Space of modular forms of level 405, weight 3, and Character 405.i

• Label: 405.3.i
• Level: $405$
• Weight: $3$
• Character orbit: 405.i
• Rep. character: $\chi_{405}(26,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $64$
• Newform subspaces: $6$
• Sturm bound: $162$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.i (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(162\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(405, [\chi])\).

	Total	New	Old
Modular forms	240	64	176
Cusp forms	192	64	128
Eisenstein series	48	0	48

Trace form

\( 64 q + 64 q^{4} - 10 q^{7} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(405, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
405.3.i.a	$405$	$3$	405.i	9.d	$6$	$4$	$2$	$11.035$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{3})q^{5}+(-12+\cdots)q^{7}+\cdots\)
405.3.i.b	$405$	$3$	405.i	9.d	$6$	$4$	$2$	$11.035$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{3})q^{5}+(6+\cdots)q^{7}+\cdots\)
405.3.i.c	$405$	$3$	405.i	9.d	$6$	$8$	$4$	$11.035$	8.0.12960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6})q^{4}+(-\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots\)
405.3.i.d	$405$	$3$	405.i	9.d	$6$	$8$	$4$	$11.035$	8.0.3317760000.8	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{3}-\beta _{4})q^{2}+(3-3\beta _{2}+2\beta _{7})q^{4}+\cdots\)
405.3.i.e	$405$	$3$	405.i	9.d	$6$	$8$	$4$	$11.035$	8.0.\(\cdots\).2	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(-5\beta _{2}+\beta _{4}+\beta _{6})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
405.3.i.f	$405$	$3$	405.i	9.d	$6$	$32$	$16$	$11.035$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(405, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(405, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)