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

Newform orbit 685.2.a.d

Trace form

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

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

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

$x^{11} - 13x^{10} + 44x^{9} + 73x^{8} - 609x^{7} + 429x^{6} + 2083x^{5} - 2972x^{4} - 1358x^{3} + 3248x^{2} + 109x - 1043$

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 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	7	−11	6	21	3	13	−5	21	17	−22	12	13	16	5	17	−3	−9	6	−8	13	−13	24	−4	−4

$\operatorname{tr} a_n$ 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)[3]   # 685.2.a.d

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

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