Atlas › Modular forms › Weight 6 › Level 74 › 74.6.a › 74.6.a.a

Newform orbit 74.6.a.a

$q$ -expansion

$q + 4q^{2} - 15q^{3} + 16q^{4} + 4q^{5} - 60q^{6} + 15q^{7} + 64q^{8} - 18q^{9} + 16q^{10} - 389q^{11} - 240q^{12} - 588q^{13} + 60q^{14} - 60q^{15} + 256q^{16} - 1002q^{17} - 72q^{18} - 1354q^{19} + 64q^{20} - 225q^{21} - 1556q^{22} - 1290q^{23} - 960q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 11$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
37	$-1$

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
$\operatorname{tr} a_p$	4	−15	4	15	−389	−588	−1002	−1354	−1290	2214	4276	1369	−2121	1024	5947	3405	4024	15460	49332	28905	−65885	23406	−39095	−69758	−42924

$\operatorname{tr} a_n$ 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 74, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 37, 74]: [1, 2, 17, 49] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(74, 6)[0]   # 74.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 74:6 1000