Space of modular forms of level 900, weight 2, and Character 900.s

• Label: 900.2.s
• Level: $900$
• Weight: $2$
• Character orbit: 900.s
• Rep. character: $\chi_{900}(49,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $36$
• Newform subspaces: $4$
• Sturm bound: $360$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.s (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(360\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(900, [\chi])\).

	Total	New	Old
Modular forms	396	36	360
Cusp forms	324	36	288
Eisenstein series	72	0	72

Trace form

\( 36 q - 14 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(900, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.2.s.a	$900$	$2$	900.s	45.j	$6$	$4$	$2$	$7.187$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}q^{7}+(-3+\cdots)q^{9}+\cdots\)
900.2.s.b	$900$	$2$	900.s	45.j	$6$	$4$	$2$	$7.187$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}q^{7}+3q^{9}+\cdots\)
900.2.s.c	$900$	$2$	900.s	45.j	$6$	$12$	$6$	$7.187$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{5}q^{3}+(-\beta _{1}-\beta _{2}+\beta _{4})q^{7}+(\beta _{3}+\cdots)q^{9}+\cdots\)
900.2.s.d	$900$	$2$	900.s	45.j	$6$	$16$	$8$	$7.187$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{3}-\beta _{15})q^{3}+(\beta _{3}+\beta _{8}-\beta _{9}-\beta _{15})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(900, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)