Atlas › Modular forms › Weight 6 › Level 7 › 7.6.a › 7.6.a.b

Newform orbit 7.6.a.b

Trace form

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

$2q + 9q^{2} - 6q^{3} + 5q^{4} - 18q^{5} - 198q^{6} + 98q^{7} - 9q^{8} + 558q^{9} + 204q^{10} + 396q^{11} - 1554q^{12} - 350q^{13} + 441q^{14} - 1656q^{15} + 113q^{16} + 1800q^{17} + 3537q^{18} - 3266q^{19} + 2520q^{20} - 294q^{21} - 1752q^{22} + 2088q^{23} - 1854q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{57})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 3x - 354$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	9	−6	−18	98	396	−350	1800	−3266	2088	6696	−20	6232	−6048	−3020	11700	9468	−43938	−64754	24784	97416	17452	51256	117558	84276	20776

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 7]: [1, 5] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(7, 6)[1]   # 7.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 7:6 1000