Atlas › Modular forms › Weight 6 › Level 69 › 69.6.a › 69.6.a.e

Newform orbit 69.6.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 8q^{2} + 45q^{3} + 118q^{4} + 94q^{5} + 72q^{6} + 272q^{7} + 258q^{8} + 405q^{9} - 172q^{10} + 1100q^{11} + 1062q^{12} - 978q^{13} - 344q^{14} + 846q^{15} + 1218q^{16} + 2522q^{17} + 648q^{18} + 2060q^{19} + 7720q^{20} + 2448q^{21} - 2572q^{22} - 2645q^{23} + 2322q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 102x^{4} - 8595x^{3} + 1056452x^{2} - 4696824x - 1084414968$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	8	45	94	272	1100	−978	2522	2060	−2645	1526	−7392	−8210	21250	−4548	536	−11482	74676	−44618	−1412	37912	46546	50544	89588	280410	90074

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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 engine's checks

dim M^+ at levels [1, 3, 23, 69]: [1, 3, 11, 42] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(69, 6)[4]   # 69.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 69:6 1000