## Defining parameters

 Level: $$N$$ = $$1694 = 2 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$696960$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1694))$$.

Total New Old
Modular forms 263280 82116 181164
Cusp forms 259440 82116 177324
Eisenstein series 3840 0 3840

## Trace form

 $$82116 q + 12 q^{3} - 24 q^{5} + 164 q^{6} - 8 q^{7} - 598 q^{9} + O(q^{10})$$ $$82116 q + 12 q^{3} - 24 q^{5} + 164 q^{6} - 8 q^{7} - 598 q^{9} - 364 q^{10} - 200 q^{11} - 272 q^{12} - 94 q^{13} + 172 q^{14} + 1628 q^{15} + 1090 q^{17} + 752 q^{18} - 960 q^{19} - 120 q^{20} - 912 q^{21} + 1266 q^{23} + 752 q^{24} + 1410 q^{25} + 1148 q^{26} + 876 q^{27} - 120 q^{28} - 424 q^{29} - 2952 q^{30} + 570 q^{31} - 640 q^{32} - 2530 q^{33} - 2808 q^{34} - 2440 q^{35} - 568 q^{36} - 918 q^{37} - 132 q^{38} - 892 q^{39} + 1104 q^{40} + 2528 q^{41} + 4684 q^{42} + 10244 q^{43} + 2120 q^{44} + 7562 q^{45} + 1272 q^{46} - 1018 q^{47} - 96 q^{48} - 2750 q^{49} - 2696 q^{50} - 1646 q^{51} - 3528 q^{52} + 2342 q^{53} - 8784 q^{54} - 740 q^{55} + 48 q^{56} - 3468 q^{57} - 7224 q^{58} - 7564 q^{59} - 2424 q^{60} - 17800 q^{61} - 10440 q^{62} - 34768 q^{63} + 384 q^{64} - 19136 q^{65} - 2320 q^{66} - 2138 q^{67} + 4840 q^{68} + 10668 q^{69} + 13164 q^{70} + 8328 q^{71} + 8288 q^{72} + 18966 q^{73} + 12712 q^{74} + 51094 q^{75} + 7544 q^{76} + 14310 q^{77} + 24832 q^{78} + 27210 q^{79} + 4736 q^{80} + 63864 q^{81} + 16816 q^{82} + 30014 q^{83} + 7568 q^{84} + 11840 q^{85} + 5600 q^{86} - 4496 q^{87} - 2240 q^{88} - 15490 q^{89} - 36732 q^{90} - 39674 q^{91} - 7152 q^{92} - 45346 q^{93} - 34816 q^{94} - 42778 q^{95} - 192 q^{96} - 17604 q^{97} - 4600 q^{98} - 15580 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1694))$$

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
1694.4.a $$\chi_{1694}(1, \cdot)$$ 1694.4.a.a 1 1
1694.4.a.b 1
1694.4.a.c 1
1694.4.a.d 1
1694.4.a.e 1
1694.4.a.f 1
1694.4.a.g 1
1694.4.a.h 2
1694.4.a.i 2
1694.4.a.j 2
1694.4.a.k 2
1694.4.a.l 2
1694.4.a.m 2
1694.4.a.n 2
1694.4.a.o 2
1694.4.a.p 2
1694.4.a.q 3
1694.4.a.r 3
1694.4.a.s 3
1694.4.a.t 3
1694.4.a.u 3
1694.4.a.v 4
1694.4.a.w 4
1694.4.a.x 4
1694.4.a.y 4
1694.4.a.z 5
1694.4.a.ba 5
1694.4.a.bb 5
1694.4.a.bc 5
1694.4.a.bd 6
1694.4.a.be 6
1694.4.a.bf 8
1694.4.a.bg 8
1694.4.a.bh 10
1694.4.a.bi 10
1694.4.a.bj 10
1694.4.a.bk 10
1694.4.a.bl 10
1694.4.a.bm 10
1694.4.c $$\chi_{1694}(1693, \cdot)$$ n/a 216 1
1694.4.e $$\chi_{1694}(485, \cdot)$$ n/a 436 2
1694.4.f $$\chi_{1694}(323, \cdot)$$ n/a 648 4
1694.4.i $$\chi_{1694}(241, \cdot)$$ n/a 432 2
1694.4.k $$\chi_{1694}(475, \cdot)$$ n/a 864 4
1694.4.m $$\chi_{1694}(155, \cdot)$$ n/a 1980 10
1694.4.n $$\chi_{1694}(9, \cdot)$$ n/a 1728 8
1694.4.o $$\chi_{1694}(153, \cdot)$$ n/a 2640 10
1694.4.r $$\chi_{1694}(215, \cdot)$$ n/a 1728 8
1694.4.u $$\chi_{1694}(23, \cdot)$$ n/a 5280 20
1694.4.v $$\chi_{1694}(15, \cdot)$$ n/a 7920 40
1694.4.x $$\chi_{1694}(87, \cdot)$$ n/a 5280 20
1694.4.bb $$\chi_{1694}(13, \cdot)$$ n/a 10560 40
1694.4.bc $$\chi_{1694}(25, \cdot)$$ n/a 21120 80
1694.4.be $$\chi_{1694}(17, \cdot)$$ n/a 21120 80

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1694))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1694)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(121))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(154))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(242))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(847))$$$$^{\oplus 2}$$