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

\( 90 q + 20 q^{29} + 28 q^{41} - 122 q^{49} - 4 q^{61} - 76 q^{89}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	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					800.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}-5 q^{11}+5 i q^{17}-5 q^{19}+\cdots\)
7200.2.f.b	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 q^{11}+\beta q^{13}-\beta q^{17}-8 q^{19}+\cdots\)
7200.2.f.c	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-4 q^{11}+3\beta q^{13}+\beta q^{17}+\cdots\)
7200.2.f.d	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}-4 q^{11}-3 i q^{13}-4 i q^{17}+\cdots\)
7200.2.f.e	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}-4 q^{11}-7 i q^{13}+4 i q^{17}+\cdots\)
7200.2.f.f	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					96.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-4 q^{11}-\beta q^{13}-3\beta q^{17}+\cdots\)
7200.2.f.g	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					160.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-4 q^{11}+3\beta q^{13}-\beta q^{17}+\cdots\)
7200.2.f.h	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					1440.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-2 q^{11}+\beta q^{17}-4 q^{19}+\cdots\)
7200.2.f.i	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					1440.2.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-2 q^{11}+\beta q^{17}+4 q^{19}+\cdots\)
7200.2.f.j	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}-5 i q^{13}-5 q^{19}-4 i q^{23}+\cdots\)
7200.2.f.k	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{13}-3\beta q^{17}-4 q^{19}-4\beta q^{23}+\cdots\)
7200.2.f.l	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}-i q^{13}-3 q^{19}+4 i q^{23}+\cdots\)
7200.2.f.m	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					32.2.a.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3\beta q^{13}+\beta q^{17}-10 q^{29}+\beta q^{37}+\cdots\)
7200.2.f.n	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					288.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3\beta q^{13}+4\beta q^{17}-4 q^{29}+\beta q^{37}+\cdots\)
7200.2.f.o	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-\beta q^{13}-3\beta q^{17}+2\beta q^{23}+\cdots\)
7200.2.f.p	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+\beta q^{13}+3\beta q^{17}+2\beta q^{23}+\cdots\)
7200.2.f.q	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					288.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3\beta q^{13}-4\beta q^{17}+4 q^{29}+\beta q^{37}+\cdots\)
7200.2.f.r	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}+i q^{13}+3 q^{19}+4 i q^{23}+\cdots\)
7200.2.f.s	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{13}+3\beta q^{17}+4 q^{19}-4\beta q^{23}+\cdots\)
7200.2.f.t	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}+5 i q^{13}+5 q^{19}-4 i q^{23}+\cdots\)
7200.2.f.u	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					1440.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+2 q^{11}-\beta q^{17}-4 q^{19}+\cdots\)
7200.2.f.v	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					1440.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+2 q^{11}-\beta q^{17}+4 q^{19}+\cdots\)
7200.2.f.w	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					160.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+4 q^{11}-3\beta q^{13}+\beta q^{17}+\cdots\)
7200.2.f.x	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					96.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+4 q^{11}+\beta q^{13}+3\beta q^{17}+\cdots\)
7200.2.f.y	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}+4 q^{11}+7 i q^{13}-4 i q^{17}+\cdots\)
7200.2.f.z	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					2400.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}+4 q^{11}+3 i q^{13}+4 i q^{17}+\cdots\)
7200.2.f.ba	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+4 q^{11}-3\beta q^{13}-\beta q^{17}+\cdots\)
7200.2.f.bb	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					480.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 q^{11}+\beta q^{13}-\beta q^{17}+8 q^{19}+\cdots\)
7200.2.f.bc	$7200$	$2$	7200.f	5.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					800.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}+5 q^{11}-5 i q^{17}+5 q^{19}+\cdots\)
7200.2.f.bd	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{13})\)	None		✓			7200.2.a.cd	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{7}-2q^{11}-3\beta _{1}q^{13}-2\beta _{2}q^{17}+\cdots\)
7200.2.f.be	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None					1440.2.a.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}-\beta _{3}q^{11}-2\beta _{1}q^{13}-\beta _{1}q^{17}+\cdots\)
7200.2.f.bf	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None		✓			7200.2.a.ci	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}-2\beta _{3}q^{11}-\beta _{1}q^{13}+2\beta _{1}q^{17}+\cdots\)
7200.2.f.bg	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None					800.2.a.k	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{2}q^{7}+\beta _{3}q^{11}+4\beta _{1}q^{13}-7\beta _{1}q^{17}+\cdots\)
7200.2.f.bh	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					160.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{7}+\beta_{3} q^{11}-\beta_1 q^{13}+\beta_1 q^{17}+\cdots\)
7200.2.f.bi	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None		✓			7200.2.a.ci	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}+2\beta _{3}q^{11}-\beta _{1}q^{13}-2\beta _{1}q^{17}+\cdots\)
7200.2.f.bj	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{5})\)	None					1440.2.a.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}+\beta _{3}q^{11}-2\beta _{1}q^{13}+\beta _{1}q^{17}+\cdots\)
7200.2.f.bk	$7200$	$2$	7200.f	5.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{13})\)	None		✓			7200.2.a.cd	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{7}+2q^{11}-3\beta _{1}q^{13}+2\beta _{2}q^{17}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(7200, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1440, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1800, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2400, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(3600, [\chi])\)\(^{\oplus 2}\)