Space of modular forms of level 51, weight 3, and Character 51.g

• Label: 51.3.g
• Level: $51$
• Weight: $3$
• Character orbit: 51.g
• Rep. character: $\chi_{51}(2,\cdot)$
• Character field: $\Q(\zeta_{8})$
• Dimension: $40$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	51.g (of order \(8\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 51 \)
Character field:			\(\Q(\zeta_{8})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(51, [\chi])\).

	Total	New	Old
Modular forms	56	56	0
Cusp forms	40	40	0
Eisenstein series	16	16	0

Trace form

\( 40 q - 4 q^{3} - 4 q^{6} - 8 q^{7} - 28 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(51, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
51.3.g.a	$51$	$3$	51.g	51.g	$8$	$4$	$1$	$1.390$	\(\Q(\zeta_{8})\)	None		✓			51.3.g.a	$2$	$0$	\(-4\)	\(12\)	\(8\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q+(-1-\zeta_{8}-\zeta_{8}^{2})q^{2}+3q^{3}+(2\zeta_{8}+\cdots)q^{4}+\cdots\)
51.3.g.b	$51$	$3$	51.g	51.g	$8$	$4$	$1$	$1.390$	\(\Q(\zeta_{8})\)	None		✓			51.3.g.a	$2$	$0$	\(4\)	\(0\)	\(-8\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q+(1+\zeta_{8}+\zeta_{8}^{2})q^{2}+3\zeta_{8}q^{3}+(2\zeta_{8}+\cdots)q^{4}+\cdots\)
51.3.g.c	$51$	$3$	51.g	51.g	$8$	$32$	$8$	$1.390$		None		✓	✓		51.3.g.c	$4$	$0$	\(0\)	\(-16\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{8}]$