## Defining parameters

 Level: $$N$$ = $$1694 = 2 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Sturm bound: $$348480$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1694))$$.

Total New Old
Modular forms 89040 27370 61670
Cusp forms 85201 27370 57831
Eisenstein series 3839 0 3839

## Trace form

 $$27370q - 2q^{2} - 6q^{3} - 6q^{5} + 18q^{6} + 20q^{7} - 2q^{8} + 68q^{9} + O(q^{10})$$ $$27370q - 2q^{2} - 6q^{3} - 6q^{5} + 18q^{6} + 20q^{7} - 2q^{8} + 68q^{9} + 34q^{10} + 20q^{11} + 34q^{12} + 22q^{13} + 18q^{14} + 96q^{15} + 68q^{17} + 6q^{18} + 42q^{19} - 6q^{20} + 34q^{21} + 56q^{23} + 18q^{24} + 124q^{25} + 70q^{26} + 144q^{27} + 20q^{28} + 124q^{29} + 96q^{30} + 84q^{31} + 18q^{32} + 110q^{33} + 56q^{34} + 114q^{35} + 8q^{36} + 44q^{37} + 98q^{38} + 152q^{39} + 34q^{40} + 124q^{41} + 58q^{42} + 164q^{43} + 30q^{44} + 122q^{45} + 56q^{46} + 100q^{47} - 6q^{48} + 120q^{49} + 54q^{50} + 176q^{51} + 62q^{52} + 152q^{53} + 116q^{54} + 140q^{55} - 2q^{56} + 256q^{57} + 136q^{58} + 194q^{59} + 56q^{60} + 106q^{61} - 28q^{62} - 132q^{63} - 44q^{65} - 120q^{66} + 48q^{67} - 52q^{68} - 256q^{69} - 286q^{70} - 232q^{71} - 174q^{72} - 192q^{73} - 200q^{74} - 614q^{75} - 98q^{76} - 100q^{77} - 384q^{78} - 232q^{79} - 86q^{80} - 552q^{81} - 148q^{82} - 190q^{83} - 166q^{84} - 148q^{85} - 152q^{86} - 308q^{87} - 40q^{88} - 16q^{89} - 198q^{90} - 178q^{91} + 96q^{92} + 40q^{93} + 84q^{94} + 200q^{95} - 2q^{96} + 272q^{97} + 18q^{98} + 260q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1694))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1694.2.a $$\chi_{1694}(1, \cdot)$$ 1694.2.a.a 1 1
1694.2.a.b 1
1694.2.a.c 1
1694.2.a.d 1
1694.2.a.e 1
1694.2.a.f 1
1694.2.a.g 1
1694.2.a.h 1
1694.2.a.i 1
1694.2.a.j 1
1694.2.a.k 2
1694.2.a.l 2
1694.2.a.m 2
1694.2.a.n 2
1694.2.a.o 2
1694.2.a.p 2
1694.2.a.q 2
1694.2.a.r 2
1694.2.a.s 2
1694.2.a.t 2
1694.2.a.u 2
1694.2.a.v 3
1694.2.a.w 3
1694.2.a.x 4
1694.2.a.y 4
1694.2.a.z 4
1694.2.a.ba 4
1694.2.c $$\chi_{1694}(1693, \cdot)$$ 1694.2.c.a 8 1
1694.2.c.b 32
1694.2.c.c 32
1694.2.e $$\chi_{1694}(485, \cdot)$$ n/a 144 2
1694.2.f $$\chi_{1694}(323, \cdot)$$ n/a 216 4
1694.2.i $$\chi_{1694}(241, \cdot)$$ n/a 144 2
1694.2.k $$\chi_{1694}(475, \cdot)$$ n/a 288 4
1694.2.m $$\chi_{1694}(155, \cdot)$$ n/a 660 10
1694.2.n $$\chi_{1694}(9, \cdot)$$ n/a 576 8
1694.2.o $$\chi_{1694}(153, \cdot)$$ n/a 880 10
1694.2.r $$\chi_{1694}(215, \cdot)$$ n/a 576 8
1694.2.u $$\chi_{1694}(23, \cdot)$$ n/a 1760 20
1694.2.v $$\chi_{1694}(15, \cdot)$$ n/a 2640 40
1694.2.x $$\chi_{1694}(87, \cdot)$$ n/a 1760 20
1694.2.bb $$\chi_{1694}(13, \cdot)$$ n/a 3520 40
1694.2.bc $$\chi_{1694}(25, \cdot)$$ n/a 7040 80
1694.2.be $$\chi_{1694}(17, \cdot)$$ n/a 7040 80

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1694))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1694)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(121))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(154))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(242))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(847))$$$$^{\oplus 2}$$