Space of modular forms of level 600, weight 8, and Character 600.f

• Label: 600.8.f
• Level: $600$
• Weight: $8$
• Character orbit: 600.f
• Rep. character: $\chi_{600}(49,\cdot)$
• Character field: $\Q$
• Dimension: $62$
• Newform subspaces: $15$
• Sturm bound: $960$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(600, [\chi])\).

	Total	New	Old
Modular forms	864	62	802
Cusp forms	816	62	754
Eisenstein series	48	0	48

Trace form

\( 62 q - 45198 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(600, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.8.f.a	$600$	$8$	600.f	5.b	$2$	$2$	$2$	$187.431$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}iq^{3}+120iq^{7}-3^{6}q^{9}-7196q^{11}+\cdots\)
600.8.f.b	$600$	$8$	600.f	5.b	$2$	$2$	$2$	$187.431$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}iq^{3}+776iq^{7}-3^{6}q^{9}+612q^{11}+\cdots\)
600.8.f.c	$600$	$8$	600.f	5.b	$2$	$2$	$2$	$187.431$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}iq^{3}+540iq^{7}-3^{6}q^{9}+3584q^{11}+\cdots\)
600.8.f.d	$600$	$8$	600.f	5.b	$2$	$2$	$2$	$187.431$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}iq^{3}+504iq^{7}-3^{6}q^{9}+3812q^{11}+\cdots\)
600.8.f.e	$600$	$8$	600.f	5.b	$2$	$2$	$2$	$187.431$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}iq^{3}+1056iq^{7}-3^{6}q^{9}+6412q^{11}+\cdots\)
600.8.f.f	$600$	$8$	600.f	5.b	$2$	$4$	$4$	$187.431$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(448\beta _{1}+\beta _{2})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.g	$600$	$8$	600.f	5.b	$2$	$4$	$4$	$187.431$	\(\Q(i, \sqrt{106})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(-412\beta _{1}-7\beta _{2})q^{7}+\cdots\)
600.8.f.h	$600$	$8$	600.f	5.b	$2$	$4$	$4$	$187.431$	\(\Q(i, \sqrt{106})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}\beta _{1}q^{3}+(352\beta _{1}-\beta _{2})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.i	$600$	$8$	600.f	5.b	$2$	$4$	$4$	$187.431$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(224\beta _{1}-\beta _{2})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.j	$600$	$8$	600.f	5.b	$2$	$4$	$4$	$187.431$	\(\Q(i, \sqrt{114})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(164\beta _{1}-\beta _{2})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.k	$600$	$8$	600.f	5.b	$2$	$4$	$4$	$187.431$	\(\Q(i, \sqrt{7849})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}\beta _{1}q^{3}+(476\beta _{1}+\beta _{2})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.l	$600$	$8$	600.f	5.b	$2$	$6$	$6$	$187.431$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}\beta _{2}q^{3}+(248\beta _{2}-\beta _{4})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.m	$600$	$8$	600.f	5.b	$2$	$6$	$6$	$187.431$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(38\beta _{1}-\beta _{4})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.n	$600$	$8$	600.f	5.b	$2$	$8$	$8$	$187.431$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(137\beta _{1}+\beta _{4})q^{7}-3^{6}q^{9}+\cdots\)
600.8.f.o	$600$	$8$	600.f	5.b	$2$	$8$	$8$	$187.431$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(-43\beta _{1}+\beta _{4})q^{7}-3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(600, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)