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

Newform orbit 69.6.a.a

Trace form

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

$2q - 18q^{3} + 168q^{4} + 94q^{5} - 118q^{7} + 162q^{9} + 116q^{10} + 320q^{11} - 1512q^{12} - 288q^{13} - 1276q^{14} - 846q^{15} + 6688q^{16} - 1810q^{17} + 730q^{19} + 7896q^{20} + 1062q^{21} + 12528q^{22} + 1058q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{29})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 94x + 1948$

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
$\operatorname{tr} a_p$	0	−18	94	−118	320	−288	−1810	730	1058	8208	1772	−23112	5516	10322	42952	−25350	18344	37224	−7482	126848	137660	62286	83120	69770	−170104

$\operatorname{tr} a_n$ 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)[0]   # 69.6.a.a

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

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