Atlas › Modular forms › Weight 4 › Level 70 › 70.4.a › 70.4.a.a

Newform orbit 70.4.a.a

$q$ -expansion

$q - 2q^{2} - 8q^{3} + 4q^{4} - 5q^{5} + 16q^{6} - 7q^{7} - 8q^{8} + 37q^{9} + 10q^{10} + 68q^{11} - 32q^{12} + 34q^{13} + 14q^{14} + 40q^{15} + 16q^{16} + 74q^{17} - 74q^{18} - 128q^{19} - 20q^{20} + 56q^{21} - 136q^{22} - 80q^{23} + 64q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{11}$ is a root of

$x - 196$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$+1$
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 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
$\operatorname{tr} a_p$	−2	−8	−5	−7	68	34	74	−128	−80	286	−24	294	66	−124	312	−34	168	170	564	616	250	−944	672	−1430	−1270

$\operatorname{tr} a_n$ 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 70, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.115 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, 2, 5, 7, 10, 14, 35, 70]: [1, 2, 3, 3, 7, 8, 14, 40] (certified at ell = 2147483629)
T = [(3, 1), (11, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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(70, 4)[0]   # 70.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 70:4 1000