## Defining parameters

 Level: $$N$$ = $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$373248$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1296))$$.

Total New Old
Modular forms 141480 62460 79020
Cusp forms 138456 61956 76500
Eisenstein series 3024 504 2520

## Trace form

 $$61956 q - 48 q^{2} - 54 q^{3} - 80 q^{4} - 60 q^{5} - 72 q^{6} - 60 q^{7} - 48 q^{8} - 18 q^{9} + O(q^{10})$$ $$61956 q - 48 q^{2} - 54 q^{3} - 80 q^{4} - 60 q^{5} - 72 q^{6} - 60 q^{7} - 48 q^{8} - 18 q^{9} - 116 q^{10} - 36 q^{11} - 72 q^{12} - 100 q^{13} - 48 q^{14} - 54 q^{15} - 80 q^{16} - 111 q^{17} - 72 q^{18} - 87 q^{19} - 48 q^{20} - 90 q^{21} - 80 q^{22} - 36 q^{23} - 72 q^{24} - 145 q^{25} - 48 q^{26} - 54 q^{27} - 116 q^{28} - 228 q^{29} - 72 q^{30} + 210 q^{31} - 48 q^{32} - 162 q^{33} - 80 q^{34} + 1227 q^{35} - 72 q^{36} + 647 q^{37} - 48 q^{38} - 54 q^{39} - 80 q^{40} + 108 q^{41} - 72 q^{42} - 438 q^{43} - 48 q^{44} - 90 q^{45} - 52 q^{46} - 1782 q^{47} - 72 q^{48} - 523 q^{49} - 48 q^{50} - 54 q^{51} - 80 q^{52} - 45 q^{53} - 72 q^{54} + 163 q^{55} + 4068 q^{56} - 18 q^{57} + 5248 q^{58} + 1494 q^{59} - 72 q^{60} + 1772 q^{61} + 2460 q^{62} - 54 q^{63} - 1196 q^{64} - 2892 q^{65} - 72 q^{66} - 1284 q^{67} - 9252 q^{68} - 90 q^{69} - 10016 q^{70} - 3003 q^{71} - 72 q^{72} - 3341 q^{73} - 10968 q^{74} - 54 q^{75} - 7424 q^{76} - 3996 q^{77} - 72 q^{78} - 420 q^{79} + 132 q^{80} - 162 q^{81} + 2584 q^{82} + 1794 q^{83} - 72 q^{84} + 6103 q^{85} + 16632 q^{86} - 54 q^{87} + 9568 q^{88} + 19725 q^{89} - 72 q^{90} + 12479 q^{91} - 48 q^{92} + 11034 q^{93} - 144 q^{94} + 13569 q^{95} - 72 q^{96} + 72 q^{97} - 48 q^{98} - 5724 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1296))$$

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
1296.4.a $$\chi_{1296}(1, \cdot)$$ 1296.4.a.a 1 1
1296.4.a.b 1
1296.4.a.c 1
1296.4.a.d 1
1296.4.a.e 1
1296.4.a.f 1
1296.4.a.g 1
1296.4.a.h 1
1296.4.a.i 2
1296.4.a.j 2
1296.4.a.k 2
1296.4.a.l 2
1296.4.a.m 2
1296.4.a.n 2
1296.4.a.o 2
1296.4.a.p 2
1296.4.a.q 2
1296.4.a.r 2
1296.4.a.s 2
1296.4.a.t 2
1296.4.a.u 2
1296.4.a.v 3
1296.4.a.w 3
1296.4.a.x 4
1296.4.a.y 4
1296.4.a.z 4
1296.4.a.ba 4
1296.4.a.bb 4
1296.4.a.bc 5
1296.4.a.bd 5
1296.4.c $$\chi_{1296}(1295, \cdot)$$ 1296.4.c.a 2 1
1296.4.c.b 2
1296.4.c.c 4
1296.4.c.d 8
1296.4.c.e 10
1296.4.c.f 10
1296.4.c.g 12
1296.4.c.h 24
1296.4.d $$\chi_{1296}(649, \cdot)$$ None 0 1
1296.4.f $$\chi_{1296}(647, \cdot)$$ None 0 1
1296.4.i $$\chi_{1296}(433, \cdot)$$ n/a 142 2
1296.4.k $$\chi_{1296}(325, \cdot)$$ n/a 568 2
1296.4.l $$\chi_{1296}(323, \cdot)$$ n/a 568 2
1296.4.p $$\chi_{1296}(215, \cdot)$$ None 0 2
1296.4.r $$\chi_{1296}(217, \cdot)$$ None 0 2
1296.4.s $$\chi_{1296}(431, \cdot)$$ n/a 144 2
1296.4.u $$\chi_{1296}(145, \cdot)$$ n/a 318 6
1296.4.v $$\chi_{1296}(107, \cdot)$$ n/a 1144 4
1296.4.y $$\chi_{1296}(109, \cdot)$$ n/a 1144 4
1296.4.bb $$\chi_{1296}(73, \cdot)$$ None 0 6
1296.4.bd $$\chi_{1296}(71, \cdot)$$ None 0 6
1296.4.be $$\chi_{1296}(143, \cdot)$$ n/a 324 6
1296.4.bg $$\chi_{1296}(49, \cdot)$$ n/a 2898 18
1296.4.bh $$\chi_{1296}(37, \cdot)$$ n/a 2568 12
1296.4.bk $$\chi_{1296}(35, \cdot)$$ n/a 2568 12
1296.4.bn $$\chi_{1296}(23, \cdot)$$ None 0 18
1296.4.bp $$\chi_{1296}(25, \cdot)$$ None 0 18
1296.4.bq $$\chi_{1296}(47, \cdot)$$ n/a 2916 18
1296.4.bs $$\chi_{1296}(11, \cdot)$$ n/a 23256 36
1296.4.bv $$\chi_{1296}(13, \cdot)$$ n/a 23256 36

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1296))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1296)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 15}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(648))$$$$^{\oplus 2}$$