Space of modular forms of level 8700, weight 2, and Character 8700.g

• Label: 8700.2.g
• Level: $8700$
• Weight: $2$
• Character orbit: 8700.g
• Rep. character: $\chi_{8700}(349,\cdot)$
• Character field: $\Q$
• Dimension: $84$
• Newform subspaces: $24$
• Sturm bound: $3600$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8700.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(3600\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(8700, [\chi])\).

	Total	New	Old
Modular forms	1836	84	1752
Cusp forms	1764	84	1680
Eisenstein series	72	0	72

Trace form

\( 84 q - 84 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(8700, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
8700.2.g.a	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+3iq^{7}-q^{9}-6q^{11}-iq^{13}+\cdots\)
8700.2.g.b	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-q^{9}-6q^{11}-6iq^{13}+4iq^{17}+\cdots\)
8700.2.g.c	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+2iq^{7}-q^{9}-4q^{11}-2iq^{13}+\cdots\)
8700.2.g.d	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}-3q^{11}-iq^{13}+\cdots\)
8700.2.g.e	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+3iq^{7}-q^{9}-3q^{11}-3iq^{13}+\cdots\)
8700.2.g.f	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+3iq^{7}-q^{9}-3q^{11}+iq^{13}+\cdots\)
8700.2.g.g	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}-2q^{11}+5iq^{13}+\cdots\)
8700.2.g.h	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+3iq^{7}-q^{9}-q^{11}-3iq^{13}+\cdots\)
8700.2.g.i	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+iq^{7}-q^{9}+q^{11}+3iq^{13}+\cdots\)
8700.2.g.j	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}+3q^{11}-5iq^{13}+\cdots\)
8700.2.g.k	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+2iq^{7}-q^{9}+3q^{11}-2iq^{13}+\cdots\)
8700.2.g.l	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+3iq^{7}-q^{9}+3q^{11}-3iq^{13}+\cdots\)
8700.2.g.m	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+2iq^{7}-q^{9}+4q^{11}-2iq^{13}+\cdots\)
8700.2.g.n	$8700$	$2$	8700.g	5.b	$2$	$2$	$2$	$69.470$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+5iq^{7}-q^{9}+5q^{11}-5iq^{13}+\cdots\)
8700.2.g.o	$8700$	$2$	8700.g	5.b	$2$	$4$	$4$	$69.470$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+2\beta _{1}q^{7}-q^{9}+(-1-\beta _{3})q^{11}+\cdots\)
8700.2.g.p	$8700$	$2$	8700.g	5.b	$2$	$4$	$4$	$69.470$	\(\Q(i, \sqrt{57})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-2\beta _{2}q^{7}-q^{9}+(-2+\beta _{3})q^{11}+\cdots\)
8700.2.g.q	$8700$	$2$	8700.g	5.b	$2$	$4$	$4$	$69.470$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(\beta _{1}+3\beta _{2})q^{7}-q^{9}-\beta _{3}q^{11}+\cdots\)
8700.2.g.r	$8700$	$2$	8700.g	5.b	$2$	$4$	$4$	$69.470$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{7}-q^{9}+(1-2\beta _{3})q^{11}+\cdots\)
8700.2.g.s	$8700$	$2$	8700.g	5.b	$2$	$4$	$4$	$69.470$	\(\Q(i, \sqrt{33})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(1+\beta _{3})q^{11}+\cdots\)
8700.2.g.t	$8700$	$2$	8700.g	5.b	$2$	$4$	$4$	$69.470$	\(\Q(i, \sqrt{33})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(3-\beta _{3})q^{11}+\cdots\)
8700.2.g.u	$8700$	$2$	8700.g	5.b	$2$	$6$	$6$	$69.470$	6.0.4227136.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(\beta _{1}-\beta _{4})q^{7}-q^{9}+\beta _{2}q^{11}+\cdots\)
8700.2.g.v	$8700$	$2$	8700.g	5.b	$2$	$6$	$6$	$69.470$	6.0.6594624.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(2\beta _{1}-\beta _{4}-\beta _{5})q^{7}-q^{9}+\cdots\)
8700.2.g.w	$8700$	$2$	8700.g	5.b	$2$	$10$	$10$	$69.470$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{3}-\beta _{9}q^{7}-q^{9}+(-2+\beta _{7}+\cdots)q^{11}+\cdots\)
8700.2.g.x	$8700$	$2$	8700.g	5.b	$2$	$10$	$10$	$69.470$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+\beta _{7}q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(8700, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(8700, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(145, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(290, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(580, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(725, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1450, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2900, [\chi])\)\(^{\oplus 2}\)