Space of modular forms of level 1350, weight 2, and Character 1350.j

• Label: 1350.2.j
• Level: $1350$
• Weight: $2$
• Character orbit: 1350.j
• Rep. character: $\chi_{1350}(199,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $36$
• Newform subspaces: $7$
• Sturm bound: $540$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.j (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(540\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1350, [\chi])\).

	Total	New	Old
Modular forms	612	36	576
Cusp forms	468	36	432
Eisenstein series	144	0	144

Trace form

\( 36 q + 18 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1350, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1350.2.j.a	$1350$	$2$	1350.j	45.j	$6$	$4$	$2$	$10.780$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+2\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots\)
1350.2.j.b	$1350$	$2$	1350.j	45.j	$6$	$4$	$2$	$10.780$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+2\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots\)
1350.2.j.c	$1350$	$2$	1350.j	45.j	$6$	$4$	$2$	$10.780$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+4\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots\)
1350.2.j.d	$1350$	$2$	1350.j	45.j	$6$	$4$	$2$	$10.780$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+4\zeta_{12}q^{7}-\zeta_{12}^{3}q^{8}+\cdots\)
1350.2.j.e	$1350$	$2$	1350.j	45.j	$6$	$4$	$2$	$10.780$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+\zeta_{12}q^{7}+\zeta_{12}^{3}q^{8}+\cdots\)
1350.2.j.f	$1350$	$2$	1350.j	45.j	$6$	$8$	$4$	$10.780$	8.0.303595776.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}-\beta _{3}q^{4}-\beta _{7}q^{7}+(\beta _{1}+\beta _{5}+\cdots)q^{8}+\cdots\)
1350.2.j.g	$1350$	$2$	1350.j	45.j	$6$	$8$	$4$	$10.780$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{24}+\zeta_{24}^{3})q^{2}+(1-\zeta_{24}^{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)