Space of modular forms of level 510, weight 2, and Character 510.d

• Label: 510.2.d
• Level: $510$
• Weight: $2$
• Character orbit: 510.d
• Rep. character: $\chi_{510}(409,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $4$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(216\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(510, [\chi])\).

	Total	New	Old
Modular forms	116	16	100
Cusp forms	100	16	84
Eisenstein series	16	0	16

Trace form

\( 16 q - 16 q^{4} - 4 q^{6} - 16 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(510, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
510.2.d.a	$510$	$2$	510.d	5.b	$2$	$2$	$2$	$4.072$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}+iq^{3}-q^{4}+(-2+i)q^{5}+\cdots\)
510.2.d.b	$510$	$2$	510.d	5.b	$2$	$4$	$4$	$4.072$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-\beta _{2}q^{5}-q^{6}+\cdots\)
510.2.d.c	$510$	$2$	510.d	5.b	$2$	$4$	$4$	$4.072$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+\beta _{2}q^{3}-q^{4}+(1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
510.2.d.d	$510$	$2$	510.d	5.b	$2$	$6$	$6$	$4.072$	6.0.5161984.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+\beta _{3}q^{3}-q^{4}+\beta _{5}q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(510, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(510, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(255, [\chi])\)\(^{\oplus 2}\)