# Properties

 Label 448.7 Level 448 Weight 7 Dimension 19762 Nonzero newspaces 16 Sturm bound 86016 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$16$$ Sturm bound: $$86016$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(448))$$.

Total New Old
Modular forms 37296 19982 17314
Cusp forms 36432 19762 16670
Eisenstein series 864 220 644

## Trace form

 $$19762 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 32 q^{7} - 80 q^{8} - 1498 q^{9} + O(q^{10})$$ $$19762 q - 32 q^{2} - 24 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 32 q^{7} - 80 q^{8} - 1498 q^{9} - 32 q^{10} + 2696 q^{11} - 32 q^{12} - 10112 q^{13} - 40 q^{14} - 56 q^{15} - 32 q^{16} + 19496 q^{17} - 32 q^{18} + 7848 q^{19} - 32 q^{20} - 16564 q^{21} + 127360 q^{22} - 26268 q^{23} - 322032 q^{24} - 51158 q^{25} + 21168 q^{26} + 71532 q^{27} + 170960 q^{28} + 132720 q^{29} + 412928 q^{30} - 4 q^{31} - 235632 q^{32} - 355580 q^{33} - 558032 q^{34} - 112448 q^{35} - 730480 q^{36} + 96448 q^{37} + 400368 q^{38} + 508772 q^{39} + 1151248 q^{40} + 350680 q^{41} + 173760 q^{42} - 536028 q^{43} - 962032 q^{44} - 640648 q^{45} - 32 q^{46} - 20 q^{47} - 32 q^{48} + 205850 q^{49} - 1410944 q^{50} - 39260 q^{51} + 2550208 q^{52} + 602176 q^{53} + 2519392 q^{54} - 1396240 q^{55} - 326576 q^{56} - 435656 q^{57} - 3540272 q^{58} + 3622856 q^{59} - 5023328 q^{60} + 393088 q^{61} - 1082368 q^{62} - 36 q^{63} + 2310256 q^{64} - 712596 q^{65} + 6204192 q^{66} - 6283992 q^{67} + 3694528 q^{68} + 1597960 q^{69} + 2271992 q^{70} - 1602112 q^{71} - 1574672 q^{72} - 271912 q^{73} - 5848336 q^{74} + 8034128 q^{75} - 9130400 q^{76} - 324852 q^{77} + 7564480 q^{78} - 3443732 q^{79} + 4180624 q^{80} - 3544402 q^{81} - 13933952 q^{82} - 576664 q^{83} - 5971544 q^{84} + 1602448 q^{85} + 1656688 q^{86} + 4059748 q^{87} + 9391168 q^{88} + 8814744 q^{89} + 22931968 q^{90} + 1714156 q^{91} + 20210752 q^{92} + 4953424 q^{93} + 5017504 q^{94} - 5378692 q^{95} - 7416112 q^{96} - 13102392 q^{97} - 10348960 q^{98} - 10469164 q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(448))$$

We only show spaces with odd 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
448.7.c $$\chi_{448}(321, \cdot)$$ 448.7.c.a 1 1
448.7.c.b 1
448.7.c.c 2
448.7.c.d 2
448.7.c.e 4
448.7.c.f 4
448.7.c.g 4
448.7.c.h 4
448.7.c.i 12
448.7.c.j 12
448.7.c.k 24
448.7.c.l 24
448.7.d $$\chi_{448}(127, \cdot)$$ 448.7.d.a 6 1
448.7.d.b 12
448.7.d.c 16
448.7.d.d 18
448.7.d.e 20
448.7.g $$\chi_{448}(351, \cdot)$$ 448.7.g.a 24 1
448.7.g.b 48
448.7.h $$\chi_{448}(97, \cdot)$$ 448.7.h.a 32 1
448.7.h.b 64
448.7.k $$\chi_{448}(15, \cdot)$$ n/a 144 2
448.7.l $$\chi_{448}(209, \cdot)$$ n/a 188 2
448.7.n $$\chi_{448}(33, \cdot)$$ n/a 192 2
448.7.o $$\chi_{448}(95, \cdot)$$ n/a 192 2
448.7.r $$\chi_{448}(191, \cdot)$$ n/a 188 2
448.7.s $$\chi_{448}(129, \cdot)$$ n/a 188 2
448.7.v $$\chi_{448}(41, \cdot)$$ None 0 4
448.7.w $$\chi_{448}(71, \cdot)$$ None 0 4
448.7.y $$\chi_{448}(79, \cdot)$$ n/a 376 4
448.7.bb $$\chi_{448}(17, \cdot)$$ n/a 376 4
448.7.be $$\chi_{448}(43, \cdot)$$ n/a 2304 8
448.7.bf $$\chi_{448}(13, \cdot)$$ n/a 3056 8
448.7.bg $$\chi_{448}(73, \cdot)$$ None 0 8
448.7.bj $$\chi_{448}(23, \cdot)$$ None 0 8
448.7.bk $$\chi_{448}(5, \cdot)$$ n/a 6112 16
448.7.bl $$\chi_{448}(11, \cdot)$$ n/a 6112 16

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

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(448))$$ into lower level spaces

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