Atlas › Modular forms › Weight 4 › Level 71 › 71.4.a › 71.4.a.a

Newform orbit 71.4.a.a

$q$ -expansion

$q + q^{2} + q^{3} - 7q^{4} - 16q^{5} + q^{6} - q^{7} - 15q^{8} - 26q^{9} - 16q^{10} + 24q^{11} - 7q^{12} + 7q^{13} - q^{14} - 16q^{15} + 41q^{16} + 72q^{17} - 26q^{18} - 153q^{19} + 112q^{20} - q^{21} + 24q^{22} - 213q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x - 2$

Atkin–Lehner signs

$p$	$w_p$
71	$-1$

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	1	−16	−1	24	7	72	−153	−213	232	149	−204	−432	71	273	−274	126	−134	−760	71	−457	112	−124	837	−1424

$\operatorname{tr} a_n$ 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 71, 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, 71]: [1, 19] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 3 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(71, 4)[0]   # 71.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 71:4 1000