Atlas › Modular forms › Weight 2 › Level 612 › 612.2.a › 612.2.a.d

Newform orbit 612.2.a.d

$q$ -expansion

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

Coefficient field

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

$x - 9$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 612$ )
3	— ( $3^2 \mid 612$ )
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 612.d (Cremona's 612a). Its optimal curve 612a1 is

$y^2 = x^3 - 456x + 3748$

of rank 0 with torsion of order 3. 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	3	2	3	−1	−1	−7	3	6	−4	2	9	−1	−12	12	−6	2	−4	0	8	14	−12	−6	−16

$\operatorname{tr} a_n$ 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 612, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.412 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, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204, 306, 612]: [0, 1, 1, 2, 3, 2, 5, 2, 5, 6, 9, 8, 12, 22, 20, 42, 58, 114] (certified at ell = 2147483137)
T = [(5, 1), (7, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 1 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(612, 2)[3]   # 612.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 612:2 1000