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

• Label: 1200.3.c
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.c
• Rep. character: $\chi_{1200}(449,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $13$
• Sturm bound: $720$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(720\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	516	74	442
Cusp forms	444	70	374
Eisenstein series	72	4	68

Trace form

\( 70 q + 2 q^{9} + O(q^{10}) \)

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.c.a	$1200$	$3$	1200.c	15.d	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3iq^{3}+11iq^{7}-9q^{9}-iq^{13}+\cdots\)
1200.3.c.b	$1200$	$3$	1200.c	15.d	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3iq^{3}+13iq^{7}-9q^{9}-23iq^{13}+\cdots\)
1200.3.c.c	$1200$	$3$	1200.c	15.d	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3iq^{3}+2iq^{7}-9q^{9}-22iq^{13}+\cdots\)
1200.3.c.d	$1200$	$3$	1200.c	15.d	$2$	$4$	$4$	$32.698$	\(\Q(i, \sqrt{11})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-3+\beta _{3})q^{9}+(-1-2\beta _{3})q^{11}+\cdots\)
1200.3.c.e	$1200$	$3$	1200.c	15.d	$2$	$4$	$4$	$32.698$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{1}q^{7}+(1+2\beta _{2}+\cdots)q^{9}+\cdots\)
1200.3.c.f	$1200$	$3$	1200.c	15.d	$2$	$4$	$4$	$32.698$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{3})q^{3}-3\beta _{1}q^{7}+(1+2\beta _{2}+\cdots)q^{9}+\cdots\)
1200.3.c.g	$1200$	$3$	1200.c	15.d	$2$	$4$	$4$	$32.698$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}q^{7}+(7-2\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1200.3.c.h	$1200$	$3$	1200.c	15.d	$2$	$4$	$4$	$32.698$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{8}+\zeta_{8}^{3})q^{3}+7\zeta_{8}q^{7}+(7-2\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1200.3.c.i	$1200$	$3$	1200.c	15.d	$2$	$4$	$4$	$32.698$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{8}-\zeta_{8}^{2})q^{3}-3\zeta_{8}q^{7}+(7+\zeta_{8}^{3})q^{9}+\cdots\)
1200.3.c.j	$1200$	$3$	1200.c	15.d	$2$	$4$	$4$	$32.698$	\(\Q(i, \sqrt{35})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-8\beta _{2}q^{7}+(9+\beta _{3})q^{9}+(3+\cdots)q^{11}+\cdots\)
1200.3.c.k	$1200$	$3$	1200.c	15.d	$2$	$8$	$8$	$32.698$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-2\beta _{1}-2\beta _{2}+\beta _{7})q^{7}+\cdots\)
1200.3.c.l	$1200$	$3$	1200.c	15.d	$2$	$12$	$12$	$32.698$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{6}-\beta _{10})q^{3}+(-2\beta _{6}-\beta _{7}+\beta _{8}+\cdots)q^{7}+\cdots\)
1200.3.c.m	$1200$	$3$	1200.c	15.d	$2$	$16$	$16$	$32.698$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{26}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{7}+(-3+\cdots)q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)