Space of modular forms of level 225, weight 8, and Character 225.b

• Label: 225.8.b
• Level: $225$
• Weight: $8$
• Character orbit: 225.b
• Rep. character: $\chi_{225}(199,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $17$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(225, [\chi])\).

	Total	New	Old
Modular forms	222	54	168
Cusp forms	198	52	146
Eisenstein series	24	2	22

Trace form

52 * q - 3274 * q^4 - 4340 * q^11 - 11880 * q^14 + 250378 * q^16 + 49948 * q^19 - 724220 * q^26 + 588340 * q^29 + 346076 * q^31 + 625230 * q^34 - 899380 * q^41 + 1617850 * q^44 - 384780 * q^46 - 6700584 * q^49 + 60180 * q^56 + 695180 * q^59 + 4966004 * q^61 - 22828186 * q^64 + 181280 * q^71 + 30513220 * q^74 - 29113366 * q^76 - 12854888 * q^79 - 20406500 * q^86 + 47332860 * q^89 + 44344060 * q^91 - 10526400 * q^94

Decomposition of \(S_{8}^{\mathrm{new}}(225, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.8.b.a	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					15.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+11\beta q^{2}-356 q^{4}-210\beta q^{7}+\cdots\)
225.8.b.b	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					5.8.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+7\beta q^{2}-68 q^{4}-822\beta q^{7}+420\beta q^{8}+\cdots\)
225.8.b.c	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					15.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+13 i q^{2}-41 q^{4}+1380 i q^{7}+\cdots\)
225.8.b.d	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					45.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+28 q^{4}+117\beta q^{7}+156\beta q^{8}+\cdots\)
225.8.b.e	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					45.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+28 q^{4}-117\beta q^{7}+156\beta q^{8}+\cdots\)
225.8.b.f	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					3.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta q^{2}+92 q^{4}+32\beta q^{7}+660\beta q^{8}+\cdots\)
225.8.b.g	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					45.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+103 q^{4}-186\beta q^{7}+231\beta q^{8}+\cdots\)
225.8.b.h	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	None					45.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+103 q^{4}+186\beta q^{7}+231\beta q^{8}+\cdots\)
225.8.b.i	$225$	$8$	225.b	5.b	$2$	$2$	$2$	$70.287$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			225.8.a.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{U}(1)[D_{2}]$	\(q+128 q^{4}-251\beta q^{7}-2521\beta q^{13}+\cdots\)
225.8.b.j	$225$	$8$	225.b	5.b	$2$	$4$	$4$	$70.287$	\(\Q(i, \sqrt{115})\)	None		✓			225.8.a.r	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-332q^{4}+297\beta _{1}q^{7}-204\beta _{3}q^{8}+\cdots\)
225.8.b.k	$225$	$8$	225.b	5.b	$2$	$4$	$4$	$70.287$	\(\Q(i, \sqrt{10})\)	None					9.8.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-232q^{4}+26\beta _{1}q^{7}-104\beta _{2}q^{8}+\cdots\)
225.8.b.l	$225$	$8$	225.b	5.b	$2$	$4$	$4$	$70.287$	\(\Q(i, \sqrt{649})\)	None					25.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+2\beta _{2})q^{2}+(-92+3\beta _{3})q^{4}+\cdots\)
225.8.b.m	$225$	$8$	225.b	5.b	$2$	$4$	$4$	$70.287$	\(\Q(i, \sqrt{19})\)	None					5.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{3})q^{2}+(-48-2\beta _{2})q^{4}+(5\beta _{1}+\cdots)q^{7}+\cdots\)
225.8.b.n	$225$	$8$	225.b	5.b	$2$	$4$	$4$	$70.287$	\(\Q(i, \sqrt{601})\)	None					15.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-2\beta _{2})q^{2}+(-38+7\beta _{3})q^{4}+\cdots\)
225.8.b.o	$225$	$8$	225.b	5.b	$2$	$4$	$4$	$70.287$	\(\Q(i, \sqrt{31})\)	None					75.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4\beta _{1}-\beta _{3})q^{2}+(-12-8\beta _{2})q^{4}+\cdots\)
225.8.b.p	$225$	$8$	225.b	5.b	$2$	$4$	$4$	$70.287$	\(\Q(i, \sqrt{130})\)	None		✓			225.8.a.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-2q^{4}-33\beta _{1}q^{7}-126\beta _{2}q^{8}+\cdots\)
225.8.b.q	$225$	$8$	225.b	5.b	$2$	$6$	$6$	$70.287$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None			✓		75.8.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{2}+2\beta _{4})q^{2}+(-51-\beta _{1}+4\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(225, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)