Atlas › Modular forms › Weight 2 › Level 420 › 420.2.a › 420.2.a.d

Newform orbit 420.2.a.d

$q$ -expansion

$q + q^{3} + q^{5} - q^{7} + q^{9} + 2q^{11} + 4q^{13} + q^{15} + 2q^{17} - 2q^{19} - q^{21} + 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{11} + 3T_{13} + 5T_{17} + 7T_{19} + 9T_{23} + 11T_{29}$ is a root of

$x - 24$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 420$ )
3	$-1$
5	$-1$
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 420.d (Cremona's 420d). Its optimal curve 420d1 is

$y^2 = x^3 + x^2 - 5x$

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
$\operatorname{tr} a_p$	0	1	1	−1	2	4	2	−2	4	−2	−6	−6	6	−4	0	8	0	−10	−12	−14	4	−8	12	−14	8

$\operatorname{tr} a_n$ 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 420, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420]: [0, 1, 1, 2, 1, 3, 1, 3, 5, 4, 4, 6, 4, 7, 10, 6, 12, 18, 16, 22, 20, 30, 56, 108] (certified at ell = 2147483629)
T = [(11, 1), (13, 3), (17, 5), (19, 7), (23, 9), (29, 11)]; h of degree 4 = 1 x 4 (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(420, 2)[3]   # 420.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 420:2 1000