Space of modular forms of level 4200, weight 2, and Character 4200.t

• Label: 4200.2.t
• Level: $4200$
• Weight: $2$
• Character orbit: 4200.t
• Rep. character: $\chi_{4200}(1849,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $23$
• Sturm bound: $1920$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.t (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(1920\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(19\), \(29\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4200, [\chi])\).

	Total	New	Old
Modular forms	1008	52	956
Cusp forms	912	52	860
Eisenstein series	96	0	96

Trace form

\( 52 q - 52 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4200, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4200.2.t.a	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}-5q^{11}-4iq^{13}+\cdots\)
4200.2.t.b	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}-4q^{11}+2iq^{13}+\cdots\)
4200.2.t.c	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+iq^{7}-q^{9}-4q^{11}-2iq^{13}+\cdots\)
4200.2.t.d	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}-4q^{11}+6iq^{13}+\cdots\)
4200.2.t.e	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}-3q^{11}-2iq^{13}+\cdots\)
4200.2.t.f	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-iq^{7}-q^{9}-3q^{11}+2iq^{13}+\cdots\)
4200.2.t.g	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}-2q^{11}+5iq^{13}+\cdots\)
4200.2.t.h	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+iq^{7}-q^{9}-q^{11}+4iq^{13}+\cdots\)
4200.2.t.i	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}-6iq^{13}-2iq^{17}+\cdots\)
4200.2.t.j	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}+6iq^{13}+2iq^{17}+\cdots\)
4200.2.t.k	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-iq^{7}-q^{9}+2iq^{13}-2iq^{17}+\cdots\)
4200.2.t.l	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+iq^{7}-q^{9}+2iq^{13}-2iq^{17}+\cdots\)
4200.2.t.m	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}+2iq^{13}+6iq^{17}+\cdots\)
4200.2.t.n	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}+2iq^{13}+6iq^{17}+\cdots\)
4200.2.t.o	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}+2q^{11}+3iq^{13}+\cdots\)
4200.2.t.p	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}+3q^{11}+2iq^{17}+\cdots\)
4200.2.t.q	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}+4q^{11}-2iq^{13}+\cdots\)
4200.2.t.r	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}+4q^{11}-2iq^{13}+\cdots\)
4200.2.t.s	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+iq^{7}-q^{9}+4q^{11}-2iq^{13}+\cdots\)
4200.2.t.t	$4200$	$2$	4200.t	5.b	$2$	$2$	$2$	$33.537$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}-iq^{7}-q^{9}+5q^{11}+2iq^{13}+\cdots\)
4200.2.t.u	$4200$	$2$	4200.t	5.b	$2$	$4$	$4$	$33.537$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}q^{3}+\zeta_{8}q^{7}-q^{9}-\zeta_{8}^{3}q^{11}+\cdots\)
4200.2.t.v	$4200$	$2$	4200.t	5.b	$2$	$4$	$4$	$33.537$	\(\Q(i, \sqrt{73})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-\beta _{2}q^{7}-q^{9}+(1-\beta _{3})q^{11}+\cdots\)
4200.2.t.w	$4200$	$2$	4200.t	5.b	$2$	$4$	$4$	$33.537$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{2}q^{7}-q^{9}+(1+\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4200, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4200, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(700, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1050, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1400, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2100, [\chi])\)\(^{\oplus 2}\)