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

• Label: 36.3.d
• Level: $36$
• Weight: $3$
• Character orbit: 36.d
• Rep. character: $\chi_{36}(19,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	36.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	16	6	10
Cusp forms	8	4	4
Eisenstein series	8	2	6

Trace form

4 * q + 2 * q^2 + 4 * q^4 + 4 * q^5 - 16 * q^8 - 28 * q^10 - 16 * q^13 + 24 * q^14 + 16 * q^16 - 20 * q^17 - 8 * q^20 + 24 * q^22 + 36 * q^25 + 4 * q^26 + 48 * q^28 + 52 * q^29 + 32 * q^32 + 44 * q^34 - 88 * q^37 - 72 * q^38 - 160 * q^40 - 116 * q^41 + 48 * q^44 - 96 * q^46 + 100 * q^49 - 42 * q^50 - 88 * q^52 + 148 * q^53 + 212 * q^58 + 8 * q^61 - 24 * q^62 + 256 * q^64 + 8 * q^65 + 40 * q^68 + 48 * q^70 + 128 * q^73 + 52 * q^74 - 144 * q^76 - 96 * q^77 - 32 * q^80 - 436 * q^82 - 296 * q^85 + 168 * q^86 - 164 * q^89 - 192 * q^92 + 240 * q^94 - 256 * q^97 + 2 * q^98

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.d.a	$36$	$3$	36.d	4.b	$2$	$1$	$1$	$0.981$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			36.3.d.a	$2$	$0$	\(-2\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}+8q^{5}-8q^{8}-2^{4}q^{10}+\cdots\)
36.3.d.b	$36$	$3$	36.d	4.b	$2$	$1$	$1$	$0.981$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			36.3.d.a	$2$	$0$	\(2\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}-8q^{5}+8q^{8}-2^{4}q^{10}+\cdots\)
36.3.d.c	$36$	$3$	36.d	4.b	$2$	$2$	$2$	$0.981$	\(\Q(\sqrt{-3}) \)	None			✓		12.3.d.a	$2$	$0$	\(2\)	\(0\)	\(4\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta+1)q^{2}+(2\beta-2)q^{4}+2 q^{5}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(36, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 2}\)