Space of modular forms of level 81 and weight 3

• Label: 81.3
• Level: 81
• Weight: 3
• Dimension: 356
• Nonzero newspaces: 4
• Newform subspaces: 7
• Sturm bound: 1458
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 81 = 3^{4} \)
Weight:	\( k \)	=	\( 3 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 7 \)
Sturm bound:			\(1458\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(81))\).

	Total	New	Old
Modular forms	540	412	128
Cusp forms	432	356	76
Eisenstein series	108	56	52

Trace form

356 * q - 12 * q^2 - 18 * q^3 - 16 * q^4 - 3 * q^5 - 18 * q^6 - 17 * q^7 - 9 * q^8 - 18 * q^9 - 33 * q^10 - 12 * q^11 - 18 * q^12 - 11 * q^13 - 3 * q^14 - 18 * q^15 - 52 * q^16 - 9 * q^17 - 90 * q^18 - 125 * q^19 - 447 * q^20 - 153 * q^21 - 156 * q^22 - 219 * q^23 - 126 * q^24 - 64 * q^25 - 27 * q^26 + 9 * q^27 + 127 * q^28 + 231 * q^29 + 198 * q^30 + 133 * q^31 + 666 * q^32 + 171 * q^33 + 198 * q^34 + 477 * q^35 + 342 * q^36 + 109 * q^37 + 174 * q^38 - 18 * q^39 - 93 * q^40 - 462 * q^41 - 513 * q^42 - 314 * q^43 - 1305 * q^44 - 450 * q^45 - 681 * q^46 - 741 * q^47 - 513 * q^48 - 258 * q^49 - 672 * q^50 - 144 * q^51 - 167 * q^52 - 27 * q^53 + 108 * q^54 + 183 * q^55 + 789 * q^56 + 198 * q^57 + 447 * q^58 + 834 * q^59 + 801 * q^60 + 505 * q^61 + 1773 * q^62 + 522 * q^63 + 833 * q^64 + 1581 * q^65 + 1926 * q^66 + 52 * q^67 + 2070 * q^68 + 1062 * q^69 - 183 * q^70 + 315 * q^71 + 1710 * q^72 - 80 * q^73 + 525 * q^74 + 432 * q^75 - 266 * q^76 + 213 * q^77 + 189 * q^78 - 125 * q^79 - 90 * q^81 + 102 * q^82 - 201 * q^83 - 945 * q^84 - 9 * q^85 - 624 * q^86 - 1026 * q^87 + 48 * q^88 - 1386 * q^89 - 2529 * q^90 - 892 * q^91 - 3957 * q^92 - 2214 * q^93 - 327 * q^94 - 3075 * q^95 - 3321 * q^96 - 368 * q^97 - 3735 * q^98 - 1566 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(81))\)

We only show spaces with odd 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
81.3.b	\(\chi_{81}(80, \cdot)\)	81.3.b.a	2	1
		81.3.b.b	4
81.3.d	\(\chi_{81}(26, \cdot)\)	81.3.d.a	2	2
		81.3.d.b	4
		81.3.d.c	8
81.3.f	\(\chi_{81}(8, \cdot)\)	81.3.f.a	30	6
81.3.h	\(\chi_{81}(2, \cdot)\)	81.3.h.a	306	18

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

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