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

• Label: 1200.3.bg
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.bg
• Rep. character: $\chi_{1200}(193,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $17$
• Sturm bound: $720$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(720\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(7\)

Dimensions

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

	Total	New	Old
Modular forms	1032	72	960
Cusp forms	888	72	816
Eisenstein series	144	0	144

Trace form

72 * q - 24 * q^13 + 8 * q^17 - 96 * q^23 - 128 * q^31 - 48 * q^33 - 8 * q^37 + 64 * q^41 + 64 * q^43 + 192 * q^47 + 192 * q^51 + 56 * q^53 - 64 * q^61 - 480 * q^67 - 512 * q^71 + 232 * q^73 - 648 * q^81 + 544 * q^83 + 288 * q^87 + 576 * q^91 - 96 * q^93 - 56 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.3.bg.a	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					150.3.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(-6+6\beta _{2}-\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.b	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					600.3.u.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(-4+4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.c	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					600.3.u.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(-4+4\beta _{2}+3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.d	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					30.3.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+(-4+4\beta _{1}-4\beta _{2})q^{7}-3\beta _{2}q^{9}+\cdots\)
1200.3.bg.e	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					120.3.u.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(-3+3\beta _{2}-2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.f	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					600.3.u.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(-2+2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.g	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					75.3.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}+5\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots\)
1200.3.bg.h	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					300.3.k.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}+2\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots\)
1200.3.bg.i	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					300.3.k.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+3\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots\)
1200.3.bg.j	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					75.3.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+6\beta _{1}q^{7}-3\beta _{2}q^{9}+18q^{11}+\cdots\)
1200.3.bg.k	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					15.3.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}+(1+2\beta _{1}+\beta _{2})q^{7}-3\beta _{2}q^{9}+\cdots\)
1200.3.bg.l	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					600.3.u.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(2-2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.m	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					600.3.u.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(4-4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.n	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					600.3.u.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(4-4\beta _{2}-3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.o	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					60.3.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(5-5\beta _{2}-2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.p	$1200$	$3$	1200.bg	5.c	$4$	$4$	$2$	$32.698$	\(\Q(i, \sqrt{6})\)	None					150.3.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(6-6\beta _{2}-\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots\)
1200.3.bg.q	$1200$	$3$	1200.bg	5.c	$4$	$8$	$4$	$32.698$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None			✓		120.3.u.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(\beta _{3}-\beta _{6})q^{7}-3\beta _{2}q^{9}+(-4+\cdots)q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)