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

• Label: 1800.2.f
• Level: $1800$
• Weight: $2$
• Character orbit: 1800.f
• Rep. character: $\chi_{1800}(649,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $11$
• Sturm bound: $720$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(720\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	408	22	386
Cusp forms	312	22	290
Eisenstein series	96	0	96

Trace form

\( 22 q - 2 q^{11} - 10 q^{19} - 16 q^{29} - 4 q^{31} - 10 q^{41} - 6 q^{49} + 16 q^{59} + 4 q^{61} + 24 q^{71} + 20 q^{79} - 18 q^{89} + 24 q^{91}+O(q^{100}) \)

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.f.a	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					40.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-4 q^{11}-\beta q^{13}+\beta q^{17}+\cdots\)
1800.2.f.b	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		✓			1800.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}-4 q^{11}+i q^{13}-4 i q^{17}+\cdots\)
1800.2.f.c	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					24.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 q^{11}-\beta q^{13}+\beta q^{17}+4 q^{19}+\cdots\)
1800.2.f.d	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					360.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-2 q^{11}-2\beta q^{13}+\beta q^{17}+\cdots\)
1800.2.f.e	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					600.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}-2 q^{11}+3 i q^{13}-6 i q^{17}+\cdots\)
1800.2.f.f	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					200.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}-q^{11}-4 i q^{13}-5 i q^{17}+\cdots\)
1800.2.f.g	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					120.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+3\beta q^{13}+\beta q^{17}-4 q^{19}+\cdots\)
1800.2.f.h	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					360.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+2 q^{11}-2\beta q^{13}-\beta q^{17}+\cdots\)
1800.2.f.i	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None		✓			1800.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{7}+4 q^{11}+i q^{13}+4 i q^{17}+\cdots\)
1800.2.f.j	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					120.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 q^{11}-3\beta q^{13}+3\beta q^{17}+4 q^{19}+\cdots\)
1800.2.f.k	$1800$	$2$	1800.f	5.b	$2$	$2$	$2$	$14.373$	\(\Q(\sqrt{-1}) \)	None					600.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{7}+6 q^{11}-3 i q^{13}-2 i q^{17}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1800, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 2}\)