Atlas › Modular forms › Weight 4 › Level 69 › 69.4.a › 69.4.a.a

Newform orbit 69.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} + 6q^{3} + 2q^{4} - 26q^{5} - 12q^{6} - 10q^{7} - 12q^{8} + 18q^{9} + 42q^{10} - 60q^{11} + 6q^{12} - 24q^{13} - 50q^{14} - 78q^{15} + 18q^{16} - 150q^{17} - 36q^{18} - 46q^{19} + 14q^{20} - 30q^{21} + 220q^{22} - 46q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 82x + 1661$

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 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
$\operatorname{tr} a_p$	−4	6	−26	−10	−60	−24	−150	−46	−46	−216	324	140	−364	−126	120	490	−32	−908	−874	488	652	−1198	−100	102	1624

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 2, 7, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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, 4)[0]   # 69.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 69:4 1000