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

Newform orbit 957.2.a.i

Trace form

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

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

Coefficient field

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

$x^{7} + 50x^{6} - 8368x^{5} - 576326x^{4} + 9047596x^{3} + 1533962121x^{2} + 41420361247x + 355326445871$

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 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
$\operatorname{tr} a_p$	−2	−7	−6	−4	−7	5	−5	3	−17	−7	−4	−5	0	−4	7	−34	−7	1	−37	−17	12	2	−33	−16	−33

$\operatorname{tr} a_n$ 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)[8]   # 957.2.a.i

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

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