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

Newform orbit 207.4.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 2q^{2} + 26q^{4} + 2q^{5} + 32q^{7} + 18q^{8} + 150q^{10} - 62q^{11} + 32q^{13} - 180q^{14} + 258q^{16} + 28q^{17} + 42q^{19} + 490q^{20} - 322q^{22} + 92q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{4} - 8x^{3} - 4335x^{2} + 30400x + 1191450$

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	2	32	−62	32	28	42	92	20	444	162	528	1230	−36	−358	−892	294	−126	−1032	−1220	1832	−1178	872	−1324

$\operatorname{tr} a_n$ 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)[5]   # 207.4.a.f

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

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