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

Newform orbit 908.2.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 6q^{3} + q^{5} + 15q^{7} + 18q^{9} + 6q^{11} - q^{13} + 14q^{15} + 9q^{17} + 9q^{19} + 3q^{21} + 15q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{12} - 7x^{11} - 59x^{10} + 485x^{9} + 839x^{8} - 10748x^{7} + 3720x^{6} + 77227x^{5} - 81226x^{4} - 79309x^{3} - 11160x^{2} + 2940x + 586$

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	6	1	15	6	−1	9	9	15	−6	18	−2	8	29	15	0	0	1	56	−1	12	16	−15	−9	30

$\operatorname{tr} a_n$ 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)[2]   # 908.2.a.c

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

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