# Properties

 Label 96.2 Level 96 Weight 2 Dimension 98 Nonzero newspaces 6 Newform subspaces 7 Sturm bound 1024 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$1024$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(96))$$.

Total New Old
Modular forms 320 118 202
Cusp forms 193 98 95
Eisenstein series 127 20 107

## Trace form

 $$98q - 2q^{3} - 8q^{4} + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} + O(q^{10})$$ $$98q - 2q^{3} - 8q^{4} + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} - 24q^{10} - 20q^{12} - 20q^{13} - 32q^{14} - 12q^{15} - 48q^{16} - 16q^{17} - 12q^{18} - 12q^{19} - 32q^{20} - 20q^{21} - 32q^{22} - 24q^{23} + 8q^{24} - 30q^{25} + 40q^{26} - 38q^{27} + 32q^{28} + 4q^{29} + 44q^{30} - 52q^{31} + 40q^{32} - 8q^{33} + 16q^{34} - 48q^{35} + 48q^{36} + 12q^{37} + 40q^{38} - 20q^{39} + 32q^{40} + 8q^{41} + 56q^{42} - 4q^{43} + 8q^{44} + 32q^{45} - 8q^{46} + 24q^{47} + 48q^{48} + 38q^{49} + 24q^{50} + 16q^{51} + 40q^{52} - 12q^{53} + 48q^{54} + 56q^{55} + 56q^{56} + 16q^{57} + 64q^{58} + 64q^{59} + 56q^{60} - 68q^{61} + 48q^{62} + 12q^{63} + 64q^{64} + 8q^{65} + 20q^{66} + 44q^{67} - 8q^{68} - 68q^{69} + 16q^{70} + 40q^{71} - 64q^{72} - 52q^{73} - 32q^{74} + 38q^{75} - 8q^{76} - 64q^{77} - 68q^{78} - 4q^{79} - 56q^{80} - 38q^{81} - 88q^{82} - 136q^{84} - 72q^{85} - 64q^{86} + 64q^{87} - 96q^{88} - 136q^{90} - 8q^{91} - 80q^{92} + 8q^{93} - 96q^{94} + 16q^{95} - 144q^{96} - 28q^{97} - 80q^{98} + 76q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(96))$$

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
96.2.a $$\chi_{96}(1, \cdot)$$ 96.2.a.a 1 1
96.2.a.b 1
96.2.c $$\chi_{96}(95, \cdot)$$ 96.2.c.a 4 1
96.2.d $$\chi_{96}(49, \cdot)$$ 96.2.d.a 2 1
96.2.f $$\chi_{96}(47, \cdot)$$ 96.2.f.a 2 1
96.2.j $$\chi_{96}(25, \cdot)$$ None 0 2
96.2.k $$\chi_{96}(23, \cdot)$$ None 0 2
96.2.n $$\chi_{96}(13, \cdot)$$ 96.2.n.a 32 4
96.2.o $$\chi_{96}(11, \cdot)$$ 96.2.o.a 56 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(96))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(96)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()()()()
$3$ ($$1 + T$$)($$1 - T$$)($$1 - 2 T^{2} + 9 T^{4}$$)($$1 + T^{2}$$)($$1 - 2 T + 3 T^{2}$$)
$5$ ($$1 - 2 T + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 + 5 T^{2} )^{2}$$)
$7$ ($$1 - 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$( 1 - 10 T^{2} + 49 T^{4} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 - 7 T^{2} )^{2}$$)
$11$ ($$1 + 4 T + 11 T^{2}$$)($$1 - 4 T + 11 T^{2}$$)($$( 1 + 14 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 11 T^{2} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$)
$13$ ($$1 + 2 T + 13 T^{2}$$)($$1 + 2 T + 13 T^{2}$$)($$( 1 + 2 T + 13 T^{2} )^{4}$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$( 1 - 13 T^{2} )^{2}$$)
$17$ ($$1 + 6 T + 17 T^{2}$$)($$1 + 6 T + 17 T^{2}$$)($$( 1 - 17 T^{2} )^{4}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$)
$19$ ($$1 - 4 T + 19 T^{2}$$)($$1 + 4 T + 19 T^{2}$$)($$( 1 - 2 T^{2} + 361 T^{4} )^{2}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)
$23$ ($$1 + 23 T^{2}$$)($$1 + 23 T^{2}$$)($$( 1 + 14 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 4 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)
$29$ ($$1 - 2 T + 29 T^{2}$$)($$1 - 2 T + 29 T^{2}$$)($$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$( 1 + 29 T^{2} )^{2}$$)
$31$ ($$1 + 4 T + 31 T^{2}$$)($$1 - 4 T + 31 T^{2}$$)($$( 1 - 58 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 - 31 T^{2} )^{2}$$)
$37$ ($$1 + 2 T + 37 T^{2}$$)($$1 + 2 T + 37 T^{2}$$)($$( 1 - 6 T + 37 T^{2} )^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$( 1 - 37 T^{2} )^{2}$$)
$41$ ($$1 - 2 T + 41 T^{2}$$)($$1 - 2 T + 41 T^{2}$$)($$( 1 - 50 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )( 1 + 6 T + 41 T^{2} )$$)
$43$ ($$1 + 4 T + 43 T^{2}$$)($$1 - 4 T + 43 T^{2}$$)($$( 1 - 82 T^{2} + 1849 T^{4} )^{2}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$( 1 - 10 T + 43 T^{2} )^{2}$$)
$47$ ($$1 + 8 T + 47 T^{2}$$)($$1 - 8 T + 47 T^{2}$$)($$( 1 - 34 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 12 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)
$53$ ($$1 - 10 T + 53 T^{2}$$)($$1 - 10 T + 53 T^{2}$$)($$( 1 - 34 T^{2} + 2809 T^{4} )^{2}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 + 53 T^{2} )^{2}$$)
$59$ ($$1 - 4 T + 59 T^{2}$$)($$1 + 4 T + 59 T^{2}$$)($$( 1 + 110 T^{2} + 3481 T^{4} )^{2}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} )$$)
$61$ ($$1 - 6 T + 61 T^{2}$$)($$1 - 6 T + 61 T^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{4}$$)($$( 1 - 61 T^{2} )^{2}$$)($$( 1 - 61 T^{2} )^{2}$$)
$67$ ($$1 + 4 T + 67 T^{2}$$)($$1 - 4 T + 67 T^{2}$$)($$( 1 - 130 T^{2} + 4489 T^{4} )^{2}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$( 1 + 14 T + 67 T^{2} )^{2}$$)
$71$ ($$1 - 16 T + 71 T^{2}$$)($$1 + 16 T + 71 T^{2}$$)($$( 1 + 110 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)
$73$ ($$1 + 6 T + 73 T^{2}$$)($$1 + 6 T + 73 T^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{4}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)
$79$ ($$1 + 4 T + 79 T^{2}$$)($$1 - 4 T + 79 T^{2}$$)($$( 1 + 38 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 10 T + 79 T^{2} )^{2}$$)($$( 1 - 79 T^{2} )^{2}$$)
$83$ ($$1 + 12 T + 83 T^{2}$$)($$1 - 12 T + 83 T^{2}$$)($$( 1 + 158 T^{2} + 6889 T^{4} )^{2}$$)($$1 + 90 T^{2} + 6889 T^{4}$$)($$( 1 - 18 T + 83 T^{2} )( 1 + 18 T + 83 T^{2} )$$)
$89$ ($$1 - 10 T + 89 T^{2}$$)($$1 - 10 T + 89 T^{2}$$)($$( 1 + 110 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 - 18 T + 89 T^{2} )( 1 + 18 T + 89 T^{2} )$$)
$97$ ($$1 + 14 T + 97 T^{2}$$)($$1 + 14 T + 97 T^{2}$$)($$( 1 - 10 T + 97 T^{2} )^{4}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 + 10 T + 97 T^{2} )^{2}$$)