Atlas › Modular forms › Weight 10 › Level 27 › 27.10.a › 27.10.a.b

Newform orbit 27.10.a.b

Trace form

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

$3q - 3q^{2} + 597q^{4} + 1983q^{5} - 3693q^{7} + 4503q^{8} - 18981q^{10} + 16863q^{11} + 116916q^{13} + 503463q^{14} - 239919q^{16} + 1014048q^{17} - 15222q^{19} + 2548407q^{20} + 305721q^{22} + 2927118q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} - 1980x^{2} - 2018439x + 3869606358$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 27$ )

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
$\operatorname{tr} a_p$	−3	0	1983	−3693	16863	116916	1014048	−15222	2927118	5768790	−6575223	−11686026	−22213518	45384414	12392034	80579637	244026660	369729960	−252614586	403193088	−406626717	265451856	121625871	377904006	−438907539

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.323 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, 3, 9, 27]: [1, 4, 10, 28] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(27, 10)[1]   # 27.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 27:10 1000