Space of modular forms of level 81 and weight 9

• Label: 81.9
• Level: 81
• Weight: 9
• Dimension: 1508
• Nonzero newspaces: 4
• Newform subspaces: 11
• Sturm bound: 4374
• Trace bound: 1

Defining parameters

Level:	\( N \)	=	\( 81 = 3^{4} \)
Weight:	\( k \)	=	\( 9 \)
Nonzero newspaces:			\( 4 \)
Newform subspaces:			\( 11 \)
Sturm bound:			\(4374\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(81))\).

	Total	New	Old
Modular forms	1998	1564	434
Cusp forms	1890	1508	382
Eisenstein series	108	56	52

Trace form

1508 * q - 12 * q^2 - 18 * q^3 + 236 * q^4 + 429 * q^5 - 18 * q^6 - 2789 * q^7 - 9 * q^8 - 18 * q^9 + 11235 * q^10 - 28686 * q^11 - 18 * q^12 + 52045 * q^13 + 120957 * q^14 - 18 * q^15 - 234376 * q^16 - 9 * q^17 - 274194 * q^18 + 732367 * q^19 - 815307 * q^20 - 1360953 * q^21 - 1124256 * q^22 + 863673 * q^23 + 3725550 * q^24 + 2283272 * q^25 - 27 * q^26 - 1933299 * q^27 - 4908329 * q^28 - 1909749 * q^29 - 1292562 * q^30 + 1720033 * q^31 + 14472126 * q^32 + 6069339 * q^33 - 3557322 * q^34 - 12450843 * q^35 - 18224658 * q^36 + 4384945 * q^37 + 36466050 * q^38 - 18 * q^39 - 7270149 * q^40 - 23149344 * q^41 + 39542067 * q^42 - 89648 * q^43 - 34682265 * q^44 - 37896786 * q^45 - 31245801 * q^46 - 8464917 * q^47 + 15217767 * q^48 + 23682702 * q^49 + 145747380 * q^50 + 54355878 * q^51 + 14270485 * q^52 - 27 * q^53 - 75728412 * q^54 - 62139633 * q^55 - 221070243 * q^56 - 65747826 * q^57 - 98272785 * q^58 - 68739708 * q^59 + 125543673 * q^60 + 99186013 * q^61 + 447419205 * q^62 + 101958462 * q^63 + 73639193 * q^64 - 65778843 * q^65 - 552090186 * q^66 - 61291730 * q^67 - 292730634 * q^68 + 273910662 * q^69 - 188308695 * q^70 + 125718795 * q^71 + 548536302 * q^72 + 33257476 * q^73 - 87410787 * q^74 - 246093768 * q^75 + 369351490 * q^76 - 230951739 * q^77 - 478793403 * q^78 + 78357691 * q^79 + 148365990 * q^81 - 136309578 * q^82 - 14587329 * q^83 + 1377777303 * q^84 - 213183261 * q^85 + 363808068 * q^86 + 61361982 * q^87 + 193071684 * q^88 - 1029546702 * q^89 - 624764277 * q^90 + 709272512 * q^91 + 597873567 * q^92 - 53952426 * q^93 - 287856975 * q^94 - 1195036359 * q^95 + 626058639 * q^96 - 915507962 * q^97 - 1883568663 * q^98 - 967332078 * q^99

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(81))\)

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
81.9.b	\(\chi_{81}(80, \cdot)\)	81.9.b.a	14	1
		81.9.b.b	16
81.9.d	\(\chi_{81}(26, \cdot)\)	81.9.d.a	2	2
		81.9.d.b	4
		81.9.d.c	4
		81.9.d.d	4
		81.9.d.e	4
		81.9.d.f	12
		81.9.d.g	32
81.9.f	\(\chi_{81}(8, \cdot)\)	81.9.f.a	138	6
81.9.h	\(\chi_{81}(2, \cdot)\)	81.9.h.a	1278	18

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

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