# Properties

 Label 112.10 Level 112 Weight 10 Dimension 1841 Nonzero newspaces 8 Sturm bound 7680 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ = $$10$$ Nonzero newspaces: $$8$$ Sturm bound: $$7680$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_1(112))$$.

Total New Old
Modular forms 3540 1885 1655
Cusp forms 3372 1841 1531
Eisenstein series 168 44 124

## Trace form

 $$1841 q - 8 q^{2} + 155 q^{3} - 348 q^{4} + 709 q^{5} + 4372 q^{6} - 1385 q^{7} + 1408 q^{8} - 11627 q^{9} + O(q^{10})$$ $$1841 q - 8 q^{2} + 155 q^{3} - 348 q^{4} + 709 q^{5} + 4372 q^{6} - 1385 q^{7} + 1408 q^{8} - 11627 q^{9} - 9380 q^{10} - 109741 q^{11} - 434324 q^{12} + 86146 q^{13} + 133528 q^{14} - 930906 q^{15} + 1634500 q^{16} - 344227 q^{17} - 4359816 q^{18} + 2559653 q^{19} + 6555964 q^{20} - 2759551 q^{21} + 68576 q^{22} + 1430565 q^{23} - 3727724 q^{24} + 2348589 q^{25} + 12695524 q^{26} - 15473230 q^{27} - 3629540 q^{28} - 1230918 q^{29} + 9355596 q^{30} + 18087423 q^{31} + 3311972 q^{32} - 642155 q^{33} + 9056396 q^{34} - 28934519 q^{35} - 47128304 q^{36} + 28758025 q^{37} + 29992692 q^{38} - 121564070 q^{39} + 23502564 q^{40} - 9595734 q^{41} + 15533180 q^{42} + 198811680 q^{43} - 72280228 q^{44} + 88731780 q^{45} + 317384496 q^{46} - 389588759 q^{47} + 24132892 q^{48} - 1290575879 q^{49} - 545233464 q^{50} + 284022509 q^{51} - 193617368 q^{52} + 294004613 q^{53} + 960636844 q^{54} - 214172066 q^{55} + 655987472 q^{56} + 322857178 q^{57} + 151101648 q^{58} + 43331451 q^{59} - 2719663236 q^{60} - 460728107 q^{61} - 683041736 q^{62} - 117147087 q^{63} + 1749389628 q^{64} + 1052934718 q^{65} + 2452878100 q^{66} - 606325321 q^{67} - 79371456 q^{68} - 1936679202 q^{69} - 2781321212 q^{70} - 654645936 q^{71} - 153538644 q^{72} - 634078251 q^{73} - 247821732 q^{74} + 2265797796 q^{75} + 906538188 q^{76} + 1629494525 q^{77} + 3470049584 q^{78} - 2381356403 q^{79} + 1400545700 q^{80} + 657170160 q^{81} + 1348575156 q^{82} - 805955192 q^{83} - 1281919924 q^{84} - 2814167990 q^{85} - 487961364 q^{86} + 3992098986 q^{87} - 3562294268 q^{88} + 1946820477 q^{89} - 6289106748 q^{90} - 820048046 q^{91} + 357523548 q^{92} - 1352778547 q^{93} + 15599499452 q^{94} - 127481037 q^{95} + 7664652572 q^{96} - 578031614 q^{97} - 6954079108 q^{98} + 6210596644 q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_1(112))$$

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
112.10.a $$\chi_{112}(1, \cdot)$$ 112.10.a.a 1 1
112.10.a.b 1
112.10.a.c 2
112.10.a.d 2
112.10.a.e 2
112.10.a.f 2
112.10.a.g 3
112.10.a.h 3
112.10.a.i 3
112.10.a.j 4
112.10.a.k 4
112.10.b $$\chi_{112}(57, \cdot)$$ None 0 1
112.10.e $$\chi_{112}(55, \cdot)$$ None 0 1
112.10.f $$\chi_{112}(111, \cdot)$$ 112.10.f.a 12 1
112.10.f.b 24
112.10.i $$\chi_{112}(65, \cdot)$$ 112.10.i.a 6 2
112.10.i.b 6
112.10.i.c 10
112.10.i.d 12
112.10.i.e 18
112.10.i.f 18
112.10.j $$\chi_{112}(27, \cdot)$$ n/a 284 2
112.10.m $$\chi_{112}(29, \cdot)$$ n/a 216 2
112.10.p $$\chi_{112}(31, \cdot)$$ 112.10.p.a 24 2
112.10.p.b 24
112.10.p.c 24
112.10.q $$\chi_{112}(87, \cdot)$$ None 0 2
112.10.t $$\chi_{112}(9, \cdot)$$ None 0 2
112.10.v $$\chi_{112}(3, \cdot)$$ n/a 568 4
112.10.w $$\chi_{112}(37, \cdot)$$ n/a 568 4

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

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_1(112))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_1(112)) \cong$$ $$S_{10}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 5}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 3}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 2}$$