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

Newform orbit 946.2.a.i

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 5q^{2} + q^{3} + 5q^{4} - 2q^{5} - q^{6} - 4q^{7} - 5q^{8} + 2q^{10} - 5q^{11} + q^{12} + 4q^{13} + 4q^{14} - 10q^{15} + 5q^{16} - 10q^{17} - 6q^{19} - 2q^{20} - 6q^{21} + 5q^{22} - 6q^{23} - q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 5x^{4} - 100x^{3} - 308x^{2} + 1815x + 1331$

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 odd 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
$\operatorname{tr} a_p$	−5	1	−2	−4	−5	4	−10	−6	−6	−9	−4	−4	−22	−5	−4	7	−14	−5	8	−22	−7	−31	−3	−6	31

$\operatorname{tr} a_n$ 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)[8]   # 946.2.a.i

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

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