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

Newform orbit 271.2.a.a

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 5x^{5} + 2x^{4} - 10x^{3} - 4x^{2} + 4x + 1$

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 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
$\operatorname{tr} a_p$	−4	−1	−8	−3	−11	−2	−10	−5	−3	−17	−6	0	−18	7	−2	−5	−11	16	12	−11	6	4	10	−50	−2

$\operatorname{tr} a_n$ 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)[0]   # 271.2.a.a

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

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