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

Newform orbit 789.2.a.c

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 11x^{13} + 5x^{12} + 323x^{11} - 997x^{10} - 1569x^{9} + 9770x^{8} - 5917x^{7} - 17230x^{6} + 13064x^{5} + 11876x^{4} - 3640x^{3} - 1312x^{2} + 364x - 13$

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
$\operatorname{tr} a_p$	4	14	7	1	20	−6	19	7	21	15	5	−9	24	1	12	14	26	−15	1	17	−30	−1	32	15	−33

$\operatorname{tr} a_n$ 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)[2]   # 789.2.a.c

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

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