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

• Label: 270.3.b
• Level: $270$
• Weight: $3$
• Character orbit: 270.b
• Rep. character: $\chi_{270}(269,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $4$
• Sturm bound: $162$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	270.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(162\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(7\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	120	16	104
Cusp forms	96	16	80
Eisenstein series	24	0	24

Trace form

16 * q + 32 * q^4 + 8 * q^10 + 64 * q^16 + 48 * q^19 + 72 * q^25 - 8 * q^31 + 176 * q^34 + 16 * q^40 + 112 * q^46 - 552 * q^49 - 136 * q^55 - 400 * q^61 + 128 * q^64 + 104 * q^70 + 96 * q^76 + 288 * q^79 - 568 * q^85 + 72 * q^91 - 224 * q^94

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.b.a	$270$	$3$	270.b	15.d	$2$	$4$	$4$	$7.357$	4.0.31744.1	None		✓			270.3.b.a	$2$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+2q^{4}+(-3-\beta _{1}-\beta _{2})q^{5}+\cdots\)
270.3.b.b	$270$	$3$	270.b	15.d	$2$	$4$	$4$	$7.357$	\(\Q(\zeta_{8})\)	None		✓			270.3.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{8}+\zeta_{8}^{3})q^{2}+2q^{4}+(3\zeta_{8}-4\zeta_{8}^{3})q^{5}+\cdots\)
270.3.b.c	$270$	$3$	270.b	15.d	$2$	$4$	$4$	$7.357$	\(\Q(\zeta_{8})\)	None		✓			270.3.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{8}-\zeta_{8}^{3})q^{2}+2q^{4}+5\zeta_{8}q^{5}+5\zeta_{8}^{2}q^{7}+\cdots\)
270.3.b.d	$270$	$3$	270.b	15.d	$2$	$4$	$4$	$7.357$	4.0.31744.1	None		✓			270.3.b.a	$2$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+2q^{4}+(3-\beta _{1}+\beta _{2})q^{5}+\cdots\)

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

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