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

Newform orbit 943.2.a.e

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 9q^{2} + 3q^{3} + 31q^{4} + 12q^{5} + 4q^{6} - 3q^{7} + 27q^{8} + 37q^{9} - 5q^{10} + 11q^{11} + 3q^{12} - 5q^{13} - 3q^{14} + 8q^{15} + 41q^{16} + 26q^{17} + 37q^{18} - 4q^{19} + 37q^{20} + 4q^{21} - q^{22} + 24q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 12x^{23} - 26x^{22} + 788x^{21} - 1030x^{20} - 20411x^{19} + 53328x^{18} + 269443x^{17} - 938833x^{16} - 1999161x^{15} + 8637467x^{14} + 9014073x^{13} - 46745018x^{12} - 27391005x^{11} + 154506755x^{10} + 65750525x^{9} - 306155081x^{8} - 133506119x^{7} + 330323615x^{6} + 186667767x^{5} - 141875132x^{4} - 121940356x^{3} - 9903763x^{2} + 10565556x + 2179933$

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
$\operatorname{tr} a_p$	9	3	12	−3	11	−5	26	−4	24	13	13	12	−24	14	36	37	27	−19	4	−5	30	−57	−6	75	−21

$\operatorname{tr} a_n$ 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)[4]   # 943.2.a.e

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

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