# Properties

 Label 416.2 Level 416 Weight 2 Dimension 2878 Nonzero newspaces 20 Newform subspaces 55 Sturm bound 21504 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$416 = 2^{5} \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$20$$ Newform subspaces: $$55$$ Sturm bound: $$21504$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(416))$$.

Total New Old
Modular forms 5760 3098 2662
Cusp forms 4993 2878 2115
Eisenstein series 767 220 547

## Trace form

 $$2878 q - 40 q^{2} - 28 q^{3} - 40 q^{4} - 36 q^{5} - 40 q^{6} - 28 q^{7} - 40 q^{8} - 58 q^{9} + O(q^{10})$$ $$2878 q - 40 q^{2} - 28 q^{3} - 40 q^{4} - 36 q^{5} - 40 q^{6} - 28 q^{7} - 40 q^{8} - 58 q^{9} - 56 q^{10} - 28 q^{11} - 72 q^{12} - 50 q^{13} - 120 q^{14} - 36 q^{15} - 80 q^{16} - 28 q^{17} - 80 q^{18} - 28 q^{19} - 72 q^{20} - 40 q^{21} - 64 q^{22} - 44 q^{23} - 16 q^{24} - 62 q^{25} - 24 q^{26} - 112 q^{27} - 20 q^{29} + 24 q^{30} - 68 q^{31} - 104 q^{33} - 16 q^{34} - 76 q^{35} + 16 q^{36} - 36 q^{37} - 56 q^{38} - 56 q^{39} - 112 q^{40} - 84 q^{41} - 80 q^{42} - 44 q^{43} - 120 q^{44} - 76 q^{45} - 104 q^{46} - 36 q^{47} - 144 q^{48} - 10 q^{49} - 120 q^{50} - 32 q^{51} - 52 q^{52} - 148 q^{53} - 64 q^{54} + 36 q^{55} - 64 q^{56} - 64 q^{57} - 32 q^{58} + 36 q^{59} - 32 q^{60} - 84 q^{61} + 60 q^{63} + 32 q^{64} - 100 q^{65} - 40 q^{66} + 52 q^{67} - 80 q^{68} - 104 q^{69} - 16 q^{70} + 36 q^{71} - 88 q^{72} - 52 q^{73} - 72 q^{74} - 40 q^{75} - 40 q^{76} - 72 q^{77} - 84 q^{78} - 72 q^{79} - 32 q^{80} - 42 q^{81} - 40 q^{82} - 108 q^{83} - 80 q^{84} - 96 q^{85} - 212 q^{87} - 48 q^{88} - 180 q^{89} - 64 q^{90} - 176 q^{91} - 16 q^{92} - 208 q^{93} - 64 q^{94} - 268 q^{95} - 48 q^{96} - 236 q^{97} - 324 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(416))$$

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
416.2.a $$\chi_{416}(1, \cdot)$$ 416.2.a.a 1 1
416.2.a.b 1
416.2.a.c 2
416.2.a.d 2
416.2.a.e 2
416.2.a.f 4
416.2.b $$\chi_{416}(209, \cdot)$$ 416.2.b.a 2 1
416.2.b.b 4
416.2.b.c 6
416.2.e $$\chi_{416}(337, \cdot)$$ 416.2.e.a 2 1
416.2.e.b 2
416.2.e.c 8
416.2.f $$\chi_{416}(129, \cdot)$$ 416.2.f.a 2 1
416.2.f.b 2
416.2.f.c 2
416.2.f.d 4
416.2.f.e 4
416.2.i $$\chi_{416}(289, \cdot)$$ 416.2.i.a 2 2
416.2.i.b 2
416.2.i.c 4
416.2.i.d 4
416.2.i.e 4
416.2.i.f 4
416.2.i.g 8
416.2.k $$\chi_{416}(31, \cdot)$$ 416.2.k.a 2 2
416.2.k.b 2
416.2.k.c 4
416.2.k.d 4
416.2.k.e 8
416.2.k.f 8
416.2.l $$\chi_{416}(343, \cdot)$$ None 0 2
416.2.n $$\chi_{416}(105, \cdot)$$ None 0 2
416.2.p $$\chi_{416}(25, \cdot)$$ None 0 2
416.2.s $$\chi_{416}(135, \cdot)$$ None 0 2
416.2.u $$\chi_{416}(47, \cdot)$$ 416.2.u.a 4 2
416.2.u.b 20
416.2.w $$\chi_{416}(225, \cdot)$$ 416.2.w.a 4 2
416.2.w.b 4
416.2.w.c 8
416.2.w.d 12
416.2.z $$\chi_{416}(81, \cdot)$$ 416.2.z.a 24 2
416.2.ba $$\chi_{416}(17, \cdot)$$ 416.2.ba.a 4 2
416.2.ba.b 4
416.2.ba.c 16
416.2.bd $$\chi_{416}(83, \cdot)$$ 416.2.bd.a 216 4
416.2.bf $$\chi_{416}(53, \cdot)$$ 416.2.bf.a 192 4
416.2.bg $$\chi_{416}(77, \cdot)$$ 416.2.bg.a 216 4
416.2.bi $$\chi_{416}(99, \cdot)$$ 416.2.bi.a 216 4
416.2.bk $$\chi_{416}(15, \cdot)$$ 416.2.bk.a 48 4
416.2.bn $$\chi_{416}(7, \cdot)$$ None 0 4
416.2.bp $$\chi_{416}(121, \cdot)$$ None 0 4
416.2.br $$\chi_{416}(9, \cdot)$$ None 0 4
416.2.bs $$\chi_{416}(71, \cdot)$$ None 0 4
416.2.bu $$\chi_{416}(63, \cdot)$$ 416.2.bu.a 4 4
416.2.bu.b 4
416.2.bu.c 8
416.2.bu.d 8
416.2.bu.e 16
416.2.bu.f 16
416.2.bx $$\chi_{416}(115, \cdot)$$ 416.2.bx.a 432 8
416.2.bz $$\chi_{416}(69, \cdot)$$ 416.2.bz.a 432 8
416.2.ca $$\chi_{416}(29, \cdot)$$ 416.2.ca.a 432 8
416.2.cc $$\chi_{416}(11, \cdot)$$ 416.2.cc.a 432 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(416))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(416)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(208))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(416))$$$$^{\oplus 1}$$