Atlas › Modular forms › Weight 2 › Level 971 › 971.2.a › 971.2.a.a

Newform orbit 971.2.a.a

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q - 4q^{2} - 6q^{3} + 8q^{4} - 6q^{5} - 7q^{6} - 12q^{7} - 9q^{8} + 2q^{9} - 12q^{10} - 14q^{11} - 6q^{12} - 34q^{13} - 5q^{14} - 12q^{15} - 20q^{16} - 11q^{17} - 8q^{18} - 22q^{19} - 8q^{20} - 35q^{21} - 13q^{22} + q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} + 10x^{25} - 13x^{24} - 401x^{23} - 469x^{22} + 6540x^{21} + 13101x^{20} - 58595x^{19} - 141229x^{18} + 326346x^{17} + 838628x^{16} - 1210783x^{15} - 2996269x^{14} + 3123471x^{13} + 6539308x^{12} - 5686530x^{11} - 8443126x^{10} + 7085586x^{9} + 5859193x^{8} - 5502018x^{7} - 1627542x^{6} + 2226749x^{5} - 109029x^{4} - 321213x^{3} + 69112x^{2} + 5755x - 1825$

Atkin–Lehner signs

$p$	$w_p$
971	$+1$

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	−6	−6	−12	−14	−34	−11	−22	1	−42	−16	−46	−19	−36	−10	−21	−12	−100	1	−1	−17	−52	18	11	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 971]: [0, 82] (certified at ell = 2147479121)
T = [(2, 1), (3, 1)]; h of degree 81 = 1 x 81 (phi(ord eps) x dim S_k^new); 8 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(971, 2)[0]   # 971.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 971:2 1000