Space of modular forms of level 525, weight 2, and trivial character

• Label: 525.2.a
• Level: $525$
• Weight: $2$
• Character orbit: 525.a
• Rep. character: $\chi_{525}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $11$
• Sturm bound: $160$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(160\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(525))\).

	Total	New	Old
Modular forms	92	20	72
Cusp forms	69	20	49
Eisenstein series	23	0	23

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

\(3\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(5\)
Plus space			\(+\)	\(7\)
Minus space			\(-\)	\(13\)

Trace form

\( 20 q + 24 q^{4} + 4 q^{6} - 2 q^{7} + 20 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(525))\) 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	7
525.2.a.a	$525$	$2$	525.a	1.a	$1$	$1$	$1$	$4.192$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}-q^{7}+3q^{8}+\cdots\)
525.2.a.b	$525$	$2$	525.a	1.a	$1$	$1$	$1$	$4.192$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}-q^{7}+3q^{8}+\cdots\)
525.2.a.c	$525$	$2$	525.a	1.a	$1$	$1$	$1$	$4.192$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
525.2.a.d	$525$	$2$	525.a	1.a	$1$	$1$	$1$	$4.192$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
525.2.a.e	$525$	$2$	525.a	1.a	$1$	$2$	$2$	$4.192$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-3\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
525.2.a.f	$525$	$2$	525.a	1.a	$1$	$2$	$2$	$4.192$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots\)
525.2.a.g	$525$	$2$	525.a	1.a	$1$	$2$	$2$	$4.192$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+3q^{4}-\beta q^{6}-q^{7}-\beta q^{8}+\cdots\)
525.2.a.h	$525$	$2$	525.a	1.a	$1$	$2$	$2$	$4.192$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(1\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+\beta q^{6}+q^{7}+\cdots\)
525.2.a.i	$525$	$2$	525.a	1.a	$1$	$2$	$2$	$4.192$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
525.2.a.j	$525$	$2$	525.a	1.a	$1$	$3$	$3$	$4.192$	3.3.148.1	None	✓	$1$	$0$	\(-1\)	\(3\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
525.2.a.k	$525$	$2$	525.a	1.a	$1$	$3$	$3$	$4.192$	3.3.148.1	None	✓	$1$	$0$	\(1\)	\(-3\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(525))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(525)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)