Space of modular forms of level 36, weight 3, and Character 36.g

• Label: 36.3.g
• Level: $36$
• Weight: $3$
• Character orbit: 36.g
• Rep. character: $\chi_{36}(5,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $4$
• Newform subspaces: $1$
• Sturm bound: $18$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	36.g (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(18\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(36, [\chi])\).

	Total	New	Old
Modular forms	30	4	26
Cusp forms	18	4	14
Eisenstein series	12	0	12

Trace form

4 * q + 3 * q^3 + 9 * q^5 - q^7 - 15 * q^9 - 36 * q^11 + 5 * q^13 - 45 * q^15 + 2 * q^19 + 99 * q^21 + 99 * q^23 + 13 * q^25 - 63 * q^29 - 7 * q^31 - 36 * q^33 - 64 * q^37 + 57 * q^39 - 18 * q^41 - 46 * q^43 - 99 * q^45 - 81 * q^47 - 51 * q^49 - 27 * q^51 + 90 * q^55 + 51 * q^57 + 126 * q^59 + 41 * q^61 + 141 * q^63 + 171 * q^65 + 116 * q^67 + 99 * q^69 + 86 * q^73 - 297 * q^75 - 279 * q^77 + 83 * q^79 - 63 * q^81 - 81 * q^83 + 18 * q^85 - 63 * q^87 - 302 * q^91 - 159 * q^93 - 144 * q^95 - 196 * q^97 + 171 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(36, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
36.3.g.a	$36$	$3$	36.g	9.d	$6$	$4$	$2$	$0.981$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓	✓	✓	36.3.g.a	$2$	$0$	\(0\)	\(3\)	\(9\)	\(-1\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{3})q^{3}+(2-2\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(36, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(36, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)