Atlas › Modular forms › Weight 6 › Level 10 › 10.6.a › 10.6.a.b

Newform orbit 10.6.a.b

$q$ -expansion

$q - 4q^{2} + 24q^{3} + 16q^{4} + 25q^{5} - 96q^{6} - 172q^{7} - 64q^{8} + 333q^{9} - 100q^{10} + 132q^{11} + 384q^{12} - 946q^{13} + 688q^{14} + 600q^{15} + 256q^{16} - 222q^{17} - 1332q^{18} + 500q^{19} + 400q^{20} - 4128q^{21} - 528q^{22} + 3564q^{23} - 1536q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{7}$ is a root of

$x + 148$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$-1$

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
$\operatorname{tr} a_p$	−4	24	25	−172	132	−946	−222	500	3564	2190	2312	−11242	1242	20624	6588	−21066	7980	16622	1808	−24528	20474	−46240	−51576	−110310	−78382

$\operatorname{tr} a_n$ 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 10, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.105 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, 10]: [1, 2, 3, 9] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 3 = 1 x 3 (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(10, 6)[1]   # 10.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 10:6 1000