# Properties

 Label 1350.2 Level 1350 Weight 2 Dimension 11798 Nonzero newspaces 18 Sturm bound 194400 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Sturm bound: $$194400$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1350))$$.

Total New Old
Modular forms 50280 11798 38482
Cusp forms 46921 11798 35123
Eisenstein series 3359 0 3359

## Trace form

 $$11798q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{4}$$ $$\mathstrut -\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 4q^{7}$$ $$\mathstrut -\mathstrut 5q^{8}$$ $$\mathstrut -\mathstrut 12q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$11798q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{4}$$ $$\mathstrut -\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 4q^{7}$$ $$\mathstrut -\mathstrut 5q^{8}$$ $$\mathstrut -\mathstrut 12q^{9}$$ $$\mathstrut -\mathstrut 8q^{10}$$ $$\mathstrut -\mathstrut 50q^{11}$$ $$\mathstrut -\mathstrut 3q^{12}$$ $$\mathstrut -\mathstrut 42q^{13}$$ $$\mathstrut -\mathstrut 36q^{14}$$ $$\mathstrut +\mathstrut q^{16}$$ $$\mathstrut -\mathstrut 54q^{17}$$ $$\mathstrut +\mathstrut 6q^{18}$$ $$\mathstrut -\mathstrut 30q^{19}$$ $$\mathstrut -\mathstrut 8q^{20}$$ $$\mathstrut +\mathstrut 24q^{21}$$ $$\mathstrut -\mathstrut 16q^{22}$$ $$\mathstrut -\mathstrut 30q^{23}$$ $$\mathstrut -\mathstrut 16q^{25}$$ $$\mathstrut +\mathstrut 32q^{26}$$ $$\mathstrut +\mathstrut 27q^{27}$$ $$\mathstrut +\mathstrut 2q^{28}$$ $$\mathstrut -\mathstrut 12q^{29}$$ $$\mathstrut -\mathstrut 36q^{31}$$ $$\mathstrut +\mathstrut q^{32}$$ $$\mathstrut +\mathstrut 75q^{33}$$ $$\mathstrut +\mathstrut 58q^{34}$$ $$\mathstrut +\mathstrut 96q^{35}$$ $$\mathstrut +\mathstrut 54q^{36}$$ $$\mathstrut +\mathstrut 60q^{37}$$ $$\mathstrut +\mathstrut 227q^{38}$$ $$\mathstrut +\mathstrut 198q^{39}$$ $$\mathstrut +\mathstrut 32q^{40}$$ $$\mathstrut +\mathstrut 298q^{41}$$ $$\mathstrut +\mathstrut 168q^{42}$$ $$\mathstrut +\mathstrut 228q^{43}$$ $$\mathstrut +\mathstrut 80q^{44}$$ $$\mathstrut +\mathstrut 120q^{45}$$ $$\mathstrut +\mathstrut 96q^{46}$$ $$\mathstrut +\mathstrut 378q^{47}$$ $$\mathstrut +\mathstrut 42q^{48}$$ $$\mathstrut +\mathstrut 213q^{49}$$ $$\mathstrut +\mathstrut 160q^{50}$$ $$\mathstrut +\mathstrut 192q^{51}$$ $$\mathstrut +\mathstrut 54q^{52}$$ $$\mathstrut +\mathstrut 336q^{53}$$ $$\mathstrut +\mathstrut 108q^{54}$$ $$\mathstrut +\mathstrut 32q^{55}$$ $$\mathstrut +\mathstrut 76q^{56}$$ $$\mathstrut +\mathstrut 135q^{57}$$ $$\mathstrut +\mathstrut 46q^{58}$$ $$\mathstrut +\mathstrut 65q^{59}$$ $$\mathstrut +\mathstrut 6q^{61}$$ $$\mathstrut +\mathstrut 56q^{62}$$ $$\mathstrut -\mathstrut 6q^{63}$$ $$\mathstrut -\mathstrut 5q^{64}$$ $$\mathstrut +\mathstrut 144q^{65}$$ $$\mathstrut +\mathstrut 114q^{67}$$ $$\mathstrut +\mathstrut 57q^{68}$$ $$\mathstrut +\mathstrut 18q^{69}$$ $$\mathstrut +\mathstrut 128q^{70}$$ $$\mathstrut +\mathstrut 16q^{71}$$ $$\mathstrut +\mathstrut 24q^{72}$$ $$\mathstrut +\mathstrut 134q^{73}$$ $$\mathstrut +\mathstrut 146q^{74}$$ $$\mathstrut +\mathstrut 72q^{75}$$ $$\mathstrut +\mathstrut 39q^{76}$$ $$\mathstrut +\mathstrut 548q^{77}$$ $$\mathstrut +\mathstrut 36q^{78}$$ $$\mathstrut +\mathstrut 364q^{79}$$ $$\mathstrut +\mathstrut 192q^{81}$$ $$\mathstrut +\mathstrut 126q^{82}$$ $$\mathstrut +\mathstrut 580q^{83}$$ $$\mathstrut +\mathstrut 208q^{85}$$ $$\mathstrut +\mathstrut 34q^{86}$$ $$\mathstrut +\mathstrut 276q^{87}$$ $$\mathstrut +\mathstrut 11q^{88}$$ $$\mathstrut +\mathstrut 505q^{89}$$ $$\mathstrut +\mathstrut 206q^{91}$$ $$\mathstrut -\mathstrut 8q^{92}$$ $$\mathstrut +\mathstrut 282q^{93}$$ $$\mathstrut -\mathstrut 34q^{94}$$ $$\mathstrut +\mathstrut 136q^{95}$$ $$\mathstrut +\mathstrut 6q^{96}$$ $$\mathstrut +\mathstrut 84q^{97}$$ $$\mathstrut -\mathstrut 105q^{98}$$ $$\mathstrut +\mathstrut 204q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1350))$$

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
1350.2.a $$\chi_{1350}(1, \cdot)$$ 1350.2.a.a 1 1
1350.2.a.b 1
1350.2.a.c 1
1350.2.a.d 1
1350.2.a.e 1
1350.2.a.f 1
1350.2.a.g 1
1350.2.a.h 1
1350.2.a.i 1
1350.2.a.j 1
1350.2.a.k 1
1350.2.a.l 1
1350.2.a.m 1
1350.2.a.n 1
1350.2.a.o 1
1350.2.a.p 1
1350.2.a.q 1
1350.2.a.r 1
1350.2.a.s 1
1350.2.a.t 1
1350.2.a.u 1
1350.2.a.v 1
1350.2.a.w 2
1350.2.a.x 2
1350.2.c $$\chi_{1350}(649, \cdot)$$ 1350.2.c.a 2 1
1350.2.c.b 2
1350.2.c.c 2
1350.2.c.d 2
1350.2.c.e 2
1350.2.c.f 2
1350.2.c.g 2
1350.2.c.h 2
1350.2.c.i 2
1350.2.c.j 2
1350.2.c.k 2
1350.2.c.l 2
1350.2.e $$\chi_{1350}(451, \cdot)$$ 1350.2.e.a 2 2
1350.2.e.b 2
1350.2.e.c 2
1350.2.e.d 2
1350.2.e.e 2
1350.2.e.f 2
1350.2.e.g 2
1350.2.e.h 2
1350.2.e.i 2
1350.2.e.j 4
1350.2.e.k 4
1350.2.e.l 4
1350.2.e.m 4
1350.2.e.n 4
1350.2.f $$\chi_{1350}(107, \cdot)$$ 1350.2.f.a 8 2
1350.2.f.b 8
1350.2.f.c 8
1350.2.f.d 8
1350.2.f.e 8
1350.2.f.f 8
1350.2.h $$\chi_{1350}(271, \cdot)$$ n/a 160 4
1350.2.j $$\chi_{1350}(199, \cdot)$$ 1350.2.j.a 4 2
1350.2.j.b 4
1350.2.j.c 4
1350.2.j.d 4
1350.2.j.e 4
1350.2.j.f 8
1350.2.j.g 8
1350.2.l $$\chi_{1350}(151, \cdot)$$ n/a 342 6
1350.2.m $$\chi_{1350}(109, \cdot)$$ n/a 160 4
1350.2.q $$\chi_{1350}(143, \cdot)$$ 1350.2.q.a 8 4
1350.2.q.b 8
1350.2.q.c 8
1350.2.q.d 8
1350.2.q.e 8
1350.2.q.f 8
1350.2.q.g 8
1350.2.q.h 16
1350.2.r $$\chi_{1350}(91, \cdot)$$ n/a 240 8
1350.2.u $$\chi_{1350}(49, \cdot)$$ n/a 324 6
1350.2.w $$\chi_{1350}(53, \cdot)$$ n/a 320 8
1350.2.z $$\chi_{1350}(19, \cdot)$$ n/a 240 8
1350.2.bb $$\chi_{1350}(257, \cdot)$$ n/a 648 12
1350.2.bc $$\chi_{1350}(31, \cdot)$$ n/a 2160 24
1350.2.bd $$\chi_{1350}(17, \cdot)$$ n/a 480 16
1350.2.bf $$\chi_{1350}(79, \cdot)$$ n/a 2160 24
1350.2.bi $$\chi_{1350}(23, \cdot)$$ n/a 4320 48

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1350))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1350)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(675))$$$$^{\oplus 2}$$