Space of modular forms of level 72, weight 12, and trivial character

• Label: 72.12.a
• Level: $72$
• Weight: $12$
• Character orbit: 72.a
• Rep. character: $\chi_{72}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $8$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	72.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 \(M_{12}(\Gamma_0(72))\).

	Total	New	Old
Modular forms	140	14	126
Cusp forms	124	14	110
Eisenstein series	16	0	16

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
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(7\)

Trace form

\( 14 q - 1872 q^{5} - 35592 q^{7} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(72))\) 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
72.12.a.a	$72$	$12$	72.a	1.a	$1$	$1$	$1$	$55.321$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1870\)	\(-72312\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-1870q^{5}-72312q^{7}-147940q^{11}+\cdots\)
72.12.a.b	$72$	$12$	72.a	1.a	$1$	$1$	$1$	$55.321$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1190\)	\(18480\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-1190q^{5}+18480q^{7}-135884q^{11}+\cdots\)
72.12.a.c	$72$	$12$	72.a	1.a	$1$	$1$	$1$	$55.321$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(3490\)	\(-55464\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3490q^{5}-55464q^{7}+597004q^{11}+\cdots\)
72.12.a.d	$72$	$12$	72.a	1.a	$1$	$1$	$1$	$55.321$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(7130\)	\(-19536\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7130q^{5}-19536q^{7}+196148q^{11}+\cdots\)
72.12.a.e	$72$	$12$	72.a	1.a	$1$	$2$	$2$	$55.321$	\(\Q(\sqrt{109}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-7868\)	\(91056\)		$-$	$-$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-3934-4\beta )q^{5}+(45528-14\beta )q^{7}+\cdots\)
72.12.a.f	$72$	$12$	72.a	1.a	$1$	$2$	$2$	$55.321$	\(\Q(\sqrt{3061}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1564\)	\(37776\)		$+$	$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-782-\beta )q^{5}+(18888-\beta )q^{7}+\cdots\)
72.12.a.g	$72$	$12$	72.a	1.a	$1$	$3$	$3$	$55.321$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1584\)	\(-17796\)		$+$	$+$	$+$	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-528-\beta _{1})q^{5}+(-5932-\beta _{1}+\cdots)q^{7}+\cdots\)
72.12.a.h	$72$	$12$	72.a	1.a	$1$	$3$	$3$	$55.321$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1584\)	\(-17796\)		$-$	$+$	$-$	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(528+\beta _{1})q^{5}+(-5932-\beta _{1}-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(72))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(72)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)