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

Newform orbit 75.6.a.i

Trace form

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

$2q + 6q^{2} + 18q^{3} + 16q^{4} + 54q^{6} + 102q^{7} + 228q^{8} + 162q^{9} - 12q^{11} + 144q^{12} + 1054q^{13} + 1794q^{14} - 200q^{16} - 1716q^{17} + 486q^{18} + 4214q^{19} + 918q^{21} - 5492q^{22} - 444q^{23} + 2052q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{31})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - 108x - 16459$

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
$\operatorname{tr} a_p$	6	18	0	102	−12	1054	−1716	4214	−444	4068	−2598	4412	11232	−8450	2460	−65064	−63924	7310	61734	98304	−26564	84000	−65772	103104	69374

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.272 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)[8]   # 75.6.a.i

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

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