Atlas › Modular forms › Weight 2 › Level 762 › 762.2.a › 762.2.a.l

Newform orbit 762.2.a.l

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ , whose minimal polynomial is

$x^{5} - 573x^{4} + 99463x^{3} - 4615075x^{2} - 256797844x + 19702848524$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
127	$+1$

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	−5	2	5	4	3	7	6	5	9	8	21	−5	11	−2	10	−17	1	−11	−26	18	8	−31	−24	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.108 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, 2, 3, 6, 127, 254, 381, 762]: [0, 1, 1, 3, 11, 34, 44, 132] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 7 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(762, 2)[11]   # 762.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 762:2 1000