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Properties

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

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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

\( 392 q + 9 q^{3} - 22 q^{4} + 66 q^{5} - 54 q^{6} + 232 q^{7} - 534 q^{8} + 531 q^{9} + O(q^{10}) \) \( 392 q + 9 q^{3} - 22 q^{4} + 66 q^{5} - 54 q^{6} + 232 q^{7} - 534 q^{8} + 531 q^{9} + 2140 q^{10} - 534 q^{11} - 246 q^{12} - 958 q^{13} - 3720 q^{14} + 918 q^{15} - 2398 q^{16} + 3252 q^{17} - 1380 q^{18} + 1504 q^{19} + 1323 q^{21} - 9828 q^{22} + 3828 q^{23} + 21894 q^{24} - 2886 q^{25} + 17970 q^{26} - 1458 q^{27} + 14394 q^{28} - 32940 q^{29} - 36636 q^{30} + 6796 q^{31} - 22530 q^{32} + 13821 q^{33} + 20944 q^{34} + 36966 q^{35} + 55422 q^{36} + 8310 q^{37} + 41238 q^{38} - 13026 q^{39} - 88904 q^{40} - 46680 q^{41} + 18324 q^{42} - 45496 q^{43} + 14520 q^{44} + 37701 q^{45} + 64164 q^{46} + 66474 q^{47} + 8886 q^{48} - 114658 q^{49} - 26250 q^{50} - 21483 q^{51} - 99992 q^{52} + 19776 q^{53} + 20856 q^{54} - 141120 q^{55} + 45378 q^{56} - 99822 q^{57} + 350608 q^{58} + 61716 q^{59} + 252768 q^{60} + 15152 q^{61} - 52797 q^{63} - 276970 q^{64} - 197418 q^{65} - 333732 q^{66} - 242556 q^{67} - 330516 q^{68} + 134100 q^{69} - 193164 q^{70} - 92736 q^{71} + 205698 q^{72} + 389042 q^{73} - 36366 q^{74} + 170658 q^{75} + 388140 q^{76} + 192024 q^{77} + 311904 q^{78} + 118740 q^{79} + 61848 q^{80} - 129681 q^{81} + 8116 q^{82} + 63156 q^{83} - 328830 q^{84} - 329644 q^{85} - 6918 q^{86} - 83556 q^{87} + 571272 q^{88} + 201732 q^{89} + 814044 q^{90} + 168916 q^{91} + 37884 q^{92} + 836175 q^{93} - 725772 q^{94} - 99882 q^{95} - 809346 q^{96} - 156256 q^{97} - 1266006 q^{98} - 297126 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

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