Space of modular forms of level 50, weight 5, and Character 50.c

• Label: 50.5.c
• Level: $50$
• Weight: $5$
• Character orbit: 50.c
• Rep. character: $\chi_{50}(7,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $37$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	50.c (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(37\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(50, [\chi])\).

	Total	New	Old
Modular forms	72	12	60
Cusp forms	48	12	36
Eisenstein series	24	0	24

Trace form

12 * q - 20 * q^3 + 128 * q^6 - 20 * q^7 - 336 * q^11 - 160 * q^12 + 60 * q^13 - 768 * q^16 + 1020 * q^17 + 640 * q^18 + 2024 * q^21 - 1280 * q^22 - 1620 * q^23 - 192 * q^26 - 320 * q^27 + 160 * q^28 - 1416 * q^31 + 1720 * q^33 + 4256 * q^36 + 1020 * q^37 - 960 * q^38 - 6576 * q^41 + 2240 * q^42 - 660 * q^43 - 4352 * q^46 - 1620 * q^47 + 1280 * q^48 + 9664 * q^51 + 480 * q^52 + 4860 * q^53 - 6144 * q^56 - 5120 * q^57 - 3520 * q^58 - 2456 * q^61 + 3840 * q^62 - 7700 * q^63 + 12416 * q^66 - 8660 * q^67 - 8160 * q^68 + 36024 * q^71 + 5120 * q^72 + 12860 * q^73 - 1600 * q^76 + 14520 * q^77 + 5760 * q^78 - 111428 * q^81 + 2560 * q^82 + 300 * q^83 + 9408 * q^86 - 11840 * q^87 + 10240 * q^88 + 79464 * q^91 - 12960 * q^92 - 2120 * q^93 - 8192 * q^96 + 13500 * q^97 - 3840 * q^98

Decomposition of \(S_{5}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.5.c.a	$50$	$5$	50.c	5.c	$4$	$2$	$1$	$5.168$	\(\Q(\sqrt{-1}) \)	None					10.5.c.b	$2$	$0$	\(-4\)	\(-2\)	\(0\)	\(38\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2 i-2)q^{2}+(i-1)q^{3}+8 i q^{4}+\cdots\)
50.5.c.b	$50$	$5$	50.c	5.c	$4$	$2$	$1$	$5.168$	\(\Q(\sqrt{-1}) \)	None					10.5.c.a	$2$	$0$	\(4\)	\(-18\)	\(0\)	\(-58\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2 i+2)q^{2}+(9 i-9)q^{3}+8 i q^{4}+\cdots\)
50.5.c.c	$50$	$5$	50.c	5.c	$4$	$4$	$2$	$5.168$	\(\Q(i, \sqrt{6})\)	None		✓			50.5.c.c	$2$	$0$	\(-8\)	\(-24\)	\(0\)	\(-144\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\beta _{2})q^{2}+(-6+\beta _{1}-6\beta _{2}+\cdots)q^{3}+\cdots\)
50.5.c.d	$50$	$5$	50.c	5.c	$4$	$4$	$2$	$5.168$	\(\Q(i, \sqrt{6})\)	None		✓			50.5.c.c	$2$	$0$	\(8\)	\(24\)	\(0\)	\(144\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2\beta _{2})q^{2}+(6+\beta _{1}+6\beta _{2})q^{3}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

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