Space of modular forms of level 1200, weight 4, and Character 1200.f

• Label: 1200.4.f
• Level: $1200$
• Weight: $4$
• Character orbit: 1200.f
• Rep. character: $\chi_{1200}(49,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $24$
• Sturm bound: $960$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(960\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1200, [\chi])\).

	Total	New	Old
Modular forms	756	54	702
Cusp forms	684	54	630
Eisenstein series	72	0	72

Trace form

54 * q - 486 * q^9 + 252 * q^19 + 284 * q^29 + 636 * q^31 - 312 * q^39 - 356 * q^41 - 2030 * q^49 + 132 * q^51 + 1376 * q^59 + 756 * q^61 - 528 * q^69 - 224 * q^71 - 2240 * q^79 + 4374 * q^81 - 1932 * q^89 - 812 * q^91

Decomposition of \(S_{4}^{\mathrm{new}}(1200, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.4.f.a	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					120.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+4 i q^{7}-9 q^{9}-72 q^{11}+\cdots\)
1200.4.f.b	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					15.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+24 i q^{7}-9 q^{9}-52 q^{11}+\cdots\)
1200.4.f.c	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					150.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+i q^{7}-9 q^{9}-42 q^{11}+\cdots\)
1200.4.f.d	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					12.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+8 i q^{7}-9 q^{9}-36 q^{11}+\cdots\)
1200.4.f.e	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					60.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+32 i q^{7}-9 q^{9}-36 q^{11}+\cdots\)
1200.4.f.f	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					600.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+19 i q^{7}-9 q^{9}-22 q^{11}+\cdots\)
1200.4.f.g	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					120.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+8 i q^{7}-9 q^{9}-20 q^{11}+\cdots\)
1200.4.f.h	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					120.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+20 i q^{7}-9 q^{9}-16 q^{11}+\cdots\)
1200.4.f.i	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					600.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+5 i q^{7}-9 q^{9}-14 q^{11}+\cdots\)
1200.4.f.j	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					6.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+16 i q^{7}-9 q^{9}-12 q^{11}+\cdots\)
1200.4.f.k	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					300.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+13 i q^{7}-9 q^{9}-6 q^{11}+\cdots\)
1200.4.f.l	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					120.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-9 q^{9}-4 q^{11}+54 i q^{13}+\cdots\)
1200.4.f.m	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					15.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+20 i q^{7}-9 q^{9}+24 q^{11}+\cdots\)
1200.4.f.n	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					60.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+28 i q^{7}-9 q^{9}+24 q^{11}+\cdots\)
1200.4.f.o	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					120.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+16 i q^{7}-9 q^{9}+28 q^{11}+\cdots\)
1200.4.f.p	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					24.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+24 i q^{7}-9 q^{9}+28 q^{11}+\cdots\)
1200.4.f.q	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					150.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+23 i q^{7}-9 q^{9}+30 q^{11}+\cdots\)
1200.4.f.r	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					30.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+4 i q^{7}-9 q^{9}+48 q^{11}+\cdots\)
1200.4.f.s	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					300.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+7 i q^{7}-9 q^{9}+54 q^{11}+\cdots\)
1200.4.f.t	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					120.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+20 i q^{7}-9 q^{9}+56 q^{11}+\cdots\)
1200.4.f.u	$1200$	$4$	1200.f	5.b	$2$	$2$	$2$	$70.802$	\(\Q(\sqrt{-1}) \)	None					30.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+32 i q^{7}-9 q^{9}+60 q^{11}+\cdots\)
1200.4.f.v	$1200$	$4$	1200.f	5.b	$2$	$4$	$4$	$70.802$	\(\Q(i, \sqrt{19})\)	None					75.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-13\beta _{1}+\beta _{3})q^{7}-9q^{9}+\cdots\)
1200.4.f.w	$1200$	$4$	1200.f	5.b	$2$	$4$	$4$	$70.802$	\(\Q(i, \sqrt{181})\)	None					600.4.a.r	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-3\beta _{1}+\beta _{2})q^{7}-9q^{9}+\cdots\)
1200.4.f.x	$1200$	$4$	1200.f	5.b	$2$	$4$	$4$	$70.802$	\(\Q(i, \sqrt{109})\)	None					600.4.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{7}-9q^{9}+(8+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

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