Space of modular forms of level 600 and weight 2

• Label: 600.2
• Level: 600
• Weight: 2
• Dimension: 3651
• Nonzero newspaces: 18
• Newform subspaces: 81
• Sturm bound: 38400
• Trace bound: 8

Defining parameters

Level:	\( N \)	=	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 18 \)
Newform subspaces:			\( 81 \)
Sturm bound:			\(38400\)
Trace bound:			\(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(600))\).

	Total	New	Old
Modular forms	10272	3815	6457
Cusp forms	8929	3651	5278
Eisenstein series	1343	164	1179

Trace form

3651 * q - 2 * q^2 - 15 * q^3 - 28 * q^4 - 2 * q^5 - 18 * q^6 - 44 * q^7 + 4 * q^8 - 31 * q^9 - 32 * q^10 - 12 * q^11 + 20 * q^12 - 10 * q^13 + 52 * q^14 - 8 * q^15 + 24 * q^16 + 6 * q^17 + 38 * q^18 + 40 * q^19 + 40 * q^20 + 32 * q^21 + 48 * q^22 + 48 * q^23 + 32 * q^24 - 58 * q^25 + 56 * q^26 + 45 * q^27 - 40 * q^28 + 30 * q^29 - 12 * q^30 + 36 * q^31 - 72 * q^32 + 28 * q^33 - 132 * q^34 + 48 * q^35 - 32 * q^36 + 72 * q^37 - 104 * q^38 + 78 * q^39 - 112 * q^40 + 46 * q^41 - 96 * q^42 + 136 * q^43 - 112 * q^44 + 70 * q^45 - 176 * q^46 + 120 * q^47 - 140 * q^48 + 55 * q^49 - 40 * q^50 - 62 * q^51 - 72 * q^52 + 64 * q^53 - 110 * q^54 + 64 * q^55 - 8 * q^56 + 16 * q^57 - 36 * q^58 + 20 * q^59 - 92 * q^60 - 50 * q^61 - 4 * q^62 - 120 * q^63 - 88 * q^64 + 6 * q^65 - 188 * q^66 - 160 * q^67 - 112 * q^68 - 24 * q^69 - 104 * q^70 - 128 * q^71 - 200 * q^72 + 74 * q^73 - 160 * q^74 - 88 * q^75 - 232 * q^76 - 188 * q^78 - 204 * q^79 - 80 * q^80 + 57 * q^81 - 348 * q^82 - 268 * q^83 - 300 * q^84 - 82 * q^85 - 184 * q^86 - 206 * q^87 - 472 * q^88 + 128 * q^89 - 172 * q^90 - 184 * q^91 - 384 * q^92 - 32 * q^93 - 496 * q^94 - 256 * q^95 - 84 * q^96 - 86 * q^97 - 378 * q^98 - 180 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(600))\)

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
600.2.a	\(\chi_{600}(1, \cdot)\)	600.2.a.a	1	1
		600.2.a.b	1
		600.2.a.c	1
		600.2.a.d	1
		600.2.a.e	1
		600.2.a.f	1
		600.2.a.g	1
		600.2.a.h	1
		600.2.a.i	1
600.2.b	\(\chi_{600}(251, \cdot)\)	600.2.b.a	2	1
		600.2.b.b	4
		600.2.b.c	4
		600.2.b.d	4
		600.2.b.e	8
		600.2.b.f	8
		600.2.b.g	12
		600.2.b.h	12
		600.2.b.i	16
600.2.d	\(\chi_{600}(349, \cdot)\)	600.2.d.a	2	1
		600.2.d.b	2
		600.2.d.c	2
		600.2.d.d	2
		600.2.d.e	6
		600.2.d.f	6
		600.2.d.g	8
		600.2.d.h	8
600.2.f	\(\chi_{600}(49, \cdot)\)	600.2.f.a	2	1
		600.2.f.b	2
		600.2.f.c	2
		600.2.f.d	2
		600.2.f.e	2
600.2.h	\(\chi_{600}(551, \cdot)\)	None	0	1
600.2.k	\(\chi_{600}(301, \cdot)\)	600.2.k.a	2	1
		600.2.k.b	2
		600.2.k.c	6
		600.2.k.d	8
		600.2.k.e	8
		600.2.k.f	12
600.2.m	\(\chi_{600}(299, \cdot)\)	600.2.m.a	4	1
		600.2.m.b	8
		600.2.m.c	16
		600.2.m.d	16
		600.2.m.e	24
600.2.o	\(\chi_{600}(599, \cdot)\)	None	0	1
600.2.r	\(\chi_{600}(257, \cdot)\)	600.2.r.a	4	2
		600.2.r.b	4
		600.2.r.c	4
		600.2.r.d	4
		600.2.r.e	4
		600.2.r.f	16
600.2.s	\(\chi_{600}(7, \cdot)\)	None	0	2
600.2.v	\(\chi_{600}(43, \cdot)\)	600.2.v.a	16	2
		600.2.v.b	24
		600.2.v.c	32
600.2.w	\(\chi_{600}(293, \cdot)\)	600.2.w.a	4	2
		600.2.w.b	4
		600.2.w.c	4
		600.2.w.d	4
		600.2.w.e	4
		600.2.w.f	4
		600.2.w.g	4
		600.2.w.h	4
		600.2.w.i	8
		600.2.w.j	32
		600.2.w.k	64
600.2.y	\(\chi_{600}(121, \cdot)\)	600.2.y.a	4	4
		600.2.y.b	4
		600.2.y.c	12
		600.2.y.d	12
		600.2.y.e	16
		600.2.y.f	16
600.2.ba	\(\chi_{600}(71, \cdot)\)	None	0	4
600.2.bc	\(\chi_{600}(169, \cdot)\)	600.2.bc.a	8	4
		600.2.bc.b	24
		600.2.bc.c	24
600.2.be	\(\chi_{600}(109, \cdot)\)	600.2.be.a	120	4
		600.2.be.b	120
600.2.bg	\(\chi_{600}(11, \cdot)\)	600.2.bg.a	464	4
600.2.bi	\(\chi_{600}(119, \cdot)\)	None	0	4
600.2.bk	\(\chi_{600}(59, \cdot)\)	600.2.bk.a	464	4
600.2.bm	\(\chi_{600}(61, \cdot)\)	600.2.bm.a	240	4
600.2.bp	\(\chi_{600}(53, \cdot)\)	600.2.bp.a	16	8
		600.2.bp.b	16
		600.2.bp.c	896
600.2.bq	\(\chi_{600}(67, \cdot)\)	600.2.bq.a	480	8
600.2.bt	\(\chi_{600}(103, \cdot)\)	None	0	8
600.2.bu	\(\chi_{600}(17, \cdot)\)	600.2.bu.a	240	8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(600))\) into lower level spaces

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