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

• Label: 50.2.e
• Level: $50$
• Weight: $2$
• Character orbit: 50.e
• Rep. character: $\chi_{50}(9,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $8$
• Newform subspaces: $1$
• Sturm bound: $15$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	50.e (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(15\)
Trace bound:			\(0\)

Dimensions

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

	Total	New	Old
Modular forms	40	8	32
Cusp forms	24	8	16
Eisenstein series	16	0	16

Trace form

8 * q + 2 * q^4 - 10 * q^5 + 2 * q^6 - 4 * q^9 - 4 * q^11 - 10 * q^12 - 4 * q^14 - 20 * q^15 - 2 * q^16 + 10 * q^17 + 10 * q^19 + 16 * q^21 + 20 * q^22 + 10 * q^23 + 8 * q^24 + 10 * q^25 - 28 * q^26 + 10 * q^28 + 10 * q^29 + 20 * q^30 + 6 * q^31 - 30 * q^33 - 4 * q^34 + 10 * q^35 + 4 * q^36 - 10 * q^37 - 8 * q^39 - 14 * q^41 - 10 * q^42 - 6 * q^44 + 10 * q^45 - 8 * q^46 - 30 * q^47 - 10 * q^48 - 16 * q^49 + 16 * q^51 - 20 * q^54 - 10 * q^55 + 4 * q^56 - 14 * q^61 + 20 * q^63 + 2 * q^64 + 50 * q^65 + 24 * q^66 + 10 * q^67 - 8 * q^69 - 34 * q^71 + 36 * q^74 + 10 * q^75 + 40 * q^77 + 20 * q^78 - 12 * q^81 + 50 * q^83 + 14 * q^84 - 20 * q^85 + 22 * q^86 + 20 * q^87 - 10 * q^88 + 20 * q^89 - 30 * q^90 - 4 * q^91 - 10 * q^92 - 24 * q^94 + 2 * q^96 - 20 * q^97 - 40 * q^98 - 28 * 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.e.a	$50$	$2$	50.e	25.e	$10$	$8$	$2$	$0.399$	\(\Q(\zeta_{20})\)	None		✓	✓	✓	50.2.e.a	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\zeta_{20}q^{2}+(-1+\zeta_{20}^{2}-\zeta_{20}^{4}+2\zeta_{20}^{6}+\cdots)q^{3}+\cdots\)

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

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