Atlas › Modular forms › Weight 4 › Level 79 › 79.4.a › 79.4.a.b

Newform orbit 79.4.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 3q^{2} - 8q^{3} + 23q^{4} - 41q^{5} - 40q^{6} - 20q^{7} - 69q^{8} + 31q^{9} - 4q^{10} + 5q^{11} - 96q^{12} - 15q^{13} - 206q^{14} - 34q^{15} - 21q^{16} - 212q^{17} + 23q^{18} - 229q^{19} - 252q^{20} - 46q^{21} + 422q^{22} + 65q^{23} + 256q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 11x^{4} - 14x^{3} - 204x^{2} - 32x + 704$

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
$\operatorname{tr} a_p$	−3	−8	−41	−20	5	−15	−212	−229	65	−366	−283	810	−1032	−368	290	−1102	104	1128	281	−1014	−2133	395	−60	−1315	605

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.099 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, 21] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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, 4)[1]   # 79.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 79:4 1000