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

• Label: 600.2.k
• Level: $600$
• Weight: $2$
• Character orbit: 600.k
• Rep. character: $\chi_{600}(301,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $6$
• Sturm bound: $240$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(240\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\)

Dimensions

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

	Total	New	Old
Modular forms	132	38	94
Cusp forms	108	38	70
Eisenstein series	24	0	24

Trace form

\( 38 q - 2 q^{2} + 4 q^{4} + 2 q^{6} - 4 q^{7} - 8 q^{8} - 38 q^{9} + O(q^{10}) \)

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	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.k.a	$600$	$2$	600.k	8.b	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-1}) \)	None					120.2.k.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i-1)q^{2}-i q^{3}-2 i q^{4}+(i+1)q^{6}+\cdots\)
600.2.k.b	$600$	$2$	600.k	8.b	$2$	$2$	$2$	$4.791$	\(\Q(\sqrt{-1}) \)	None					24.2.d.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+1)q^{2}+i q^{3}+2 i q^{4}+(i-1)q^{6}+\cdots\)
600.2.k.c	$600$	$2$	600.k	8.b	$2$	$6$	$6$	$4.791$	6.0.399424.1	None					120.2.k.b	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{2}-\beta _{3})q^{4}+\beta _{4}q^{6}+\cdots\)
600.2.k.d	$600$	$2$	600.k	8.b	$2$	$8$	$8$	$4.791$	8.0.214798336.3	None		✓			600.2.k.d	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+\beta _{2}q^{3}+(-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
600.2.k.e	$600$	$2$	600.k	8.b	$2$	$8$	$8$	$4.791$	8.0.214798336.3	None		✓			600.2.k.d	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-\beta _{2}q^{3}+(-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
600.2.k.f	$600$	$2$	600.k	8.b	$2$	$12$	$12$	$4.791$	12.0.\(\cdots\).1	None			✓		120.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-\beta _{5}q^{6}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)