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

Newform orbit 789.2.a.a

Trace form

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

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

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

$x^{7} + 12x^{6} + 48x^{5} + 54x^{4} - 76x^{3} - 136x^{2} + 33x + 3$

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
$\operatorname{tr} a_p$	−3	7	−9	−3	−20	−4	−19	−3	−15	−13	−3	1	−10	−11	−20	−14	−20	7	1	−21	2	−7	0	−23	1

$\operatorname{tr} a_n$ 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)[0]   # 789.2.a.a

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

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