Space of modular forms of level 816 and weight 2

• Label: 816.2
• Level: 816
• Weight: 2
• Dimension: 7586
• Nonzero newspaces: 26
• Newform subspaces: 84
• Sturm bound: 73728
• Trace bound: 19

Defining parameters

Level:	\( N \)	=	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 26 \)
Newform subspaces:			\( 84 \)
Sturm bound:			\(73728\)
Trace bound:			\(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(816))\).

	Total	New	Old
Modular forms	19328	7858	11470
Cusp forms	17537	7586	9951
Eisenstein series	1791	272	1519

Trace form

7586 * q - 22 * q^3 - 48 * q^4 + 4 * q^5 - 16 * q^6 - 32 * q^7 + 24 * q^8 + 2 * q^9 - 48 * q^10 + 24 * q^11 - 32 * q^12 - 60 * q^13 - 24 * q^14 - 4 * q^15 - 96 * q^16 - 2 * q^17 - 80 * q^18 - 16 * q^19 - 32 * q^20 - 48 * q^21 - 96 * q^22 - 16 * q^23 - 88 * q^24 - 34 * q^25 - 40 * q^26 - 46 * q^27 - 64 * q^28 + 20 * q^29 - 72 * q^30 - 80 * q^31 - 48 * q^33 - 56 * q^34 - 48 * q^35 - 64 * q^36 - 12 * q^37 + 16 * q^38 - 60 * q^39 - 16 * q^40 + 12 * q^41 + 56 * q^42 - 48 * q^43 + 80 * q^44 - 12 * q^45 + 32 * q^46 + 96 * q^48 - 54 * q^49 + 72 * q^50 - 58 * q^51 - 64 * q^52 + 4 * q^53 + 64 * q^54 - 16 * q^55 + 72 * q^57 - 96 * q^58 + 40 * q^59 - 48 * q^60 - 60 * q^61 + 24 * q^62 + 104 * q^63 - 192 * q^64 + 152 * q^65 - 136 * q^66 - 32 * q^68 - 8 * q^69 - 208 * q^70 + 112 * q^71 - 104 * q^72 + 52 * q^73 - 104 * q^74 + 158 * q^75 - 192 * q^76 + 96 * q^77 - 128 * q^78 + 80 * q^79 - 16 * q^80 - 22 * q^81 - 112 * q^82 + 168 * q^83 - 16 * q^84 - 60 * q^85 + 32 * q^86 + 84 * q^87 + 12 * q^89 + 16 * q^90 + 80 * q^91 + 32 * q^92 - 8 * q^93 + 16 * q^94 + 112 * q^95 + 80 * q^96 - 76 * q^97 + 80 * q^98 + 104 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(816))\)

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
816.2.a	\(\chi_{816}(1, \cdot)\)	816.2.a.a	1	1
		816.2.a.b	1
		816.2.a.c	1
		816.2.a.d	1
		816.2.a.e	1
		816.2.a.f	1
		816.2.a.g	1
		816.2.a.h	1
		816.2.a.i	1
		816.2.a.j	1
		816.2.a.k	2
		816.2.a.l	2
		816.2.a.m	2
816.2.c	\(\chi_{816}(577, \cdot)\)	816.2.c.a	2	1
		816.2.c.b	2
		816.2.c.c	2
		816.2.c.d	2
		816.2.c.e	4
		816.2.c.f	6
816.2.e	\(\chi_{816}(239, \cdot)\)	816.2.e.a	4	1
		816.2.e.b	8
		816.2.e.c	20
816.2.f	\(\chi_{816}(409, \cdot)\)	None	0	1
816.2.h	\(\chi_{816}(407, \cdot)\)	None	0	1
816.2.j	\(\chi_{816}(647, \cdot)\)	None	0	1
816.2.l	\(\chi_{816}(169, \cdot)\)	None	0	1
816.2.o	\(\chi_{816}(815, \cdot)\)	816.2.o.a	4	1
		816.2.o.b	4
		816.2.o.c	4
		816.2.o.d	8
		816.2.o.e	16
816.2.r	\(\chi_{816}(395, \cdot)\)	816.2.r.a	4	2
		816.2.r.b	4
		816.2.r.c	272
816.2.s	\(\chi_{816}(157, \cdot)\)	816.2.s.a	2	2
		816.2.s.b	2
		816.2.s.c	68
		816.2.s.d	72
816.2.u	\(\chi_{816}(205, \cdot)\)	816.2.u.a	64	2
		816.2.u.b	64
816.2.w	\(\chi_{816}(203, \cdot)\)	816.2.w.a	280	2
816.2.y	\(\chi_{816}(455, \cdot)\)	None	0	2
816.2.ba	\(\chi_{816}(217, \cdot)\)	None	0	2
816.2.bd	\(\chi_{816}(625, \cdot)\)	816.2.bd.a	4	2
		816.2.bd.b	4
		816.2.bd.c	4
		816.2.bd.d	4
		816.2.bd.e	8
		816.2.bd.f	12
816.2.bf	\(\chi_{816}(47, \cdot)\)	816.2.bf.a	8	2
		816.2.bf.b	8
		816.2.bf.c	8
		816.2.bf.d	24
		816.2.bf.e	24
816.2.bh	\(\chi_{816}(35, \cdot)\)	816.2.bh.a	4	2
		816.2.bh.b	4
		816.2.bh.c	124
		816.2.bh.d	124
816.2.bj	\(\chi_{816}(373, \cdot)\)	816.2.bj.a	144	2
816.2.bl	\(\chi_{816}(13, \cdot)\)	816.2.bl.a	2	2
		816.2.bl.b	2
		816.2.bl.c	68
		816.2.bl.d	72
816.2.bm	\(\chi_{816}(251, \cdot)\)	816.2.bm.a	4	2
		816.2.bm.b	4
		816.2.bm.c	272
816.2.bq	\(\chi_{816}(49, \cdot)\)	816.2.bq.a	8	4
		816.2.bq.b	8
		816.2.bq.c	8
		816.2.bq.d	16
		816.2.bq.e	16
		816.2.bq.f	16
816.2.br	\(\chi_{816}(287, \cdot)\)	816.2.br.a	48	4
		816.2.br.b	96
816.2.bs	\(\chi_{816}(155, \cdot)\)	816.2.bs.a	560	4
816.2.bt	\(\chi_{816}(229, \cdot)\)	816.2.bt.a	288	4
816.2.bw	\(\chi_{816}(59, \cdot)\)	816.2.bw.a	560	4
816.2.bx	\(\chi_{816}(325, \cdot)\)	816.2.bx.a	288	4
816.2.ca	\(\chi_{816}(25, \cdot)\)	None	0	4
816.2.cb	\(\chi_{816}(263, \cdot)\)	None	0	4
816.2.cf	\(\chi_{816}(29, \cdot)\)	816.2.cf.a	1120	8
816.2.cg	\(\chi_{816}(91, \cdot)\)	816.2.cg.a	576	8
816.2.cj	\(\chi_{816}(65, \cdot)\)	816.2.cj.a	24	8
		816.2.cj.b	24
		816.2.cj.c	32
		816.2.cj.d	48
		816.2.cj.e	72
		816.2.cj.f	72
816.2.ck	\(\chi_{816}(31, \cdot)\)	816.2.ck.a	48	8
		816.2.ck.b	96
816.2.cn	\(\chi_{816}(7, \cdot)\)	None	0	8
816.2.co	\(\chi_{816}(41, \cdot)\)	None	0	8
816.2.cr	\(\chi_{816}(5, \cdot)\)	816.2.cr.a	1120	8
816.2.cs	\(\chi_{816}(139, \cdot)\)	816.2.cs.a	576	8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(816))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(816)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(34))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(51))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(204))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(272))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(408))\)\(^{\oplus 2}\)