Atlas › Modular forms › Weight 12 › Level 6 › 6.12.a › 6.12.a.a

Newform orbit 6.12.a.a

$q$ -expansion

$q - 32q^{2} - 243q^{3} + 1024q^{4} + 5766q^{5} + 7776q^{6} + 72464q^{7} - 32768q^{8} + 59049q^{9} - 184512q^{10} - 408948q^{11} - 248832q^{12} + 1367558q^{13} - 2318848q^{14} - 1401138q^{15} + 1048576q^{16} + 5422914q^{17} - 1889568q^{18} + 15166100q^{19} + 5904384q^{20} - 17608752q^{21} + 13086336q^{22} - 52194072q^{23} + 7962624q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{7}$ is a root of

$x - 78230$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$

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
$\operatorname{tr} a_p$	−32	−243	5766	72464	−408948	1367558	5422914	15166100	−52194072	118581150	−57652408	−375985186	856316202	−1245189172	−1306762656	409556358	−2882866260	5731767302	3893272244	−9075890088	−15571822822	−30196762600	23135252628	−25614819990	−61937553406

$\operatorname{tr} a_n$ 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 6, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 3, 6]: [2, 4, 5, 13] (certified at ell = 2147483647)
T = [(5, 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(6, 12)[0]   # 6.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 6:12 1000