Atlas › Modular forms › Weight 12 › Level 14 › 14.12.a › 14.12.a.a

Newform orbit 14.12.a.a

$q$ -expansion

$q - 32q^{2} - 396q^{3} + 1024q^{4} + 7350q^{5} + 12672q^{6} + 16807q^{7} - 32768q^{8} - 20331q^{9} - 235200q^{10} - 108780q^{11} - 405504q^{12} - 635842q^{13} - 537824q^{14} - 2910600q^{15} + 1048576q^{16} - 9225918q^{17} + 650592q^{18} - 7555372q^{19} + 7526400q^{20} - 6655572q^{21} + 3480960q^{22} + 26489400q^{23} + 12976128q^{24} + O(q^{25})$

Coefficient field

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

$x - 6954$

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
$\operatorname{tr} a_p$	−32	−396	7350	16807	−108780	−635842	−9225918	−7555372	26489400	−169827594	−51362704	−251605906	−928817814	−1818895756	523343136	4199520078	9140129196	−6639312802	−2878139188	−4345596360	23450332826	−28761853648	−5577757548	78002173386	−26685859630

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

Sato–Tate

The normalized $a_p / 2p^{11/2}$ 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 0.341 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]: [2, 4, 9, 24] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 3 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, 12)[0]   # 14.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 14:12 1000