Existence of CM Newforms in Level p

Question

If $p$ is a prime and $k \geq 2$ is an even integer, what can we say about the existence of CM forms in the space $S_k^\text{new}(\Gamma_0(p))$? If it helps at all, I'm specifically interested in the primes $2,3,5$ and $7$, and I'm looking at spaces without a character.

Answer 1

No such newform exists.

If $\psi$ is a Groessencharacter of an imaginary quadratic field $K$, the level of the associated newform is $N_{K/\mathbf{Q}}(\mathfrak{f}) \cdot \operatorname{disc}(K/\mathbf{Q})$, where $\mathfrak{f}$ is the conductor of $\psi$. So a CM-type newform of prime level would have to come from an imaginary quadratic field $K$ of prime discriminant (which automatically rules out levels 2 and 5), and a Grossencharacter of $K$ with conductor 1, and infinity-type $(1-k, 0)$. Since you insist on small primes, $K$ will have class number 1, so such a Grossencharacter is unique if it exists and sends a fractional ideal to the $(k-1)$-st power of a generator. So we are in trouble if $k-1$ is not a multiple of the order of the unit group of $K$. But the unit group always contains $\pm 1$, so $k$ must be odd.

This also proves that an $f$ of level $\Gamma_1(p)$ exists when $p=3$ and $k = 1 \bmod 6$, or when $p = 7$ and $k$ is odd (and its Nebentypus is the quadratic character modulo $p$). I'll leave you to work out what the story is for larger prime levels.