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 - 90 * q^11 + 288 * q^14 + 314 * q^16 - 154 * q^19 + 516 * q^26 + 708 * q^29 + 396 * q^31 + 194 * q^34 - 1326 * q^41 + 402 * q^44 - 324 * q^46 - 1742 * q^49 + 660 * q^56 + 936 * q^59 + 268 * q^61 - 1066 * q^64 - 2064 * q^71 - 852 * q^74 - 286 * q^76 + 4860 * q^79 + 4860 * q^86 - 1206 * q^89 - 1584 * q^91 + 4160 * q^94

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.4.b.a	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None					45.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-17 q^{4}+6\beta q^{7}-9\beta q^{8}+\cdots\)
225.4.b.b	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None					45.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-17 q^{4}-6\beta q^{7}-9\beta q^{8}+\cdots\)
225.4.b.c	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None					5.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{2}-8 q^{4}+3\beta q^{7}-32 q^{11}+\cdots\)
225.4.b.d	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None					15.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}-q^{4}-20 i q^{7}+21 i q^{8}+\cdots\)
225.4.b.e	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None					15.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+7 q^{4}+24 i q^{7}+15 i q^{8}+\cdots\)
225.4.b.f	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	None					25.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+7 q^{4}-6 i q^{7}+15 i q^{8}+\cdots\)
225.4.b.g	$225$	$4$	225.b	5.b	$2$	$2$	$2$	$13.275$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					9.4.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q+8 q^{4}+2\beta q^{7}+7\beta q^{13}+64 q^{16}+\cdots\)
225.4.b.h	$225$	$4$	225.b	5.b	$2$	$4$	$4$	$13.275$	\(\Q(i, \sqrt{19})\)	None					75.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{3})q^{2}+(-12+2\beta _{2})q^{4}+\cdots\)
225.4.b.i	$225$	$4$	225.b	5.b	$2$	$4$	$4$	$13.275$	\(\Q(i, \sqrt{10})\)	None		✓			225.4.a.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-2q^{4}+3\beta _{1}q^{7}-6\beta _{2}q^{8}+\cdots\)

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

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