Atlas › Modular forms › Weight 2 › Level 271 › 271.2.a › 271.2.a.b

Newform orbit 271.2.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 5q^{2} + q^{3} + 17q^{4} + 10q^{5} - 4q^{6} - 3q^{7} + 12q^{8} + 35q^{9} - q^{10} + 17q^{11} - 8q^{12} - 4q^{13} + 10q^{14} + 3q^{15} + 19q^{16} + 12q^{17} - 3q^{18} - 3q^{19} + 12q^{20} + 9q^{21} - 3q^{22} + 5q^{23} - 30q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 6x^{15} - 44x^{14} + 322x^{13} + 488x^{12} - 6008x^{11} + 1457x^{10} + 48747x^{9} - 48149x^{8} - 184577x^{7} + 260329x^{6} + 303900x^{5} - 543667x^{4} - 152811x^{3} + 409517x^{2} + 12955x - 102211$

Atkin–Lehner signs

$p$	$w_p$
271	$-1$

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
$\operatorname{tr} a_p$	5	1	10	−3	17	−4	12	−3	5	21	−10	2	52	−15	2	21	15	−14	−12	31	2	−4	−2	92	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.169 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, 271]: [0, 23] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(271, 2)[1]   # 271.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 271:2 1000