Atlas › Modular forms › Weight 2 › Level 962 › 962.2.a › 962.2.a.j

Newform orbit 962.2.a.j

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{17}$ , whose minimal polynomial is

$x^{7} - 177x^{6} + 8620x^{5} + 116348x^{4} - 19978002x^{3} + 457713727x^{2} - 522754196x - 53735943172$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
13	$+1$
37	$+1$

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
$\operatorname{tr} a_p$	7	6	4	8	8	−7	7	−9	−3	17	−26	−7	−6	12	−13	12	4	−9	10	−17	27	−3	−3	11	−21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.477 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, 2, 13, 26, 37, 74, 481, 962]: [0, 1, 1, 5, 3, 11, 44, 136] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (17, 9)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 9 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(962, 2)[9]   # 962.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 962:2 1000