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

• Label: 735.2.d
• Level: $735$
• Weight: $2$
• Character orbit: 735.d
• Rep. character: $\chi_{735}(589,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $6$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(224\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(19\)

Dimensions

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

	Total	New	Old
Modular forms	128	40	88
Cusp forms	96	40	56
Eisenstein series	32	0	32

Trace form

40 * q - 40 * q^4 - 4 * q^5 + 4 * q^6 - 40 * q^9 + 8 * q^10 - 4 * q^15 + 24 * q^16 + 28 * q^20 - 12 * q^24 + 8 * q^25 - 24 * q^26 - 8 * q^29 + 8 * q^30 + 16 * q^31 - 32 * q^34 + 40 * q^36 + 8 * q^39 - 16 * q^40 - 8 * q^41 + 56 * q^44 + 4 * q^45 - 32 * q^50 - 8 * q^51 - 4 * q^54 + 8 * q^55 + 16 * q^59 + 16 * q^60 + 16 * q^61 - 8 * q^64 - 68 * q^65 - 8 * q^66 - 8 * q^69 - 24 * q^71 + 32 * q^74 - 16 * q^75 - 16 * q^76 + 32 * q^79 - 44 * q^80 + 40 * q^81 - 56 * q^85 - 32 * q^86 + 40 * q^89 - 8 * q^90 - 32 * q^94 - 36 * q^95 + 68 * q^96

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
735.2.d.a	$735$	$2$	735.d	5.b	$2$	$2$	$2$	$5.869$	\(\Q(\sqrt{-1}) \)	None					105.2.d.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}+q^{4}+(2 i-1)q^{5}+\cdots\)
735.2.d.b	$735$	$2$	735.d	5.b	$2$	$6$	$6$	$5.869$	6.0.350464.1	None					105.2.d.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-\beta _{4}q^{3}+(-1+\beta _{3}+\beta _{5})q^{4}+\cdots\)
735.2.d.c	$735$	$2$	735.d	5.b	$2$	$8$	$8$	$5.869$	8.0.309760000.3	None		✓			735.2.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}-\beta _{4}q^{3}+(-2+\beta _{6}-\beta _{7})q^{4}+\cdots\)
735.2.d.d	$735$	$2$	735.d	5.b	$2$	$8$	$8$	$5.869$	8.0.2058981376.2	None					105.2.q.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}-\beta _{4}q^{3}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
735.2.d.e	$735$	$2$	735.d	5.b	$2$	$8$	$8$	$5.869$	8.0.2058981376.2	None					105.2.q.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}+\beta _{4}q^{3}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
735.2.d.f	$735$	$2$	735.d	5.b	$2$	$8$	$8$	$5.869$	\(\Q(\zeta_{24})\)	None		✓			735.2.d.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{4} q^{2}+\beta_1 q^{3}+\beta_{3} q^{4}+(\beta_{6}+\beta_{5}-\beta_{2})q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(735, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)