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

Newform orbit 75.4.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 3q^{2} + 6q^{3} + 9q^{4} + 9q^{6} + 6q^{7} + 51q^{8} + 18q^{9} - 42q^{11} + 27q^{12} + 78q^{13} - 114q^{14} + 25q^{16} + 102q^{17} + 27q^{18} + 56q^{19} + 18q^{21} - 186q^{22} - 48q^{23} + 153q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 21x - 2852$

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	6	0	6	−42	78	102	56	−48	−318	52	306	−408	120	180	402	−186	340	732	−36	1332	380	−984	1116	−768

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

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)[5]   # 75.4.a.f

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

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