Space of modular forms of level 1200, weight 2, and Character 1200.f

• Label: 1200.2.f
• Level: $1200$
• Weight: $2$
• Character orbit: 1200.f
• Rep. character: $\chi_{1200}(49,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $9$
• Sturm bound: $480$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(480\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1200, [\chi])\).

	Total	New	Old
Modular forms	276	18	258
Cusp forms	204	18	186
Eisenstein series	72	0	72

Trace form

\( 18 q - 18 q^{9} - 20 q^{19} + 4 q^{29} - 4 q^{31} + 8 q^{39} + 20 q^{41} - 26 q^{49} + 20 q^{51} - 32 q^{59} - 20 q^{61} + 16 q^{69} + 32 q^{71} + 18 q^{81} - 36 q^{89} + 68 q^{91}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1200, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.2.f.a	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					300.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+i q^{7}-q^{9}-6 q^{11}-5 i q^{13}+\cdots\)
1200.2.f.b	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					24.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}-q^{9}-4 q^{11}+2 i q^{13}+2 i q^{17}+\cdots\)
1200.2.f.c	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					600.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+3 i q^{7}-q^{9}-2 q^{11}-3 i q^{13}+\cdots\)
1200.2.f.d	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					75.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+3 i q^{7}-q^{9}-2 q^{11}-i q^{13}+\cdots\)
1200.2.f.e	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					30.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+4 i q^{7}-q^{9}-2 i q^{13}+\cdots\)
1200.2.f.f	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					120.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+4 i q^{7}-q^{9}-6 i q^{13}+\cdots\)
1200.2.f.g	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					120.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-q^{9}+4 q^{11}-6 i q^{13}-6 i q^{17}+\cdots\)
1200.2.f.h	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					15.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-q^{9}+4 q^{11}-2 i q^{13}-2 i q^{17}+\cdots\)
1200.2.f.i	$1200$	$2$	1200.f	5.b	$2$	$2$	$2$	$9.582$	\(\Q(\sqrt{-1}) \)	None					600.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+5 i q^{7}-q^{9}+6 q^{11}+3 i q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

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