Atlas › Modular forms › Weight 4 › Level 152 › 152.4.a › 152.4.a.b

Newform orbit 152.4.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 5q^{3} + 2q^{5} - 35q^{7} + 48q^{9} - 28q^{11} - 109q^{13} - 228q^{15} - 123q^{17} + 57q^{19} + 25q^{21} - 193q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - x^{2} - 1909x + 29197$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 152$ )
19	$-1$

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	−5	2	−35	−28	−109	−123	57	−193	−297	−140	38	736	−514	134	311	199	56	−509	−874	203	242	−62	1764	−2178

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.128 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, 19, 38, 76, 152]: [1, 2, 3, 5, 6, 17, 33, 64] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 13 = 1 x 13 (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(152, 4)[1]   # 152.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 152:4 1000