Atlas › Modular forms › Weight 2 › Level 908 › 908.2.a › 908.2.a.b

Newform orbit 908.2.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 5q^{3} - q^{5} - 6q^{7} + 6q^{9} - 5q^{11} + 5q^{13} - 6q^{15} + 7q^{17} - 4q^{19} + 3q^{21} - 8q^{23} + O(q^{25})$

Coefficient field

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

$x^{5} + 6x^{4} + 7x^{3} - 5x^{2} - 8x - 2$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 908$ )
227	$-1$

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	−1	−6	−5	5	7	−4	−8	−11	−28	0	−16	−26	−8	3	1	5	−12	−6	−19	−33	−1	−2	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.344 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 227, 454, 908]: [0, 1, 2, 20, 59, 117] (certified at ell = 2147479573)
T = [(3, 1), (5, 1)]; h of degree 19 = 1 x 19 (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(908, 2)[1]   # 908.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 908:2 1000