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I would like to know if
Spells per Day: When a new arcane trickster level is gained, the character gains new spells per day as if she had also gained a level in a spellcasting class she belonged to before adding the prestige class. She does not, however, gain other benefits a character of that class would have gained, except for additional spells per day, spells known (if she is a spontaneous spellcaster), and an increased effective level of spellcasting. If a character had more than one spellcasting class before becoming an arcane trickster, she must decide to which class she adds the new level for purposes of determining spells per day.
would effect this..
Hex: Witches learn a number of magic tricks, called hexes, that grant them powers or weaken foes. At 1st level, a witch gains one hex of her choice. She gains an additional hex at 2nd level and for every 2 levels attained after 2nd level, as noted on Table 2–10. A witch cannot select an individual hex more than once.
Unless otherwise noted, using a hex is a standard action that does not provoke an attack of opportunity. The save to resist a hex is equal to 10 + 1/2 the witch's level + the witch's Intelligence modifier.
Basically I am wondering if the increased spell casting level gained would also increase the DC of the HEX?
Not only that, but the spells gained from your patron are considered a class feature, not a part of your spellcasting, so you lose those on top of losing hex DC. (You keep the ones you already learned from your real witch levels, but you don't get new ones from PrCs.)
Witches can't be arcane tricksters they don't have Mage Hand on their spell list.
Huh, that's true. Technically a gnome with the Gnome Trickster racial feat can qualify for the prestige class as a single-class witch. The requirements don't say that you have to be able to cast it as an arcane spell. (And I can't imagine any DM calling "flavor" on that one.) More straightforward would be the two-world magic trait, which lets you grab any zero-level spell for your own list.