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

Newform orbit 974.2.a.n

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. 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

$x^{6} + x^{5} - 15932x^{4} + 409152x^{3} + 51065868x^{2} - 1806043072x - 437451356$

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 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
$\operatorname{tr} a_p$	−6	1	−2	2	−5	0	−6	−4	10	−4	14	−2	−15	24	10	4	−7	16	−20	28	10	2	−14	6	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 0.706 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)[13]   # 974.2.a.n

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

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