Atlas › Modular forms › Weight 2 › Level 283 › 283.2.a › 283.2.a.b

Newform orbit 283.2.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 10x^{13} + x^{12} + 273x^{11} - 641x^{10} - 2202x^{9} + 8767x^{8} + 2226x^{7} - 39214x^{6} + 31066x^{5} + 47094x^{4} - 66957x^{3} + 3026x^{2} + 25656x - 8968$

Atkin–Lehner signs

$p$	$w_p$
283	$-1$

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	4	14	0	5	9	13	−7	22	33	−13	16	6	−12	−8	73	−4	0	−14	6	−16	−9	−7	−1	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.168 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, 283]: [0, 24] (certified at ell = 2147480197)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 3 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(283, 2)[1]   # 283.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 283:2 1000