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

• Label: 840.2.g
• Level: $840$
• Weight: $2$
• Character orbit: 840.g
• Rep. character: $\chi_{840}(421,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $4$
• Sturm bound: $384$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(384\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(11\)

Dimensions

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

	Total	New	Old
Modular forms	200	48	152
Cusp forms	184	48	136
Eisenstein series	16	0	16

Trace form

48 * q - 4 * q^2 - 4 * q^6 - 8 * q^7 - 4 * q^8 - 48 * q^9 + 4 * q^10 + 4 * q^14 + 8 * q^15 + 8 * q^16 + 4 * q^18 + 16 * q^23 + 4 * q^24 - 48 * q^25 - 4 * q^28 - 16 * q^31 + 36 * q^32 + 40 * q^34 - 40 * q^38 - 4 * q^40 - 40 * q^44 + 24 * q^46 - 32 * q^48 + 48 * q^49 + 4 * q^50 - 8 * q^52 + 4 * q^54 + 4 * q^56 - 32 * q^58 - 4 * q^60 + 8 * q^62 + 8 * q^63 - 48 * q^64 - 32 * q^66 + 48 * q^68 - 16 * q^71 + 4 * q^72 - 40 * q^74 + 24 * q^76 - 40 * q^78 - 32 * q^80 + 48 * q^81 - 8 * q^82 + 32 * q^86 + 48 * q^88 - 4 * q^90 + 56 * q^92 + 32 * q^94 + 36 * q^96 - 4 * q^98

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.g.a	$840$	$2$	840.g	8.b	$2$	$8$	$8$	$6.707$	\(\Q(\zeta_{16})\)	None		✓			840.2.g.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{5} q^{2}+\beta_{2} q^{3}-\beta_1 q^{4}+\beta_{2} q^{5}+\cdots\)
840.2.g.b	$840$	$2$	840.g	8.b	$2$	$12$	$12$	$6.707$	12.0.\(\cdots\).1	None		✓			840.2.g.b	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-12\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-\beta _{4}q^{3}-\beta _{7}q^{4}-\beta _{4}q^{5}+\cdots\)
840.2.g.c	$840$	$2$	840.g	8.b	$2$	$12$	$12$	$6.707$	12.0.\(\cdots\).1	None		✓			840.2.g.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+\beta _{4}q^{3}+\beta _{7}q^{4}-\beta _{4}q^{5}+\cdots\)
840.2.g.d	$840$	$2$	840.g	8.b	$2$	$16$	$16$	$6.707$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		840.2.g.d	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-16\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}-\beta _{5}q^{3}-\beta _{2}q^{4}+\beta _{5}q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(840, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)