Atlas › Modular forms › Weight 2 › Level 247 › 247.2.a › 247.2.a.e

Newform orbit 247.2.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 4q^{2} + 3q^{3} + 6q^{4} + 3q^{5} - q^{7} + 12q^{8} + 2q^{9} + 8q^{10} - 2q^{11} - 14q^{12} - 5q^{13} - 5q^{14} + 4q^{15} + 8q^{16} + 14q^{17} + q^{18} + 5q^{19} + 7q^{20} + 10q^{21} - 11q^{22} - 6q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 7x^{4} + 8x^{3} + 34x^{2} - 84x + 49$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$
19	$-1$

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
$\operatorname{tr} a_p$	4	3	3	−1	−2	−5	14	5	−6	−9	−3	6	21	−13	15	5	14	−10	5	11	35	6	2	11	24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 13, 19, 247]: [0, 1, 2, 24] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (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(247, 2)[4]   # 247.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 247:2 1000