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

• Label: 1800.2.b
• Level: $1800$
• Weight: $2$
• Character orbit: 1800.b
• Rep. character: $\chi_{1800}(251,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $9$
• Sturm bound: $720$
• Trace bound: $22$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(720\)
Trace bound:			\(22\)
Distinguishing \(T_p\):			\(7\), \(23\), \(43\)

Dimensions

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

	Total	New	Old
Modular forms	384	76	308
Cusp forms	336	76	260
Eisenstein series	48	0	48

Trace form

\( 76 q - 4 q^{4} + 16 q^{19} + 24 q^{22} - 8 q^{28} + 28 q^{34} - 32 q^{43} - 92 q^{49} + 32 q^{52} + 28 q^{58} + 56 q^{64} - 16 q^{67} - 16 q^{73} + 24 q^{76} - 84 q^{82} + 48 q^{91} + 32 q^{94}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1800.2.b.a	$1800$	$2$	1800.b	24.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-2}) \)	None					360.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-2q^{4}+3\beta q^{7}-2\beta q^{8}+\beta q^{11}+\cdots\)
1800.2.b.b	$1800$	$2$	1800.b	24.f	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-2}) \)	None					360.2.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{2}-2q^{4}+3\beta q^{7}+2\beta q^{8}-\beta q^{11}+\cdots\)
1800.2.b.c	$1800$	$2$	1800.b	24.f	$2$	$4$	$4$	$14.373$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					72.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(1+\beta _{3})q^{4}+2\beta _{3}q^{7}-2\beta _{2}q^{8}+\cdots\)
1800.2.b.d	$1800$	$2$	1800.b	24.f	$2$	$6$	$6$	$14.373$	6.0.2580992.1	None					360.2.b.c	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(\beta _{2}-\beta _{3})q^{4}+\beta _{3}q^{7}+(-\beta _{2}+\cdots)q^{8}+\cdots\)
1800.2.b.e	$1800$	$2$	1800.b	24.f	$2$	$6$	$6$	$14.373$	6.0.2580992.1	None					360.2.b.c	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(\beta _{2}-\beta _{3})q^{4}+\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{8}+\cdots\)
1800.2.b.f	$1800$	$2$	1800.b	24.f	$2$	$8$	$8$	$14.373$	8.0.40960000.1	\(\Q(\sqrt{-10}) \)					360.2.m.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{2}+2q^{4}-\beta _{5}q^{7}+2\beta _{2}q^{8}+\cdots\)
1800.2.b.g	$1800$	$2$	1800.b	24.f	$2$	$16$	$16$	$14.373$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					360.2.m.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(-1-\beta _{5})q^{4}+(\beta _{2}+\beta _{12}+\cdots)q^{7}+\cdots\)
1800.2.b.h	$1800$	$2$	1800.b	24.f	$2$	$16$	$16$	$14.373$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			1800.2.b.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{2}+\beta _{5}q^{4}-\beta _{15}q^{7}+\beta _{13}q^{8}+\cdots\)
1800.2.b.i	$1800$	$2$	1800.b	24.f	$2$	$16$	$16$	$14.373$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			1800.2.b.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{9}q^{2}+\beta _{2}q^{4}+\beta _{12}q^{7}+\beta _{15}q^{8}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1800, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)