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

Newform orbit 755.2.a.i

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 9x^{4} - 34x^{3} - 344x^{2} + 155x + 2000$

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 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	−3	−5	−7	−5	−4	11	−3	6	−20	−7	−27	−14	3	18	−2	−16	−25	15	−5	−6	−16	4	−7	−13

$\operatorname{tr} a_n$ 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)[8]   # 755.2.a.i

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

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