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

• Label: 270.2.e
• Level: $270$
• Weight: $2$
• Character orbit: 270.e
• Rep. character: $\chi_{270}(91,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $108$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(108\)
Trace bound:			\(2\)
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	8	124
Cusp forms	84	8	76
Eisenstein series	48	0	48

Trace form

8 * q - 2 * q^2 - 4 * q^4 - 2 * q^5 + 4 * q^7 + 4 * q^8 + 6 * q^11 + 4 * q^13 + 2 * q^14 - 4 * q^16 + 12 * q^17 + 4 * q^19 - 2 * q^20 - 6 * q^22 + 12 * q^23 - 4 * q^25 - 16 * q^26 - 8 * q^28 + 6 * q^29 + 4 * q^31 - 2 * q^32 - 6 * q^34 + 8 * q^35 - 8 * q^37 - 10 * q^38 - 12 * q^41 - 2 * q^43 - 12 * q^44 + 12 * q^46 - 12 * q^47 - 6 * q^49 - 2 * q^50 + 4 * q^52 - 24 * q^53 + 2 * q^56 + 6 * q^59 + 22 * q^61 + 8 * q^62 + 8 * q^64 - 4 * q^65 + 22 * q^67 - 6 * q^68 - 6 * q^70 + 24 * q^71 - 44 * q^73 + 20 * q^74 - 2 * q^76 - 36 * q^77 - 8 * q^79 + 4 * q^80 + 12 * q^82 + 12 * q^83 - 12 * q^85 + 14 * q^86 - 6 * q^88 - 60 * q^89 - 40 * q^91 + 12 * q^92 - 6 * q^94 + 8 * q^95 - 2 * q^97 + 36 * q^98

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
270.2.e.a	$270$	$2$	270.e	9.c	$3$	$2$	$1$	$2.156$	\(\Q(\sqrt{-3}) \)	None					90.2.e.b	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(4+\cdots)q^{7}+\cdots\)
270.2.e.b	$270$	$2$	270.e	9.c	$3$	$2$	$1$	$2.156$	\(\Q(\sqrt{-3}) \)	None					90.2.e.a	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots\)
270.2.e.c	$270$	$2$	270.e	9.c	$3$	$4$	$2$	$2.156$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None			✓		90.2.e.c	$2$	$0$	\(-2\)	\(0\)	\(-2\)	\(-1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(-1+\beta _{1})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(270, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)