Space of modular forms of level 1200, weight 3, and Character 1200.l

• Label: 1200.3.l
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.l
• Rep. character: $\chi_{1200}(401,\cdot)$
• Character field: $\Q$
• Dimension: $73$
• Newform subspaces: $25$
• Sturm bound: $720$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.l (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(720\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1200, [\chi])\).

	Total	New	Old
Modular forms	516	79	437
Cusp forms	444	73	371
Eisenstein series	72	6	66

Trace form

\( 73 q - q^{3} + 6 q^{7} - 3 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1200, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.3.l.a	$1200$	$3$	1200.l	3.b	$2$	$1$	$1$	$32.698$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(-13\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}-13q^{7}+9q^{9}-23q^{13}-11q^{19}+\cdots\)
1200.3.l.b	$1200$	$3$	1200.l	3.b	$2$	$1$	$1$	$32.698$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(2\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+2q^{7}+9q^{9}+22q^{13}-26q^{19}+\cdots\)
1200.3.l.c	$1200$	$3$	1200.l	3.b	$2$	$1$	$1$	$32.698$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(11\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+11q^{7}+9q^{9}+q^{13}+37q^{19}+\cdots\)
1200.3.l.d	$1200$	$3$	1200.l	3.b	$2$	$1$	$1$	$32.698$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-11\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}-11q^{7}+9q^{9}-q^{13}+37q^{19}+\cdots\)
1200.3.l.e	$1200$	$3$	1200.l	3.b	$2$	$1$	$1$	$32.698$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(3\)	\(0\)	\(13\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}+13q^{7}+9q^{9}+23q^{13}-11q^{19}+\cdots\)
1200.3.l.f	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-11}) \)	None		$2$	$0$	\(0\)	\(-5\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3+\beta )q^{3}+(6-5\beta )q^{9}+(5-10\beta )q^{11}+\cdots\)
1200.3.l.g	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-5}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta )q^{3}-6q^{7}+(-1-4\beta )q^{9}+\cdots\)
1200.3.l.h	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-5}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta )q^{3}+8q^{7}+(-1-4\beta )q^{9}+\cdots\)
1200.3.l.i	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-14\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta )q^{3}-7q^{7}+(-7-2\beta )q^{9}+\cdots\)
1200.3.l.j	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta )q^{3}+q^{7}+(-7+2\beta )q^{9}+\cdots\)
1200.3.l.k	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-35}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}-8q^{7}+(-9+\beta )q^{9}+(3-6\beta )q^{11}+\cdots\)
1200.3.l.l	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-15}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3iq^{3}-9q^{9}+14iq^{17}-22q^{19}+\cdots\)
1200.3.l.m	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-35}) \)	None		$2$	$0$	\(0\)	\(1\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+8q^{7}+(-9+\beta )q^{9}+(3-6\beta )q^{11}+\cdots\)
1200.3.l.n	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(-12\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta )q^{3}-6q^{7}+(-7+2\beta )q^{9}+\cdots\)
1200.3.l.o	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta )q^{3}-q^{7}+(-7-2\beta )q^{9}+3\beta q^{11}+\cdots\)
1200.3.l.p	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(14\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta )q^{3}+7q^{7}+(-7-2\beta )q^{9}+\cdots\)
1200.3.l.q	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-5}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta )q^{3}-8q^{7}+(-1-4\beta )q^{9}+\cdots\)
1200.3.l.r	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-5}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta )q^{3}+2q^{7}+(-1+4\beta )q^{9}+\cdots\)
1200.3.l.s	$1200$	$3$	1200.l	3.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-11}) \)	None		$2$	$0$	\(0\)	\(5\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3-\beta )q^{3}+(6-5\beta )q^{9}+(5-10\beta )q^{11}+\cdots\)
1200.3.l.t	$1200$	$3$	1200.l	3.b	$2$	$4$	$4$	$32.698$	\(\Q(\sqrt{-2}, \sqrt{-17})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{3}+(-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
1200.3.l.u	$1200$	$3$	1200.l	3.b	$2$	$4$	$4$	$32.698$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{3})q^{3}+(2+\beta _{1}-2\beta _{2}-2\beta _{3})q^{7}+\cdots\)
1200.3.l.v	$1200$	$3$	1200.l	3.b	$2$	$6$	$6$	$32.698$	6.0.574198272.1	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(10\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2})q^{3}+(2+\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
1200.3.l.w	$1200$	$3$	1200.l	3.b	$2$	$6$	$6$	$32.698$	6.0.574198272.1	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(-10\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{2})q^{3}+(-2+\beta _{1})q^{7}+(3+\beta _{1}+\cdots)q^{9}+\cdots\)
1200.3.l.x	$1200$	$3$	1200.l	3.b	$2$	$8$	$8$	$32.698$	8.0.\(\cdots\).5	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(16\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(2+\beta _{3}-\beta _{4}-\beta _{7})q^{7}+(2+\cdots)q^{9}+\cdots\)
1200.3.l.y	$1200$	$3$	1200.l	3.b	$2$	$12$	$12$	$32.698$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}-\beta _{5}q^{7}+(-1-\beta _{9})q^{9}+(\beta _{7}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)