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}$