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

Newform orbit 957.2.a.h

Trace form

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

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

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

$x^{7} + 150x^{6} + 3098x^{5} - 360922x^{4} - 12501530x^{3} + 130699329x^{2} + 7431188263x + 47507385779$

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
$\operatorname{tr} a_p$	−6	−7	−2	−8	7	−5	−7	−7	−15	7	14	−13	−18	−30	−25	2	−13	11	−21	−21	2	−18	−21	0	39

$\operatorname{tr} a_n$ 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)[7]   # 957.2.a.h

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

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