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

• Label: 1800.2.m
• Level: $1800$
• Weight: $2$
• Character orbit: 1800.m
• Rep. character: $\chi_{1800}(899,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $6$
• Sturm bound: $720$
• Trace bound: $26$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.m (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 120 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(720\)
Trace bound:			\(26\)
Distinguishing \(T_p\):			\(7\), \(29\)

Dimensions

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

	Total	New	Old
Modular forms	384	72	312
Cusp forms	336	72	264
Eisenstein series	48	0	48

Trace form

\( 72 q + 8 q^{4} - 24 q^{16} - 32 q^{19} + 32 q^{34} + 56 q^{46} + 72 q^{49} + 8 q^{64} + 88 q^{76} - 88 q^{94}+O(q^{100}) \)

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.m.a	$1800$	$2$	1800.m	120.m	$2$	$4$	$4$	$14.373$	\(\Q(\zeta_{8})\)	None					360.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{2}+2 q^{4}+3\beta_{3} q^{7}-2\beta_{3} q^{8}+\cdots\)
1800.2.m.b	$1800$	$2$	1800.m	120.m	$2$	$4$	$4$	$14.373$	\(\Q(\zeta_{8})\)	None					360.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{3} q^{2}+2 q^{4}+3\beta_{3} q^{7}+2\beta_{3} q^{8}+\cdots\)
1800.2.m.c	$1800$	$2$	1800.m	120.m	$2$	$8$	$8$	$14.373$	\(\Q(\zeta_{24})\)	None					72.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{7} q^{2}+(-\beta_{4}-1)q^{4}+(\beta_{2}-2\beta_1)q^{7}+\cdots\)
1800.2.m.d	$1800$	$2$	1800.m	120.m	$2$	$12$	$12$	$14.373$	12.0.\(\cdots\).1	None					360.2.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(\beta _{3}+\beta _{7})q^{4}-\beta _{5}q^{7}+(\beta _{5}+\cdots)q^{8}+\cdots\)
1800.2.m.e	$1800$	$2$	1800.m	120.m	$2$	$12$	$12$	$14.373$	12.0.\(\cdots\).1	None					360.2.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(\beta _{3}-\beta _{9})q^{4}-\beta _{7}q^{7}+(-\beta _{5}+\cdots)q^{8}+\cdots\)
1800.2.m.f	$1800$	$2$	1800.m	120.m	$2$	$32$	$32$	$14.373$		None		✓	✓		1800.2.b.h	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

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