Atlas › Modular forms › Weight 2 › Level 759 › 759.2.a › 759.2.a.i

Newform orbit 759.2.a.i

Trace form

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

$8q + 2q^{2} - 8q^{3} + 14q^{4} + 6q^{5} - 2q^{6} - 6q^{7} + 12q^{8} + 8q^{9} + 8q^{11} - 14q^{12} - 2q^{13} + 8q^{14} - 6q^{15} + 30q^{16} + 10q^{17} + 2q^{18} - 14q^{19} + 22q^{20} + 6q^{21} + 2q^{22} - 8q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{8} - 20x^{7} - 121x^{6} + 4168x^{5} - 9032x^{4} - 211978x^{3} + 1118781x^{2} + 33110x - 5755613$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
11	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	2	−8	6	−6	8	−2	10	−14	−8	10	16	−10	28	−26	12	12	12	−24	−12	−4	4	−8	−6	26	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.388 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, 3, 11, 23, 33, 69, 253, 759]: [0, 1, 2, 3, 6, 10, 26, 100] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 39 = 1 x 39 (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(759, 2)[8]   # 759.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 759:2 1000