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4 added 130 characters in body

It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.

In their 1995 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the time of writing, very little was known about what sort of number fields could arise as some $K_f$. They do claim, however, that $K_f$ must be totally real or CM. This claim is made just before Lemma 1.37, on page 40 of the copy I linked to.

This is probably standard knowledge among experts, but I'm having trouble finding a reference, so my questions are:

1) Can someone please provide a reference for this claim?

2) Is this still the state of the art, or do we now know more about what types of fields can appear as $K_f$ for some $f$? What if we restrict our attention to weight $k=2$?

Thank you!

Edit: In my question, I originally just wrote "modular form" instead of "normalized eigenform". Thanks to @Stopple for pointing this out! Also, I originally claimed the paper was published in 2007, but Kevin Buzzard pointed out it was published in 1995. Thanks Kevin!

3 Paper was published in 1995, not 2007!

It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.

In their 2007 1995 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the time of writing, very little was known about what sort of number fields could arise as some $K_f$. They do claim, however, that $K_f$ must be totally real or CM. This claim is made just before Lemma 1.37, on page 40 of the copy I linked to.

This is probably standard knowledge among experts, but I'm having trouble finding a reference, so my questions are:

1) Can someone please provide a reference for this claim?

2) Is this still the state of the art, or do we now know more about what types of fields can appear as $K_f$ for some $f$? What if we restrict our attention to weight $k=2$?

Thank you!

Edit: In my question, I originally just wrote "modular form" instead of "normalized eigenform". Thanks to @Stopple for pointing this out!

2 Changed "modular form" to "normalized eigenform".

# Number Fields Arising from ModularFormsNewforms

It is well-known that, given a modular form normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.

In their 2007 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the time of writing, very little was known about what sort of number fields could arise as some $K_f$. They do claim, however, that $K_f$ must be totally real or CM. This claim is made just before Lemma 1.37, on page 40 of the copy I linked to.

This is probably standard knowledge among experts, but I'm having trouble finding a reference, so my questions are:

1) Can someone please provide a reference for this claim?

2) Is this still the state of the art, or do we now know more about what types of fields can appear as $K_f$ for some modular form?$f$? What if we restrict our attention to weight $k=2$?

Thank you!

Edit: In my question, I originally just wrote "modular form" instead of "normalized eigenform". Thanks to @Stopple for pointing this out!

1