Space of modular forms of level 50, weight 6, and Character 50.b

• Label: 50.6.b
• Level: $50$
• Weight: $6$
• Character orbit: 50.b
• Rep. character: $\chi_{50}(49,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $45$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(45\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(50, [\chi])\).

	Total	New	Old
Modular forms	44	8	36
Cusp forms	32	8	24
Eisenstein series	12	0	12

Trace form

\( 8 q - 128 q^{4} + 152 q^{6} - 874 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.6.b.a	$50$	$6$	50.b	5.b	$2$	$2$	$2$	$8.019$	\(\Q(\sqrt{-1}) \)	None					10.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{2}+12iq^{3}-2^{4}q^{4}-96q^{6}+\cdots\)
50.6.b.b	$50$	$6$	50.b	5.b	$2$	$2$	$2$	$8.019$	\(\Q(\sqrt{-1}) \)	None					10.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{2}-3iq^{3}-2^{4}q^{4}+24q^{6}+\cdots\)
50.6.b.c	$50$	$6$	50.b	5.b	$2$	$2$	$2$	$8.019$	\(\Q(\sqrt{-1}) \)	None		✓			50.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{2}-11iq^{3}-2^{4}q^{4}+44q^{6}+\cdots\)
50.6.b.d	$50$	$6$	50.b	5.b	$2$	$2$	$2$	$8.019$	\(\Q(\sqrt{-1}) \)	None					10.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{2}-13iq^{3}-2^{4}q^{4}+104q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)