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

• Label: 7200.2.d
• Level: $7200$
• Weight: $2$
• Character orbit: 7200.d
• Rep. character: $\chi_{7200}(2449,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $21$
• Sturm bound: $2880$
• Trace bound: $53$

Defining parameters

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(2880\)
Trace bound:			\(53\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(37\), \(41\), \(53\)

Dimensions

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

	Total	New	Old
Modular forms	1536	92	1444
Cusp forms	1344	88	1256
Eisenstein series	192	4	188

Trace form

\( 88 q + O(q^{10}) \)

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	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.d.a	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					360.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-\beta q^{11}-6 q^{13}-3\beta q^{17}+\cdots\)
7200.2.d.b	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					360.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+\beta q^{11}-6 q^{13}+3\beta q^{17}+\cdots\)
7200.2.d.c	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}-5 i q^{11}-6 q^{13}-3 i q^{17}+\cdots\)
7200.2.d.d	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					24.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-4 q^{13}+\beta q^{17}+2\beta q^{19}+\cdots\)
7200.2.d.e	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					120.2.k.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-2\beta q^{11}-3\beta q^{17}-2\beta q^{19}+\cdots\)
7200.2.d.f	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					120.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+2\beta q^{11}-3\beta q^{17}+2\beta q^{19}+\cdots\)
7200.2.d.g	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					24.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+4 q^{13}+\beta q^{17}-2\beta q^{19}+\cdots\)
7200.2.d.h	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					360.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+\beta q^{11}+6 q^{13}-3\beta q^{17}+\cdots\)
7200.2.d.i	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					360.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-\beta q^{11}+6 q^{13}+3\beta q^{17}+\cdots\)
7200.2.d.j	$7200$	$2$	7200.d	40.f	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}+5 i q^{11}+6 q^{13}-3 i q^{17}+\cdots\)
7200.2.d.k	$7200$	$2$	7200.d	40.f	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{7})\)	None					360.2.k.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{7}-\beta _{1}q^{11}-\beta _{3}q^{19}-\beta _{3}q^{23}+\cdots\)
7200.2.d.l	$7200$	$2$	7200.d	40.f	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{7})\)	None					360.2.k.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{7}+\beta _{1}q^{11}-\beta _{3}q^{19}+\beta _{3}q^{23}+\cdots\)
7200.2.d.m	$7200$	$2$	7200.d	40.f	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{7})\)	None					200.2.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{7}+\beta _{3}q^{11}-3\beta _{1}q^{17}+\beta _{3}q^{19}+\cdots\)
7200.2.d.n	$7200$	$2$	7200.d	40.f	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{12})\)	None					40.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{7}+\beta_{2} q^{11}+2\beta_1 q^{13}+(-2\beta_{3}-\beta_{2})q^{17}+\cdots\)
7200.2.d.o	$7200$	$2$	7200.d	40.f	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{12})\)	None					40.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{7}-\beta_{2} q^{11}-2\beta_1 q^{13}+(-2\beta_{3}-\beta_{2})q^{17}+\cdots\)
7200.2.d.p	$7200$	$2$	7200.d	40.f	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-6}) \)					72.2.d.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta_1 q^{7}+2\beta_{2} q^{11}-\beta_{2} q^{29}+10 q^{31}+\cdots\)
7200.2.d.q	$7200$	$2$	7200.d	40.f	$2$	$6$	$6$	$57.492$	6.0.399424.1	None					120.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{7}+(\beta _{2}-\beta _{3}-\beta _{5})q^{11}-\beta _{1}q^{13}+\cdots\)
7200.2.d.r	$7200$	$2$	7200.d	40.f	$2$	$6$	$6$	$57.492$	6.0.399424.1	None					120.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{7}+(-\beta _{2}+\beta _{3}+\beta _{5})q^{11}+\beta _{1}q^{13}+\cdots\)
7200.2.d.s	$7200$	$2$	7200.d	40.f	$2$	$8$	$8$	$57.492$	8.0.214798336.3	None					600.2.k.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{2}+\beta _{7})q^{7}-\beta _{5}q^{11}+(-\beta _{1}-\beta _{3}+\cdots)q^{13}+\cdots\)
7200.2.d.t	$7200$	$2$	7200.d	40.f	$2$	$8$	$8$	$57.492$	8.0.214798336.3	None					600.2.k.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{2}+\beta _{7})q^{7}+\beta _{5}q^{11}+(\beta _{1}+\beta _{3}+\cdots)q^{13}+\cdots\)
7200.2.d.u	$7200$	$2$	7200.d	40.f	$2$	$16$	$16$	$57.492$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			✓		1800.2.k.r	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{8}q^{7}+\beta _{4}q^{11}-\beta _{3}q^{13}+(\beta _{5}-\beta _{7}+\cdots)q^{17}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(7200, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1440, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1800, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2400, [\chi])\)\(^{\oplus 2}\)