It is known that a Jewish king has to be male, as quoted by an answer to this question.
What halachik powers did the queen (i.e. wife of the king) have within the kingdom?
Also, what historical documentation is there on this topic?
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The queen had no official power, but they could influence the king. Often the mother of the king was a position of influence, for example:
וַתָּבֹא בַת־שֶׁבַע אֶל־הַמֶּלֶךְ שְׁלֹמֹה לְדַבֶּר־לוֹ עַל־אֲדֹנִיָּהוּ וַיָּקָם הַמֶּלֶךְ לִקְרָאתָהּ וַיִּשְׁתַּחוּ לָהּ וַיֵּשֶׁב עַל־כִּסְאוֹ וַיָּשֶׂם כִּסֵּא לְאֵם הַמֶּלֶךְ וַתֵּשֶׁב לִימִינוֹ׃
So Bathsheba went to King Solomon to speak to him about Adonijah. The king rose to greet her and bowed down to her. He sat on his throne; and he had a throne placed for the queen mother, and she sat on his right.
In addition, the mother of the king is often mentioned by the Bible. Here is a list.