Space of modular forms of level 7200, weight 2, and Character 7200.f

• Label: 7200.2.f
• Level: $7200$
• Weight: $2$
• Character orbit: 7200.f
• Rep. character: $\chi_{7200}(6049,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $37$
• Sturm bound: $2880$
• Trace bound: $49$

Defining parameters

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 37 \)
Sturm bound:			\(2880\)
Trace bound:			\(49\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(19\), \(29\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(7200, [\chi])\).

	Total	New	Old
Modular forms	1536	90	1446
Cusp forms	1344	90	1254
Eisenstein series	192	0	192

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(7200, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
7200.2.f.a	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}-5q^{11}+5iq^{17}-5q^{19}+\cdots\)
7200.2.f.b	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4q^{11}+iq^{13}-iq^{17}-8q^{19}+\cdots\)
7200.2.f.c	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}-4q^{11}+3iq^{13}+iq^{17}+\cdots\)
7200.2.f.d	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}-4q^{11}-3iq^{13}-4iq^{17}+\cdots\)
7200.2.f.e	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{7}-4q^{11}-7iq^{13}+4iq^{17}+\cdots\)
7200.2.f.f	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}-4q^{11}-iq^{13}-3iq^{17}+\cdots\)
7200.2.f.g	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}-4q^{11}+3iq^{13}-iq^{17}+\cdots\)
7200.2.f.h	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}-2q^{11}+iq^{17}-4q^{19}+2q^{29}+\cdots\)
7200.2.f.i	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}-2q^{11}+iq^{17}+4q^{19}-2q^{29}+\cdots\)
7200.2.f.j	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{7}-5iq^{13}-5q^{19}-4iq^{23}+\cdots\)
7200.2.f.k	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{13}-3iq^{17}-4q^{19}-4iq^{23}+\cdots\)
7200.2.f.l	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}-iq^{13}-3q^{19}+4iq^{23}+\cdots\)
7200.2.f.m	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3iq^{13}+iq^{17}-10q^{29}+iq^{37}+\cdots\)
7200.2.f.n	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3iq^{13}+4iq^{17}-4q^{29}+iq^{37}+\cdots\)
7200.2.f.o	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}-iq^{13}-3iq^{17}+2iq^{23}+\cdots\)
7200.2.f.p	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}+iq^{13}+3iq^{17}+2iq^{23}+\cdots\)
7200.2.f.q	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3iq^{13}-4iq^{17}+4q^{29}+iq^{37}+\cdots\)
7200.2.f.r	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}+iq^{13}+3q^{19}+4iq^{23}+\cdots\)
7200.2.f.s	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{13}+3iq^{17}+4q^{19}-4iq^{23}+\cdots\)
7200.2.f.t	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{7}+5iq^{13}+5q^{19}-4iq^{23}+\cdots\)
7200.2.f.u	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}+2q^{11}-iq^{17}-4q^{19}-2q^{29}+\cdots\)
7200.2.f.v	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}+2q^{11}-iq^{17}+4q^{19}+2q^{29}+\cdots\)
7200.2.f.w	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}+4q^{11}-3iq^{13}+iq^{17}+\cdots\)
7200.2.f.x	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}+4q^{11}+iq^{13}+3iq^{17}+\cdots\)
7200.2.f.y	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{7}+4q^{11}+7iq^{13}-4iq^{17}+\cdots\)
7200.2.f.z	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{7}+4q^{11}+3iq^{13}+4iq^{17}+\cdots\)
7200.2.f.ba	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}+4q^{11}-3iq^{13}-iq^{17}+\cdots\)
7200.2.f.bb	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4q^{11}+iq^{13}-iq^{17}+8q^{19}+\cdots\)
7200.2.f.bc	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}+5q^{11}-5iq^{17}+5q^{19}+\cdots\)
7200.2.f.bd	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{13})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{7}-2q^{11}-3\beta _{1}q^{13}-2\beta _{2}q^{17}+\cdots\)
7200.2.f.be	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}-\beta _{3}q^{11}-2\beta _{1}q^{13}-\beta _{1}q^{17}+\cdots\)
7200.2.f.bf	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}-2\beta _{3}q^{11}-\beta _{1}q^{13}+2\beta _{1}q^{17}+\cdots\)
7200.2.f.bg	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{2}q^{7}+\beta _{3}q^{11}+4\beta _{1}q^{13}-7\beta _{1}q^{17}+\cdots\)
7200.2.f.bh	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{2}q^{7}+\zeta_{8}^{3}q^{11}-\zeta_{8}q^{13}+\zeta_{8}q^{17}+\cdots\)
7200.2.f.bi	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}+2\beta _{3}q^{11}-\beta _{1}q^{13}-2\beta _{1}q^{17}+\cdots\)
7200.2.f.bj	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}+\beta _{3}q^{11}-2\beta _{1}q^{13}+\beta _{1}q^{17}+\cdots\)
7200.2.f.bk	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{13})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{7}+2q^{11}-3\beta _{1}q^{13}+2\beta _{2}q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(7200, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(7200, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 3}\)