Atlas › Modular forms › Weight 10 › Level 2 › 2.10.a › 2.10.a.a

Newform orbit 2.10.a.a

$q$ -expansion

$q + 16q^{2} - 156q^{3} + 256q^{4} + 870q^{5} - 2496q^{6} - 952q^{7} + 4096q^{8} + 4653q^{9} + 13920q^{10} - 56148q^{11} - 39936q^{12} + 178094q^{13} - 15232q^{14} - 135720q^{15} + 65536q^{16} - 247662q^{17} + 74448q^{18} + 315380q^{19} + 222720q^{20} + 148512q^{21} - 898368q^{22} + 204504q^{23} - 638976q^{24} + O(q^{25})$

Coefficient field

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

$x - 714$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$

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
$\operatorname{tr} a_p$	16	−156	870	−952	−56148	178094	−247662	315380	204504	−3840450	−1309408	4307078	1512042	33670604	−10581072	16616214	112235100	−33197218	−121372252	−387172728	255240074	492101840	−457420236	−31809510	−673532062

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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]: [1, 3] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 1 = 1 x 1 (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(2, 10)[0]   # 2.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 2:10 1000