Space of modular forms of level 50, weight 8, and Character 50.b

• Label: 50.8.b
• Level: $50$
• Weight: $8$
• Character orbit: 50.b
• Rep. character: $\chi_{50}(49,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $5$
• Sturm bound: $60$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(60\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(50, [\chi])\).

	Total	New	Old
Modular forms	58	10	48
Cusp forms	46	10	36
Eisenstein series	12	0	12

Trace form

10 * q - 640 * q^4 - 1360 * q^6 - 5320 * q^9 - 25230 * q^11 + 2080 * q^14 + 40960 * q^16 - 16850 * q^19 + 217820 * q^21 + 87040 * q^24 + 79040 * q^26 - 591000 * q^29 + 368420 * q^31 + 460080 * q^34 + 340480 * q^36 - 487640 * q^39 - 666330 * q^41 + 1614720 * q^44 - 881760 * q^46 - 2236530 * q^49 - 3229230 * q^51 + 5954800 * q^54 - 133120 * q^56 - 6702000 * q^59 - 5419180 * q^61 - 2621440 * q^64 + 5948880 * q^66 + 8060460 * q^69 + 6441720 * q^71 + 13571680 * q^74 + 1078400 * q^76 - 12812300 * q^79 + 13236010 * q^81 - 13940480 * q^84 - 13773760 * q^86 + 7227150 * q^89 + 2351920 * q^91 - 25389120 * q^94 - 5570560 * q^96 + 17796660 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.8.b.a	$50$	$8$	50.b	5.b	$2$	$2$	$2$	$15.619$	\(\Q(\sqrt{-1}) \)	None		✓			50.8.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8 i q^{2}+87 i q^{3}-64 q^{4}-696 q^{6}+\cdots\)
50.8.b.b	$50$	$8$	50.b	5.b	$2$	$2$	$2$	$15.619$	\(\Q(\sqrt{-1}) \)	None		✓			50.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8 i q^{2}+57 i q^{3}-64 q^{4}-456 q^{6}+\cdots\)
50.8.b.c	$50$	$8$	50.b	5.b	$2$	$2$	$2$	$15.619$	\(\Q(\sqrt{-1}) \)	None					2.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{2}+6\beta q^{3}-64 q^{4}-96 q^{6}+\cdots\)
50.8.b.d	$50$	$8$	50.b	5.b	$2$	$2$	$2$	$15.619$	\(\Q(\sqrt{-1}) \)	None					10.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}+14\beta q^{3}-64 q^{4}+224 q^{6}+\cdots\)
50.8.b.e	$50$	$8$	50.b	5.b	$2$	$2$	$2$	$15.619$	\(\Q(\sqrt{-1}) \)	None		✓			50.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-8 i q^{2}+43 i q^{3}-64 q^{4}+344 q^{6}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)