# Properties

 Label 3969.1 Level 3969 Weight 1 Dimension 164 Nonzero newspaces 15 Newform subspaces 33 Sturm bound 1143072 Trace bound 13

## Defining parameters

 Level: $$N$$ = $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$15$$ Newform subspaces: $$33$$ Sturm bound: $$1143072$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(3969))$$.

Total New Old
Modular forms 6840 2880 3960
Cusp forms 360 164 196
Eisenstein series 6480 2716 3764

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 164 0 0 0

## Trace form

 $$164 q - q^{4} + O(q^{10})$$ $$164 q - q^{4} - 2 q^{13} - q^{16} + 4 q^{19} + 12 q^{22} - q^{25} - 2 q^{31} + 4 q^{37} + 4 q^{43} + 12 q^{46} + 6 q^{49} + 10 q^{52} + 10 q^{61} - 64 q^{64} + 10 q^{67} + 4 q^{73} + 10 q^{76} + 10 q^{79} - 2 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(3969))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
3969.1.b $$\chi_{3969}(3725, \cdot)$$ 3969.1.b.a 4 1
3969.1.d $$\chi_{3969}(244, \cdot)$$ None 0 1
3969.1.j $$\chi_{3969}(863, \cdot)$$ 3969.1.j.a 2 2
3969.1.j.b 4
3969.1.j.c 8
3969.1.k $$\chi_{3969}(460, \cdot)$$ 3969.1.k.a 2 2
3969.1.k.b 2
3969.1.k.c 2
3969.1.k.d 2
3969.1.k.e 4
3969.1.l $$\chi_{3969}(1567, \cdot)$$ 3969.1.l.a 2 2
3969.1.l.b 2
3969.1.m $$\chi_{3969}(325, \cdot)$$ 3969.1.m.a 2 2
3969.1.m.b 2
3969.1.m.c 4
3969.1.n $$\chi_{3969}(2321, \cdot)$$ 3969.1.n.a 2 2
3969.1.n.b 4
3969.1.n.c 8
3969.1.q $$\chi_{3969}(2186, \cdot)$$ 3969.1.q.a 8 2
3969.1.r $$\chi_{3969}(1079, \cdot)$$ 3969.1.r.a 2 2
3969.1.r.b 2
3969.1.r.c 4
3969.1.r.d 8
3969.1.t $$\chi_{3969}(2971, \cdot)$$ 3969.1.t.a 2 2
3969.1.t.b 2
3969.1.t.c 2
3969.1.t.d 2
3969.1.t.e 4
3969.1.y $$\chi_{3969}(811, \cdot)$$ None 0 6
3969.1.ba $$\chi_{3969}(323, \cdot)$$ None 0 6
3969.1.bb $$\chi_{3969}(901, \cdot)$$ None 0 6
3969.1.bc $$\chi_{3969}(685, \cdot)$$ None 0 6
3969.1.bd $$\chi_{3969}(19, \cdot)$$ None 0 6
3969.1.bf $$\chi_{3969}(197, \cdot)$$ None 0 6
3969.1.bg $$\chi_{3969}(557, \cdot)$$ None 0 6
3969.1.bj $$\chi_{3969}(116, \cdot)$$ None 0 6
3969.1.br $$\chi_{3969}(136, \cdot)$$ 3969.1.br.a 12 12
3969.1.bt $$\chi_{3969}(134, \cdot)$$ 3969.1.bt.a 12 12
3969.1.bu $$\chi_{3969}(242, \cdot)$$ None 0 12
3969.1.bx $$\chi_{3969}(53, \cdot)$$ 3969.1.bx.a 12 12
3969.1.by $$\chi_{3969}(82, \cdot)$$ None 0 12
3969.1.bz $$\chi_{3969}(55, \cdot)$$ 3969.1.bz.a 12 12
3969.1.ca $$\chi_{3969}(514, \cdot)$$ 3969.1.ca.a 12 12
3969.1.cb $$\chi_{3969}(296, \cdot)$$ 3969.1.cb.a 12 12
3969.1.cd $$\chi_{3969}(263, \cdot)$$ None 0 18
3969.1.ce $$\chi_{3969}(166, \cdot)$$ None 0 18
3969.1.ch $$\chi_{3969}(97, \cdot)$$ None 0 18
3969.1.ci $$\chi_{3969}(31, \cdot)$$ None 0 18
3969.1.cj $$\chi_{3969}(50, \cdot)$$ None 0 18
3969.1.ck $$\chi_{3969}(128, \cdot)$$ None 0 18
3969.1.cp $$\chi_{3969}(44, \cdot)$$ None 0 36
3969.1.cs $$\chi_{3969}(170, \cdot)$$ None 0 36
3969.1.ct $$\chi_{3969}(8, \cdot)$$ None 0 36
3969.1.cv $$\chi_{3969}(10, \cdot)$$ None 0 36
3969.1.cw $$\chi_{3969}(118, \cdot)$$ None 0 36
3969.1.cx $$\chi_{3969}(73, \cdot)$$ None 0 36
3969.1.dc $$\chi_{3969}(29, \cdot)$$ None 0 108
3969.1.dd $$\chi_{3969}(2, \cdot)$$ None 0 108
3969.1.de $$\chi_{3969}(13, \cdot)$$ None 0 108
3969.1.df $$\chi_{3969}(61, \cdot)$$ None 0 108
3969.1.di $$\chi_{3969}(40, \cdot)$$ None 0 108
3969.1.dj $$\chi_{3969}(11, \cdot)$$ None 0 108

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(3969))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(3969)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(189))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(441))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(567))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1323))$$$$^{\oplus 2}$$