Space of modular forms of level 1800, weight 4, and Character 1800.f

• Label: 1800.4.f
• Level: $1800$
• Weight: $4$
• Character orbit: 1800.f
• Rep. character: $\chi_{1800}(649,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $28$
• Sturm bound: $1440$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(1440\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1800, [\chi])\).

	Total	New	Old
Modular forms	1128	68	1060
Cusp forms	1032	68	964
Eisenstein series	96	0	96

Trace form

\( 68 q + 38 q^{11} + 46 q^{19} - 188 q^{29} + 492 q^{31} - 234 q^{41} - 2676 q^{49} - 1144 q^{59} + 104 q^{61} + 984 q^{71} + 692 q^{79} + 4374 q^{89} - 40 q^{91}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1800, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1800.4.f.a	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					120.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}-72 q^{11}+3\beta q^{13}-19\beta q^{17}+\cdots\)
1800.4.f.b	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					72.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+6\beta q^{7}-64 q^{11}+29\beta q^{13}+16\beta q^{17}+\cdots\)
1800.4.f.c	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					200.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}-39 q^{11}-84 i q^{13}+61 i q^{17}+\cdots\)
1800.4.f.d	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					40.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+8\beta q^{7}-36 q^{11}+21\beta q^{13}+55\beta q^{17}+\cdots\)
1800.4.f.e	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					360.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta q^{7}-34 q^{11}+6\beta q^{13}-51\beta q^{17}+\cdots\)
1800.4.f.f	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					360.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}-34 q^{11}+34\beta q^{13}-19\beta q^{17}+\cdots\)
1800.4.f.g	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					600.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+19 i q^{7}-22 q^{11}+i q^{13}-58 i q^{17}+\cdots\)
1800.4.f.h	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					120.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{7}-20 q^{11}-11\beta q^{13}+7\beta q^{17}+\cdots\)
1800.4.f.i	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					360.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+17\beta q^{7}-18 q^{11}-6\beta q^{13}+53\beta q^{17}+\cdots\)
1800.4.f.j	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					40.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+17\beta q^{7}-16 q^{11}+29\beta q^{13}-35\beta q^{17}+\cdots\)
1800.4.f.k	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					120.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+10\beta q^{7}-16 q^{11}-29\beta q^{13}-19\beta q^{17}+\cdots\)
1800.4.f.l	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					600.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{7}-14 q^{11}-i q^{13}-46 i q^{17}+\cdots\)
1800.4.f.m	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					120.4.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 q^{11}-27\beta q^{13}-57\beta q^{17}-44 q^{19}+\cdots\)
1800.4.f.n	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					40.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta q^{7}+16 q^{11}-3\beta q^{13}-3\beta q^{17}+\cdots\)
1800.4.f.o	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					360.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+17\beta q^{7}+18 q^{11}-6\beta q^{13}-53\beta q^{17}+\cdots\)
1800.4.f.p	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					200.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+6 i q^{7}+19 q^{11}+12 i q^{13}-75 i q^{17}+\cdots\)
1800.4.f.q	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					24.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+12\beta q^{7}+28 q^{11}-37\beta q^{13}+41\beta q^{17}+\cdots\)
1800.4.f.r	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					120.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+8\beta q^{7}+28 q^{11}-13\beta q^{13}-31\beta q^{17}+\cdots\)
1800.4.f.s	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					360.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta q^{7}+34 q^{11}+6\beta q^{13}+51\beta q^{17}+\cdots\)
1800.4.f.t	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					360.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+34 q^{11}+34\beta q^{13}+19\beta q^{17}+\cdots\)
1800.4.f.u	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					8.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+12\beta q^{7}+44 q^{11}-11\beta q^{13}-25\beta q^{17}+\cdots\)
1800.4.f.v	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					120.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+10\beta q^{7}+56 q^{11}+43\beta q^{13}+53\beta q^{17}+\cdots\)
1800.4.f.w	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					200.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+26 i q^{7}+59 q^{11}-28 i q^{13}-5 i q^{17}+\cdots\)
1800.4.f.x	$1800$	$4$	1800.f	5.b	$2$	$2$	$2$	$106.203$	\(\Q(\sqrt{-1}) \)	None					72.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+6\beta q^{7}+64 q^{11}+29\beta q^{13}-16\beta q^{17}+\cdots\)
1800.4.f.y	$1800$	$4$	1800.f	5.b	$2$	$4$	$4$	$106.203$	\(\Q(i, \sqrt{181})\)	None					600.4.a.r	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3\beta _{1}+\beta _{2})q^{7}+(-4+\beta _{3})q^{11}+\cdots\)
1800.4.f.z	$1800$	$4$	1800.f	5.b	$2$	$4$	$4$	$106.203$	\(\Q(i, \sqrt{109})\)	None					600.4.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{7}+(8+3\beta _{3})q^{11}+(-41\beta _{1}+\cdots)q^{13}+\cdots\)
1800.4.f.ba	$1800$	$4$	1800.f	5.b	$2$	$6$	$6$	$106.203$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			1800.4.a.br	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3\beta _{1}-\beta _{4})q^{7}+(-3+\beta _{2}-\beta _{3}+\cdots)q^{11}+\cdots\)
1800.4.f.bb	$1800$	$4$	1800.f	5.b	$2$	$6$	$6$	$106.203$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			1800.4.a.br	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3\beta _{1}-\beta _{4})q^{7}+(3-\beta _{2}+\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1800, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1800, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 2}\)