Atlas › Modular forms › Weight 6 › Level 21 › 21.6.a › 21.6.a.b

Newform orbit 21.6.a.b

$q$ -expansion

$q + q^{2} - 9q^{3} - 31q^{4} - 34q^{5} - 9q^{6} - 49q^{7} - 63q^{8} + 81q^{9} - 34q^{10} - 340q^{11} + 279q^{12} + 454q^{13} - 49q^{14} + 306q^{15} + 929q^{16} - 798q^{17} + 81q^{18} + 892q^{19} + 1054q^{20} + 441q^{21} - 340q^{22} - 3192q^{23} + 567q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 33$

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
$\operatorname{tr} a_p$	1	−9	−34	−49	−340	454	−798	892	−3192	−8242	−2496	9798	19834	−17236	8928	150	−42396	14758	−1676	14568	78378	−2272	−37764	−117286	10002

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 21, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 3, 5, 16] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 6)[1]   # 21.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 21:6 1000