Space of modular forms of level 180, weight 9, and Character 180.c

• Label: 180.9.c
• Level: $180$
• Weight: $9$
• Character orbit: 180.c
• Rep. character: $\chi_{180}(91,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $3$
• Sturm bound: $324$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	180.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(324\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(180, [\chi])\).

	Total	New	Old
Modular forms	296	80	216
Cusp forms	280	80	200
Eisenstein series	16	0	16

Trace form

80 * q + 6 * q^2 + 424 * q^4 + 5724 * q^8 - 8750 * q^10 - 51392 * q^13 + 33132 * q^14 - 76220 * q^16 - 27552 * q^17 + 172500 * q^20 - 836992 * q^22 + 6250000 * q^25 + 462480 * q^26 - 374072 * q^28 + 1499808 * q^29 + 1002636 * q^32 - 4176068 * q^34 + 431552 * q^37 + 9784680 * q^38 + 192500 * q^40 + 5179392 * q^41 - 11636316 * q^44 - 7185352 * q^46 - 49107536 * q^49 + 468750 * q^50 - 5765600 * q^52 + 2452032 * q^53 - 62659716 * q^56 - 3047868 * q^58 - 7991936 * q^61 + 55030152 * q^62 + 62444860 * q^64 + 9060000 * q^65 - 155662080 * q^68 - 36765000 * q^70 - 126700128 * q^73 + 166343304 * q^74 + 145123944 * q^76 - 110190144 * q^77 - 96180000 * q^80 - 182007092 * q^82 + 106960000 * q^85 + 382888104 * q^86 + 283784864 * q^88 - 155731680 * q^89 - 716984328 * q^92 - 582859536 * q^94 + 146565216 * q^97 + 667407294 * q^98

Decomposition of \(S_{9}^{\mathrm{new}}(180, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
180.9.c.a	$180$	$9$	180.c	4.b	$2$	$16$	$16$	$73.328$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					20.9.b.a	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$2^{58}\cdot 3^{4}\cdot 5^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-3+\beta _{3})q^{4}+(1-2\beta _{1}+\cdots)q^{5}+\cdots\)
180.9.c.b	$180$	$9$	180.c	4.b	$2$	$32$	$32$	$73.328$		None		✓			180.9.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
180.9.c.c	$180$	$9$	180.c	4.b	$2$	$32$	$32$	$73.328$		None					60.9.c.a	$2$	$0$	\(12\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{9}^{\mathrm{old}}(180, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(180, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)