Space of modular forms of level 1800, weight 2, and Character 1800.s

• Label: 1800.2.s
• Level: $1800$
• Weight: $2$
• Character orbit: 1800.s
• Rep. character: $\chi_{1800}(593,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $6$
• Sturm bound: $720$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.s (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(720\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1800, [\chi])\).

	Total	New	Old
Modular forms	816	36	780
Cusp forms	624	36	588
Eisenstein series	192	0	192

Trace form

\( 36 q + 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1800, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1800.2.s.a	$1800$	$2$	1800.s	15.e	$4$	$4$	$2$	$14.373$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8})q^{7}-\zeta_{8}^{2}q^{11}+(3+3\zeta_{8}+\cdots)q^{13}+\cdots\)
1800.2.s.b	$1800$	$2$	1800.s	15.e	$4$	$4$	$2$	$14.373$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+4\zeta_{8}^{2}q^{11}+(-3+3\zeta_{8})q^{13}-4\zeta_{8}q^{19}+\cdots\)
1800.2.s.c	$1800$	$2$	1800.s	15.e	$4$	$4$	$2$	$14.373$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\zeta_{8})q^{7}-\zeta_{8}^{2}q^{11}+(-3-3\zeta_{8}+\cdots)q^{13}+\cdots\)
1800.2.s.d	$1800$	$2$	1800.s	15.e	$4$	$8$	$4$	$14.373$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\zeta_{24}^{2}-\zeta_{24}^{3})q^{7}-\zeta_{24}^{6}q^{11}+\cdots\)
1800.2.s.e	$1800$	$2$	1800.s	15.e	$4$	$8$	$4$	$14.373$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1}-\beta _{4})q^{7}+(\beta _{3}+\beta _{6})q^{11}+\cdots\)
1800.2.s.f	$1800$	$2$	1800.s	15.e	$4$	$8$	$4$	$14.373$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-\zeta_{24}+2\zeta_{24}^{2})q^{7}-\zeta_{24}^{6}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1800, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1800, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 2}\)