Space of modular forms of level 390, weight 2, and Character 390.bd

• Label: 390.2.bd
• Level: $390$
• Weight: $2$
• Character orbit: 390.bd
• Rep. character: $\chi_{390}(7,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $56$
• Newform subspaces: $3$
• Sturm bound: $168$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.bd (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(168\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(390, [\chi])\).

	Total	New	Old
Modular forms	368	56	312
Cusp forms	304	56	248
Eisenstein series	64	0	64

Trace form

56 * q + 28 * q^4 + 4 * q^5 - 8 * q^11 + 12 * q^13 + 8 * q^15 - 28 * q^16 + 8 * q^17 + 8 * q^18 - 40 * q^19 - 4 * q^20 + 16 * q^21 + 16 * q^23 + 36 * q^25 + 24 * q^31 - 24 * q^33 + 36 * q^34 + 48 * q^37 + 8 * q^39 - 12 * q^41 - 8 * q^43 + 8 * q^44 + 8 * q^46 - 48 * q^47 - 36 * q^49 + 32 * q^50 + 12 * q^53 - 88 * q^55 - 24 * q^56 - 20 * q^58 - 8 * q^59 - 8 * q^60 - 32 * q^61 - 72 * q^62 - 56 * q^64 + 24 * q^65 + 16 * q^66 - 72 * q^67 + 4 * q^68 + 16 * q^69 - 16 * q^70 - 32 * q^71 + 4 * q^72 - 60 * q^74 - 32 * q^76 - 48 * q^77 - 16 * q^78 - 8 * q^80 + 28 * q^81 + 44 * q^82 - 48 * q^83 - 16 * q^84 + 4 * q^85 + 36 * q^89 - 4 * q^90 - 48 * q^91 + 32 * q^92 - 24 * q^94 - 48 * q^95 + 144 * q^97 + 96 * q^98 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(390, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
390.2.bd.a	$390$	$2$	390.bd	65.t	$12$	$8$	$2$	$3.114$	\(\Q(\zeta_{24})\)	None		✓			390.2.bd.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{2}q^{2}+\zeta_{24}^{5}q^{3}+\zeta_{24}^{4}q^{4}+\cdots\)
390.2.bd.b	$390$	$2$	390.bd	65.t	$12$	$16$	$4$	$3.114$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			390.2.bd.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{5}-\beta _{13})q^{2}+\beta _{2}q^{3}+(1+\beta _{14}+\cdots)q^{4}+\cdots\)
390.2.bd.c	$390$	$2$	390.bd	65.t	$12$	$32$	$8$	$3.114$		None		✓	✓		390.2.bd.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(390, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(390, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)