Space of modular forms of level 360 and weight 2

• Label: 360.2
• Level: 360
• Weight: 2
• Dimension: 1357
• Nonzero newspaces: 18
• Newform subspaces: 57
• Sturm bound: 13824
• Trace bound: 10

Defining parameters

Level:	\( N \)	=	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 18 \)
Newform subspaces:			\( 57 \)
Sturm bound:			\(13824\)
Trace bound:			\(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(360))\).

	Total	New	Old
Modular forms	3840	1465	2375
Cusp forms	3073	1357	1716
Eisenstein series	767	108	659

Trace form

1357 * q - 8 * q^2 - 10 * q^3 - 8 * q^4 - 5 * q^5 - 16 * q^6 - 12 * q^7 + 16 * q^8 - 14 * q^9 - 12 * q^10 + 6 * q^11 + 12 * q^12 + 2 * q^13 + 28 * q^14 + 8 * q^15 + 12 * q^16 + 18 * q^17 - 16 * q^18 + 24 * q^19 - 2 * q^20 + 32 * q^21 - 20 * q^22 + 60 * q^23 - 64 * q^24 + q^25 - 64 * q^26 + 8 * q^27 - 56 * q^28 + 30 * q^29 - 62 * q^30 + 48 * q^31 - 108 * q^32 + 6 * q^33 - 16 * q^34 + 12 * q^35 - 132 * q^36 + 26 * q^37 - 140 * q^38 - 72 * q^39 - 62 * q^40 - 24 * q^41 - 140 * q^42 - 38 * q^43 - 172 * q^44 - 2 * q^45 - 160 * q^46 - 116 * q^47 - 112 * q^48 + 47 * q^49 - 170 * q^50 - 98 * q^51 - 96 * q^52 - 14 * q^53 - 96 * q^54 - 84 * q^55 - 128 * q^56 - 22 * q^57 - 136 * q^58 - 150 * q^59 - 70 * q^60 - 6 * q^61 - 64 * q^62 - 152 * q^63 - 152 * q^64 - 64 * q^65 + 64 * q^66 - 74 * q^67 - 28 * q^68 - 68 * q^69 - 122 * q^70 - 152 * q^71 + 108 * q^72 - 30 * q^73 + 80 * q^74 - 122 * q^75 + 20 * q^76 - 72 * q^77 + 100 * q^78 - 40 * q^79 - 16 * q^80 - 150 * q^81 - 108 * q^82 - 172 * q^83 + 80 * q^84 + 50 * q^85 + 92 * q^86 - 156 * q^87 - 36 * q^88 - 94 * q^89 - 2 * q^90 - 96 * q^91 - 68 * q^92 - 76 * q^93 - 24 * q^94 - 204 * q^95 - 128 * q^96 - 76 * q^97 - 76 * q^98 - 228 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(360))\)

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
360.2.a	\(\chi_{360}(1, \cdot)\)	360.2.a.a	1	1
		360.2.a.b	1
		360.2.a.c	1
		360.2.a.d	1
		360.2.a.e	1
360.2.b	\(\chi_{360}(251, \cdot)\)	360.2.b.a	2	1
		360.2.b.b	2
		360.2.b.c	6
		360.2.b.d	6
360.2.d	\(\chi_{360}(109, \cdot)\)	360.2.d.a	4	1
		360.2.d.b	4
		360.2.d.c	4
		360.2.d.d	4
		360.2.d.e	6
		360.2.d.f	6
360.2.f	\(\chi_{360}(289, \cdot)\)	360.2.f.a	2	1
		360.2.f.b	2
		360.2.f.c	2
		360.2.f.d	2
360.2.h	\(\chi_{360}(71, \cdot)\)	None	0	1
360.2.k	\(\chi_{360}(181, \cdot)\)	360.2.k.a	2	1
		360.2.k.b	2
		360.2.k.c	2
		360.2.k.d	4
		360.2.k.e	4
		360.2.k.f	6
360.2.m	\(\chi_{360}(179, \cdot)\)	360.2.m.a	4	1
		360.2.m.b	4
		360.2.m.c	16
360.2.o	\(\chi_{360}(359, \cdot)\)	None	0	1
360.2.q	\(\chi_{360}(121, \cdot)\)	360.2.q.a	2	2
		360.2.q.b	4
		360.2.q.c	4
		360.2.q.d	6
		360.2.q.e	8
360.2.s	\(\chi_{360}(17, \cdot)\)	360.2.s.a	4	2
		360.2.s.b	8
360.2.t	\(\chi_{360}(127, \cdot)\)	None	0	2
360.2.w	\(\chi_{360}(163, \cdot)\)	360.2.w.a	4	2
		360.2.w.b	4
		360.2.w.c	8
		360.2.w.d	16
		360.2.w.e	24
360.2.x	\(\chi_{360}(53, \cdot)\)	360.2.x.a	48	2
360.2.bb	\(\chi_{360}(119, \cdot)\)	None	0	2
360.2.bd	\(\chi_{360}(59, \cdot)\)	360.2.bd.a	8	2
		360.2.bd.b	128
360.2.bf	\(\chi_{360}(61, \cdot)\)	360.2.bf.a	4	2
		360.2.bf.b	92
360.2.bg	\(\chi_{360}(191, \cdot)\)	None	0	2
360.2.bi	\(\chi_{360}(49, \cdot)\)	360.2.bi.a	4	2
		360.2.bi.b	32
360.2.bk	\(\chi_{360}(229, \cdot)\)	360.2.bk.a	136	2
360.2.bm	\(\chi_{360}(11, \cdot)\)	360.2.bm.a	48	2
		360.2.bm.b	48
360.2.bo	\(\chi_{360}(43, \cdot)\)	360.2.bo.a	272	4
360.2.br	\(\chi_{360}(77, \cdot)\)	360.2.br.a	4	4
		360.2.br.b	4
		360.2.br.c	4
		360.2.br.d	4
		360.2.br.e	256
360.2.bs	\(\chi_{360}(113, \cdot)\)	360.2.bs.a	72	4
360.2.bv	\(\chi_{360}(7, \cdot)\)	None	0	4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(360))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(360)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 2}\)