Space of modular forms of level 270, weight 3, and Character 270.g

• Label: 270.3.g
• Level: $270$
• Weight: $3$
• Character orbit: 270.g
• Rep. character: $\chi_{270}(163,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $32$
• Newform subspaces: $6$
• Sturm bound: $162$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	270.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(162\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(270, [\chi])\).

	Total	New	Old
Modular forms	240	32	208
Cusp forms	192	32	160
Eisenstein series	48	0	48

Trace form

32 * q - 8 * q^7 - 16 * q^10 - 72 * q^13 - 128 * q^16 + 16 * q^22 - 32 * q^25 + 16 * q^28 + 128 * q^31 + 96 * q^37 + 64 * q^40 - 24 * q^43 - 64 * q^46 - 144 * q^52 - 72 * q^55 - 352 * q^58 - 96 * q^61 - 80 * q^67 - 112 * q^70 + 1064 * q^73 + 32 * q^76 + 160 * q^82 + 616 * q^85 - 32 * q^88 - 192 * q^91 - 96 * q^97

Decomposition of \(S_{3}^{\mathrm{new}}(270, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
270.3.g.a	$270$	$3$	270.g	5.c	$4$	$4$	$2$	$7.357$	\(\Q(i, \sqrt{6})\)	None		✓			270.3.g.a	$2$	$0$	\(-4\)	\(0\)	\(-12\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{2})q^{2}+2\beta _{2}q^{4}+(-3-2\beta _{1}+\cdots)q^{5}+\cdots\)
270.3.g.b	$270$	$3$	270.g	5.c	$4$	$4$	$2$	$7.357$	\(\Q(i, \sqrt{6})\)	None		✓			270.3.g.b	$2$	$0$	\(-4\)	\(0\)	\(12\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{2}-2\beta _{2}q^{4}+(3+2\beta _{1}+\cdots)q^{5}+\cdots\)
270.3.g.c	$270$	$3$	270.g	5.c	$4$	$4$	$2$	$7.357$	\(\Q(i, \sqrt{6})\)	None		✓			270.3.g.b	$2$	$0$	\(4\)	\(0\)	\(-12\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{2})q^{2}-2\beta _{2}q^{4}+(-3+2\beta _{1}+\cdots)q^{5}+\cdots\)
270.3.g.d	$270$	$3$	270.g	5.c	$4$	$4$	$2$	$7.357$	\(\Q(i, \sqrt{6})\)	None		✓			270.3.g.a	$2$	$0$	\(4\)	\(0\)	\(12\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{2})q^{2}+2\beta _{2}q^{4}+(3+2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
270.3.g.e	$270$	$3$	270.g	5.c	$4$	$8$	$4$	$7.357$	8.0.\(\cdots\).38	None		✓			270.3.g.e	$2$	$0$	\(-8\)	\(0\)	\(12\)	\(-8\)	$3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1})q^{2}+2\beta _{1}q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
270.3.g.f	$270$	$3$	270.g	5.c	$4$	$8$	$4$	$7.357$	8.0.\(\cdots\).38	None		✓			270.3.g.e	$2$	$0$	\(8\)	\(0\)	\(-12\)	\(-8\)	$3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1})q^{2}+2\beta _{1}q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(270, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(270, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)