Space of modular forms of level 84 and weight 8

• Label: 84.8
• Level: 84
• Weight: 8
• Dimension: 548
• Nonzero newspaces: 8
• Newform subspaces: 16
• Sturm bound: 3072
• Trace bound: 3

Defining parameters

Level:	\( N \)	=	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 8 \)
Newform subspaces:			\( 16 \)
Sturm bound:			\(3072\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(84))\).

	Total	New	Old
Modular forms	1404	564	840
Cusp forms	1284	548	736
Eisenstein series	120	16	104

Trace form

548 * q - 27 * q^3 - 54 * q^4 + 222 * q^5 + 426 * q^6 - 944 * q^7 + 1722 * q^8 - 2163 * q^9 - 17028 * q^10 - 4254 * q^11 + 12258 * q^12 + 16830 * q^13 + 39288 * q^14 - 40410 * q^15 - 66846 * q^16 + 55716 * q^17 - 9396 * q^18 - 25872 * q^19 + 97557 * q^21 - 57492 * q^22 + 26484 * q^23 - 56082 * q^24 + 74282 * q^25 - 409470 * q^26 + 39366 * q^27 - 665190 * q^28 + 631476 * q^29 + 1162284 * q^30 - 65988 * q^31 + 1093710 * q^32 - 1097745 * q^33 - 1035984 * q^34 - 931746 * q^35 - 772626 * q^36 + 2616386 * q^37 - 301626 * q^38 + 651750 * q^39 + 4287624 * q^40 + 380688 * q^41 + 971148 * q^42 + 2531720 * q^43 - 3098568 * q^44 - 1102125 * q^45 - 8405436 * q^46 - 2239974 * q^47 - 1748106 * q^48 - 10575922 * q^49 + 4593750 * q^50 + 283401 * q^51 + 11027208 * q^52 + 4529832 * q^53 + 8181264 * q^54 - 896688 * q^55 - 2613726 * q^56 - 1364106 * q^57 - 10918032 * q^58 - 238332 * q^59 - 21751032 * q^60 - 4590096 * q^61 + 72699 * q^63 + 25594230 * q^64 + 8473962 * q^65 + 29170980 * q^66 + 8422900 * q^67 + 3654972 * q^68 - 12171660 * q^69 + 5069844 * q^70 + 7141248 * q^71 - 32734038 * q^72 - 3484626 * q^73 - 13236366 * q^74 + 393498 * q^75 + 21146028 * q^76 - 32526360 * q^77 + 31189776 * q^78 - 28412636 * q^79 - 26342136 * q^80 - 14906655 * q^81 + 1968132 * q^82 + 41856324 * q^83 + 41097474 * q^84 + 56859612 * q^85 - 18685110 * q^86 - 14918796 * q^87 + 6635208 * q^88 - 47578956 * q^89 - 6990132 * q^90 - 34593036 * q^91 + 8202012 * q^92 + 10276137 * q^93 - 64748316 * q^94 + 19024614 * q^95 - 50501778 * q^96 - 64009272 * q^97 + 58686474 * q^98 + 40506642 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(84))\)

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
84.8.a	\(\chi_{84}(1, \cdot)\)	84.8.a.a	1	1
		84.8.a.b	1
		84.8.a.c	2
		84.8.a.d	2
84.8.b	\(\chi_{84}(55, \cdot)\)	84.8.b.a	28	1
		84.8.b.b	28
84.8.e	\(\chi_{84}(71, \cdot)\)	84.8.e.a	84	1
84.8.f	\(\chi_{84}(41, \cdot)\)	84.8.f.a	2	1
		84.8.f.b	16
84.8.i	\(\chi_{84}(25, \cdot)\)	84.8.i.a	8	2
		84.8.i.b	10
84.8.k	\(\chi_{84}(5, \cdot)\)	84.8.k.a	2	2
		84.8.k.b	36
84.8.n	\(\chi_{84}(11, \cdot)\)	84.8.n.a	216	2
84.8.o	\(\chi_{84}(19, \cdot)\)	84.8.o.a	56	2
		84.8.o.b	56

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(84))\) into lower level spaces

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