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

Newform orbit 249.4.a.a

Trace form

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

$9q - 7q^{2} + 27q^{3} + 19q^{4} - 47q^{5} - 21q^{6} - 43q^{7} - 69q^{8} + 81q^{9} - 3q^{10} - 114q^{11} + 57q^{12} - 95q^{13} - 233q^{14} - 141q^{15} - 17q^{16} - 79q^{17} - 63q^{18} - 295q^{19} - 445q^{20} - 129q^{21} + 182q^{22} - 537q^{23} - 207q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} + 54x^{8} + 777x^{7} - 3971x^{6} - 177770x^{5} - 1265583x^{4} - 1436876x^{3} + 12288840x^{2} + 34631760x + 24074784$

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 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
$\operatorname{tr} a_p$	−7	27	−47	−43	−114	−95	−79	−295	−537	−174	−359	−28	−727	−455	−1304	−2033	−2040	−416	47	−2184	−300	−830	−747	−1936	440

$\operatorname{tr} a_n$ 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)[0]   # 249.4.a.a

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

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