Atlas › Modular forms › Weight 2 › Level 982 › 982.2.a › 982.2.a.c

Newform orbit 982.2.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 24x^{4} + 204x^{3} + 720x^{2} + 832x - 224$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
491	$-1$

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
$\operatorname{tr} a_p$	5	−3	−7	−6	−7	−7	−8	−17	1	10	−17	−15	−13	5	−23	−1	−18	8	13	−19	2	32	−23	−10	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.304 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, 491, 982]: [0, 1, 42, 125] (certified at ell = 2147474491)
T = [(3, 1), (5, 3)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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(982, 2)[2]   # 982.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 982:2 1000