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

Newform orbit 10.6.a.a

$q$ -expansion

$q - 4q^{2} - 26q^{3} + 16q^{4} - 25q^{5} + 104q^{6} - 22q^{7} - 64q^{8} + 433q^{9} + 100q^{10} - 768q^{11} - 416q^{12} - 46q^{13} + 88q^{14} + 650q^{15} + 256q^{16} + 378q^{17} - 1732q^{18} + 1100q^{19} - 400q^{20} + 572q^{21} + 3072q^{22} - 1986q^{23} + 1664q^{24} + O(q^{25})$

Coefficient field

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

$x + 48$

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 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	−26	−25	−22	−768	−46	378	1100	−1986	−5610	−3988	−142	1542	−5026	24738	−14166	28380	5522	−24742	42372	−52126	−39640	−59826	57690	−144382

$\operatorname{tr} a_n$ 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)[0]   # 10.6.a.a

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

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