## Defining parameters

 Level: $$N$$ = $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$16$$ Sturm bound: $$129024$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(576))$$.

Total New Old
Modular forms 55872 24723 31149
Cusp forms 54720 24525 30195
Eisenstein series 1152 198 954

## Trace form

 $$24525 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 32 q^{6} - 16 q^{7} - 24 q^{8} - 40 q^{9} + O(q^{10})$$ $$24525 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 32 q^{6} - 16 q^{7} - 24 q^{8} - 40 q^{9} - 72 q^{10} - 1378 q^{11} - 32 q^{12} + 5016 q^{13} - 24 q^{14} - 24 q^{15} - 24 q^{16} - 9818 q^{17} - 32 q^{18} - 3990 q^{19} - 24 q^{20} - 32 q^{21} - 63744 q^{22} + 13096 q^{23} - 32 q^{24} + 25529 q^{25} - 10624 q^{26} - 24 q^{27} - 171072 q^{28} - 66424 q^{29} - 32 q^{30} - 28 q^{31} + 117776 q^{32} - 2940 q^{33} + 278976 q^{34} - 12600 q^{35} - 32 q^{36} + 173448 q^{37} - 200224 q^{38} + 254376 q^{39} - 575664 q^{40} - 342430 q^{41} - 32 q^{42} - 513090 q^{43} + 480976 q^{44} - 515648 q^{45} - 72 q^{46} - 12 q^{47} - 32 q^{48} + 500069 q^{49} + 705408 q^{50} + 767176 q^{51} - 1275144 q^{52} + 903288 q^{53} - 32 q^{54} + 232652 q^{55} + 326512 q^{56} - 977000 q^{57} + 1770096 q^{58} - 3427138 q^{59} - 32 q^{60} + 589656 q^{61} + 543192 q^{62} + 470572 q^{63} - 1155240 q^{64} + 481188 q^{65} - 32 q^{66} + 3141966 q^{67} - 1847304 q^{68} + 5800 q^{69} - 2272056 q^{70} + 801012 q^{71} - 32 q^{72} + 135846 q^{73} + 2924128 q^{74} + 65392 q^{75} + 4565160 q^{76} + 2207172 q^{77} + 9469648 q^{78} + 3443692 q^{79} - 10442352 q^{80} - 4842936 q^{81} - 20900952 q^{82} - 5860498 q^{83} - 11943040 q^{84} - 3777288 q^{85} + 2485056 q^{86} + 2029864 q^{87} + 14086776 q^{88} + 14397826 q^{89} + 22931968 q^{90} + 7099148 q^{91} + 30316224 q^{92} + 4699168 q^{93} + 7526280 q^{94} + 12 q^{95} - 7416112 q^{96} - 13434786 q^{97} - 31046784 q^{98} - 5970712 q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(576))$$

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
576.7.b $$\chi_{576}(415, \cdot)$$ 576.7.b.a 4 1
576.7.b.b 4
576.7.b.c 4
576.7.b.d 8
576.7.b.e 8
576.7.b.f 8
576.7.b.g 8
576.7.b.h 16
576.7.e $$\chi_{576}(449, \cdot)$$ 576.7.e.a 2 1
576.7.e.b 2
576.7.e.c 2
576.7.e.d 2
576.7.e.e 2
576.7.e.f 2
576.7.e.g 2
576.7.e.h 2
576.7.e.i 2
576.7.e.j 2
576.7.e.k 2
576.7.e.l 2
576.7.e.m 4
576.7.e.n 4
576.7.e.o 4
576.7.e.p 4
576.7.e.q 4
576.7.e.r 4
576.7.g $$\chi_{576}(127, \cdot)$$ 576.7.g.a 1 1
576.7.g.b 1
576.7.g.c 1
576.7.g.d 2
576.7.g.e 2
576.7.g.f 2
576.7.g.g 2
576.7.g.h 2
576.7.g.i 2
576.7.g.j 2
576.7.g.k 4
576.7.g.l 4
576.7.g.m 4
576.7.g.n 4
576.7.g.o 6
576.7.g.p 6
576.7.g.q 6
576.7.g.r 8
576.7.h $$\chi_{576}(161, \cdot)$$ 576.7.h.a 16 1
576.7.h.b 32
576.7.j $$\chi_{576}(17, \cdot)$$ 576.7.j.a 96 2
576.7.m $$\chi_{576}(271, \cdot)$$ n/a 118 2
576.7.n $$\chi_{576}(353, \cdot)$$ n/a 288 2
576.7.o $$\chi_{576}(319, \cdot)$$ n/a 284 2
576.7.q $$\chi_{576}(65, \cdot)$$ n/a 284 2
576.7.t $$\chi_{576}(31, \cdot)$$ n/a 288 2
576.7.u $$\chi_{576}(55, \cdot)$$ None 0 4
576.7.x $$\chi_{576}(89, \cdot)$$ None 0 4
576.7.z $$\chi_{576}(79, \cdot)$$ n/a 568 4
576.7.ba $$\chi_{576}(113, \cdot)$$ n/a 568 4
576.7.bc $$\chi_{576}(53, \cdot)$$ n/a 1536 8
576.7.bf $$\chi_{576}(19, \cdot)$$ n/a 1912 8
576.7.bh $$\chi_{576}(7, \cdot)$$ None 0 8
576.7.bi $$\chi_{576}(41, \cdot)$$ None 0 8
576.7.bk $$\chi_{576}(43, \cdot)$$ n/a 9184 16
576.7.bn $$\chi_{576}(5, \cdot)$$ n/a 9184 16

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

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(576))$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(\Gamma_1(576)) \cong$$ $$S_{7}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 14}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 15}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 7}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 10}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 9}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 5}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 2}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 2}$$