Atlas › Modular forms › Weight 2 › Level 921 › 921.2.a › 921.2.a.f

Newform orbit 921.2.a.f

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 4q^{2} - 11q^{3} + 10q^{4} + 7q^{5} - 4q^{6} + 2q^{7} + 9q^{8} + 11q^{9} + q^{10} + 7q^{11} - 10q^{12} - 8q^{13} + 16q^{14} - 7q^{15} + 16q^{16} + 5q^{17} + 4q^{18} + 10q^{19} + 16q^{20} - 2q^{21} + 8q^{22} + 18q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{11} - 11x^{10} + 11x^{9} + 226x^{8} - 666x^{7} - 898x^{6} + 4972x^{5} - 1329x^{4} - 11302x^{3} + 8166x^{2} + 7187x - 6008$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
307	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	4	−11	7	2	7	−8	5	10	18	2	38	−36	23	−19	47	22	48	−13	−6	48	−7	19	15	−2	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.275 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, 3, 307, 921]: [0, 1, 26, 104] (certified at ell = 2147478013)
T = [(2, 1), (5, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 6 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(921, 2)[5]   # 921.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 921:2 1000