Atlas › Modular forms › Weight 2 › Level 527 › 527.2.a › 527.2.a.b

Newform orbit 527.2.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 5x^{4} - 6x^{3} - 35x^{2} - 3x + 29$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
31	$-1$

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
$\operatorname{tr} a_p$	−2	−3	−2	−4	−2	−15	5	1	−4	−10	5	−6	−4	−5	−8	10	3	−8	−34	21	−34	15	3	−25	−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 0.204 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, 17, 31, 527]: [0, 2, 3, 50] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 4 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(527, 2)[1]   # 527.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 527:2 1000