# Properties

 Label 270.4 Level 270 Weight 4 Dimension 1408 Nonzero newspaces 9 Sturm bound 15552 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$270 = 2 \cdot 3^{3} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$9$$ Sturm bound: $$15552$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(270))$$.

Total New Old
Modular forms 6072 1408 4664
Cusp forms 5592 1408 4184
Eisenstein series 480 0 480

## Trace form

 $$1408 q + 4 q^{2} - 8 q^{4} + 42 q^{5} - 24 q^{6} - 112 q^{7} - 80 q^{8} - 192 q^{9} + O(q^{10})$$ $$1408 q + 4 q^{2} - 8 q^{4} + 42 q^{5} - 24 q^{6} - 112 q^{7} - 80 q^{8} - 192 q^{9} - 12 q^{10} + 148 q^{11} + 24 q^{12} + 92 q^{13} + 240 q^{14} + 72 q^{15} - 32 q^{16} - 756 q^{17} - 552 q^{18} + 56 q^{19} + 8 q^{20} - 408 q^{21} + 240 q^{22} + 1464 q^{23} - 38 q^{25} + 1088 q^{26} + 2754 q^{27} - 160 q^{28} + 132 q^{29} + 504 q^{30} - 904 q^{31} + 64 q^{32} - 2382 q^{33} - 312 q^{34} - 5412 q^{35} - 1872 q^{36} + 140 q^{37} - 460 q^{38} + 288 q^{39} + 528 q^{40} + 1492 q^{41} + 1824 q^{42} + 3320 q^{43} + 608 q^{44} + 3606 q^{45} + 672 q^{46} + 6732 q^{47} + 384 q^{48} + 2646 q^{49} + 3396 q^{50} + 6564 q^{51} + 752 q^{52} + 2496 q^{53} - 1296 q^{54} - 3486 q^{55} - 1664 q^{56} - 1782 q^{57} - 4008 q^{58} - 6458 q^{59} - 1008 q^{60} + 1292 q^{61} - 4336 q^{62} - 7872 q^{63} + 2176 q^{64} - 8794 q^{65} - 8640 q^{66} - 8440 q^{67} - 2520 q^{68} - 12240 q^{69} - 3036 q^{70} - 4896 q^{71} + 384 q^{72} - 844 q^{73} + 4920 q^{74} - 3714 q^{75} + 2216 q^{76} + 6960 q^{77} + 11376 q^{78} + 13376 q^{79} + 384 q^{80} + 13944 q^{81} + 3768 q^{82} + 18408 q^{83} + 4320 q^{84} + 13908 q^{85} + 11752 q^{86} + 14316 q^{87} + 2256 q^{88} + 18734 q^{89} - 1260 q^{90} + 11500 q^{91} - 3936 q^{92} - 8304 q^{93} - 13800 q^{94} - 5014 q^{95} - 1920 q^{96} - 18676 q^{97} - 8484 q^{98} - 12684 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(270))$$

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
270.4.a $$\chi_{270}(1, \cdot)$$ 270.4.a.a 1 1
270.4.a.b 1
270.4.a.c 1
270.4.a.d 1
270.4.a.e 1
270.4.a.f 1
270.4.a.g 1
270.4.a.h 1
270.4.a.i 1
270.4.a.j 1
270.4.a.k 1
270.4.a.l 1
270.4.a.m 2
270.4.a.n 2
270.4.c $$\chi_{270}(109, \cdot)$$ 270.4.c.a 2 1
270.4.c.b 2
270.4.c.c 6
270.4.c.d 6
270.4.c.e 8
270.4.e $$\chi_{270}(91, \cdot)$$ 270.4.e.a 2 2
270.4.e.b 4
270.4.e.c 4
270.4.e.d 6
270.4.e.e 8
270.4.f $$\chi_{270}(53, \cdot)$$ 270.4.f.a 24 2
270.4.f.b 24
270.4.i $$\chi_{270}(19, \cdot)$$ 270.4.i.a 36 2
270.4.k $$\chi_{270}(31, \cdot)$$ n/a 216 6
270.4.m $$\chi_{270}(17, \cdot)$$ 270.4.m.a 72 4
270.4.p $$\chi_{270}(49, \cdot)$$ n/a 324 6
270.4.r $$\chi_{270}(23, \cdot)$$ n/a 648 12

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(270))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(270)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 2}$$