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

• Label: 405.3.l
• Level: $405$
• Weight: $3$
• Character orbit: 405.l
• Rep. character: $\chi_{405}(28,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $184$
• Newform subspaces: $15$
• Sturm bound: $162$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.l (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(162\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	480	200	280
Cusp forms	384	184	200
Eisenstein series	96	16	80

Trace form

\( 184 q + 4 q^{7} + O(q^{10}) \)

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.l.a	$405$	$3$	405.l	45.k	$12$	$4$	$1$	$11.035$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-4\)	\(0\)	\(20\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1+\zeta_{12})q^{2}+(1+2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
405.3.l.b	$405$	$3$	405.l	45.k	$12$	$4$	$1$	$11.035$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-2\)	\(0\)	\(10\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1+\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}+\cdots)q^{4}+\cdots\)
405.3.l.c	$405$	$3$	405.l	45.k	$12$	$4$	$1$	$11.035$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(2\)	\(0\)	\(-10\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}+\cdots)q^{4}+\cdots\)
405.3.l.d	$405$	$3$	405.l	45.k	$12$	$4$	$1$	$11.035$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(4\)	\(0\)	\(-20\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12})q^{2}+(1+2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
405.3.l.e	$405$	$3$	405.l	45.k	$12$	$8$	$2$	$11.035$	8.0.49787136.1	None		$4$	$0$	\(-6\)	\(0\)	\(-6\)	\(26\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1-\beta _{4}+\beta _{5})q^{2}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
405.3.l.f	$405$	$3$	405.l	45.k	$12$	$8$	$2$	$11.035$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(-4\)	\(0\)	\(-4\)	\(-4\)	$3^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\zeta_{24}-\zeta_{24}^{2}+\zeta_{24}^{4}-\zeta_{24}^{7})q^{2}+\cdots\)
405.3.l.g	$405$	$3$	405.l	45.k	$12$	$8$	$2$	$11.035$	8.0.3317760000.2	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{7}q^{2}-\beta _{2}q^{4}+(2\beta _{3}+\beta _{5}-2\beta _{7})q^{5}+\cdots\)
405.3.l.h	$405$	$3$	405.l	45.k	$12$	$8$	$2$	$11.035$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(4\)	\(0\)	\(4\)	\(-4\)	$3^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{24}+\zeta_{24}^{2}-\zeta_{24}^{4}+\zeta_{24}^{7})q^{2}+\cdots\)
405.3.l.i	$405$	$3$	405.l	45.k	$12$	$8$	$2$	$11.035$	8.0.49787136.1	None		$4$	$0$	\(6\)	\(0\)	\(6\)	\(26\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\beta _{4}-\beta _{5})q^{2}+(-\beta _{1}-\beta _{2}-\beta _{5}+\cdots)q^{4}+\cdots\)
405.3.l.j	$405$	$3$	405.l	45.k	$12$	$16$	$4$	$11.035$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(-6\)	\(0\)	\(12\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{2}q^{2}+(-\beta _{1}+\beta _{4}-\beta _{7}-2\beta _{8}+\cdots)q^{4}+\cdots\)
405.3.l.k	$405$	$3$	405.l	45.k	$12$	$16$	$4$	$11.035$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(-2\)	\(0\)	\(-2\)	\(-26\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{1}q^{2}+(\beta _{3}+3\beta _{5}-\beta _{6}+\beta _{12})q^{4}+\cdots\)
405.3.l.l	$405$	$3$	405.l	45.k	$12$	$16$	$4$	$11.035$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(2\)	\(0\)	\(2\)	\(-26\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{1}q^{2}+(\beta _{3}+3\beta _{5}-\beta _{6}+\beta _{12})q^{4}+\cdots\)
405.3.l.m	$405$	$3$	405.l	45.k	$12$	$16$	$4$	$11.035$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(6\)	\(0\)	\(-12\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{1}-\beta _{2})q^{2}+(\beta _{1}-\beta _{3}-\beta _{5}+\beta _{7}+\cdots)q^{4}+\cdots\)
405.3.l.n	$405$	$3$	405.l	45.k	$12$	$32$	$8$	$11.035$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)		$\mathrm{SU}(2)[C_{12}]$
405.3.l.o	$405$	$3$	405.l	45.k	$12$	$32$	$8$	$11.035$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$\mathrm{SU}(2)[C_{12}]$

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

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