Space of modular forms of level 1800, weight 3, and Character 1800.v

• Label: 1800.3.v
• Level: $1800$
• Weight: $3$
• Character orbit: 1800.v
• Rep. character: $\chi_{1800}(793,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $90$
• Newform subspaces: $22$
• Sturm bound: $1080$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.v (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(1080\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1800, [\chi])\).

	Total	New	Old
Modular forms	1536	90	1446
Cusp forms	1344	90	1254
Eisenstein series	192	0	192

Trace form

\( 90 q + 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1800, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1800.3.v.a	$1800$	$3$	1800.v	5.c	$4$	$2$	$1$	$49.046$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-7-7i)q^{7}+4q^{11}+(-12+12i)q^{13}+\cdots\)
1800.3.v.b	$1800$	$3$	1800.v	5.c	$4$	$2$	$1$	$49.046$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4-4i)q^{7}-2^{4}q^{11}+(12-12i)q^{13}+\cdots\)
1800.3.v.c	$1800$	$3$	1800.v	5.c	$4$	$2$	$1$	$49.046$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4-4i)q^{7}+2^{4}q^{11}+(12-12i)q^{13}+\cdots\)
1800.3.v.d	$1800$	$3$	1800.v	5.c	$4$	$2$	$1$	$49.046$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3+3i)q^{7}+14q^{11}+(3-3i)q^{13}+\cdots\)
1800.3.v.e	$1800$	$3$	1800.v	5.c	$4$	$2$	$1$	$49.046$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4i)q^{7}-2^{4}q^{11}+(-12+12i)q^{13}+\cdots\)
1800.3.v.f	$1800$	$3$	1800.v	5.c	$4$	$2$	$1$	$49.046$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4i)q^{7}+2^{4}q^{11}+(-12+12i)q^{13}+\cdots\)
1800.3.v.g	$1800$	$3$	1800.v	5.c	$4$	$2$	$1$	$49.046$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(7+7i)q^{7}+4q^{11}+(12-12i)q^{13}+\cdots\)
1800.3.v.h	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4\beta _{1}-4\beta _{2})q^{7}+(-5-6\beta _{1}+\cdots)q^{11}+\cdots\)
1800.3.v.i	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4\beta _{2}+3\beta _{3})q^{7}+(2-4\beta _{1}+\cdots)q^{11}+\cdots\)
1800.3.v.j	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4\beta _{2}+\beta _{3})q^{7}+(2+2\beta _{1}-2\beta _{3})q^{11}+\cdots\)
1800.3.v.k	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+3\beta _{1}+\beta _{3})q^{7}+(-6+\beta _{1}+\cdots)q^{11}+\cdots\)
1800.3.v.l	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\beta _{2}+\beta _{3})q^{7}+(2+2\beta _{1}-2\beta _{3})q^{11}+\cdots\)
1800.3.v.m	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2\beta _{2}+\beta _{3})q^{7}+(2-2\beta _{1}+2\beta _{3})q^{11}+\cdots\)
1800.3.v.n	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-3\beta _{1}+\beta _{2}-\beta _{3})q^{7}+(4-\beta _{3})q^{11}+\cdots\)
1800.3.v.o	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4\beta _{1}+4\beta _{2})q^{7}+(-5+6\beta _{1}+\cdots)q^{11}+\cdots\)
1800.3.v.p	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4-4\beta _{2}-3\beta _{3})q^{7}+(2-4\beta _{1}+4\beta _{3})q^{11}+\cdots\)
1800.3.v.q	$1800$	$3$	1800.v	5.c	$4$	$4$	$2$	$49.046$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4-4\beta _{2}+\beta _{3})q^{7}+(2-2\beta _{1}+2\beta _{3})q^{11}+\cdots\)
1800.3.v.r	$1800$	$3$	1800.v	5.c	$4$	$6$	$3$	$49.046$	6.0.4315964416.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{7}+(-4+\beta _{4})q^{11}+\cdots\)
1800.3.v.s	$1800$	$3$	1800.v	5.c	$4$	$6$	$3$	$49.046$	6.0.4315964416.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{7}+(4-\beta _{4})q^{11}+(2+\cdots)q^{13}+\cdots\)
1800.3.v.t	$1800$	$3$	1800.v	5.c	$4$	$8$	$4$	$49.046$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{4})q^{7}+(-4+\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{11}+\cdots\)
1800.3.v.u	$1800$	$3$	1800.v	5.c	$4$	$8$	$4$	$49.046$	8.0.\(\cdots\).28	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\beta _{1}+\beta _{3})q^{7}+(-3\beta _{1}+\beta _{3}+3\beta _{5}+\cdots)q^{11}+\cdots\)
1800.3.v.v	$1800$	$3$	1800.v	5.c	$4$	$8$	$4$	$49.046$	8.0.\(\cdots\).28	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2\beta _{1}+\beta _{3})q^{7}+(3\beta _{1}-\beta _{3}-3\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1800, [\chi])\) into lower level spaces

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