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 + 666 * q^11 - 1744 * q^14 + 2048 * q^16 - 10 * q^19 - 5404 * q^21 - 2432 * q^24 + 9712 * q^26 + 4860 * q^29 - 13484 * q^31 - 5064 * q^34 + 13984 * q^36 + 49192 * q^39 - 25434 * q^41 - 10656 * q^44 - 14928 * q^46 + 6144 * q^49 - 20574 * q^51 + 31480 * q^54 + 27904 * q^56 - 70680 * q^59 + 119416 * q^61 - 32768 * q^64 - 107496 * q^66 - 252348 * q^69 + 161016 * q^71 + 58496 * q^74 + 160 * q^76 + 11060 * q^79 + 27448 * q^81 + 86464 * q^84 - 207008 * q^86 + 89430 * q^89 - 184624 * q^91 + 91776 * q^94 + 38912 * q^96 + 846252 * q^99

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	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{2}+12\beta q^{3}-16 q^{4}-96 q^{6}+\cdots\)
50.6.b.b	$50$	$6$	50.b	5.b	$2$	$2$	$2$	$8.019$	\(\Q(\sqrt{-1}) \)	None					10.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{2}-3\beta q^{3}-16 q^{4}+24 q^{6}+\cdots\)
50.6.b.c	$50$	$6$	50.b	5.b	$2$	$2$	$2$	$8.019$	\(\Q(\sqrt{-1}) \)	None		✓			50.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 i q^{2}-11 i q^{3}-16 q^{4}+44 q^{6}+\cdots\)
50.6.b.d	$50$	$6$	50.b	5.b	$2$	$2$	$2$	$8.019$	\(\Q(\sqrt{-1}) \)	None					10.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{2}-13\beta q^{3}-16 q^{4}+104 q^{6}+\cdots\)

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

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