Atlas › Modular forms › Weight 10 › Level 10 › 10.10.a › 10.10.a.b

Newform orbit 10.10.a.b

$q$ -expansion

$q - 16q^{2} + 46q^{3} + 256q^{4} - 625q^{5} - 736q^{6} - 10318q^{7} - 4096q^{8} - 17567q^{9} + 10000q^{10} - 5568q^{11} + 11776q^{12} + 45986q^{13} + 165088q^{14} - 28750q^{15} + 65536q^{16} - 381318q^{17} + 281072q^{18} + 610460q^{19} - 160000q^{20} - 474628q^{21} + 89088q^{22} - 1447914q^{23} - 188416q^{24} + O(q^{25})$

Coefficient field

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

$x + 10272$

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
$\operatorname{tr} a_p$	−16	46	−625	−10318	−5568	45986	−381318	610460	−1447914	5385510	3053852	12889442	−33786618	−36886234	−44163798	29746266	−65575380	40183202	−115706158	−231681708	358691906	−486017080	251168886	−526039110	−1075981438

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{9/2}$ 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 0.285 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, 3, 5, 15] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 3 = 1 x 3 (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(10, 10)[1]   # 10.10.a.b

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

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