Atlas › Modular forms › Weight 4 › Level 247 › 247.4.a › 247.4.a.a

Newform orbit 247.4.a.a

$q$ -expansion

$q - 3q^{2} - 5q^{3} + q^{4} + 7q^{5} + 15q^{6} - 27q^{7} + 21q^{8} - 2q^{9} - 21q^{10} + 60q^{11} - 5q^{12} + 13q^{13} + 81q^{14} - 35q^{15} - 71q^{16} + 21q^{17} + 6q^{18} - 19q^{19} + 7q^{20} + 135q^{21} - 180q^{22} + 202q^{23} - 105q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ is a root of

$x - 368$

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 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	−5	7	−27	60	13	21	−19	202	36	−96	−155	−254	−405	−567	−382	746	−206	−448	457	−312	688	44	1352	−1674

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 247, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.254 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, 13, 19, 247]: [1, 5, 6, 72] (certified at ell = 2147483629)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 18 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, 4)[0]   # 247.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 247:4 1000