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

• Label: 36.3.f
• Level: $36$
• Weight: $3$
• Character orbit: 36.f
• Rep. character: $\chi_{36}(7,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $20$
• 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.f (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 36 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	28	28	0
Cusp forms	20	20	0
Eisenstein series	8	8	0

Trace form

20 * q - q^2 - q^4 - 2 * q^5 - 9 * q^6 - 22 * q^8 + 4 * q^10 - 36 * q^12 - 2 * q^13 - 24 * q^14 - q^16 - 32 * q^17 + 12 * q^18 + 52 * q^20 - 30 * q^21 - 9 * q^22 + 129 * q^24 - 12 * q^25 + 160 * q^26 - 36 * q^28 - 26 * q^29 + 156 * q^30 + 119 * q^32 - 42 * q^33 - 11 * q^34 - 45 * q^36 - 8 * q^37 - 153 * q^38 + 4 * q^40 + 58 * q^41 - 354 * q^42 - 390 * q^44 + 102 * q^45 - 96 * q^46 - 333 * q^48 - 16 * q^49 - 237 * q^50 + 22 * q^52 + 136 * q^53 + 57 * q^54 + 270 * q^56 + 204 * q^57 + 52 * q^58 + 576 * q^60 - 2 * q^61 + 564 * q^62 + 2 * q^64 + 146 * q^65 + 510 * q^66 + 331 * q^68 - 54 * q^69 + 102 * q^70 - 189 * q^72 + 16 * q^73 - 404 * q^74 + 93 * q^76 - 246 * q^77 - 690 * q^78 - 848 * q^80 - 264 * q^81 + 202 * q^82 - 642 * q^84 - 52 * q^85 - 447 * q^86 + 75 * q^88 - 392 * q^89 + 180 * q^90 + 426 * q^92 - 486 * q^93 + 48 * q^94 + 900 * q^96 - 62 * q^97 + 1022 * 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.f.a	$36$	$3$	36.f	36.f	$6$	$2$	$1$	$0.981$	\(\Q(\sqrt{-3}) \)	None		✓			36.3.f.a	$2$	$0$	\(-2\)	\(3\)	\(-4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+2\zeta_{6})q^{2}+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots\)
36.3.f.b	$36$	$3$	36.f	36.f	$6$	$2$	$1$	$0.981$	\(\Q(\sqrt{-3}) \)	None		✓			36.3.f.a	$2$	$0$	\(4\)	\(-3\)	\(-4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2q^{2}-3\zeta_{6}q^{3}+4q^{4}+(-4+4\zeta_{6})q^{5}+\cdots\)
36.3.f.c	$36$	$3$	36.f	36.f	$6$	$16$	$8$	$0.981$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		36.3.f.c	$4$	$0$	\(-3\)	\(0\)	\(6\)	\(0\)	$2^{8}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{2}+\beta _{14}q^{3}+(-\beta _{2}+\beta _{3})q^{4}+\cdots\)