Atlas › Modular forms › Weight 4 › Level 21 › 21.4.a › 21.4.a.a

Newform orbit 21.4.a.a

$q$ -expansion

$q - 3q^{2} - 3q^{3} + q^{4} - 18q^{5} + 9q^{6} + 7q^{7} + 21q^{8} + 9q^{9} + 54q^{10} - 36q^{11} - 3q^{12} - 34q^{13} - 21q^{14} + 54q^{15} - 71q^{16} + 42q^{17} - 27q^{18} - 124q^{19} - 18q^{20} - 21q^{21} + 108q^{22} - 63q^{24} + O(q^{25})$

Coefficient field

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

$x + 21$

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
$\operatorname{tr} a_p$	−3	−3	−18	7	−36	−34	42	−124	0	102	−160	398	−318	−268	240	−498	−132	398	92	−720	−502	−1024	−204	354	−286

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.096 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, 2, 3, 10] (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, 4)[0]   # 21.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 21:4 1000