Space of modular forms of level 405, weight 2, and Character 405.e

• Label: 405.2.e
• Level: $405$
• Weight: $2$
• Character orbit: 405.e
• Rep. character: $\chi_{405}(136,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $32$
• Newform subspaces: $12$
• Sturm bound: $108$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(108\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(405, [\chi])\).

	Total	New	Old
Modular forms	132	32	100
Cusp forms	84	32	52
Eisenstein series	48	0	48

Trace form

\( 32 q - 16 q^{4} + 10 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(405, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
405.2.e.a	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(-2\)	\(0\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-2\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots\)
405.2.e.b	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(-1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-2\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots\)
405.2.e.c	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(0\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+\zeta_{6}q^{5}-3q^{8}+\cdots\)
405.2.e.d	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+\cdots\)
405.2.e.e	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+\cdots\)
405.2.e.f	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(0\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}-\zeta_{6}q^{5}+3q^{8}+\cdots\)
405.2.e.g	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(2\)	\(0\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+\zeta_{6}q^{5}+2q^{10}+\cdots\)
405.2.e.h	$405$	$2$	405.e	9.c	$3$	$2$	$1$	$3.234$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(2\)	\(0\)	\(1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(3+\cdots)q^{7}+\cdots\)
405.2.e.i	$405$	$2$	405.e	9.c	$3$	$4$	$2$	$3.234$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-2\)	\(0\)	\(2\)	\(6\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots\)
405.2.e.j	$405$	$2$	405.e	9.c	$3$	$4$	$2$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		$2$	$0$	\(-1\)	\(0\)	\(2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
405.2.e.k	$405$	$2$	405.e	9.c	$3$	$4$	$2$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		$2$	$0$	\(1\)	\(0\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
405.2.e.l	$405$	$2$	405.e	9.c	$3$	$4$	$2$	$3.234$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(2\)	\(0\)	\(-2\)	\(6\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-2+2\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(405, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(405, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)