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

Newform orbit 249.4.a.b

Trace form

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

$9q + 3q^{2} - 27q^{3} + 23q^{4} - q^{5} - 9q^{6} - 43q^{7} + 21q^{8} + 81q^{9} - 71q^{10} - 54q^{11} - 69q^{12} - 43q^{13} - 79q^{14} + 3q^{15} + 15q^{16} - 47q^{17} + 27q^{18} - 193q^{19} - 313q^{20} + 129q^{21} - 284q^{22} - 87q^{23} - 63q^{24} + O(q^{25})$

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

$x^{9} - 2x^{8} - 497x^{7} - 241x^{6} + 55194x^{5} + 102681x^{4} - 1584424x^{3} - 4638440x^{2} + 1193968x + 121632$

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
$\operatorname{tr} a_p$	3	−27	−1	−43	−54	−43	−47	−193	−87	−180	−827	−496	−323	−661	−784	993	−596	−1760	−219	−116	−452	−420	747	−1376	−1436

$\operatorname{tr} a_n$ 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)[1]   # 249.4.a.b

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

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