Atlas › Modular forms › Weight 6 › Level 75 › 75.6.a › 75.6.a.b

Newform orbit 75.6.a.b

$q$ -expansion

$q - 4q^{2} - 9q^{3} - 16q^{4} + 36q^{6} + 225q^{7} + 192q^{8} + 81q^{9} - 434q^{11} + 144q^{12} - 613q^{13} - 900q^{14} - 256q^{16} + 878q^{17} - 324q^{18} - 731q^{19} - 2025q^{21} + 1736q^{22} - 2850q^{23} - 1728q^{24} + O(q^{25})$

Coefficient field

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

$x - 221$

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
$\operatorname{tr} a_p$	−4	−9	0	225	−434	−613	878	−731	−2850	−7582	2175	9310	−12040	1121	−29878	−5740	−5174	−38717	−31707	64472	19790	−105000	3318	−65376	89143

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.272 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, 3, 3, 12, 13, 52] (certified at ell = 2147482921)
T = [(2, 1), (7, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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, 6)[1]   # 75.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 75:6 1000