Space of modular forms of level 390, weight 2, and trivial character

• Label: 390.2.a
• Level: $390$
• Weight: $2$
• Character orbit: 390.a
• Rep. character: $\chi_{390}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $8$
• Sturm bound: $168$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(168\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(390))\).

	Total	New	Old
Modular forms	92	9	83
Cusp forms	77	9	68
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(1\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(2\)
Plus space				\(+\)	\(1\)
Minus space				\(-\)	\(8\)

Trace form

\( 9 q + q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 9 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(390))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5	13
390.2.a.a	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
390.2.a.b	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(1\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
390.2.a.c	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
390.2.a.d	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.e	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.f	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
390.2.a.g	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
390.2.a.h	$390$	$2$	390.a	1.a	$1$	$2$	$2$	$3.114$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(390))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(390)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 2}\)