Space of modular forms of level 600, weight 4, and Character 600.f

• Label: 600.4.f
• Level: $600$
• Weight: $4$
• Character orbit: 600.f
• Rep. character: $\chi_{600}(49,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $11$
• Sturm bound: $480$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(480\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(600, [\chi])\).

	Total	New	Old
Modular forms	384	26	358
Cusp forms	336	26	310
Eisenstein series	48	0	48

Trace form

\( 26 q - 234 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(600, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.4.f.a	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None					120.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+20iq^{7}-9q^{9}-56q^{11}+\cdots\)
600.4.f.b	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None					24.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+24iq^{7}-9q^{9}-28q^{11}+\cdots\)
600.4.f.c	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None					120.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+2^{4}iq^{7}-9q^{9}-28q^{11}+\cdots\)
600.4.f.d	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None					120.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}-9q^{9}+4q^{11}-54iq^{13}+\cdots\)
600.4.f.e	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None		✓			600.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+5iq^{7}-9q^{9}+14q^{11}+\cdots\)
600.4.f.f	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None					120.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+20iq^{7}-9q^{9}+2^{4}q^{11}+\cdots\)
600.4.f.g	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None					120.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+8iq^{7}-9q^{9}+20q^{11}+\cdots\)
600.4.f.h	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None		✓			600.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+19iq^{7}-9q^{9}+22q^{11}+\cdots\)
600.4.f.i	$600$	$4$	600.f	5.b	$2$	$2$	$2$	$35.401$	\(\Q(\sqrt{-1}) \)	None					120.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+4iq^{7}-9q^{9}+72q^{11}+\cdots\)
600.4.f.j	$600$	$4$	600.f	5.b	$2$	$4$	$4$	$35.401$	\(\Q(i, \sqrt{109})\)	None		✓			600.4.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{7}-9q^{9}+(-8+\cdots)q^{11}+\cdots\)
600.4.f.k	$600$	$4$	600.f	5.b	$2$	$4$	$4$	$35.401$	\(\Q(i, \sqrt{181})\)	None		✓			600.4.a.r	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-3\beta _{1}+\beta _{2})q^{7}-9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(600, [\chi])\) into lower level spaces

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