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

• Label: 600.2.b
• Level: $600$
• Weight: $2$
• Character orbit: 600.b
• Rep. character: $\chi_{600}(251,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $9$
• Sturm bound: $240$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(240\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(7\), \(11\), \(23\), \(43\)

Dimensions

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

	Total	New	Old
Modular forms	132	82	50
Cusp forms	108	70	38
Eisenstein series	24	12	12

Trace form

\( 70 q + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{9} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.2.b.a	$600$	$2$	600.b	24.f	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}+(1-\beta )q^{3}-2q^{4}+(2+\beta )q^{6}+\cdots\)
600.2.b.b	$600$	$2$	600.b	24.f	$2$	$4$	$4$	$4.791$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{2}+(-1-\beta _{2})q^{3}-2q^{4}+(-1+\cdots)q^{6}+\cdots\)
600.2.b.c	$600$	$2$	600.b	24.f	$2$	$4$	$4$	$4.791$	\(\Q(\sqrt{-3}, \sqrt{5})\)	\(\Q(\sqrt{-15}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{2}+(\beta _{1}-\beta _{2})q^{3}+\beta _{3}q^{4}+(-2+\cdots)q^{6}+\cdots\)
600.2.b.d	$600$	$2$	600.b	24.f	$2$	$4$	$4$	$4.791$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{3}-2q^{4}+(-1+\cdots)q^{6}+\cdots\)
600.2.b.e	$600$	$2$	600.b	24.f	$2$	$8$	$8$	$4.791$	8.0.1649659456.5	None		$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(\beta _{1}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
600.2.b.f	$600$	$2$	600.b	24.f	$2$	$8$	$8$	$4.791$	8.0.1649659456.5	None		$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(\beta _{1}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
600.2.b.g	$600$	$2$	600.b	24.f	$2$	$12$	$12$	$4.791$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{8}q^{2}-\beta _{5}q^{3}+(1+\beta _{4})q^{4}+(-\beta _{4}+\cdots)q^{6}+\cdots\)
600.2.b.h	$600$	$2$	600.b	24.f	$2$	$12$	$12$	$4.791$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
600.2.b.i	$600$	$2$	600.b	24.f	$2$	$16$	$16$	$4.791$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{15}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{10}q^{2}+\beta _{5}q^{3}-\beta _{14}q^{4}+(\beta _{7}+\beta _{12}+\cdots)q^{6}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)