Properties

Label 84.12
Level 84
Weight 12
Dimension 868
Nonzero newspaces 8
Newform subspaces 17
Sturm bound 4608
Trace bound 3

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Defining parameters

Level: \( N \) = \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 17 \)
Sturm bound: \(4608\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(84))\).

Total New Old
Modular forms 2172 884 1288
Cusp forms 2052 868 1184
Eisenstein series 120 16 104

Trace form

\( 868 q - 243 q^{3} - 3958 q^{4} - 10518 q^{5} - 26262 q^{6} + 145104 q^{7} + 336858 q^{8} - 584325 q^{9} + O(q^{10}) \) \( 868 q - 243 q^{3} - 3958 q^{4} - 10518 q^{5} - 26262 q^{6} + 145104 q^{7} + 336858 q^{8} - 584325 q^{9} + 950332 q^{10} - 229422 q^{11} - 2214318 q^{12} + 591858 q^{13} + 1254264 q^{14} - 4823010 q^{15} - 19598494 q^{16} + 20152164 q^{17} + 22848108 q^{18} + 4513088 q^{19} + 34025355 q^{21} - 169597812 q^{22} + 52181364 q^{23} + 119837310 q^{24} - 84163858 q^{25} - 168696030 q^{26} + 28697814 q^{27} - 415778214 q^{28} - 169620660 q^{29} + 603487164 q^{30} - 700190548 q^{31} - 356184210 q^{32} - 332588631 q^{33} - 1871054864 q^{34} + 3956574 q^{35} - 80420274 q^{36} - 1010042186 q^{37} - 2854889946 q^{38} - 140979666 q^{39} + 1172256424 q^{40} - 1247385696 q^{41} - 2049861060 q^{42} - 264802968 q^{43} + 1696833336 q^{44} + 4720301865 q^{45} + 5825150724 q^{46} + 4609267938 q^{47} - 17269986570 q^{48} - 39577276646 q^{49} + 9433593750 q^{50} + 765425817 q^{51} + 21956168776 q^{52} + 8260519728 q^{53} - 17624085504 q^{54} + 17627802192 q^{55} - 2560736286 q^{56} - 26800446990 q^{57} - 42107277008 q^{58} - 17467426140 q^{59} + 43993446168 q^{60} + 44576636136 q^{61} + 22931847159 q^{63} - 125713739722 q^{64} - 29743038138 q^{65} + 16354525236 q^{66} - 33619335692 q^{67} + 135465717852 q^{68} + 91627834068 q^{69} - 1119530796 q^{70} + 5393138112 q^{71} - 40255154502 q^{72} + 23061247674 q^{73} - 152264887566 q^{74} - 142049245662 q^{75} + 86854692012 q^{76} + 7179223968 q^{77} + 144635110128 q^{78} + 220885308292 q^{79} + 184062380904 q^{80} + 155127000231 q^{81} - 487857311708 q^{82} - 225420751932 q^{83} - 158088678270 q^{84} + 242580817652 q^{85} + 465025488618 q^{86} + 264714914628 q^{87} + 562912649544 q^{88} + 58587421164 q^{89} + 133742821548 q^{90} - 227135983180 q^{91} - 310267383972 q^{92} - 800117106693 q^{93} - 1218728471868 q^{94} + 464081155614 q^{95} - 223345212978 q^{96} + 704138505264 q^{97} + 658567234122 q^{98} + 92236928994 q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(84))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
84.12.a \(\chi_{84}(1, \cdot)\) 84.12.a.a 1 1
84.12.a.b 2
84.12.a.c 2
84.12.a.d 2
84.12.a.e 3
84.12.b \(\chi_{84}(55, \cdot)\) 84.12.b.a 44 1
84.12.b.b 44
84.12.e \(\chi_{84}(71, \cdot)\) 84.12.e.a 132 1
84.12.f \(\chi_{84}(41, \cdot)\) 84.12.f.a 2 1
84.12.f.b 28
84.12.i \(\chi_{84}(25, \cdot)\) 84.12.i.a 14 2
84.12.i.b 16
84.12.k \(\chi_{84}(5, \cdot)\) 84.12.k.a 2 2
84.12.k.b 56
84.12.n \(\chi_{84}(11, \cdot)\) 84.12.n.a 344 2
84.12.o \(\chi_{84}(19, \cdot)\) 84.12.o.a 88 2
84.12.o.b 88

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(84))\) into lower level spaces

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