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

Newform orbit 957.2.a.j

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{10} - 35x^{9} - 9967x^{8} + 163585x^{7} + 37982716x^{6} + 37117919x^{5} - 60387659068x^{4} - 888674569773x^{3} + 26296411247571x^{2} + 669396299263152x + 3381182440119579$

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
$\operatorname{tr} a_p$	1	10	2	10	−10	11	−11	17	15	−10	10	19	−12	8	5	4	3	15	47	21	−12	22	3	−8	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 0.492 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)[9]   # 957.2.a.j

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

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