Space of modular forms of level 50 and weight 8

• Label: 50.8
• Level: 50
• Weight: 8
• Dimension: 162
• Nonzero newspaces: 4
• Newform subspaces: 18
• Sturm bound: 1200
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 18 \)
Sturm bound:			\(1200\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(50))\).

	Total	New	Old
Modular forms	553	162	391
Cusp forms	497	162	335
Eisenstein series	56	0	56

Trace form

162 * q + 8 * q^2 - 92 * q^3 + 192 * q^4 + 25 * q^5 - 1952 * q^6 - 3256 * q^7 + 512 * q^8 + 22511 * q^9 - 2040 * q^10 - 28596 * q^11 - 5888 * q^12 + 13058 * q^13 + 35776 * q^14 + 33980 * q^15 - 53248 * q^16 - 212326 * q^17 - 172384 * q^18 + 43260 * q^19 + 53760 * q^20 + 356584 * q^21 + 79456 * q^22 - 426272 * q^23 - 172032 * q^24 - 709515 * q^25 + 215568 * q^26 + 921940 * q^27 + 222976 * q^28 + 576010 * q^29 + 327840 * q^30 + 195304 * q^31 - 131072 * q^32 - 2063304 * q^33 + 282696 * q^34 + 1594300 * q^35 - 297024 * q^36 + 719049 * q^37 - 1686560 * q^38 - 5763648 * q^39 - 212480 * q^40 - 1686406 * q^41 + 246016 * q^42 + 6999028 * q^43 + 2728704 * q^44 + 6607445 * q^45 + 509248 * q^46 - 4449416 * q^47 - 376832 * q^48 - 9335106 * q^49 - 3383800 * q^50 - 6841516 * q^51 + 835712 * q^52 + 4047393 * q^53 + 10622400 * q^54 + 8490540 * q^55 + 618496 * q^56 + 11466400 * q^57 - 6165840 * q^58 + 3731400 * q^59 - 1853440 * q^60 - 14306106 * q^61 - 14813984 * q^62 - 20532092 * q^63 + 786432 * q^64 + 24853085 * q^65 + 15557536 * q^66 + 31637044 * q^67 + 12003456 * q^68 + 14522412 * q^69 - 1138880 * q^70 + 11218424 * q^71 - 1701376 * q^72 - 23922002 * q^73 - 32091184 * q^74 - 53853940 * q^75 - 2269440 * q^76 - 28949312 * q^77 + 5628512 * q^78 + 2089120 * q^79 + 102400 * q^80 - 1522213 * q^81 + 39976016 * q^82 + 56747188 * q^83 + 15554304 * q^84 + 108685 * q^85 - 21817472 * q^86 - 73470860 * q^87 + 6948864 * q^88 - 11090115 * q^89 - 39668280 * q^90 + 9535104 * q^91 + 24004352 * q^92 + 104128876 * q^93 - 26547584 * q^94 - 50318460 * q^95 - 7995392 * q^96 - 29442206 * q^97 + 18012744 * q^98 - 27750808 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(50))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
50.8.a	\(\chi_{50}(1, \cdot)\)	50.8.a.a	1	1
		50.8.a.b	1
		50.8.a.c	1
		50.8.a.d	1
		50.8.a.e	1
		50.8.a.f	1
		50.8.a.g	1
		50.8.a.h	1
		50.8.a.i	2
		50.8.a.j	2
50.8.b	\(\chi_{50}(49, \cdot)\)	50.8.b.a	2	1
		50.8.b.b	2
		50.8.b.c	2
		50.8.b.d	2
		50.8.b.e	2
50.8.d	\(\chi_{50}(11, \cdot)\)	50.8.d.a	32	4
		50.8.d.b	36
50.8.e	\(\chi_{50}(9, \cdot)\)	50.8.e.a	72	4

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(50))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(50)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)