Space of modular forms of level 450, weight 2, and Character 450.h

• Label: 450.2.h
• Level: $450$
• Weight: $2$
• Character orbit: 450.h
• Rep. character: $\chi_{450}(91,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $52$
• Newform subspaces: $7$
• Sturm bound: $180$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.h (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(180\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

	Total	New	Old
Modular forms	392	52	340
Cusp forms	328	52	276
Eisenstein series	64	0	64

Trace form

\( 52 q - q^{2} - 13 q^{4} - 11 q^{5} - q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.2.h.a	$450$	$2$	450.h	25.d	$5$	$4$	$1$	$3.593$	\(\Q(\zeta_{10})\)	None		$2$	$1$	\(-1\)	\(0\)	\(-5\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
450.2.h.b	$450$	$2$	450.h	25.d	$5$	$4$	$1$	$3.593$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(-1\)	\(0\)	\(-5\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
450.2.h.c	$450$	$2$	450.h	25.d	$5$	$4$	$1$	$3.593$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(1\)	\(0\)	\(-5\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{3}q^{4}+\cdots\)
450.2.h.d	$450$	$2$	450.h	25.d	$5$	$8$	$2$	$3.593$	8.0.1064390625.3	None		$2$	$0$	\(-2\)	\(0\)	\(4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{3}q^{2}+(-1-\beta _{2}+\beta _{3}-\beta _{5})q^{4}+\cdots\)
450.2.h.e	$450$	$2$	450.h	25.d	$5$	$8$	$2$	$3.593$	8.0.58140625.2	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{6}q^{2}-\beta _{3}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
450.2.h.f	$450$	$2$	450.h	25.d	$5$	$12$	$3$	$3.593$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-3\)	\(0\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{7}q^{2}+(-1+\beta _{4}-\beta _{6}+\beta _{7})q^{4}+\cdots\)
450.2.h.g	$450$	$2$	450.h	25.d	$5$	$12$	$3$	$3.593$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(3\)	\(0\)	\(-1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{7}q^{2}+(-1+\beta _{4}-\beta _{6}+\beta _{7})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)