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

Newform orbit 21.12.a.a

$q$ -expansion

$q - 62q^{2} + 243q^{3} + 1796q^{4} - 3310q^{5} - 15066q^{6} - 16807q^{7} + 15624q^{8} + 59049q^{9} + 205220q^{10} + 628904q^{11} + 436428q^{12} + 176854q^{13} + 1042034q^{14} - 804330q^{15} - 4646896q^{16} + 566958q^{17} - 3661038q^{18} - 12916124q^{19} - 5944760q^{20} - 4084101q^{21} - 38992048q^{22} - 25664100q^{23} + 3796632q^{24} + O(q^{25})$

Coefficient field

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

$x + 3372$

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
$\operatorname{tr} a_p$	−62	243	−3310	−16807	628904	176854	566958	−12916124	−25664100	−47411458	13942680	−641657298	−600859298	−1417753612	860414040	3221420478	−6082959012	−864141122	11897667268	−14077803900	−18814150398	17021361416	47613135564	61562070254	−166479510534

$\operatorname{tr} a_n$ 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)[0]   # 21.12.a.a

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

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