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

Newform orbit 69.4.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 4q^{2} + 12q^{3} + 26q^{4} + 4q^{5} + 12q^{6} - 14q^{7} + 84q^{8} + 36q^{9} - 100q^{10} + 70q^{11} + 78q^{12} - 12q^{13} - 18q^{14} + 12q^{15} + 130q^{16} + 178q^{17} + 36q^{18} + 96q^{19} - 296q^{20} - 42q^{21} - 326q^{22} + 92q^{23} + 252q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{4} - 16x^{3} - 1491x^{2} + 42164x - 268738$

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
$\operatorname{tr} a_p$	4	12	4	−14	70	−12	178	96	92	−24	−400	−358	152	−196	260	968	1236	−482	620	−88	−948	254	−550	202	−1800

$\operatorname{tr} a_n$ 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)[3]   # 69.4.a.d

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

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