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

• Label: 399.2.bd
• Level: $399$
• Weight: $2$
• Character orbit: 399.bd
• Rep. character: $\chi_{399}(68,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $100$
• Newform subspaces: $3$
• Sturm bound: $106$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$399 = 3 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	399.bd (of order $6$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$399$
Character field:			$\Q(\zeta_{6})$
Newform subspaces:			$3$
Sturm bound:			$106$
Trace bound:			$7$
Distinguishing $T_p$:			$2$, $13$

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 + 50 * q^4 - 12 * q^6 - 11 * q^7 - 2 * q^9 - 6 * q^10 + 3 * q^13 - 7 * q^15 - 46 * q^16 + 10 * q^18 - 9 * q^19 - 10 * q^21 - 12 * q^22 - 32 * q^25 - 4 * q^28 + 11 * q^30 - 39 * q^31 - 3 * q^33 - 12 * q^34 - 22 * q^36 - 17 * q^37 - 7 * q^39 + 12 * q^40 - 60 * q^42 - 17 * q^43 + 36 * q^45 + 12 * q^46 + 51 * q^48 - 13 * q^49 - 42 * q^51 + 9 * q^54 + 30 * q^55 - 6 * q^57 - 36 * q^58 - 4 * q^60 + 38 * q^63 - 144 * q^64 + 51 * q^66 + 7 * q^67 - 62 * q^70 + 142 * q^72 - 9 * q^75 + 66 * q^76 + 9 * q^78 + 9 * q^79 + 54 * q^81 - 15 * q^84 - 44 * q^85 - 21 * q^87 + 44 * q^88 - 171 * q^90 - 9 * q^91 - 21 * q^93 + 144 * q^94 + 132 * q^96 - 54 * 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.bd.a	$399$	$2$	399.bd	399.ad	$6$	$2$	$1$	$3.186$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$		✓			399.2.p.a	$4$	$1$	$0$	$0$	$0$	$-5$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(-1+2\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-3+\zeta_{6})q^{7}+\cdots$
399.2.bd.b	$399$	$2$	399.bd	399.ad	$6$	$2$	$1$	$3.186$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$		✓			399.2.p.b	$4$	$0$	$0$	$0$	$0$	$4$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(-1+2\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(1+2\zeta_{6})q^{7}+\cdots$
399.2.bd.c	$399$	$2$	399.bd	399.ad	$6$	$96$	$48$	$3.186$		None		✓	✓		399.2.p.c	$4$	$0$	$0$	$0$	$0$	$-10$		$\mathrm{SU}(2)[C_{6}]$