Space of modular forms of level 5850, weight 2, and Character 5850.e

• Label: 5850.2.e
• Level: $5850$
• Weight: $2$
• Character orbit: 5850.e
• Rep. character: $\chi_{5850}(5149,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $39$
• Sturm bound: $2520$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 39 \)
Sturm bound:			\(2520\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(19\), \(29\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(5850, [\chi])\).

	Total	New	Old
Modular forms	1308	90	1218
Cusp forms	1212	90	1122
Eisenstein series	96	0	96

Trace form

\( 90 q - 90 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(5850, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
5850.2.e.a	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+iq^{7}+iq^{8}-6q^{11}+\cdots\)
5850.2.e.b	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-6q^{11}+\cdots\)
5850.2.e.c	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+4iq^{7}-iq^{8}-4q^{11}+\cdots\)
5850.2.e.d	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-4q^{11}+\cdots\)
5850.2.e.e	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+iq^{8}-4q^{11}-iq^{13}+\cdots\)
5850.2.e.f	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}-iq^{8}-4q^{11}+iq^{13}+\cdots\)
5850.2.e.g	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-4q^{11}+\cdots\)
5850.2.e.h	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-4q^{11}+\cdots\)
5850.2.e.i	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-4q^{11}+\cdots\)
5850.2.e.j	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+iq^{7}+iq^{8}-3q^{11}+\cdots\)
5850.2.e.k	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-2q^{11}+\cdots\)
5850.2.e.l	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+4iq^{7}-iq^{8}-q^{11}+\cdots\)
5850.2.e.m	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+iq^{13}+\cdots\)
5850.2.e.n	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+iq^{13}+\cdots\)
5850.2.e.o	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-iq^{13}+\cdots\)
5850.2.e.p	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}-iq^{8}+iq^{13}+q^{16}+\cdots\)
5850.2.e.q	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+iq^{8}-iq^{13}+q^{16}+\cdots\)
5850.2.e.r	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-iq^{13}+\cdots\)
5850.2.e.s	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+4iq^{7}+iq^{8}+iq^{13}+\cdots\)
5850.2.e.t	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+4iq^{7}+iq^{8}+iq^{13}+\cdots\)
5850.2.e.u	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+2q^{11}+\cdots\)
5850.2.e.v	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+iq^{7}+iq^{8}+2q^{11}+\cdots\)
5850.2.e.w	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}+2q^{11}+\cdots\)
5850.2.e.x	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+iq^{7}-iq^{8}+3q^{11}+\cdots\)
5850.2.e.y	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}-iq^{8}+3q^{11}+iq^{13}+\cdots\)
5850.2.e.z	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+3iq^{7}+iq^{8}+3q^{11}+\cdots\)
5850.2.e.ba	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+5iq^{7}+iq^{8}+3q^{11}+\cdots\)
5850.2.e.bb	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+4q^{11}+\cdots\)
5850.2.e.bc	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+4q^{11}+\cdots\)
5850.2.e.bd	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}+4q^{11}+\cdots\)
5850.2.e.be	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+iq^{7}-iq^{8}+5q^{11}+\cdots\)
5850.2.e.bf	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+6q^{11}+\cdots\)
5850.2.e.bg	$5850$	$2$	5850.e	5.b	$2$	$2$	$2$	$46.712$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+4iq^{7}+iq^{8}+6q^{11}+\cdots\)
5850.2.e.bh	$5850$	$2$	5850.e	5.b	$2$	$4$	$4$	$46.712$	\(\Q(i, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{4}+(-2\beta _{1}-\beta _{3})q^{7}-\beta _{1}q^{8}+\cdots\)
5850.2.e.bi	$5850$	$2$	5850.e	5.b	$2$	$4$	$4$	$46.712$	\(\Q(i, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-q^{4}+(\beta _{1}-\beta _{2})q^{7}-\beta _{2}q^{8}+\cdots\)
5850.2.e.bj	$5850$	$2$	5850.e	5.b	$2$	$4$	$4$	$46.712$	\(\Q(i, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-q^{4}+(-\beta _{1}+\beta _{2})q^{7}+\beta _{1}q^{8}+\cdots\)
5850.2.e.bk	$5850$	$2$	5850.e	5.b	$2$	$4$	$4$	$46.712$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}q^{2}-q^{4}+\zeta_{8}^{2}q^{7}+\zeta_{8}q^{8}+2\zeta_{8}^{3}q^{11}+\cdots\)
5850.2.e.bl	$5850$	$2$	5850.e	5.b	$2$	$4$	$4$	$46.712$	\(\Q(i, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{4}+(-\beta _{1}+\beta _{2})q^{7}-\beta _{1}q^{8}+\cdots\)
5850.2.e.bm	$5850$	$2$	5850.e	5.b	$2$	$4$	$4$	$46.712$	\(\Q(i, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-q^{4}+(-2\beta _{1}-\beta _{3})q^{7}+\beta _{1}q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(5850, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(5850, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 3}\)