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

Newform orbit 79.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - q^{2} - 2q^{3} - 7q^{4} + q^{6} + 8q^{7} + 3q^{8} - 52q^{9} - 51q^{10} - 90q^{11} + 7q^{12} - 32q^{13} + 47q^{14} - 39q^{16} - 28q^{17} + 26q^{18} + 68q^{19} + 51q^{20} - 8q^{21} - 40q^{22} - 300q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 3x - 2$

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
$\operatorname{tr} a_p$	−1	−2	0	8	−90	−32	−28	68	−300	144	344	−474	6	380	−476	164	−474	−1338	406	−1092	696	158	1370	−1374	−146

$\operatorname{tr} a_n$ 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)[0]   # 79.4.a.a

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

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