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

Newform orbit 503.2.a.f

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + 4q^{2} + 4q^{3} + 36q^{4} + 9q^{5} - 4q^{6} + 11q^{7} + 18q^{8} + 42q^{9} + 4q^{10} - 17q^{11} + 19q^{12} + 14q^{13} + q^{14} + 18q^{15} + 48q^{16} + 17q^{17} - 10q^{18} - 22q^{19} - 19q^{20} - 16q^{21} + 38q^{22} + 27q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 8x^{25} - 68x^{24} + 656x^{23} + 1624x^{22} - 22208x^{21} - 12260x^{20} + 410522x^{19} - 147661x^{18} - 4604083x^{17} + 4062371x^{16} + 32724437x^{15} - 40208191x^{14} - 148387452x^{13} + 222867363x^{12} + 416913685x^{11} - 743947956x^{10} - 665018660x^{9} + 1479426047x^{8} + 460903078x^{7} - 1634532440x^{6} + 54910880x^{5} + 871942090x^{4} - 163245078x^{3} - 195350329x^{2} + 36711663x + 14483700$

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
$\operatorname{tr} a_p$	4	4	9	11	−17	14	17	−22	27	13	26	55	24	20	−25	30	−26	15	−20	−35	38	21	−48	−5	142

$\operatorname{tr} a_n$ 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)[5]   # 503.2.a.f

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

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