Space of modular forms of level 50, weight 4, and Character 50.d

• Label: 50.4.d
• Level: $50$
• Weight: $4$
• Character orbit: 50.d
• Rep. character: $\chi_{50}(11,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $28$
• Newform subspaces: $2$
• Sturm bound: $30$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
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:			\(30\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(50, [\chi])\).

	Total	New	Old
Modular forms	100	28	72
Cusp forms	84	28	56
Eisenstein series	16	0	16

Trace form

28 * q - 2 * q^2 + 8 * q^3 - 28 * q^4 + 35 * q^5 + 12 * q^6 + 4 * q^7 - 8 * q^8 - 109 * q^9 - 10 * q^10 + 106 * q^11 - 48 * q^12 + 58 * q^13 + 56 * q^14 - 180 * q^15 - 112 * q^16 + 254 * q^17 + 376 * q^18 + 110 * q^19 + 80 * q^20 + 36 * q^21 - 384 * q^22 - 642 * q^23 - 192 * q^24 - 265 * q^25 - 588 * q^26 + 200 * q^27 - 104 * q^28 + 120 * q^29 + 200 * q^30 + 486 * q^31 + 128 * q^32 - 544 * q^33 + 646 * q^34 + 880 * q^35 - 436 * q^36 - 111 * q^37 + 200 * q^38 + 1012 * q^39 - 40 * q^40 - 134 * q^41 + 956 * q^42 - 312 * q^43 - 136 * q^44 - 775 * q^45 + 832 * q^46 - 2556 * q^47 - 192 * q^48 + 864 * q^49 - 50 * q^50 - 884 * q^51 + 232 * q^52 + 663 * q^53 - 120 * q^54 - 1820 * q^55 + 224 * q^56 - 240 * q^57 + 180 * q^58 + 2250 * q^59 + 2360 * q^60 - 1764 * q^61 + 416 * q^62 + 6608 * q^63 - 448 * q^64 + 1415 * q^65 - 16 * q^66 + 3364 * q^67 - 2184 * q^68 - 4708 * q^69 - 2360 * q^70 + 5096 * q^71 - 296 * q^72 - 5522 * q^73 - 3804 * q^74 - 7540 * q^75 - 800 * q^76 - 6512 * q^77 - 488 * q^78 - 2800 * q^79 - 80 * q^80 + 5333 * q^81 - 564 * q^82 - 942 * q^83 + 624 * q^84 + 8025 * q^85 + 692 * q^86 + 370 * q^87 + 1584 * q^88 + 7345 * q^89 - 550 * q^90 - 3764 * q^91 + 2832 * q^92 + 4296 * q^93 + 656 * q^94 + 2800 * q^95 + 192 * q^96 - 4676 * q^97 - 2626 * q^98 - 2988 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.4.d.a	$50$	$4$	50.d	25.d	$5$	$12$	$3$	$2.950$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			50.4.d.a	$2$	$0$	\(6\)	\(1\)	\(20\)	\(58\)	$5^{3}$	$\mathrm{SU}(2)[C_{5}]$	\(q+2\beta _{5}q^{2}+(-\beta _{2}+\beta _{3}-\beta _{4}+\beta _{5}+\cdots)q^{3}+\cdots\)
50.4.d.b	$50$	$4$	50.d	25.d	$5$	$16$	$4$	$2.950$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓		50.4.d.b	$2$	$0$	\(-8\)	\(7\)	\(15\)	\(-54\)	$5^{4}$	$\mathrm{SU}(2)[C_{5}]$	\(q+2\beta _{6}q^{2}+(1-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

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