Atlas › Modular forms › Weight 2 › Level 979 › 979.2.a › 979.2.a.e

Newform orbit 979.2.a.e

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 4q^{2} - q^{3} + 18q^{4} + 11q^{5} + 10q^{6} + 3q^{7} + 6q^{8} + 27q^{9} + 6q^{10} - 18q^{11} + 13q^{12} + 13q^{13} + 15q^{14} - 17q^{15} + 6q^{16} + 15q^{17} + 24q^{18} + q^{19} + 29q^{20} + 17q^{21} - 4q^{22} + 13q^{23} + 28q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{18} - 77x^{17} - 3502x^{16} + 384027x^{15} + 2103612x^{14} - 738718801x^{13} + 5626739810x^{12} + 697234937024x^{11} - 10007674775261x^{10} - 334130490862850x^{9} + 6448000183200078x^{8} + 72467654071175696x^{7} - 1881207461876597119x^{6} - 3527278427567334568x^{5} + 226847101713296352301x^{4} - 591894352049510353527x^{3} - 8257209133352728803133x^{2} + 49447806279951105927828x - 76063429295826416917139$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
89	$-1$

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	−1	11	3	−18	13	15	1	13	53	−32	10	17	20	−4	66	3	34	−39	38	9	−3	21	18	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.366 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, 11, 89, 979]: [0, 2, 8, 92] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 16 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(979, 2)[4]   # 979.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 979:2 1000