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

Newform orbit 69.6.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} - 510q^{8} + 243q^{9} + 282q^{10} - 376q^{11} + 198q^{12} - 858q^{13} + 588q^{14} - 504q^{15} + 2738q^{16} - 2548q^{17} - 648q^{18} - 2846q^{19} - 4618q^{20} - 1026q^{21} - 5050q^{22} + 1587q^{23} - 4590q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 64x^{2} - 3357x - 213192$

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
$\operatorname{tr} a_p$	−8	27	−56	−114	−376	−858	−2548	−2846	1587	−16370	−14756	15874	12606	3154	29928	−44084	−29300	54010	43390	23424	−91402	−49398	−103936	96112	−135318

$\operatorname{tr} a_n$ 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)[1]   # 69.6.a.b

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

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