Space of modular forms of level 225, weight 6, and trivial character

• Label: 225.6.a
• Level: $225$
• Weight: $6$
• Character orbit: 225.a
• Rep. character: $\chi_{225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $22$
• Sturm bound: $180$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(180\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(225))\).

	Total	New	Old
Modular forms	162	41	121
Cusp forms	138	38	100
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.
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(11\)
\(-\)	\(-\)	\(+\)	\(11\)
Plus space		\(+\)	\(18\)
Minus space		\(-\)	\(20\)

Trace form

\( 38 q + 578 q^{4} - 80 q^{7} - 180 q^{8} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\) 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
225.6.a.a	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	None	✓	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(40\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}+4q^{4}+40q^{7}+168q^{8}+564q^{11}+\cdots\)
225.6.a.b	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-225\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-2^{4}q^{4}-15^{2}q^{7}+192q^{8}+\cdots\)
225.6.a.c	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(132\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-28q^{4}+132q^{7}+120q^{8}+\cdots\)
225.6.a.d	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-25\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}-5^{2}q^{7}-775q^{13}+2^{10}q^{16}+\cdots\)
225.6.a.e	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(25\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}+5^{2}q^{7}+775q^{13}+2^{10}q^{16}+\cdots\)
225.6.a.f	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-192\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}-192q^{7}-120q^{8}+\cdots\)
225.6.a.g	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(0\)	\(225\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-2^{4}q^{4}+15^{2}q^{7}-192q^{8}+\cdots\)
225.6.a.h	$225$	$6$	225.a	1.a	$1$	$1$	$1$	$36.086$	\(\Q\)	None	✓	$1$	$0$	\(7\)	\(0\)	\(0\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{2}+17q^{4}-12q^{7}-105q^{8}+\cdots\)
225.6.a.i	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{89}) \)	None	✓	$1$	$1$	\(-9\)	\(0\)	\(0\)	\(108\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{2}+(6+9\beta )q^{4}+(42+24\beta )q^{7}+\cdots\)
225.6.a.j	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{31}) \)	None	✓	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(102\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta )q^{2}+(8-6\beta )q^{4}+(51-24\beta )q^{7}+\cdots\)
225.6.a.k	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{145}) \)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(0\)	\(-80\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(8+5\beta )q^{4}+(-30+\cdots)q^{7}+\cdots\)
225.6.a.l	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(0\)	\(200\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(2^{5}+5\beta )q^{4}+(102+\cdots)q^{7}+\cdots\)
225.6.a.m	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(112\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(70+\beta )q^{4}+(2^{6}-2^{4}\beta )q^{7}+\cdots\)
225.6.a.n	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{11}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+12q^{4}+9\beta q^{7}-20\beta q^{8}+\cdots\)
225.6.a.o	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{70}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-90\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+38q^{4}-45q^{7}+6\beta q^{8}-40\beta q^{11}+\cdots\)
225.6.a.p	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{70}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(90\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+38q^{4}+45q^{7}+6\beta q^{8}+40\beta q^{11}+\cdots\)
225.6.a.q	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2\cdot 5$	$N(\mathrm{U}(1))$	\(q-\beta q^{2}+93q^{4}-61\beta q^{8}+4649q^{16}+\cdots\)
225.6.a.r	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{145}) \)	None	✓	$1$	$1$	\(5\)	\(0\)	\(0\)	\(-80\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(13-5\beta )q^{4}+(-50+20\beta )q^{7}+\cdots\)
225.6.a.s	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-200\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(37-5\beta )q^{4}+(-98-4\beta )q^{7}+\cdots\)
225.6.a.t	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{31}) \)	None	✓	$1$	$1$	\(6\)	\(0\)	\(0\)	\(-102\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(8+6\beta )q^{4}+(-51-24\beta )q^{7}+\cdots\)
225.6.a.u	$225$	$6$	225.a	1.a	$1$	$2$	$2$	$36.086$	\(\Q(\sqrt{89}) \)	None	✓	$1$	$1$	\(9\)	\(0\)	\(0\)	\(-108\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{2}+(15-9\beta )q^{4}+(-66+24\beta )q^{7}+\cdots\)
225.6.a.v	$225$	$6$	225.a	1.a	$1$	$4$	$4$	$36.086$	\(\Q(\sqrt{5}, \sqrt{14})\)	None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-12q^{4}-\beta _{2}q^{7}+44\beta _{1}q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

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