Atlas › Modular forms › Weight 2 › Level 687 › 687.2.a › 687.2.a.e

Newform orbit 687.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 2q^{2} + 15q^{3} + 20q^{4} + 7q^{5} + 2q^{6} + 12q^{7} + 15q^{9} - 2q^{11} + 20q^{12} + 17q^{13} - 7q^{14} + 7q^{15} + 30q^{16} + 20q^{17} + 2q^{18} + 23q^{19} + 3q^{20} + 12q^{21} - 7q^{22} + q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{15} - 9x^{14} - 36x^{13} + 557x^{12} - 613x^{11} - 9394x^{10} + 26869x^{9} + 46448x^{8} - 256909x^{7} + 115750x^{6} + 715069x^{5} - 1077996x^{4} + 401490x^{3} - 2937x^{2} - 3035x + 67$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
229	$+1$

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
$\operatorname{tr} a_p$	2	15	7	12	−2	17	20	23	1	−8	7	7	5	14	−2	−8	−8	20	24	−16	60	10	5	−11	39

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.225 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, 229, 687]: [0, 1, 19, 78] (certified at ell = 2147478649)
T = [(2, 1), (5, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 4 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(687, 2)[4]   # 687.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 687:2 1000