Space of modular forms of level 225, weight 12, and trivial character

• Label: 225.12.a
• Level: $225$
• Weight: $12$
• Character orbit: 225.a
• Rep. character: $\chi_{225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $85$
• Newform subspaces: $26$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(225))\).

	Total	New	Old
Modular forms	342	88	254
Cusp forms	318	85	233
Eisenstein series	24	3	21

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(17\)
\(+\)	\(-\)	$-$	\(17\)
\(-\)	\(+\)	$-$	\(24\)
\(-\)	\(-\)	$+$	\(27\)
Plus space		\(+\)	\(44\)
Minus space		\(-\)	\(41\)

Trace form

\( 85 q - 24 q^{2} + 80450 q^{4} + 1824 q^{7} - 70500 q^{8} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(225))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
225.12.a.a	$225$	$12$	225.a	1.a	$1$	$1$	$1$	$172.877$	\(\Q\)	None	✓	$1$	$1$	\(-56\)	\(0\)	\(0\)	\(-27984\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-56q^{2}+1088q^{4}-27984q^{7}+53760q^{8}+\cdots\)
225.12.a.b	$225$	$12$	225.a	1.a	$1$	$1$	$1$	$172.877$	\(\Q\)	None	✓	$1$	$1$	\(-24\)	\(0\)	\(0\)	\(16744\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-24q^{2}-1472q^{4}+16744q^{7}+84480q^{8}+\cdots\)
225.12.a.c	$225$	$12$	225.a	1.a	$1$	$1$	$1$	$172.877$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-76885\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{11}q^{4}-76885q^{7}-2248615q^{13}+\cdots\)
225.12.a.d	$225$	$12$	225.a	1.a	$1$	$1$	$1$	$172.877$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(76885\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{11}q^{4}+76885q^{7}+2248615q^{13}+\cdots\)
225.12.a.e	$225$	$12$	225.a	1.a	$1$	$1$	$1$	$172.877$	\(\Q\)	None	✓	$1$	$1$	\(34\)	\(0\)	\(0\)	\(17556\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+34q^{2}-892q^{4}+17556q^{7}-99960q^{8}+\cdots\)
225.12.a.f	$225$	$12$	225.a	1.a	$1$	$1$	$1$	$172.877$	\(\Q\)	None	✓	$1$	$1$	\(78\)	\(0\)	\(0\)	\(27760\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+78q^{2}+4036q^{4}+27760q^{7}+155064q^{8}+\cdots\)
225.12.a.g	$225$	$12$	225.a	1.a	$1$	$2$	$2$	$172.877$	\(\Q(\sqrt{1609}) \)	None	✓	$1$	$1$	\(-22\)	\(0\)	\(0\)	\(10864\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-11-\beta )q^{2}+(-318+22\beta )q^{4}+\cdots\)
225.12.a.h	$225$	$12$	225.a	1.a	$1$	$2$	$2$	$172.877$	\(\Q(\sqrt{151}) \)	None	✓	$1$	$1$	\(-20\)	\(0\)	\(0\)	\(-57900\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-10+\beta )q^{2}+(3488-20\beta )q^{4}+\cdots\)
225.12.a.i	$225$	$12$	225.a	1.a	$1$	$2$	$2$	$172.877$	\(\Q(\sqrt{1801}) \)	None	✓	$1$	$1$	\(-13\)	\(0\)	\(0\)	\(-7784\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-6-\beta )q^{2}+(-1562+13\beta )q^{4}+\cdots\)
225.12.a.j	$225$	$12$	225.a	1.a	$1$	$2$	$2$	$172.877$	\(\Q(\sqrt{70}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-116200\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+472q^{4}-58100q^{7}-1576\beta q^{8}+\cdots\)
225.12.a.k	$225$	$12$	225.a	1.a	$1$	$2$	$2$	$172.877$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q-23\beta q^{2}+597q^{4}+33373\beta q^{8}+\cdots\)
225.12.a.l	$225$	$12$	225.a	1.a	$1$	$3$	$3$	$172.877$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(14608\)		$-$	$+$	$-$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1585+3\beta _{1}+\beta _{2})q^{4}+(4708+\cdots)q^{7}+\cdots\)
225.12.a.m	$225$	$12$	225.a	1.a	$1$	$3$	$3$	$172.877$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-53129\)		$-$	$+$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(867+8\beta _{1}+\beta _{2})q^{4}+(-17713+\cdots)q^{7}+\cdots\)
225.12.a.n	$225$	$12$	225.a	1.a	$1$	$3$	$3$	$172.877$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(53129\)		$-$	$-$	$+$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(867+8\beta _{1}+\beta _{2})q^{4}+(17713+\cdots)q^{7}+\cdots\)
225.12.a.o	$225$	$12$	225.a	1.a	$1$	$4$	$4$	$172.877$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-75\)	\(0\)	\(0\)	\(62080\)		$+$	$+$	$+$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-19+\beta _{1})q^{2}+(1807-15\beta _{1}-\beta _{3})q^{4}+\cdots\)
225.12.a.p	$225$	$12$	225.a	1.a	$1$	$4$	$4$	$172.877$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-55\)	\(0\)	\(0\)	\(96400\)		$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-14+\beta _{1})q^{2}+(1393-14\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
225.12.a.q	$225$	$12$	225.a	1.a	$1$	$4$	$4$	$172.877$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-46\)	\(0\)	\(0\)	\(68372\)		$-$	$-$	$+$	$2^{3}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{2}+(1145+35\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
225.12.a.r	$225$	$12$	225.a	1.a	$1$	$4$	$4$	$172.877$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{5}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(18-\beta _{3})q^{4}+(-156\beta _{1}+\cdots)q^{7}+\cdots\)
225.12.a.s	$225$	$12$	225.a	1.a	$1$	$4$	$4$	$172.877$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(46\)	\(0\)	\(0\)	\(-68372\)		$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{2}+(1145+35\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
225.12.a.t	$225$	$12$	225.a	1.a	$1$	$4$	$4$	$172.877$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(55\)	\(0\)	\(0\)	\(-96400\)		$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(14-\beta _{1})q^{2}+(1393-14\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
225.12.a.u	$225$	$12$	225.a	1.a	$1$	$4$	$4$	$172.877$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(75\)	\(0\)	\(0\)	\(62080\)		$+$	$+$	$+$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(19-\beta _{1})q^{2}+(1807-15\beta _{1}-\beta _{3})q^{4}+\cdots\)
225.12.a.v	$225$	$12$	225.a	1.a	$1$	$6$	$6$	$172.877$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-9\)	\(0\)	\(0\)	\(64368\)		$-$	$-$	$+$	$2^{6}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(1359+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
225.12.a.w	$225$	$12$	225.a	1.a	$1$	$6$	$6$	$172.877$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-63930\)		$+$	$+$	$+$	$2^{7}\cdot 3^{8}\cdot 5^{5}\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(522+\beta _{2})q^{4}+(-10655+\cdots)q^{7}+\cdots\)
225.12.a.x	$225$	$12$	225.a	1.a	$1$	$6$	$6$	$172.877$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(63930\)		$+$	$-$	$-$	$2^{7}\cdot 3^{8}\cdot 5^{5}\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(522+\beta _{2})q^{4}+(10655-7\beta _{2}+\cdots)q^{7}+\cdots\)
225.12.a.y	$225$	$12$	225.a	1.a	$1$	$6$	$6$	$172.877$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(9\)	\(0\)	\(0\)	\(-64368\)		$-$	$-$	$+$	$2^{6}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(1359+\beta _{1}+\beta _{2})q^{4}+\cdots\)
225.12.a.z	$225$	$12$	225.a	1.a	$1$	$8$	$8$	$172.877$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{24}\cdot 3^{12}\cdot 5^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1122+\beta _{3})q^{4}+\beta _{4}q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(225)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 9}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)