# Properties

 Label 1339.2 Level 1339 Weight 2 Dimension 72245 Nonzero newspaces 30 Sturm bound 297024 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$1339 = 13 \cdot 103$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$30$$ Sturm bound: $$297024$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1339))$$.

Total New Old
Modular forms 75480 74469 1011
Cusp forms 73033 72245 788
Eisenstein series 2447 2224 223

## Trace form

 $$72245q - 507q^{2} - 510q^{3} - 519q^{4} - 516q^{5} - 534q^{6} - 518q^{7} - 525q^{8} - 521q^{9} + O(q^{10})$$ $$72245q - 507q^{2} - 510q^{3} - 519q^{4} - 516q^{5} - 534q^{6} - 518q^{7} - 525q^{8} - 521q^{9} - 522q^{10} - 522q^{11} - 530q^{12} - 552q^{13} - 1146q^{14} - 546q^{15} - 551q^{16} - 534q^{17} - 549q^{18} - 518q^{19} - 558q^{20} - 542q^{21} - 546q^{22} - 546q^{23} - 558q^{24} - 549q^{25} - 570q^{26} - 1146q^{27} - 566q^{28} - 534q^{29} - 570q^{30} - 542q^{31} - 555q^{32} - 558q^{33} - 564q^{34} - 558q^{35} - 603q^{36} - 578q^{37} - 594q^{38} - 593q^{39} - 1224q^{40} - 534q^{41} - 606q^{42} - 538q^{43} - 594q^{44} - 582q^{45} - 546q^{46} - 558q^{47} - 654q^{48} - 553q^{49} - 591q^{50} - 558q^{51} - 572q^{52} - 1164q^{53} - 654q^{54} - 582q^{55} - 630q^{56} - 590q^{57} - 618q^{58} - 594q^{59} - 678q^{60} - 570q^{61} - 618q^{62} - 614q^{63} - 633q^{64} - 549q^{65} - 1266q^{66} - 590q^{67} - 630q^{68} - 630q^{69} - 654q^{70} - 594q^{71} - 711q^{72} - 584q^{73} - 594q^{74} - 626q^{75} - 638q^{76} - 606q^{77} - 525q^{78} - 1218q^{79} - 666q^{80} - 653q^{81} - 654q^{82} - 594q^{83} - 598q^{84} - 432q^{85} - 438q^{86} - 414q^{87} + 126q^{88} - 372q^{89} + 60q^{90} - 348q^{91} - 906q^{92} - 206q^{93} - 234q^{94} - 336q^{95} + 462q^{96} - 190q^{97} + 93q^{98} - 156q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1339))$$

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
1339.2.a $$\chi_{1339}(1, \cdot)$$ 1339.2.a.a 1 1
1339.2.a.b 1
1339.2.a.c 3
1339.2.a.d 19
1339.2.a.e 21
1339.2.a.f 28
1339.2.a.g 30
1339.2.c $$\chi_{1339}(207, \cdot)$$ n/a 118 1
1339.2.e $$\chi_{1339}(458, \cdot)$$ n/a 238 2
1339.2.f $$\chi_{1339}(516, \cdot)$$ n/a 240 2
1339.2.g $$\chi_{1339}(365, \cdot)$$ n/a 208 2
1339.2.h $$\chi_{1339}(159, \cdot)$$ n/a 238 2
1339.2.i $$\chi_{1339}(411, \cdot)$$ n/a 236 2
1339.2.k $$\chi_{1339}(355, \cdot)$$ n/a 238 2
1339.2.q $$\chi_{1339}(413, \cdot)$$ n/a 236 2
1339.2.r $$\chi_{1339}(56, \cdot)$$ n/a 238 2
1339.2.u $$\chi_{1339}(571, \cdot)$$ n/a 240 2
1339.2.w $$\chi_{1339}(47, \cdot)$$ n/a 480 4
1339.2.ba $$\chi_{1339}(665, \cdot)$$ n/a 476 4
1339.2.bb $$\chi_{1339}(150, \cdot)$$ n/a 476 4
1339.2.bc $$\chi_{1339}(102, \cdot)$$ n/a 480 4
1339.2.be $$\chi_{1339}(14, \cdot)$$ n/a 1664 16
1339.2.bg $$\chi_{1339}(64, \cdot)$$ n/a 1888 16
1339.2.bi $$\chi_{1339}(107, \cdot)$$ n/a 3808 32
1339.2.bj $$\chi_{1339}(92, \cdot)$$ n/a 3328 32
1339.2.bk $$\chi_{1339}(9, \cdot)$$ n/a 3840 32
1339.2.bl $$\chi_{1339}(16, \cdot)$$ n/a 3808 32
1339.2.bn $$\chi_{1339}(31, \cdot)$$ n/a 3776 32
1339.2.bp $$\chi_{1339}(25, \cdot)$$ n/a 3840 32
1339.2.bs $$\chi_{1339}(4, \cdot)$$ n/a 3808 32
1339.2.bt $$\chi_{1339}(23, \cdot)$$ n/a 3840 32
1339.2.bz $$\chi_{1339}(36, \cdot)$$ n/a 3808 32
1339.2.cb $$\chi_{1339}(24, \cdot)$$ n/a 7680 64
1339.2.cc $$\chi_{1339}(45, \cdot)$$ n/a 7616 64
1339.2.cd $$\chi_{1339}(6, \cdot)$$ n/a 7616 64
1339.2.ch $$\chi_{1339}(5, \cdot)$$ n/a 7680 64

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1339))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1339)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(103))$$$$^{\oplus 2}$$