Atlas › Modular forms › Weight 10 › Level 14 › 14.10.a › 14.10.a.a

Newform orbit 14.10.a.a

$q$ -expansion

$q - 16q^{2} - 6q^{3} + 256q^{4} + 560q^{5} + 96q^{6} - 2401q^{7} - 4096q^{8} - 19647q^{9} - 8960q^{10} - 54152q^{11} - 1536q^{12} - 113172q^{13} + 38416q^{14} - 3360q^{15} + 65536q^{16} + 6262q^{17} + 314352q^{18} + 257078q^{19} + 143360q^{20} + 14406q^{21} + 866432q^{22} - 266000q^{23} + 24576q^{24} + O(q^{25})$

Coefficient field

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

$x - 1674$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$+1$

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	−6	560	−2401	−54152	−113172	6262	257078	−266000	1574714	−4637484	−11946238	21909126	27520592	52927836	16221222	−140509618	−202963560	153734572	279655936	−404022830	−130689816	420134014	−469542390	−872501690

$\operatorname{tr} a_n$ 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 14, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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, 7, 14]: [1, 3, 7, 20] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 4 = 1 x 4 (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(14, 10)[0]   # 14.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 14:10 1000