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Properties

Label 539.8
Level 539
Weight 8
Dimension 77242
Nonzero newspaces 16
Sturm bound 188160
Trace bound 2

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Defining parameters

Level: \( N \) = \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(188160\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(539))\).

Total New Old
Modular forms 82920 78114 4806
Cusp forms 81720 77242 4478
Eisenstein series 1200 872 328

Trace form

\( 77242 q - 125 q^{2} - 17 q^{3} - 1149 q^{4} - 137 q^{5} + 6767 q^{6} + 520 q^{7} - 12009 q^{8} + 129 q^{9} + O(q^{10}) \) \( 77242 q - 125 q^{2} - 17 q^{3} - 1149 q^{4} - 137 q^{5} + 6767 q^{6} + 520 q^{7} - 12009 q^{8} + 129 q^{9} - 26478 q^{10} + 5657 q^{11} + 64572 q^{12} - 31958 q^{13} + 92808 q^{14} + 40047 q^{15} - 296737 q^{16} - 307220 q^{17} - 370228 q^{18} + 180380 q^{19} + 1642970 q^{20} + 283614 q^{21} - 217838 q^{22} - 1015447 q^{23} - 1807911 q^{24} - 432435 q^{25} + 592288 q^{26} + 1656361 q^{27} + 1175680 q^{28} + 703326 q^{29} - 2967932 q^{30} - 2297277 q^{31} + 1341068 q^{32} + 1713360 q^{33} + 1953344 q^{34} - 203334 q^{35} + 1874944 q^{36} - 292183 q^{37} - 10788852 q^{38} - 12034386 q^{39} - 682148 q^{40} + 10379386 q^{41} + 14061714 q^{42} + 8263650 q^{43} + 15154707 q^{44} + 10822204 q^{45} - 16684338 q^{46} - 22469572 q^{47} - 67539374 q^{48} - 22267628 q^{49} - 10020943 q^{50} + 1646026 q^{51} + 39890774 q^{52} + 32928660 q^{53} + 71282850 q^{54} + 17934995 q^{55} + 25583520 q^{56} + 16526130 q^{57} - 6090284 q^{58} - 23010871 q^{59} - 87099760 q^{60} - 18980176 q^{61} - 49052834 q^{62} - 19013574 q^{63} - 39777213 q^{64} - 10500824 q^{65} + 47487823 q^{66} + 48353655 q^{67} + 119722610 q^{68} + 150708665 q^{69} + 82506186 q^{70} + 23605797 q^{71} - 16423049 q^{72} - 28425882 q^{73} - 132310698 q^{74} - 180518768 q^{75} - 227995266 q^{76} - 46116909 q^{77} - 197927266 q^{78} - 90417134 q^{79} + 92840078 q^{80} + 78296730 q^{81} + 8261123 q^{82} + 69835146 q^{83} + 28005696 q^{84} + 214706330 q^{85} + 180720177 q^{86} + 171480384 q^{87} + 142756761 q^{88} + 62646471 q^{89} + 40513380 q^{90} - 37046596 q^{91} - 341340044 q^{92} - 268337523 q^{93} - 98007472 q^{94} - 16358796 q^{95} - 304830462 q^{96} - 81183245 q^{97} - 401433324 q^{98} - 95572085 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(539))\)

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
539.8.a \(\chi_{539}(1, \cdot)\) 539.8.a.a 2 1
539.8.a.b 4
539.8.a.c 7
539.8.a.d 9
539.8.a.e 9
539.8.a.f 11
539.8.a.g 16
539.8.a.h 20
539.8.a.i 23
539.8.a.j 23
539.8.a.k 23
539.8.a.l 23
539.8.a.m 30
539.8.a.n 38
539.8.b \(\chi_{539}(538, \cdot)\) n/a 276 1
539.8.e \(\chi_{539}(67, \cdot)\) n/a 468 2
539.8.f \(\chi_{539}(148, \cdot)\) n/a 1128 4
539.8.i \(\chi_{539}(362, \cdot)\) n/a 552 2
539.8.j \(\chi_{539}(78, \cdot)\) n/a 1968 6
539.8.m \(\chi_{539}(195, \cdot)\) n/a 1104 4
539.8.p \(\chi_{539}(76, \cdot)\) n/a 2340 6
539.8.q \(\chi_{539}(214, \cdot)\) n/a 2208 8
539.8.r \(\chi_{539}(23, \cdot)\) n/a 3912 12
539.8.s \(\chi_{539}(19, \cdot)\) n/a 2208 8
539.8.v \(\chi_{539}(15, \cdot)\) n/a 9360 24
539.8.w \(\chi_{539}(10, \cdot)\) n/a 4680 12
539.8.z \(\chi_{539}(6, \cdot)\) n/a 9360 24
539.8.bc \(\chi_{539}(4, \cdot)\) n/a 18720 48
539.8.bf \(\chi_{539}(17, \cdot)\) n/a 18720 48

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(539))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(539)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(77))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(539))\)\(^{\oplus 1}\)