# Imaginary quadratic field data (2013 updates)

## Numbers of newforms (rational, cuspidal, weight 2)

These counts are expected to equal the number of isogeny classes of elliptic curves defined over the field $K$ in question with conductor equal to the level of the newform, with two provisos:
1. Elliptic curves with CM by an order in $K$ will not be counted. These are characterised as those curves having specific $j$-invariants, as listed. (These $j$-invariants my be listed using the Sage command cm_j_invariants_and_orders(QQ))
2. Newforms which are base-changes of newforms over Q with extra twist may not have associated elliptic curves defined over $K$: see Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields
Journal of the London Mathematical Society 45 (1992) 402-416

For each field (currently just the five Euclidean fields) we give

1. basic data about the field, specifically the primes of norm ${}<1000$ in standard order (norm, label, characteristic, degree, ramification degree);
2. a list giving the number of rational weight 2 newforms at each level in some range;
3. a list of these newforms $F$ (label, level, sign of functional equation of $L(F,s)$, ratio $L(F,1)/\Omega\in\mathbb{Q}$ showing whether or not $L(F,1)=0$, Atkin-Lehner eigenvalues, first 25 Fourier coefficients $a_{\mathfrak{p}}$ indexed by all prime ideals in standard order).

• $\mathbb{Q}(\sqrt{-1})$: field data; exceptional CM $j$-invariants: 1728, 287496.
counts for levels of norm ${}<10^4$, rational newforms of levels of norm ${}<10^4$.
• $\mathbb{Q}(\sqrt{-2})$: field data; exceptional CM $j$-invariant: 8000.
counts for levels of norm ${}<10^4$, rational newforms of levels of norm ${}<10^4$.
• $\mathbb{Q}(\sqrt{-3})$: field data; exceptional CM $j$-invariants: 0, 54000, -12288000.
counts for levels of norm ${}<10^4$, rational newforms of levels of norm ${}<10^4$.
• $\mathbb{Q}(\sqrt{-7})$: field data; exceptional CM $j$-invariants: -3375, 16581375.
counts for levels of norm ${}<10^4$, rational newforms of levels of norm ${}<10^4$.
• $\mathbb{Q}(\sqrt{-11})$: field data; exceptional CM $j$-invariant: -32768.
counts for levels of norm ${}<10^4$, rational newforms of levels of norm ${}<10^4$.

Plain text index of ancient data