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

Newform orbit 946.2.a.h

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{4} - 4x^{3} - 53x^{2} + 260x - 104$

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
$\operatorname{tr} a_p$	4	1	1	0	−4	10	5	5	9	7	1	7	19	−4	−5	22	−14	−4	−8	10	−10	5	4	0	13

$\operatorname{tr} a_n$ 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)[7]   # 946.2.a.h

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

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