Atlas › Modular forms › Weight 2 › Level 526 › 526.2.a › 526.2.a.e

Newform orbit 526.2.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 8x^{5} + 9x^{4} + 69x^{3} - 213x^{2} + 189x - 31$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
263	$+1$

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
$\operatorname{tr} a_p$	6	3	5	5	4	0	1	6	5	5	−1	−6	5	−3	13	−10	10	−5	−5	−1	−14	−14	15	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 0.298 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, 2, 263, 526]: [0, 1, 23, 68] (certified at ell = 2147482739)
T = [(3, 1), (5, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 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(526, 2)[4]   # 526.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 526:2 1000