Space of modular forms of level 50, weight 12, and Character 50.d

• Label: 50.12.d
• Level: $50$
• Weight: $12$
• Character orbit: 50.d
• Rep. character: $\chi_{50}(11,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $108$
• Newform subspaces: $2$
• Sturm bound: $90$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
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:			\(90\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(50, [\chi])\).

	Total	New	Old
Modular forms	340	108	232
Cusp forms	324	108	216
Eisenstein series	16	0	16

Trace form

108 * q - 32 * q^2 - 1012 * q^3 - 27648 * q^4 + 6665 * q^5 + 15552 * q^6 + 45224 * q^7 - 32768 * q^8 - 1544029 * q^9 - 100000 * q^10 + 1791796 * q^11 + 1554432 * q^12 + 2191418 * q^13 + 2151296 * q^14 + 14618280 * q^15 - 28311552 * q^16 - 16943236 * q^17 + 48666496 * q^18 - 27928990 * q^19 - 23142400 * q^20 + 74693616 * q^21 + 54424576 * q^22 + 44600118 * q^23 - 63700992 * q^24 - 10915615 * q^25 - 369371328 * q^26 + 183448820 * q^27 + 126969856 * q^28 - 41210940 * q^29 - 745578880 * q^30 - 219968094 * q^31 + 134217728 * q^32 - 143684614 * q^33 - 413081504 * q^34 - 1346987390 * q^35 - 1581085696 * q^36 + 833337509 * q^37 + 247560320 * q^38 + 919311592 * q^39 - 102400000 * q^40 + 1738857136 * q^41 - 1131187264 * q^42 - 1470108692 * q^43 - 673478656 * q^44 + 3293952785 * q^45 + 4311304192 * q^46 + 10825826934 * q^47 + 1591738368 * q^48 + 20851602884 * q^49 - 312500000 * q^50 - 14327333984 * q^51 + 2244012032 * q^52 + 22605765783 * q^53 + 901092480 * q^54 + 15204424990 * q^55 + 2202927104 * q^56 - 43124245920 * q^57 - 4214028480 * q^58 + 9219690000 * q^59 - 6831308800 * q^60 + 18737584736 * q^61 + 30273422336 * q^62 - 28958480512 * q^63 - 28991029248 * q^64 - 57755551375 * q^65 + 23551917824 * q^66 + 30981614554 * q^67 + 41099175936 * q^68 + 76651064492 * q^69 - 4582614400 * q^70 - 127232954134 * q^71 + 1461551104 * q^72 + 55108572078 * q^73 - 77511305664 * q^74 - 138321589990 * q^75 + 2162688000 * q^76 - 11646859112 * q^77 + 156013869952 * q^78 + 136435848000 * q^79 - 3276800000 * q^80 - 237335655037 * q^81 - 97286306624 * q^82 - 330681697182 * q^83 - 72908199936 * q^84 + 190984182355 * q^85 + 114058455872 * q^86 + 76515405160 * q^87 - 45129465856 * q^88 + 279358368895 * q^89 - 239562563680 * q^90 + 261678778196 * q^91 - 96349267968 * q^92 + 1130106566436 * q^93 + 188681861376 * q^94 + 30629596300 * q^95 + 16307453952 * q^96 + 8257169494 * q^97 - 238983547936 * q^98 - 393806817828 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.12.d.a	$50$	$12$	50.d	25.d	$5$	$52$	$13$	$38.417$		None		✓			50.12.d.a	$2$	$0$	\(416\)	\(-749\)	\(4895\)	\(157068\)		$\mathrm{SU}(2)[C_{5}]$
50.12.d.b	$50$	$12$	50.d	25.d	$5$	$56$	$14$	$38.417$		None		✓	✓		50.12.d.b	$2$	$0$	\(-448\)	\(-263\)	\(1770\)	\(-111844\)		$\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{12}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{12}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)