Atlas › Modular forms › Weight 2 › Level 755 › 755.2.a › 755.2.a.j

Newform orbit 755.2.a.j

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 2q^{2} + 5q^{3} + 18q^{4} + 15q^{5} - 4q^{6} + 11q^{7} + 12q^{8} + 10q^{9} - 2q^{10} + 2q^{11} + 4q^{12} + 11q^{13} - 9q^{14} + 5q^{15} + 40q^{16} + 25q^{17} - 15q^{18} - 3q^{19} + 18q^{20} - 11q^{21} + 13q^{22} + 8q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{15} - 13x^{14} - 175x^{13} + 2703x^{12} + 9447x^{11} - 205392x^{10} - 152495x^{9} + 7473756x^{8} - 1184684x^{7} - 139849744x^{6} + 23791312x^{5} + 1313783744x^{4} + 523747008x^{3} - 4834572288x^{2} - 6878601216x - 2675624960$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
151	$+1$

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	5	15	11	2	11	25	−3	8	24	−11	35	−20	−13	6	38	−15	33	24	19	4	−4	19	−35	55

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 5, 151, 755]: [0, 1, 13, 78] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 7 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(755, 2)[9]   # 755.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 755:2 1000