Space of modular forms of level 81 and weight 8

• Label: 81.8
• Level: 81
• Weight: 8
• Dimension: 1316
• Nonzero newspaces: 4
• Newform subspaces: 18
• Sturm bound: 3888
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 81 = 3^{4} \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 18 \)
Sturm bound:			\(3888\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(81))\).

	Total	New	Old
Modular forms	1755	1372	383
Cusp forms	1647	1316	331
Eisenstein series	108	56	52

Trace form

1316 * q - 12 * q^2 - 18 * q^3 - 148 * q^4 + 201 * q^5 - 18 * q^6 + 229 * q^7 + 4593 * q^8 - 18 * q^9 - 4269 * q^10 - 9417 * q^11 - 18 * q^12 - 3377 * q^13 - 16665 * q^14 - 18 * q^15 + 16160 * q^16 + 58941 * q^17 + 108774 * q^18 - 136175 * q^19 - 403845 * q^20 + 185985 * q^21 + 140592 * q^22 + 355017 * q^23 - 100242 * q^24 - 177238 * q^25 - 1392315 * q^26 - 416979 * q^27 - 168665 * q^28 + 162339 * q^29 + 1355166 * q^30 + 372697 * q^31 + 2136006 * q^32 + 154233 * q^33 + 239670 * q^34 - 1284339 * q^35 - 2486034 * q^36 + 295267 * q^37 + 2295570 * q^38 - 18 * q^39 - 1302891 * q^40 + 1907103 * q^41 - 4341303 * q^42 - 1842899 * q^43 - 2846127 * q^44 + 1459008 * q^45 + 8979063 * q^46 + 11416839 * q^47 + 9962649 * q^48 - 245196 * q^49 - 3438534 * q^50 - 4677327 * q^51 - 8174321 * q^52 - 13671117 * q^53 - 17669880 * q^54 - 8034825 * q^55 - 8809887 * q^56 + 3123720 * q^57 + 10004385 * q^58 + 25818213 * q^59 + 28050615 * q^60 + 11662843 * q^61 + 22534941 * q^62 - 1006956 * q^63 - 12361543 * q^64 - 50738727 * q^65 - 11249946 * q^66 - 20656847 * q^67 + 33243264 * q^68 + 31645980 * q^69 + 14376999 * q^70 - 2625813 * q^71 - 87478290 * q^72 + 740524 * q^73 - 13362765 * q^74 - 18281268 * q^75 + 13653838 * q^76 + 15550743 * q^77 + 48218193 * q^78 - 42151955 * q^79 + 27943374 * q^80 + 35458470 * q^81 - 20001522 * q^82 + 13604307 * q^83 + 4745169 * q^84 + 33886521 * q^85 + 5931552 * q^86 - 92159586 * q^87 + 69730620 * q^88 + 40335318 * q^89 - 99084321 * q^90 - 49367662 * q^91 - 182315715 * q^92 - 51873804 * q^93 - 75451731 * q^94 + 71483307 * q^95 + 241882443 * q^96 + 18083317 * q^97 + 245015973 * q^98 + 10370646 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(81))\)

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
81.8.a	\(\chi_{81}(1, \cdot)\)	81.8.a.a	4	1
		81.8.a.b	4
		81.8.a.c	6
		81.8.a.d	6
		81.8.a.e	6
81.8.c	\(\chi_{81}(28, \cdot)\)	81.8.c.a	2	2
		81.8.c.b	2
		81.8.c.c	2
		81.8.c.d	4
		81.8.c.e	4
		81.8.c.f	4
		81.8.c.g	4
		81.8.c.h	4
		81.8.c.i	8
		81.8.c.j	8
		81.8.c.k	12
81.8.e	\(\chi_{81}(10, \cdot)\)	81.8.e.a	120	6
81.8.g	\(\chi_{81}(4, \cdot)\)	81.8.g.a	1116	18

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(81)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)