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From the spell description of mage hand:

You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.

Some other examples of manipulating an object can be found here in the basic rules, here are a few of them:

The spell description of mage hand says (emphasis mine):

You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.

Here, the PHB outlines the rules for interacting with objects in combat. Some notable examples of things you can do with your object interaction are:

It is important note the simplicity of these actions. Actions requiring more dexterity are not intended to be doable with mage hand - this is evident in the Arcane Trickster's ability Mage Hand Legerdemain:

By using the phrase "additional tasks", this ability description implies that the list given in the mage hand spell description is intended to be a more restrictive list. A broad reading of mage hand would seem to imply anyone could do these things anyway, yet the Arcane Trickster has an entire class feature dedicated to them.

Keeping in mind the intended limits of mage hand, it's going to be up to the DM to determine if we can fully operate the ballista with mage hand. Load and Fire seem to be, at least to this DM, simple enough manipulations that a mage hand could perform, assuming that the ammunition and reset force is less than 10 pounds and the firing mechanism is simple like a trigger or button - more complex machinery will likely fall afoul of the arguments made in the previous section.

It is important note the simplicity of these actions - these actions typically would not require any sort of ability check to perform them without the use of mage hand. Actions requiring more dexterity are not intended to be doable with mage hand - this is evident in the Arcane Trickster's ability Mage Hand Legerdemain:

By using the phrase "additional tasks", this ability description implies that the list given in the mage hand spell description is intended to be a more restrictive list, and this seems quite natural. These additional tasks, if performed without mage hand, would also require the associated ability check to be made, as described in Mage Hand Legerdemain:

You can perform one of these tasks without being noticed by a creature if you succeed on a Dexterity (Sleight of Hand) check contested by the creature’s Wisdom (Perception) check.

These particular tasks notably require more dexterity than the more vanilla object manipulations, as evidenced by the required dexterity check to perform them unnoticed. This points to an intentional lack of dexterity of the vanilla mage hand that Mage Hand Legerdemain makes up for. 

A broad reading of mage hand would seem to imply anyone could do these things anyway, as this apparent lack of dexterity is not sufficiently communicated in the spell description of mage hand, yet the Arcane Trickster has an entire class feature dedicated to them.

Keeping in mind the intended limits of mage hand, it's going to be up to the DM to determine if we can fully operate the ballista with mage hand. Load and Fire seem to be, at least to this DM, simple enough manipulations that a mage hand could perform, assuming that the ammunition and reset force is less than 10 pounds and the firing mechanism is simple like a trigger or button - more complex machinery will likely fall afoul of the arguments made in the previous section. A possible rule of thumb (rule of mage thumb?) I would suggest would be:

Could the tasks of loading and firing be reasonably performed with one hand without requiring any sort of dexterity check?

If it's simple enough to be performed with one real hand with no issue, it's probably simple enough to be performed with one mage hand with no issue.

Mage hand is intended to be able to perform simple manual tasks.

From the spell description of mage hand:

You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.

Some other examples of manipulating an object can be found here in the basic rules, here are a few of them:

• draw or sheathe a sword
• open or close a door
• withdraw a potion from your backpack
• pick up a dropped axe
• take a bauble from a table
• throw a lever or a switch
• stuff some food into your mouth

It is important note the simplicity of these actions. Actions requiring more dexterity are not intended to be doable with mage hand - this is evident in the Arcane Trickster's ability Mage Hand Legerdemain:

Starting at 3rd level, when you cast mage hand, you can make the spectral hand invisible, and you can perform the following additional tasks with it:

• You can stow one object the hand is holding in a container worn or carried by another creature.
• You can retrieve an object in a container worn or carried by another creature.
• You can use thieves’ tools to pick locks and disarm traps at range.

By using the phrase "additional tasks", this ability description implies that the list given in the mage hand spell description is intended to be a more restrictive list. A broad reading of mage hand would seem to imply anyone could do these things anyway, yet the Arcane Trickster has an entire class feature dedicated to them.

But what about operating siege equipment?

It's up to the DM.

Keeping in mind the intended limits of mage hand, it's going to be up to the DM to determine if we can fully operate the ballista with mage hand. Load and Fire seem to be, at least to this DM, simple enough manipulations that a mage hand could perform, assuming that the ammunition and reset force is less than 10 pounds and the firing mechanism is simple like a trigger or button - more complex machinery will likely fall afoul of the arguments made in the previous section.

But, I would rule that aiming is beyond the limitations of mage hand for a few reasons. First, siege equipment is quite large, and often lacks fine manipulations for aiming, and where there are fine aiming instruments, they may require a dexterity that mage hand is not intended to offer. I would rule that it takes more strength than a mage hand can muster to aim siege equipment. Further to this, there is also the matter of what aiming entails. It requires a field vision that mage hand doesn't afford. If you are in a position with field of view to meaningfully be able aim the siege equipment, you're probably not using mage hand anyway, you're moving it yourself.

Conclusion: You might be able to load and fire, but probably can't aim with mage hand.

Mage hand is intended to be able to perform simple manual tasks.

From the spell description of mage hand:

You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.

Some other examples of manipulating an object can be found here in the basic rules, here are a few of them:

• draw or sheathe a sword
• open or close a door
• withdraw a potion from your backpack
• pick up a dropped axe
• take a bauble from a table
• throw a lever or a switch
• stuff some food into your mouth

It is important note the simplicity of these actions. Actions requiring more dexterity are not intended to be doable with mage hand - this is evident in the Arcane Trickster's ability Mage Hand Legerdemain:

Starting at 3rd level, when you cast mage hand, you can make the spectral hand invisible, and you can perform the following additional tasks with it:

• You can stow one object the hand is holding in a container worn or carried by another creature.
• You can retrieve an object in a container worn or carried by another creature.
• You can use thieves’ tools to pick locks and disarm traps at range.

By using the phrase "additional tasks", this ability description implies that the list given in the mage hand spell description is intended to be a more restrictive list. A broad reading of mage hand would seem to imply anyone could do these things anyway, yet the Arcane Trickster has an entire class feature dedicated to them.

But what about operating siege equipment?

It's up to the DM.

Keeping in mind the intended limits of mage hand, it's going to be up to the DM to determine if we can fully operate the ballista with mage hand. Load and Fire seem to be, at least to this DM, simple enough manipulations that a mage hand could perform, assuming that the ammunition and reset force is less than 10 pounds and the firing mechanism is simple like a trigger or button - more complex machinery will likely fall afoul of the arguments made in the previous section.

But, I would rule that aiming is beyond the limitations of mage hand for a few reasons. First, siege equipment is quite large, and often lacks fine manipulations for aiming, and where there are fine aiming instruments, they may require a dexterity that mage hand is not intended to offer. I would rule that it takes more strength than a mage hand can muster to aim siege equipment. Further to this, there is also the matter of what aiming entails. It requires a field vision that mage hand doesn't afford. If you are in a position with field of view to meaningfully be able aim the siege equipment, you're probably not using mage hand anyway, you're moving it yourself.

Conclusion: You might be able to load and fire, but probably can't aim with mage hand.