Atlas › Modular forms › Weight 4 › Level 207 › 207.4.a › 207.4.a.a

Newform orbit 207.4.a.a

$q$ -expansion

$q + 2q^{2} - 4q^{4} + 6q^{5} - 8q^{7} - 24q^{8} + 12q^{10} - 34q^{11} - 57q^{13} - 16q^{14} - 16q^{16} + 80q^{17} - 70q^{19} - 24q^{20} - 68q^{22} - 23q^{23} + O(q^{25})$

Coefficient field

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

$x - 20$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 207$ )
23	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	0	6	−8	−34	−57	80	−70	−23	−245	103	−298	−95	88	357	414	408	822	926	−335	−899	−1322	36	460	−964

$\operatorname{tr} a_n$ 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 207, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.15 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, 9, 23, 69, 207]: [1, 2, 4, 7, 26, 74] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 6 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(207, 4)[0]   # 207.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 207:4 1000