Atlas › Modular forms › Weight 4 › Level 127 › 127.4.a › 127.4.a.a

Newform orbit 127.4.a.a

$q$ -expansion

$q - q^{2} - 8q^{3} - 7q^{4} - 15q^{5} + 8q^{6} - 25q^{7} + 15q^{8} + 37q^{9} + 15q^{10} - 51q^{11} + 56q^{12} + 2q^{13} + 25q^{14} + 120q^{15} + 41q^{16} + 31q^{17} - 37q^{18} - 123q^{19} + 105q^{20} + 200q^{21} + 51q^{22} - 149q^{23} - 120q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 9$

Atkin–Lehner signs

$p$	$w_p$
127	$+1$

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
$\operatorname{tr} a_p$	−1	−8	−15	−25	−51	2	31	−123	−149	6	10	−348	−387	−80	266	347	−656	−158	−314	312	−646	−846	1352	1242	632

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 127, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 127]: [1, 33] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 5 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(127, 4)[0]   # 127.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 127:4 1000