Atlas › Modular forms › Weight 2 › Level 771 › 771.2.a › 771.2.a.c

Newform orbit 771.2.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} + 6x^{7} + x^{6} - 30x^{5} + 4x^{4} + 45x^{3} - 38x^{2} + 11x - 1$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
257	$+1$

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
$\operatorname{tr} a_p$	1	−8	−7	−6	0	−15	−2	−16	17	−17	−17	−22	−23	−18	23	6	−14	−46	−1	19	2	−11	26	−24	−23

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 3, 257, 771]: [0, 1, 22, 88] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 5 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(771, 2)[2]   # 771.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 771:2 1000