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

Newform orbit 6.12.a.c

$q$ -expansion

$q + 32q^{2} + 243q^{3} + 1024q^{4} + 3630q^{5} + 7776q^{6} + 32936q^{7} + 32768q^{8} + 59049q^{9} + 116160q^{10} - 758748q^{11} + 248832q^{12} - 2482858q^{13} + 1053952q^{14} + 882090q^{15} + 1048576q^{16} + 8290386q^{17} + 1889568q^{18} - 10867300q^{19} + 3717120q^{20} + 8003448q^{21} - 24279936q^{22} + 20539272q^{23} + 7962624q^{24} + O(q^{25})$

Coefficient field

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

$x - 36566$

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	3630	32936	−758748	−2482858	8290386	−10867300	20539272	28814550	150501392	−319891714	−368008998	620469572	2763110256	−268284258	1672894740	−7787197498	18706694156	−8346990888	19641746522	−5873807200	8492558172	75527864010	−82356782494

$\operatorname{tr} a_n$ 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)[2]   # 6.12.a.c

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

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