Space of modular forms of level 225, weight 6, and Character 225.b

• Label: 225.6.b
• Level: $225$
• Weight: $6$
• Character orbit: 225.b
• Rep. character: $\chi_{225}(199,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $12$
• Sturm bound: $180$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(180\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(225, [\chi])\).

	Total	New	Old
Modular forms	162	38	124
Cusp forms	138	36	102
Eisenstein series	24	2	22

Trace form

36 * q - 522 * q^4 + 1044 * q^11 - 2664 * q^14 + 2442 * q^16 + 1812 * q^19 - 1884 * q^26 - 16476 * q^29 + 6060 * q^31 + 19830 * q^34 + 15492 * q^41 - 96342 * q^44 - 37452 * q^46 + 10656 * q^49 + 121380 * q^56 - 174132 * q^59 - 88020 * q^61 + 248838 * q^64 + 214896 * q^71 - 572124 * q^74 - 183174 * q^76 - 38664 * q^79 - 69876 * q^86 - 95868 * q^89 - 166452 * q^91 + 175008 * q^94

Decomposition of \(S_{6}^{\mathrm{new}}(225, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.6.b.a	$225$	$6$	225.b	5.b	$2$	$2$	$2$	$36.086$	\(\Q(\sqrt{-1}) \)	None					15.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+7 i q^{2}-17 q^{4}-12 i q^{7}+105 i q^{8}+\cdots\)
225.6.b.b	$225$	$6$	225.b	5.b	$2$	$2$	$2$	$36.086$	\(\Q(\sqrt{-1}) \)	None					3.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta q^{2}-4 q^{4}-20\beta q^{7}+84\beta q^{8}+\cdots\)
225.6.b.c	$225$	$6$	225.b	5.b	$2$	$2$	$2$	$36.086$	\(\Q(\sqrt{-1}) \)	None					75.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}+16 q^{4}+225 i q^{7}+192 i q^{8}+\cdots\)
225.6.b.d	$225$	$6$	225.b	5.b	$2$	$2$	$2$	$36.086$	\(\Q(\sqrt{-1}) \)	None					15.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+28 q^{4}-66\beta q^{7}+60\beta q^{8}+\cdots\)
225.6.b.e	$225$	$6$	225.b	5.b	$2$	$2$	$2$	$36.086$	\(\Q(\sqrt{-1}) \)	None					5.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+28 q^{4}-96\beta q^{7}+60\beta q^{8}+\cdots\)
225.6.b.f	$225$	$6$	225.b	5.b	$2$	$2$	$2$	$36.086$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			225.6.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+32 q^{4}-\beta q^{7}+31\beta q^{13}+1024 q^{16}+\cdots\)
225.6.b.g	$225$	$6$	225.b	5.b	$2$	$4$	$4$	$36.086$	\(\Q(i, \sqrt{409})\)	None					15.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-71+\beta _{3})q^{4}+(2^{4}\beta _{1}+2^{5}\beta _{2}+\cdots)q^{7}+\cdots\)
225.6.b.h	$225$	$6$	225.b	5.b	$2$	$4$	$4$	$36.086$	\(\Q(i, \sqrt{70})\)	None		✓			225.6.a.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-38q^{4}+9\beta _{1}q^{7}+6\beta _{2}q^{8}+\cdots\)
225.6.b.i	$225$	$6$	225.b	5.b	$2$	$4$	$4$	$36.086$	\(\Q(i, \sqrt{241})\)	None					25.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{2}+(-35+\beta _{3})q^{4}+\cdots\)
225.6.b.j	$225$	$6$	225.b	5.b	$2$	$4$	$4$	$36.086$	\(\Q(i, \sqrt{145})\)	None					45.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-11+\beta _{3})q^{4}+(19\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
225.6.b.k	$225$	$6$	225.b	5.b	$2$	$4$	$4$	$36.086$	\(\Q(i, \sqrt{145})\)	None					45.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-11+\beta _{3})q^{4}+(-19\beta _{1}+\cdots)q^{7}+\cdots\)
225.6.b.l	$225$	$6$	225.b	5.b	$2$	$4$	$4$	$36.086$	\(\Q(i, \sqrt{31})\)	None					75.6.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3\beta _{1}+\beta _{3})q^{2}+(-8+6\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

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