Atlas › Modular forms › Weight 2 › Level 699 › 699.2.a › 699.2.a.e

Newform orbit 699.2.a.e

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + q^{2} - 13q^{3} + 19q^{4} + 10q^{5} - q^{6} + 2q^{7} + 6q^{8} + 13q^{9} + 5q^{10} - 5q^{11} - 19q^{12} + 19q^{13} + q^{14} - 10q^{15} + 47q^{16} + 7q^{17} + q^{18} + 5q^{19} + 16q^{20} - 2q^{21} - q^{22} - 17q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{13} - 31x^{12} + 11x^{11} + 7885x^{10} - 50827x^{9} - 653149x^{8} + 5788786x^{7} + 22872192x^{6} - 221191998x^{5} - 516188202x^{4} + 2803313839x^{3} + 8351523380x^{2} + 7234680932x + 2018087653$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
233	$-1$

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
$\operatorname{tr} a_p$	1	−13	10	2	−5	19	7	5	−17	28	6	27	25	−2	−27	13	−6	50	21	−12	12	6	−23	4	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 233, 699]: [0, 1, 20, 80] (certified at ell = 2147482481)
T = [(2, 1), (5, 3)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 6 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(699, 2)[4]   # 699.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 699:2 1000