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

• Label: 450.3.d
• Level: $450$
• Weight: $3$
• Character orbit: 450.d
• Rep. character: $\chi_{450}(251,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $7$
• Sturm bound: $270$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(270\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	204	14	190
Cusp forms	156	14	142
Eisenstein series	48	0	48

Trace form

14 * q - 28 * q^4 + 8 * q^7 - 16 * q^13 + 56 * q^16 + 56 * q^19 + 48 * q^22 - 16 * q^28 + 96 * q^31 - 68 * q^34 + 68 * q^37 + 80 * q^43 + 16 * q^46 - 102 * q^49 + 32 * q^52 + 12 * q^58 + 100 * q^61 - 112 * q^64 - 16 * q^67 + 32 * q^73 - 112 * q^76 + 536 * q^79 + 132 * q^82 - 96 * q^88 - 728 * q^91 - 400 * q^94 - 352 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.3.d.a	$450$	$3$	450.d	3.b	$2$	$2$	$2$	$12.262$	\(\Q(\sqrt{-2}) \)	None		✓			450.3.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-22\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-2q^{4}-11q^{7}-2\beta q^{8}+3\beta q^{11}+\cdots\)
450.3.d.b	$450$	$3$	450.d	3.b	$2$	$2$	$2$	$12.262$	\(\Q(\sqrt{-2}) \)	None					90.3.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{2}-2q^{4}-4q^{7}+2\beta q^{8}+8\beta q^{11}+\cdots\)
450.3.d.c	$450$	$3$	450.d	3.b	$2$	$2$	$2$	$12.262$	\(\Q(\sqrt{-2}) \)	None		✓			450.3.d.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-2q^{4}-q^{7}-2\beta q^{8}+3\beta q^{11}+\cdots\)
450.3.d.d	$450$	$3$	450.d	3.b	$2$	$2$	$2$	$12.262$	\(\Q(\sqrt{-2}) \)	None		✓			450.3.d.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-2q^{4}+q^{7}-2\beta q^{8}-3\beta q^{11}+\cdots\)
450.3.d.e	$450$	$3$	450.d	3.b	$2$	$2$	$2$	$12.262$	\(\Q(\sqrt{-2}) \)	None					90.3.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{2}-2q^{4}+4q^{7}+2\beta q^{8}-8\beta q^{11}+\cdots\)
450.3.d.f	$450$	$3$	450.d	3.b	$2$	$2$	$2$	$12.262$	\(\Q(\sqrt{-2}) \)	None					18.3.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{2}-2q^{4}+4q^{7}+2\beta q^{8}+12\beta q^{11}+\cdots\)
450.3.d.g	$450$	$3$	450.d	3.b	$2$	$2$	$2$	$12.262$	\(\Q(\sqrt{-2}) \)	None		✓			450.3.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{2}-2q^{4}+11q^{7}+2\beta q^{8}+3\beta q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(450, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)