Space of modular forms of level 5000, weight 2, and trivial character

• Label: 5000.2.a
• Level: $5000$
• Weight: $2$
• Character orbit: 5000.a
• Rep. character: $\chi_{5000}(1,\cdot)$
• Character field: $\Q$
• Dimension: $120$
• Newform subspaces: $18$
• Sturm bound: $1500$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 5000 = 2^{3} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5000.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(1500\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5000))\).

	Total	New	Old
Modular forms	810	120	690
Cusp forms	691	120	571
Eisenstein series	119	0	119

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(195\)	\(26\)	\(169\)		\(166\)	\(26\)	\(140\)		\(29\)	\(0\)	\(29\)
\(+\)	\(-\)	\(-\)		\(210\)	\(34\)	\(176\)		\(180\)	\(34\)	\(146\)		\(30\)	\(0\)	\(30\)
\(-\)	\(+\)	\(-\)		\(210\)	\(32\)	\(178\)		\(180\)	\(32\)	\(148\)		\(30\)	\(0\)	\(30\)
\(-\)	\(-\)	\(+\)		\(195\)	\(28\)	\(167\)		\(165\)	\(28\)	\(137\)		\(30\)	\(0\)	\(30\)
Plus space		\(+\)		\(390\)	\(54\)	\(336\)		\(331\)	\(54\)	\(277\)		\(59\)	\(0\)	\(59\)
Minus space		\(-\)		\(420\)	\(66\)	\(354\)		\(360\)	\(66\)	\(294\)		\(60\)	\(0\)	\(60\)

Trace form

\( 120 q + 120 q^{9} + 10 q^{29} + 10 q^{41} + 120 q^{49} - 30 q^{51} - 30 q^{59} + 10 q^{61} + 60 q^{79} + 120 q^{81} + 10 q^{89} + 60 q^{91} - 30 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5000))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
5000.2.a.a	$5000$	$2$	5000.a	1.a	$1$	$2$	$2$	$39.925$	\(\Q(\sqrt{5}) \)	None	✓				200.2.m.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{3}+(-2+\beta )q^{7}+2q^{9}+\cdots\)
5000.2.a.b	$5000$	$2$	5000.a	1.a	$1$	$2$	$2$	$39.925$	\(\Q(\sqrt{5}) \)	None	✓	✓			5000.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{3}+(-1-\beta )q^{7}+2q^{9}+\cdots\)
5000.2.a.c	$5000$	$2$	5000.a	1.a	$1$	$2$	$2$	$39.925$	\(\Q(\sqrt{5}) \)	None	✓				200.2.m.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{3}+(1+\beta )q^{7}+2q^{9}+(-2+\cdots)q^{11}+\cdots\)
5000.2.a.d	$5000$	$2$	5000.a	1.a	$1$	$2$	$2$	$39.925$	\(\Q(\sqrt{5}) \)	None	✓	✓			5000.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{3}+(2-\beta )q^{7}+2q^{9}+(5+\cdots)q^{11}+\cdots\)
5000.2.a.e	$5000$	$2$	5000.a	1.a	$1$	$4$	$4$	$39.925$	\(\Q(\sqrt{70 +2 \sqrt{5}})\)	None	✓	✓			5000.2.a.e	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{2}q^{7}+(3-\beta _{1}+\beta _{3})q^{9}+\cdots\)
5000.2.a.f	$5000$	$2$	5000.a	1.a	$1$	$4$	$4$	$39.925$	\(\Q(\sqrt{70 +2 \sqrt{5}})\)	None	✓				200.2.m.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{3}q^{7}+(-2-\beta _{2})q^{9}+(2+\cdots)q^{11}+\cdots\)
5000.2.a.g	$5000$	$2$	5000.a	1.a	$1$	$4$	$4$	$39.925$	\(\Q(\sqrt{270 +26 \sqrt{5}})\)	None	✓	✓			5000.2.a.g	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(\beta _{1}-\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
5000.2.a.h	$5000$	$2$	5000.a	1.a	$1$	$4$	$4$	$39.925$	\(\Q(\sqrt{270 +26 \sqrt{5}})\)	None	✓	✓			5000.2.a.g	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-1+\beta _{1})q^{7}+(-1+\cdots)q^{9}+\cdots\)
5000.2.a.i	$5000$	$2$	5000.a	1.a	$1$	$4$	$4$	$39.925$	\(\Q(\sqrt{70 +2 \sqrt{5}})\)	None	✓				200.2.m.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{3}q^{7}+(-2-\beta _{2})q^{9}+(2+\cdots)q^{11}+\cdots\)
5000.2.a.j	$5000$	$2$	5000.a	1.a	$1$	$4$	$4$	$39.925$	\(\Q(\sqrt{70 +2 \sqrt{5}})\)	None	✓	✓			5000.2.a.e	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+\beta _{2}q^{7}+(3-\beta _{1}+\beta _{3})q^{9}+\cdots\)
5000.2.a.k	$5000$	$2$	5000.a	1.a	$1$	$8$	$8$	$39.925$	8.8.3266578125.1	None	✓	✓			5000.2.a.k	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{3})q^{3}+(\beta _{4}-\beta _{5})q^{7}+\cdots\)
5000.2.a.l	$5000$	$2$	5000.a	1.a	$1$	$8$	$8$	$39.925$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				200.2.m.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{2}q^{7}+(\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
5000.2.a.m	$5000$	$2$	5000.a	1.a	$1$	$8$	$8$	$39.925$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				200.2.m.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{2}q^{7}+(\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
5000.2.a.n	$5000$	$2$	5000.a	1.a	$1$	$8$	$8$	$39.925$	8.8.3266578125.1	None	✓	✓			5000.2.a.k	$1$	$1$	\(0\)	\(3\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{3})q^{3}+(-\beta _{4}+\beta _{5})q^{7}+\cdots\)
5000.2.a.o	$5000$	$2$	5000.a	1.a	$1$	$12$	$12$	$39.925$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			5000.2.a.o	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$-$	$5$	$\mathrm{SU}(2)$	\(q-\beta _{6}q^{3}+(1+\beta _{1}-\beta _{2}+\beta _{5})q^{7}+(2+\cdots)q^{9}+\cdots\)
5000.2.a.p	$5000$	$2$	5000.a	1.a	$1$	$12$	$12$	$39.925$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		5000.2.a.o	$1$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$+$	$-$	$5$	$\mathrm{SU}(2)$	\(q+\beta _{6}q^{3}+(-1-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots\)
5000.2.a.q	$5000$	$2$	5000.a	1.a	$1$	$16$	$16$	$39.925$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		200.2.q.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-8\)		$-$	$-$	$+$	$5^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
5000.2.a.r	$5000$	$2$	5000.a	1.a	$1$	$16$	$16$	$39.925$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		200.2.q.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(8\)		$+$	$-$	$-$	$5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5000))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5000)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1000))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1250))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2500))\)\(^{\oplus 2}\)