Space of modular forms of level 50, weight 3, and Character 50.c

• Label: 50.3.c
• Level: $50$
• Weight: $3$
• Character orbit: 50.c
• Rep. character: $\chi_{50}(7,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $6$
• Newform subspaces: $3$
• Sturm bound: $22$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	50.c (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(22\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(50, [\chi])\).

	Total	New	Old
Modular forms	42	6	36
Cusp forms	18	6	12
Eisenstein series	24	0	24

Trace form

6 * q + 2 * q^2 + 4 * q^3 - 16 * q^6 - 4 * q^7 - 4 * q^8 + 32 * q^11 + 8 * q^12 - 6 * q^13 - 24 * q^16 - 14 * q^17 - 2 * q^18 - 88 * q^21 - 16 * q^22 + 4 * q^23 + 84 * q^26 + 40 * q^27 + 8 * q^28 + 72 * q^31 - 8 * q^32 - 32 * q^33 + 68 * q^36 + 6 * q^37 + 40 * q^38 - 208 * q^41 - 16 * q^42 + 84 * q^43 - 16 * q^46 + 36 * q^47 - 16 * q^48 + 232 * q^51 - 12 * q^52 - 106 * q^53 - 32 * q^56 - 80 * q^57 - 80 * q^58 + 32 * q^61 + 104 * q^62 + 4 * q^63 - 352 * q^66 - 124 * q^67 + 28 * q^68 - 248 * q^71 - 4 * q^72 + 94 * q^73 - 80 * q^76 + 32 * q^77 - 24 * q^78 + 466 * q^81 - 16 * q^82 - 36 * q^83 + 384 * q^86 + 160 * q^87 + 32 * q^88 + 312 * q^91 + 8 * q^92 + 208 * q^93 + 64 * q^96 + 126 * q^97 + 82 * q^98

Decomposition of \(S_{3}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.3.c.a	$50$	$3$	50.c	5.c	$4$	$2$	$1$	$1.362$	\(\Q(\sqrt{-1}) \)	None		✓			50.3.c.a	$2$	$0$	\(-2\)	\(6\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+(-3 i+3)q^{3}+2 i q^{4}+\cdots\)
50.3.c.b	$50$	$3$	50.c	5.c	$4$	$2$	$1$	$1.362$	\(\Q(\sqrt{-1}) \)	None		✓			50.3.c.a	$2$	$0$	\(2\)	\(-6\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{2}+(3 i-3)q^{3}+2 i q^{4}+\cdots\)
50.3.c.c	$50$	$3$	50.c	5.c	$4$	$2$	$1$	$1.362$	\(\Q(\sqrt{-1}) \)	None					10.3.c.a	$2$	$0$	\(2\)	\(4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{2}+(-2 i+2)q^{3}+2 i q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)