Space of modular forms of level 2500, weight 2, and trivial character

• Label: 2500.2.a
• Level: $2500$
• Weight: $2$
• Character orbit: 2500.a
• Rep. character: $\chi_{2500}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $7$
• Sturm bound: $750$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2500.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(750\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2500))\).

	Total	New	Old
Modular forms	420	40	380
Cusp forms	331	40	291
Eisenstein series	89	0	89

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

\(2\)	\(5\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(22\)
\(-\)	\(-\)	$+$	\(18\)
Plus space		\(+\)	\(18\)
Minus space		\(-\)	\(22\)

Trace form

\( 40 q + 40 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2500))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
2500.2.a.a	$2500$	$2$	2500.a	1.a	$1$	$2$	$2$	$19.963$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+\beta q^{7}+(-2+\beta )q^{9}+(2+\beta )q^{11}+\cdots\)
2500.2.a.b	$2500$	$2$	2500.a	1.a	$1$	$2$	$2$	$19.963$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+(-2+\beta )q^{9}+(2+\beta )q^{11}+\cdots\)
2500.2.a.c	$2500$	$2$	2500.a	1.a	$1$	$6$	$6$	$19.963$	6.6.103238125.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{3}+\beta _{4})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
2500.2.a.d	$2500$	$2$	2500.a	1.a	$1$	$6$	$6$	$19.963$	6.6.103238125.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{3}-\beta _{4})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
2500.2.a.e	$2500$	$2$	2500.a	1.a	$1$	$8$	$8$	$19.963$	8.8.\(\cdots\).1	None	✓	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-\beta _{2}-\beta _{4}-\beta _{7})q^{7}+\cdots\)
2500.2.a.f	$2500$	$2$	2500.a	1.a	$1$	$8$	$8$	$19.963$	8.8.\(\cdots\).2	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{6})q^{7}+\beta _{2}q^{9}+\cdots\)
2500.2.a.g	$2500$	$2$	2500.a	1.a	$1$	$8$	$8$	$19.963$	8.8.\(\cdots\).1	None	✓	$1$	$0$	\(0\)	\(5\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(\beta _{2}+\beta _{4}+\beta _{7})q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2500))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2500)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1250))\)\(^{\oplus 2}\)