# Properties

 Label 1764.3 Level 1764 Weight 3 Dimension 74608 Nonzero newspaces 40 Sturm bound 508032 Trace bound 13

## Defining parameters

 Level: $$N$$ = $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$40$$ Sturm bound: $$508032$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(1764))$$.

Total New Old
Modular forms 171744 75482 96262
Cusp forms 166944 74608 92336
Eisenstein series 4800 874 3926

## Trace form

 $$74608 q - 44 q^{2} + 3 q^{3} - 42 q^{4} - 85 q^{5} - 69 q^{6} + 4 q^{7} - 89 q^{8} - 135 q^{9} + O(q^{10})$$ $$74608 q - 44 q^{2} + 3 q^{3} - 42 q^{4} - 85 q^{5} - 69 q^{6} + 4 q^{7} - 89 q^{8} - 135 q^{9} - 105 q^{10} + 18 q^{11} - 96 q^{12} - 35 q^{13} - 66 q^{14} + 75 q^{15} - 102 q^{16} - 28 q^{17} - 48 q^{18} + 4 q^{19} - 271 q^{20} - 162 q^{21} - 270 q^{22} - 15 q^{23} + 69 q^{24} - 245 q^{25} + 17 q^{26} - 252 q^{27} - 138 q^{28} - 679 q^{29} - 204 q^{30} - 533 q^{31} - 554 q^{32} - 582 q^{33} - 510 q^{34} - 363 q^{35} - 753 q^{36} - 724 q^{37} - 612 q^{38} - 87 q^{39} - 231 q^{40} + 92 q^{41} - 6 q^{42} + 276 q^{43} - 57 q^{44} + 267 q^{45} + 303 q^{46} + 675 q^{47} + 447 q^{48} + 150 q^{49} + 987 q^{50} - 99 q^{51} + 1011 q^{52} + 74 q^{53} + 825 q^{54} + 363 q^{55} + 564 q^{56} - 441 q^{57} + 897 q^{58} - 234 q^{59} + 816 q^{60} - 162 q^{61} + 1269 q^{62} + 84 q^{63} + 753 q^{64} - 221 q^{65} + 990 q^{66} - 418 q^{67} + 2132 q^{68} - 243 q^{69} + 1017 q^{70} - 384 q^{71} + 1203 q^{72} - 2798 q^{73} + 1541 q^{74} + 687 q^{75} + 918 q^{76} - 603 q^{77} + 42 q^{78} - 1147 q^{79} - 1258 q^{80} - 975 q^{81} - 2130 q^{82} - 1257 q^{83} - 180 q^{84} - 840 q^{85} - 2787 q^{86} + 837 q^{87} - 2919 q^{88} + 2732 q^{89} - 1476 q^{90} + 520 q^{91} - 2514 q^{92} + 1599 q^{93} - 3150 q^{94} + 2442 q^{95} - 1344 q^{96} + 2614 q^{97} - 1560 q^{98} + 1419 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(1764))$$

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
1764.3.c $$\chi_{1764}(197, \cdot)$$ 1764.3.c.a 2 1
1764.3.c.b 2
1764.3.c.c 2
1764.3.c.d 2
1764.3.c.e 4
1764.3.c.f 4
1764.3.c.g 4
1764.3.c.h 8
1764.3.d $$\chi_{1764}(685, \cdot)$$ 1764.3.d.a 2 1
1764.3.d.b 2
1764.3.d.c 2
1764.3.d.d 4
1764.3.d.e 4
1764.3.d.f 4
1764.3.d.g 8
1764.3.d.h 8
1764.3.g $$\chi_{1764}(883, \cdot)$$ n/a 200 1
1764.3.h $$\chi_{1764}(1763, \cdot)$$ n/a 160 1
1764.3.m $$\chi_{1764}(569, \cdot)$$ n/a 160 2
1764.3.p $$\chi_{1764}(313, \cdot)$$ n/a 160 2
1764.3.q $$\chi_{1764}(215, \cdot)$$ n/a 320 2
1764.3.r $$\chi_{1764}(227, \cdot)$$ n/a 944 2
1764.3.s $$\chi_{1764}(587, \cdot)$$ n/a 944 2
1764.3.u $$\chi_{1764}(655, \cdot)$$ n/a 944 2
1764.3.v $$\chi_{1764}(295, \cdot)$$ n/a 964 2
1764.3.y $$\chi_{1764}(667, \cdot)$$ n/a 392 2
1764.3.z $$\chi_{1764}(325, \cdot)$$ 1764.3.z.a 2 2
1764.3.z.b 2
1764.3.z.c 2
1764.3.z.d 2
1764.3.z.e 2
1764.3.z.f 2
1764.3.z.g 2
1764.3.z.h 4
1764.3.z.i 4
1764.3.z.j 4
1764.3.z.k 8
1764.3.z.l 8
1764.3.z.m 8
1764.3.z.n 16
1764.3.bc $$\chi_{1764}(97, \cdot)$$ n/a 160 2
1764.3.bd $$\chi_{1764}(1489, \cdot)$$ n/a 160 2
1764.3.bg $$\chi_{1764}(785, \cdot)$$ n/a 164 2
1764.3.bh $$\chi_{1764}(1145, \cdot)$$ n/a 160 2
1764.3.bk $$\chi_{1764}(557, \cdot)$$ 1764.3.bk.a 4 2
1764.3.bk.b 4
1764.3.bk.c 4
1764.3.bk.d 8
1764.3.bk.e 8
1764.3.bk.f 8
1764.3.bk.g 16
1764.3.bl $$\chi_{1764}(67, \cdot)$$ n/a 944 2
1764.3.bn $$\chi_{1764}(803, \cdot)$$ n/a 944 2
1764.3.bp $$\chi_{1764}(251, \cdot)$$ n/a 1344 6
1764.3.bq $$\chi_{1764}(127, \cdot)$$ n/a 1668 6
1764.3.bt $$\chi_{1764}(181, \cdot)$$ n/a 276 6
1764.3.bu $$\chi_{1764}(449, \cdot)$$ n/a 216 6
1764.3.ca $$\chi_{1764}(47, \cdot)$$ n/a 8016 12
1764.3.cc $$\chi_{1764}(319, \cdot)$$ n/a 8016 12
1764.3.cd $$\chi_{1764}(53, \cdot)$$ n/a 456 12
1764.3.cg $$\chi_{1764}(137, \cdot)$$ n/a 1344 12
1764.3.ch $$\chi_{1764}(29, \cdot)$$ n/a 1344 12
1764.3.ck $$\chi_{1764}(229, \cdot)$$ n/a 1344 12
1764.3.cl $$\chi_{1764}(13, \cdot)$$ n/a 1344 12
1764.3.co $$\chi_{1764}(73, \cdot)$$ n/a 564 12
1764.3.cp $$\chi_{1764}(163, \cdot)$$ n/a 3336 12
1764.3.cs $$\chi_{1764}(43, \cdot)$$ n/a 8016 12
1764.3.ct $$\chi_{1764}(151, \cdot)$$ n/a 8016 12
1764.3.cv $$\chi_{1764}(83, \cdot)$$ n/a 8016 12
1764.3.cw $$\chi_{1764}(131, \cdot)$$ n/a 8016 12
1764.3.cx $$\chi_{1764}(143, \cdot)$$ n/a 2688 12
1764.3.cy $$\chi_{1764}(61, \cdot)$$ n/a 1344 12
1764.3.db $$\chi_{1764}(65, \cdot)$$ n/a 1344 12

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(1764))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(1764)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 18}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(252))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(294))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(441))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(588))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(882))$$$$^{\oplus 2}$$