Atlas › Modular forms › Weight 2 › Level 789 › 789.2.a › 789.2.a.d

Newform orbit 789.2.a.d

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - q^{2} - 15q^{3} + 25q^{4} - q^{5} + q^{6} + 7q^{7} - 6q^{8} + 15q^{9} + 5q^{10} + 22q^{11} - 25q^{12} + 4q^{13} + q^{14} + q^{15} + 49q^{16} - 21q^{17} - q^{18} + 17q^{19} - q^{20} - 7q^{21} + 14q^{22} + 13q^{23} + 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{15} + 2x^{14} - 84x^{13} - 138x^{12} + 2574x^{11} + 2708x^{10} - 38401x^{9} - 16513x^{8} + 295363x^{7} - 32292x^{6} - 1101856x^{5} + 563092x^{4} + 1600662x^{3} - 1275396x^{2} - 201013x + 222995$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
263	$-1$

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
$\operatorname{tr} a_p$	−1	−15	−1	7	22	4	−21	17	13	11	17	21	−16	29	−16	−18	22	19	21	39	18	15	2	−39	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 263, 789]: [0, 1, 23, 90] (certified at ell = 2147482739)
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(789, 2)[3]   # 789.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 789:2 1000