Space of modular forms of level 81, weight 10, and Character 81.c

• Label: 81.10.c
• Level: $81$
• Weight: $10$
• Character orbit: 81.c
• Rep. character: $\chi_{81}(28,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $70$
• Newform subspaces: $12$
• Sturm bound: $90$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(90\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(81, [\chi])\).

	Total	New	Old
Modular forms	174	74	100
Cusp forms	150	70	80
Eisenstein series	24	4	20

Trace form

70 * q - 8702 * q^4 + 1712 * q^7 - 2052 * q^10 - 162178 * q^13 - 2096126 * q^16 - 329044 * q^19 - 1055646 * q^22 - 10647917 * q^25 - 4552708 * q^28 + 16432778 * q^31 - 19250784 * q^34 - 52213756 * q^37 + 77162814 * q^40 - 78604252 * q^43 + 26624592 * q^46 - 207810519 * q^49 - 58523812 * q^52 + 130260204 * q^55 + 43645824 * q^58 + 36087050 * q^61 + 2527530388 * q^64 - 327047398 * q^67 + 580554810 * q^70 - 457726648 * q^73 - 146743396 * q^76 - 765317230 * q^79 - 485698464 * q^82 - 382001994 * q^85 - 3738784986 * q^88 - 3050515988 * q^91 + 5947108884 * q^94 - 1560307930 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(81, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.10.c.a	$81$	$10$	81.c	9.c	$3$	$2$	$1$	$41.718$	\(\Q(\sqrt{-3}) \)	None					3.10.a.a	$2$	$0$	\(-36\)	\(0\)	\(-1314\)	\(4480\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6^{2}+6^{2}\zeta_{6})q^{2}-28^{2}\zeta_{6}q^{4}-1314\zeta_{6}q^{5}+\cdots\)
81.10.c.b	$81$	$10$	81.c	9.c	$3$	$2$	$1$	$41.718$	\(\Q(\sqrt{-3}) \)	None					3.10.a.b	$2$	$0$	\(-18\)	\(0\)	\(1530\)	\(-9128\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-18+18\zeta_{6})q^{2}+188\zeta_{6}q^{4}+1530\zeta_{6}q^{5}+\cdots\)
81.10.c.c	$81$	$10$	81.c	9.c	$3$	$2$	$1$	$41.718$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					9.10.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12580\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2^{9}\zeta_{6}q^{4}+(12580-12580\zeta_{6})q^{7}+\cdots\)
81.10.c.d	$81$	$10$	81.c	9.c	$3$	$2$	$1$	$41.718$	\(\Q(\sqrt{-3}) \)	None					3.10.a.b	$2$	$0$	\(18\)	\(0\)	\(-1530\)	\(-9128\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(18-18\zeta_{6})q^{2}+188\zeta_{6}q^{4}-1530\zeta_{6}q^{5}+\cdots\)
81.10.c.e	$81$	$10$	81.c	9.c	$3$	$2$	$1$	$41.718$	\(\Q(\sqrt{-3}) \)	None					3.10.a.a	$2$	$0$	\(36\)	\(0\)	\(1314\)	\(4480\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(6^{2}-6^{2}\zeta_{6})q^{2}-28^{2}\zeta_{6}q^{4}+1314\zeta_{6}q^{5}+\cdots\)
81.10.c.f	$81$	$10$	81.c	9.c	$3$	$4$	$2$	$41.718$	\(\Q(\sqrt{-3}, \sqrt{14})\)	None					27.10.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1526\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}-8\beta _{2}q^{4}+(22\beta _{1}+22\beta _{3})q^{5}+\cdots\)
81.10.c.g	$81$	$10$	81.c	9.c	$3$	$6$	$3$	$41.718$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					27.10.a.b	$2$	$0$	\(-3\)	\(0\)	\(1983\)	\(3693\)	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2}-\beta _{3})q^{2}+(2\beta _{1}+199\beta _{2}+\cdots)q^{4}+\cdots\)
81.10.c.h	$81$	$10$	81.c	9.c	$3$	$6$	$3$	$41.718$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					27.10.a.b	$2$	$0$	\(3\)	\(0\)	\(-1983\)	\(3693\)	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2}+\beta _{3})q^{2}+(2\beta _{1}+199\beta _{2}+\cdots)q^{4}+\cdots\)
81.10.c.i	$81$	$10$	81.c	9.c	$3$	$8$	$4$	$41.718$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			81.10.a.a	$2$	$0$	\(-33\)	\(0\)	\(-570\)	\(3238\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\beta _{3}+\beta _{4})q^{2}+(-6\beta _{2}-212\beta _{3}+\cdots)q^{4}+\cdots\)
81.10.c.j	$81$	$10$	81.c	9.c	$3$	$8$	$4$	$41.718$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					27.10.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-11852\)	$2^{6}\cdot 3^{16}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{2}q^{2}+(-559+559\beta _{1}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
81.10.c.k	$81$	$10$	81.c	9.c	$3$	$8$	$4$	$41.718$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			81.10.a.a	$2$	$0$	\(33\)	\(0\)	\(570\)	\(3238\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+8\beta _{3})q^{2}+(-212+\beta _{1}+212\beta _{3}+\cdots)q^{4}+\cdots\)
81.10.c.l	$81$	$10$	81.c	9.c	$3$	$20$	$10$	$41.718$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓	✓		81.10.a.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-5108\)	$2^{24}\cdot 3^{65}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{9}-\beta _{10})q^{2}+(-290-290\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(81, [\chi]) \simeq \) \(S_{10}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)