Space of modular forms of level 225, weight 3, and Character 225.j

• Label: 225.3.j
• Level: $225$
• Weight: $3$
• Character orbit: 225.j
• Rep. character: $\chi_{225}(101,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $70$
• Newform subspaces: $5$
• Sturm bound: $90$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.j (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(90\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(225, [\chi])\).

	Total	New	Old
Modular forms	132	82	50
Cusp forms	108	70	38
Eisenstein series	24	12	12

Trace form

\( 70 q + 3 q^{2} - q^{3} + 65 q^{4} - 3 q^{6} - 3 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(225, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.3.j.a	$225$	$3$	225.j	9.d	$6$	$2$	$1$	$6.131$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(3\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
225.3.j.b	$225$	$3$	225.j	9.d	$6$	$16$	$8$	$6.131$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-2\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(-\beta _{8}-\beta _{11})q^{3}+(2+\cdots)q^{4}+\cdots\)
225.3.j.c	$225$	$3$	225.j	9.d	$6$	$16$	$8$	$6.131$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-1\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{2})q^{2}+\beta _{3}q^{3}+(-\beta _{3}-2\beta _{4}+\cdots)q^{4}+\cdots\)
225.3.j.d	$225$	$3$	225.j	9.d	$6$	$16$	$8$	$6.131$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(1\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{2})q^{2}-\beta _{3}q^{3}+(-\beta _{3}-2\beta _{4}+\cdots)q^{4}+\cdots\)
225.3.j.e	$225$	$3$	225.j	9.d	$6$	$20$	$10$	$6.131$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{13}q^{2}+(\beta _{9}+\beta _{12})q^{3}+(2+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)