Atlas › Modular forms › Weight 2 › Level 350 › 350.2.a › 350.2.a.f

Newform orbit 350.2.a.f

$q$ -expansion

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

Coefficient field

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

$x + 72$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	— ( $5^2 \mid 350$ )
7	$+1$

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 350.f (Cremona's 350d). Its optimal curve 350d1 is

$y^2 + xy + y = x^3 + x^2 - 13x + 31$

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	0	−1	0	4	−6	2	0	−6	−4	−2	6	−8	12	−6	−6	8	4	0	−2	8	6	−6	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 350, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.44 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, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350]: [0, 1, 1, 1, 3, 4, 3, 6, 9, 16, 22, 64] (certified at ell = 2147482921)
T = [(3, 1), (11, 2), (19, 3), (23, 4), (29, 5), (13, 6), (17, 7), (31, 8)]; h of degree 10 = 1 x 10 (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(350, 2)[5]   # 350.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 350:2 1000