Atlas › Modular forms › Weight 10 › Level 28 › 28.10.a › 28.10.a.a

Newform orbit 28.10.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 224q^{3} + 1596q^{5} + 4802q^{7} + 22210q^{9} - 50592q^{11} - 184772q^{13} - 507144q^{15} - 152124q^{17} - 605024q^{19} - 537824q^{21} - 2503608q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{4561})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 1372x - 697020$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 28$ )
7	$-1$

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	−224	1596	4802	−50592	−184772	−152124	−605024	−2503608	−2727492	−6655040	−6860228	14154756	−1133840	35486640	−33311940	29419488	35951692	−170286704	−463615488	−100806860	−197493824	51427488	649287492	2781293620

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.317 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, 7, 14, 28]: [1, 3, 6, 7, 20, 39] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(28, 10)[0]   # 28.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 28:10 1000