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

Newform orbit 538.2.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{4} - 18x^{3} + 45x^{2} + 256x - 16$

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 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	3	5	−1	7	4	4	2	16	0	−1	−2	−1	9	13	28	19	−20	15	15	−7	−17	6	20	3

$\operatorname{tr} a_n$ 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)[2]   # 538.2.a.c

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

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