Space of modular forms of level 3600, weight 3, and Character 3600.j

• Label: 3600.3.j
• Level: $3600$
• Weight: $3$
• Character orbit: 3600.j
• Rep. character: $\chi_{3600}(1999,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $16$
• Sturm bound: $2160$
• Trace bound: $49$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.j (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(2160\)
Trace bound:			\(49\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(3600, [\chi])\).

	Total	New	Old
Modular forms	1512	90	1422
Cusp forms	1368	90	1278
Eisenstein series	144	0	144

Trace form

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

Decomposition of \(S_{3}^{\mathrm{new}}(3600, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.3.j.a	$3600$	$3$	3600.j	20.d	$2$	$2$	$2$	$98.093$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q+iq^{13}-3iq^{17}+42q^{29}+7iq^{37}+\cdots\)
3600.3.j.b	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(i, \sqrt{15})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{7}-\beta _{3}q^{11}-4\beta _{2}q^{13}-3\beta _{2}q^{17}+\cdots\)
3600.3.j.c	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-3\zeta_{12}^{3}q^{7}+23\zeta_{12}q^{13}-21\zeta_{12}^{2}q^{19}+\cdots\)
3600.3.j.d	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{12}^{3}q^{7}-\zeta_{12}q^{13}-\zeta_{12}^{2}q^{19}+\cdots\)
3600.3.j.e	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{12}^{2}q^{7}+11\zeta_{12}q^{13}-2\zeta_{12}^{3}q^{19}+\cdots\)
3600.3.j.f	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\zeta_{12}^{3}q^{7}+7\zeta_{12}^{2}q^{11}-2^{4}\zeta_{12}q^{13}+\cdots\)
3600.3.j.g	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\zeta_{12}^{3}q^{7}+2\zeta_{12}^{2}q^{11}-13\zeta_{12}q^{13}+\cdots\)
3600.3.j.h	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}^{3}q^{7}+2\zeta_{12}^{2}q^{11}+11\zeta_{12}q^{13}+\cdots\)
3600.3.j.i	$3600$	$3$	3600.j	20.d	$2$	$4$	$4$	$98.093$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}^{2}q^{7}-3\zeta_{12}^{3}q^{11}-7\zeta_{12}q^{13}+\cdots\)
3600.3.j.j	$3600$	$3$	3600.j	20.d	$2$	$8$	$8$	$98.093$	8.0.303595776.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{5})q^{7}+\beta _{4}q^{11}-\beta _{2}q^{13}+\cdots\)
3600.3.j.k	$3600$	$3$	3600.j	20.d	$2$	$8$	$8$	$98.093$	8.0.12960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{7}+\beta _{5}q^{11}+(2\beta _{4}+\beta _{7})q^{13}+\cdots\)
3600.3.j.l	$3600$	$3$	3600.j	20.d	$2$	$8$	$8$	$98.093$	8.0.12960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{3}+\beta _{5})q^{7}-\beta _{2}q^{11}+(-4\beta _{1}-\beta _{7})q^{13}+\cdots\)
3600.3.j.m	$3600$	$3$	3600.j	20.d	$2$	$8$	$8$	$98.093$	8.0.12960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{7}+\beta _{5}q^{11}-4\beta _{2}q^{13}+\beta _{7}q^{17}+\cdots\)
3600.3.j.n	$3600$	$3$	3600.j	20.d	$2$	$8$	$8$	$98.093$	8.0.3317760000.2	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{4}q^{7}-\beta _{3}q^{11}-13\beta _{1}q^{13}-\beta _{7}q^{17}+\cdots\)
3600.3.j.o	$3600$	$3$	3600.j	20.d	$2$	$8$	$8$	$98.093$	8.0.12960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{7}+\beta _{4}q^{11}+2\beta _{1}q^{13}+\beta _{6}q^{17}+\cdots\)
3600.3.j.p	$3600$	$3$	3600.j	20.d	$2$	$8$	$8$	$98.093$	8.0.12960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{7}+\beta _{1}q^{11}+(-2\beta _{2}+\beta _{7})q^{13}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

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