Space of modular forms of level 216, weight 6, and trivial character

• Label: 216.6.a
• Level: $216$
• Weight: $6$
• Character orbit: 216.a
• Rep. character: $\chi_{216}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $10$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	216.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(216\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(216))\).

	Total	New	Old
Modular forms	192	20	172
Cusp forms	168	20	148
Eisenstein series	24	0	24

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
\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(11\)

Trace form

\( 20 q - 24 q^{7} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(216))\) 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
216.6.a.a	$216$	$6$	216.a	1.a	$1$	$1$	$1$	$34.643$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-29\)	\(201\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-29q^{5}+201q^{7}+43q^{11}+244q^{13}+\cdots\)
216.6.a.b	$216$	$6$	216.a	1.a	$1$	$1$	$1$	$34.643$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(29\)	\(201\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+29q^{5}+201q^{7}-43q^{11}+244q^{13}+\cdots\)
216.6.a.c	$216$	$6$	216.a	1.a	$1$	$2$	$2$	$34.643$	\(\Q(\sqrt{185}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-38\)	\(-294\)		$+$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-19-\beta )q^{5}+(-147+\beta )q^{7}+(65+\cdots)q^{11}+\cdots\)
216.6.a.d	$216$	$6$	216.a	1.a	$1$	$2$	$2$	$34.643$	\(\Q(\sqrt{85}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-28\)	\(54\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-14-\beta )q^{5}+(3^{3}+2\beta )q^{7}+(-74+\cdots)q^{11}+\cdots\)
216.6.a.e	$216$	$6$	216.a	1.a	$1$	$2$	$2$	$34.643$	\(\Q(\sqrt{209}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(12\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-10-\beta )q^{5}+(6+\beta )q^{7}+(119+12\beta )q^{11}+\cdots\)
216.6.a.f	$216$	$6$	216.a	1.a	$1$	$2$	$2$	$34.643$	\(\Q(\sqrt{209}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(20\)	\(12\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(10+\beta )q^{5}+(6+\beta )q^{7}+(-119-12\beta )q^{11}+\cdots\)
216.6.a.g	$216$	$6$	216.a	1.a	$1$	$2$	$2$	$34.643$	\(\Q(\sqrt{85}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(28\)	\(54\)		$+$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(14+\beta )q^{5}+(3^{3}+2\beta )q^{7}+(74+3\beta )q^{11}+\cdots\)
216.6.a.h	$216$	$6$	216.a	1.a	$1$	$2$	$2$	$34.643$	\(\Q(\sqrt{185}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(38\)	\(-294\)		$-$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(19+\beta )q^{5}+(-147+\beta )q^{7}+(-65+\cdots)q^{11}+\cdots\)
216.6.a.i	$216$	$6$	216.a	1.a	$1$	$3$	$3$	$34.643$	3.3.2292.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-36\)	\(15\)		$-$	$+$	$-$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-12-\beta _{1})q^{5}+(5-\beta _{1}+\beta _{2})q^{7}+\cdots\)
216.6.a.j	$216$	$6$	216.a	1.a	$1$	$3$	$3$	$34.643$	3.3.2292.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(36\)	\(15\)		$+$	$+$	$+$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(12+\beta _{1})q^{5}+(5-\beta _{1}+\beta _{2})q^{7}+(-148+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(216))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(216)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)