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

Newform orbit 789.2.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} - 2x^{6} - 12x^{5} + 18x^{4} + 34x^{3} - 20x^{2} - 3x + 1$

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 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	−7	1	−5	−14	−8	11	−13	−7	−7	−23	−7	8	−27	0	4	−16	−21	−35	−11	−4	−19	6	17	−7

$\operatorname{tr} a_n$ 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)[1]   # 789.2.a.b

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

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