Atlas › Modular forms › Weight 2 › Level 943 › 943.2.a › 943.2.a.b

Newform orbit 943.2.a.b

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} + 11x^{12} + 22x^{11} - 124x^{10} - 413x^{9} + 444x^{8} + 1943x^{7} - 1139x^{6} - 3526x^{5} + 2669x^{4} + 1725x^{3} - 2253x^{2} + 721x - 73$

Atkin–Lehner signs

$p$	$w_p$
23	$-1$
41	$-1$

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
$\operatorname{tr} a_p$	−6	−5	−6	−7	−5	−13	−26	−2	13	−21	−11	2	13	6	−40	−29	3	19	−2	−69	−20	−1	14	−33	−31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.283 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, 23, 41, 943]: [0, 3, 4, 86] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 8 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(943, 2)[1]   # 943.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 943:2 1000