Atlas › Modular forms › Weight 2 › Level 206 › 206.2.a › 206.2.a.a

Newform orbit 206.2.a.a

$q$ -expansion

$q - q^{2} + 2q^{3} + q^{4} + 4q^{5} - 2q^{6} - q^{8} + q^{9} - 4q^{10} - 6q^{11} + 2q^{12} - 2q^{13} + 8q^{15} + q^{16} + 2q^{17} - q^{18} - 4q^{19} + 4q^{20} + 6q^{22} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5}$ is a root of

$x - 14$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
103	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Elliptic curve

By modularity this newform is the one attached to the isogeny class 206.a (Cremona's 206a). Its optimal curve 206a1 is

$y^2 + xy = x^3 + x^2 + 2x$

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

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−1	2	4	0	−6	−2	2	−4	0	−6	8	8	2	2	−8	−12	12	10	−2	0	10	0	−4	2	14

$\operatorname{tr} a_n$ 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 206, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.091 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.

The engine's checks

dim M^+ at levels [1, 2, 103, 206]: [0, 1, 9, 28] (certified at ell = 2147482501)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(206, 2)[0]   # 206.2.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 206:2 1000