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

Newform orbit 207.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 2q^{2} + 2q^{4} - 8q^{5} - 28q^{7} + 18q^{8} - 72q^{10} + 72q^{11} - 60q^{13} - 44q^{14} - 78q^{16} - 96q^{17} - 148q^{19} - 136q^{20} + 184q^{22} - 46q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 22x - 847$

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	−8	−28	72	−60	−96	−148	−46	204	−168	116	−4	−420	48	−32	40	−764	−988	224	820	1772	−1480	−744	−260

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.15 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)[1]   # 207.4.a.b

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

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