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Properties

Label 84.4
Level 84
Weight 4
Dimension 230
Nonzero newspaces 8
Newform subspaces 14
Sturm bound 1536
Trace bound 5

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Defining parameters

Level: \( N \) = \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 14 \)
Sturm bound: \(1536\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(84))\).

Total New Old
Modular forms 636 246 390
Cusp forms 516 230 286
Eisenstein series 120 16 104

Trace form

\( 230 q + 10 q^{4} + 12 q^{5} + 42 q^{6} - 56 q^{7} - 102 q^{8} - 42 q^{9} + O(q^{10}) \) \( 230 q + 10 q^{4} + 12 q^{5} + 42 q^{6} - 56 q^{7} - 102 q^{8} - 42 q^{9} - 148 q^{10} + 96 q^{11} - 78 q^{12} + 304 q^{13} + 312 q^{14} + 180 q^{15} + 610 q^{16} - 24 q^{17} + 372 q^{18} - 184 q^{19} - 438 q^{21} - 756 q^{22} - 60 q^{23} - 930 q^{24} + 322 q^{25} - 750 q^{26} - 1062 q^{28} + 1620 q^{29} + 324 q^{30} + 8 q^{31} - 210 q^{32} + 576 q^{33} - 112 q^{34} - 816 q^{35} - 114 q^{36} - 116 q^{37} + 1494 q^{38} - 696 q^{39} + 104 q^{40} + 12 q^{41} - 948 q^{42} - 952 q^{43} + 1848 q^{44} - 1500 q^{45} + 2388 q^{46} - 684 q^{47} + 1014 q^{48} - 3922 q^{49} - 1050 q^{50} - 360 q^{51} - 2872 q^{52} - 1752 q^{53} - 2112 q^{54} - 468 q^{55} - 5022 q^{56} + 1788 q^{57} - 2320 q^{58} + 1128 q^{59} - 72 q^{60} + 6208 q^{61} + 2508 q^{63} + 694 q^{64} + 2832 q^{65} + 732 q^{66} + 3536 q^{67} + 7452 q^{68} + 1572 q^{69} + 9444 q^{70} + 1224 q^{71} + 2970 q^{72} - 2192 q^{73} + 3234 q^{74} - 2592 q^{75} - 2388 q^{76} + 468 q^{77} + 336 q^{78} - 3544 q^{79} - 5976 q^{80} - 3006 q^{81} - 7228 q^{82} - 4272 q^{83} - 1470 q^{84} - 6008 q^{85} - 6342 q^{86} - 4320 q^{87} + 3144 q^{88} + 708 q^{89} + 5148 q^{90} + 1472 q^{91} + 7644 q^{92} - 1452 q^{93} + 14292 q^{94} + 684 q^{95} + 4878 q^{96} + 3256 q^{97} + 11130 q^{98} + 9144 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(84))\)

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
84.4.a \(\chi_{84}(1, \cdot)\) 84.4.a.a 1 1
84.4.a.b 1
84.4.b \(\chi_{84}(55, \cdot)\) 84.4.b.a 12 1
84.4.b.b 12
84.4.e \(\chi_{84}(71, \cdot)\) 84.4.e.a 36 1
84.4.f \(\chi_{84}(41, \cdot)\) 84.4.f.a 8 1
84.4.i \(\chi_{84}(25, \cdot)\) 84.4.i.a 4 2
84.4.i.b 4
84.4.k \(\chi_{84}(5, \cdot)\) 84.4.k.a 2 2
84.4.k.b 2
84.4.k.c 12
84.4.n \(\chi_{84}(11, \cdot)\) 84.4.n.a 88 2
84.4.o \(\chi_{84}(19, \cdot)\) 84.4.o.a 24 2
84.4.o.b 24

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(84))\) into lower level spaces

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