Space of modular forms of level 39, weight 3, and Character 39.g

• Label: 39.3.g
• Level: $39$
• Weight: $3$
• Character orbit: 39.g
• Rep. character: $\chi_{39}(31,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $8$
• Newform subspaces: $1$
• Sturm bound: $14$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	39.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(14\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(39, [\chi])\).

	Total	New	Old
Modular forms	24	8	16
Cusp forms	16	8	8
Eisenstein series	8	0	8

Trace form

8 * q + 4 * q^2 - 20 * q^5 + 8 * q^7 - 24 * q^8 + 24 * q^9 - 20 * q^11 + 8 * q^13 - 16 * q^14 + 12 * q^15 + 56 * q^16 + 12 * q^18 - 40 * q^19 + 44 * q^20 - 48 * q^21 - 128 * q^22 + 36 * q^24 - 32 * q^26 + 32 * q^28 + 56 * q^29 + 32 * q^31 + 148 * q^32 - 36 * q^33 + 96 * q^34 + 104 * q^35 - 16 * q^37 + 24 * q^39 + 48 * q^40 - 116 * q^41 - 216 * q^42 - 92 * q^44 - 60 * q^45 - 264 * q^46 + 100 * q^47 + 20 * q^50 + 72 * q^52 - 232 * q^53 - 176 * q^55 + 120 * q^57 - 104 * q^58 + 316 * q^59 + 12 * q^60 + 160 * q^61 + 24 * q^63 - 92 * q^65 + 312 * q^66 + 176 * q^67 + 168 * q^68 + 184 * q^70 + 4 * q^71 - 72 * q^72 - 64 * q^73 - 64 * q^74 + 112 * q^76 + 180 * q^78 + 208 * q^79 - 332 * q^80 + 72 * q^81 - 212 * q^83 - 240 * q^84 - 168 * q^85 - 24 * q^87 - 452 * q^89 - 400 * q^91 - 720 * q^92 - 24 * q^93 + 16 * q^94 + 24 * q^96 + 128 * q^97 + 724 * q^98 - 60 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(39, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
39.3.g.a	$39$	$3$	39.g	13.d	$4$	$8$	$4$	$1.063$	8.0.\(\cdots\).10	None		✓	✓	✓	39.3.g.a	$2$	$0$	\(4\)	\(0\)	\(-20\)	\(8\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{2}-\beta _{2}q^{3}+(-1-\beta _{1}-\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(39, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(39, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)