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

Newform orbit 943.2.a.d

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 6q^{2} + 3q^{3} + 20q^{4} + 14q^{5} + 2q^{6} + 3q^{7} + 24q^{8} + 36q^{9} + 11q^{10} + 9q^{11} + 5q^{12} + 3q^{13} - q^{14} + 12q^{15} + 14q^{16} + 24q^{17} - 10q^{18} + 10q^{19} - q^{20} + 4q^{21} + 13q^{22} - 19q^{23} - 34q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 9x^{18} - 37x^{17} + 531x^{16} - 128x^{15} - 10731x^{14} + 17487x^{13} + 91134x^{12} - 215276x^{11} - 322498x^{10} + 981673x^{9} + 397383x^{8} - 1744971x^{7} - 196916x^{6} + 1204106x^{5} + 153175x^{4} - 289094x^{3} - 73538x^{2} + 4815x + 1597$

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 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
$\operatorname{tr} a_p$	6	3	14	3	9	3	24	10	−19	11	−27	10	19	12	8	49	−17	7	28	19	−4	9	30	27	57

$\operatorname{tr} a_n$ 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)[3]   # 943.2.a.d

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

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