Atlas › Modular forms › Weight 12 › Level 21 › 21.12.a › 21.12.a.b

Newform orbit 21.12.a.b

$q$ -expansion

$q + 8q^{2} + 243q^{3} - 1984q^{4} + 4390q^{5} + 1944q^{6} - 16807q^{7} - 32256q^{8} + 59049q^{9} + 35120q^{10} - 804836q^{11} - 482112q^{12} + 358294q^{13} - 134456q^{14} + 1066770q^{15} + 3805184q^{16} - 5657862q^{17} + 472392q^{18} - 14602004q^{19} - 8709760q^{20} - 4084101q^{21} - 6438688q^{22} - 36724800q^{23} - 7838208q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x - 4398$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
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
$\operatorname{tr} a_p$	8	243	4390	−16807	−804836	358294	−5657862	−14602004	−36724800	51126982	−208102080	652145982	951188402	858607748	−1336554720	1497595998	7067944068	−7643926442	−5086757252	2801411040	−7844280438	−21156661264	−10894949316	70788775714	82223797746

$\operatorname{tr} a_n$ 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 21, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.534 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, 3, 7, 21]: [2, 5, 9, 32] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 6 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(21, 12)[1]   # 21.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 21:12 1000