Space of modular forms of level 180, weight 8, and Character 180.d

• Label: 180.8.d
• Level: $180$
• Weight: $8$
• Character orbit: 180.d
• Rep. character: $\chi_{180}(109,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $4$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	180.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(180, [\chi])\).

	Total	New	Old
Modular forms	264	18	246
Cusp forms	240	18	222
Eisenstein series	24	0	24

Trace form

18 * q - 14 * q^5 - 2056 * q^11 + 30288 * q^19 + 26466 * q^25 + 17052 * q^29 + 194856 * q^31 - 291568 * q^35 - 399724 * q^41 - 1755186 * q^49 + 1044360 * q^55 + 2920088 * q^59 + 942132 * q^61 - 459768 * q^65 + 1147488 * q^71 + 241848 * q^79 - 4404744 * q^85 - 5646596 * q^89 - 6830688 * q^91 + 6314736 * q^95

Decomposition of \(S_{8}^{\mathrm{new}}(180, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
180.8.d.a	$180$	$8$	180.d	5.b	$2$	$2$	$2$	$56.229$	\(\Q(\sqrt{-1}) \)	None					60.8.d.a	$2$	$0$	\(0\)	\(0\)	\(-500\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-125 i-250)q^{5}+722 i q^{7}+\cdots\)
180.8.d.b	$180$	$8$	180.d	5.b	$2$	$4$	$4$	$56.229$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					20.8.c.a	$2$	$0$	\(0\)	\(0\)	\(156\)	\(0\)	$2^{7}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+(39-\beta _{1}-\beta _{2})q^{5}+(5\beta _{1}-2\beta _{2}+\beta _{3})q^{7}+\cdots\)
180.8.d.c	$180$	$8$	180.d	5.b	$2$	$4$	$4$	$56.229$	\(\Q(i, \sqrt{1129})\)	None					60.8.d.b	$2$	$0$	\(0\)	\(0\)	\(330\)	\(0\)	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(82+8\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+(-114\beta _{1}+\cdots)q^{7}+\cdots\)
180.8.d.d	$180$	$8$	180.d	5.b	$2$	$8$	$8$	$56.229$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓	✓		180.8.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{4}\cdot 5^{6}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}-\beta _{3}q^{7}+(-\beta _{2}+\beta _{5})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(180, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(180, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)