Atlas › Modular forms › Weight 2 › Level 603 › 603.2.a › 603.2.a.b

Newform orbit 603.2.a.b

$q$ -expansion

$q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - 4q^{14} - q^{16} + 4q^{19} - 2q^{20} + 4q^{22} - 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ is a root of

$x + 11$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 603$ )
67	$-1$

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 603.b (Cremona's 603b). Its optimal curve 603b1 is

$y^2 + xy + y = x^3 - x^2 - 29x + 28$

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
$\operatorname{tr} a_p$	−1	0	2	4	−4	2	0	4	−6	4	8	2	6	4	2	10	−6	−2	1	6	−10	−8	−2	16	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 603, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.352 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, 3, 9, 67, 201, 603]: [0, 1, 2, 6, 24, 70] (certified at ell = 2147483647)
T = [(2, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 28 = 1 x 28 (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(603, 2)[1]   # 603.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 603:2 1000