\)	\(12\)		\(8\)	\(2\)	\(6\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(11\)	\(0\)	\(11\)		\(5\)	\(0\)	\(5\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(21\)	\(1\)	\(20\)		\(10\)	\(1\)	\(9\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(27\)	\(3\)	\(24\)		\(15\)	\(3\)	\(12\)		\(12\)	\(0\)	\(12\)

Trace form

\( 4 q + 10 q^{13} + 2 q^{19} + 14 q^{25} - 22 q^{31} - 30 q^{37} - 20 q^{43} + 8 q^{49} + 22 q^{55} - 26 q^{61} + 2 q^{67} + 4 q^{73} + 22 q^{79} - 16 q^{85} + 18 q^{91} + 8 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(216))\) 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
216.2.a.a	$216$	$2$	216.a	1.a	$1$	$1$	$1$	$1.725$	\(\Q\)	None	✓	✓	✓	✓	216.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{7}-4q^{11}+q^{13}+4q^{17}+\cdots\)
216.2.a.b	$216$	$2$	216.a	1.a	$1$	$1$	$1$	$1.725$	\(\Q\)	None	✓	✓	✓	✓	216.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}+5q^{11}+4q^{13}-8q^{17}+\cdots\)
216.2.a.c	$216$	$2$	216.a	1.a	$1$	$1$	$1$	$1.725$	\(\Q\)	None	✓	✓			216.2.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-5q^{11}+4q^{13}+8q^{17}+\cdots\)
216.2.a.d	$216$	$2$	216.a	1.a	$1$	$1$	$1$	$1.725$	\(\Q\)	None	✓	✓			216.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{7}+4q^{11}+q^{13}-4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(216))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(216)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)