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

• Label: 390.2.f
• Level: $390$
• Weight: $2$
• Character orbit: 390.f
• Rep. character: $\chi_{390}(259,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $168$
• Trace bound: $2$

Defining parameters

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

Dimensions

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

	Total	New	Old
Modular forms	92	12	80
Cusp forms	76	12	64
Eisenstein series	16	0	16

Trace form

12 * q + 12 * q^4 - 12 * q^9 + 4 * q^10 - 8 * q^14 + 12 * q^16 - 4 * q^25 + 16 * q^26 + 8 * q^29 + 32 * q^35 - 12 * q^36 + 4 * q^39 + 4 * q^40 - 4 * q^49 + 16 * q^51 - 64 * q^55 - 8 * q^56 - 8 * q^61 + 12 * q^64 - 40 * q^65 - 24 * q^66 - 8 * q^69 - 48 * q^74 + 16 * q^75 - 32 * q^79 + 12 * q^81 - 4 * q^90 - 40 * q^91 + 64 * q^94 + 24 * q^95

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.f.a	$390$	$2$	390.f	65.d	$2$	$6$	$6$	$3.114$	6.0.350464.1	None		✓			390.2.f.a	$2$	$0$	\(-6\)	\(0\)	\(-2\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-\beta _{4}q^{3}+q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots\)
390.2.f.b	$390$	$2$	390.f	65.d	$2$	$6$	$6$	$3.114$	6.0.350464.1	None		✓			390.2.f.a	$2$	$0$	\(6\)	\(0\)	\(2\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}+\beta _{4}q^{3}+q^{4}+(-\beta _{1}-\beta _{5})q^{5}+\cdots\)

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

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