Space of modular forms of level 84 and weight 6

• Label: 84.6
• Level: 84
• Weight: 6
• Dimension: 392
• Nonzero newspaces: 8
• Newform subspaces: 18
• Sturm bound: 2304
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	=	\( 6 \)
Nonzero newspaces:			\( 8 \)
Newform subspaces:			\( 18 \)
Sturm bound:			\(2304\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(84))\).

	Total	New	Old
Modular forms	1020	408	612
Cusp forms	900	392	508
Eisenstein series	120	16	104

Trace form

392 * q + 9 * q^3 - 22 * q^4 + 66 * q^5 - 54 * q^6 + 232 * q^7 - 534 * q^8 + 531 * q^9 + 2140 * q^10 - 534 * q^11 - 246 * q^12 - 958 * q^13 - 3720 * q^14 + 918 * q^15 - 2398 * q^16 + 3252 * q^17 - 1380 * q^18 + 1504 * q^19 + 1323 * q^21 - 9828 * q^22 + 3828 * q^23 + 21894 * q^24 - 2886 * q^25 + 17970 * q^26 - 1458 * q^27 + 14394 * q^28 - 32940 * q^29 - 36636 * q^30 + 6796 * q^31 - 22530 * q^32 + 13821 * q^33 + 20944 * q^34 + 36966 * q^35 + 55422 * q^36 + 8310 * q^37 + 41238 * q^38 - 13026 * q^39 - 88904 * q^40 - 46680 * q^41 + 18324 * q^42 - 45496 * q^43 + 14520 * q^44 + 37701 * q^45 + 64164 * q^46 + 66474 * q^47 + 8886 * q^48 - 114658 * q^49 - 26250 * q^50 - 21483 * q^51 - 99992 * q^52 + 19776 * q^53 + 20856 * q^54 - 141120 * q^55 + 45378 * q^56 - 99822 * q^57 + 350608 * q^58 + 61716 * q^59 + 252768 * q^60 + 15152 * q^61 - 52797 * q^63 - 276970 * q^64 - 197418 * q^65 - 333732 * q^66 - 242556 * q^67 - 330516 * q^68 + 134100 * q^69 - 193164 * q^70 - 92736 * q^71 + 205698 * q^72 + 389042 * q^73 - 36366 * q^74 + 170658 * q^75 + 388140 * q^76 + 192024 * q^77 + 311904 * q^78 + 118740 * q^79 + 61848 * q^80 - 129681 * q^81 + 8116 * q^82 + 63156 * q^83 - 328830 * q^84 - 329644 * q^85 - 6918 * q^86 - 83556 * q^87 + 571272 * q^88 + 201732 * q^89 + 814044 * q^90 + 168916 * q^91 + 37884 * q^92 + 836175 * q^93 - 725772 * q^94 - 99882 * q^95 - 809346 * q^96 - 156256 * q^97 - 1266006 * q^98 - 297126 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(84))\)

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
84.6.a	\(\chi_{84}(1, \cdot)\)	84.6.a.a	1	1
		84.6.a.b	1
		84.6.a.c	2
		84.6.a.d	2
84.6.b	\(\chi_{84}(55, \cdot)\)	84.6.b.a	20	1
		84.6.b.b	20
84.6.e	\(\chi_{84}(71, \cdot)\)	84.6.e.a	60	1
84.6.f	\(\chi_{84}(41, \cdot)\)	84.6.f.a	2	1
		84.6.f.b	4
		84.6.f.c	8
84.6.i	\(\chi_{84}(25, \cdot)\)	84.6.i.a	2	2
		84.6.i.b	4
		84.6.i.c	8
84.6.k	\(\chi_{84}(5, \cdot)\)	84.6.k.a	2	2
		84.6.k.b	24
84.6.n	\(\chi_{84}(11, \cdot)\)	84.6.n.a	152	2
84.6.o	\(\chi_{84}(19, \cdot)\)	84.6.o.a	40	2
		84.6.o.b	40

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(84))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(84)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 2}\)