Space of modular forms of level 729, weight 2, and Character 729.e

• Label: 729.2.e
• Level: $729$
• Weight: $2$
• Character orbit: 729.e
• Rep. character: $\chi_{729}(82,\cdot)$
• Character field: $\Q(\zeta_{9})$
• Dimension: $198$
• Newform subspaces: $21$
• Sturm bound: $162$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.e (of order \(9\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 27 \)
Character field:			\(\Q(\zeta_{9})\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(162\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(729, [\chi])\).

	Total	New	Old
Modular forms	594	234	360
Cusp forms	378	198	180
Eisenstein series	216	36	180

Trace form

\( 198 q + 18 q^{10} + 18 q^{19} - 36 q^{28} + 18 q^{37} + 18 q^{46} - 36 q^{55} - 18 q^{64} - 117 q^{73} - 36 q^{82} - 117 q^{91}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(729, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
729.2.e.a	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	None					243.2.a.e	$2$	$1$	\(-6\)	\(0\)	\(-3\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-1-\zeta_{18}+\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)
729.2.e.b	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	None					243.2.a.e	$2$	$0$	\(-3\)	\(0\)	\(-6\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-1+\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
729.2.e.c	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	None					243.2.a.e	$2$	$0$	\(-3\)	\(0\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}-\zeta_{18}^{4})q^{2}+\cdots\)
729.2.e.d	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	\(\Q(\sqrt{-3}) \)					243.2.a.b	$12$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{9}]$	\(q+2\zeta_{18}^{5}q^{4}+5\zeta_{18}^{4}q^{7}+2\zeta_{18}^{2}q^{13}+\cdots\)
729.2.e.e	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	\(\Q(\sqrt{-3}) \)					243.2.a.a	$12$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{9}]$	\(q+2\zeta_{18}^{5}q^{4}-4\zeta_{18}^{4}q^{7}-7\zeta_{18}^{2}q^{13}+\cdots\)
729.2.e.f	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	\(\Q(\sqrt{-3}) \)					27.2.a.a	$12$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{9}]$	\(q+2\zeta_{18}^{5}q^{4}-\zeta_{18}^{4}q^{7}+5\zeta_{18}^{2}q^{13}+\cdots\)
729.2.e.g	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	None					243.2.a.e	$2$	$0$	\(3\)	\(0\)	\(6\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
729.2.e.h	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	None					243.2.a.e	$2$	$0$	\(3\)	\(0\)	\(-3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
729.2.e.i	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	\(\Q(\zeta_{18})\)	None					243.2.a.e	$2$	$0$	\(6\)	\(0\)	\(3\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(1+\zeta_{18}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+(1+\cdots)q^{4}+\cdots\)
729.2.e.j	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			729.2.a.b	$2$	$0$	\(-6\)	\(0\)	\(6\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-1-\beta _{6}-\beta _{7}+\beta _{11})q^{2}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
729.2.e.k	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			729.2.a.b	$2$	$0$	\(-3\)	\(0\)	\(-6\)	\(6\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\beta _{5}-\beta _{7}+\beta _{10})q^{2}+(-1+\beta _{4}+\cdots)q^{4}+\cdots\)
729.2.e.l	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			729.2.a.b	$2$	$0$	\(-3\)	\(0\)	\(12\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+\beta _{2}q^{2}+(-2\beta _{1}-\beta _{2}+\beta _{3}-\beta _{5}+\cdots)q^{4}+\cdots\)
729.2.e.m	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\Q(\zeta_{36})\)	None		✓			729.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\beta_{11}-\beta_{9}+\cdots-\beta_{2})q^{2}+\cdots+(-\beta_{5}-1)q^{4}+\cdots\)
729.2.e.n	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\Q(\zeta_{36})\)	None					243.2.a.c	$12$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta_{8} q^{2}+(\beta_{7}-\beta_{2})q^{4}-2\beta_{4} q^{5}+\cdots\)
729.2.e.o	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\Q(\zeta_{36})\)	None					81.2.a.a	$12$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta_{8} q^{2}+(\beta_{7}-\beta_{2})q^{4}+\beta_{4} q^{5}+\cdots\)
729.2.e.p	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	12.0.\(\cdots\).1	None					243.2.a.d	$12$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{9}]$	\(q+\beta _{4}q^{2}+4\beta _{5}q^{4}+\beta _{11}q^{5}-2\beta _{7}q^{7}+\cdots\)
729.2.e.q	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\Q(\zeta_{36})\)	None		✓			729.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+\beta_{9} q^{2}+(-\beta_{4}+\beta_1+1)q^{4}+\cdots\)
729.2.e.r	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\Q(\zeta_{36})\)	None		✓			729.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\beta_{10}-\beta_{9})q^{2}+(-\beta_{6}-\beta_{4}+\beta_{3}+1)q^{4}+\cdots\)
729.2.e.s	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			729.2.a.b	$2$	$0$	\(3\)	\(0\)	\(-12\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(1-\beta _{5}+\beta _{6}+\beta _{10}-\beta _{11})q^{2}+(\beta _{1}+\cdots)q^{4}+\cdots\)
729.2.e.t	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			729.2.a.b	$2$	$0$	\(3\)	\(0\)	\(6\)	\(6\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\beta _{1}+\beta _{2}+\beta _{5}-\beta _{6}-\beta _{8})q^{2}+(1+\cdots)q^{4}+\cdots\)
729.2.e.u	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			729.2.a.b	$2$	$0$	\(6\)	\(0\)	\(-6\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\beta _{1}+\beta _{5}-\beta _{6}-\beta _{8})q^{2}+(-1+2\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(729, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(729, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(243, [\chi])\)\(^{\oplus 2}\)