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Given the stated source of the Warlock's power - their spells are granted by their patrons - I'm unsure whether a Warlock could actually research/create new spells in her/his downtime as Wizards can. I mention the Pact of the Tome specifically because it seems as if this flavor of Warlock specifically could justify doing so more easily since this type of Warlock can already learn ritual spells from other classes.

Note: I am new. I am hoping to go to a gaming store and begin playing. I am looking through the book and trying to decide on a class. I am not looking to necessarily make a character that is maximally powerful or a "broken" class. I do like the idea of having the versatility of being able to do research to create something that uniquely fits the character, and I quite like the flavor of the warlock over that of the wizard, but I am unclear about this particular point. I realize that ultimately these decisions are up to the DM, but I thought that there might be a standard for something like this.

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    \$\begingroup\$ Is this question about spell research in D&D 5e, as in coming up with new spells that are not in the PHB or other official material, or is this about pulling spells from other spell lists (already published) into the Necromancer's collection of spells? \$\endgroup\$ – KorvinStarmast Aug 23 '16 at 11:08
  • \$\begingroup\$ @KorvinStarmast wouldn't "research new spells", as in creating them, be just plain homebrew? Or are there rules for that? \$\endgroup\$ – Chepelink Aug 23 '16 at 11:19
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    \$\begingroup\$ There is some guidance in the DMG (IIRC p 261 or so called "creating a new spell") which means that it's either made by the DM, or a player and a DM put their heads together and come up with a new spell based on the guidance therein. That's why I asked for further clarification. \$\endgroup\$ – KorvinStarmast Aug 23 '16 at 12:23
  • \$\begingroup\$ Let the OP clarify then answer in answers. \$\endgroup\$ – mxyzplk Aug 24 '16 at 0:43
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Some clarification first.

Warlocks

Does the Pact of the Tome Warlocks learn new ritual spells from other classes?

No

(PHB 108) When you gain this feature, choose three cantrips from any class’s spell list. While the book is on your person, you can cast those cantrips at will.

You can only get 3 cantrips as stated. Another reason why it does not allow you to research new spells is because of the Incantation "Book of Ancient Secrets", and that send us to the following point.

Book of Ancient Secrets

From the description (PHB 110):

You can now inscribe magical rituals in your Book of Shadows.

With your Book of Shadows in hand, you can cast the chosen spells as rituals.

and

On your adventures, you can add other ritual spells to your Book of Shadows. When you find such a spell, you can add it to the book if the spell’s level is equal to or less than half your warlock level (rounded up) and if you can spare the time to transcribe the spell.

With "Book of Ancient Secrets" invocation you can learn new ritual spells. These spells can only be cast as rituals.

Research

There are two meanings to the word "research" referent to wizards and spells; the homebrew type of research, where it is implied that you create new and custom spells, and the to add new (but existing in the game) spells to the wizard spell book.

Homebrew research

This type of "research" allows to create custom spells, as in literally new spells and modifications of existing spells. In the first instance you create spells based on an idea or an existing spell from other editions or games. In the second instance you take one spell, let say magic missile, and modified it to be more fun or align with a thematic; e.g. "Elemental magic missile". There exist guidelines in the DMG for homebrewing spells and has to be worked with your DM.

The way you create custom spells vary from table to table, some might just add to the wizard spell list, other would ask you to use your downtime time to "research" the spell and add to the list, or it can be the reason for an entire adventure. On the last point, just to add a bit of history, there are prominent wizards (that might or might not be relevant or exist in your game) through the history that left their work for future generation of wizards, two of the most common examples are Mordenkainen and his disciple Bigby.

Typical Research

From PHB 114, emphasis mine.

Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it.

This segment feels like a typical day for a researcher. You pick an article (and if interest you), some times you need to decipher the notation, you keep reading and reading until you are sure that you understand the article, and then you transcribe it into your own article (or thesis). If in doubt of the results, you do the often undervalued action of trying to reproduce the experiment.

Downtime research

In PHB page 187 it is describe the action of researching as a downtime activity, emphasis mine. This can be ruled as a way to create homebrew spells as a mean of "research", or use the downtime to find spells and transcribe them in you spell book. This is up to the DM, though.

The time between adventures is a great chance to perform research, gaining insight into mysteries that have unfurled over the course of the campaign. Research can include poring over dusty tomes and crumbling scrolls in a library or buying drinks for the locals to pry rumors and gossip from their lips.

Note: Store playing

Bear in mind that store playing, as in Adventure League, might be against of homebrewing since they try to keep thing as balance and uniform as possible so you can move your character from table to table.

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Now that you have edited your question a bit my answer will be:

Why not?

The "Book of Ancient Secrets" doesn't allow for this in anyway however...

Let's not combine two ideas here. I will address the research of spells as a whole first.

The DMG allows for the DM and possibly the players to research new spells for their particular world, these rules are entirely optional and you would need to work with your DM to implement them and the limitations on an invented spell. This implicitly would be down time. The rules don't call out this option in the same section as crafting magic items on page 128 but they could easily be adapted to this purpose.

The research of spells is not limited to any single class and to my knowledge never has in previous editions, Clerics, Bards etc can research spells and have done so to serve the public or the greater glory of their deity.

I grant you that Warlock and Sorcerer are sort of precluded by flavor but I would say the latter more than the former. After all who is to say what the purpose of the pact is for the Warlock? Especially for the newly released Seeker in the UA. The Warlock (regardless of pact boon) could be travelling to find new things for their patron but researching something entirely new should not be out of the question (albeit they are not as suited to it as a Wizard obviously). Warlocks learn from their Patrons but they also serve them in some capacity in order to gain their power (at least that is how I run them, of all the classes they are the most beholden to their source of power imho). So obviously a patron in search of knowledge could definitely have a Warlock do research into a new spell.

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No, you cannot.

The description of the ability clearly states (emphasis mine):

choose three cantrips from any class's spell list.

This does not mean that your character can create a brand new spell or modify an existing one. It means that you can choose an existing cantrip from the spell lists and learn it.

Of course, a generous DM may allow you to do this, but I suspect that their answer will be no.

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