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

• Label: 390.2.p
• Level: $390$
• Weight: $2$
• Character orbit: 390.p
• Rep. character: $\chi_{390}(161,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $32$
• Newform subspaces: $7$
• Sturm bound: $168$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.p (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(168\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	184	32	152
Cusp forms	152	32	120
Eisenstein series	32	0	32

Trace form

32 * q + 24 * q^7 - 8 * q^15 - 32 * q^16 - 16 * q^18 + 24 * q^19 + 8 * q^21 + 48 * q^27 - 24 * q^28 - 16 * q^31 - 8 * q^33 - 8 * q^34 - 40 * q^37 - 8 * q^39 - 48 * q^42 + 16 * q^45 + 8 * q^46 - 24 * q^52 + 48 * q^54 - 16 * q^55 + 48 * q^57 + 16 * q^58 - 8 * q^60 - 16 * q^61 - 8 * q^63 - 16 * q^66 + 72 * q^67 - 16 * q^72 + 72 * q^73 + 24 * q^76 + 24 * q^78 - 80 * q^79 - 64 * q^81 - 8 * q^84 + 48 * q^85 - 64 * q^87 + 40 * q^91 + 8 * q^93 + 80 * q^94 + 8 * q^97 - 48 * 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.p.a	$390$	$2$	390.p	39.f	$4$	$4$	$2$	$3.114$	\(\Q(\zeta_{8})\)	None		✓			390.2.p.a	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(-1+\zeta_{8}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.p.b	$390$	$2$	390.p	39.f	$4$	$4$	$2$	$3.114$	\(\Q(\zeta_{8})\)	None		✓			390.2.p.b	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(-1-\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.p.c	$390$	$2$	390.p	39.f	$4$	$4$	$2$	$3.114$	\(\Q(\zeta_{8})\)	None		✓			390.2.p.c	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(-1-\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.p.d	$390$	$2$	390.p	39.f	$4$	$4$	$2$	$3.114$	\(\Q(\zeta_{8})\)	None		✓			390.2.p.d	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(1+\zeta_{8}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.p.e	$390$	$2$	390.p	39.f	$4$	$4$	$2$	$3.114$	\(\Q(\zeta_{8})\)	None		✓			390.2.p.e	$2$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.p.f	$390$	$2$	390.p	39.f	$4$	$4$	$2$	$3.114$	\(\Q(\zeta_{8})\)	None		✓			390.2.p.e	$2$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(1+\zeta_{8}-\zeta_{8}^{2})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
390.2.p.g	$390$	$2$	390.p	39.f	$4$	$8$	$4$	$3.114$	8.0.40960000.1	None		✓	✓		390.2.p.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{7}q^{2}-\beta _{6}q^{3}-\beta _{2}q^{4}-\beta _{7}q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(390, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)