# Properties

 Label 64.2 Level 64 Weight 2 Dimension 61 Nonzero newspaces 4 Newform subspaces 4 Sturm bound 512 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$64 = 2^{6}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$4$$ Sturm bound: $$512$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(64))$$.

Total New Old
Modular forms 164 83 81
Cusp forms 93 61 32
Eisenstein series 71 22 49

## Trace form

 $$61q - 8q^{2} - 6q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 13q^{9} + O(q^{10})$$ $$61q - 8q^{2} - 6q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 13q^{9} - 8q^{10} - 10q^{11} - 8q^{12} - 16q^{13} - 8q^{14} - 12q^{15} - 8q^{16} - 22q^{17} - 8q^{18} - 14q^{19} - 8q^{20} - 4q^{21} - 8q^{23} + 32q^{24} + q^{25} + 32q^{26} + 32q^{28} + 8q^{29} + 72q^{30} + 16q^{31} + 32q^{32} + 20q^{33} + 32q^{34} - 4q^{35} + 72q^{36} + 32q^{38} - 8q^{39} + 32q^{40} - 10q^{41} + 32q^{42} - 18q^{43} - 12q^{45} - 8q^{46} - 24q^{47} - 8q^{48} - 23q^{49} - 32q^{50} + 20q^{51} - 56q^{52} - 32q^{53} - 72q^{54} + 56q^{55} - 64q^{56} - 16q^{57} - 80q^{58} + 62q^{59} - 104q^{60} - 16q^{61} - 40q^{62} + 60q^{63} - 104q^{64} - 24q^{65} - 88q^{66} + 82q^{67} - 56q^{68} - 20q^{69} - 104q^{70} + 56q^{71} - 80q^{72} - 10q^{73} - 64q^{74} + 50q^{75} - 72q^{76} - 4q^{77} - 32q^{78} + 24q^{79} + 32q^{80} - 11q^{81} + 72q^{82} - 6q^{83} + 104q^{84} + 96q^{86} - 8q^{87} + 72q^{88} + 38q^{89} + 136q^{90} - 12q^{91} + 144q^{92} + 32q^{93} + 88q^{94} + 12q^{95} + 128q^{96} + 34q^{97} + 128q^{98} + 18q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$

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
64.2.a $$\chi_{64}(1, \cdot)$$ 64.2.a.a 1 1
64.2.b $$\chi_{64}(33, \cdot)$$ 64.2.b.a 2 1
64.2.e $$\chi_{64}(17, \cdot)$$ 64.2.e.a 2 2
64.2.g $$\chi_{64}(9, \cdot)$$ None 0 4
64.2.i $$\chi_{64}(5, \cdot)$$ 64.2.i.a 56 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(64))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(64)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 3 T^{2}$$)($$1 - 2 T^{2} + 9 T^{4}$$)($$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$)
$5$ ($$1 - 2 T + 5 T^{2}$$)($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)
$7$ ($$1 + 7 T^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$1 - 10 T^{2} + 49 T^{4}$$)
$11$ ($$1 + 11 T^{2}$$)($$1 + 14 T^{2} + 121 T^{4}$$)($$1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4}$$)
$13$ ($$1 + 6 T + 13 T^{2}$$)($$( 1 - 13 T^{2} )^{2}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)
$17$ ($$1 - 2 T + 17 T^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)
$19$ ($$1 + 19 T^{2}$$)($$1 - 34 T^{2} + 361 T^{4}$$)($$1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4}$$)
$23$ ($$1 + 23 T^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$1 - 10 T^{2} + 529 T^{4}$$)
$29$ ($$1 - 10 T + 29 T^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 10 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} )$$)
$31$ ($$1 + 31 T^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)
$37$ ($$1 - 2 T + 37 T^{2}$$)($$( 1 - 37 T^{2} )^{2}$$)($$1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4}$$)
$41$ ($$1 - 10 T + 41 T^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 41 T^{2} )^{2}$$)
$43$ ($$1 + 43 T^{2}$$)($$1 + 14 T^{2} + 1849 T^{4}$$)($$1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4}$$)
$47$ ($$1 + 47 T^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)
$53$ ($$1 + 14 T + 53 T^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 4 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$)
$59$ ($$1 + 59 T^{2}$$)($$1 - 82 T^{2} + 3481 T^{4}$$)($$1 - 6 T + 18 T^{2} - 354 T^{3} + 3481 T^{4}$$)
$61$ ($$1 - 10 T + 61 T^{2}$$)($$( 1 - 61 T^{2} )^{2}$$)($$1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4}$$)
$67$ ($$1 + 67 T^{2}$$)($$1 + 62 T^{2} + 4489 T^{4}$$)($$1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4}$$)
$71$ ($$1 + 71 T^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$1 - 42 T^{2} + 5041 T^{4}$$)
$73$ ($$1 + 6 T + 73 T^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)
$79$ ($$1 + 79 T^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)
$83$ ($$1 + 83 T^{2}$$)($$1 + 158 T^{2} + 6889 T^{4}$$)($$1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4}$$)
$89$ ($$1 - 10 T + 89 T^{2}$$)($$( 1 - 18 T + 89 T^{2} )^{2}$$)($$1 - 162 T^{2} + 7921 T^{4}$$)
$97$ ($$1 - 18 T + 97 T^{2}$$)($$( 1 - 10 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)