Space of modular forms of level 50, weight 9, and Character 50.c

• Label: 50.9.c
• Level: $50$
• Weight: $9$
• Character orbit: 50.c
• Rep. character: $\chi_{50}(7,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $24$
• Newform subspaces: $6$
• Sturm bound: $67$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	50.c (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(67\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(50, [\chi])\).

	Total	New	Old
Modular forms	132	24	108
Cusp forms	108	24	84
Eisenstein series	24	0	24

Trace form

24 * q - 140 * q^3 - 1024 * q^6 - 4540 * q^7 + 69168 * q^11 - 17920 * q^12 - 119280 * q^13 - 393216 * q^16 - 202560 * q^17 - 158720 * q^18 + 598328 * q^21 - 40960 * q^22 - 174540 * q^23 - 1841664 * q^26 + 704560 * q^27 + 581120 * q^28 + 1696008 * q^31 + 3190840 * q^33 + 12434432 * q^36 + 6531240 * q^37 + 5076480 * q^38 - 26032272 * q^41 - 13150720 * q^42 - 12059820 * q^43 + 34820096 * q^46 + 8748660 * q^47 + 2293760 * q^48 - 61195472 * q^51 - 15267840 * q^52 - 30279480 * q^53 + 28311552 * q^56 + 21729760 * q^57 + 24194560 * q^58 - 10823032 * q^61 - 56017920 * q^62 - 21701900 * q^63 + 55903232 * q^66 + 47676980 * q^67 + 25927680 * q^68 + 97401288 * q^71 - 20316160 * q^72 + 72484120 * q^73 - 33336320 * q^76 + 200406840 * q^77 + 100792320 * q^78 - 420356576 * q^81 - 120186880 * q^82 - 135315900 * q^83 + 353865216 * q^86 - 66654080 * q^87 + 5242880 * q^88 + 111682008 * q^91 - 22341120 * q^92 - 115691240 * q^93 + 16777216 * q^96 - 173745000 * q^97 - 115169280 * q^98

Decomposition of \(S_{9}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.9.c.a	$50$	$9$	50.c	5.c	$4$	$4$	$2$	$20.369$	\(\Q(i, \sqrt{29})\)	None		✓			50.9.c.a	$2$	$0$	\(-32\)	\(-96\)	\(0\)	\(5664\)	$2^{3}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-8-8\beta _{1})q^{2}+(-24+24\beta _{1}-\beta _{3})q^{3}+\cdots\)
50.9.c.b	$50$	$9$	50.c	5.c	$4$	$4$	$2$	$20.369$	\(\Q(i, \sqrt{601})\)	None					10.9.c.b	$2$	$0$	\(-32\)	\(-86\)	\(0\)	\(-5726\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-8+8\beta _{1})q^{2}+(-22-22\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
50.9.c.c	$50$	$9$	50.c	5.c	$4$	$4$	$2$	$20.369$	\(\Q(i, \sqrt{6})\)	None		✓			50.9.c.c	$2$	$0$	\(-32\)	\(144\)	\(0\)	\(4704\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-8+8\beta _{2})q^{2}+(6^{2}-11\beta _{1}+6^{2}\beta _{2}+\cdots)q^{3}+\cdots\)
50.9.c.d	$50$	$9$	50.c	5.c	$4$	$4$	$2$	$20.369$	\(\Q(i, \sqrt{6})\)	None		✓			50.9.c.c	$2$	$0$	\(32\)	\(-144\)	\(0\)	\(-4704\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(8-8\beta _{2})q^{2}+(-6^{2}-11\beta _{1}-6^{2}\beta _{2}+\cdots)q^{3}+\cdots\)
50.9.c.e	$50$	$9$	50.c	5.c	$4$	$4$	$2$	$20.369$	\(\Q(i, \sqrt{249})\)	None					10.9.c.a	$2$	$0$	\(32\)	\(-54\)	\(0\)	\(1186\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(8+8\beta _{1})q^{2}+(-14+13\beta _{1}-\beta _{3})q^{3}+\cdots\)
50.9.c.f	$50$	$9$	50.c	5.c	$4$	$4$	$2$	$20.369$	\(\Q(i, \sqrt{29})\)	None		✓			50.9.c.a	$2$	$0$	\(32\)	\(96\)	\(0\)	\(-5664\)	$2^{3}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(8+8\beta _{1})q^{2}+(24-24\beta _{1}+\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)