Atlas › Modular forms › Weight 2 › Level 719 › 719.2.a › 719.2.a.b

Newform orbit 719.2.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 6q^{3} + 2q^{4} - 5q^{5} + q^{6} - 10q^{7} - 6q^{8} - 6q^{9} - 2q^{10} + 2q^{11} + 2q^{12} - 2q^{13} + 6q^{15} - 6q^{16} - 5q^{17} - 9q^{18} - 8q^{19} - 20q^{20} - 9q^{21} - 26q^{22} + 4q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 6x^{9} - 6x^{8} - 73x^{7} - 5x^{6} + 279x^{5} + 14x^{4} - 399x^{3} + 45x^{2} + 152x - 13$

Atkin–Lehner signs

$p$	$w_p$
719	$+1$

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
$\operatorname{tr} a_p$	0	−6	−5	−10	2	−2	−5	−8	4	−20	−14	−4	−7	−13	−10	16	18	−54	−5	−28	−26	−19	−12	−8	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 719]: [0, 61] (certified at ell = 2147481997)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (phi(ord eps) x dim S_k^new); 6 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(719, 2)[1]   # 719.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 719:2 1000