Atlas › Modular forms › Weight 4 › Level 150 › 150.4.a › 150.4.a.f

Newform orbit 150.4.a.f

$q$ -expansion

$q + 2q^{2} - 3q^{3} + 4q^{4} - 6q^{6} - 2q^{7} + 8q^{8} + 9q^{9} + 70q^{11} - 12q^{12} + 54q^{13} - 4q^{14} + 16q^{16} - 22q^{17} + 18q^{18} + 24q^{19} + 6q^{21} + 140q^{22} - 100q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + 3T_{11}$ is a root of

$x - 208$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
5	— ( $5^2 \mid 150$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	−3	0	−2	70	54	−22	24	−100	216	208	−254	−206	292	−320	−402	−370	−550	728	−540	604	792	404	−938	56

$\operatorname{tr} a_n$ 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 150, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150]: [1, 2, 2, 3, 5, 7, 8, 9, 22, 25, 32, 94] (certified at ell = 2147482921)
T = [(7, 1), (11, 3)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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(150, 4)[5]   # 150.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 150:4 1000