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

Newform orbit 959.2.a.e

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + 4q^{2} + 3q^{3} + 42q^{4} - 3q^{5} + 6q^{6} + 26q^{7} + 12q^{8} + 43q^{9} + 3q^{10} + 10q^{11} + 3q^{12} - 6q^{13} + 4q^{14} + 15q^{15} + 62q^{16} + 11q^{17} + 13q^{18} + 4q^{19} - 3q^{20} + 3q^{21} + 16q^{22} - 7q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 7x^{25} - 89x^{24} + 689x^{23} + 3072x^{22} - 27945x^{21} - 50042x^{20} + 606737x^{19} + 333527x^{18} - 7714351x^{17} + 675991x^{16} + 59702004x^{15} - 23399959x^{14} - 287591753x^{13} + 140549782x^{12} + 874695699x^{11} - 404927025x^{10} - 1677628214x^{9} + 640484108x^{8} + 1998594054x^{7} - 559786313x^{6} - 1419016552x^{5} + 239038013x^{4} + 543485091x^{3} - 23491387x^{2} - 85123544x - 9413891$

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 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
$\operatorname{tr} a_p$	4	3	−3	26	10	−6	11	4	−7	−7	14	59	13	29	−18	−1	−10	47	37	−9	13	67	−18	24	26

$\operatorname{tr} a_n$ 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)[4]   # 959.2.a.e

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

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