Space of modular forms of level 420, weight 2, and Character 420.bi

• Label: 420.2.bi
• Level: $420$
• Weight: $2$
• Character orbit: 420.bi
• Rep. character: $\chi_{420}(31,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $64$
• Newform subspaces: $4$
• Sturm bound: $192$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.bi (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(192\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(420, [\chi])\).

	Total	New	Old
Modular forms	208	64	144
Cusp forms	176	64	112
Eisenstein series	32	0	32

Trace form

64 * q - 4 * q^2 + 4 * q^4 + 8 * q^8 - 32 * q^9 + 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 + 24 * q^22 - 36 * q^24 + 32 * q^25 - 60 * q^26 + 8 * q^28 + 64 * q^29 - 24 * q^32 - 8 * q^36 - 24 * q^37 - 12 * q^42 - 8 * q^44 - 20 * q^46 + 8 * q^49 - 8 * q^50 + 36 * q^52 + 16 * q^53 - 60 * q^56 - 48 * q^57 - 20 * q^58 - 48 * q^61 - 8 * q^64 - 8 * q^65 + 36 * q^66 - 60 * q^68 - 4 * q^70 - 4 * q^72 + 72 * q^73 + 8 * q^74 + 16 * q^77 - 24 * q^78 - 32 * q^81 - 12 * q^82 + 36 * q^84 - 20 * q^86 + 32 * q^88 + 40 * q^92 - 8 * q^93 + 36 * q^94

Decomposition of \(S_{2}^{\mathrm{new}}(420, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
420.2.bi.a	$420$	$2$	420.bi	28.f	$6$	$4$	$2$	$3.354$	\(\Q(\zeta_{12})\)	None		✓			420.2.bi.a	$2$	$0$	\(-4\)	\(2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2})q^{3}+\cdots\)
420.2.bi.b	$420$	$2$	420.bi	28.f	$6$	$4$	$2$	$3.354$	\(\Q(\zeta_{12})\)	None		✓			420.2.bi.a	$2$	$0$	\(2\)	\(-2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}-\zeta_{12}^{2}q^{3}+\cdots\)
420.2.bi.c	$420$	$2$	420.bi	28.f	$6$	$28$	$14$	$3.354$		None		✓			420.2.bi.c	$2$	$0$	\(-4\)	\(-14\)	\(0\)	\(-6\)		$\mathrm{SU}(2)[C_{6}]$
420.2.bi.d	$420$	$2$	420.bi	28.f	$6$	$28$	$14$	$3.354$		None		✓			420.2.bi.c	$2$	$0$	\(2\)	\(14\)	\(0\)	\(6\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(420, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(420, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 2}\)