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

• Label: 1800.2.d
• Level: $1800$
• Weight: $2$
• Character orbit: 1800.d
• Rep. character: $\chi_{1800}(1549,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $21$
• Sturm bound: $720$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(720\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(37\), \(41\), \(53\)

Dimensions

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

	Total	New	Old
Modular forms	384	92	292
Cusp forms	336	88	248
Eisenstein series	48	4	44

Trace form

\( 88 q - 2 q^{4} + O(q^{10}) \)

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.d.a	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+i)q^{2}-2iq^{4}+4iq^{7}+(2+\cdots)q^{8}+\cdots\)
1800.2.d.b	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-i)q^{2}+2iq^{4}+2iq^{7}+(2+\cdots)q^{8}+\cdots\)
1800.2.d.c	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-i)q^{2}+2iq^{4}+2iq^{7}+(2+\cdots)q^{8}+\cdots\)
1800.2.d.d	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-i)q^{2}+2iq^{4}+2iq^{7}+(2+\cdots)q^{8}+\cdots\)
1800.2.d.e	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-i)q^{2}+2iq^{4}+4iq^{7}+(2+\cdots)q^{8}+\cdots\)
1800.2.d.f	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-i)q^{2}-2iq^{4}+2iq^{7}+(-2+\cdots)q^{8}+\cdots\)
1800.2.d.g	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-i)q^{2}-2iq^{4}+4iq^{7}+(-2+\cdots)q^{8}+\cdots\)
1800.2.d.h	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-i)q^{2}-2iq^{4}+2iq^{7}+(-2+\cdots)q^{8}+\cdots\)
1800.2.d.i	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-i)q^{2}-2iq^{4}+2iq^{7}+(-2+\cdots)q^{8}+\cdots\)
1800.2.d.j	$1800$	$2$	1800.d	40.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+i)q^{2}+2iq^{4}+4iq^{7}+(-2+\cdots)q^{8}+\cdots\)
1800.2.d.k	$1800$	$2$	1800.d	40.f	$2$	$4$	$4$	$14.373$	\(\Q(i, \sqrt{7})\)	None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{2})q^{2}+(-1-\beta _{2})q^{4}-\beta _{3}q^{7}+\cdots\)
1800.2.d.l	$1800$	$2$	1800.d	40.f	$2$	$4$	$4$	$14.373$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{4}+\cdots\)
1800.2.d.m	$1800$	$2$	1800.d	40.f	$2$	$4$	$4$	$14.373$	\(\Q(i, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(2-\beta _{2})q^{4}+4\beta _{3}q^{7}+\cdots\)
1800.2.d.n	$1800$	$2$	1800.d	40.f	$2$	$4$	$4$	$14.373$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-\zeta_{8}^{2}q^{2}+2q^{4}+\zeta_{8}q^{7}-2\zeta_{8}^{2}q^{8}+\cdots\)
1800.2.d.o	$1800$	$2$	1800.d	40.f	$2$	$4$	$4$	$14.373$	\(\Q(i, \sqrt{7})\)	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{2})q^{2}+(-1-\beta _{2})q^{4}-\beta _{3}q^{7}+\cdots\)
1800.2.d.p	$1800$	$2$	1800.d	40.f	$2$	$4$	$4$	$14.373$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\zeta_{12})q^{2}+(-\zeta_{12}+2\zeta_{12}^{2}-\zeta_{12}^{3})q^{4}+\cdots\)
1800.2.d.q	$1800$	$2$	1800.d	40.f	$2$	$6$	$6$	$14.373$	6.0.399424.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-\beta _{1}q^{4}+(-\beta _{1}-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
1800.2.d.r	$1800$	$2$	1800.d	40.f	$2$	$6$	$6$	$14.373$	6.0.399424.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-\beta _{2}q^{4}+(-\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
1800.2.d.s	$1800$	$2$	1800.d	40.f	$2$	$8$	$8$	$14.373$	8.0.214798336.3	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}+(\beta _{4}+\beta _{5}-\beta _{6})q^{4}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
1800.2.d.t	$1800$	$2$	1800.d	40.f	$2$	$8$	$8$	$14.373$	8.0.214798336.3	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}+(-1-\beta _{1}+\beta _{3}+\beta _{6})q^{4}+\cdots\)
1800.2.d.u	$1800$	$2$	1800.d	40.f	$2$	$16$	$16$	$14.373$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}+\beta _{8}q^{4}-\beta _{5}q^{7}+\beta _{9}q^{8}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1800, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)