Atlas › Modular forms › Weight 2 › Level 959 › 959.2.a › 959.2.a.c

Newform orbit 959.2.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 2x^{9} - 17x^{8} - 17x^{7} + 110x^{6} - 10x^{5} - 247x^{4} + 258x^{3} - 102x^{2} + 17x - 1$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
137	$+1$

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
$\operatorname{tr} a_p$	1	−3	−3	−10	−6	−6	−11	−12	5	−9	−16	−13	−17	3	0	−3	−4	−35	3	−7	−15	−1	4	−12	−56

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 7, 137, 959]: [0, 1, 12, 94] (certified at ell = 2147477809)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (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(959, 2)[2]   # 959.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 959:2 1000