Space of modular forms of level 216, weight 4, and trivial character

• Label: 216.4.a
• Level: $216$
• Weight: $4$
• Character orbit: 216.a
• Rep. character: $\chi_{216}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $8$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	216.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(216))\).

	Total	New	Old
Modular forms	120	12	108
Cusp forms	96	12	84
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
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(5\)

Trace form

\( 12 q + 24 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(216))\) 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
216.4.a.a	$216$	$4$	216.a	1.a	$1$	$1$	$1$	$12.744$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+3q^{7}-28q^{11}-11q^{13}+\cdots\)
216.4.a.b	$216$	$4$	216.a	1.a	$1$	$1$	$1$	$12.744$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-9q^{7}+17q^{11}-44q^{13}-56q^{17}+\cdots\)
216.4.a.c	$216$	$4$	216.a	1.a	$1$	$1$	$1$	$12.744$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-9q^{7}-17q^{11}-44q^{13}+56q^{17}+\cdots\)
216.4.a.d	$216$	$4$	216.a	1.a	$1$	$1$	$1$	$12.744$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+3q^{7}+28q^{11}-11q^{13}+\cdots\)
216.4.a.e	$216$	$4$	216.a	1.a	$1$	$2$	$2$	$12.744$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(24\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(12+\beta )q^{7}-q^{11}+\cdots\)
216.4.a.f	$216$	$4$	216.a	1.a	$1$	$2$	$2$	$12.744$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-6\)		$+$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{5}+(-3+2\beta )q^{7}+(-26+\cdots)q^{11}+\cdots\)
216.4.a.g	$216$	$4$	216.a	1.a	$1$	$2$	$2$	$12.744$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-6\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-3+2\beta )q^{7}+(26-3\beta )q^{11}+\cdots\)
216.4.a.h	$216$	$4$	216.a	1.a	$1$	$2$	$2$	$12.744$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(8\)	\(24\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(12+\beta )q^{7}+q^{11}+(2^{4}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(216))\) into lower level spaces

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