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

Newform orbit 962.2.a.d

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{17}$ , whose minimal polynomial is

$x^{2} - 23x - 1587$

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
$\operatorname{tr} a_p$	2	1	−6	5	6	2	−3	8	8	−2	10	2	−6	12	−2	−4	15	−6	6	−6	−25	−2	−7	−14	−8

$\operatorname{tr} a_n$ 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)[3]   # 962.2.a.d

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

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