Space of modular forms of level 1000, weight 4, and trivial character

• Label: 1000.4.a
• Level: $1000$
• Weight: $4$
• Character orbit: 1000.a
• Rep. character: $\chi_{1000}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $8$
• Sturm bound: $600$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1000.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(600\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1000))\).

	Total	New	Old
Modular forms	470	72	398
Cusp forms	430	72	358
Eisenstein series	40	0	40

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
\(+\)	\(+\)	\(+\)		\(120\)	\(20\)	\(100\)		\(110\)	\(20\)	\(90\)		\(10\)	\(0\)	\(10\)
\(+\)	\(-\)	\(-\)		\(115\)	\(16\)	\(99\)		\(105\)	\(16\)	\(89\)		\(10\)	\(0\)	\(10\)
\(-\)	\(+\)	\(-\)		\(115\)	\(18\)	\(97\)		\(105\)	\(18\)	\(87\)		\(10\)	\(0\)	\(10\)
\(-\)	\(-\)	\(+\)		\(120\)	\(18\)	\(102\)		\(110\)	\(18\)	\(92\)		\(10\)	\(0\)	\(10\)
Plus space		\(+\)		\(240\)	\(38\)	\(202\)		\(220\)	\(38\)	\(182\)		\(20\)	\(0\)	\(20\)
Minus space		\(-\)		\(230\)	\(34\)	\(196\)		\(210\)	\(34\)	\(176\)		\(20\)	\(0\)	\(20\)

Trace form

72 * q + 634 * q^9 + 14 * q^11 - 34 * q^19 - 188 * q^21 + 42 * q^29 + 108 * q^31 - 264 * q^39 - 428 * q^41 + 3648 * q^49 - 2100 * q^51 - 1378 * q^59 - 18 * q^61 + 812 * q^69 + 1144 * q^71 + 5752 * q^79 + 5128 * q^81 + 616 * q^89 + 2996 * q^91 + 7170 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1000))\) 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
1000.4.a.a	$1000$	$4$	1000.a	1.a	$1$	$8$	$8$	$59.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1000.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(8\)		$+$	$-$	$-$	$2^{6}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{3}+(1-\beta _{3}-\beta _{7})q^{7}+\cdots\)
1000.4.a.b	$1000$	$4$	1000.a	1.a	$1$	$8$	$8$	$59.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1000.4.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-23\)		$-$	$+$	$-$	$2^{6}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-3-\beta _{7})q^{7}+(6+\beta _{2}-\beta _{5}+\cdots)q^{9}+\cdots\)
1000.4.a.c	$1000$	$4$	1000.a	1.a	$1$	$8$	$8$	$59.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1000.4.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(23\)		$+$	$-$	$-$	$2^{6}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(3+\beta _{7})q^{7}+(6+\beta _{2}-\beta _{5}+\cdots)q^{9}+\cdots\)
1000.4.a.d	$1000$	$4$	1000.a	1.a	$1$	$8$	$8$	$59.002$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1000.4.a.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(-8\)		$-$	$-$	$+$	$2^{6}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{3}+(-1+\beta _{3}+\beta _{7})q^{7}+\cdots\)
1000.4.a.e	$1000$	$4$	1000.a	1.a	$1$	$10$	$10$	$59.002$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		1000.4.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-27\)		$-$	$+$	$-$	$2^{10}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-3-\beta _{9})q^{7}+(8-\beta _{3}+\beta _{5}+\cdots)q^{9}+\cdots\)
1000.4.a.f	$1000$	$4$	1000.a	1.a	$1$	$10$	$10$	$59.002$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		1000.4.a.f	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(48\)		$-$	$-$	$+$	$2^{8}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+(4-\beta _{1}+\beta _{3}-\beta _{4})q^{7}+(13+\cdots)q^{9}+\cdots\)
1000.4.a.g	$1000$	$4$	1000.a	1.a	$1$	$10$	$10$	$59.002$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			1000.4.a.f	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-48\)		$+$	$+$	$+$	$2^{8}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+(-4+\beta _{1}-\beta _{3}+\beta _{4})q^{7}+\cdots\)
1000.4.a.h	$1000$	$4$	1000.a	1.a	$1$	$10$	$10$	$59.002$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			1000.4.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(27\)		$+$	$+$	$+$	$2^{10}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(3+\beta _{9})q^{7}+(8-\beta _{3}+\beta _{5}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1000))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1000)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 2}\)