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

• Label: 735.2.i
• Level: $735$
• Weight: $2$
• Character orbit: 735.i
• Rep. character: $\chi_{735}(226,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $52$
• Newform subspaces: $14$
• Sturm bound: $224$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(224\)
Trace bound:			\(4\)
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	52	204
Cusp forms	192	52	140
Eisenstein series	64	0	64

Trace form

52 * q - 8 * q^2 - 2 * q^3 - 32 * q^4 + 48 * q^8 - 26 * q^9 - 4 * q^10 - 12 * q^11 - 4 * q^12 + 4 * q^13 - 36 * q^16 + 8 * q^17 - 8 * q^18 + 6 * q^19 - 16 * q^20 - 8 * q^22 - 16 * q^23 + 12 * q^24 - 26 * q^25 + 8 * q^26 + 4 * q^27 + 40 * q^29 + 6 * q^31 - 36 * q^32 - 8 * q^33 + 16 * q^34 + 64 * q^36 + 2 * q^37 - 32 * q^38 + 22 * q^39 - 12 * q^40 - 8 * q^41 + 20 * q^43 - 28 * q^44 - 8 * q^46 + 8 * q^47 + 16 * q^48 + 16 * q^50 - 12 * q^51 + 24 * q^55 - 68 * q^57 + 4 * q^58 - 4 * q^59 + 4 * q^61 - 24 * q^62 + 56 * q^64 + 12 * q^65 + 12 * q^66 - 2 * q^67 - 12 * q^68 - 24 * q^69 - 24 * q^72 + 10 * q^73 - 12 * q^74 - 2 * q^75 - 40 * q^76 + 8 * q^78 + 6 * q^79 + 16 * q^80 - 26 * q^81 + 28 * q^82 - 8 * q^83 - 8 * q^85 + 132 * q^86 - 24 * q^87 + 136 * q^88 - 16 * q^89 + 8 * q^90 - 200 * q^92 + 34 * q^93 + 24 * q^94 - 8 * q^95 + 24 * q^96 + 32 * q^97 + 24 * q^99

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.i.a	$735$	$2$	735.i	7.c	$3$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None					105.2.a.a	$2$	$1$	\(-1\)	\(-1\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.b	$735$	$2$	735.i	7.c	$3$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None					105.2.a.a	$2$	$0$	\(-1\)	\(1\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.c	$735$	$2$	735.i	7.c	$3$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None					105.2.i.a	$2$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots\)
735.2.i.d	$735$	$2$	735.i	7.c	$3$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None					15.2.a.a	$2$	$0$	\(1\)	\(-1\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.e	$735$	$2$	735.i	7.c	$3$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None					15.2.a.a	$2$	$0$	\(1\)	\(1\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.f	$735$	$2$	735.i	7.c	$3$	$2$	$1$	$5.869$	\(\Q(\sqrt{-3}) \)	None					105.2.i.b	$2$	$0$	\(2\)	\(1\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
735.2.i.g	$735$	$2$	735.i	7.c	$3$	$4$	$2$	$5.869$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			735.2.a.l	$2$	$0$	\(-2\)	\(-2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{2}+\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.h	$735$	$2$	735.i	7.c	$3$	$4$	$2$	$5.869$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			735.2.a.l	$2$	$0$	\(-2\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{2}-\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.i	$735$	$2$	735.i	7.c	$3$	$4$	$2$	$5.869$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					105.2.a.b	$2$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+\beta _{1}q^{3}+3\beta _{1}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
735.2.i.j	$735$	$2$	735.i	7.c	$3$	$4$	$2$	$5.869$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					105.2.i.c	$2$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+\beta _{3}q^{6}+\cdots\)
735.2.i.k	$735$	$2$	735.i	7.c	$3$	$4$	$2$	$5.869$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					105.2.a.b	$2$	$0$	\(0\)	\(2\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}-\beta _{1}q^{3}+3\beta _{1}q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
735.2.i.l	$735$	$2$	735.i	7.c	$3$	$4$	$2$	$5.869$	\(\Q(\zeta_{12})\)	None					105.2.i.d	$2$	$0$	\(2\)	\(-2\)	\(2\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta_{2}+\beta_1)q^{2}+(\beta_1-1)q^{3}+\cdots\)
735.2.i.m	$735$	$2$	735.i	7.c	$3$	$8$	$4$	$5.869$	8.0.\(\cdots\).10	None		✓			735.2.a.n	$2$	$0$	\(-4\)	\(-4\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{5})q^{2}+\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.n	$735$	$2$	735.i	7.c	$3$	$8$	$4$	$5.869$	8.0.\(\cdots\).10	None		✓			735.2.a.n	$2$	$0$	\(-4\)	\(4\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{5})q^{2}-\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(735, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)