## Defining parameters

 Level: $$N$$ = $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$40$$ Sturm bound: $$1362816$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3380))$$.

Total New Old
Modular forms 345264 181261 164003
Cusp forms 336145 178805 157340
Eisenstein series 9119 2456 6663

## Trace form

 $$178805 q - 134 q^{2} - 2 q^{3} - 132 q^{4} - 401 q^{5} - 396 q^{6} - 6 q^{7} - 128 q^{8} - 295 q^{9} + O(q^{10})$$ $$178805 q - 134 q^{2} - 2 q^{3} - 132 q^{4} - 401 q^{5} - 396 q^{6} - 6 q^{7} - 128 q^{8} - 295 q^{9} - 192 q^{10} - 24 q^{11} - 108 q^{12} - 312 q^{13} - 252 q^{14} - 22 q^{15} - 404 q^{16} - 276 q^{17} - 66 q^{18} + 12 q^{19} - 154 q^{20} - 708 q^{21} - 12 q^{22} + 54 q^{23} + 60 q^{24} - 365 q^{25} - 372 q^{26} + 148 q^{27} - 12 q^{28} - 246 q^{29} - 102 q^{30} - 12 q^{31} - 4 q^{32} - 240 q^{33} - 36 q^{34} - 38 q^{35} - 384 q^{36} - 320 q^{37} - 108 q^{38} - 40 q^{39} - 430 q^{40} - 766 q^{41} - 204 q^{42} - 2 q^{43} - 252 q^{44} - 361 q^{45} - 636 q^{46} - 30 q^{47} - 324 q^{48} - 19 q^{49} - 296 q^{50} + 108 q^{51} - 252 q^{52} - 300 q^{53} - 300 q^{54} + 156 q^{55} - 612 q^{56} + 168 q^{57} - 316 q^{58} + 132 q^{59} - 258 q^{60} - 538 q^{61} - 180 q^{62} + 330 q^{63} - 156 q^{64} - 291 q^{65} - 612 q^{66} + 210 q^{67} + 168 q^{68} + 36 q^{69} - 114 q^{70} + 228 q^{71} + 168 q^{72} - 28 q^{73} + 204 q^{74} + 290 q^{75} - 156 q^{76} - 24 q^{77} + 84 q^{78} + 56 q^{79} + 46 q^{80} - 541 q^{81} + 124 q^{82} + 174 q^{83} + 36 q^{84} - 390 q^{85} - 180 q^{86} + 36 q^{87} + 36 q^{88} - 270 q^{89} - 180 q^{90} - 4 q^{91} - 156 q^{92} - 312 q^{93} - 180 q^{94} + 4 q^{95} - 756 q^{96} - 460 q^{97} - 166 q^{98} - 216 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3380))$$

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
3380.2.a $$\chi_{3380}(1, \cdot)$$ 3380.2.a.a 1 1
3380.2.a.b 1
3380.2.a.c 1
3380.2.a.d 1
3380.2.a.e 1
3380.2.a.f 1
3380.2.a.g 1
3380.2.a.h 1
3380.2.a.i 1
3380.2.a.j 1
3380.2.a.k 3
3380.2.a.l 3
3380.2.a.m 3
3380.2.a.n 3
3380.2.a.o 3
3380.2.a.p 4
3380.2.a.q 4
3380.2.a.r 9
3380.2.a.s 9
3380.2.c $$\chi_{3380}(2029, \cdot)$$ 3380.2.c.a 6 1
3380.2.c.b 6
3380.2.c.c 6
3380.2.c.d 8
3380.2.c.e 16
3380.2.c.f 18
3380.2.c.g 18
3380.2.d $$\chi_{3380}(1689, \cdot)$$ 3380.2.d.a 6 1
3380.2.d.b 6
3380.2.d.c 12
3380.2.d.d 16
3380.2.d.e 36
3380.2.f $$\chi_{3380}(3041, \cdot)$$ 3380.2.f.a 2 1
3380.2.f.b 2
3380.2.f.c 2
3380.2.f.d 2
3380.2.f.e 2
3380.2.f.f 2
3380.2.f.g 6
3380.2.f.h 6
3380.2.f.i 8
3380.2.f.j 18
3380.2.i $$\chi_{3380}(1881, \cdot)$$ n/a 104 2
3380.2.j $$\chi_{3380}(1451, \cdot)$$ n/a 616 2
3380.2.m $$\chi_{3380}(577, \cdot)$$ n/a 154 2
3380.2.o $$\chi_{3380}(2367, \cdot)$$ n/a 886 2
3380.2.p $$\chi_{3380}(2027, \cdot)$$ n/a 884 2
3380.2.r $$\chi_{3380}(437, \cdot)$$ n/a 154 2
3380.2.u $$\chi_{3380}(99, \cdot)$$ n/a 884 2
3380.2.x $$\chi_{3380}(361, \cdot)$$ n/a 104 2
3380.2.z $$\chi_{3380}(2389, \cdot)$$ n/a 152 2
3380.2.ba $$\chi_{3380}(529, \cdot)$$ n/a 156 2
3380.2.bc $$\chi_{3380}(19, \cdot)$$ n/a 1768 4
3380.2.bf $$\chi_{3380}(657, \cdot)$$ n/a 308 4
3380.2.bg $$\chi_{3380}(23, \cdot)$$ n/a 1768 4
3380.2.bj $$\chi_{3380}(867, \cdot)$$ n/a 1768 4
3380.2.bk $$\chi_{3380}(357, \cdot)$$ n/a 308 4
3380.2.bn $$\chi_{3380}(1371, \cdot)$$ n/a 1232 4
3380.2.bo $$\chi_{3380}(261, \cdot)$$ n/a 744 12
3380.2.br $$\chi_{3380}(181, \cdot)$$ n/a 744 12
3380.2.bt $$\chi_{3380}(129, \cdot)$$ n/a 1104 12
3380.2.bu $$\chi_{3380}(209, \cdot)$$ n/a 1080 12
3380.2.bw $$\chi_{3380}(61, \cdot)$$ n/a 1440 24
3380.2.bx $$\chi_{3380}(359, \cdot)$$ n/a 13008 24
3380.2.ca $$\chi_{3380}(177, \cdot)$$ n/a 2184 24
3380.2.cc $$\chi_{3380}(103, \cdot)$$ n/a 13008 24
3380.2.cd $$\chi_{3380}(27, \cdot)$$ n/a 13008 24
3380.2.cf $$\chi_{3380}(57, \cdot)$$ n/a 2184 24
3380.2.ci $$\chi_{3380}(31, \cdot)$$ n/a 8736 24
3380.2.ck $$\chi_{3380}(9, \cdot)$$ n/a 2160 24
3380.2.cl $$\chi_{3380}(49, \cdot)$$ n/a 2208 24
3380.2.cn $$\chi_{3380}(101, \cdot)$$ n/a 1440 24
3380.2.cq $$\chi_{3380}(11, \cdot)$$ n/a 17472 48
3380.2.ct $$\chi_{3380}(33, \cdot)$$ n/a 4368 48
3380.2.cu $$\chi_{3380}(3, \cdot)$$ n/a 26016 48
3380.2.cx $$\chi_{3380}(43, \cdot)$$ n/a 26016 48
3380.2.cy $$\chi_{3380}(37, \cdot)$$ n/a 4368 48
3380.2.db $$\chi_{3380}(59, \cdot)$$ n/a 26016 48

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3380))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3380)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(130))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(260))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(338))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(676))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(845))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1690))$$$$^{\oplus 2}$$