Atlas › Modular forms › Weight 2 › Level 503 › 503.2.a › 503.2.a.d

Newform orbit 503.2.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{3} + 4q^{4} + q^{6} + q^{7} - 9q^{8} + 11q^{11} + 9q^{12} + 5q^{13} - 9q^{14} + 2q^{16} + 12q^{17} + 10q^{18} + 12q^{19} + 8q^{21} - q^{22} - 12q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{3} - x^{2} - 10x - 9$

Atkin–Lehner signs

$p$	$w_p$
503	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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	0	1	11	5	12	12	−12	−2	−10	2	2	5	19	14	−5	6	35	12	35	−7	−1	4	−23

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 503]: [0, 43] (certified at ell = 2147476183)
T = [(2, 1), (3, 1)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 5 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(503, 2)[3]   # 503.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 503:2 1000