Atlas › Modular forms › Weight 2 › Level 249 › 249.2.a › 249.2.a.d

Newform orbit 249.2.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{4} - 20x^{3} + 128x^{2} - 304x + 224$

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
$\operatorname{tr} a_p$	2	4	6	0	4	−6	0	−2	8	0	8	−16	−8	−10	16	12	20	−12	−20	14	−22	6	−4	−4	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 3, 83, 249]: [0, 1, 8, 30] (certified at ell = 2147483323)
T = [(2, 1), (5, 3)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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, 2)[3]   # 249.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 249:2 1000