Space of modular forms of level 600, weight 2, and Character 600.f

• Label: 600.2.f
• Level: $600$
• Weight: $2$
• Character orbit: 600.f
• Rep. character: $\chi_{600}(49,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $5$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(600, [\chi])\).

	Total	New	Old
Modular forms	144	10	134
Cusp forms	96	10	86
Eisenstein series	48	0	48

Trace form

\( 10 q - 10 q^{9} - 8 q^{11} + 20 q^{19} + 4 q^{21} - 4 q^{29} - 4 q^{31} + 8 q^{39} + 28 q^{41} - 30 q^{49} - 12 q^{51} - 8 q^{59} + 16 q^{61} - 8 q^{69} + 32 q^{79} + 10 q^{81} - 12 q^{89} - 36 q^{91} + 8 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(600, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.2.f.a	$600$	$2$	600.f	5.b	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-1}) \)	None		✓			600.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+5 i q^{7}-q^{9}-6 q^{11}-3 i q^{13}+\cdots\)
600.2.f.b	$600$	$2$	600.f	5.b	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-1}) \)	None					120.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-q^{9}-4 q^{11}+6 i q^{13}+6 i q^{17}+\cdots\)
600.2.f.c	$600$	$2$	600.f	5.b	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-1}) \)	None					120.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-4 i q^{7}-q^{9}-6 i q^{13}+\cdots\)
600.2.f.d	$600$	$2$	600.f	5.b	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-1}) \)	None		✓			600.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-3 i q^{7}-q^{9}+2 q^{11}-3 i q^{13}+\cdots\)
600.2.f.e	$600$	$2$	600.f	5.b	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-1}) \)	None					24.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-q^{9}+4 q^{11}+2 i q^{13}+2 i q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(600, [\chi])\) into lower level spaces

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