Jury duty is far from over for the jurors who sat through more than a month of testimony to convict a former Hawaii-based soldier of murder in the death of his 5-year-old daughter.

This week, they began hearing more testimony on whether Naeem Williams is eligible for a death sentence. This process is unfamiliar in Hawaii, where capital punishment was abolished in 1957.

But the death penalty is still possible in federal cases, and Williams is being tried in federal court because the 2005 beating death of his daughter Talia occurred on military property.

Part of the prosecution’s burden is to prove the crime was so heinous that a death sentence is warranted, said Richard Dieter, executive director of the Death Penalty Information Center in Washington, D.C.

"This is a question about whether he’s also guilty of a capital murder, of a death-eligible murder," he said.

The prosecution needs to prove there’s at least one so-called aggravating factor that makes this murder especially heinous. In his opening statement for the penalty phase of the case, Assistant U.S. Attorney Darren Ching alluded to Talia’s age and vulnerability in highlighting the terrible nature of her death.

The defense is presenting what it views as mitigating factors, arguing Williams shouldn’t be sentenced to death because of his low IQ. Mental health experts have been testifying this week about his low level of intelligence, using test scores to make their case.

The jurors will then deliberate, weighing the aggravating and mitigating factors, Dieter said. If they reach a verdict unanimously agreeing he’s eligible for the death penalty, they will then deliberate and render a verdict on his sentence.

If they don’t agree he can be sentenced to death, the judge will sentence Williams to life in prison without possibility for release.

Death penalty cases are contentious, partly because jurors who have any moral opposition to the death penalty can’t serve on such cases.

"There’s a major difference in the way the jury gets constituted because of that," said University of Hawaii Professor Justin Levinson, who conducts research on criminal law and death-penalty cases. "You’re stacking the jury for death, basically."

Research has shown that the jurors who are eligible to serve on death penalty cases "have higher levels of racial bias," he said. Williams is black.

There have been calls for bifurcated juries — separate panels to handle the separate phases of death-penalty cases — Levinson noted. But the associated costs and time are deterrents, he said.