Atlas › Modular forms › Weight 2 › Level 79 › 79.2.a › 79.2.a.b

Newform orbit 79.2.a.b

Trace form

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

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

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

$x^{5} - x^{4} - 17x^{3} + 45x^{2} - 35x + 8$

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
$\operatorname{tr} a_p$	0	1	7	−5	2	−3	10	−4	2	6	2	0	30	−14	5	2	5	−6	−16	3	−12	5	−30	47	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 79]: [0, 7] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 1 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, 2)[1]   # 79.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 79:2 1000