Space of modular forms of level 50, weight 2, and Character 50.d

• Label: 50.2.d
• Level: $50$
• Weight: $2$
• Character orbit: 50.d
• Rep. character: $\chi_{50}(11,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $15$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	50.d (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(15\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(50, [\chi])\).

	Total	New	Old
Modular forms	36	12	24
Cusp forms	20	12	8
Eisenstein series	16	0	16

Trace form

12 * q - q^2 - 4 * q^3 - 3 * q^4 + 5 * q^5 - 2 * q^6 - 8 * q^7 - q^8 - 9 * q^9 - 5 * q^10 + 4 * q^11 + 6 * q^12 - 14 * q^13 - 4 * q^14 - 3 * q^16 - 8 * q^17 + 12 * q^18 + 10 * q^19 - 16 * q^21 + 8 * q^22 + 6 * q^23 + 8 * q^24 + 25 * q^25 + 18 * q^26 + 20 * q^27 + 2 * q^28 - 6 * q^31 + 4 * q^32 + 22 * q^33 + 11 * q^34 - 30 * q^35 - 9 * q^36 - 23 * q^37 - 20 * q^38 - 8 * q^39 - 5 * q^40 + 4 * q^41 - 22 * q^42 - 4 * q^43 - 6 * q^44 + 5 * q^45 + 8 * q^46 + 22 * q^47 + 6 * q^48 + 4 * q^49 - 25 * q^50 - 16 * q^51 - 14 * q^52 - 9 * q^53 - 20 * q^54 - 10 * q^55 - 4 * q^56 - 30 * q^58 + 4 * q^61 + 28 * q^62 + 16 * q^63 - 3 * q^64 - 15 * q^65 - 24 * q^66 + 2 * q^67 + 42 * q^68 + 52 * q^69 + 40 * q^70 - 6 * q^71 - 13 * q^72 + 6 * q^73 + 26 * q^74 + 50 * q^75 + 24 * q^77 + 24 * q^78 - 5 * q^80 - 33 * q^81 + 38 * q^82 + 26 * q^83 + 14 * q^84 + 35 * q^85 - 22 * q^86 - 2 * q^88 - 5 * q^89 + 25 * q^90 + 4 * q^91 + 6 * q^92 - 68 * q^93 - 24 * q^94 - 60 * q^95 - 2 * q^96 - 58 * q^97 - 17 * q^98 - 68 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.2.d.a	$50$	$2$	50.d	25.d	$5$	$4$	$1$	$0.399$	\(\Q(\zeta_{10})\)	None		✓			50.2.d.a	$2$	$0$	\(1\)	\(-1\)	\(5\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-1+\cdots)q^{3}+\cdots\)
50.2.d.b	$50$	$2$	50.d	25.d	$5$	$8$	$2$	$0.399$	8.0.58140625.2	None		✓	✓		50.2.d.b	$2$	$0$	\(-2\)	\(-3\)	\(0\)	\(4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+\beta _{2}+\beta _{3}+\beta _{6})q^{2}+\beta _{4}q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

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