Space of modular forms of level 450, weight 6, and Character 450.f

• Label: 450.6.f
• Level: $450$
• Weight: $6$
• Character orbit: 450.f
• Rep. character: $\chi_{450}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $60$
• Newform subspaces: $7$
• Sturm bound: $540$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(540\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(450, [\chi])\).

	Total	New	Old
Modular forms	948	60	888
Cusp forms	852	60	792
Eisenstein series	96	0	96

Trace form

\( 60 q + 152 q^{7} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(450, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.6.f.a	$450$	$6$	450.f	15.e	$4$	$4$	$2$	$72.173$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-132\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{2}+2^{4}\zeta_{8}q^{4}+\cdots\)
450.6.f.b	$450$	$6$	450.f	15.e	$4$	$4$	$2$	$72.173$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(88\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{2}-2^{4}\zeta_{8}q^{4}+(22+\cdots)q^{7}+\cdots\)
450.6.f.c	$450$	$6$	450.f	15.e	$4$	$4$	$2$	$72.173$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(132\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\zeta_{8}^{2}+2\zeta_{8}^{3})q^{2}-2^{4}\zeta_{8}q^{4}+\cdots\)
450.6.f.d	$450$	$6$	450.f	15.e	$4$	$4$	$2$	$72.173$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(208\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{2}+2^{4}\zeta_{8}q^{4}+\cdots\)
450.6.f.e	$450$	$6$	450.f	15.e	$4$	$12$	$6$	$72.173$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-144\)	$2^{12}\cdot 3^{4}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2\beta _{3}+2\beta _{4})q^{2}+2^{4}\beta _{2}q^{4}+(-12+\cdots)q^{7}+\cdots\)
450.6.f.f	$450$	$6$	450.f	15.e	$4$	$16$	$8$	$72.173$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-528\)	$2^{20}\cdot 3^{8}\cdot 5^{16}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\beta _{1}+2\beta _{5})q^{2}-2^{4}\beta _{4}q^{4}+(-33+\cdots)q^{7}+\cdots\)
450.6.f.g	$450$	$6$	450.f	15.e	$4$	$16$	$8$	$72.173$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(528\)	$2^{20}\cdot 3^{8}\cdot 5^{16}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2\beta _{1}+2\beta _{5})q^{2}+2^{4}\beta _{4}q^{4}+(33-33\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)