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

Newform orbit 79.4.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 4q^{2} + 8q^{3} + 56q^{4} + 39q^{5} + 21q^{6} + 16q^{7} + 30q^{8} + 204q^{9} + 25q^{10} + 91q^{11} + 127q^{12} + 57q^{13} + 177q^{14} + 86q^{15} + 260q^{16} + 338q^{17} + 265q^{18} + 67q^{19} + 304q^{20} - 182q^{21} - 867q^{22} + q^{23} - 897q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 12x^{11} - 289x^{10} + 3358x^{9} + 28458x^{8} - 323239x^{7} - 1220466x^{6} + 13292403x^{5} + 23870122x^{4} - 231587208x^{3} - 158130872x^{2} + 1278743984x - 790227904$

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
$\operatorname{tr} a_p$	4	8	39	16	91	57	338	67	1	522	−471	−396	1388	−134	870	26	1904	1554	1107	890	1047	−948	−1294	3491	335

$\operatorname{tr} a_n$ 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)[2]   # 79.4.a.c

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

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