## Defining parameters

 Level: $$N$$ = $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$40$$ Sturm bound: $$338688$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1764))$$.

Total New Old
Modular forms 87072 37959 49113
Cusp forms 82273 37087 45186
Eisenstein series 4799 872 3927

## Trace form

 $$37087q - 48q^{2} - 50q^{4} - 93q^{5} - 63q^{6} + 4q^{7} - 87q^{8} - 132q^{9} + O(q^{10})$$ $$37087q - 48q^{2} - 50q^{4} - 93q^{5} - 63q^{6} + 4q^{7} - 87q^{8} - 132q^{9} - 157q^{10} - 9q^{11} - 54q^{12} - 125q^{13} - 66q^{14} - 15q^{15} - 62q^{16} - 144q^{17} - 42q^{18} - 58q^{19} - 9q^{20} - 162q^{21} - 54q^{22} - 63q^{23} - 57q^{24} - 171q^{25} + 21q^{26} - 36q^{27} - 138q^{28} - 231q^{29} - 18q^{30} + 17q^{31} + 42q^{32} - 57q^{33} + 80q^{34} + 57q^{35} - 21q^{36} - 238q^{37} + 126q^{38} + 69q^{39} + 167q^{40} + 93q^{41} - 6q^{42} + 79q^{43} + 177q^{44} - 9q^{45} + 87q^{46} + 93q^{47} + 81q^{48} - 144q^{49} + 99q^{50} + 144q^{51} + 125q^{52} + 42q^{53} + 15q^{54} - 51q^{55} + 18q^{56} - 114q^{57} - 7q^{58} + 39q^{59} - 42q^{60} - 204q^{61} - 63q^{62} + 84q^{63} - 263q^{64} + 183q^{65} - 120q^{66} - 99q^{67} - 186q^{68} + 33q^{69} - 159q^{70} + 120q^{71} - 213q^{72} - 146q^{73} - 393q^{74} + 180q^{75} - 342q^{76} + 27q^{77} - 288q^{78} + 51q^{79} - 396q^{80} + 132q^{81} - 286q^{82} + 189q^{83} - 180q^{84} + 136q^{85} - 231q^{86} + 27q^{87} - 165q^{88} + 324q^{89} - 282q^{90} + 100q^{91} - 246q^{92} - 15q^{93} - 108q^{94} + 126q^{95} - 216q^{96} + 247q^{97} + 120q^{98} - 39q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1764))$$

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
1764.2.a $$\chi_{1764}(1, \cdot)$$ 1764.2.a.a 1 1
1764.2.a.b 1
1764.2.a.c 1
1764.2.a.d 1
1764.2.a.e 1
1764.2.a.f 1
1764.2.a.g 1
1764.2.a.h 1
1764.2.a.i 1
1764.2.a.j 1
1764.2.a.k 1
1764.2.a.l 2
1764.2.a.m 4
1764.2.b $$\chi_{1764}(1567, \cdot)$$ 1764.2.b.a 4 1
1764.2.b.b 4
1764.2.b.c 4
1764.2.b.d 4
1764.2.b.e 4
1764.2.b.f 4
1764.2.b.g 4
1764.2.b.h 4
1764.2.b.i 8
1764.2.b.j 8
1764.2.b.k 8
1764.2.b.l 12
1764.2.b.m 12
1764.2.b.n 16
1764.2.e $$\chi_{1764}(1079, \cdot)$$ 1764.2.e.a 2 1
1764.2.e.b 2
1764.2.e.c 2
1764.2.e.d 4
1764.2.e.e 4
1764.2.e.f 8
1764.2.e.g 12
1764.2.e.h 16
1764.2.e.i 16
1764.2.e.j 16
1764.2.f $$\chi_{1764}(881, \cdot)$$ 1764.2.f.a 4 1
1764.2.f.b 8
1764.2.i $$\chi_{1764}(373, \cdot)$$ 1764.2.i.a 2 2
1764.2.i.b 2
1764.2.i.c 2
1764.2.i.d 6
1764.2.i.e 6
1764.2.i.f 6
1764.2.i.g 6
1764.2.i.h 12
1764.2.i.i 14
1764.2.i.j 24
1764.2.j $$\chi_{1764}(589, \cdot)$$ 1764.2.j.a 2 2
1764.2.j.b 2
1764.2.j.c 2
1764.2.j.d 6
1764.2.j.e 6
1764.2.j.f 12
1764.2.j.g 14
1764.2.j.h 14
1764.2.j.i 24
1764.2.k $$\chi_{1764}(361, \cdot)$$ 1764.2.k.a 2 2
1764.2.k.b 2
1764.2.k.c 2
1764.2.k.d 2
1764.2.k.e 2
1764.2.k.f 2
1764.2.k.g 2
1764.2.k.h 2
1764.2.k.i 2
1764.2.k.j 2
1764.2.k.k 2
1764.2.k.l 4
1764.2.k.m 8
1764.2.l $$\chi_{1764}(949, \cdot)$$ 1764.2.l.a 2 2
1764.2.l.b 2
1764.2.l.c 2
1764.2.l.d 6
1764.2.l.e 6
1764.2.l.f 6
1764.2.l.g 6
1764.2.l.h 12
1764.2.l.i 14
1764.2.l.j 24
1764.2.n $$\chi_{1764}(31, \cdot)$$ n/a 464 2
1764.2.o $$\chi_{1764}(851, \cdot)$$ n/a 464 2
1764.2.t $$\chi_{1764}(521, \cdot)$$ 1764.2.t.a 4 2
1764.2.t.b 8
1764.2.t.c 16
1764.2.w $$\chi_{1764}(509, \cdot)$$ 1764.2.w.a 16 2
1764.2.w.b 16
1764.2.w.c 48
1764.2.x $$\chi_{1764}(293, \cdot)$$ 1764.2.x.a 16 2
1764.2.x.b 16
1764.2.x.c 48
1764.2.ba $$\chi_{1764}(491, \cdot)$$ n/a 472 2
1764.2.bb $$\chi_{1764}(263, \cdot)$$ n/a 464 2
1764.2.be $$\chi_{1764}(863, \cdot)$$ n/a 160 2
1764.2.bf $$\chi_{1764}(19, \cdot)$$ n/a 192 2
1764.2.bi $$\chi_{1764}(391, \cdot)$$ n/a 464 2
1764.2.bj $$\chi_{1764}(607, \cdot)$$ n/a 464 2
1764.2.bm $$\chi_{1764}(1685, \cdot)$$ 1764.2.bm.a 16 2
1764.2.bm.b 16
1764.2.bm.c 48
1764.2.bo $$\chi_{1764}(253, \cdot)$$ n/a 144 6
1764.2.br $$\chi_{1764}(125, \cdot)$$ n/a 120 6
1764.2.bs $$\chi_{1764}(71, \cdot)$$ n/a 672 6
1764.2.bv $$\chi_{1764}(55, \cdot)$$ n/a 828 6
1764.2.bw $$\chi_{1764}(193, \cdot)$$ n/a 672 12
1764.2.bx $$\chi_{1764}(37, \cdot)$$ n/a 276 12
1764.2.by $$\chi_{1764}(85, \cdot)$$ n/a 672 12
1764.2.bz $$\chi_{1764}(25, \cdot)$$ n/a 672 12
1764.2.cb $$\chi_{1764}(173, \cdot)$$ n/a 672 12
1764.2.ce $$\chi_{1764}(103, \cdot)$$ n/a 3984 12
1764.2.cf $$\chi_{1764}(139, \cdot)$$ n/a 3984 12
1764.2.ci $$\chi_{1764}(199, \cdot)$$ n/a 1656 12
1764.2.cj $$\chi_{1764}(107, \cdot)$$ n/a 1344 12
1764.2.cm $$\chi_{1764}(11, \cdot)$$ n/a 3984 12
1764.2.cn $$\chi_{1764}(155, \cdot)$$ n/a 3984 12
1764.2.cq $$\chi_{1764}(41, \cdot)$$ n/a 672 12
1764.2.cr $$\chi_{1764}(5, \cdot)$$ n/a 672 12
1764.2.cu $$\chi_{1764}(17, \cdot)$$ n/a 216 12
1764.2.cz $$\chi_{1764}(95, \cdot)$$ n/a 3984 12
1764.2.da $$\chi_{1764}(187, \cdot)$$ n/a 3984 12

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1764))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1764)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(252))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(294))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(441))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(588))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(882))$$$$^{\oplus 2}$$