Atlas › Modular forms › Weight 2 › Level 879 › 879.2.a › 879.2.a.a

Newform orbit 879.2.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{7} + 15x^{6} + 80x^{5} + 168x^{4} + 64x^{3} - 144x^{2} - 56x + 41$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
293	$-1$

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	7	−10	−3	−15	−5	−12	−4	−12	−4	−6	−5	−34	1	−14	−13	−23	0	0	−19	−2	10	−29	−31	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.351 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, 3, 293, 879]: [0, 1, 25, 100] (certified at ell = 2147478961)
T = [(2, 1), (5, 1)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 5 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(879, 2)[0]   # 879.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 879:2 1000