Space of modular forms of level 72, weight 6, and Character 72.d

• Label: 72.6.d
• Level: $72$
• Weight: $6$
• Character orbit: 72.d
• Rep. character: $\chi_{72}(37,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $4$
• Sturm bound: $72$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
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:			\(72\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(72, [\chi])\).

	Total	New	Old
Modular forms	64	26	38
Cusp forms	56	24	32
Eisenstein series	8	2	6

Trace form

24 * q + 96 * q^7 + 180 * q^8 - 36 * q^10 - 1308 * q^14 + 1296 * q^16 + 204 * q^17 + 1752 * q^20 - 1236 * q^22 - 4008 * q^23 - 10944 * q^25 + 912 * q^26 - 10632 * q^28 + 1392 * q^31 - 1320 * q^32 + 9816 * q^34 - 11268 * q^38 + 4920 * q^40 + 7044 * q^41 + 22200 * q^44 + 36408 * q^46 - 10536 * q^47 + 28584 * q^49 + 40104 * q^50 + 25920 * q^52 - 24576 * q^55 - 51576 * q^56 - 67332 * q^58 - 59796 * q^62 + 69168 * q^64 - 32016 * q^65 + 132312 * q^68 + 58872 * q^70 + 65544 * q^71 - 65160 * q^73 + 197400 * q^74 - 155880 * q^76 + 7104 * q^79 - 351936 * q^80 - 214224 * q^82 - 385188 * q^86 + 216624 * q^88 - 90948 * q^89 + 448800 * q^92 + 261672 * q^94 + 16560 * q^95 + 179976 * q^97 + 576888 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(72, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
72.6.d.a	$72$	$6$	72.d	8.b	$2$	$2$	$2$	$11.548$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-6}) \)		✓			72.6.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(484\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2\beta q^{2}-2^{5}q^{4}-11\beta q^{5}+242q^{7}+\cdots\)
72.6.d.b	$72$	$6$	72.d	8.b	$2$	$4$	$4$	$11.548$	\(\Q(\sqrt{-54 +2 \sqrt{73}})\)	None					8.6.b.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(96\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(5+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+(2+\cdots)q^{5}+\cdots\)
72.6.d.c	$72$	$6$	72.d	8.b	$2$	$8$	$8$	$11.548$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			72.6.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-288\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(10+\beta _{2})q^{4}+(\beta _{1}-\beta _{4})q^{5}+\cdots\)
72.6.d.d	$72$	$6$	72.d	8.b	$2$	$10$	$10$	$11.548$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None			✓		24.6.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-196\)	$2^{20}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(-3-\beta _{4})q^{4}+(-2\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(72, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(72, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)