Atlas › Modular forms › Weight 4 › Level 75 › 75.4.a › 75.4.a.a

Newform orbit 75.4.a.a

$q$ -expansion

$q - 3q^{2} + 3q^{3} + q^{4} - 9q^{6} - 20q^{7} + 21q^{8} + 9q^{9} - 24q^{11} + 3q^{12} - 74q^{13} + 60q^{14} - 71q^{16} - 54q^{17} - 27q^{18} - 124q^{19} - 60q^{21} + 72q^{22} + 120q^{23} + 63q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{7}$ is a root of

$x + 63$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	— ( $5^2 \mid 75$ )

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
$\operatorname{tr} a_p$	−3	3	0	−20	−24	−74	−54	−124	120	−78	200	70	330	−92	24	−450	24	−322	196	−288	430	−520	−156	1026	286

$\operatorname{tr} a_n$ 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 75, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 5, 15, 25, 75]: [1, 2, 3, 8, 9, 32] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 10 = 1 x 10 (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(75, 4)[0]   # 75.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 75:4 1000