Space of modular forms of level 180, weight 3, and Character 180.f

• Label: 180.3.f
• Level: $180$
• Weight: $3$
• Character orbit: 180.f
• Rep. character: $\chi_{180}(19,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $8$
• Sturm bound: $108$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	180.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(108\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(7\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(180, [\chi])\).

	Total	New	Old
Modular forms	80	32	48
Cusp forms	64	28	36
Eisenstein series	16	4	12

Trace form

\( 28 q - 4 q^{4} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(180, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
180.3.f.a	$180$	$3$	180.f	20.d	$2$	$1$	$1$	$4.905$	\(\Q\)	\(\Q(\sqrt{-5}) \)	✓		20.3.d.a	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(4\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}+5q^{5}+4q^{7}-8q^{8}+\cdots\)
180.3.f.b	$180$	$3$	180.f	20.d	$2$	$1$	$1$	$4.905$	\(\Q\)	\(\Q(\sqrt{-5}) \)	✓		20.3.d.a	$2$	$0$	\(2\)	\(0\)	\(5\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}+5q^{5}-4q^{7}+8q^{8}+\cdots\)
180.3.f.c	$180$	$3$	180.f	20.d	$2$	$2$	$2$	$4.905$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)			20.3.d.c	$4$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+iq^{2}-4q^{4}+(-3-2i)q^{5}-4iq^{8}+\cdots\)
180.3.f.d	$180$	$3$	180.f	20.d	$2$	$4$	$4$	$4.905$	\(\Q(\sqrt{-3}, \sqrt{22})\)	None		✓	180.3.f.d	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2})q^{2}+(-2+2\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
180.3.f.e	$180$	$3$	180.f	20.d	$2$	$4$	$4$	$4.905$	\(\Q(\zeta_{12})\)	None			60.3.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{12}+\zeta_{12}^{3})q^{2}+(2+2\zeta_{12}^{2})q^{4}+\cdots\)
180.3.f.f	$180$	$3$	180.f	20.d	$2$	$4$	$4$	$4.905$	\(\Q(i, \sqrt{15})\)	\(\Q(\sqrt{-15}) \)		✓	180.3.f.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(4+\beta _{3})q^{4}+5\beta _{2}q^{5}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
180.3.f.g	$180$	$3$	180.f	20.d	$2$	$4$	$4$	$4.905$	\(\Q(\sqrt{-3}, \sqrt{22})\)	None		✓	180.3.f.d	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{2})q^{2}+(-2+2\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
180.3.f.h	$180$	$3$	180.f	20.d	$2$	$8$	$8$	$4.905$	8.0.\(\cdots\).4	None			60.3.f.b	$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(180, [\chi])\) into lower level spaces

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