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 + q^10 + 12 * q^11 + q^12 - 3 * q^13 + q^15 + 9 * q^16 - 6 * q^17 + q^18 + 4 * q^19 + q^20 + 8 * q^21 + 4 * q^22 + 8 * q^23 + q^24 + 9 * q^25 + q^26 + q^27 + 8 * q^28 - 10 * q^29 + q^30 + q^32 - 12 * q^33 + 2 * q^34 - 8 * q^35 + 9 * q^36 - 18 * q^37 - 4 * q^38 + q^39 + q^40 - 6 * q^41 - 8 * q^42 + 12 * q^43 + 12 * q^44 + q^45 + 8 * q^46 - 16 * q^47 + q^48 - 15 * q^49 + q^50 - 6 * q^51 - 3 * q^52 - 18 * q^53 + q^54 + 4 * q^55 + 12 * q^57 - 10 * q^58 - 20 * q^59 + q^60 - 2 * q^61 - 8 * q^62 + 8 * q^63 + 9 * q^64 + q^65 - 4 * q^66 - 4 * q^67 - 6 * q^68 - 16 * q^69 + 8 * q^71 + q^72 - 22 * q^73 - 26 * q^74 + q^75 + 4 * q^76 - 32 * q^77 - 3 * q^78 - 16 * q^79 + q^80 + 9 * q^81 - 30 * q^82 - 12 * q^83 + 8 * q^84 - 6 * q^85 - 4 * q^86 - 10 * q^87 + 4 * q^88 - 54 * q^89 + q^90 + 8 * q^92 + 8 * q^93 - 16 * q^94 + 4 * q^95 + q^96 - 30 * q^97 - 7 * q^98 + 12 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	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	✓	✓	✓	✓	390.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
390.2.a.b	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	✓	✓	✓	390.2.a.b	$1$	$0$	\(-1\)	\(-1\)	\(1\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
390.2.a.c	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	✓	✓	✓	390.2.a.c	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
390.2.a.d	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	✓	✓	✓	390.2.a.d	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.e	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	✓	✓	✓	390.2.a.e	$1$	$0$	\(1\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
390.2.a.f	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	✓	✓	✓	390.2.a.f	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
390.2.a.g	$390$	$2$	390.a	1.a	$1$	$1$	$1$	$3.114$	\(\Q\)	None	✓	✓	✓	✓	390.2.a.g	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
390.2.a.h	$390$	$2$	390.a	1.a	$1$	$2$	$2$	$3.114$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	390.2.a.h	$1$	$0$	\(2\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)

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

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