Space of modular forms of level 72, weight 4, and Character 72.d

• Label: 72.4.d
• Level: $72$
• Weight: $4$
• Character orbit: 72.d
• Rep. character: $\chi_{72}(37,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(72, [\chi])\).

	Total	New	Old
Modular forms	40	16	24
Cusp forms	32	14	18
Eisenstein series	8	2	6

Trace form

\( 14 q - 16 q^{7} + 36 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(72, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
72.4.d.a	$72$	$4$	72.d	8.b	$2$	$2$	$2$	$4.248$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-6}) \)		✓			72.4.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-68\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}-8q^{4}-7\beta q^{5}-34q^{7}-8\beta q^{8}+\cdots\)
72.4.d.b	$72$	$4$	72.d	8.b	$2$	$2$	$2$	$4.248$	\(\Q(\sqrt{-7}) \)	None					8.4.b.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-16\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta )q^{2}+(-6+2\beta )q^{4}+4\beta q^{5}+\cdots\)
72.4.d.c	$72$	$4$	72.d	8.b	$2$	$4$	$4$	$4.248$	\(\Q(\sqrt{-10}, \sqrt{22})\)	None		✓			72.4.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(40\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)
72.4.d.d	$72$	$4$	72.d	8.b	$2$	$6$	$6$	$4.248$	6.0.8248384.1	None			✓		24.4.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(28\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(3+\beta _{5})q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(72, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(72, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)