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

Newform orbit 771.2.a.d

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{13} - 16x^{12} + 68x^{11} + 169x^{10} - 1989x^{9} + 3502x^{8} + 9306x^{7} - 39222x^{6} + 32856x^{5} + 38237x^{4} - 78164x^{3} + 35313x^{2} + 4538x - 4658$

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 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	13	11	−2	4	7	22	2	21	21	5	2	15	−12	15	20	6	32	−23	0	−12	−13	31	30	−9

$\operatorname{tr} a_n$ 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)[3]   # 771.2.a.d

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

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