Atlas › Modular forms › Weight 10 › Level 21 › 21.10.a › 21.10.a.b

Newform orbit 21.10.a.b

Trace form

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

$2q + 9q^{2} - 162q^{3} + 193q^{4} + 1170q^{5} - 729q^{6} - 4802q^{7} + 6849q^{8} + 13122q^{9} - 77090q^{10} - 145746q^{11} - 15633q^{12} + 86528q^{13} - 21609q^{14} - 94770q^{15} - 509183q^{16} - 229842q^{17} + 59049q^{18} - 221224q^{19} - 628290q^{20} + 388962q^{21} - 1302932q^{22} - 2035782q^{23} - 554769q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2353})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 1179x - 2453148$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$+1$

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 odd 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
$\operatorname{tr} a_p$	9	−162	1170	−4802	−145746	86528	−229842	−221224	−2035782	9756252	204000	−13959816	−42362550	−4763912	−48278484	−108980352	−188376804	19722092	70274396	−382044186	191785896	−72592148	187994232	42930954	1726854096

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 7, 21]: [1, 4, 7, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 4 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(21, 10)[1]   # 21.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 21:10 1000