For higher weight newforms, it seems that the conjecture you might want to look into is the Bloch-Kato conjecture. Here are a few links that might help you get started. The first is a survey article from 2003 (although a lot of progress has been made since then, this could be a good starting point):

https://jtnb.centre-mersenne.org/item/?id=JTNB_2003__15_1_179_0

Here is a fun paper on the distribution of zeros of certain polynomials (period polynomials) whose coefficients are built using the critical values you wish to study:

https://www.pnas.org/doi/10.1073/pnas.1600569113

Here is a high-level overview that requires a bit of background in algebraic number theory, representation theory of finite groups, and group cohomology:

https://www.claymath.org/sites/default/files/bellaiche.pdf

Here are some related Wikipedia articles:

https://en.wikipedia.org/wiki/Norm_residue_isomorphism_theorem https://en.wikipedia.org/wiki/Special_values_of_L-functions

For higher weight newforms, it seems that the conjecture you might want to look into is the Bloch-Kato conjecture. Here are a few links that might help you get started. The first is a survey article from 2003 (although a lot of progress has been made since then, this could be a good starting point):

https://jtnb.centre-mersenne.org/item/?id=JTNB_2003__15_1_179_0

Here is a fun paper on the distribution of zeros of certain polynomials (period polynomials) whose coefficients are built using the critical values you wish to study:

https://www.pnas.org/doi/10.1073/pnas.1600569113

Here is a high-level overview that requires a bit of background in algebraic number theory, representation theory of finite groups, and group cohomology:

https://www.claymath.org/sites/default/files/bellaiche.pdf

Here is a related Wikipedia article:

https://en.wikipedia.org/wiki/Special_values_of_L-functions