Newform orbit 550.2.a.f

$q$-expansion

$q - q^{2} + q^{3} + q^{4} - q^{6} - 3q^{7} - q^{8} - 2q^{9} + q^{11} + q^{12} + 6q^{13} + 3q^{14} + q^{16} + 7q^{17} + 2q^{18} + 5q^{19} - 3q^{21} - q^{22} + 6q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$. The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ is a root of

$x - 86$

Atkin–Lehner signs

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$.

Elliptic curve

By modularity this newform is the one attached to the isogeny class 550.f (Cremona's 550b). Its optimal curve 550b1 is

$y^2 + xy + y = x^3 + 249x - 6102$

of rank 0 with torsion of order 1. Its $a_p$, computed by Sagebrush's point-counting engine, equal the newform's for every prime $p < 1000$.

Traces of Hecke eigenvalues

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 550, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$:

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.796 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

newform_orbits(550, 2)[5]   # 550.2.a.f

cargo run --release -p sagebrush-web --example atlas -- 550:2 1000