Atlas › Modular forms › Weight 2 › Level 408 › 408.2.a › 408.2.a.c

Newform orbit 408.2.a.c

$q$ -expansion

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

Coefficient field

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

$x - 78$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 408$ )
3	$-1$
17	$+1$

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 408.c (Cremona's 408a). Its optimal curve 408a1 is

$y^2 = x^3 + x^2 - 48x - 144$

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	0	2	0	2	−1	4	2	0	6	0	−10	4	−4	−2	−4	0	4	−2	−14	6	−12	−2	−2

$\operatorname{tr} a_n$ 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 408, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.457 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, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408]: [0, 1, 1, 2, 3, 3, 5, 2, 8, 6, 8, 12, 22, 22, 42, 80] (certified at ell = 2147483489)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (19, 9), (23, 11)]; h of degree 8 = 1 x 8 (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(408, 2)[2]   # 408.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 408:2 1000