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

• Label: 450.5.g
• Level: $450$
• Weight: $5$
• Character orbit: 450.g
• Rep. character: $\chi_{450}(307,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $60$
• Newform subspaces: $16$
• Sturm bound: $450$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(450\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	768	60	708
Cusp forms	672	60	612
Eisenstein series	96	0	96

Trace form

60 * q + 140 * q^7 + 912 * q^11 + 300 * q^13 - 3840 * q^16 - 780 * q^17 + 1280 * q^22 + 1860 * q^23 - 192 * q^26 - 1120 * q^28 - 3672 * q^31 + 300 * q^37 + 4800 * q^38 + 19344 * q^41 - 5460 * q^43 + 2816 * q^46 - 16620 * q^47 + 2400 * q^52 - 780 * q^53 - 2240 * q^58 - 25384 * q^61 - 1920 * q^62 + 9740 * q^67 + 6240 * q^68 + 4872 * q^71 + 3820 * q^73 + 3136 * q^76 - 9480 * q^77 - 21760 * q^82 - 11340 * q^83 - 26304 * q^86 - 10240 * q^88 + 34680 * q^91 + 14880 * q^92 + 50700 * q^97 + 42240 * q^98

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.g.a	$450$	$5$	450.g	5.c	$4$	$2$	$1$	$46.516$	\(\Q(\sqrt{-1}) \)	None					10.5.c.a	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(-58\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2 i-2)q^{2}+8 i q^{4}+(-29 i-29)q^{7}+\cdots\)
450.5.g.b	$450$	$5$	450.g	5.c	$4$	$2$	$1$	$46.516$	\(\Q(\sqrt{-1}) \)	None					10.5.c.b	$2$	$0$	\(4\)	\(0\)	\(0\)	\(38\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2 i+2)q^{2}+8 i q^{4}+(19 i+19)q^{7}+\cdots\)
450.5.g.c	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					150.5.f.c	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(-192\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\beta _{2})q^{2}-8\beta _{2}q^{4}+(-48+\cdots)q^{7}+\cdots\)
450.5.g.d	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{19})\)	None		✓			450.5.g.d	$4$	$0$	\(-8\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-2\beta _{1})q^{2}+8\beta _{1}q^{4}-\beta _{2}q^{7}+\cdots\)
450.5.g.e	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None		✓			450.5.g.e	$4$	$0$	\(-8\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-2\beta _{2})q^{2}+8\beta _{2}q^{4}+7\beta _{1}q^{7}+\cdots\)
450.5.g.f	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					30.5.f.a	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(28\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-2\beta _{1})q^{2}+8\beta _{1}q^{4}+(7+7\beta _{1}+\cdots)q^{7}+\cdots\)
450.5.g.g	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					150.5.f.b	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(48\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-2\beta _{2})q^{2}+8\beta _{2}q^{4}+(12-\beta _{1}+\cdots)q^{7}+\cdots\)
450.5.g.h	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{26})\)	None					90.5.g.d	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(100\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-2\beta _{1})q^{2}+8\beta _{1}q^{4}+(5^{2}+5^{2}\beta _{1}+\cdots)q^{7}+\cdots\)
450.5.g.i	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					50.5.c.c	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(144\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\beta _{2})q^{2}-8\beta _{2}q^{4}+(6^{2}-6^{2}\beta _{2}+\cdots)q^{7}+\cdots\)
450.5.g.j	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					50.5.c.c	$2$	$0$	\(8\)	\(0\)	\(0\)	\(-144\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2\beta _{2})q^{2}-8\beta _{2}q^{4}+(-6^{2}+6^{2}\beta _{2}+\cdots)q^{7}+\cdots\)
450.5.g.k	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					30.5.f.b	$2$	$0$	\(8\)	\(0\)	\(0\)	\(-68\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2\beta _{1})q^{2}-8\beta _{1}q^{4}+(-17+17\beta _{1}+\cdots)q^{7}+\cdots\)
450.5.g.l	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					150.5.f.b	$2$	$0$	\(8\)	\(0\)	\(0\)	\(-48\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2+2\beta _{2})q^{2}+8\beta _{2}q^{4}+(-12+\beta _{1}+\cdots)q^{7}+\cdots\)
450.5.g.m	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None		✓			450.5.g.e	$4$	$0$	\(8\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2+2\beta _{2})q^{2}+8\beta _{2}q^{4}+7\beta _{1}q^{7}+\cdots\)
450.5.g.n	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{19})\)	None		✓			450.5.g.d	$4$	$0$	\(8\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2+2\beta _{1})q^{2}+8\beta _{1}q^{4}-\beta _{2}q^{7}+(-2^{4}+\cdots)q^{8}+\cdots\)
450.5.g.o	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{26})\)	None					90.5.g.d	$2$	$0$	\(8\)	\(0\)	\(0\)	\(100\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2+2\beta _{1})q^{2}+8\beta _{1}q^{4}+(5^{2}+5^{2}\beta _{1}+\cdots)q^{7}+\cdots\)
450.5.g.p	$450$	$5$	450.g	5.c	$4$	$4$	$2$	$46.516$	\(\Q(i, \sqrt{6})\)	None					150.5.f.c	$2$	$0$	\(8\)	\(0\)	\(0\)	\(192\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2\beta _{2})q^{2}-8\beta _{2}q^{4}+(48-48\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{5}^{\mathrm{old}}(450, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)