Atlas › Modular forms › Weight 4 › Level 236 › 236.4.a › 236.4.a.b

Newform orbit 236.4.a.b

Trace form

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

$4q + 10q^{5} + 35q^{7} + 10q^{9} - 57q^{11} + 115q^{13} + 210q^{15} - 115q^{17} + 10q^{19} + 104q^{21} + 240q^{23} + O(q^{25})$

Coefficient field

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

$x^{4} - 10x^{3} - 544x^{2} + 6720x - 9216$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 236$ )
59	$-1$

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
$\operatorname{tr} a_p$	0	0	10	35	−57	115	−115	10	240	100	692	695	157	185	150	−20	236	708	1120	321	−1640	−635	−385	−500	−1880

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 2, 4, 59, 118, 236]: [1, 2, 3, 16, 47, 93] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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(236, 4)[1]   # 236.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 236:4 1000