Space of modular forms of level 3600, weight 2, and Character 3600.x

• Label: 3600.2.x
• Level: $3600$
• Weight: $2$
• Character orbit: 3600.x
• Rep. character: $\chi_{3600}(2143,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $90$
• Newform subspaces: $15$
• Sturm bound: $1440$
• Trace bound: $101$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.x (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(1440\)
Trace bound:			\(101\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3600, [\chi])\).

	Total	New	Old
Modular forms	1584	90	1494
Cusp forms	1296	90	1206
Eisenstein series	288	0	288

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(3600, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.2.x.a	$3600$	$2$	3600.x	20.e	$4$	$2$	$1$	$28.746$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1+i)q^{13}+(-5-5i)q^{17}+10iq^{29}+\cdots\)
3600.2.x.b	$3600$	$2$	3600.x	20.e	$4$	$2$	$1$	$28.746$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1+i)q^{13}+(5+5i)q^{17}-10iq^{29}+\cdots\)
3600.2.x.c	$3600$	$2$	3600.x	20.e	$4$	$2$	$1$	$28.746$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(5-5i)q^{13}+(-5-5i)q^{17}+4iq^{29}+\cdots\)
3600.2.x.d	$3600$	$2$	3600.x	20.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{7}+(\zeta_{8}+\zeta_{8}^{3})q^{11}+(-3-3\zeta_{8}^{2}+\cdots)q^{13}+\cdots\)
3600.2.x.e	$3600$	$2$	3600.x	20.e	$4$	$4$	$2$	$28.746$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{12}^{2}q^{7}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{11}+(-1+\cdots)q^{13}+\cdots\)
3600.2.x.f	$3600$	$2$	3600.x	20.e	$4$	$4$	$2$	$28.746$	\(\Q(i, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{3}q^{7}+3\beta _{2}q^{23}-6\beta _{1}q^{29}+12q^{41}+\cdots\)
3600.2.x.g	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}^{5}q^{7}-2\zeta_{24}^{4}q^{11}+3\zeta_{24}^{7}q^{13}+\cdots\)
3600.2.x.h	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}^{4}q^{7}-\zeta_{24}^{6}q^{11}-\zeta_{24}^{5}q^{13}+\cdots\)
3600.2.x.i	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-3}) \)		$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{4}$	$\mathrm{U}(1)[D_{4}]$	\(q+\zeta_{24}^{6}q^{7}+3\zeta_{24}^{4}q^{13}-\zeta_{24}^{7}q^{19}+\cdots\)
3600.2.x.j	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-3}) \)		$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{4}]$	\(q+\zeta_{24}q^{7}-\zeta_{24}^{2}q^{13}-3\zeta_{24}^{6}q^{19}+\cdots\)
3600.2.x.k	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}^{5}q^{7}-2\zeta_{24}^{4}q^{11}+\zeta_{24}^{7}q^{13}+\cdots\)
3600.2.x.l	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-3}) \)		$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{U}(1)[D_{4}]$	\(q+\zeta_{24}q^{7}-\zeta_{24}^{2}q^{13}+\zeta_{24}^{7}q^{19}+\cdots\)
3600.2.x.m	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+4\zeta_{24}^{5}q^{7}-3\zeta_{24}^{4}q^{11}-2\zeta_{24}^{7}q^{13}+\cdots\)
3600.2.x.n	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\zeta_{24}^{2}+\zeta_{24}^{4}-\zeta_{24}^{5})q^{7}-\zeta_{24}^{2}q^{11}+\cdots\)
3600.2.x.o	$3600$	$2$	3600.x	20.e	$4$	$8$	$4$	$28.746$	8.0.3317760000.2	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{7}+\beta _{7}q^{11}+(2+2\beta _{1})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1800, [\chi])\)\(^{\oplus 2}\)