## Defining parameters

 Level: $$N$$ = $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$94080$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(539))$$.

Total New Old
Modular forms 35880 33726 2154
Cusp forms 34680 32854 1826
Eisenstein series 1200 872 328

## Trace form

 $$32854 q - 125 q^{2} - 149 q^{3} - 125 q^{4} - 77 q^{5} - 43 q^{6} - 96 q^{7} - 289 q^{8} - 213 q^{9} + O(q^{10})$$ $$32854 q - 125 q^{2} - 149 q^{3} - 125 q^{4} - 77 q^{5} - 43 q^{6} - 96 q^{7} - 289 q^{8} - 213 q^{9} - 138 q^{10} - 90 q^{11} + 60 q^{12} - 93 q^{13} - 96 q^{14} - 243 q^{15} - 265 q^{16} + 37 q^{17} - 202 q^{18} - 524 q^{19} - 350 q^{20} - 600 q^{21} + 370 q^{22} + 460 q^{23} + 249 q^{24} + 655 q^{25} - 288 q^{26} - 98 q^{27} + 240 q^{28} - 223 q^{29} - 752 q^{30} - 1043 q^{31} - 1172 q^{32} - 237 q^{33} - 1160 q^{34} - 264 q^{35} + 3676 q^{36} + 2871 q^{37} + 5528 q^{38} + 4941 q^{39} + 7612 q^{40} + 4597 q^{41} - 390 q^{42} - 2894 q^{43} - 4705 q^{44} - 9026 q^{45} - 8722 q^{46} - 4191 q^{47} - 14654 q^{48} - 7908 q^{49} - 9743 q^{50} - 7418 q^{51} - 13354 q^{52} - 6305 q^{53} - 7974 q^{54} - 2760 q^{55} + 144 q^{56} + 4038 q^{57} + 8124 q^{58} + 10070 q^{59} + 37400 q^{60} + 20331 q^{61} + 32122 q^{62} + 19104 q^{63} + 31835 q^{64} + 13066 q^{65} + 17599 q^{66} + 1578 q^{67} - 2046 q^{68} - 14476 q^{69} - 7014 q^{70} - 17295 q^{71} - 39521 q^{72} - 12539 q^{73} - 19146 q^{74} - 35228 q^{75} - 43946 q^{76} - 6768 q^{77} - 31330 q^{78} - 11081 q^{79} - 12562 q^{80} + 6132 q^{81} + 12369 q^{82} + 16804 q^{83} + 31344 q^{84} + 13875 q^{85} + 16607 q^{86} + 17226 q^{87} + 22609 q^{88} + 11318 q^{89} + 21840 q^{90} + 1506 q^{91} + 20724 q^{92} - 10563 q^{93} - 10424 q^{94} - 21501 q^{95} - 39774 q^{96} - 21392 q^{97} - 55116 q^{98} - 14837 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(539))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
539.4.a $$\chi_{539}(1, \cdot)$$ 539.4.a.a 1 1
539.4.a.b 1
539.4.a.c 1
539.4.a.d 2
539.4.a.e 2
539.4.a.f 4
539.4.a.g 4
539.4.a.h 5
539.4.a.i 6
539.4.a.j 9
539.4.a.k 9
539.4.a.l 10
539.4.a.m 10
539.4.a.n 10
539.4.a.o 10
539.4.a.p 18
539.4.b $$\chi_{539}(538, \cdot)$$ n/a 116 1
539.4.e $$\chi_{539}(67, \cdot)$$ n/a 200 2
539.4.f $$\chi_{539}(148, \cdot)$$ n/a 472 4
539.4.i $$\chi_{539}(362, \cdot)$$ n/a 232 2
539.4.j $$\chi_{539}(78, \cdot)$$ n/a 840 6
539.4.m $$\chi_{539}(195, \cdot)$$ n/a 464 4
539.4.p $$\chi_{539}(76, \cdot)$$ n/a 996 6
539.4.q $$\chi_{539}(214, \cdot)$$ n/a 928 8
539.4.r $$\chi_{539}(23, \cdot)$$ n/a 1680 12
539.4.s $$\chi_{539}(19, \cdot)$$ n/a 928 8
539.4.v $$\chi_{539}(15, \cdot)$$ n/a 3984 24
539.4.w $$\chi_{539}(10, \cdot)$$ n/a 1992 12
539.4.z $$\chi_{539}(6, \cdot)$$ n/a 3984 24
539.4.bc $$\chi_{539}(4, \cdot)$$ n/a 7968 48
539.4.bf $$\chi_{539}(17, \cdot)$$ n/a 7968 48

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(539))$$ into lower level spaces

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