Space of modular forms of level 2000, weight 2, and trivial character

• Label: 2000.2.a
• Level: $2000$
• Weight: $2$
• Character orbit: 2000.a
• Rep. character: $\chi_{2000}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $18$
• Sturm bound: $600$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2000.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(600\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2000))\).

	Total	New	Old
Modular forms	330	48	282
Cusp forms	271	48	223
Eisenstein series	59	0	59

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
\(+\)	\(+\)	$+$	\(10\)
\(+\)	\(-\)	$-$	\(14\)
\(-\)	\(+\)	$-$	\(14\)
\(-\)	\(-\)	$+$	\(10\)
Plus space		\(+\)	\(20\)
Minus space		\(-\)	\(28\)

Trace form

\( 48 q + 48 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2000))\) 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
2000.2.a.a	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-3q^{7}+(-1+3\beta )q^{9}+\cdots\)
2000.2.a.b	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-2+3\beta )q^{7}+(-1+\cdots)q^{9}+\cdots\)
2000.2.a.c	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2\beta q^{3}+\beta q^{7}+(1+4\beta )q^{9}+(-5+\cdots)q^{11}+\cdots\)
2000.2.a.d	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-3+2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
2000.2.a.e	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
2000.2.a.f	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
2000.2.a.g	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1-2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
2000.2.a.h	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{7}+(-2+\beta )q^{9}-2\beta q^{11}+\cdots\)
2000.2.a.i	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(3-2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
2000.2.a.j	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}-\beta q^{7}+(1+4\beta )q^{9}+(-5+\cdots)q^{11}+\cdots\)
2000.2.a.k	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(2-3\beta )q^{7}+(-1+3\beta )q^{9}+\cdots\)
2000.2.a.l	$2000$	$2$	2000.a	1.a	$1$	$2$	$2$	$15.970$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+3q^{7}+(-1+3\beta )q^{9}+\cdots\)
2000.2.a.m	$2000$	$2$	2000.a	1.a	$1$	$4$	$4$	$15.970$	4.4.4400.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-2-\beta _{2}-\beta _{3})q^{7}+\cdots\)
2000.2.a.n	$2000$	$2$	2000.a	1.a	$1$	$4$	$4$	$15.970$	4.4.10025.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-9\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2+\beta _{2}+\beta _{3})q^{7}+\cdots\)
2000.2.a.o	$2000$	$2$	2000.a	1.a	$1$	$4$	$4$	$15.970$	4.4.4400.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
2000.2.a.p	$2000$	$2$	2000.a	1.a	$1$	$4$	$4$	$15.970$	4.4.12400.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(2-3\beta _{2}+\cdots)q^{9}+\cdots\)
2000.2.a.q	$2000$	$2$	2000.a	1.a	$1$	$4$	$4$	$15.970$	4.4.10025.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{3})q^{7}+(2-\beta _{2}+\beta _{3})q^{9}+\cdots\)
2000.2.a.r	$2000$	$2$	2000.a	1.a	$1$	$4$	$4$	$15.970$	4.4.4400.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(2+\beta _{2}-\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2000))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2000)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1000))\)\(^{\oplus 2}\)