Atlas › Modular forms › Weight 2 › Level 974 › 974.2.a › 974.2.a.j

Newform orbit 974.2.a.j

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ , whose minimal polynomial is

$x^{2} + 111x + 2988$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
487	$-1$

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	2	−6	−4	−5	−1	2	−4	−6	0	−8	−12	9	−1	6	0	13	−14	12	9	7	−2	12	7	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.706 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, 487, 974]: [0, 1, 41, 124] (certified at ell = 2147481883)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 11 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(974, 2)[9]   # 974.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 974:2 1000