Space of modular forms of level 180, weight 2, and Character 180.k

• Label: 180.2.k
• Level: $180$
• Weight: $2$
• Character orbit: 180.k
• Rep. character: $\chi_{180}(127,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $26$
• Newform subspaces: $5$
• Sturm bound: $72$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.k (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(72\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(180, [\chi])\).

	Total	New	Old
Modular forms	88	34	54
Cusp forms	56	26	30
Eisenstein series	32	8	24

Trace form

26 * q + 2 * q^2 + 4 * q^5 + 8 * q^8 - 10 * q^10 - 2 * q^13 - 16 * q^16 + 14 * q^17 - 16 * q^20 - 16 * q^22 + 6 * q^25 - 20 * q^26 - 32 * q^28 - 28 * q^32 - 6 * q^37 - 16 * q^38 + 20 * q^40 + 16 * q^46 + 30 * q^50 + 52 * q^52 - 22 * q^53 + 64 * q^56 + 24 * q^58 - 40 * q^61 + 56 * q^62 - 22 * q^65 + 28 * q^68 + 16 * q^70 - 38 * q^73 - 48 * q^76 - 48 * q^77 - 20 * q^80 - 32 * q^82 - 26 * q^85 - 64 * q^86 + 32 * q^88 - 56 * q^92 - 54 * q^97 - 38 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(180, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
180.2.k.a	$180$	$2$	180.k	20.e	$4$	$2$	$1$	$1.437$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			180.2.k.a	$4$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-i-1)q^{2}+2 i q^{4}+(-2 i+1)q^{5}+\cdots\)
180.2.k.b	$180$	$2$	180.k	20.e	$4$	$2$	$1$	$1.437$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			180.2.k.a	$4$	$0$	\(2\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i+1)q^{2}+2 i q^{4}+(2 i-1)q^{5}+\cdots\)
180.2.k.c	$180$	$2$	180.k	20.e	$4$	$2$	$1$	$1.437$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					20.2.e.a	$4$	$0$	\(2\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i+1)q^{2}+2 i q^{4}+(-i+2)q^{5}+\cdots\)
180.2.k.d	$180$	$2$	180.k	20.e	$4$	$8$	$4$	$1.437$	8.0.157351936.1	None		✓			180.2.k.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{2}+\beta _{4})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(2\beta _{4}+\cdots)q^{5}+\cdots\)
180.2.k.e	$180$	$2$	180.k	20.e	$4$	$12$	$6$	$1.437$	12.0.\(\cdots\).1	None			✓		60.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{1}+\beta _{7}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(180, [\chi])\) into lower level spaces

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