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

• Label: 405.2.b
• Level: $405$
• Weight: $2$
• Character orbit: 405.b
• Rep. character: $\chi_{405}(244,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $5$
• Sturm bound: $108$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(108\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	66	28	38
Cusp forms	42	20	22
Eisenstein series	24	8	16

Trace form

20 * q - 12 * q^4 - 2 * q^10 + 4 * q^16 - 8 * q^19 + 14 * q^25 - 8 * q^31 - 28 * q^34 - 10 * q^40 + 8 * q^46 - 28 * q^49 + 12 * q^55 + 52 * q^61 - 4 * q^64 - 84 * q^70 + 120 * q^76 + 16 * q^79 - 26 * q^85 - 48 * q^91 + 80 * q^94

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
405.2.b.a	$405$	$2$	405.b	5.b	$2$	$4$	$4$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓			405.2.b.a	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(-2+\beta _{1}+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
405.2.b.b	$405$	$2$	405.b	5.b	$2$	$4$	$4$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓			405.2.b.a	$2$	$0$	\(0\)	\(0\)	\(3\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(-2+\beta _{1}+\beta _{3})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
405.2.b.c	$405$	$2$	405.b	5.b	$2$	$4$	$4$	$3.234$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None					45.2.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{2}-\beta _{3})q^{5}+\cdots\)
405.2.b.d	$405$	$2$	405.b	5.b	$2$	$4$	$4$	$3.234$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None					45.2.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
405.2.b.e	$405$	$2$	405.b	5.b	$2$	$4$	$4$	$3.234$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		✓			405.2.b.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{5}+\beta _{3}q^{7}-2\beta _{1}q^{8}+\cdots\)

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

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