Space of modular forms of level 450, weight 3, and Character 450.i

• Label: 450.3.i
• Level: $450$
• Weight: $3$
• Character orbit: 450.i
• Rep. character: $\chi_{450}(101,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $76$
• Newform subspaces: $7$
• Sturm bound: $270$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.i (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(270\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(450, [\chi])\).

	Total	New	Old
Modular forms	384	76	308
Cusp forms	336	76	260
Eisenstein series	48	0	48

Trace form

\( 76 q + 4 q^{3} + 76 q^{4} + 4 q^{6} + 2 q^{7} - 8 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(450, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.3.i.a	$450$	$3$	450.i	9.d	$6$	$4$	$2$	$12.262$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}-3\beta _{2}q^{3}+2\beta _{2}q^{4}-3\beta _{3}q^{6}+\cdots\)
450.3.i.b	$450$	$3$	450.i	9.d	$6$	$4$	$2$	$12.262$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(1-2\beta _{1}-2\beta _{2}+\beta _{3})q^{3}+\cdots\)
450.3.i.c	$450$	$3$	450.i	9.d	$6$	$4$	$2$	$12.262$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{2}+3\beta _{2}q^{3}+2\beta _{2}q^{4}-3\beta _{3}q^{6}+\cdots\)
450.3.i.d	$450$	$3$	450.i	9.d	$6$	$16$	$8$	$12.262$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)	$3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{2}+\beta _{8})q^{2}+(\beta _{1}-\beta _{7})q^{3}+(2+\cdots)q^{4}+\cdots\)
450.3.i.e	$450$	$3$	450.i	9.d	$6$	$16$	$8$	$12.262$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{9}q^{2}-\beta _{13}q^{3}+(2-2\beta _{5})q^{4}+(-1+\cdots)q^{6}+\cdots\)
450.3.i.f	$450$	$3$	450.i	9.d	$6$	$16$	$8$	$12.262$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(2\)	$3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{2}-\beta _{8})q^{2}+(-\beta _{1}+\beta _{7})q^{3}+(2+\cdots)q^{4}+\cdots\)
450.3.i.g	$450$	$3$	450.i	9.d	$6$	$16$	$8$	$12.262$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$2^{4}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{7}q^{2}+(-\beta _{3}-\beta _{7})q^{3}+2\beta _{10}q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)