Atlas › Modular forms › Weight 4 › Level 136 › 136.4.a › 136.4.a.d

Newform orbit 136.4.a.d

Trace form

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

$4q - 2q^{3} + 8q^{5} - 22q^{7} + 100q^{9} + 70q^{11} + 120q^{13} + 140q^{15} - 68q^{17} - 44q^{19} + 488q^{21} + 158q^{23} + O(q^{25})$

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

$x^{4} - 6x^{3} - 750x^{2} + 7048x + 32520$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 136$ )
17	$+1$

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	−2	8	−22	70	120	−68	−44	158	264	122	256	240	−1100	−800	432	−148	−728	−1032	798	−1544	758	244	1440	−1344

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.121 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, 8, 17, 34, 68, 136]: [1, 2, 3, 5, 6, 16, 30, 58] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(136, 4)[3]   # 136.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 136:4 1000