## Defining parameters

 Level: $$N$$ = $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$40$$ Sturm bound: $$207360$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1386))$$.

Total New Old
Modular forms 53760 12890 40870
Cusp forms 49921 12890 37031
Eisenstein series 3839 0 3839

## Trace form

 $$12890 q - 4 q^{2} - 12 q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} - 30 q^{7} + 8 q^{8} + 12 q^{9} + O(q^{10})$$ $$12890 q - 4 q^{2} - 12 q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} - 30 q^{7} + 8 q^{8} + 12 q^{9} - 32 q^{10} - 26 q^{11} - 32 q^{13} + 22 q^{14} + 48 q^{15} + 8 q^{16} + 32 q^{17} + 24 q^{18} + 30 q^{19} + 36 q^{20} + 84 q^{21} + 42 q^{22} + 160 q^{23} + 32 q^{24} + 160 q^{25} + 196 q^{26} + 192 q^{27} + 38 q^{28} + 184 q^{29} + 144 q^{30} + 124 q^{31} + 26 q^{32} + 206 q^{33} + 108 q^{34} + 200 q^{35} + 32 q^{36} + 104 q^{37} + 148 q^{38} + 192 q^{39} + 48 q^{40} + 220 q^{41} + 40 q^{42} + 56 q^{43} - 14 q^{44} + 136 q^{45} - 136 q^{46} + 16 q^{47} + 12 q^{48} - 4 q^{49} - 140 q^{50} - 16 q^{51} - 28 q^{52} + 12 q^{53} - 108 q^{54} + 20 q^{55} - 16 q^{56} - 20 q^{57} - 56 q^{58} - 90 q^{59} - 144 q^{60} + 92 q^{61} - 200 q^{62} - 280 q^{63} + 4 q^{64} - 36 q^{65} - 96 q^{66} + 312 q^{67} - 88 q^{68} - 56 q^{69} + 176 q^{70} + 160 q^{71} + 12 q^{72} + 268 q^{73} - 48 q^{74} - 88 q^{75} + 40 q^{76} + 194 q^{77} - 120 q^{78} + 232 q^{79} + 28 q^{80} - 92 q^{81} + 170 q^{82} - 90 q^{83} - 92 q^{84} + 60 q^{85} - 110 q^{86} - 184 q^{87} + 58 q^{88} - 72 q^{89} - 304 q^{90} + 452 q^{91} - 108 q^{92} - 184 q^{93} + 196 q^{94} - 192 q^{95} + 24 q^{96} + 78 q^{97} + 122 q^{98} - 484 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1386))$$

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
1386.2.a $$\chi_{1386}(1, \cdot)$$ 1386.2.a.a 1 1
1386.2.a.b 1
1386.2.a.c 1
1386.2.a.d 1
1386.2.a.e 1
1386.2.a.f 1
1386.2.a.g 1
1386.2.a.h 1
1386.2.a.i 1
1386.2.a.j 1
1386.2.a.k 1
1386.2.a.l 1
1386.2.a.m 2
1386.2.a.n 2
1386.2.a.o 2
1386.2.a.p 2
1386.2.a.q 3
1386.2.a.r 3
1386.2.c $$\chi_{1386}(197, \cdot)$$ 1386.2.c.a 12 1
1386.2.c.b 12
1386.2.e $$\chi_{1386}(307, \cdot)$$ 1386.2.e.a 8 1
1386.2.e.b 8
1386.2.e.c 8
1386.2.e.d 8
1386.2.e.e 8
1386.2.g $$\chi_{1386}(881, \cdot)$$ 1386.2.g.a 16 1
1386.2.g.b 16
1386.2.i $$\chi_{1386}(529, \cdot)$$ n/a 160 2
1386.2.j $$\chi_{1386}(463, \cdot)$$ n/a 120 2
1386.2.k $$\chi_{1386}(793, \cdot)$$ 1386.2.k.a 2 2
1386.2.k.b 2
1386.2.k.c 2
1386.2.k.d 2
1386.2.k.e 2
1386.2.k.f 2
1386.2.k.g 2
1386.2.k.h 2
1386.2.k.i 2
1386.2.k.j 2
1386.2.k.k 2
1386.2.k.l 2
1386.2.k.m 2
1386.2.k.n 2
1386.2.k.o 2
1386.2.k.p 2
1386.2.k.q 4
1386.2.k.r 4
1386.2.k.s 4
1386.2.k.t 4
1386.2.k.u 4
1386.2.k.v 6
1386.2.k.w 6
1386.2.l $$\chi_{1386}(67, \cdot)$$ n/a 160 2
1386.2.m $$\chi_{1386}(379, \cdot)$$ n/a 120 4
1386.2.n $$\chi_{1386}(439, \cdot)$$ n/a 192 2
1386.2.p $$\chi_{1386}(65, \cdot)$$ n/a 192 2
1386.2.r $$\chi_{1386}(89, \cdot)$$ 1386.2.r.a 8 2
1386.2.r.b 8
1386.2.r.c 8
1386.2.r.d 24
1386.2.w $$\chi_{1386}(353, \cdot)$$ n/a 160 2
1386.2.y $$\chi_{1386}(419, \cdot)$$ n/a 160 2
1386.2.ba $$\chi_{1386}(989, \cdot)$$ 1386.2.ba.a 32 2
1386.2.ba.b 32
1386.2.bd $$\chi_{1386}(241, \cdot)$$ n/a 192 2
1386.2.bf $$\chi_{1386}(769, \cdot)$$ n/a 192 2
1386.2.bh $$\chi_{1386}(263, \cdot)$$ n/a 192 2
1386.2.bj $$\chi_{1386}(659, \cdot)$$ n/a 144 2
1386.2.bk $$\chi_{1386}(703, \cdot)$$ 1386.2.bk.a 16 2
1386.2.bk.b 16
1386.2.bk.c 16
1386.2.bk.d 32
1386.2.bn $$\chi_{1386}(551, \cdot)$$ n/a 160 2
1386.2.bq $$\chi_{1386}(125, \cdot)$$ n/a 128 4
1386.2.bs $$\chi_{1386}(811, \cdot)$$ n/a 160 4
1386.2.bu $$\chi_{1386}(701, \cdot)$$ 1386.2.bu.a 48 4
1386.2.bu.b 48
1386.2.bw $$\chi_{1386}(445, \cdot)$$ n/a 768 8
1386.2.bx $$\chi_{1386}(37, \cdot)$$ n/a 320 8
1386.2.by $$\chi_{1386}(169, \cdot)$$ n/a 576 8
1386.2.bz $$\chi_{1386}(25, \cdot)$$ n/a 768 8
1386.2.cb $$\chi_{1386}(47, \cdot)$$ n/a 768 8
1386.2.ce $$\chi_{1386}(19, \cdot)$$ n/a 320 8
1386.2.cf $$\chi_{1386}(29, \cdot)$$ n/a 576 8
1386.2.ch $$\chi_{1386}(149, \cdot)$$ n/a 768 8
1386.2.cj $$\chi_{1386}(13, \cdot)$$ n/a 768 8
1386.2.cl $$\chi_{1386}(481, \cdot)$$ n/a 768 8
1386.2.co $$\chi_{1386}(107, \cdot)$$ n/a 256 8
1386.2.cq $$\chi_{1386}(335, \cdot)$$ n/a 768 8
1386.2.cs $$\chi_{1386}(5, \cdot)$$ n/a 768 8
1386.2.cx $$\chi_{1386}(269, \cdot)$$ n/a 256 8
1386.2.cz $$\chi_{1386}(95, \cdot)$$ n/a 768 8
1386.2.db $$\chi_{1386}(61, \cdot)$$ n/a 768 8

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1386))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1386)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(66))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(99))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(154))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(198))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(231))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(462))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(693))$$$$^{\oplus 2}$$