Space of modular forms of level 1500, weight 4, and trivial character

• Label: 1500.4.a
• Level: $1500$
• Weight: $4$
• Character orbit: 1500.a
• Rep. character: $\chi_{1500}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $8$
• Sturm bound: $1200$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1500.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(1200\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1500))\).

	Total	New	Old
Modular forms	930	48	882
Cusp forms	870	48	822
Eisenstein series	60	0	60

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

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(12\)
\(-\)	\(+\)	\(-\)	$+$	\(12\)
\(-\)	\(-\)	\(+\)	$+$	\(12\)
\(-\)	\(-\)	\(-\)	$-$	\(12\)
Plus space			\(+\)	\(24\)
Minus space			\(-\)	\(24\)

Trace form

\( 48 q + 432 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1500))\) 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	3	5
1500.4.a.a	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-18\)	\(0\)	\(-28\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4-\beta _{1}+\beta _{2})q^{7}+9q^{9}+\cdots\)
1500.4.a.b	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-18\)	\(0\)	\(-23\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4+\beta _{1}+\beta _{2})q^{7}+9q^{9}+\cdots\)
1500.4.a.c	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-18\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+9q^{9}+\cdots\)
1500.4.a.d	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-18\)	\(0\)	\(12\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+\beta _{1}-\beta _{5})q^{7}+9q^{9}+(-6+\cdots)q^{11}+\cdots\)
1500.4.a.e	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(18\)	\(0\)	\(-12\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-\beta _{1}+\beta _{5})q^{7}+9q^{9}+\cdots\)
1500.4.a.f	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(18\)	\(0\)	\(-7\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
1500.4.a.g	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(18\)	\(0\)	\(23\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(4-\beta _{1}-\beta _{2})q^{7}+9q^{9}+(10+\cdots)q^{11}+\cdots\)
1500.4.a.h	$1500$	$4$	1500.a	1.a	$1$	$6$	$6$	$88.503$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(18\)	\(0\)	\(28\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(4+\beta _{1}-\beta _{2})q^{7}+9q^{9}+(-1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1500))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1500)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(750))\)\(^{\oplus 2}\)