Newform orbit 800.2.a.f

$q$-expansion

$q - 3q^{9} + 4q^{13} + 8q^{17} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$. The eigenvalue of $T = 3990T_{3} + 2855T_{7} + 5427T_{11} + 9524T_{13} + 7538T_{17} + 6229T_{19} + 7256T_{23} + 6862T_{29}$ is a root of

$x - 167020$

Complex multiplication by $\mathbb{Q}(\sqrt{-4})$: $a_p = 0$ for every conjugate at the 6 primes $p \le 31$ inert in it (exactly, from $\operatorname{tr} a_{p^2} = -\dim \cdot p^{k-1}$), and $\operatorname{tr} a_p = 0$ at all 87 inert primes $p < 1000$. This is evidence from the stored coefficients, not a proof.

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 800.f (Cremona's 800d). Its optimal curve 800d1 is

$y^2 = x^3 - 125x$

of rank 0 with torsion of order 2. 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

With CM the normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 800, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ (CM forms follow a different law):

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.312 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(800, 2)[5]   # 800.2.a.f

cargo run --release -p sagebrush-web --example atlas -- 800:2 1000