Space of modular forms of level 81 and weight 2

• Label: 81.2
• Level: 81
• Weight: 2
• Dimension: 164
• Nonzero newspaces: 4
• Newform subspaces: 5
• Sturm bound: 972
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 81 = 3^{4} \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 5 \)
Sturm bound:			\(972\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(81))\).

	Total	New	Old
Modular forms	297	220	77
Cusp forms	190	164	26
Eisenstein series	107	56	51

Trace form

164 * q - 12 * q^2 - 18 * q^3 - 22 * q^4 - 15 * q^5 - 18 * q^6 - 23 * q^7 - 24 * q^8 - 18 * q^9 - 39 * q^10 - 21 * q^11 - 18 * q^12 - 29 * q^13 - 33 * q^14 - 18 * q^15 - 22 * q^16 - 27 * q^17 - 9 * q^18 - 23 * q^19 + 21 * q^20 + 9 * q^21 - 15 * q^22 + 21 * q^23 + 36 * q^24 - 10 * q^25 + 75 * q^26 + 9 * q^27 - 5 * q^28 + 15 * q^29 + 36 * q^30 - 11 * q^31 + 36 * q^32 + 9 * q^33 - 9 * q^34 - 3 * q^35 + 18 * q^36 - 41 * q^37 - 51 * q^38 - 18 * q^39 + 3 * q^40 - 15 * q^41 + 27 * q^42 - 23 * q^43 + 51 * q^44 + 36 * q^45 + 15 * q^46 + 51 * q^47 + 81 * q^48 + 132 * q^50 + 45 * q^51 + 19 * q^52 + 63 * q^53 + 108 * q^54 + 15 * q^55 + 159 * q^56 + 36 * q^57 + 3 * q^58 + 57 * q^59 + 99 * q^60 - 5 * q^61 + 93 * q^62 + 36 * q^63 - 16 * q^64 - 3 * q^65 - 18 * q^66 - 11 * q^67 - 108 * q^68 - 72 * q^69 + 21 * q^70 - 117 * q^71 - 234 * q^72 - 68 * q^73 - 195 * q^74 - 108 * q^75 + q^76 - 177 * q^77 - 135 * q^78 - 23 * q^79 - 330 * q^80 - 90 * q^81 - 120 * q^82 - 129 * q^83 - 243 * q^84 + 9 * q^85 - 213 * q^86 - 162 * q^87 + 33 * q^88 - 90 * q^89 - 99 * q^90 - 34 * q^91 - 105 * q^92 + 21 * q^94 + 51 * q^95 - 27 * q^96 + 31 * q^97 + 126 * q^98 + 54 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(81))\)

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
81.2.a	\(\chi_{81}(1, \cdot)\)	81.2.a.a	2	1
81.2.c	\(\chi_{81}(28, \cdot)\)	81.2.c.a	2	2
		81.2.c.b	4
81.2.e	\(\chi_{81}(10, \cdot)\)	81.2.e.a	12	6
81.2.g	\(\chi_{81}(4, \cdot)\)	81.2.g.a	144	18

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

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