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

• Label: 399.2.w
• Level: $399$
• Weight: $2$
• Character orbit: 399.w
• Rep. character: $\chi_{399}(170,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $100$
• Newform subspaces: $4$
• Sturm bound: $106$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$399 = 3 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	399.w (of order $6$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$399$
Character field:			$\Q(\zeta_{6})$
Newform subspaces:			$4$
Sturm bound:			$106$
Trace bound:			$3$
Distinguishing $T_p$:			$2$, $5$, $31$

Dimensions

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

	Total	New	Old
Modular forms	116	116	0
Cusp forms	100	100	0
Eisenstein series	16	16	0

Trace form

100 * q - 52 * q^4 - 2 * q^7 - 2 * q^9 - 56 * q^16 - 3 * q^19 - 8 * q^24 + 42 * q^25 + 8 * q^28 - 24 * q^36 + 2 * q^39 - 104 * q^42 + 20 * q^43 - 44 * q^45 + 10 * q^49 - 44 * q^54 - 88 * q^55 - 6 * q^57 + 4 * q^58 + 4 * q^61 + 2 * q^63 + 104 * q^64 + 28 * q^66 + 6 * q^73 + 124 * q^76 + 46 * q^81 - 92 * q^82 - 72 * q^85 - 48 * q^87 - 46 * q^93 + 100 * q^96 + 104 * 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.w.a	$399$	$2$	399.w	399.w	$6$	$2$	$1$	$3.186$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$		✓			399.2.w.a	$4$	$0$	$0$	$-3$	$0$	$1$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(-2+\zeta_{6})q^{3}+2\zeta_{6}q^{4}+(-1+3\zeta_{6})q^{7}+\cdots$
399.2.w.b	$399$	$2$	399.w	399.w	$6$	$2$	$1$	$3.186$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$		✓			399.2.w.a	$4$	$0$	$0$	$3$	$0$	$1$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(2-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+(-1+3\zeta_{6})q^{7}+\cdots$
399.2.w.c	$399$	$2$	399.w	399.w	$6$	$8$	$4$	$3.186$	8.0.3317760000.8	None		✓			399.2.w.c	$8$	$0$	$0$	$0$	$0$	$-16$	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{1}q^{3}+2\beta _{4}q^{4}+\beta _{6}q^{5}+(-1-2\beta _{4}+\cdots)q^{7}+\cdots$
399.2.w.d	$399$	$2$	399.w	399.w	$6$	$88$	$44$	$3.186$		None		✓	✓		399.2.w.d	$8$	$0$	$0$	$0$	$0$	$12$		$\mathrm{SU}(2)[C_{6}]$