Atlas › Modular forms › Weight 2 › Level 319 › 319.2.a › 319.2.a.e

Newform orbit 319.2.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 4q^{3} + 10q^{4} + 10q^{5} + 14q^{6} - 7q^{7} + 3q^{8} + 14q^{9} - 9q^{10} - 8q^{11} - 9q^{12} - 4q^{13} - 2q^{14} + 10q^{15} + 14q^{16} + 12q^{17} - 25q^{18} - 10q^{19} + 7q^{20} + 10q^{21} + 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} - 12x^{7} - 154x^{6} + 2372x^{5} + 722x^{4} - 105053x^{3} + 301902x^{2} + 648040x - 2564168$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
29	$-1$

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	4	10	−7	−8	−4	12	−10	0	8	5	9	29	−11	5	−10	24	−31	11	8	−11	−45	−22	66	28

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.183 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, 11, 29, 319]: [0, 2, 3, 32] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (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(319, 2)[4]   # 319.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 319:2 1000