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

Newform orbit 75.6.a.h

Trace form

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

$2q + q^{2} + 18q^{3} + 141q^{4} + 9q^{6} + 112q^{7} + 243q^{8} + 162q^{9} + 248q^{11} + 1269q^{12} - 876q^{13} - 3216q^{14} + 3585q^{16} - 2036q^{17} + 81q^{18} + 1464q^{19} + 1008q^{21} + 6668q^{22} + 3216q^{23} + 2187q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{409})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - 113x - 19814$

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
$\operatorname{tr} a_p$	1	18	0	112	248	−876	−2036	1464	3216	1948	2672	−8668	−7628	16440	19360	14356	−904	20220	12904	−40976	−59124	107600	122088	103764	24764

$\operatorname{tr} a_n$ 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)[7]   # 75.6.a.h

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

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