Space of modular forms of level 225, weight 4, and Character 225.b

• Label: 225.4.b
• Level: $225$
• Weight: $4$
• Character orbit: 225.b
• Rep. character: $\chi_{225}(199,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $9$
• Sturm bound: $120$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(120\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(225, [\chi])\).

	Total	New	Old
Modular forms	102	24	78
Cusp forms	78	22	56
Eisenstein series	24	2	22

Trace form

\( 22 q - 98 q^{4} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.4.b.a	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-17q^{4}+6iq^{7}-9iq^{8}-50q^{11}+\cdots\)
225.4.b.b	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-17q^{4}-6iq^{7}-9iq^{8}+50q^{11}+\cdots\)
225.4.b.c	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{2}-8q^{4}+3iq^{7}-2^{5}q^{11}+\cdots\)
225.4.b.d	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{2}-q^{4}-20iq^{7}+21iq^{8}+\cdots\)
225.4.b.e	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}+7q^{4}+24iq^{7}+15iq^{8}+\cdots\)
225.4.b.f	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}+7q^{4}-6iq^{7}+15iq^{8}+43q^{11}+\cdots\)
225.4.b.g	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q+8q^{4}+2iq^{7}+7iq^{13}+2^{6}q^{16}+\cdots\)
225.4.b.h	$225$	$4$	225.b	5.b	$2$	$4$	$4$	$13.275$	\(\Q(i, \sqrt{19})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{3})q^{2}+(-12+2\beta _{2})q^{4}+\cdots\)
225.4.b.i	$225$	$4$	225.b	5.b	$2$	$4$	$4$	$13.275$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-2q^{4}+3\beta _{1}q^{7}-6\beta _{2}q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)