Space of modular forms of level 600 and weight 4

• Label: 600.4
• Level: 600
• Weight: 4
• Dimension: 11115
• Nonzero newspaces: 18
• Sturm bound: 76800
• Trace bound: 8

Defining parameters

Level:	\( N \)	=	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 18 \)
Sturm bound:			\(76800\)
Trace bound:			\(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(600))\).

	Total	New	Old
Modular forms	29472	11279	18193
Cusp forms	28128	11115	17013
Eisenstein series	1344	164	1180

Trace form

11115 * q + 2 * q^2 - 11 * q^3 - 4 * q^4 - 22 * q^5 - 34 * q^6 + 28 * q^7 - 76 * q^8 - 35 * q^9 - 32 * q^10 + 84 * q^11 + 148 * q^12 + 142 * q^13 + 428 * q^14 - 88 * q^15 - 584 * q^16 - 210 * q^17 - 798 * q^18 - 856 * q^19 - 760 * q^20 - 136 * q^21 - 616 * q^22 + 576 * q^24 + 2 * q^25 + 1528 * q^26 + 217 * q^27 + 1672 * q^28 + 1566 * q^29 - 332 * q^30 + 1516 * q^31 - 1608 * q^32 - 508 * q^33 - 204 * q^34 - 912 * q^35 - 616 * q^36 - 1308 * q^37 + 3368 * q^38 + 14 * q^39 + 3088 * q^40 - 1258 * q^41 + 3232 * q^42 - 3384 * q^43 + 3280 * q^44 - 1910 * q^45 + 1344 * q^46 - 600 * q^47 - 3916 * q^48 + 375 * q^49 - 2840 * q^50 + 2746 * q^51 - 5768 * q^52 - 412 * q^53 + 890 * q^54 - 1376 * q^55 - 1864 * q^56 + 2120 * q^57 - 2500 * q^58 + 1252 * q^59 + 3268 * q^60 - 362 * q^61 - 2108 * q^62 + 3568 * q^63 - 9304 * q^64 + 5586 * q^65 - 3356 * q^66 - 5456 * q^67 - 2960 * q^68 + 2232 * q^69 - 824 * q^70 + 2192 * q^71 - 1048 * q^72 - 398 * q^73 + 10192 * q^74 + 3192 * q^75 + 10360 * q^76 - 2208 * q^77 + 7940 * q^78 + 12188 * q^79 + 6320 * q^80 - 9555 * q^81 - 13716 * q^82 - 6860 * q^83 - 11308 * q^84 - 19302 * q^85 - 8 * q^86 - 6342 * q^87 - 10872 * q^88 + 1228 * q^89 - 1012 * q^90 - 8488 * q^91 + 960 * q^92 - 792 * q^93 + 18688 * q^94 + 5824 * q^95 - 3860 * q^96 + 10770 * q^97 + 23322 * q^98 - 7828 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(600))\)

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
600.4.a	\(\chi_{600}(1, \cdot)\)	600.4.a.a	1	1
		600.4.a.b	1
		600.4.a.c	1
		600.4.a.d	1
		600.4.a.e	1
		600.4.a.f	1
		600.4.a.g	1
		600.4.a.h	1
		600.4.a.i	1
		600.4.a.j	1
		600.4.a.k	1
		600.4.a.l	1
		600.4.a.m	1
		600.4.a.n	1
		600.4.a.o	1
		600.4.a.p	1
		600.4.a.q	1
		600.4.a.r	2
		600.4.a.s	2
		600.4.a.t	2
		600.4.a.u	2
		600.4.a.v	2
		600.4.a.w	2
600.4.b	\(\chi_{600}(251, \cdot)\)	n/a	222	1
600.4.d	\(\chi_{600}(349, \cdot)\)	n/a	108	1
600.4.f	\(\chi_{600}(49, \cdot)\)	600.4.f.a	2	1
		600.4.f.b	2
		600.4.f.c	2
		600.4.f.d	2
		600.4.f.e	2
		600.4.f.f	2
		600.4.f.g	2
		600.4.f.h	2
		600.4.f.i	2
		600.4.f.j	4
		600.4.f.k	4
600.4.h	\(\chi_{600}(551, \cdot)\)	None	0	1
600.4.k	\(\chi_{600}(301, \cdot)\)	n/a	114	1
600.4.m	\(\chi_{600}(299, \cdot)\)	n/a	212	1
600.4.o	\(\chi_{600}(599, \cdot)\)	None	0	1
600.4.r	\(\chi_{600}(257, \cdot)\)	n/a	108	2
600.4.s	\(\chi_{600}(7, \cdot)\)	None	0	2
600.4.v	\(\chi_{600}(43, \cdot)\)	n/a	216	2
600.4.w	\(\chi_{600}(293, \cdot)\)	n/a	424	2
600.4.y	\(\chi_{600}(121, \cdot)\)	n/a	176	4
600.4.ba	\(\chi_{600}(71, \cdot)\)	None	0	4
600.4.bc	\(\chi_{600}(169, \cdot)\)	n/a	184	4
600.4.be	\(\chi_{600}(109, \cdot)\)	n/a	720	4
600.4.bg	\(\chi_{600}(11, \cdot)\)	n/a	1424	4
600.4.bi	\(\chi_{600}(119, \cdot)\)	None	0	4
600.4.bk	\(\chi_{600}(59, \cdot)\)	n/a	1424	4
600.4.bm	\(\chi_{600}(61, \cdot)\)	n/a	720	4
600.4.bp	\(\chi_{600}(53, \cdot)\)	n/a	2848	8
600.4.bq	\(\chi_{600}(67, \cdot)\)	n/a	1440	8
600.4.bt	\(\chi_{600}(103, \cdot)\)	None	0	8
600.4.bu	\(\chi_{600}(17, \cdot)\)	n/a	720	8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(600))\) into lower level spaces

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