Problem:

There is an isomorphism of

with

in which

. Find the element in

to which

must correspond for

.

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After arriving at an answer, I noticed that I said there was no answer for all of the even m values, so I'm not sure if I am correct.

To figure out the isomorphism, I used the example in the problem,

.

I reasoned that

, in which that 1 corresponds to the exponent of zeta.

I used this to look for the other values asked in the problem.

For m=0,

, but

, so I said that for m=0, there is no corresponding zeta function.

For m=3 (skipping ahead),

,

, so

.

I think I'm doing something incorrectly. Because since this is an isomorphism, and therefore a one-to-one correspondence, shouldn't all m=0,1,2,3,4,5,6 in

map to a member of

? My method is producing answers for only the odd numbers.

Any help is extremely appreciated.