Space of modular forms of level 39, weight 4, and trivial character

• Label: 39.4.a
• Level: $39$
• Weight: $4$
• Character orbit: 39.a
• Rep. character: $\chi_{39}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(39))\).

	Total	New	Old
Modular forms	16	6	10
Cusp forms	12	6	6
Eisenstein series	4	0	4

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
\(+\)	\(+\)	\(+\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(3\)	\(2\)		\(4\)	\(3\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(10\)	\(5\)	\(5\)		\(8\)	\(5\)	\(3\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)

Trace form

6 * q + 4 * q^2 + 16 * q^4 + 16 * q^5 + 32 * q^7 + 48 * q^8 + 54 * q^9 - 36 * q^10 - 96 * q^11 + 12 * q^12 + 26 * q^13 - 232 * q^14 - 24 * q^15 - 76 * q^16 - 60 * q^17 + 36 * q^18 + 216 * q^19 - 92 * q^20 + 84 * q^21 - 436 * q^22 - 136 * q^23 - 180 * q^24 + 314 * q^25 - 48 * q^28 + 420 * q^29 + 84 * q^30 + 128 * q^31 + 28 * q^32 + 192 * q^33 + 440 * q^34 + 168 * q^35 + 144 * q^36 - 12 * q^37 + 416 * q^38 + 156 * q^39 + 76 * q^40 + 296 * q^41 - 360 * q^42 - 352 * q^43 + 20 * q^44 + 144 * q^45 + 248 * q^46 - 48 * q^47 - 432 * q^48 - 466 * q^49 - 380 * q^50 - 696 * q^51 - 156 * q^52 - 284 * q^53 - 976 * q^55 + 40 * q^56 - 84 * q^57 - 896 * q^58 - 1072 * q^59 - 1188 * q^60 + 932 * q^61 + 1264 * q^62 + 288 * q^63 - 1456 * q^64 - 416 * q^65 + 972 * q^66 - 360 * q^67 + 1536 * q^68 + 120 * q^69 + 656 * q^70 - 976 * q^71 + 432 * q^72 - 1804 * q^73 + 2408 * q^74 - 72 * q^75 + 2248 * q^76 + 712 * q^77 + 156 * q^78 + 248 * q^79 - 700 * q^80 + 486 * q^81 + 3396 * q^82 - 3072 * q^83 + 624 * q^84 + 2432 * q^85 + 2640 * q^86 - 1272 * q^87 + 84 * q^88 + 1376 * q^89 - 324 * q^90 + 416 * q^91 + 2216 * q^92 + 1428 * q^93 - 1348 * q^94 - 4456 * q^95 + 840 * q^96 + 2820 * q^97 - 5644 * q^98 - 864 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(39))\) 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
39.4.a.a	$39$	$4$	39.a	1.a	$1$	$1$	$1$	$2.301$	\(\Q\)	None	✓	✓	✓	✓	39.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(-12\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{4}-12q^{5}+2q^{7}+9q^{9}+\cdots\)
39.4.a.b	$39$	$4$	39.a	1.a	$1$	$2$	$2$	$2.301$	\(\Q(\sqrt{14}) \)	None	✓	✓	✓	✓	39.4.a.b	$1$	$0$	\(2\)	\(-6\)	\(24\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-3q^{3}+(7+2\beta )q^{4}+(12+\cdots)q^{5}+\cdots\)
39.4.a.c	$39$	$4$	39.a	1.a	$1$	$3$	$3$	$2.301$	3.3.3144.1	None	✓	✓	✓	✓	39.4.a.c	$1$	$0$	\(2\)	\(9\)	\(4\)	\(30\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(3+\beta _{2})q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(39))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(39)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)