Atlas › Modular forms › Weight 2 › Level 946 › 946.2.a › 946.2.a.j

Newform orbit 946.2.a.j

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 6q^{2} + 2q^{3} + 6q^{4} + 4q^{5} - 2q^{6} - 6q^{8} + 12q^{9} - 4q^{10} + 6q^{11} + 2q^{12} - 4q^{13} + 10q^{15} + 6q^{16} + 8q^{17} - 12q^{18} - 4q^{19} + 4q^{20} - 6q^{21} - 6q^{22} + 16q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} - 14x^{5} - 118x^{4} + 1596x^{3} + 3483x^{2} - 26894x - 37706$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
11	$-1$
43	$+1$

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
$\operatorname{tr} a_p$	−6	2	4	0	6	−4	8	−4	16	−4	22	−12	10	−6	18	12	18	−12	4	24	−34	10	2	26	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.444 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 11, 22, 43, 86, 473, 946]: [0, 1, 2, 5, 4, 13, 46, 136] (certified at ell = 2147482681)
T = [(3, 1), (5, 3)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 5 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(946, 2)[9]   # 946.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 946:2 1000