Space of modular forms of level 1125, weight 2, and trivial character

• Label: 1125.2.a
• Level: $1125$
• Weight: $2$
• Character orbit: 1125.a
• Rep. character: $\chi_{1125}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $13$
• Sturm bound: $300$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1125 = 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1125.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(300\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1125))\).

	Total	New	Old
Modular forms	170	40	130
Cusp forms	131	40	91
Eisenstein series	39	0	39

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
\(+\)	\(+\)	$+$	\(8\)
\(+\)	\(-\)	$-$	\(8\)
\(-\)	\(+\)	$-$	\(14\)
\(-\)	\(-\)	$+$	\(10\)
Plus space		\(+\)	\(18\)
Minus space		\(-\)	\(22\)

Trace form

\( 40 q + 42 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1125))\) 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
1125.2.a.a	$1125$	$2$	1125.a	1.a	$1$	$2$	$2$	$8.983$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}+(3-\beta )q^{7}+\cdots\)
1125.2.a.b	$1125$	$2$	1125.a	1.a	$1$	$2$	$2$	$8.983$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+\beta q^{7}+(-1+\cdots)q^{8}+\cdots\)
1125.2.a.c	$1125$	$2$	1125.a	1.a	$1$	$2$	$2$	$8.983$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+3q^{7}+(-1+\cdots)q^{8}+\cdots\)
1125.2.a.d	$1125$	$2$	1125.a	1.a	$1$	$2$	$2$	$8.983$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}-3q^{7}+(1-2\beta )q^{8}+\cdots\)
1125.2.a.e	$1125$	$2$	1125.a	1.a	$1$	$2$	$2$	$8.983$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}-\beta q^{7}+(1-2\beta )q^{8}+\cdots\)
1125.2.a.f	$1125$	$2$	1125.a	1.a	$1$	$2$	$2$	$8.983$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+(-3+\beta )q^{7}+\cdots\)
1125.2.a.g	$1125$	$2$	1125.a	1.a	$1$	$4$	$4$	$8.983$	\(\Q(\zeta_{15})^+\)	\(\Q(\sqrt{-15}) \)	✓	$2$	$1$	\(-5\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-1+\beta _{2}+\beta _{3})q^{2}+(1+\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots\)
1125.2.a.h	$1125$	$2$	1125.a	1.a	$1$	$4$	$4$	$8.983$	4.4.2525.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)
1125.2.a.i	$1125$	$2$	1125.a	1.a	$1$	$4$	$4$	$8.983$	\(\Q(\zeta_{20})^+\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}-2\beta _{3})q^{7}+\cdots\)
1125.2.a.j	$1125$	$2$	1125.a	1.a	$1$	$4$	$4$	$8.983$	\(\Q(\zeta_{20})^+\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}+2\beta _{3})q^{7}+\cdots\)
1125.2.a.k	$1125$	$2$	1125.a	1.a	$1$	$4$	$4$	$8.983$	4.4.4400.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(1-2\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{7}+\cdots\)
1125.2.a.l	$1125$	$2$	1125.a	1.a	$1$	$4$	$4$	$8.983$	4.4.2525.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2}-\beta _{3})q^{2}+(2-\beta _{1}-\beta _{3})q^{4}+\cdots\)
1125.2.a.m	$1125$	$2$	1125.a	1.a	$1$	$4$	$4$	$8.983$	\(\Q(\zeta_{15})^+\)	\(\Q(\sqrt{-15}) \)	✓	$2$	$0$	\(5\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(1-\beta _{2}-\beta _{3})q^{2}+(1+\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1125))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1125)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 2}\)