Atlas › Modular forms › Weight 2 › Level 312 › 312.2.a › 312.2.a.f

Newform orbit 312.2.a.f

$q$ -expansion

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

Coefficient field

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

$x + 20$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 312$ )
3	$-1$
13	$-1$

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 312.f (Cremona's 312c). Its optimal curve 312c1 is

$y^2 = x^3 + x^2 - 7x + 2$

of rank 0 with torsion of order 4. 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	2	0	0	1	2	−4	0	6	0	−2	6	−12	−4	6	−8	−2	4	−12	−14	0	8	−18	−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 312, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.286 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, 13, 24, 26, 39, 52, 78, 104, 156, 312]: [0, 1, 1, 2, 3, 3, 5, 1, 8, 5, 6, 10, 18, 18, 34, 64] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (17, 7), (19, 9), (23, 11)]; h of degree 6 = 1 x 6 (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(312, 2)[5]   # 312.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 312:2 1000