Atlas › Modular forms › Weight 2 › Level 979 › 979.2.a › 979.2.a.c

Newform orbit 979.2.a.c

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{11} + 60x^{10} - 533x^{9} - 63839x^{8} + 30105x^{7} + 19577771x^{6} + 15225415x^{5} - 1957633640x^{4} - 2096951120x^{3} + 53448702771x^{2} + 463369961x - 4814151707$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
89	$-1$

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	−5	−10	1	11	−6	−7	3	−25	−48	−30	−24	−10	11	−7	−15	−51	−35	−11	−18	−9	19	16	11	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.366 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, 11, 89, 979]: [0, 2, 8, 92] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 16 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(979, 2)[2]   # 979.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 979:2 1000