Atlas › Modular forms › Weight 2 › Level 685 › 685.2.a › 685.2.a.c

Newform orbit 685.2.a.c

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 6q^{2} - 7q^{3} + 16q^{4} - 11q^{5} + 2q^{6} - 9q^{7} - 15q^{8} + 14q^{9} + 6q^{10} - 13q^{11} - 21q^{12} - 10q^{13} - 19q^{14} + 7q^{15} + 26q^{16} - 15q^{17} - 19q^{18} + 13q^{19} - 16q^{20} - q^{22} - 21q^{23} - 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 13x^{10} + 40x^{9} - 105x^{8} - 627x^{7} - 177x^{6} + 1889x^{5} + 846x^{4} - 1770x^{3} - 578x^{2} + 413x - 43$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
137	$+1$

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
$\operatorname{tr} a_p$	−6	−7	−11	−9	−13	−10	−15	13	−21	−17	21	−12	5	−38	−13	−32	−17	4	−34	−30	11	12	−38	−4	−19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.221 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, 5, 137, 685]: [0, 1, 12, 70] (certified at ell = 2147483249)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(685, 2)[2]   # 685.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 685:2 1000