Space of modular forms of level 225, weight 7, and Character 225.d

• Label: 225.7.d
• Level: $225$
• Weight: $7$
• Character orbit: 225.d
• Rep. character: $\chi_{225}(224,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $3$
• Sturm bound: $210$
• Trace bound: $16$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(210\)
Trace bound:			\(16\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(225, [\chi])\).

	Total	New	Old
Modular forms	192	36	156
Cusp forms	168	36	132
Eisenstein series	24	0	24

Trace form

\( 36 q + 888 q^{4} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(225, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.7.d.a	$225$	$7$	225.d	15.d	$2$	$4$	$4$	$51.762$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{3}q^{2}+98q^{4}-262\zeta_{8}q^{7}+34\zeta_{8}^{3}q^{8}+\cdots\)
225.7.d.b	$225$	$7$	225.d	15.d	$2$	$16$	$16$	$51.762$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{23}\cdot 3^{20}\cdot 5^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(15-\beta _{1})q^{4}+(-2\beta _{3}-\beta _{11}+\cdots)q^{7}+\cdots\)
225.7.d.c	$225$	$7$	225.d	15.d	$2$	$16$	$16$	$51.762$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{21}\cdot 3^{18}\cdot 5^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(2^{4}+\beta _{3})q^{4}+(-2^{4}\beta _{8}-\beta _{12}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(225, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)