Decidability of checking whether all words accepted by a TM have an even number of 1s Let $\Sigma = \{0,1\}$, and consider the language
$$L = \{\langle M \rangle \mid 
M \text{ is a TM and } L(M) \text{ is a subset of } Σ^* \text{ and all strings in this set have an even number of 1s}\}.$$

I think $L$ is undecidable. But I am stuck on proving it. I think it is equivalent to prove $L = \{\langle M \rangle \mid M \text{ is a TM and } L(M) \text{ is a specific set with an even number of 1s}\}$ is undecidable. How can I do some reduction for it?
First, it can be shown easily by Rice's theorem.

Alternatively, you can show a reduction from $\overline{HP}$ as follows:

$$f(\left&lt; M \right&gt;,x)=\left&lt;M_x\right&gt;$$

$M_x$ on input $w$:


run $M$ on $x$ for $|w|$ steps. 
if M stopped then accept.
if the number of 1's in $w$ is even then accept, otherwise reject.


I'll let you verify the correctness.