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

Newform orbit 79.6.a.b

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 8q^{2} + 17q^{3} + 352q^{4} + 199q^{5} + 201q^{6} + 39q^{7} + 294q^{8} + 2238q^{9} + 381q^{10} + 1370q^{11} + 439q^{12} + 701q^{13} + 1893q^{14} + 167q^{15} + 6788q^{16} + 4962q^{17} + 2821q^{18} + 1596q^{19} + 11256q^{20} + 14607q^{21} + 22553q^{22} + 9674q^{23} + 32023q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 25x^{18} - 3796x^{17} + 87572x^{16} + 5911257x^{15} - 118699364x^{14} - 5086486233x^{13} + 81870294760x^{12} + 2692793282610x^{11} - 30831667632887x^{10} - 904085700208615x^{9} + 6058750661275016x^{8} + 187607222495222040x^{7} - 454950677654686656x^{6} - 21960253942720551712x^{5} - 21276571156911119904x^{4} + 1168489366510480801856x^{3} + 3730518629110530777472x^{2} - 15428410480253436636672x - 60090647579363652243456$

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 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
$\operatorname{tr} a_p$	8	17	199	39	1370	701	4962	1596	9674	11496	8502	−12080	42270	1674	15351	4306	28691	−57770	−106648	4263	83698	118579	398834	597611	204743

$\operatorname{tr} a_n$ 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)[1]   # 79.6.a.b

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

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