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

• Label: 225.4.f
• Level: $225$
• Weight: $4$
• Character orbit: 225.f
• Rep. character: $\chi_{225}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	204	36	168
Cusp forms	156	36	120
Eisenstein series	48	0	48

Trace form

\( 36 q - 24 q^{7} + O(q^{10}) \)

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.f.a	$225$	$4$	225.f	15.e	$4$	$4$	$2$	$13.275$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-36\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-9+9\zeta_{8}^{2})q^{7}+\cdots\)
225.4.f.b	$225$	$4$	225.f	15.e	$4$	$4$	$2$	$13.275$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(36\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(9-9\zeta_{8}^{2})q^{7}+\cdots\)
225.4.f.c	$225$	$4$	225.f	15.e	$4$	$12$	$6$	$13.275$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$2^{11}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}+(-6\beta _{1}+\beta _{5})q^{4}+(-2-2\beta _{1}+\cdots)q^{7}+\cdots\)
225.4.f.d	$225$	$4$	225.f	15.e	$4$	$16$	$8$	$13.275$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{13}q^{2}+(\beta _{9}+\beta _{15})q^{4}+(7\beta _{3}+2\beta _{14}+\cdots)q^{7}+\cdots\)

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

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