Space of modular forms of level 7500, weight 2, and Character 7500.d

• Label: 7500.2.d
• Level: $7500$
• Weight: $2$
• Character orbit: 7500.d
• Rep. character: $\chi_{7500}(1249,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $7$
• Sturm bound: $3000$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(3000\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(7500, [\chi])\).

	Total	New	Old
Modular forms	1590	80	1510
Cusp forms	1410	80	1330
Eisenstein series	180	0	180

Trace form

\( 80 q - 80 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(7500, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
7500.2.d.a	$7500$	$2$	7500.d	5.b	$2$	$8$	$8$	$59.888$	8.0.324000000.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(-\beta _{3}-\beta _{7})q^{7}-q^{9}+(-1+\cdots)q^{11}+\cdots\)
7500.2.d.b	$7500$	$2$	7500.d	5.b	$2$	$8$	$8$	$59.888$	8.0.\(\cdots\).17	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+\beta _{1}q^{7}-q^{9}+(-2+\beta _{4}+\cdots)q^{11}+\cdots\)
7500.2.d.c	$7500$	$2$	7500.d	5.b	$2$	$8$	$8$	$59.888$	8.0.\(\cdots\).12	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{1}-\beta _{3}-\beta _{4})q^{7}-q^{9}+\cdots\)
7500.2.d.d	$7500$	$2$	7500.d	5.b	$2$	$8$	$8$	$59.888$	8.0.324000000.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+(\beta _{1}+2\beta _{3}+2\beta _{5}+\beta _{7})q^{7}+\cdots\)
7500.2.d.e	$7500$	$2$	7500.d	5.b	$2$	$8$	$8$	$59.888$	8.0.\(\cdots\).2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{3}+\beta _{5}q^{7}-q^{9}+(1+\beta _{3}-\beta _{6}+\cdots)q^{11}+\cdots\)
7500.2.d.f	$7500$	$2$	7500.d	5.b	$2$	$16$	$16$	$59.888$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{3}+(\beta _{2}+\beta _{7}+\beta _{10})q^{7}-q^{9}+\cdots\)
7500.2.d.g	$7500$	$2$	7500.d	5.b	$2$	$24$	$24$	$59.888$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{2}^{\mathrm{old}}(7500, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(7500, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(500, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(625, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1250, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2500, [\chi])\)\(^{\oplus 2}\)