Space of modular forms of level 27, weight 10, and trivial character

• Label: 27.10.a
• Level: $27$
• Weight: $10$
• Character orbit: 27.a
• Rep. character: $\chi_{27}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $30$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(30\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(27))\).

	Total	New	Old
Modular forms	30	12	18
Cusp forms	24	12	12
Eisenstein series	6	0	6

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(14\)	\(5\)	\(9\)		\(11\)	\(5\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)		\(16\)	\(7\)	\(9\)		\(13\)	\(7\)	\(6\)		\(3\)	\(0\)	\(3\)

Trace form

12 * q + 3414 * q^4 + 2940 * q^7 - 101898 * q^10 + 267486 * q^13 + 1352874 * q^16 - 215274 * q^19 + 1530738 * q^22 + 3403146 * q^25 + 12734682 * q^28 - 15909510 * q^31 - 1360800 * q^34 + 1367142 * q^37 - 130379382 * q^40 + 117867660 * q^43 - 70071804 * q^46 - 115298280 * q^49 + 322229364 * q^52 + 21445398 * q^55 - 822158100 * q^58 + 653809506 * q^61 + 950398770 * q^64 - 756172074 * q^67 + 1236623922 * q^70 + 517624044 * q^73 - 2190446688 * q^76 + 350003130 * q^79 - 2196835020 * q^82 + 836759808 * q^85 + 1113537942 * q^88 + 554521998 * q^91 - 2582298468 * q^94 + 587305752 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(27))\) 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					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3
27.10.a.a	$27$	$10$	27.a	1.a	$1$	$2$	$2$	$13.906$	\(\Q(\sqrt{14}) \)	None	✓	✓			27.10.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1526\)		$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-8q^{4}-22\beta q^{5}-763q^{7}+\cdots\)
27.10.a.b	$27$	$10$	27.a	1.a	$1$	$3$	$3$	$13.906$	3.3.177113.1	None	✓	✓			27.10.a.b	$1$	$0$	\(-3\)	\(0\)	\(1983\)	\(-3693\)		$-$	$-$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(199-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
27.10.a.c	$27$	$10$	27.a	1.a	$1$	$3$	$3$	$13.906$	3.3.177113.1	None	✓	✓	✓		27.10.a.b	$1$	$1$	\(3\)	\(0\)	\(-1983\)	\(-3693\)		$+$	$+$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(199-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
27.10.a.d	$27$	$10$	27.a	1.a	$1$	$4$	$4$	$13.906$	\(\Q(\sqrt{166 +2 \sqrt{1129}})\)	None	✓	✓	✓		27.10.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(11852\)		$-$	$-$	$2^{3}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(559+\beta _{3})q^{4}+(-10\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(27))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(27)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)