Atlas › Modular forms › Weight 2 › Level 306 › 306.2.a › 306.2.a.a

Newform orbit 306.2.a.a

$q$ -expansion

$q - q^{2} + q^{4} - 4q^{7} - q^{8} - 6q^{11} + 2q^{13} + 4q^{14} + q^{16} + q^{17} - 4q^{19} + 6q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ is a root of

$x + 78$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 306$ )
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 306.a (Cremona's 306b). Its optimal curve 306b1 is

$y^2 + xy = x^3 - x^2 - 27x - 27$

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

$\operatorname{tr} a_n$ 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 306, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.415 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, 6, 9, 17, 18, 34, 51, 102, 153, 306]: [0, 1, 1, 3, 2, 2, 5, 6, 8, 22, 20, 58] (certified at ell = 2147483137)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; 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(306, 2)[0]   # 306.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 306:2 1000