Atlas › Modular forms › Weight 4 › Level 151 › 151.4.a › 151.4.a.b

Newform orbit 151.4.a.b

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + 6q^{2} + 8q^{3} + 100q^{4} + 59q^{5} + 48q^{6} + 30q^{7} + 51q^{8} + 330q^{9} - 11q^{10} + 125q^{11} + 21q^{12} + 116q^{13} + 200q^{14} + 116q^{15} + 504q^{16} + 391q^{17} + 205q^{18} + 141q^{19} + 699q^{20} + 270q^{21} + 84q^{22} + 204q^{23} + 517q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} - 14x^{21} - 550x^{20} + 7775x^{19} + 121884x^{18} - 1704187x^{17} - 14995995x^{16} + 198234727x^{15} + 1171912322x^{14} - 13622784462x^{13} - 62008140723x^{12} + 576247397266x^{11} + 2254637430557x^{10} - 15070026017873x^{9} - 55032035299736x^{8} + 237694354846122x^{7} + 857433194268028x^{6} - 2128562205760434x^{5} - 7903527758972180x^{4} + 9447245633001024x^{3} + 37727290344398272x^{2} - 14288741498091520x - 67372030867991488$

Atkin–Lehner signs

$p$	$w_p$
151	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	6	8	59	30	125	116	391	141	204	485	261	101	1746	−1043	1013	956	981	524	818	1494	−290	238	2180	6774	2101

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 151]: [1, 39] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 5 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(151, 4)[1]   # 151.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 151:4 1000