Space of modular forms of level 600 and weight 8

• Label: 600.8
• Level: 600
• Weight: 8
• Dimension: 26045
• Nonzero newspaces: 18
• Sturm bound: 153600
• Trace bound: 8

Defining parameters

Level:	\( N \)	=	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 18 \)
Sturm bound:			\(153600\)
Trace bound:			\(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(600))\).

	Total	New	Old
Modular forms	67872	26209	41663
Cusp forms	66528	26045	40483
Eisenstein series	1344	164	1180

Trace form

26045 * q - 14 * q^2 + 13 * q^3 - 260 * q^4 + 58 * q^5 - 34 * q^6 - 2364 * q^7 - 428 * q^8 - 5129 * q^9 - 32 * q^10 - 9084 * q^11 + 12340 * q^12 - 63830 * q^13 - 65876 * q^14 + 58712 * q^15 + 196280 * q^16 - 35838 * q^17 - 222318 * q^18 - 258472 * q^19 - 227960 * q^20 - 81520 * q^21 + 373560 * q^22 + 329424 * q^23 + 628896 * q^24 + 24122 * q^25 - 427048 * q^26 - 578591 * q^27 - 1090296 * q^28 + 618162 * q^29 + 989428 * q^30 - 2236172 * q^31 - 1036104 * q^32 + 1191932 * q^33 - 4787308 * q^34 + 1633008 * q^35 - 148264 * q^36 + 919344 * q^37 + 4372648 * q^38 - 7332610 * q^39 + 4367888 * q^40 - 1083350 * q^41 + 351616 * q^42 + 10490904 * q^43 - 8058928 * q^44 + 3793330 * q^45 - 13333824 * q^46 - 1022760 * q^47 + 12601268 * q^48 - 13538527 * q^49 + 7947560 * q^50 - 10635446 * q^51 - 657736 * q^52 + 9581512 * q^53 - 13122886 * q^54 + 9121664 * q^55 + 1643704 * q^56 + 17010512 * q^57 - 2993540 * q^58 - 3990076 * q^59 + 11616388 * q^60 - 15724974 * q^61 + 5767172 * q^62 - 15944552 * q^63 - 18863960 * q^64 + 6541866 * q^65 - 9372380 * q^66 + 45922128 * q^67 + 14608304 * q^68 + 7666488 * q^69 - 19691384 * q^70 - 40089824 * q^71 + 1065224 * q^72 + 61216806 * q^73 - 41765776 * q^74 - 20361048 * q^75 + 10814008 * q^76 - 31805760 * q^77 + 57599588 * q^78 + 62356740 * q^79 + 37919920 * q^80 - 43739529 * q^81 - 85816116 * q^82 + 48410276 * q^83 + 41817812 * q^84 + 129107498 * q^85 - 46716040 * q^86 - 92768718 * q^87 - 210401528 * q^88 + 97169048 * q^89 - 95595412 * q^90 - 71784952 * q^91 - 89423808 * q^92 - 44918736 * q^93 + 45180544 * q^94 + 100893824 * q^95 + 74331628 * q^96 + 126726390 * q^97 + 203689866 * q^98 - 59889316 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(600))\)

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
600.8.a	\(\chi_{600}(1, \cdot)\)	600.8.a.a	1	1
		600.8.a.b	1
		600.8.a.c	1
		600.8.a.d	1
		600.8.a.e	1
		600.8.a.f	2
		600.8.a.g	2
		600.8.a.h	2
		600.8.a.i	2
		600.8.a.j	2
		600.8.a.k	2
		600.8.a.l	3
		600.8.a.m	3
		600.8.a.n	3
		600.8.a.o	3
		600.8.a.p	4
		600.8.a.q	4
		600.8.a.r	4
		600.8.a.s	4
		600.8.a.t	5
		600.8.a.u	5
		600.8.a.v	6
		600.8.a.w	6
600.8.b	\(\chi_{600}(251, \cdot)\)	n/a	526	1
600.8.d	\(\chi_{600}(349, \cdot)\)	n/a	252	1
600.8.f	\(\chi_{600}(49, \cdot)\)	600.8.f.a	2	1
		600.8.f.b	2
		600.8.f.c	2
		600.8.f.d	2
		600.8.f.e	2
		600.8.f.f	4
		600.8.f.g	4
		600.8.f.h	4
		600.8.f.i	4
		600.8.f.j	4
		600.8.f.k	4
		600.8.f.l	6
		600.8.f.m	6
		600.8.f.n	8
		600.8.f.o	8
600.8.h	\(\chi_{600}(551, \cdot)\)	None	0	1
600.8.k	\(\chi_{600}(301, \cdot)\)	n/a	266	1
600.8.m	\(\chi_{600}(299, \cdot)\)	n/a	500	1
600.8.o	\(\chi_{600}(599, \cdot)\)	None	0	1
600.8.r	\(\chi_{600}(257, \cdot)\)	n/a	252	2
600.8.s	\(\chi_{600}(7, \cdot)\)	None	0	2
600.8.v	\(\chi_{600}(43, \cdot)\)	n/a	504	2
600.8.w	\(\chi_{600}(293, \cdot)\)	n/a	1000	2
600.8.y	\(\chi_{600}(121, \cdot)\)	n/a	416	4
600.8.ba	\(\chi_{600}(71, \cdot)\)	None	0	4
600.8.bc	\(\chi_{600}(169, \cdot)\)	n/a	424	4
600.8.be	\(\chi_{600}(109, \cdot)\)	n/a	1680	4
600.8.bg	\(\chi_{600}(11, \cdot)\)	n/a	3344	4
600.8.bi	\(\chi_{600}(119, \cdot)\)	None	0	4
600.8.bk	\(\chi_{600}(59, \cdot)\)	n/a	3344	4
600.8.bm	\(\chi_{600}(61, \cdot)\)	n/a	1680	4
600.8.bp	\(\chi_{600}(53, \cdot)\)	n/a	6688	8
600.8.bq	\(\chi_{600}(67, \cdot)\)	n/a	3360	8
600.8.bt	\(\chi_{600}(103, \cdot)\)	None	0	8
600.8.bu	\(\chi_{600}(17, \cdot)\)	n/a	1680	8

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(600))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(600)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 24}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 18}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(300))\)\(^{\oplus 2}\)