Atlas › Modular forms › Weight 2 › Level 538 › 538.2.a › 538.2.a.d

Newform orbit 538.2.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{7} + 22x^{6} + 48x^{5} - 1842x^{4} - 11679x^{3} + 14023x^{2} + 202484x + 189675$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
269	$+1$

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
$\operatorname{tr} a_p$	−7	−4	−6	−3	−12	3	−8	−7	−22	−7	−3	−13	−11	−14	−23	−21	−20	13	−30	−23	9	11	−3	16	21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.205 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, 269, 538]: [0, 1, 23, 69] (certified at ell = 2147482661)
T = [(3, 1), (5, 3)]; h of degree 22 = 1 x 22 (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(538, 2)[3]   # 538.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 538:2 1000