Space of modular forms of level 840, weight 2, and Character 840.bg

• Label: 840.2.bg
• Level: $840$
• Weight: $2$
• Character orbit: 840.bg
• Rep. character: $\chi_{840}(121,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $32$
• Newform subspaces: $10$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.bg (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(840, [\chi])\).

	Total	New	Old
Modular forms	416	32	384
Cusp forms	352	32	320
Eisenstein series	64	0	64

Trace form

32 * q - 4 * q^3 - 4 * q^7 - 16 * q^9 - 4 * q^11 - 24 * q^13 - 8 * q^17 + 8 * q^21 - 16 * q^25 + 8 * q^27 + 16 * q^29 + 4 * q^31 + 8 * q^33 + 4 * q^35 - 4 * q^37 + 4 * q^39 - 24 * q^41 - 24 * q^43 + 12 * q^49 + 8 * q^51 + 24 * q^53 - 8 * q^57 + 8 * q^59 + 16 * q^61 - 4 * q^63 + 12 * q^65 + 12 * q^67 + 16 * q^69 - 16 * q^71 + 20 * q^73 - 4 * q^75 + 12 * q^79 - 16 * q^81 + 32 * q^83 - 40 * q^89 - 20 * q^91 + 4 * q^93 + 8 * q^95 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(840, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
840.2.bg.a	$840$	$2$	840.bg	7.c	$3$	$2$	$1$	$6.707$	\(\Q(\sqrt{-3}) \)	None		✓			840.2.bg.a	$2$	$0$	\(0\)	\(1\)	\(-1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
840.2.bg.b	$840$	$2$	840.bg	7.c	$3$	$2$	$1$	$6.707$	\(\Q(\sqrt{-3}) \)	None		✓			840.2.bg.b	$2$	$0$	\(0\)	\(1\)	\(-1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
840.2.bg.c	$840$	$2$	840.bg	7.c	$3$	$2$	$1$	$6.707$	\(\Q(\sqrt{-3}) \)	None		✓			840.2.bg.c	$2$	$0$	\(0\)	\(1\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
840.2.bg.d	$840$	$2$	840.bg	7.c	$3$	$2$	$1$	$6.707$	\(\Q(\sqrt{-3}) \)	None		✓			840.2.bg.d	$2$	$0$	\(0\)	\(1\)	\(1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
840.2.bg.e	$840$	$2$	840.bg	7.c	$3$	$2$	$1$	$6.707$	\(\Q(\sqrt{-3}) \)	None		✓			840.2.bg.e	$2$	$0$	\(0\)	\(1\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
840.2.bg.f	$840$	$2$	840.bg	7.c	$3$	$2$	$1$	$6.707$	\(\Q(\sqrt{-3}) \)	None		✓			840.2.bg.f	$2$	$0$	\(0\)	\(1\)	\(1\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
840.2.bg.g	$840$	$2$	840.bg	7.c	$3$	$4$	$2$	$6.707$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			840.2.bg.g	$2$	$0$	\(0\)	\(-2\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{3}+\beta _{2}q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
840.2.bg.h	$840$	$2$	840.bg	7.c	$3$	$4$	$2$	$6.707$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		✓			840.2.bg.h	$2$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
840.2.bg.i	$840$	$2$	840.bg	7.c	$3$	$6$	$3$	$6.707$	6.0.38363328.2	None		✓			840.2.bg.i	$2$	$0$	\(0\)	\(-3\)	\(-3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{3})q^{3}+\beta _{3}q^{5}+(-\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)
840.2.bg.j	$840$	$2$	840.bg	7.c	$3$	$6$	$3$	$6.707$	6.0.29428272.1	None		✓			840.2.bg.j	$2$	$0$	\(0\)	\(-3\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{3}+(1+\beta _{3})q^{5}+(\beta _{1}+\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(840, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(840, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 2}\)