Space of modular forms of level 399, weight 2, and Character 399.bq

• Label: 399.2.bq
• Level: $399$
• Weight: $2$
• Character orbit: 399.bq
• Rep. character: $\chi_{399}(4,\cdot)$
• Character field: $\Q(\zeta_{9})$
• Dimension: $162$
• Newform subspaces: $2$
• Sturm bound: $106$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$399 = 3 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	399.bq (of order $9$ and degree $6$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$133$
Character field:			$\Q(\zeta_{9})$
Newform subspaces:			$2$
Sturm bound:			$106$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(399, [\chi])$.

	Total	New	Old
Modular forms	342	162	180
Cusp forms	294	162	132
Eisenstein series	48	0	48

Trace form

162 * q + 12 * q^7 - 36 * q^10 - 12 * q^11 + 48 * q^12 + 6 * q^13 - 30 * q^14 - 24 * q^17 - 3 * q^19 - 48 * q^20 - 15 * q^21 + 24 * q^23 + 12 * q^25 + 24 * q^26 - 9 * q^27 - 36 * q^28 - 36 * q^29 - 72 * q^31 - 42 * q^32 + 12 * q^34 - 12 * q^37 + 138 * q^38 - 72 * q^40 + 36 * q^41 - 12 * q^42 - 60 * q^43 - 72 * q^44 + 6 * q^45 + 36 * q^46 + 36 * q^47 - 12 * q^49 - 18 * q^50 + 6 * q^52 + 36 * q^53 + 36 * q^56 - 12 * q^57 + 36 * q^58 - 48 * q^59 + 21 * q^61 + 18 * q^62 - 6 * q^63 - 102 * q^64 - 36 * q^65 + 27 * q^67 + 36 * q^68 - 48 * q^69 - 12 * q^70 - 24 * q^71 + 18 * q^72 + 27 * q^73 + 60 * q^74 - 63 * q^75 - 36 * q^76 - 102 * q^77 - 12 * q^78 + 93 * q^79 - 72 * q^80 - 96 * q^82 + 24 * q^83 - 72 * q^84 + 36 * q^85 + 18 * q^86 + 18 * q^90 - 51 * q^91 + 96 * q^92 - 30 * q^93 - 108 * q^94 - 24 * q^95 - 6 * q^97 - 72 * q^98 + 12 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(399, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
399.2.bq.a	$399$	$2$	399.bq	133.w	$9$	$72$	$12$	$3.186$		None		✓			399.2.bp.a	$2$	$0$	$0$	$0$	$0$	$6$		$\mathrm{SU}(2)[C_{9}]$
399.2.bq.b	$399$	$2$	399.bq	133.w	$9$	$90$	$15$	$3.186$		None		✓	✓		399.2.bp.b	$2$	$0$	$0$	$0$	$0$	$6$		$\mathrm{SU}(2)[C_{9}]$

Decomposition of $S_{2}^{\mathrm{old}}(399, [\chi])$ into lower level spaces

$S_{2}^{\mathrm{old}}(399, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(133, [\chi])$$^{\oplus 2}$