Atlas › Modular forms › Weight 4 › Level 154 › 154.4.a › 154.4.a.a

Newform orbit 154.4.a.a

$q$ -expansion

$q - 2q^{2} - 5q^{3} + 4q^{4} - q^{5} + 10q^{6} - 7q^{7} - 8q^{8} - 2q^{9} + 2q^{10} - 11q^{11} - 20q^{12} - 8q^{13} + 14q^{14} + 5q^{15} + 16q^{16} + 22q^{17} + 4q^{18} + 54q^{19} - 4q^{20} + 35q^{21} + 22q^{22} + 213q^{23} + 40q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ is a root of

$x - 592$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$+1$
11	$+1$

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
$\operatorname{tr} a_p$	−2	−5	−1	−7	−11	−8	22	54	213	190	163	31	110	4	−80	−566	645	634	−729	431	−918	−254	904	901	−89

$\operatorname{tr} a_n$ 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 154, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.213 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, 7, 11, 14, 22, 77, 154]: [1, 2, 3, 4, 8, 11, 26, 76] (certified at ell = 2147482951)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 14 = 1 x 14 (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(154, 4)[0]   # 154.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 154:4 1000