Space of modular forms of level 27, weight 8, and trivial character

• Label: 27.8.a
• Level: $27$
• Weight: $8$
• Character orbit: 27.a
• Rep. character: $\chi_{27}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $5$
• Sturm bound: $24$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(24\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(27))\).

	Total	New	Old
Modular forms	24	9	15
Cusp forms	18	9	9
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
\(+\)		\(13\)	\(5\)	\(8\)		\(10\)	\(5\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)		\(11\)	\(4\)	\(7\)		\(8\)	\(4\)	\(4\)		\(3\)	\(0\)	\(3\)

Trace form

9 * q + 486 * q^4 - 477 * q^7 + 10566 * q^10 + 3465 * q^13 - 14166 * q^16 - 62379 * q^19 + 276930 * q^22 - 116289 * q^25 - 306054 * q^28 + 459468 * q^31 - 1531440 * q^34 - 293553 * q^37 + 3203082 * q^40 - 1441656 * q^43 - 1755612 * q^46 + 4729986 * q^49 - 688860 * q^52 - 178056 * q^55 + 580140 * q^58 - 974637 * q^61 - 4297662 * q^64 - 1320615 * q^67 - 8772174 * q^70 + 6787089 * q^73 - 16525728 * q^76 + 30330405 * q^79 + 26821620 * q^82 - 34065792 * q^85 + 11245590 * q^88 - 33990489 * q^91 - 8233524 * q^94 + 20656197 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(27))\) 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
27.8.a.a	$27$	$8$	27.a	1.a	$1$	$1$	$1$	$8.434$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			27.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(1763\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{7}q^{4}+1763q^{7}+12605q^{13}+\cdots\)
27.8.a.b	$27$	$8$	27.a	1.a	$1$	$2$	$2$	$8.434$	\(\Q(\sqrt{65}) \)	None	✓	✓			27.8.a.b	$1$	$1$	\(-9\)	\(0\)	\(-180\)	\(700\)		$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{2}+(43+9\beta )q^{4}+(-91+\cdots)q^{5}+\cdots\)
27.8.a.c	$27$	$8$	27.a	1.a	$1$	$2$	$2$	$8.434$	\(\Q(\sqrt{3}) \)	None	✓	✓			27.8.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1118\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-20q^{4}-34\beta q^{5}-559q^{7}+\cdots\)
27.8.a.d	$27$	$8$	27.a	1.a	$1$	$2$	$2$	$8.434$	\(\Q(\sqrt{42}) \)	None	✓	✓			27.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2522\)		$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+250q^{4}+20\beta q^{5}-1261q^{7}+\cdots\)
27.8.a.e	$27$	$8$	27.a	1.a	$1$	$2$	$2$	$8.434$	\(\Q(\sqrt{65}) \)	None	✓	✓			27.8.a.b	$1$	$0$	\(9\)	\(0\)	\(180\)	\(700\)		$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(5+\beta )q^{2}+(43+9\beta )q^{4}+(91+2\beta )q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(27))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(27)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)