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

Newform orbit 979.2.a.d

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q - 4q^{2} - q^{3} + 14q^{4} - 14q^{5} - 6q^{6} - 5q^{7} - 18q^{8} + 6q^{9} - 6q^{10} - 17q^{11} + q^{12} - 6q^{13} - 17q^{14} - 3q^{15} + 8q^{16} - 19q^{17} - 9q^{19} - 29q^{20} - 35q^{21} + 4q^{22} - 13q^{23} - 20q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{17} + 112x^{16} + 3444x^{15} - 35759x^{14} - 3288975x^{13} - 19908032x^{12} + 1086422346x^{11} + 11163033040x^{10} - 183873168441x^{9} - 2051191025195x^{8} + 18304874495285x^{7} + 172348894245979x^{6} - 1062946265549308x^{5} - 6234288449629378x^{4} + 28584877669322783x^{3} + 61270800485537220x^{2} - 88524445861712275x - 95808647635531819$

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
$\operatorname{tr} a_p$	−4	−1	−14	−5	−17	−6	−19	−9	−13	−54	8	0	−16	−21	−3	−43	−1	−29	17	6	3	−43	−14	−17	−6

$\operatorname{tr} a_n$ 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)[3]   # 979.2.a.d

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

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