Atlas › Modular forms › Weight 2 › Level 303 › 303.2.a › 303.2.a.b

Newform orbit 303.2.a.b

$q$ -expansion

$q + q^{3} - 2q^{4} - 3q^{5} + q^{9} - 2q^{11} - 2q^{12} - 3q^{13} - 3q^{15} + 4q^{16} - 7q^{17} - 5q^{19} + 6q^{20} - 5q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5}$ is a root of

$x + 9$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
101	$-1$

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 odd analytic rank.

Elliptic curve

By modularity this newform is the one attached to the isogeny class 303.b (Cremona's 303a). Its optimal curve 303a1 is

$y^2 + y = x^3 + x^2 - 197x - 208$

of rank 1 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

$p$	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
$\operatorname{tr} a_p$	0	1	−3	0	−2	−3	−7	−5	−5	6	7	10	6	4	−7	−4	−10	−2	10	−9	−8	7	2	−8	−10

$\operatorname{tr} a_n$ 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 303, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 3, 101, 303]: [0, 1, 9, 36] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 3 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(303, 2)[1]   # 303.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 303:2 1000