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

• Label: 7200.2.k
• Level: $7200$
• Weight: $2$
• Character orbit: 7200.k
• Rep. character: $\chi_{7200}(3601,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $21$
• Sturm bound: $2880$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(2880\)
Trace bound:			\(31\)
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	98	1438
Cusp forms	1344	92	1252
Eisenstein series	192	6	186

Trace form

\( 92 q - 4 q^{7} + O(q^{10}) \)

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.k.a	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4q^{7}+iq^{11}+3iq^{13}-6q^{17}+\cdots\)
7200.2.k.b	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-7}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4q^{7}-\beta q^{11}+3q^{17}+\beta q^{19}+4q^{23}+\cdots\)
7200.2.k.c	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4q^{7}-iq^{11}+3iq^{13}+6q^{17}+\cdots\)
7200.2.k.d	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2q^{7}+2iq^{13}-2q^{17}-2iq^{19}+\cdots\)
7200.2.k.e	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2q^{7}+5iq^{11}-6iq^{13}+3q^{17}+\cdots\)
7200.2.k.f	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2q^{7}+2iq^{11}-6q^{17}-2iq^{19}+\cdots\)
7200.2.k.g	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2q^{7}+5iq^{11}+6iq^{13}-3q^{17}+\cdots\)
7200.2.k.h	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-6}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{7}-2\beta q^{11}-\beta q^{29}+10q^{31}+\cdots\)
7200.2.k.i	$7200$	$2$	7200.k	8.b	$2$	$2$	$2$	$57.492$	\(\Q(\sqrt{-7}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4q^{7}-\beta q^{11}-3q^{17}+\beta q^{19}-4q^{23}+\cdots\)
7200.2.k.j	$7200$	$2$	7200.k	8.b	$2$	$4$	$4$	$57.492$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{12})q^{7}+\zeta_{12}^{2}q^{11}+(\zeta_{12}^{2}+\cdots)q^{13}+\cdots\)
7200.2.k.k	$7200$	$2$	7200.k	8.b	$2$	$4$	$4$	$57.492$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{7}-\beta _{3}q^{11}-3\beta _{3}q^{29}-10q^{31}+\cdots\)
7200.2.k.l	$7200$	$2$	7200.k	8.b	$2$	$4$	$4$	$57.492$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{7}-\beta _{3}q^{11}+2\beta _{2}q^{17}+\beta _{3}q^{19}+\cdots\)
7200.2.k.m	$7200$	$2$	7200.k	8.b	$2$	$4$	$4$	$57.492$	\(\Q(\sqrt{2}, \sqrt{-5})\)	\(\Q(\sqrt{-30}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{3}q^{11}-\beta _{2}q^{13}-\beta _{1}q^{17}+2\beta _{1}q^{23}+\cdots\)
7200.2.k.n	$7200$	$2$	7200.k	8.b	$2$	$4$	$4$	$57.492$	\(\Q(\sqrt{3}, \sqrt{-5})\)	\(\Q(\sqrt{-15}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q+2\beta _{2}q^{17}+\beta _{3}q^{19}+\beta _{2}q^{23}+8q^{31}+\cdots\)
7200.2.k.o	$7200$	$2$	7200.k	8.b	$2$	$4$	$4$	$57.492$	\(\Q(i, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2q^{7}+\beta _{1}q^{11}+\beta _{3}q^{19}+\beta _{2}q^{23}+\cdots\)
7200.2.k.p	$7200$	$2$	7200.k	8.b	$2$	$6$	$6$	$57.492$	6.0.399424.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{7}+(\beta _{2}-\beta _{4}-\beta _{5})q^{11}+\cdots\)
7200.2.k.q	$7200$	$2$	7200.k	8.b	$2$	$8$	$8$	$57.492$	8.0.\(\cdots\).29	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{6})q^{7}-\beta _{1}q^{11}+\beta _{4}q^{13}+\cdots\)
7200.2.k.r	$7200$	$2$	7200.k	8.b	$2$	$8$	$8$	$57.492$	8.0.214798336.3	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{7}+\beta _{5}q^{11}+(\beta _{6}-\beta _{7})q^{13}+\cdots\)
7200.2.k.s	$7200$	$2$	7200.k	8.b	$2$	$8$	$8$	$57.492$	8.0.214798336.3	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{7}-\beta _{5}q^{11}+(\beta _{6}-\beta _{7})q^{13}+\cdots\)
7200.2.k.t	$7200$	$2$	7200.k	8.b	$2$	$8$	$8$	$57.492$	8.0.\(\cdots\).29	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{6})q^{7}+\beta _{1}q^{11}-\beta _{4}q^{13}+(-\beta _{5}+\cdots)q^{17}+\cdots\)
7200.2.k.u	$7200$	$2$	7200.k	8.b	$2$	$12$	$12$	$57.492$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{7}+\beta _{8}q^{11}+\beta _{9}q^{13}+(\beta _{4}+\beta _{5}+\cdots)q^{17}+\cdots\)

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

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