Space of modular forms of level 840 and weight 2

• Label: 840.2
• Level: 840
• Weight: 2
• Dimension: 6712
• Nonzero newspaces: 36
• Newform subspaces: 110
• Sturm bound: 73728
• Trace bound: 12

Defining parameters

Level:	\( N \)	=	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 36 \)
Newform subspaces:			\( 110 \)
Sturm bound:			\(73728\)
Trace bound:			\(12\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(840))\).

	Total	New	Old
Modular forms	19584	6952	12632
Cusp forms	17281	6712	10569
Eisenstein series	2303	240	2063

Trace form

6712 * q - 8 * q^2 - 16 * q^3 - 24 * q^4 - 8 * q^5 - 12 * q^6 - 32 * q^7 + 16 * q^8 - 36 * q^9 - 12 * q^10 - 24 * q^11 + 52 * q^12 - 40 * q^13 + 32 * q^14 - 12 * q^15 + 24 * q^16 - 24 * q^17 + 28 * q^18 + 56 * q^19 + 100 * q^20 + 16 * q^21 + 136 * q^22 + 120 * q^23 + 76 * q^24 - 20 * q^25 + 152 * q^26 + 104 * q^27 + 168 * q^28 + 48 * q^29 + 38 * q^30 + 168 * q^31 + 72 * q^32 + 116 * q^33 + 88 * q^34 + 84 * q^35 - 32 * q^36 + 104 * q^37 - 8 * q^38 + 112 * q^39 - 104 * q^40 - 116 * q^42 + 48 * q^43 - 208 * q^44 + 18 * q^45 - 320 * q^46 + 112 * q^47 - 276 * q^48 + 48 * q^49 - 128 * q^50 - 224 * q^52 + 40 * q^53 - 276 * q^54 + 116 * q^55 - 184 * q^56 + 48 * q^57 - 176 * q^58 + 192 * q^59 - 264 * q^60 + 48 * q^61 - 232 * q^62 + 56 * q^63 - 312 * q^64 + 176 * q^65 - 356 * q^66 + 168 * q^67 - 232 * q^68 + 48 * q^69 - 156 * q^70 + 64 * q^71 - 384 * q^72 + 208 * q^73 - 112 * q^74 - 84 * q^75 - 136 * q^76 + 72 * q^77 - 336 * q^78 - 56 * q^79 - 160 * q^80 - 36 * q^81 - 280 * q^82 - 160 * q^83 - 300 * q^84 + 152 * q^85 + 32 * q^86 - 240 * q^87 + 72 * q^88 + 32 * q^89 - 204 * q^90 - 224 * q^91 - 8 * q^92 - 68 * q^93 + 88 * q^94 - 168 * q^95 - 100 * q^96 + 168 * q^97 - 40 * q^98 - 416 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(840))\)

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
840.2.a	\(\chi_{840}(1, \cdot)\)	840.2.a.a	1	1
		840.2.a.b	1
		840.2.a.c	1
		840.2.a.d	1
		840.2.a.e	1
		840.2.a.f	1
		840.2.a.g	1
		840.2.a.h	1
		840.2.a.i	1
		840.2.a.j	1
		840.2.a.k	2
840.2.d	\(\chi_{840}(391, \cdot)\)	None	0	1
840.2.e	\(\chi_{840}(491, \cdot)\)	840.2.e.a	4	1
		840.2.e.b	4
		840.2.e.c	44
		840.2.e.d	44
840.2.f	\(\chi_{840}(41, \cdot)\)	840.2.f.a	16	1
		840.2.f.b	16
840.2.g	\(\chi_{840}(421, \cdot)\)	840.2.g.a	8	1
		840.2.g.b	12
		840.2.g.c	12
		840.2.g.d	16
840.2.j	\(\chi_{840}(589, \cdot)\)	840.2.j.a	2	1
		840.2.j.b	2
		840.2.j.c	2
		840.2.j.d	2
		840.2.j.e	32
		840.2.j.f	32
840.2.k	\(\chi_{840}(209, \cdot)\)	840.2.k.a	24	1
		840.2.k.b	24
840.2.p	\(\chi_{840}(659, \cdot)\)	840.2.p.a	72	1
		840.2.p.b	72
840.2.q	\(\chi_{840}(559, \cdot)\)	None	0	1
840.2.t	\(\chi_{840}(169, \cdot)\)	840.2.t.a	2	1
		840.2.t.b	2
		840.2.t.c	4
		840.2.t.d	6
		840.2.t.e	6
840.2.u	\(\chi_{840}(629, \cdot)\)	840.2.u.a	4	1
		840.2.u.b	4
		840.2.u.c	8
		840.2.u.d	8
		840.2.u.e	160
840.2.v	\(\chi_{840}(239, \cdot)\)	None	0	1
840.2.w	\(\chi_{840}(139, \cdot)\)	840.2.w.a	48	1
		840.2.w.b	48
840.2.z	\(\chi_{840}(811, \cdot)\)	840.2.z.a	4	1
		840.2.z.b	4
		840.2.z.c	28
		840.2.z.d	28
840.2.ba	\(\chi_{840}(71, \cdot)\)	None	0	1
840.2.bf	\(\chi_{840}(461, \cdot)\)	840.2.bf.a	8	1
		840.2.bf.b	120
840.2.bg	\(\chi_{840}(121, \cdot)\)	840.2.bg.a	2	2
		840.2.bg.b	2
		840.2.bg.c	2
		840.2.bg.d	2
		840.2.bg.e	2
		840.2.bg.f	2
		840.2.bg.g	4
		840.2.bg.h	4
		840.2.bg.i	6
		840.2.bg.j	6
840.2.bj	\(\chi_{840}(13, \cdot)\)	840.2.bj.a	8	2
		840.2.bj.b	184
840.2.bk	\(\chi_{840}(113, \cdot)\)	840.2.bk.a	4	2
		840.2.bk.b	4
		840.2.bk.c	32
		840.2.bk.d	32
840.2.bl	\(\chi_{840}(127, \cdot)\)	None	0	2
840.2.bm	\(\chi_{840}(83, \cdot)\)	840.2.bm.a	368	2
840.2.br	\(\chi_{840}(43, \cdot)\)	840.2.br.a	72	2
		840.2.br.b	72
840.2.bs	\(\chi_{840}(167, \cdot)\)	None	0	2
840.2.bt	\(\chi_{840}(97, \cdot)\)	840.2.bt.a	24	2
		840.2.bt.b	24
840.2.bu	\(\chi_{840}(197, \cdot)\)	840.2.bu.a	288	2
840.2.bz	\(\chi_{840}(19, \cdot)\)	840.2.bz.a	96	2
		840.2.bz.b	96
840.2.ca	\(\chi_{840}(359, \cdot)\)	None	0	2
840.2.cb	\(\chi_{840}(269, \cdot)\)	840.2.cb.a	8	2
		840.2.cb.b	8
		840.2.cb.c	352
840.2.cc	\(\chi_{840}(289, \cdot)\)	840.2.cc.a	4	2
		840.2.cc.b	20
		840.2.cc.c	24
840.2.cf	\(\chi_{840}(101, \cdot)\)	840.2.cf.a	256	2
840.2.ck	\(\chi_{840}(191, \cdot)\)	None	0	2
840.2.cl	\(\chi_{840}(451, \cdot)\)	840.2.cl.a	64	2
		840.2.cl.b	64
840.2.co	\(\chi_{840}(541, \cdot)\)	840.2.co.a	4	2
		840.2.co.b	4
		840.2.co.c	60
		840.2.co.d	60
840.2.cp	\(\chi_{840}(521, \cdot)\)	840.2.cp.a	32	2
		840.2.cp.b	32
840.2.cq	\(\chi_{840}(11, \cdot)\)	840.2.cq.a	128	2
		840.2.cq.b	128
840.2.cr	\(\chi_{840}(31, \cdot)\)	None	0	2
840.2.cu	\(\chi_{840}(199, \cdot)\)	None	0	2
840.2.cv	\(\chi_{840}(179, \cdot)\)	840.2.cv.a	368	2
840.2.da	\(\chi_{840}(89, \cdot)\)	840.2.da.a	48	2
		840.2.da.b	48
840.2.db	\(\chi_{840}(109, \cdot)\)	840.2.db.a	4	2
		840.2.db.b	4
		840.2.db.c	4
		840.2.db.d	4
		840.2.db.e	88
		840.2.db.f	88
840.2.dc	\(\chi_{840}(53, \cdot)\)	840.2.dc.a	8	4
		840.2.dc.b	8
		840.2.dc.c	8
		840.2.dc.d	8
		840.2.dc.e	704
840.2.dd	\(\chi_{840}(73, \cdot)\)	840.2.dd.a	48	4
		840.2.dd.b	48
840.2.di	\(\chi_{840}(47, \cdot)\)	None	0	4
840.2.dj	\(\chi_{840}(67, \cdot)\)	840.2.dj.a	384	4
840.2.dk	\(\chi_{840}(227, \cdot)\)	840.2.dk.a	736	4
840.2.dl	\(\chi_{840}(247, \cdot)\)	None	0	4
840.2.dq	\(\chi_{840}(137, \cdot)\)	840.2.dq.a	192	4
840.2.dr	\(\chi_{840}(157, \cdot)\)	840.2.dr.a	384	4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(840)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 32}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(210))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(420))\)\(^{\oplus 2}\)