Atlas › Modular forms › Weight 6 › Level 79 › 79.6.a › 79.6.a.a

Newform orbit 79.6.a.a

Trace form

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

$14q - 8q^{2} - 37q^{3} + 192q^{4} - 201q^{5} - 159q^{6} - 157q^{7} - 474q^{8} + 213q^{9} - 419q^{10} - 566q^{11} - 2153q^{12} - 651q^{13} - 2811q^{14} - 1633q^{15} + 1668q^{16} - 4864q^{17} - 1067q^{18} - 2736q^{19} - 14249q^{20} - 6089q^{21} - 19094q^{22} - 6182q^{23} - 23643q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 45x^{13} - 956x^{12} - 55232x^{11} + 320017x^{10} + 24302749x^{9} - 69043350x^{8} - 4743555482x^{7} + 13321554304x^{6} + 411504328944x^{5} - 1521265929888x^{4} - 12781417798336x^{3} + 63423242231488x^{2} - 5920808092928x - 188859596097536$

Atkin–Lehner signs

$p$	$w_p$
79	$+1$

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
$\operatorname{tr} a_p$	−8	−37	−201	−157	−566	−651	−4864	−2736	−6182	−20034	−7238	3944	−53860	16800	−17691	−4126	−37693	−83846	4252	−128821	29398	−87374	−223066	−493717	−272397

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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 engine's checks

dim M^+ at levels [1, 79]: [1, 35] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 6 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(79, 6)[0]   # 79.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 79:6 1000