Space of modular forms of level 243, weight 3, and Character 243.d

• Label: 243.3.d
• Level: $243$
• Weight: $3$
• Character orbit: 243.d
• Rep. character: $\chi_{243}(80,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $48$
• Newform subspaces: $10$
• Sturm bound: $81$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	243.d (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(81\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(243, [\chi])\).

	Total	New	Old
Modular forms	126	48	78
Cusp forms	90	48	42
Eisenstein series	36	0	36

Trace form

48 * q + 48 * q^4 - 3 * q^7 + 15 * q^13 - 96 * q^16 + 42 * q^19 + 120 * q^25 - 48 * q^28 - 48 * q^31 - 18 * q^34 - 66 * q^37 + 90 * q^40 + 96 * q^43 + 252 * q^46 - 207 * q^49 - 120 * q^52 - 72 * q^55 - 72 * q^58 + 24 * q^61 - 780 * q^64 - 120 * q^67 + 144 * q^70 - 336 * q^73 - 66 * q^76 - 129 * q^79 + 288 * q^82 + 288 * q^85 - 36 * q^88 + 912 * q^91 + 180 * q^94 - 633 * q^97

Decomposition of \(S_{3}^{\mathrm{new}}(243, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
243.3.d.a	$243$	$3$	243.d	9.d	$6$	$2$	$1$	$6.621$	\(\Q(\sqrt{-3}) \)	None		✓			243.3.b.g	$2$	$0$	\(-3\)	\(0\)	\(-12\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-8+4\zeta_{6})q^{5}+\cdots\)
243.3.d.b	$243$	$3$	243.d	9.d	$6$	$2$	$1$	$6.621$	\(\Q(\sqrt{-3}) \)	None		✓			243.3.b.f	$2$	$0$	\(-3\)	\(0\)	\(15\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(10-5\zeta_{6})q^{5}+\cdots\)
243.3.d.c	$243$	$3$	243.d	9.d	$6$	$2$	$1$	$6.621$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			243.3.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-4\zeta_{6}q^{4}+(-11+11\zeta_{6})q^{7}+22\zeta_{6}q^{13}+\cdots\)
243.3.d.d	$243$	$3$	243.d	9.d	$6$	$2$	$1$	$6.621$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			243.3.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-4\zeta_{6}q^{4}+(-2+2\zeta_{6})q^{7}-23\zeta_{6}q^{13}+\cdots\)
243.3.d.e	$243$	$3$	243.d	9.d	$6$	$2$	$1$	$6.621$	\(\Q(\sqrt{-3}) \)	None		✓			243.3.b.f	$2$	$0$	\(3\)	\(0\)	\(-15\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-10+5\zeta_{6})q^{5}+\cdots\)
243.3.d.f	$243$	$3$	243.d	9.d	$6$	$2$	$1$	$6.621$	\(\Q(\sqrt{-3}) \)	None		✓			243.3.b.g	$2$	$0$	\(3\)	\(0\)	\(12\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(8-4\zeta_{6})q^{5}+\cdots\)
243.3.d.g	$243$	$3$	243.d	9.d	$6$	$4$	$2$	$6.621$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			243.3.b.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}+(2+2\beta _{1})q^{4}+(-\beta _{2}+\beta _{3})q^{5}+\cdots\)
243.3.d.h	$243$	$3$	243.d	9.d	$6$	$4$	$2$	$6.621$	\(\Q(\zeta_{12})\)	None		✓			243.3.b.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-22\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_1 q^{2}+5\beta_{2} q^{4}+(2\beta_{3}-2\beta_1)q^{5}+\cdots\)
243.3.d.i	$243$	$3$	243.d	9.d	$6$	$4$	$2$	$6.621$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		✓			243.3.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{2}+(11+11\beta _{1})q^{4}+(-\beta _{2}+\beta _{3})q^{5}+\cdots\)
243.3.d.j	$243$	$3$	243.d	9.d	$6$	$24$	$12$	$6.621$		None		✓	✓		243.3.b.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(243, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(243, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)