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

• Label: 7200.2.o
• Level: $7200$
• Weight: $2$
• Character orbit: 7200.o
• Rep. character: $\chi_{7200}(7199,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $16$
• Sturm bound: $2880$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.o (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 60 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(2880\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	1536	72	1464
Cusp forms	1344	72	1272
Eisenstein series	192	0	192

Trace form

\( 72 q + 104 q^{49} + 16 q^{61}+O(q^{100}) \)

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.o.a	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					288.2.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 q^{7}+4\beta_{3} q^{11}+2\beta_1 q^{13}+3\beta_{3} q^{17}+\cdots\)
7200.2.o.b	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}-2)q^{7}+(-\beta_{3}-4)q^{11}+(-\beta_{2}+\beta_1)q^{13}+\cdots\)
7200.2.o.c	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}-2)q^{7}-\beta_{3} q^{11}+(-3\beta_{2}+\beta_1)q^{13}+\cdots\)
7200.2.o.d	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}-2)q^{7}+\beta_{3} q^{11}+(3\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.e	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{3}-2)q^{7}+(-\beta_{3}+4)q^{11}+\cdots\)
7200.2.o.f	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None		✓			7200.2.h.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}-1)q^{7}+(\beta_{3}-4)q^{11}+(4\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.g	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None		✓			7200.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}-1)q^{7}+(-\beta_{3}+4)q^{11}+(4\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.h	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None		✓			7200.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}+1)q^{7}+(-\beta_{3}-4)q^{11}+(-4\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.i	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None		✓			7200.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}+1)q^{7}+(\beta_{3}+4)q^{11}+(-4\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.j	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}+2)q^{7}+(\beta_{3}-4)q^{11}+(-\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.k	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{3}+2)q^{7}+\beta_{3} q^{11}+(-3\beta_{2}+\beta_1)q^{13}+\cdots\)
7200.2.o.l	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{3}+2)q^{7}-\beta_{3} q^{11}+(3\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.m	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					1440.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{3}+2)q^{7}+(-\beta_{3}+4)q^{11}+(-\beta_{2}-\beta_1)q^{13}+\cdots\)
7200.2.o.n	$7200$	$2$	7200.o	60.h	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{8})\)	None					288.2.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 q^{7}+4\beta_{3} q^{11}+2\beta_1 q^{13}-3\beta_{3} q^{17}+\cdots\)
7200.2.o.o	$7200$	$2$	7200.o	60.h	$2$	$8$	$8$	$57.492$	\(\Q(i, \sqrt{2}, \sqrt{7})\)	None		✓			7200.2.h.j	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{7})q^{7}-\beta _{3}q^{11}-\beta _{2}q^{13}+\cdots\)
7200.2.o.p	$7200$	$2$	7200.o	60.h	$2$	$8$	$8$	$57.492$	\(\Q(i, \sqrt{2}, \sqrt{7})\)	None		✓			7200.2.h.j	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{7})q^{7}+\beta _{3}q^{11}-\beta _{2}q^{13}+(\beta _{3}+\cdots)q^{17}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(7200, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1440, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2400, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(3600, [\chi])\)\(^{\oplus 2}\)