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

• Label: 450.3.g
• Level: $450$
• Weight: $3$
• Character orbit: 450.g
• Rep. character: $\chi_{450}(307,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $30$
• Newform subspaces: $10$
• Sturm bound: $270$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(270\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	408	30	378
Cusp forms	312	30	282
Eisenstein series	96	0	96

Trace form

30 * q + 2 * q^2 - 20 * q^7 - 4 * q^8 - 64 * q^11 - 6 * q^13 - 120 * q^16 + 58 * q^17 + 16 * q^22 - 20 * q^23 + 84 * q^26 + 40 * q^28 + 24 * q^31 - 8 * q^32 + 126 * q^37 - 104 * q^38 - 208 * q^41 - 108 * q^43 - 272 * q^46 + 60 * q^47 - 12 * q^52 + 206 * q^53 + 160 * q^56 + 8 * q^58 + 688 * q^61 - 184 * q^62 - 284 * q^67 - 116 * q^68 + 376 * q^71 + 62 * q^73 - 112 * q^76 - 160 * q^77 + 232 * q^82 + 228 * q^83 - 192 * q^86 - 32 * q^88 - 600 * q^91 - 40 * q^92 + 222 * q^97 - 206 * q^98

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.g.a	$450$	$3$	450.g	5.c	$4$	$2$	$1$	$12.262$	\(\Q(\sqrt{-1}) \)	None					90.3.g.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+2 i q^{4}+(-8 i-8)q^{7}+\cdots\)
450.3.g.b	$450$	$3$	450.g	5.c	$4$	$2$	$1$	$12.262$	\(\Q(\sqrt{-1}) \)	None					10.3.c.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+2 i q^{4}+(-2 i-2)q^{7}+\cdots\)
450.3.g.c	$450$	$3$	450.g	5.c	$4$	$2$	$1$	$12.262$	\(\Q(\sqrt{-1}) \)	None					50.3.c.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+2 i q^{4}+(3 i+3)q^{7}+\cdots\)
450.3.g.d	$450$	$3$	450.g	5.c	$4$	$2$	$1$	$12.262$	\(\Q(\sqrt{-1}) \)	None					90.3.g.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{2}+2 i q^{4}+(-8 i-8)q^{7}+\cdots\)
450.3.g.e	$450$	$3$	450.g	5.c	$4$	$2$	$1$	$12.262$	\(\Q(\sqrt{-1}) \)	None					50.3.c.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{2}+2 i q^{4}+(-3 i-3)q^{7}+\cdots\)
450.3.g.f	$450$	$3$	450.g	5.c	$4$	$4$	$2$	$12.262$	\(\Q(i, \sqrt{6})\)	None		✓			450.3.g.f	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{2})q^{2}+2\beta _{2}q^{4}+\beta _{1}q^{7}+\cdots\)
450.3.g.g	$450$	$3$	450.g	5.c	$4$	$4$	$2$	$12.262$	\(\Q(i, \sqrt{6})\)	None					150.3.f.a	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{2}-2\beta _{2}q^{4}+(6-6\beta _{2}+\cdots)q^{7}+\cdots\)
450.3.g.h	$450$	$3$	450.g	5.c	$4$	$4$	$2$	$12.262$	\(\Q(i, \sqrt{6})\)	None					150.3.f.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{2})q^{2}-2\beta _{2}q^{4}+(-6+6\beta _{2}+\cdots)q^{7}+\cdots\)
450.3.g.i	$450$	$3$	450.g	5.c	$4$	$4$	$2$	$12.262$	\(\Q(i, \sqrt{6})\)	None		✓			450.3.g.f	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{2})q^{2}+2\beta _{2}q^{4}+\beta _{1}q^{7}+(-2+\cdots)q^{8}+\cdots\)
450.3.g.j	$450$	$3$	450.g	5.c	$4$	$4$	$2$	$12.262$	\(\Q(i, \sqrt{6})\)	None					30.3.f.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(16\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{2})q^{2}+2\beta _{2}q^{4}+(4+\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(450, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)