Space of modular forms of level 735, weight 2, and Character 735.s

• Label: 735.2.s
• Level: $735$
• Weight: $2$
• Character orbit: 735.s
• Rep. character: $\chi_{735}(521,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $108$
• Newform subspaces: $14$
• Sturm bound: $224$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.s (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(224\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(735, [\chi])\).

	Total	New	Old
Modular forms	256	108	148
Cusp forms	192	108	84
Eisenstein series	64	0	64

Trace form

\( 108 q + 56 q^{4} + 6 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(735, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
735.2.s.a	$735$	$2$	735.s	21.g	$6$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(-3\)	\(-3\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}+(1+\cdots)q^{4}+\cdots\)
735.2.s.b	$735$	$2$	735.s	21.g	$6$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-3\)	\(3\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.c	$735$	$2$	735.s	21.g	$6$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-3\)	\(3\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.d	$735$	$2$	735.s	21.g	$6$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(-3\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.e	$735$	$2$	735.s	21.g	$6$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(0\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.f	$735$	$2$	735.s	21.g	$6$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(3\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.g	$735$	$2$	735.s	21.g	$6$	$4$	$2$	$5.869$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$1$	\(-3\)	\(-2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{3})q^{2}+(-1+\beta _{1}-\beta _{3})q^{3}+\cdots\)
735.2.s.h	$735$	$2$	735.s	21.g	$6$	$4$	$2$	$5.869$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(-3\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{3})q^{2}+(1-\beta _{1}+\beta _{3})q^{3}+\cdots\)
735.2.s.i	$735$	$2$	735.s	21.g	$6$	$4$	$2$	$5.869$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(3\)	\(-1\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1})q^{2}-\beta _{1}q^{3}+(1-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.s.j	$735$	$2$	735.s	21.g	$6$	$4$	$2$	$5.869$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(3\)	\(1\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1})q^{2}+\beta _{1}q^{3}+(1-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.s.k	$735$	$2$	735.s	21.g	$6$	$8$	$4$	$5.869$	8.0.856615824.2	None		$2$	$0$	\(-3\)	\(-1\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{3})q^{2}+(-1+\beta _{1}-\beta _{4}-\beta _{6}+\cdots)q^{3}+\cdots\)
735.2.s.l	$735$	$2$	735.s	21.g	$6$	$8$	$4$	$5.869$	8.0.856615824.2	None		$2$	$0$	\(3\)	\(-2\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(-1-\beta _{3}-\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots\)
735.2.s.m	$735$	$2$	735.s	21.g	$6$	$32$	$16$	$5.869$		None		$4$	$0$	\(0\)	\(-4\)	\(16\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
735.2.s.n	$735$	$2$	735.s	21.g	$6$	$32$	$16$	$5.869$		None		$4$	$0$	\(0\)	\(4\)	\(-16\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(735, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(735, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)