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

Newform orbit 503.2.a.e

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 12x^{9} + 48x^{8} + 59x^{7} - 44x^{6} - 91x^{5} + 28x^{4} + 41x^{3} - 16x^{2} - 2x + 1$

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 odd 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
$\operatorname{tr} a_p$	−4	−8	−1	−5	−3	−18	−11	0	−2	−9	−22	−35	−4	−20	7	−24	17	−4	−6	−1	−31	−10	22	1	−57

$\operatorname{tr} a_n$ 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)[4]   # 503.2.a.e

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

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