Space of modular forms of level 72, weight 3, and Character 72.b

• Label: 72.3.b
• Level: $72$
• Weight: $3$
• Character orbit: 72.b
• Rep. character: $\chi_{72}(19,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $3$
• Sturm bound: $36$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	72.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(36\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(72, [\chi])\).

	Total	New	Old
Modular forms	28	11	17
Cusp forms	20	9	11
Eisenstein series	8	2	6

Trace form

9 * q + 12 * q^8 - 12 * q^10 + 18 * q^11 - 36 * q^14 - 48 * q^16 + 6 * q^17 + 30 * q^19 - 72 * q^20 + 36 * q^22 - 15 * q^25 + 96 * q^26 + 72 * q^28 + 120 * q^32 - 120 * q^34 - 96 * q^35 - 108 * q^38 - 24 * q^40 + 6 * q^41 - 114 * q^43 - 120 * q^44 + 120 * q^46 - 39 * q^49 + 168 * q^50 + 192 * q^52 + 168 * q^56 + 36 * q^58 + 210 * q^59 - 204 * q^62 - 144 * q^64 - 96 * q^65 + 126 * q^67 - 120 * q^68 + 264 * q^70 + 18 * q^73 + 120 * q^74 - 120 * q^76 + 96 * q^80 - 192 * q^82 - 318 * q^83 - 60 * q^86 - 240 * q^88 + 54 * q^89 - 192 * q^91 - 96 * q^92 - 360 * q^94 + 162 * q^97 - 72 * q^98

Decomposition of \(S_{3}^{\mathrm{new}}(72, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
72.3.b.a	$72$	$3$	72.b	8.d	$2$	$1$	$1$	$1.962$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				8.3.d.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}+8q^{8}-14q^{11}+2^{4}q^{16}+\cdots\)
72.3.b.b	$72$	$3$	72.b	8.d	$2$	$4$	$4$	$1.962$	4.0.4752.1	None					24.3.b.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-2+\beta _{1}-\beta _{3})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
72.3.b.c	$72$	$3$	72.b	8.d	$2$	$4$	$4$	$1.962$	\(\Q(\sqrt{-6}, \sqrt{10})\)	None		✓			72.3.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(1-\beta _{2})q^{4}+(\beta _{1}+2\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(72, [\chi])\) into lower level spaces

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