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Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:

$$(i)\ 1 + 80q^3 + 270q^4 + 432q^5 + 960q^6 + 2160q^7 + 3240q^8 + 5360q^9 + 8640q^{10}+\dots$$

$$(ii)\ 1 + 270q^4 + 960q^6 + 3240q^8 + 8640q^{10} + 17790q^{12} + 25920q^{14} + 62910q^{16} + \dots$$

The second theta series is given by taking only the even coefficients from the first series. The source of these series are ten dimensional lattices known as $O_{10}$ and $(C6\times SU(4,2)):C2$ from Nebe and Sloane's database of lattices.

Another example is $E_7$ and its dual lattice, having theta series given below (in reverse order)

$1 + 56q^3 + 126q^4 + 576q^7 + 756q^8 + 1512q^{11} + 2072q^{12} + \dots $

$1 + 126q^4 + 756q^8 + 2072q^{12} + 4158q^{16} + 7560q^{20} + \dots$

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  • $\begingroup$ Is there a direct algebraic relation between the two lattices you have mentioned? $\endgroup$
    – user35360
    Feb 15, 2019 at 6:39
  • $\begingroup$ @ramanujan_dirac Sure, the order of the automorphism group of the second is three times that of the first. Another fact about the first is that it is alternatively known as the shorter Coxeter-Todd lattice and its dual is equivalent to itself. $\endgroup$ Feb 15, 2019 at 6:43
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    $\begingroup$ A Magma keyword (in this case) is simply EvenSublattice. But obviously that's a special case. $\endgroup$ Feb 15, 2019 at 9:32
  • $\begingroup$ @literature-searcher Yes, but that explains this example well. The maximal even sublattice of $O_{10}$ is (equivalent to) $(C6\times SU(4,2)):C2$. $\endgroup$ Feb 15, 2019 at 9:36

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