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

• Label: 405.2.j
• Level: $405$
• Weight: $2$
• Character orbit: 405.j
• Rep. character: $\chi_{405}(109,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $44$
• Newform subspaces: $8$
• Sturm bound: $108$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.j (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(108\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	132	52	80
Cusp forms	84	44	40
Eisenstein series	48	8	40

Trace form

44 * q + 24 * q^4 + 4 * q^10 - 20 * q^16 - 8 * q^19 + 8 * q^25 - 8 * q^31 + 26 * q^34 - 4 * q^40 + 44 * q^46 + 2 * q^49 - 96 * q^55 + 16 * q^61 - 100 * q^64 + 24 * q^70 - 18 * q^76 - 68 * q^79 - 14 * q^85 + 24 * q^91 - 136 * 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.j.a	$405$	$2$	405.j	45.j	$6$	$4$	$2$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓			405.2.b.a	$2$	$1$	\(-3\)	\(0\)	\(-6\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{3})q^{2}+(2-\beta _{1}-\beta _{2}+2\beta _{3})q^{4}+\cdots\)
405.2.j.b	$405$	$2$	405.j	45.j	$6$	$4$	$2$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓			405.2.b.a	$2$	$0$	\(-3\)	\(0\)	\(-3\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{3})q^{2}+(2-\beta _{1}-\beta _{2}+2\beta _{3})q^{4}+\cdots\)
405.2.j.c	$405$	$2$	405.j	45.j	$6$	$4$	$2$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	\(\Q(\sqrt{-15}) \)					45.2.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(\beta _{1}-\beta _{3})q^{5}+\beta _{3}q^{8}+\cdots\)
405.2.j.d	$405$	$2$	405.j	45.j	$6$	$4$	$2$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓			405.2.b.a	$2$	$0$	\(3\)	\(0\)	\(3\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
405.2.j.e	$405$	$2$	405.j	45.j	$6$	$4$	$2$	$3.234$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓			405.2.b.a	$2$	$0$	\(3\)	\(0\)	\(6\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
405.2.j.f	$405$	$2$	405.j	45.j	$6$	$8$	$4$	$3.234$	\(\Q(\zeta_{24})\)	None					135.2.b.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_{6} q^{2}-\beta_{4} q^{5}-\beta_1 q^{7}+(-2\beta_{6}+2\beta_{5})q^{8}+\cdots\)
405.2.j.g	$405$	$2$	405.j	45.j	$6$	$8$	$4$	$3.234$	\(\Q(\zeta_{24})\)	None		✓			405.2.b.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_{5} q^{2}+(\beta_{3}+\beta_{2})q^{5}+\beta_{6} q^{7}+\cdots\)
405.2.j.h	$405$	$2$	405.j	45.j	$6$	$8$	$4$	$3.234$	\(\Q(i, \sqrt{3}, \sqrt{5})\)	\(\Q(\sqrt{-15}) \)					135.2.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{U}(1)[D_{6}]$	\(q-\beta _{1}q^{2}+(2-\beta _{3}+2\beta _{5}+\beta _{6})q^{4}+\cdots\)

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

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