Space of modular forms of level 27, weight 36, and trivial character

• Label: 27.36.a
• Level: $27$
• Weight: $36$
• Character orbit: 27.a
• Rep. character: $\chi_{27}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $5$
• Sturm bound: $108$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(108\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_0(27))\).

	Total	New	Old
Modular forms	108	47	61
Cusp forms	102	47	55
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.

\(3\)	Dim
\(+\)	\(24\)
\(-\)	\(23\)

Trace form

\( 47 q + 794969376902 q^{4} - 60249767623001 q^{7} + O(q^{10}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(27))\) 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
27.36.a.a	$27$	$36$	27.a	1.a	$1$	$1$	$1$	$209.507$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			27.36.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(44\!\cdots\!73\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{35}q^{4}+446525205377873q^{7}+\cdots\)
27.36.a.b	$27$	$36$	27.a	1.a	$1$	$10$	$10$	$209.507$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			27.36.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-80\!\cdots\!98\)		$-$	$-$	$2^{45}\cdot 3^{66}\cdot 5^{5}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(13839718900+\beta _{2})q^{4}+\cdots\)
27.36.a.c	$27$	$36$	27.a	1.a	$1$	$12$	$12$	$209.507$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		27.36.a.c	$1$	$1$	\(-209817\)	\(0\)	\(-237590493300\)	\(-26\!\cdots\!56\)		$-$	$-$	$2^{53}\cdot 3^{96}\cdot 5^{6}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+(-17485-\beta _{1})q^{2}+(19847540011+\cdots)q^{4}+\cdots\)
27.36.a.d	$27$	$36$	27.a	1.a	$1$	$12$	$12$	$209.507$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			27.36.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-37\!\cdots\!64\)		$+$	$+$	$2^{59}\cdot 3^{98}\cdot 5^{5}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(17882581963+\beta _{2})q^{4}+\cdots\)
27.36.a.e	$27$	$36$	27.a	1.a	$1$	$12$	$12$	$209.507$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			27.36.a.c	$1$	$0$	\(209817\)	\(0\)	\(237590493300\)	\(-26\!\cdots\!56\)		$+$	$+$	$2^{53}\cdot 3^{96}\cdot 5^{6}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+(17485+\beta _{1})q^{2}+(19847540011+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(27))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(27)) \simeq \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)