Space of modular forms of level 39, weight 2, and trivial character

• Label: 39.2.a
• Level: $39$
• Weight: $2$
• Character orbit: 39.a
• Rep. character: $\chi_{39}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $9$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(9\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(39))\).

	Total	New	Old
Modular forms	6	3	3
Cusp forms	3	3	0
Eisenstein series	3	0	3

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

\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(-\)	\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(5\)	\(3\)	\(2\)		\(3\)	\(3\)	\(0\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q - q^2 + q^3 + q^4 + 2 * q^5 - 3 * q^6 - 4 * q^7 - 9 * q^8 + 3 * q^9 - 6 * q^10 + 3 * q^12 - q^13 + 4 * q^14 - 2 * q^15 + 5 * q^16 + 6 * q^17 - q^18 + 14 * q^20 + 4 * q^21 + 8 * q^22 - 8 * q^23 - 3 * q^24 + 5 * q^25 + 3 * q^26 + q^27 - 12 * q^28 - 6 * q^29 - 10 * q^30 - 4 * q^31 + 11 * q^32 - 8 * q^33 + 14 * q^34 - 24 * q^35 + q^36 - 6 * q^37 - 8 * q^38 - 3 * q^39 - 14 * q^40 + 22 * q^41 + 12 * q^42 - 4 * q^43 - 8 * q^44 + 2 * q^45 + 8 * q^46 - 12 * q^47 + 7 * q^48 + 11 * q^49 - 7 * q^50 + 2 * q^51 - 3 * q^52 + 2 * q^53 - 3 * q^54 + 8 * q^55 + 20 * q^56 - 14 * q^58 + 16 * q^59 + 18 * q^60 + 2 * q^61 + 20 * q^62 - 4 * q^63 - 7 * q^64 + 2 * q^65 - 30 * q^68 - 8 * q^69 + 8 * q^70 + 4 * q^71 - 9 * q^72 + 14 * q^73 - 14 * q^74 + 7 * q^75 + 16 * q^76 - 16 * q^77 + q^78 + 8 * q^79 - 2 * q^80 + 3 * q^81 - 18 * q^82 - 20 * q^84 - 28 * q^85 - 36 * q^86 + 14 * q^87 + 22 * q^89 - 6 * q^90 - 4 * q^91 - 8 * q^92 - 12 * q^93 - 4 * q^94 + 16 * q^95 + q^96 + 6 * q^97 + 7 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	13
39.2.a.a	$39$	$2$	39.a	1.a	$1$	$1$	$1$	$0.311$	\(\Q\)	None	✓	✓	✓	✓	39.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots\)
39.2.a.b	$39$	$2$	39.a	1.a	$1$	$2$	$2$	$0.311$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	39.2.a.b	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-2\beta q^{5}+\cdots\)