Space of modular forms of level 270, weight 2, and Character 270.f

• Label: 270.2.f
• Level: $270$
• Weight: $2$
• Character orbit: 270.f
• Rep. character: $\chi_{270}(53,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $16$
• Newform subspaces: $2$
• Sturm bound: $108$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(108\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(270, [\chi])\).

	Total	New	Old
Modular forms	132	16	116
Cusp forms	84	16	68
Eisenstein series	48	0	48

Trace form

\( 16 q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(270, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
270.2.f.a	$270$	$2$	270.f	15.e	$4$	$8$	$4$	$2.156$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}^{5}q^{2}-\zeta_{24}^{3}q^{4}+(\zeta_{24}-\zeta_{24}^{4}+\cdots)q^{5}+\cdots\)
270.2.f.b	$270$	$2$	270.f	15.e	$4$	$8$	$4$	$2.156$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\zeta_{24}-\zeta_{24}^{5})q^{2}-\zeta_{24}^{6}q^{4}+(2\zeta_{24}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(270, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(270, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)