Space of modular forms of level 13, weight 8, and Character 13.b

• Label: 13.8.b
• Level: $13$
• Weight: $8$
• Character orbit: 13.b
• Rep. character: $\chi_{13}(12,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $1$
• Sturm bound: $9$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	13.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(9\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(13, [\chi])\).

	Total	New	Old
Modular forms	8	8	0
Cusp forms	6	6	0
Eisenstein series	2	2	0

Trace form

6 * q - 56 * q^3 - 130 * q^4 - 1150 * q^9 - 406 * q^10 + 1898 * q^12 - 5018 * q^13 + 9558 * q^14 + 7778 * q^16 + 13152 * q^17 - 125080 * q^22 + 27264 * q^23 - 18262 * q^25 - 54210 * q^26 + 194560 * q^27 + 42924 * q^29 - 60114 * q^30 - 546720 * q^35 + 747772 * q^36 - 243492 * q^38 + 511160 * q^39 + 792058 * q^40 - 1138134 * q^42 - 1005576 * q^43 - 2807426 * q^48 + 3246846 * q^49 + 297984 * q^51 + 3252964 * q^52 + 1705524 * q^53 - 5320576 * q^55 - 2282730 * q^56 - 8237572 * q^61 + 9625800 * q^62 + 5070862 * q^64 + 7139028 * q^65 + 9289056 * q^66 - 16801086 * q^68 - 6017400 * q^69 - 28760826 * q^74 + 20426072 * q^75 + 4093560 * q^77 + 14088594 * q^78 + 15147200 * q^79 - 21318442 * q^81 - 9130240 * q^82 - 36474240 * q^87 + 38008000 * q^88 + 4769676 * q^90 + 21281832 * q^91 + 36462348 * q^92 - 34229770 * q^94 - 22075632 * q^95

Decomposition of \(S_{8}^{\mathrm{new}}(13, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
13.8.b.a	$13$	$8$	13.b	13.b	$2$	$6$	$6$	$4.061$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓	✓	✓	13.8.b.a	$2$	$0$	\(0\)	\(-56\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-9-\beta _{2})q^{3}+(-22-\beta _{4}+\cdots)q^{4}+\cdots\)