Atlas › Modular forms › Weight 4 › Level 249 › 249.4.a › 249.4.a.c

Newform orbit 249.4.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q - 3q^{2} - 36q^{3} + 59q^{4} - 11q^{5} + 9q^{6} + 41q^{7} - 51q^{8} + 108q^{9} + 9q^{10} + 56q^{11} - 177q^{12} + 61q^{13} + 5q^{14} + 33q^{15} + 351q^{16} - 13q^{17} - 27q^{18} + 187q^{19} - 233q^{20} - 123q^{21} + 502q^{22} + 51q^{23} + 153q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{12} + 14x^{11} - 1002x^{10} - 13239x^{9} + 339931x^{8} + 3950150x^{7} - 47252928x^{6} - 384977251x^{5} + 3168919168x^{4} + 9185779304x^{3} - 80181311120x^{2} + 98584386576x + 20397960000$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
83	$+1$

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
$\operatorname{tr} a_p$	−3	−36	−11	41	56	61	−13	187	51	78	1077	500	87	255	632	−841	798	56	−379	852	550	1318	−996	−1020	−714

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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 engine's checks

dim M^+ at levels [1, 3, 83, 249]: [1, 2, 22, 86] (certified at ell = 2147483323)
T = [(2, 1), (5, 1)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 7 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(249, 4)[2]   # 249.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 249:4 1000