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

Newform orbit 249.4.a.d

Trace form

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

$12q + 7q^{2} + 36q^{3} + 55q^{4} + 43q^{5} + 21q^{6} + 41q^{7} + 99q^{8} + 108q^{9} + 77q^{10} + 128q^{11} + 165q^{12} + 9q^{13} + 19q^{14} + 129q^{15} + 319q^{16} + 227q^{17} + 63q^{18} + 85q^{19} + 435q^{20} + 123q^{21} - 252q^{22} + 245q^{23} + 297q^{24} + O(q^{25})$

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

$x^{12} - 50x^{11} + 200x^{10} + 23519x^{9} - 288403x^{8} - 2866856x^{7} + 53090580x^{6} + 22306625x^{5} - 2730712876x^{4} + 8047983704x^{3} + 9816117824x^{2} - 25381033840x - 16369604160$

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
$\operatorname{tr} a_p$	7	36	43	41	128	9	227	85	245	380	713	−136	995	−339	1416	1073	1286	696	−753	1952	382	668	996	276	−3582

$\operatorname{tr} a_n$ 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)[3]   # 249.4.a.d

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

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