# Properties

 Label 448.6 Level 448 Weight 6 Dimension 16450 Nonzero newspaces 16 Sturm bound 73728 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$16$$ Sturm bound: $$73728$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(448))$$.

Total New Old
Modular forms 31152 16670 14482
Cusp forms 30288 16450 13838
Eisenstein series 864 220 644

## Trace form

 $$16450 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 28 q^{7} - 80 q^{8} + 446 q^{9} + O(q^{10})$$ $$16450 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 28 q^{7} - 80 q^{8} + 446 q^{9} - 32 q^{10} - 1232 q^{11} - 32 q^{12} + 432 q^{13} - 40 q^{14} + 3536 q^{15} - 32 q^{16} + 1560 q^{17} - 32 q^{18} - 4744 q^{19} - 32 q^{20} - 5260 q^{21} - 27264 q^{22} - 20 q^{23} + 44048 q^{24} + 31138 q^{25} + 51888 q^{26} + 8412 q^{27} - 4400 q^{28} - 16368 q^{29} - 129792 q^{30} - 23108 q^{31} - 74352 q^{32} - 44428 q^{33} - 25552 q^{34} + 8608 q^{35} + 124560 q^{36} + 46864 q^{37} + 139248 q^{38} - 20 q^{39} + 124688 q^{40} - 26664 q^{41} - 46400 q^{42} - 30820 q^{43} - 131056 q^{44} + 47016 q^{45} - 32 q^{46} + 88332 q^{47} - 32 q^{48} + 29194 q^{49} + 274048 q^{50} - 89036 q^{51} - 147008 q^{52} - 77744 q^{53} - 466592 q^{54} + 440160 q^{55} - 150960 q^{56} - 70808 q^{57} - 52016 q^{58} - 175328 q^{59} + 395680 q^{60} + 100272 q^{61} + 350720 q^{62} - 329696 q^{63} + 749680 q^{64} - 27604 q^{65} + 509216 q^{66} - 395080 q^{67} + 14272 q^{68} - 287048 q^{69} - 286984 q^{70} + 575304 q^{71} - 828176 q^{72} + 169944 q^{73} - 755472 q^{74} + 1082424 q^{75} - 536992 q^{76} + 53236 q^{77} - 665408 q^{78} - 816468 q^{79} + 599696 q^{80} + 408286 q^{81} + 1006208 q^{82} + 455656 q^{83} + 985000 q^{84} + 497888 q^{85} + 1443696 q^{86} - 20 q^{87} + 446528 q^{88} - 230408 q^{89} - 568832 q^{90} - 432876 q^{91} - 1842624 q^{92} - 1837664 q^{93} - 1642592 q^{94} - 616068 q^{95} - 2428720 q^{96} - 146312 q^{97} - 957344 q^{98} + 870236 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(448))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
448.6.a $$\chi_{448}(1, \cdot)$$ 448.6.a.a 1 1
448.6.a.b 1
448.6.a.c 1
448.6.a.d 1
448.6.a.e 1
448.6.a.f 1
448.6.a.g 1
448.6.a.h 1
448.6.a.i 1
448.6.a.j 1
448.6.a.k 1
448.6.a.l 1
448.6.a.m 1
448.6.a.n 1
448.6.a.o 1
448.6.a.p 1
448.6.a.q 2
448.6.a.r 2
448.6.a.s 2
448.6.a.t 2
448.6.a.u 2
448.6.a.v 2
448.6.a.w 2
448.6.a.x 2
448.6.a.y 2
448.6.a.z 2
448.6.a.ba 3
448.6.a.bb 3
448.6.a.bc 4
448.6.a.bd 4
448.6.a.be 5
448.6.a.bf 5
448.6.b $$\chi_{448}(225, \cdot)$$ 448.6.b.a 10 1
448.6.b.b 10
448.6.b.c 20
448.6.b.d 20
448.6.e $$\chi_{448}(223, \cdot)$$ 448.6.e.a 8 1
448.6.e.b 16
448.6.e.c 56
448.6.f $$\chi_{448}(447, \cdot)$$ 448.6.f.a 2 1
448.6.f.b 8
448.6.f.c 12
448.6.f.d 16
448.6.f.e 40
448.6.i $$\chi_{448}(65, \cdot)$$ n/a 156 2
448.6.j $$\chi_{448}(111, \cdot)$$ n/a 156 2
448.6.m $$\chi_{448}(113, \cdot)$$ n/a 120 2
448.6.p $$\chi_{448}(255, \cdot)$$ n/a 156 2
448.6.q $$\chi_{448}(31, \cdot)$$ n/a 160 2
448.6.t $$\chi_{448}(289, \cdot)$$ n/a 160 2
448.6.u $$\chi_{448}(57, \cdot)$$ None 0 4
448.6.x $$\chi_{448}(55, \cdot)$$ None 0 4
448.6.z $$\chi_{448}(47, \cdot)$$ n/a 312 4
448.6.ba $$\chi_{448}(81, \cdot)$$ n/a 312 4
448.6.bc $$\chi_{448}(29, \cdot)$$ n/a 1920 8
448.6.bd $$\chi_{448}(27, \cdot)$$ n/a 2544 8
448.6.bh $$\chi_{448}(9, \cdot)$$ None 0 8
448.6.bi $$\chi_{448}(87, \cdot)$$ None 0 8
448.6.bm $$\chi_{448}(3, \cdot)$$ n/a 5088 16
448.6.bn $$\chi_{448}(37, \cdot)$$ n/a 5088 16

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(448))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(448)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 14}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 10}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 7}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(448))$$$$^{\oplus 1}$$