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:
- 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))
- 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
- basic data about the field, specifically the primes of norm ${}<1000$ in
standard order (norm, label, characteristic, degree, ramification
degree);
- a list giving the number of rational weight 2 newforms at each level in
some range;
- 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