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

Newform orbit 207.4.a.c

Trace form

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

$2q + 4q^{2} + 2q^{4} + 26q^{5} - 10q^{7} + 12q^{8} + 42q^{10} + 60q^{11} - 24q^{13} + 50q^{14} + 18q^{16} + 150q^{17} - 46q^{19} - 14q^{20} + 220q^{22} + 46q^{23} + O(q^{25})$

Coefficient field

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

$x^{2} - 82x + 1661$

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
$\operatorname{tr} a_p$	4	0	26	−10	60	−24	150	−46	46	216	324	140	364	−126	−120	−490	32	−908	−874	−488	652	−1198	100	−102	1624

$\operatorname{tr} a_n$ 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)[2]   # 207.4.a.c

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

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