Space of modular forms of level 450, weight 8, and Character 450.c

• Label: 450.8.c
• Level: $450$
• Weight: $8$
• Character orbit: 450.c
• Rep. character: $\chi_{450}(199,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $22$
• Sturm bound: $720$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(720\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(450, [\chi])\).

	Total	New	Old
Modular forms	654	52	602
Cusp forms	606	52	554
Eisenstein series	48	0	48

Trace form

52 * q - 3328 * q^4 + 3726 * q^11 + 15904 * q^14 + 212992 * q^16 + 1510 * q^19 + 289952 * q^26 - 442620 * q^29 + 261764 * q^31 - 238704 * q^34 + 1576206 * q^41 - 238464 * q^44 + 1726368 * q^46 - 4722636 * q^49 - 1017856 * q^56 - 1669200 * q^59 + 541184 * q^61 - 13631488 * q^64 + 8157576 * q^71 - 5564096 * q^74 - 96640 * q^76 + 8385460 * q^79 + 21978752 * q^86 - 29206770 * q^89 - 59804096 * q^91 + 21391296 * q^94

Decomposition of \(S_{8}^{\mathrm{new}}(450, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.8.c.a	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					6.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{2}-64 q^{4}+788\beta q^{7}-256\beta q^{8}+\cdots\)
450.8.c.b	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					30.8.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}-64 q^{4}+542\beta q^{7}+256\beta q^{8}+\cdots\)
450.8.c.c	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					30.8.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}-64 q^{4}+256\beta q^{7}+256\beta q^{8}+\cdots\)
450.8.c.d	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					150.8.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-8 i q^{2}-64 q^{4}+1427 i q^{7}+512 i q^{8}+\cdots\)
450.8.c.e	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					150.8.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8 i q^{2}-64 q^{4}+713 i q^{7}-512 i q^{8}+\cdots\)
450.8.c.f	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					150.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-8 i q^{2}-64 q^{4}+349 i q^{7}+512 i q^{8}+\cdots\)
450.8.c.g	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					2.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{2}-64 q^{4}+508\beta q^{7}-256\beta q^{8}+\cdots\)
450.8.c.h	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					90.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{2}-64 q^{4}+143\beta q^{7}-256\beta q^{8}+\cdots\)
450.8.c.i	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					50.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-8 i q^{2}-64 q^{4}+974 i q^{7}+512 i q^{8}+\cdots\)
450.8.c.j	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					30.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}-64 q^{4}+116\beta q^{7}+256\beta q^{8}+\cdots\)
450.8.c.k	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					90.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}-64 q^{4}+143\beta q^{7}+256\beta q^{8}+\cdots\)
450.8.c.l	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					50.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8 i q^{2}-64 q^{4}+1366 i q^{7}-512 i q^{8}+\cdots\)
450.8.c.m	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					30.8.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}-64 q^{4}+802\beta q^{7}+256\beta q^{8}+\cdots\)
450.8.c.n	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					30.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{2}-64 q^{4}+208\beta q^{7}-256\beta q^{8}+\cdots\)
450.8.c.o	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					150.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8 i q^{2}-64 q^{4}+391 i q^{7}-512 i q^{8}+\cdots\)
450.8.c.p	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					10.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}-64 q^{4}+52\beta q^{7}+256\beta q^{8}+\cdots\)
450.8.c.q	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					50.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-8 i q^{2}-64 q^{4}+1174 i q^{7}+512 i q^{8}+\cdots\)
450.8.c.r	$450$	$8$	450.c	5.b	$2$	$2$	$2$	$140.573$	\(\Q(\sqrt{-1}) \)	None					30.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{2}-64 q^{4}+494\beta q^{7}-256\beta q^{8}+\cdots\)
450.8.c.s	$450$	$8$	450.c	5.b	$2$	$4$	$4$	$140.573$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					90.8.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}-2^{6}q^{4}+(10^{2}\beta _{1}+\beta _{2})q^{7}+\cdots\)
450.8.c.t	$450$	$8$	450.c	5.b	$2$	$4$	$4$	$140.573$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None		✓			450.8.a.bb	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+8\beta _{1}q^{2}-2^{6}q^{4}+(-350\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
450.8.c.u	$450$	$8$	450.c	5.b	$2$	$4$	$4$	$140.573$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None		✓			450.8.a.bb	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-8\beta _{1}q^{2}-2^{6}q^{4}+(-350\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
450.8.c.v	$450$	$8$	450.c	5.b	$2$	$4$	$4$	$140.573$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					90.8.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}-2^{6}q^{4}+(10^{2}\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

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