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

Newform orbit 685.2.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 8x^{7} + 15x^{6} - 21x^{5} - 65x^{4} + 14x^{3} + 57x^{2} - 10x - 1$

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
$\operatorname{tr} a_p$	−3	−5	8	−4	−23	−3	−3	−25	−13	−11	−22	−2	−13	−8	9	7	−35	−13	6	−12	17	−37	−26	−18	24

$\operatorname{tr} a_n$ 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)[1]   # 685.2.a.b

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

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