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Properties

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

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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

\( 548 q - 27 q^{3} - 54 q^{4} + 222 q^{5} + 426 q^{6} - 944 q^{7} + 1722 q^{8} - 2163 q^{9} + O(q^{10}) \) \( 548 q - 27 q^{3} - 54 q^{4} + 222 q^{5} + 426 q^{6} - 944 q^{7} + 1722 q^{8} - 2163 q^{9} - 17028 q^{10} - 4254 q^{11} + 12258 q^{12} + 16830 q^{13} + 39288 q^{14} - 40410 q^{15} - 66846 q^{16} + 55716 q^{17} - 9396 q^{18} - 25872 q^{19} + 97557 q^{21} - 57492 q^{22} + 26484 q^{23} - 56082 q^{24} + 74282 q^{25} - 409470 q^{26} + 39366 q^{27} - 665190 q^{28} + 631476 q^{29} + 1162284 q^{30} - 65988 q^{31} + 1093710 q^{32} - 1097745 q^{33} - 1035984 q^{34} - 931746 q^{35} - 772626 q^{36} + 2616386 q^{37} - 301626 q^{38} + 651750 q^{39} + 4287624 q^{40} + 380688 q^{41} + 971148 q^{42} + 2531720 q^{43} - 3098568 q^{44} - 1102125 q^{45} - 8405436 q^{46} - 2239974 q^{47} - 1748106 q^{48} - 10575922 q^{49} + 4593750 q^{50} + 283401 q^{51} + 11027208 q^{52} + 4529832 q^{53} + 8181264 q^{54} - 896688 q^{55} - 2613726 q^{56} - 1364106 q^{57} - 10918032 q^{58} - 238332 q^{59} - 21751032 q^{60} - 4590096 q^{61} + 72699 q^{63} + 25594230 q^{64} + 8473962 q^{65} + 29170980 q^{66} + 8422900 q^{67} + 3654972 q^{68} - 12171660 q^{69} + 5069844 q^{70} + 7141248 q^{71} - 32734038 q^{72} - 3484626 q^{73} - 13236366 q^{74} + 393498 q^{75} + 21146028 q^{76} - 32526360 q^{77} + 31189776 q^{78} - 28412636 q^{79} - 26342136 q^{80} - 14906655 q^{81} + 1968132 q^{82} + 41856324 q^{83} + 41097474 q^{84} + 56859612 q^{85} - 18685110 q^{86} - 14918796 q^{87} + 6635208 q^{88} - 47578956 q^{89} - 6990132 q^{90} - 34593036 q^{91} + 8202012 q^{92} + 10276137 q^{93} - 64748316 q^{94} + 19024614 q^{95} - 50501778 q^{96} - 64009272 q^{97} + 58686474 q^{98} + 40506642 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

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