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

Newform orbit 14.12.a.b

$q$ -expansion

$q + 32q^{2} - 90q^{3} + 1024q^{4} - 7480q^{5} - 2880q^{6} - 16807q^{7} + 32768q^{8} - 169047q^{9} - 239360q^{10} - 294536q^{11} - 92160q^{12} - 210588q^{13} - 537824q^{14} + 673200q^{15} + 1048576q^{16} - 6962906q^{17} - 5409504q^{18} - 9346390q^{19} - 7659520q^{20} + 1512630q^{21} - 9425152q^{22} + 51172000q^{23} - 2949120q^{24} + O(q^{25})$

Coefficient field

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

$x + 7570$

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	−90	−7480	−16807	−294536	−210588	−6962906	−9346390	51172000	166196354	119000988	−275545510	−197988378	−809489728	−2600196204	733631454	−4657126942	−5135837424	8810564836	−3849006656	−18686748254	−29850061992	−5875980446	83056539450	149400800374

$\operatorname{tr} a_n$ 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)[1]   # 14.12.a.b

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

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