## Defining parameters

 Level: $$N$$ = $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Sturm bound: $$186624$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1296))$$.

Total New Old
Modular forms 48168 20988 27180
Cusp forms 45145 20484 24661
Eisenstein series 3023 504 2519

## Trace form

 $$20484 q - 48 q^{2} - 54 q^{3} - 80 q^{4} - 60 q^{5} - 72 q^{6} - 60 q^{7} - 48 q^{8} - 18 q^{9} + O(q^{10})$$ $$20484 q - 48 q^{2} - 54 q^{3} - 80 q^{4} - 60 q^{5} - 72 q^{6} - 60 q^{7} - 48 q^{8} - 18 q^{9} - 116 q^{10} - 36 q^{11} - 72 q^{12} - 100 q^{13} - 48 q^{14} - 54 q^{15} - 80 q^{16} - 111 q^{17} - 72 q^{18} - 87 q^{19} - 48 q^{20} - 90 q^{21} - 80 q^{22} - 36 q^{23} - 72 q^{24} - 25 q^{25} - 48 q^{26} - 54 q^{27} - 116 q^{28} - 84 q^{29} - 72 q^{30} - 78 q^{31} - 48 q^{32} - 162 q^{33} - 80 q^{34} - 69 q^{35} - 72 q^{36} - 169 q^{37} - 48 q^{38} - 54 q^{39} - 80 q^{40} - 36 q^{41} - 72 q^{42} - 78 q^{43} - 48 q^{44} - 90 q^{45} - 100 q^{46} - 54 q^{47} - 72 q^{48} - 187 q^{49} - 48 q^{50} - 54 q^{51} - 80 q^{52} - 45 q^{53} - 72 q^{54} - 77 q^{55} + 36 q^{56} - 18 q^{57} - 32 q^{58} - 18 q^{59} - 72 q^{60} - 52 q^{61} + 84 q^{62} - 54 q^{63} - 44 q^{64} - 12 q^{65} - 72 q^{66} - 36 q^{67} + 108 q^{68} - 90 q^{69} + 64 q^{70} + 21 q^{71} - 72 q^{72} + 19 q^{73} + 120 q^{74} - 54 q^{75} + 64 q^{76} + 36 q^{77} - 72 q^{78} - 36 q^{79} + 132 q^{80} - 162 q^{81} - 152 q^{82} - 6 q^{83} - 72 q^{84} - 17 q^{85} + 72 q^{86} - 54 q^{87} - 32 q^{88} + 69 q^{89} - 72 q^{90} - q^{91} - 48 q^{92} + 18 q^{93} - 96 q^{94} + 249 q^{95} - 72 q^{96} - 96 q^{97} - 48 q^{98} + 108 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1296))$$

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
1296.2.a $$\chi_{1296}(1, \cdot)$$ 1296.2.a.a 1 1
1296.2.a.b 1
1296.2.a.c 1
1296.2.a.d 1
1296.2.a.e 1
1296.2.a.f 1
1296.2.a.g 1
1296.2.a.h 1
1296.2.a.i 1
1296.2.a.j 1
1296.2.a.k 1
1296.2.a.l 1
1296.2.a.m 2
1296.2.a.n 2
1296.2.a.o 2
1296.2.a.p 2
1296.2.a.q 2
1296.2.c $$\chi_{1296}(1295, \cdot)$$ 1296.2.c.a 2 1
1296.2.c.b 2
1296.2.c.c 2
1296.2.c.d 2
1296.2.c.e 4
1296.2.c.f 4
1296.2.c.g 8
1296.2.d $$\chi_{1296}(649, \cdot)$$ None 0 1
1296.2.f $$\chi_{1296}(647, \cdot)$$ None 0 1
1296.2.i $$\chi_{1296}(433, \cdot)$$ 1296.2.i.a 2 2
1296.2.i.b 2
1296.2.i.c 2
1296.2.i.d 2
1296.2.i.e 2
1296.2.i.f 2
1296.2.i.g 2
1296.2.i.h 2
1296.2.i.i 2
1296.2.i.j 2
1296.2.i.k 2
1296.2.i.l 2
1296.2.i.m 2
1296.2.i.n 2
1296.2.i.o 2
1296.2.i.p 2
1296.2.i.q 2
1296.2.i.r 4
1296.2.i.s 4
1296.2.i.t 4
1296.2.k $$\chi_{1296}(325, \cdot)$$ n/a 184 2
1296.2.l $$\chi_{1296}(323, \cdot)$$ n/a 184 2
1296.2.p $$\chi_{1296}(215, \cdot)$$ None 0 2
1296.2.r $$\chi_{1296}(217, \cdot)$$ None 0 2
1296.2.s $$\chi_{1296}(431, \cdot)$$ 1296.2.s.a 2 2
1296.2.s.b 2
1296.2.s.c 2
1296.2.s.d 2
1296.2.s.e 2
1296.2.s.f 2
1296.2.s.g 4
1296.2.s.h 4
1296.2.s.i 4
1296.2.s.j 8
1296.2.s.k 8
1296.2.s.l 8
1296.2.u $$\chi_{1296}(145, \cdot)$$ n/a 102 6
1296.2.v $$\chi_{1296}(107, \cdot)$$ n/a 376 4
1296.2.y $$\chi_{1296}(109, \cdot)$$ n/a 376 4
1296.2.bb $$\chi_{1296}(73, \cdot)$$ None 0 6
1296.2.bd $$\chi_{1296}(71, \cdot)$$ None 0 6
1296.2.be $$\chi_{1296}(143, \cdot)$$ n/a 108 6
1296.2.bg $$\chi_{1296}(49, \cdot)$$ n/a 954 18
1296.2.bh $$\chi_{1296}(37, \cdot)$$ n/a 840 12
1296.2.bk $$\chi_{1296}(35, \cdot)$$ n/a 840 12
1296.2.bn $$\chi_{1296}(23, \cdot)$$ None 0 18
1296.2.bp $$\chi_{1296}(25, \cdot)$$ None 0 18
1296.2.bq $$\chi_{1296}(47, \cdot)$$ n/a 972 18
1296.2.bs $$\chi_{1296}(11, \cdot)$$ n/a 7704 36
1296.2.bv $$\chi_{1296}(13, \cdot)$$ n/a 7704 36

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1296))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1296)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(648))$$$$^{\oplus 2}$$