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

• Label: 51.3.f
• Level: $51$
• Weight: $3$
• Character orbit: 51.f
• Rep. character: $\chi_{51}(38,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $20$
• Newform subspaces: $1$
• Sturm bound: $18$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	51.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 51 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(18\)
Trace bound:			\(0\)

Dimensions

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

	Total	New	Old
Modular forms	28	28	0
Cusp forms	20	20	0
Eisenstein series	8	8	0

Trace form

20 * q - 6 * q^3 + 24 * q^4 - 2 * q^6 - 4 * q^7 - 16 * q^10 - 42 * q^12 - 12 * q^13 - 64 * q^16 - 4 * q^18 + 88 * q^21 - 40 * q^22 - 82 * q^24 + 54 * q^27 - 160 * q^28 + 48 * q^31 + 264 * q^33 + 152 * q^34 + 32 * q^37 + 56 * q^39 + 52 * q^40 + 64 * q^45 + 388 * q^46 - 230 * q^48 - 298 * q^51 + 64 * q^52 - 254 * q^54 - 204 * q^55 - 24 * q^57 + 396 * q^58 - 336 * q^61 - 400 * q^63 - 480 * q^64 - 408 * q^67 - 132 * q^69 + 276 * q^72 - 88 * q^73 - 358 * q^75 + 380 * q^78 - 172 * q^79 + 388 * q^81 + 372 * q^82 + 1320 * q^84 + 648 * q^85 - 156 * q^88 + 672 * q^90 - 124 * q^91 + 194 * q^96 + 688 * q^97 - 864 * q^99

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.f.a	$51$	$3$	51.f	51.f	$4$	$20$	$10$	$1.390$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		✓	✓	✓	51.3.f.a	$4$	$0$	\(0\)	\(-6\)	\(0\)	\(-4\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{2}-\beta _{11}q^{3}+(1-\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)