Space of modular forms of level 225, weight 12, and Character 225.b

• Label: 225.12.b
• Level: $225$
• Weight: $12$
• Character orbit: 225.b
• Rep. character: $\chi_{225}(199,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $16$
• Sturm bound: $360$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(360\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(225, [\chi])\).

	Total	New	Old
Modular forms	342	84	258
Cusp forms	318	82	236
Eisenstein series	24	2	22

Trace form

\( 82 q - 80010 q^{4} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(225, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
225.12.b.a	$225$	$12$	225.b	5.b	$2$	$2$	$2$	$172.877$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+39iq^{2}-4036q^{4}+13880iq^{7}+\cdots\)
225.12.b.b	$225$	$12$	225.b	5.b	$2$	$2$	$2$	$172.877$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+28iq^{2}-1088q^{4}+13992iq^{7}+\cdots\)
225.12.b.c	$225$	$12$	225.b	5.b	$2$	$2$	$2$	$172.877$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+17iq^{2}+892q^{4}+8778iq^{7}+49980iq^{8}+\cdots\)
225.12.b.d	$225$	$12$	225.b	5.b	$2$	$2$	$2$	$172.877$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+12iq^{2}+1472q^{4}-8372iq^{7}+\cdots\)
225.12.b.e	$225$	$12$	225.b	5.b	$2$	$2$	$2$	$172.877$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{U}(1)[D_{2}]$	\(q+2^{11}q^{4}+15377iq^{7}-449723iq^{13}+\cdots\)
225.12.b.f	$225$	$12$	225.b	5.b	$2$	$4$	$4$	$172.877$	\(\Q(i, \sqrt{151})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(-3488+2\beta _{2})q^{4}+\cdots\)
225.12.b.g	$225$	$12$	225.b	5.b	$2$	$4$	$4$	$172.877$	\(\Q(i, \sqrt{70})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-472q^{4}+1162\beta _{1}q^{7}+1576\beta _{2}q^{8}+\cdots\)
225.12.b.h	$225$	$12$	225.b	5.b	$2$	$4$	$4$	$172.877$	\(\Q(i, \sqrt{1609})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-5\beta _{2})q^{2}+(318+11\beta _{3})q^{4}+\cdots\)
225.12.b.i	$225$	$12$	225.b	5.b	$2$	$4$	$4$	$172.877$	\(\Q(i, \sqrt{1801})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-3\beta _{2})q^{2}+(1549+13\beta _{3})q^{4}+\cdots\)
225.12.b.j	$225$	$12$	225.b	5.b	$2$	$6$	$6$	$172.877$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1586-2\beta _{3}+\beta _{4})q^{4}+\cdots\)
225.12.b.k	$225$	$12$	225.b	5.b	$2$	$6$	$6$	$172.877$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-867-8\beta _{3}+\beta _{4})q^{4}+\cdots\)
225.12.b.l	$225$	$12$	225.b	5.b	$2$	$8$	$8$	$172.877$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{8}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-1803-\beta _{3})q^{4}+(725\beta _{1}+\cdots)q^{7}+\cdots\)
225.12.b.m	$225$	$12$	225.b	5.b	$2$	$8$	$8$	$172.877$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{8}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-1803-\beta _{3})q^{4}+(-725\beta _{1}+\cdots)q^{7}+\cdots\)
225.12.b.n	$225$	$12$	225.b	5.b	$2$	$8$	$8$	$172.877$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 5^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}+(-1389-\beta _{1})q^{4}+(-34\beta _{4}+\cdots)q^{7}+\cdots\)
225.12.b.o	$225$	$12$	225.b	5.b	$2$	$8$	$8$	$172.877$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-12\beta _{1}-\beta _{2})q^{2}+(-1180-35\beta _{3}+\cdots)q^{4}+\cdots\)
225.12.b.p	$225$	$12$	225.b	5.b	$2$	$12$	$12$	$172.877$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{16}\cdot 5^{22}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-522-\beta _{2})q^{4}+(17\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{12}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)