# Properties

 Label 1764.1 Level 1764 Weight 1 Dimension 69 Nonzero newspaces 7 Newform subspaces 14 Sturm bound 169344 Trace bound 19

## Defining parameters

 Level: $$N$$ = $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$7$$ Newform subspaces: $$14$$ Sturm bound: $$169344$$ Trace bound: $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1764))$$.

Total New Old
Modular forms 2614 506 2108
Cusp forms 214 69 145
Eisenstein series 2400 437 1963

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 69 0 0 0

## Trace form

 $$69 q - q^{7} + 3 q^{8} + O(q^{10})$$ $$69 q - q^{7} + 3 q^{8} + 4 q^{16} + 6 q^{19} + 4 q^{22} + 6 q^{25} + 6 q^{29} + 6 q^{31} + 9 q^{37} + 2 q^{43} - 4 q^{46} + q^{49} - 3 q^{50} - 4 q^{58} + 7 q^{61} - 15 q^{64} - 2 q^{67} - 6 q^{73} - 2 q^{79} - 24 q^{85} - 4 q^{88} - 3 q^{91} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1764))$$

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
1764.1.c $$\chi_{1764}(197, \cdot)$$ None 0 1
1764.1.d $$\chi_{1764}(685, \cdot)$$ 1764.1.d.a 2 1
1764.1.g $$\chi_{1764}(883, \cdot)$$ 1764.1.g.a 1 1
1764.1.g.b 2
1764.1.g.c 2
1764.1.g.d 2
1764.1.h $$\chi_{1764}(1763, \cdot)$$ 1764.1.h.a 8 1
1764.1.m $$\chi_{1764}(569, \cdot)$$ None 0 2
1764.1.p $$\chi_{1764}(313, \cdot)$$ None 0 2
1764.1.q $$\chi_{1764}(215, \cdot)$$ 1764.1.q.a 8 2
1764.1.q.b 16
1764.1.r $$\chi_{1764}(227, \cdot)$$ None 0 2
1764.1.s $$\chi_{1764}(587, \cdot)$$ None 0 2
1764.1.u $$\chi_{1764}(655, \cdot)$$ None 0 2
1764.1.v $$\chi_{1764}(295, \cdot)$$ None 0 2
1764.1.y $$\chi_{1764}(667, \cdot)$$ 1764.1.y.a 2 2
1764.1.y.b 4
1764.1.y.c 4
1764.1.y.d 4
1764.1.z $$\chi_{1764}(325, \cdot)$$ 1764.1.z.a 2 2
1764.1.bc $$\chi_{1764}(97, \cdot)$$ None 0 2
1764.1.bd $$\chi_{1764}(1489, \cdot)$$ None 0 2
1764.1.bg $$\chi_{1764}(785, \cdot)$$ None 0 2
1764.1.bh $$\chi_{1764}(1145, \cdot)$$ None 0 2
1764.1.bk $$\chi_{1764}(557, \cdot)$$ None 0 2
1764.1.bl $$\chi_{1764}(67, \cdot)$$ None 0 2
1764.1.bn $$\chi_{1764}(803, \cdot)$$ None 0 2
1764.1.bp $$\chi_{1764}(251, \cdot)$$ None 0 6
1764.1.bq $$\chi_{1764}(127, \cdot)$$ None 0 6
1764.1.bt $$\chi_{1764}(181, \cdot)$$ None 0 6
1764.1.bu $$\chi_{1764}(449, \cdot)$$ None 0 6
1764.1.ca $$\chi_{1764}(47, \cdot)$$ None 0 12
1764.1.cc $$\chi_{1764}(319, \cdot)$$ None 0 12
1764.1.cd $$\chi_{1764}(53, \cdot)$$ None 0 12
1764.1.cg $$\chi_{1764}(137, \cdot)$$ None 0 12
1764.1.ch $$\chi_{1764}(29, \cdot)$$ None 0 12
1764.1.ck $$\chi_{1764}(229, \cdot)$$ None 0 12
1764.1.cl $$\chi_{1764}(13, \cdot)$$ None 0 12
1764.1.co $$\chi_{1764}(73, \cdot)$$ 1764.1.co.a 12 12
1764.1.cp $$\chi_{1764}(163, \cdot)$$ None 0 12
1764.1.cs $$\chi_{1764}(43, \cdot)$$ None 0 12
1764.1.ct $$\chi_{1764}(151, \cdot)$$ None 0 12
1764.1.cv $$\chi_{1764}(83, \cdot)$$ None 0 12
1764.1.cw $$\chi_{1764}(131, \cdot)$$ None 0 12
1764.1.cx $$\chi_{1764}(143, \cdot)$$ None 0 12
1764.1.cy $$\chi_{1764}(61, \cdot)$$ None 0 12
1764.1.db $$\chi_{1764}(65, \cdot)$$ None 0 12

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1764))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1764)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(252))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(441))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(588))$$$$^{\oplus 2}$$