Space of modular forms of level 450, weight 5, and Character 450.b

• Label: 450.5.b
• Level: $450$
• Weight: $5$
• Character orbit: 450.b
• Rep. character: $\chi_{450}(449,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $4$
• Sturm bound: $450$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	450.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(450\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(450, [\chi])\).

	Total	New	Old
Modular forms	384	24	360
Cusp forms	336	24	312
Eisenstein series	48	0	48

Trace form

\( 24 q + 192 q^{4} + 1536 q^{16} - 744 q^{19} + 456 q^{31} - 4608 q^{34} - 2304 q^{46} - 15312 q^{49} + 35880 q^{61} + 12288 q^{64} - 5952 q^{76} - 7584 q^{79} + 4152 q^{91} + 34560 q^{94}+O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.5.b.a	$450$	$5$	450.b	15.d	$2$	$4$	$4$	$46.516$	\(\Q(\zeta_{8})\)	None		✓			450.5.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta_{3} q^{2}+8 q^{4}+13\beta_1 q^{7}+16\beta_{3} q^{8}+\cdots\)
450.5.b.b	$450$	$5$	450.b	15.d	$2$	$4$	$4$	$46.516$	\(\Q(\zeta_{8})\)	None		✓			450.5.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta_{3} q^{2}+8 q^{4}-83\beta_1 q^{7}-16\beta_{3} q^{8}+\cdots\)
450.5.b.c	$450$	$5$	450.b	15.d	$2$	$8$	$8$	$46.516$	\(\Q(i, \sqrt{2}, \sqrt{5})\)	None					90.5.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+8q^{4}+(2^{4}\beta _{1}+\beta _{6})q^{7}-8\beta _{3}q^{8}+\cdots\)
450.5.b.d	$450$	$5$	450.b	15.d	$2$	$8$	$8$	$46.516$	\(\Q(i, \sqrt{2}, \sqrt{5})\)	None					90.5.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+8q^{4}+(4\beta _{1}+\beta _{6})q^{7}+8\beta _{3}q^{8}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(450, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)