# Properties

 Label 138.8 Level 138 Weight 8 Dimension 922 Nonzero newspaces 4 Sturm bound 8448 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$4$$ Sturm bound: $$8448$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(138))$$.

Total New Old
Modular forms 3784 922 2862
Cusp forms 3608 922 2686
Eisenstein series 176 0 176

## Trace form

 $$922 q - 16 q^{2} - 54 q^{3} - 128 q^{4} + 228 q^{5} - 432 q^{6} + 3152 q^{7} - 1024 q^{8} - 1458 q^{9} + O(q^{10})$$ $$922 q - 16 q^{2} - 54 q^{3} - 128 q^{4} + 228 q^{5} - 432 q^{6} + 3152 q^{7} - 1024 q^{8} - 1458 q^{9} + 1824 q^{10} - 14664 q^{11} - 3456 q^{12} + 7604 q^{13} + 25216 q^{14} - 1522 q^{15} - 8192 q^{16} - 28016 q^{17} - 290096 q^{18} - 299156 q^{19} + 11776 q^{20} + 528558 q^{21} + 351200 q^{22} + 759968 q^{23} - 27648 q^{24} - 637894 q^{25} - 737152 q^{26} - 1419888 q^{27} - 353024 q^{28} + 418984 q^{29} + 1530112 q^{30} + 2057092 q^{31} - 65536 q^{32} - 2826334 q^{33} + 105696 q^{34} + 2187172 q^{35} - 93312 q^{36} + 1883356 q^{37} - 397760 q^{38} - 2573424 q^{39} + 116736 q^{40} - 660968 q^{41} + 680832 q^{42} + 3701440 q^{43} - 938496 q^{44} + 166212 q^{45} - 331584 q^{46} + 5663816 q^{47} - 221184 q^{48} - 10477902 q^{49} + 1042064 q^{50} - 6038280 q^{51} + 486656 q^{52} - 6257408 q^{53} + 9122368 q^{54} + 10592696 q^{55} + 1613824 q^{56} - 3079626 q^{57} + 665760 q^{58} - 5234372 q^{59} + 4788352 q^{60} + 3321236 q^{61} - 530432 q^{62} + 20343638 q^{63} - 524288 q^{64} - 866856 q^{65} - 19726208 q^{66} + 6581672 q^{67} + 845568 q^{68} - 34150666 q^{69} - 2874624 q^{70} - 11432304 q^{71} + 852992 q^{72} - 5319796 q^{73} + 583456 q^{74} + 63460410 q^{75} - 3182080 q^{76} + 23110464 q^{77} + 19848432 q^{78} + 5829364 q^{79} + 933888 q^{80} - 71160382 q^{81} + 10225248 q^{82} - 32438228 q^{83} - 5610368 q^{84} - 20968732 q^{85} + 2502592 q^{86} + 4830378 q^{87} - 7507968 q^{88} + 10982104 q^{89} + 1329696 q^{90} + 54330344 q^{91} - 2652672 q^{92} - 1790208 q^{93} + 6940416 q^{94} - 153489732 q^{95} - 1769472 q^{96} + 112179460 q^{97} + 107625008 q^{98} - 16593294 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(138))$$

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
138.8.a $$\chi_{138}(1, \cdot)$$ 138.8.a.a 1 1
138.8.a.b 3
138.8.a.c 3
138.8.a.d 3
138.8.a.e 4
138.8.a.f 4
138.8.a.g 4
138.8.a.h 4
138.8.d $$\chi_{138}(137, \cdot)$$ 138.8.d.a 56 1
138.8.e $$\chi_{138}(13, \cdot)$$ n/a 280 10
138.8.f $$\chi_{138}(5, \cdot)$$ n/a 560 10

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

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(138))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(138)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(69))$$$$^{\oplus 2}$$