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

Newform orbit 283.2.a.a

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 12x^{8} + 44x^{7} + 29x^{6} - 119x^{5} - 209x^{4} - 78x^{3} + 29x^{2} + 14x - 1$

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 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	−6	−14	−2	−5	−9	−13	3	−24	−33	−1	−24	0	−2	−8	−55	10	14	12	−3	−5	11	2	1	−4

$\operatorname{tr} a_n$ 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)[0]   # 283.2.a.a

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

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