Space of modular forms of level 270, weight 2, and trivial character

• Label: 270.2.a
• Level: $270$
• Weight: $2$
• Character orbit: 270.a
• Rep. character: $\chi_{270}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $108$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(108\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(270))\).

	Total	New	Old
Modular forms	66	4	62
Cusp forms	43	4	39
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(1\)
Plus space			\(+\)	\(0\)
Minus space			\(-\)	\(4\)

Trace form

\( 4 q + 4 q^{4} + 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(270))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
270.2.a.a	$270$	$2$	270.a	1.a	$1$	$1$	$1$	$2.156$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
270.2.a.b	$270$	$2$	270.a	1.a	$1$	$1$	$1$	$2.156$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
270.2.a.c	$270$	$2$	270.a	1.a	$1$	$1$	$1$	$2.156$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
270.2.a.d	$270$	$2$	270.a	1.a	$1$	$1$	$1$	$2.156$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(270))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(270)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)