Space of modular forms of level 81 and weight 6

• Label: 81.6
• Level: 81
• Weight: 6
• Dimension: 932
• Nonzero newspaces: 4
• Newform subspaces: 16
• Sturm bound: 2916
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 81 = 3^{4} \)
Weight:	\( k \)	=	\( 6 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 16 \)
Sturm bound:			\(2916\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(81))\).

	Total	New	Old
Modular forms	1269	988	281
Cusp forms	1161	932	229
Eisenstein series	108	56	52

Trace form

932 * q - 12 * q^2 - 18 * q^3 - 52 * q^4 + 75 * q^5 - 18 * q^6 - 107 * q^7 - 591 * q^8 - 18 * q^9 - 69 * q^10 - 129 * q^11 - 18 * q^12 + 1717 * q^13 + 1623 * q^14 - 18 * q^15 - 3328 * q^16 - 3483 * q^17 + 8658 * q^18 - 1295 * q^19 - 26709 * q^20 - 13761 * q^21 - 228 * q^22 + 7401 * q^23 + 16398 * q^24 + 8900 * q^25 + 44301 * q^26 + 17505 * q^27 + 31303 * q^28 + 24153 * q^29 - 15354 * q^30 - 12287 * q^31 - 112554 * q^32 - 36765 * q^33 - 51318 * q^34 - 55299 * q^35 + 17838 * q^36 + 23995 * q^37 + 95934 * q^38 - 18 * q^39 + 44061 * q^40 + 163749 * q^41 + 52137 * q^42 - 15011 * q^43 - 268815 * q^44 - 136044 * q^45 - 181401 * q^46 - 188409 * q^47 - 66447 * q^48 - 14586 * q^49 + 249522 * q^50 + 126297 * q^51 + 250807 * q^52 + 358083 * q^53 + 347112 * q^54 + 210855 * q^55 + 362337 * q^56 + 41508 * q^57 - 114159 * q^58 - 216987 * q^59 - 329841 * q^60 - 113039 * q^61 - 707523 * q^62 - 266292 * q^63 - 324967 * q^64 - 543729 * q^65 - 324666 * q^66 + 16873 * q^67 + 974808 * q^68 + 717912 * q^69 - 233481 * q^70 + 127323 * q^71 + 981486 * q^72 - 81926 * q^73 - 58029 * q^74 - 56268 * q^75 + 213502 * q^76 - 220947 * q^77 - 741447 * q^78 + 438589 * q^79 - 688722 * q^80 - 454986 * q^81 + 200838 * q^82 + 29895 * q^83 - 824391 * q^84 + 188397 * q^85 - 30876 * q^86 + 405918 * q^87 - 332196 * q^88 + 1200708 * q^89 + 2137815 * q^90 + 44702 * q^91 + 727437 * q^92 + 30024 * q^93 - 1180083 * q^94 - 1983621 * q^95 - 1414989 * q^96 - 546425 * q^97 - 3162735 * q^98 - 1736586 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(81))\)

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
81.6.a	\(\chi_{81}(1, \cdot)\)	81.6.a.a	2	1
		81.6.a.b	2
		81.6.a.c	4
		81.6.a.d	4
		81.6.a.e	6
81.6.c	\(\chi_{81}(28, \cdot)\)	81.6.c.a	2	2
		81.6.c.b	2
		81.6.c.c	2
		81.6.c.d	4
		81.6.c.e	4
		81.6.c.f	4
		81.6.c.g	4
		81.6.c.h	4
		81.6.c.i	12
81.6.e	\(\chi_{81}(10, \cdot)\)	81.6.e.a	84	6
81.6.g	\(\chi_{81}(4, \cdot)\)	81.6.g.a	792	18

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(81)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)