Atlas › Modular forms › Weight 2 › Level 448 › 448.2.a › 448.2.a.d

Newform orbit 448.2.a.d

$q$ -expansion

$q - 2q^{5} - q^{7} - 3q^{9} + 4q^{11} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 8661T_{3} + 7879T_{5} + 8882T_{11} + 1822T_{13} + 7435T_{17} + 9474T_{19} + 6076T_{23} + 2656T_{29}$ is a root of

$x + 120212$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 448$ )
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 448.d (Cremona's 448a). Its optimal curve 448a1 is

$y^2 = x^3 + 4x + 16$

of rank 1 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	0	−2	−1	4	−2	−6	−8	0	−6	8	2	2	4	−8	−6	0	6	4	−8	10	16	−8	−6	−6

$\operatorname{tr} a_n$ 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 448, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.704 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, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448]: [0, 1, 2, 1, 3, 4, 4, 7, 6, 12, 10, 20, 36, 68] (certified at ell = 2147483137)
T = [(3, 8661), (5, 7879), (11, 8882), (13, 1822), (17, 7435), (19, 9474), (23, 6076), (29, 2656)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 8 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(448, 2)[3]   # 448.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 448:2 1000