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

Newform orbit 879.2.a.d

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - 3q^{2} - 18q^{3} + 23q^{4} + 6q^{5} + 3q^{6} + 5q^{7} - 9q^{8} + 18q^{9} + 13q^{10} - 9q^{11} - 23q^{12} + 11q^{13} + 11q^{14} - 6q^{15} + 41q^{16} + 8q^{17} - 3q^{18} + 14q^{19} + 19q^{20} - 5q^{21} + 7q^{22} - 12q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{18} - 3x^{17} - 107x^{16} + 297x^{15} + 4563x^{14} - 11694x^{13} - 99662x^{12} + 237981x^{11} + 1182893x^{10} - 2701703x^{9} - 7371793x^{8} + 16872495x^{7} + 20595152x^{6} - 50900971x^{5} - 14530501x^{4} + 49051062x^{3} + 6967696x^{2} - 15916765x - 3550737$

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 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
$\operatorname{tr} a_p$	−3	−18	6	5	−9	11	8	14	−12	6	30	23	50	17	−12	−3	19	32	24	21	62	2	−23	63	55

$\operatorname{tr} a_n$ 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)[3]   # 879.2.a.d

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

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