Atlas › Modular forms › Weight 2 › Level 957 › 957.2.a › 957.2.a.f

Newform orbit 957.2.a.f

Trace form

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

$4q + 2q^{2} - 4q^{3} + 6q^{4} + 4q^{5} - 2q^{6} + 4q^{7} + 9q^{8} + 4q^{9} + 5q^{10} + 4q^{11} - 6q^{12} + q^{13} + 5q^{14} - 4q^{15} + 2q^{16} + 5q^{17} + 2q^{18} - q^{19} - 5q^{20} - 4q^{21} + 2q^{22} + 5q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{4} - 86x^{3} + 1708x^{2} + 12934x - 437341$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
11	$-1$
29	$+1$

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
$\operatorname{tr} a_p$	2	−4	4	4	4	1	5	−1	5	−4	2	−1	4	14	3	0	19	−19	19	21	−2	−8	−5	0	−19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.492 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, 3, 11, 29, 33, 87, 319, 957]: [0, 1, 2, 3, 6, 12, 32, 124] (certified at ell = 2147482681)
T = [(2, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 12 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(957, 2)[5]   # 957.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 957:2 1000