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

Newform orbit 946.2.a.k

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 7q^{2} + 3q^{3} + 7q^{4} + 8q^{5} + 3q^{6} - 2q^{7} + 7q^{8} + 10q^{9} + 8q^{10} + 7q^{11} + 3q^{12} + 4q^{13} - 2q^{14} + 2q^{15} + 7q^{16} + 10q^{17} + 10q^{18} - 2q^{19} + 8q^{20} + 7q^{22} - 4q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 27x^{6} + 136x^{5} + 2038x^{4} - 21357x^{3} + 9563x^{2} + 463920x - 1218506$

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
$\operatorname{tr} a_p$	7	3	8	−2	7	4	10	−2	−4	5	−2	6	4	7	−6	1	−4	9	−6	0	13	−31	−11	10	23

$\operatorname{tr} a_n$ 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)[10]   # 946.2.a.k

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

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