Atlas › Modular forms › Weight 12 › Level 3 › 3.12.a › 3.12.a.a

Newform orbit 3.12.a.a

$q$ -expansion

$q + 78q^{2} - 243q^{3} + 4036q^{4} - 5370q^{5} - 18954q^{6} - 27760q^{7} + 155064q^{8} + 59049q^{9} - 418860q^{10} + 637836q^{11} - 980748q^{12} + 766214q^{13} - 2165280q^{14} + 1304910q^{15} + 3829264q^{16} + 3084354q^{17} + 4605822q^{18} - 19511404q^{19} - 21673320q^{20} + 6745680q^{21} + 49751208q^{22} + 15312360q^{23} - 37680552q^{24} + O(q^{25})$

Coefficient field

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

$x + 5292$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	78	−243	−5370	−27760	637836	766214	3084354	−19511404	15312360	10751262	−50937400	664740830	898833450	−957947188	−1555741344	3792417030	555306924	4950420998	5292399284	−14831086248	13971005210	3720542360	8768454036	−25472769174	−39092494846

$\operatorname{tr} a_n$ 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 3, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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]: [2, 5] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 1 = 1 x 1 (phi(ord eps) x dim S_k^new); 1 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(3, 12)[0]   # 3.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 3:12 1000