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

Newform orbit 879.2.a.c

Trace form

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

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

Coefficient field

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

$x^{17} - 18x^{16} + 77x^{15} + 426x^{14} - 4321x^{13} + 5743x^{12} + 48969x^{11} - 172234x^{10} - 47395x^{9} + 986562x^{8} - 1035247x^{7} - 1548354x^{6} + 3110486x^{5} - 238555x^{4} - 2036184x^{3} + 680726x^{2} + 345214x - 94139$

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
$\operatorname{tr} a_p$	6	17	12	−1	19	1	12	0	18	10	14	−11	44	−27	36	13	39	−6	−32	25	8	−14	27	55	37

$\operatorname{tr} a_n$ 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)[2]   # 879.2.a.c

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

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