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

• Label: 420.2.d
• Level: $420$
• Weight: $2$
• Character orbit: 420.d
• Rep. character: $\chi_{420}(41,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(11\), \(17\), \(41\)

Dimensions

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

	Total	New	Old
Modular forms	108	12	96
Cusp forms	84	12	72
Eisenstein series	24	0	24

Trace form

\( 12 q - 4 q^{7} + 6 q^{9} + 2 q^{15} + 2 q^{21} + 12 q^{25} - 32 q^{37} - 30 q^{39} + 4 q^{49} + 22 q^{51} - 12 q^{57} + 32 q^{63} + 32 q^{67} - 4 q^{79} - 10 q^{81} - 12 q^{85} + 4 q^{91} - 20 q^{93} + 6 q^{99}+O(q^{100}) \)

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.d.a	$420$	$2$	420.d	21.c	$2$	$2$	$2$	$3.354$	\(\Q(\sqrt{-3}) \)	None		✓			420.2.d.a	$2$	$0$	\(0\)	\(-3\)	\(-2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{6})q^{3}-q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
420.2.d.b	$420$	$2$	420.d	21.c	$2$	$2$	$2$	$3.354$	\(\Q(\sqrt{-3}) \)	None		✓			420.2.d.a	$2$	$0$	\(0\)	\(3\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\zeta_{6})q^{3}+q^{5}+(3-2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
420.2.d.c	$420$	$2$	420.d	21.c	$2$	$4$	$4$	$3.354$	\(\Q(i, \sqrt{5})\)	None		✓			420.2.d.c	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(-6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{3})q^{3}+q^{5}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
420.2.d.d	$420$	$2$	420.d	21.c	$2$	$4$	$4$	$3.354$	\(\Q(i, \sqrt{5})\)	None		✓			420.2.d.c	$2$	$0$	\(0\)	\(2\)	\(-4\)	\(-6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(420, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)